4.8 Test file Number [173] 6-Hyperbolic-functions/6.3-Hyperbolic-tangent/6.3.7-d-hyper-^m-a+b-c-tanh-^n-^p

4.8.1 Mathematica

Integral number [74] $$\int \frac{{\sinh }^3(c+dx)}{a+b{\tanh }^3(c+dx)}dx$$

[B]   time = 0.483965 (sec), size = 826 ,normalized size = 25.81 $$\frac{\cosh (3(c+dx))a^3+27b\sinh (c+dx)a^2-b\sinh (3(c+dx))a^2-9(a^2+3b^2)\cosh (c+dx)a-b^2\cosh (3(c+dx))a-2b\text{RootSum}[a{\mathrm{\#}\text{1}}^6+b{\mathrm{\#}\text{1}}^6+3a{\mathrm{\#}\text{1}}^4-3b{\mathrm{\#}\text{1}}^4+3a{\mathrm{\#}\text{1}}^2+3b{\mathrm{\#}\text{1}}^2+a-b\mathrm{\&},\frac{3a^{2}c{\mathrm{\#}\text{1}}^4+3b^{2}c{\mathrm{\#}\text{1}}^4-3abc{\mathrm{\#}\text{1}}^4+3a^{2}dx{\mathrm{\#}\text{1}}^4+3b^{2}dx{\mathrm{\#}\text{1}}^4-3abdx{\mathrm{\#}\text{1}}^4+6a^2\log (\mathrm{\#}\text{1}\cosh (\frac{1}{2}(c+dx))-\cosh (\frac{1}{2}(c+dx))-\sinh (\frac{1}{2}(c+dx))-\sinh (\frac{1}{2}(c+dx))\mathrm{\#}\text{1}){\mathrm{\#}\text{1}}^4+6b^2\log (\mathrm{\#}\text{1}\cosh (\frac{1}{2}(c+dx))-\cosh (\frac{1}{2}(c+dx))-\sinh (\frac{1}{2}(c+dx))-\sinh (\frac{1}{2}(c+dx))\mathrm{\#}\text{1}){\mathrm{\#}\text{1}}^4-6ab\log (\mathrm{\#}\text{1}\cosh (\frac{1}{2}(c+dx))-\cosh (\frac{1}{2}(c+dx))-\sinh (\frac{1}{2}(c+dx))-\sinh (\frac{1}{2}(c+dx))\mathrm{\#}\text{1}){\mathrm{\#}\text{1}}^4+2a^{2}c{\mathrm{\#}\text{1}}^2-2b^{2}c{\mathrm{\#}\text{1}}^2+2a^{2}dx{\mathrm{\#}\text{1}}^2-2b^{2}dx{\mathrm{\#}\text{1}}^2+4a^2\log (\mathrm{\#}\text{1}\cosh (\frac{1}{2}(c+dx))-\cosh (\frac{1}{2}(c+dx))-\sinh (\frac{1}{2}(c+dx))-\sinh (\frac{1}{2}(c+dx))\mathrm{\#}\text{1}){\mathrm{\#}\text{1}}^2-4b^2\log (\mathrm{\#}\text{1}\cosh (\frac{1}{2}(c+dx))-\cosh (\frac{1}{2}(c+dx))-\sinh (\frac{1}{2}(c+dx))-\sinh (\frac{1}{2}(c+dx))\mathrm{\#}\text{1}){\mathrm{\#}\text{1}}^2+3a^{2}c+3b^{2}c+3abc+3a^{2}dx+3b^{2}dx+3abdx+6a^2\log (\mathrm{\#}\text{1}\cosh (\frac{1}{2}(c+dx))-\cosh (\frac{1}{2}(c+dx))-\sinh (\frac{1}{2}(c+dx))-\sinh (\frac{1}{2}(c+dx))\mathrm{\#}\text{1})+6b^2\log (\mathrm{\#}\text{1}\cosh (\frac{1}{2}(c+dx))-\cosh (\frac{1}{2}(c+dx))-\sinh (\frac{1}{2}(c+dx))-\sinh (\frac{1}{2}(c+dx))\mathrm{\#}\text{1})+6ab\log (\mathrm{\#}\text{1}\cosh (\frac{1}{2}(c+dx))-\cosh (\frac{1}{2}(c+dx))-\sinh (\frac{1}{2}(c+dx))-\sinh (\frac{1}{2}(c+dx))\mathrm{\#}\text{1})}{a{\mathrm{\#}\text{1}}^5+b{\mathrm{\#}\text{1}}^5+2a{\mathrm{\#}\text{1}}^3-2b{\mathrm{\#}\text{1}}^3+a\mathrm{\#}\text{1}+b\mathrm{\#}\text{1}}\mathrm{\&}]a+9b^3\sinh (c+dx)+b^3\sinh (3(c+dx))}{12(a-b{)}^2(a+b{)}^{2}d}$$

