diff --git a/changelog b/changelog index 80e94f3..314b281 100644 --- a/changelog +++ b/changelog @@ -1,3 +1,4 @@ +20080424 tpd src/input/schaum23.input show Schaums-Axiom equivalence 20080423 tpd src/input/schaum22.input show Schaums-Axiom equivalence 20080423 tpd src/input/schaum21.input show Schaums-Axiom equivalence 20080423 tpd src/input/schaum20.input show Schaums-Axiom equivalence diff --git a/src/input/schaum23.input.pamphlet b/src/input/schaum23.input.pamphlet index 679e775..c24ef73 100644 --- a/src/input/schaum23.input.pamphlet +++ b/src/input/schaum23.input.pamphlet @@ -19,7 +19,7 @@ $$ )set message auto off )clear all ---S 1 of 10 +--S 1 aa:=integrate(csc(a*x),x) --R --R @@ -30,6 +30,148 @@ aa:=integrate(csc(a*x),x) --R a --R Type: Union(Expression Integer,...) --E + +--S 2 +bb1:=1/a*log(csc(a*x)-cot(a*x)) +--R +--R log(csc(a x) - cot(a x)) +--R (2) ------------------------ +--R a +--R Type: Expression Integer +--E + +--S 3 +bb2:=1/a*log(tan((a*x)/2)) +--R +--R a x +--R log(tan(---)) +--R 2 +--R (3) ------------- +--R a +--R Type: Expression Integer +--E + +--S 4 +cc1:=aa-bb1 +--R +--R sin(a x) +--R log(------------) - log(csc(a x) - cot(a x)) +--R cos(a x) + 1 +--R (4) -------------------------------------------- +--R a +--R Type: Expression Integer +--E + +--S 5 +cotrule:=rule(cot(a) == cos(a)/sin(a)) +--R +--R cos(a) +--R (5) cot(a) == ------ +--R sin(a) +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 6 +dd1:=cotrule cc1 +--R +--R sin(a x) csc(a x)sin(a x) - cos(a x) +--R log(------------) - log(---------------------------) +--R cos(a x) + 1 sin(a x) +--R (6) ---------------------------------------------------- +--R a +--R Type: Expression Integer +--E + +--S 7 +cscrule:=rule(csc(a) == 1/sin(a)) +--R +--R 1 +--R (7) csc(a) == ------ +--R sin(a) +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 8 +ee1:=cscrule dd1 +--R +--R sin(a x) - cos(a x) + 1 +--R log(------------) - log(--------------) +--R cos(a x) + 1 sin(a x) +--R (8) --------------------------------------- +--R a +--R Type: Expression Integer +--E + +--S 9 +ff1:=expandLog ee1 +--R +--R 2log(sin(a x)) - log(cos(a x) + 1) - log(cos(a x) - 1) - log(- 1) +--R (9) ----------------------------------------------------------------- +--R a +--R Type: Expression Integer +--E + +--S 10 +gg1:=complexNormalize ff1 +--R +--R 2log(- 1) +--R (10) - --------- +--R a +--R Type: Expression Integer +--E + +--S 11 +cc2:=aa-bb2 +--R +--R a x sin(a x) +--R - log(tan(---)) + log(------------) +--R 2 cos(a x) + 1 +--R (11) ----------------------------------- +--R a +--R Type: Expression Integer +--E + +--S 12 +tanrule:=rule(tan(a) == sin(a)/cos(a)) +--R +--R sin(a) +--R (12) tan(a) == ------ +--R cos(a) +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 13 +dd2:=tanrule cc2 +--R +--R a x +--R sin(---) +--R sin(a x) 2 +--R log(------------) - log(--------) +--R cos(a x) + 1 a x +--R cos(---) +--R 2 +--R (13) --------------------------------- +--R a +--R Type: Expression Integer +--E + +--S 14 +ee2:=expandLog dd2 +--R +--R a x a x +--R log(sin(a x)) - log(sin(---)) - log(cos(a x) + 1) + log(cos(---)) +--R 2 2 +--R (14) ----------------------------------------------------------------- +--R a +--R