[In]

Integrate[Sinh[c + d*x]^3/(a + b*Tanh[c + d*x]^3),x]

[Out]

(-9*a*(a^2 + 3*b^2)*Cosh[c + d*x] + a^3*Cosh[3*(c + d*x)] - a*b^2*Cosh[3*(c + d*x)] - 2*a*b*RootSum[a - b + 3*
a*#1^2 + 3*b*#1^2 + 3*a*#1^4 - 3*b*#1^4 + a*#1^6 + b*#1^6 & , (3*a^2*c + 3*a*b*c + 3*b^2*c + 3*a^2*d*x + 3*a*b
*d*x + 3*b^2*d*x + 6*a^2*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]
*#1] + 6*a*b*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1] + 6*b^2
*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1] + 2*a^2*c*#1^2 - 2*
b^2*c*#1^2 + 2*a^2*d*x*#1^2 - 2*b^2*d*x*#1^2 + 4*a^2*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*
x)/2]*#1 - Sinh[(c + d*x)/2]*#1]*#1^2 - 4*b^2*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#
1 - Sinh[(c + d*x)/2]*#1]*#1^2 + 3*a^2*c*#1^4 - 3*a*b*c*#1^4 + 3*b^2*c*#1^4 + 3*a^2*d*x*#1^4 - 3*a*b*d*x*#1^4
+ 3*b^2*d*x*#1^4 + 6*a^2*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]
*#1]*#1^4 - 6*a*b*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1]*#1
^4 + 6*b^2*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1]*#1^4)/(a*
#1 + b*#1 + 2*a*#1^3 - 2*b*#1^3 + a*#1^5 + b*#1^5) & ] + 27*a^2*b*Sinh[c + d*x] + 9*b^3*Sinh[c + d*x] - a^2*b*
Sinh[3*(c + d*x)] + b^3*Sinh[3*(c + d*x)])/(12*(a - b)^2*(a + b)^2*d)

Integral number [76] $$\int \frac{\sinh (c+dx)}{a+b{\tanh }^3(c+dx)}dx$$

[B]   time = 0.226157 (sec), size = 409 ,normalized size = 13.63 $$\frac{b\text{RootSum}[{\mathrm{\#}\text{1}}^{6}a+3{\mathrm{\#}\text{1}}^{4}a+3{\mathrm{\#}\text{1}}^{2}a+{\mathrm{\#}\text{1}}^{6}b-3{\mathrm{\#}\text{1}}^{4}b+3{\mathrm{\#}\text{1}}^{2}b+a-b\mathrm{\&},\frac{4{\mathrm{\#}\text{1}}^{4}a\log (-\mathrm{\#}\text{1}\sinh (\frac{1}{2}(c+dx))+\mathrm{\#}\text{1}\cosh (\frac{1}{2}(c+dx))-\sinh (\frac{1}{2}(c+dx))-\cosh (\frac{1}{2}(c+dx)))+2{\mathrm{\#}\text{1}}^{4}ac+2{\mathrm{\#}\text{1}}^{4}adx-2{\mathrm{\#}\text{1}}^{4}b\log (-\mathrm{\#}\text{1}\sinh (\frac{1}{2}(c+dx))+\mathrm{\#}\text{1}\cosh (\frac{1}{2}(c+dx))-\sinh (\frac{1}{2}(c+dx))-\cosh (\frac{1}{2}(c+dx)))-{\mathrm{\#}\text{1}}^{4}bc-{\mathrm{\#}\text{1}}^{4}bdx+4a\log (-\mathrm{\#}\text{1}\sinh (\frac{1}{2}(c+dx))+\mathrm{\#}\text{1}\cosh (\frac{1}{2}(c+dx))-\sinh (\frac{1}{2}(c+dx))-\cosh (\frac{1}{2}(c+dx)))+2b\log (-\mathrm{\#}\text{1}\sinh (\frac{1}{2}(c+dx))+\mathrm{\#}\text{1}\cosh (\frac{1}{2}(c+dx))-\sinh (\frac{1}{2}(c+dx))-\cosh (\frac{1}{2}(c+dx)))+2ac+2adx+bc+bdx}{{\mathrm{\#}\text{1}}^{5}a+2{\mathrm{\#}\text{1}}^{3}a+{\mathrm{\#}\text{1}}^{5}b-2{\mathrm{\#}\text{1}}^{3}b+\mathrm{\#}\text{1}a+\mathrm{\#}\text{1}b}\mathrm{\&}]+6a\cosh (c+dx)-6b\sinh (c+dx)}{6d(a-b)(a+b)}$$

[In]

Integrate[Sinh[c + d*x]/(a + b*Tanh[c + d*x]^3),x]

[Out]

(6*a*Cosh[c + d*x] + b*RootSum[a - b + 3*a*#1^2 + 3*b*#1^2 + 3*a*#1^4 - 3*b*#1^4 + a*#1^6 + b*#1^6 & , (2*a*c
+ b*c + 2*a*d*x + b*d*x + 4*a*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*
x)/2]*#1] + 2*b*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1] + 2*
a*c*#1^4 - b*c*#1^4 + 2*a*d*x*#1^4 - b*d*x*#1^4 + 4*a*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d
*x)/2]*#1 - Sinh[(c + d*x)/2]*#1]*#1^4 - 2*b*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1
- Sinh[(c + d*x)/2]*#1]*#1^4)/(a*#1 + b*#1 + 2*a*#1^3 - 2*b*#1^3 + a*#1^5 + b*#1^5) & ] - 6*b*Sinh[c + d*x])/
(6*(a - b)*(a + b)*d)

Integral number [77] $$\int \frac{\text{csch}(c+dx)}{a+b{\tanh }^3(c+dx)}dx$$

[B]   time = 0.158439 (sec), size = 319 ,normalized size = 10.63 $$\frac{6\log (\tanh (\frac{1}{2}(c+dx)))-b\text{RootSum}[{\mathrm{\#}\text{1}}^{6}a+3{\mathrm{\#}\text{1}}^{4}a+3{\mathrm{\#}\text{1}}^{2}a+{\mathrm{\#}\text{1}}^{6}b-3{\mathrm{\#}\text{1}}^{4}b+3{\mathrm{\#}\text{1}}^{2}b+a-b\mathrm{\&},\frac{2{\mathrm{\#}\text{1}}^4\log (-\mathrm{\#}\text{1}\sinh (\frac{1}{2}(c+dx))+\mathrm{\#}\text{1}\cosh (\frac{1}{2}(c+dx))-\sinh (\frac{1}{2}(c+dx))-\cosh (\frac{1}{2}(c+dx)))-4{\mathrm{\#}\text{1}}^2\log (-\mathrm{\#}\text{1}\sinh (\frac{1}{2}(c+dx))+\mathrm{\#}\text{1}\cosh (\frac{1}{2}(c+dx))-\sinh (\frac{1}{2}(c+dx))-\cosh (\frac{1}{2}(c+dx)))+{\mathrm{\#}\text{1}}^{4}c-2{\mathrm{\#}\text{1}}^{2}c+{\mathrm{\#}\text{1}}^{4}dx-2{\mathrm{\#}\text{1}}^{2}dx+2\log (-\mathrm{\#}\text{1}\sinh (\frac{1}{2}(c+dx))+\mathrm{\#}\text{1}\cosh (\frac{1}{2}(c+dx))-\sinh (\frac{1}{2}(c+dx))-\cosh (\frac{1}{2}(c+dx)))+c+dx}{{\mathrm{\#}\text{1}}^{5}a+2{\mathrm{\#}\text{1}}^{3}a+{\mathrm{\#}\text{1}}^{5}b-2{\mathrm{\#}\text{1}}^{3}b+\mathrm{\#}\text{1}a+\mathrm{\#}\text{1}b}\mathrm{\&}]}{6ad}$$

[In]

Integrate[Csch[c + d*x]/(a + b*Tanh[c + d*x]^3),x]

[Out]