Type: Expression Integer +--E + +--S 15 14:461 Schaums and Axiom agree +ff2:=complexNormalize ee2 +--R +--R (15) 0 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.462~~~~~$\displaystyle @@ -40,7 +182,7 @@ $$ <<*>>= )clear all ---S 2 of 10 +--S 16 aa:=integrate(csc(a*x)^2,x) --R --R @@ -49,6 +191,40 @@ aa:=integrate(csc(a*x)^2,x) --R a sin(a x) --R Type: Union(Expression Integer,...) --E + +--S 17 +bb:=-cot(a*x)/a +--R +--R cot(a x) +--R (2) - -------- +--R a +--R Type: Expression Integer +--E + +--S 18 +cc:=aa-bb +--R +--R cot(a x)sin(a x) - cos(a x) +--R (3) --------------------------- +--R a sin(a x) +--R Type: Expression Integer +--E + +--S 19 +cotrule:=rule(cot(a) == cos(a)/sin(a)) +--R +--R cos(a) +--R (4) cot(a) == ------ +--R sin(a) +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 20 14:462 Schaums and Axiom agree +dd:=cotrule cc +--R +--R (5) 0 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.463~~~~~$\displaystyle @@ -59,7 +235,7 @@ $$ <<*>>= )clear all ---S 3 of 10 +--S 21 aa:=integrate(csc(a*x)^3,x) --R --R @@ -71,6 +247,159 @@ aa:=integrate(csc(a*x)^3,x) --R 2a cos(a x) - 2a --R Type: Union(Expression Integer,...) --E + +--S 22 +bb:=-(csc(a*x)*cot(a*x))/(2*a)+1/(2*a)*log(tan((a*x)/2)) +--R +--R a x +--R log(tan(---)) - cot(a x)csc(a x) +--R 2 +--R (2) -------------------------------- +--R 2a +--R Type: Expression Integer +--E + +--S 23 +cc:=aa-bb +--R +--R (3) +--R 2 a x 2 sin(a x) +--R (- cos(a x) + 1)log(tan(---)) + (cos(a x) - 1)log(------------) +--R 2 cos(a x) + 1 +--R + +--R 2 +--R (cos(a x) - 1)cot(a x)csc(a x) + cos(a x) +--R / +--R 2 +--R 2a cos(a x) - 2a +--R Type: Expression Integer +--E + +--S 24 +cotrule:=rule(cot(a) == cos(a)/sin(a)) +--R +--R cos(a) +--R (4) cot(a) == ------ +--R sin(a) +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 25 +dd:=cotrule cc +--R +--R (5) +--R 2 a x +--R (- cos(a x) + 1)sin(a x)log(tan(---)) +--R 2 +--R + +--R 2 sin(a x) +--R (cos(a x) - 1)sin(a x)log(------------) + cos(a x)sin(a x) +--R cos(a x) + 1 +--R + +--R 3 +--R (cos(a x) - cos(a x))csc(a x) +--R / +--R 2 +--R (2a cos(a x) - 2a)sin(a x) +--R Type: Expression Integer +--E + +--S 26 +tanrule:=rule(tan(a) == sin(a)/cos(a)) +--R +--R sin(a) +--R (6) tan(a) == ------ +--R cos(a) +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 27 +ee:=tanrule dd +--R +--R (7) +--R 2 sin(a x) +--R (cos(a x) - 1)sin(a x)log(------------) +--R cos(a x) + 1 +--R + +--R a x +--R sin(---) +--R 2 2 +--R (- cos(a x) + 1)sin(a x)log(--------) + cos(a x)sin(a x) +--R a x +--R cos(---) +--R 2 +--R + +--R 3 +--R (cos(a x) - cos(a x))csc(a x) +--R / +--R 2 +--R (2a cos(a x) - 2a)sin(a x) +--R Type: Expression Integer +--E + +--S 28 +cscrule:=rule(csc(a) == 1/sin(a)) +--R +--R 1 +--R (8) csc(a) == ------ +--R sin(a) +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 29 +ff:=cscrule ee +--R +--R (9) +--R 2 2 sin(a x) +--R (cos(a x) - 1)sin(a x) log(------------) +--R cos(a x) + 1 +--R + +--R a x +--R sin(---) +--R 2 2 2 2 3 +--R (- cos(a x) + 1)sin(a x) log(--------) + cos(a x)sin(a x) + cos(a x) +--R a x +--R cos(---) +--R 2 +--R + +--R - cos(a x) +--R / +--R 2 2 +--R (2a cos(a x) - 2a)sin(a x) +--R Type: Expression Integer +--E + +--S 30 +gg:=expandLog ff +--R +--R (10) +--R 2 2 +--R (cos(a x) - 1)sin(a x) log(sin(a x)) +--R + +--R 2 2 a x +--R (- cos(a x) + 1)sin(a x) log(sin(---)) +--R 2 +--R + +--R 2 2 +--R (- cos(a x) + 1)sin(a x) log(cos(a x) + 1) +--R + +--R 2 2 a x 2 3 +--R (cos(a x) - 1)sin(a x) log(cos(---)) + cos(a x)sin(a x) + cos(a x) +--R 2 +--R + +--R - cos(a x) +--R / +--R 2 2 +--R (2a cos(a x) - 2a)sin(a x) +--R Type: Expression Integer +--E + +--S 31 14:463 Schaums and Axiom agree +hh:=complexNormalize gg +--R +--R (11) 0 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.464~~~~~$\displaystyle @@ -81,7 +410,7 @@ $$ <<*>>= )clear all ---S 4 of 10 +--S 32 aa:=integrate(csc(a*x)^n*cot(a*x),x) --R --R @@ -96,6 +425,31 @@ aa:=integrate(csc(a*x)^n*cot(a*x),x) --R a n --R Type: Union(Expression Integer,...) --E + +--S 33 +bb:=-csc(a*x)^n/(n*a) +--R +--R n +--R csc(a x) +--R (2) - --------- +--R a n +--R Type: Expression Integer +--E + +--S 34 14:464 Axiom cannot simplify this expression +cc:=aa-bb +--R +--R 1 +--R n log(- -------------) +--R 2 +--R cos(a x) - 1 +--R ---------------------- +--R 2 n +--R - %e + csc(a x) +--R (3) -------------------------------------- +--R a n +--R Type: Expression Integer +--E @ \section{\cite{1}:14.465~~~~~$\displaystyle @@ -106,7 +460,7 @@ $$ <<*>>= )clear all ---S 5 of 10 +--S 35 aa:=integrate(1/csc(a*x),x) --R --R @@ -114,7 +468,23 @@ aa:=integrate(1/csc(a*x),x) --R (1) - -------- --R a --R Type: Union(Expression Integer,...) +--E + +--S 36 +bb:=-cos(a*x)/a +--R +--R cos(a x) +--R (2) - -------- +--R a +--R Type: Expression Integer --E + +--S 37 14:465 Schaums and Axiom agree +cc:=aa-bb +--R +--R (3) 0 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.466~~~~~$\displaystyle @@ -126,7 +496,7 @@ $$ <<*>>= )clear all ---S 6 of 10 +--S 38 14:466 Axiom cannot compute this integral aa:=integrate(x*csc(a*x),x) --R --R @@ -147,7 +517,7 @@ $$ <<*>>= )clear all ---S 7 of 10 +--S 39 14:467 Axiom cannot compute this integral aa:=integrate(csc(a*x)/x,x) --R --R @@ -167,7 +537,7 @@ $$ <<*>>= )clear all ---S 8 of 10 +--S 40 aa:=integrate(x*csc(a*x)^2,x) --R --R @@ -179,6 +549,64 @@ aa:=integrate(x*csc(a*x)^2,x) --R a sin(a x) --R Type: Union(Expression Integer,...) --E + +--S 41 +bb:=-(x*cot(a*x))/a+1/a^2*log(sin(a*x)) +--R +--R log(sin(a x)) - a x cot(a x) +--R (2) ---------------------------- +--R 2 +--R a +--R Type: Expression Integer +--E + +--S 42 +cc:=aa-bb +--R +--R (3) +--R sin(a x) +--R - sin(a x)log(sin(a x)) + sin(a x)log(------------) +--R cos(a x) + 1 +--R + +--R 2 +--R - sin(a x)log(------------) + a x cot(a x)sin(a x) - a x cos(a x) +--R cos(a x) + 1 +--R / +--R 2 +--R a sin(a x) +--R Type: Expression Integer +--E + +--S 43 +cotrule:=rule(cot(a) == cos(a)/sin(a)) +--R +--R cos(a) +--R (4) cot(a) == ------ +--R sin(a) +--R Type: RewriteRule(Integer,Integer,Expression Integer) +--E + +--S 44 +dd:=cotrule cc +--R +--R sin(a x) 2 +--R - log(sin(a x)) + log(------------) - log(------------) +--R cos(a x) + 1 cos(a x) + 1 +--R (5) ------------------------------------------------------- +--R 2 +--R a +--R Type: Expression Integer +--E + +--S 45 14:468 Schaums and Axiom differ by a constant +ee:=expandLog dd +--R +--R log(2) +--R (6) - ------ +--R 2 +--R a +--R Type: Expression Integer +--E @ \section{\cite{1}:14.469~~~~~$\displaystyle @@ -189,7 +617,7 @@ $$ <<*>>= )clear all ---S 9 of 10 +--S 46 aa:=integrate(1/(q+p*csc(a*x)),x) --R --R @@ -227,6 +655,156 @@ aa:=integrate(1/(q+p*csc(a*x)),x) --R a q\|- q + p --R Type: Union(List Expression Integer,...) --E + +--S 47 +t1:=integrate(1/(p+q*sin(a*x)),x) +--E + +--S 48 +bb1:=x/q-p/q*t1.1 +--R +--R (3) +--R - +--R p +--R * +--R log +--R +-------+ +--R 2 2 2 | 2 2 +--R (p q sin(a x) + (q - p )cos(a x) + q )\|q - p +--R + +--R 2 3 3 2 3 2 +--R (- p q + p )sin(a x) + (- q + p q)cos(a x) - q + p q +--R / +--R q sin(a x) + p +--R + +--R +-------+ +--R | 2 2 +--R a x\|q - p +--R / +--R +-------+ +--R | 2 2 +--R a q\|q - p +--R Type: Expression Integer +--E + +--S 49 +bb2:=x/q-p/q*t1.2 +--R +--R +---------+ +--R | 2 2 +---------+ +--R (p sin(a x) + q cos(a x) + q)\|- q + p | 2 2 +--R 2p atan(-----------------------------------------) + a x\|- q + p +--R 2 2 2 2 +--R (q - p )cos(a x) + q - p +--R (4) -------------------------------------------------------------------- +--R +---------+ +--R | 2 2 +--R a q\|- q + p +--R Type: Expression Integer +--E + +--S 50 +cc1:=aa.1-bb1 +--R +--R (5) +--R p +--R * +--R log +--R +-------+ +--R 2 2 2 | 2 2 +--R (p q sin(a x) + (q - p )cos(a x) + q )\|q - p +--R + +--R 2 3 3 2 3 2 +--R (p q - p )sin(a x) + (q - p q)cos(a x) + q - p q +--R / +--R q sin(a x) + p +--R + +--R p +--R * +--R log +--R +-------+ +--R 2 2 2 | 2 2 +--R (p q sin(a x) + (q - p )cos(a x) + q )\|q - p +--R + +--R 2 3 3 2 3 2 +--R (- p q + p )sin(a x) + (- q + p q)cos(a x) - q + p q +--R / +--R q sin(a x) + p +--R / +--R +-------+ +--R | 2 2 +--R a q\|q - p +--R Type: Expression Integer +--E + +--S 51 +cc2:=aa.2-bb1 +--R +--R (6) +--R +---------+ +--R | 2 2 +--R p\|- q + p +--R * +--R log +--R +-------+ +--R 2 2 2 | 2 2 +--R (p q sin(a x) + (q - p )cos(a x) + q )\|q - p +--R + +--R 2 3 3 2 3 2 +--R (- p q + p )sin(a x) + (- q + p q)cos(a x) - q + p q +--R / +--R q sin(a x) + p +--R + +--R +---------+ +--R +-------+ | 2 2 +--R | 2 2 (p sin(a x) + q cos(a x) + q)\|- q + p +--R 2p\|q - p atan(-----------------------------------------) +--R 2 2 2 2 +--R (q - p )cos(a x) + q - p +--R / +--R +---------+ +-------+ +--R | 2 2 | 2 2 +--R a q\|- q + p \|q - p +--R Type: Expression Integer +--E + +--S 52 +cc3:=aa.1-bb2 +--R +--R (7) +--R +---------+ +--R | 2 2 +--R p\|- q + p +--R * +--R log +--R +-------+ +--R 2 2 2 | 2 2 +--R (p q sin(a x) + (q - p )cos(a x) + q )\|q - p +--R + +--R 2 3 3 2 3 2 +--R (p q - p )sin(a x) + (q - p q)cos(a x) + q - p q +--R / +--R q sin(a x) + p +--R + +--R +---------+ +--R +-------+ | 2 2 +--R | 2 2 (p sin(a x) + q cos(a x) + q)\|- q + p +--R - 2p\|q - p atan(-----------------------------------------) +--R 2 2 2 2 +--R (q - p )cos(a x) + q - p +--R / +--R +---------+ +-------+ +--R | 2 2 | 2 2 +--R a q\|- q + p \|q - p +--R Type: Expression Integer +--E + +--S 53 14:469 Schaums and Axiom agree +cc4:=aa.2-bb2 +--R +--R (8) 0 +--R Type: Expression Integer +--E @ \section{\cite{1}:14.470~~~~~$\displaystyle @@ -238,7 +816,7 @@ $$ <<*>>= )clear all ---S 10 of 10 +--S 54 14:470 Axiom cannot compute this integral aa:=integrate(csc(a*x)^n,x) --R --R