(6*Log[Tanh[(c + d*x)/2]] - b*RootSum[a - b + 3*a*#1^2 + 3*b*#1^2 + 3*a*#1^4 - 3*b*#1^4 + a*#1^6 + b*#1^6 & ,
(c + d*x + 2*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1] - 2*c*#
1^2 - 2*d*x*#1^2 - 4*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1]
*#1^2 + c*#1^4 + d*x*#1^4 + 2*Log[-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*
x)/2]*#1]*#1^4)/(a*#1 + b*#1 + 2*a*#1^3 - 2*b*#1^3 + a*#1^5 + b*#1^5) & ])/(6*a*d)

Integral number [79] $$\int \frac{{\text{csch}}^3(c+dx)}{a+b{\tanh }^3(c+dx)}dx$$

[B]   time = 0.361888 (sec), size = 201 ,normalized size = 6.28 $$-\frac{16b\text{RootSum}[{\mathrm{\#}\text{1}}^{6}a+3{\mathrm{\#}\text{1}}^{4}a+3{\mathrm{\#}\text{1}}^{2}a+{\mathrm{\#}\text{1}}^{6}b-3{\mathrm{\#}\text{1}}^{4}b+3{\mathrm{\#}\text{1}}^{2}b+a-b\mathrm{\&},\frac{2\mathrm{\#}\text{1}\log (-\mathrm{\#}\text{1}\sinh (\frac{1}{2}(c+dx))+\mathrm{\#}\text{1}\cosh (\frac{1}{2}(c+dx))-\sinh (\frac{1}{2}(c+dx))-\cosh (\frac{1}{2}(c+dx)))+\mathrm{\#}\text{1}c+\mathrm{\#}\text{1}dx}{{\mathrm{\#}\text{1}}^{4}a+2{\mathrm{\#}\text{1}}^{2}a+{\mathrm{\#}\text{1}}^{4}b-2{\mathrm{\#}\text{1}}^{2}b+a+b}\mathrm{\&}]+3({\text{csch}}^2(\frac{1}{2}(c+dx))+{\text{sech}}^2(\frac{1}{2}(c+dx))+4\log (\tanh (\frac{1}{2}(c+dx))))}{24ad}$$

[In]

Integrate[Csch[c + d*x]^3/(a + b*Tanh[c + d*x]^3),x]

[Out]

-(16*b*RootSum[a - b + 3*a*#1^2 + 3*b*#1^2 + 3*a*#1^4 - 3*b*#1^4 + a*#1^6 + b*#1^6 & , (c*#1 + d*x*#1 + 2*Log[
-Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2] + Cosh[(c + d*x)/2]*#1 - Sinh[(c + d*x)/2]*#1]*#1)/(a + b + 2*a*#1^2 -
2*b*#1^2 + a*#1^4 + b*#1^4) & ] + 3*(Csch[(c + d*x)/2]^2 + 4*Log[Tanh[(c + d*x)/2]] + Sech[(c + d*x)/2]^2))/(2
4*a*d)

4.8.2 Maple

Integral number [74] $$\int \frac{{\sinh }^3(c+dx)}{a+b{\tanh }^3(c+dx)}dx$$

[B]   time = 0.11 (sec), size = 346 ,normalized size = 10.81 $$-8\frac{1}{d(16a-16b){(\tanh (1/2dx+c/2)+1)}^2}+\frac{16}{3d(16a-16b)}{(\tanh (\frac{dx}{2}+\frac{c}{2})+1)}^{-3}-\frac{a}{2d{(a-b)}^2}{(\tanh (\frac{dx}{2}+\frac{c}{2})+1)}^{-1}-\frac{b}{d{(a-b)}^2}{(\tanh (\frac{dx}{2}+\frac{c}{2})+1)}^{-1}-\frac{16}{3d(16a+16b)}{(\tanh (\frac{dx}{2}+\frac{c}{2})-1)}^{-3}-8\frac{1}{d(16a+16b){(\tanh (1/2dx+c/2)-1)}^2}+\frac{a}{2d{(a+b)}^2}{(\tanh (\frac{dx}{2}+\frac{c}{2})-1)}^{-1}-\frac{b}{d{(a+b)}^2}{(\tanh (\frac{dx}{2}+\frac{c}{2})-1)}^{-1}-\frac{ab}{3d{(a+b)}^2{(a-b)}^2}\sum _{\mathit{\_}\mathit{R}=\mathit{R}\mathit{o}\mathit{o}\mathit{t}\mathit{O}\mathit{f}(a{\mathit{\_}\mathit{Z}}^6+3a{\mathit{\_}\mathit{Z}}^4+8b{\mathit{\_}\mathit{Z}}^3+3a{\mathit{\_}\mathit{Z}}^2+a)}\frac{(2a^2+b^2){\mathit{\_}\mathit{R}}^4-6{\mathit{\_}\mathit{R}}^{3}ab+2(4a^2+5b^2){\mathit{\_}\mathit{R}}^2-6ab\mathit{\_}\mathit{R}+2a^2+b^2}{{\mathit{\_}\mathit{R}}^{5}a+2{\mathit{\_}\mathit{R}}^{3}a+4{\mathit{\_}\mathit{R}}^{2}b+\mathit{\_}\mathit{R}a}\ln (\tanh (\frac{dx}{2}+\frac{c}{2})-\mathit{\_}\mathit{R})$$

[In]

int(sinh(d*x+c)^3/(a+b*tanh(d*x+c)^3),x)

[Out]

-8/d/(16*a-16*b)/(tanh(1/2*d*x+1/2*c)+1)^2+16/3/d/(tanh(1/2*d*x+1/2*c)+1)^3/(16*a-16*b)-1/2/d/(a-b)^2/(tanh(1/
2*d*x+1/2*c)+1)*a-1/d/(a-b)^2/(tanh(1/2*d*x+1/2*c)+1)*b-16/3/d/(tanh(1/2*d*x+1/2*c)-1)^3/(16*a+16*b)-8/d/(16*a
+16*b)/(tanh(1/2*d*x+1/2*c)-1)^2+1/2/d/(a+b)^2/(tanh(1/2*d*x+1/2*c)-1)*a-1/d/(a+b)^2/(tanh(1/2*d*x+1/2*c)-1)*b
-1/3/d*a*b/(a+b)^2/(a-b)^2*sum(((2*a^2+b^2)*_R^4-6*_R^3*a*b+2*(4*a^2+5*b^2)*_R^2-6*a*b*_R+2*a^2+b^2)/(_R^5*a+2
*_R^3*a+4*_R^2*b+_R*a)*ln(tanh(1/2*d*x+1/2*c)-_R),_R=RootOf(_Z^6*a+3*_Z^4*a+8*_Z^3*b+3*_Z^2*a+a))

Integral number [76] $$\int \frac{\sinh (c+dx)}{a+b{\tanh }^3(c+dx)}dx$$

[B]   time = 0.108 (sec), size = 164 ,normalized size = 5.47 $$-4\frac{1}{d(4a+4b)(\tanh (1/2dx+c/2)-1)}+\frac{b}{3d(a-b)(a+b)}\sum _{\mathit{\_}\mathit{R}=\mathit{R}\mathit{o}\mathit{o}\mathit{t}\mathit{O}\mathit{f}(a{\mathit{\_}\mathit{Z}}^6+3a{\mathit{\_}\mathit{Z}}^4+8b{\mathit{\_}\mathit{Z}}^3+3a{\mathit{\_}\mathit{Z}}^2+a)}\frac{{\mathit{\_}\mathit{R}}^{4}a-2{\mathit{\_}\mathit{R}}^{3}b+6{\mathit{\_}\mathit{R}}^{2}a-2\mathit{\_}\mathit{R}b+a}{{\mathit{\_}\mathit{R}}^{5}a+2{\mathit{\_}\mathit{R}}^{3}a+4{\mathit{\_}\mathit{R}}^{2}b+\mathit{\_}\mathit{R}a}\ln (\tanh (\frac{dx}{2}+\frac{c}{2})-\mathit{\_}\mathit{R})+4\frac{1}{d(4a-4b)(\tanh (1/2dx+c/2)+1)}$$

[In]

int(sinh(d*x+c)/(a+b*tanh(d*x+c)^3),x)

[Out]

-4/d/(4*a+4*b)/(tanh(1/2*d*x+1/2*c)-1)+1/3/d*b/(a-b)/(a+b)*sum((_R^4*a-2*_R^3*b+6*_R^2*a-2*_R*b+a)/(_R^5*a+2*_
R^3*a+4*_R^2*b+_R*a)*ln(tanh(1/2*d*x+1/2*c)-_R),_R=RootOf(_Z^6*a+3*_Z^4*a+8*_Z^3*b+3*_Z^2*a+a))+4/d/(4*a-4*b)/
(tanh(1/2*d*x+1/2*c)+1)

Integral number [77] $$\int \frac{\text{csch}(c+dx)}{a+b{\tanh }^3(c+dx)}dx$$

[B]   time = 0.105 (sec), size = 98 ,normalized size = 3.27 $$\frac{1}{da}\ln (\tanh (\frac{dx}{2}+\frac{c}{2}))-\frac{4b}{3da}\sum _{\mathit{\_}\mathit{R}=\mathit{R}\mathit{o}\mathit{o}\mathit{t}\mathit{O}\mathit{f}(a{\mathit{\_}\mathit{Z}}^6+3a{\mathit{\_}\mathit{Z}}^4+8b{\mathit{\_}\mathit{Z}}^3+3a{\mathit{\_}\mathit{Z}}^2+a)}\frac{{\mathit{\_}\mathit{R}}^2}{{\mathit{\_}\mathit{R}}^{5}a+2{\mathit{\_}\mathit{R}}^{3}a+4{\mathit{\_}\mathit{R}}^{2}b+\mathit{\_}\mathit{R}a}\ln (\tanh (\frac{dx}{2}+\frac{c}{2})-\mathit{\_}\mathit{R})$$

[In]

int(csch(d*x+c)/(a+b*tanh(d*x+c)^3),x)

[Out]

1/d/a*ln(tanh(1/2*d*x+1/2*c))-4/3/d/a*b*sum(_R^2/(_R^5*a+2*_R^3*a+4*_R^2*b+_R*a)*ln(tanh(1/2*d*x+1/2*c)-_R),_R
=RootOf(_Z^6*a+3*_Z^4*a+8*_Z^3*b+3*_Z^2*a+a))

Integral number [79] $$\int \frac{{\text{csch}}^3(c+dx)}{a+b{\tanh }^3(c+dx)}dx$$

[B]   time = 0.125 (sec), size = 144 ,normalized size = 4.5 $$\frac{1}{8da}{(\tanh (\frac{dx}{2}+\frac{c}{2}))}^2-\frac{1}{8da}{(\tanh (\frac{dx}{2}+\frac{c}{2}))}^{-2}-\frac{1}{2da}\ln (\tanh (\frac{dx}{2}+\frac{c}{2}))-\frac{b}{3da}\sum _{\mathit{\_}\mathit{R}=\mathit{R}\mathit{o}\mathit{o}\mathit{t}\mathit{O}\mathit{f}(a{\mathit{\_}\mathit{Z}}^6+3a{\mathit{\_}\mathit{Z}}^4+8b{\mathit{\_}\mathit{Z}}^3+3a{\mathit{\_}\mathit{Z}}^2+a)}\frac{{\mathit{\_}\mathit{R}}^4-2{\mathit{\_}\mathit{R}}^2+1}{{\mathit{\_}\mathit{R}}^{5}a+2{\mathit{\_}\mathit{R}}^{3}a+4{\mathit{\_}\mathit{R}}^{2}b+\mathit{\_}\mathit{R}a}\ln (\tanh (\frac{dx}{2}+\frac{c}{2})-\mathit{\_}\mathit{R})$$

[In]

int(csch(d*x+c)^3/(a+b*tanh(d*x+c)^3),x)

[Out]

1/8/d/a*tanh(1/2*d*x+1/2*c)^2-1/8/d/a/tanh(1/2*d*x+1/2*c)^2-1/2/d/a*ln(tanh(1/2*d*x+1/2*c))-1/3/d/a*b*sum((_R^
4-2*_R^2+1)/(_R^5*a+2*_R^3*a+4*_R^2*b+_R*a)*ln(tanh(1/2*d*x+1/2*c)-_R),_R=RootOf(_Z^6*a+3*_Z^4*a+8*_Z^3*b+3*_Z
^2*a+a))

4.8.3 Giac

Integral number [74] $$\int \frac{{\sinh }^3(c+dx)}{a+b{\tanh }^3(c+dx)}dx$$

[B]   time = 2.05765 (sec), size = 473 ,normalized size = 14.78 $$-\frac{\frac{(9a{e}^{(2dx+2c)}+9b{e}^{(2dx+2c)}-a+b)e^{(-3dx)}}{a^2{e}^{\left(3c\right)}-2ab{e}^{\left(3c\right)}+b^2{e}^{\left(3c\right)}}-\frac{a^2{e}^{(3dx+30c)}+2ab{e}^{(3dx+30c)}+b^2{e}^{(3dx+30c)}-9a^2{e}^{(dx+28c)}+9b^2{e}^{(dx+28c)}}{a^3{e}^{\left(27c\right)}+3a^{2}b{e}^{\left(27c\right)}+3a{b}^2{e}^{\left(27c\right)}+b^3{e}^{\left(27c\right)}}}{24d}-\frac{\frac{6(a^{3}b{e}^c+a^2{b}^2{e}^c+a{b}^3{e}^c)dx}{ad-bd}-\frac{(a^{3}b{e}^c+a^2{b}^2{e}^c+a{b}^3{e}^c)\log \left(|a{e}^{(6dx+6c)}+b{e}^{(6dx+6c)}+3a{e}^{(4dx+4c)}-3b{e}^{(4dx+4c)}+3a{e}^{(2dx+2c)}+3b{e}^{(2dx+2c)}+a-b|\right)}{ad-bd}}{(a^4-2a^2{b}^2+b^4)d}$$

[In]

integrate(sinh(d*x+c)^3/(a+b*tanh(d*x+c)^3),x, algorithm="giac")

[Out]

-1/24*((9*a*e^(2*d*x + 2*c) + 9*b*e^(2*d*x + 2*c) - a + b)*e^(-3*d*x)/(a^2*e^(3*c) - 2*a*b*e^(3*c) + b^2*e^(3*
c)) - (a^2*e^(3*d*x + 30*c) + 2*a*b*e^(3*d*x + 30*c) + b^2*e^(3*d*x + 30*c) - 9*a^2*e^(d*x + 28*c) + 9*b^2*e^(
d*x + 28*c))/(a^3*e^(27*c) + 3*a^2*b*e^(27*c) + 3*a*b^2*e^(27*c) + b^3*e^(27*c)))/d - (6*(a^3*b*e^c + a^2*b^2*
e^c + a*b^3*e^c)*d*x/(a*d - b*d) - (a^3*b*e^c + a^2*b^2*e^c + a*b^3*e^c)*log(abs(a*e^(6*d*x + 6*c) + b*e^(6*d*
x + 6*c) + 3*a*e^(4*d*x + 4*c) - 3*b*e^(4*d*x + 4*c) + 3*a*e^(2*d*x + 2*c) + 3*b*e^(2*d*x + 2*c) + a - b))/(a*
d - b*d))/((a^4 - 2*a^2*b^2 + b^4)*d)

Integral number [76] $$\int \frac{\sinh (c+dx)}{a+b{\tanh }^3(c+dx)}dx$$

[B]   time = 1.62107 (sec), size = 263 ,normalized size = 8.77 $$\frac{\frac{e^{(dx+8c)}}{a{e}^{\left(7c\right)}+b{e}^{\left(7c\right)}}+\frac{e^{(-dx)}}{a{e}^c-b{e}^c}}{2d}+\frac{\frac{6(2ab{e}^c+b^2{e}^c)dx}{ad-bd}-\frac{(2ab{e}^c+b^2{e}^c)\log \left(|a{e}^{(6dx+6c)}+b{e}^{(6dx+6c)}+3a{e}^{(4dx+4c)}-3b{e}^{(4dx+4c)}+3a{e}^{(2dx+2c)}+3b{e}^{(2dx+2c)}+a-b|\right)}{ad-bd}}{3(a^2-b^2)d}$$

[In]

integrate(sinh(d*x+c)/(a+b*tanh(d*x+c)^3),x, algorithm="giac")

[Out]

1/2*(e^(d*x + 8*c)/(a*e^(7*c) + b*e^(7*c)) + e^(-d*x)/(a*e^c - b*e^c))/d + 1/3*(6*(2*a*b*e^c + b^2*e^c)*d*x/(a
*d - b*d) - (2*a*b*e^c + b^2*e^c)*log(abs(a*e^(6*d*x + 6*c) + b*e^(6*d*x + 6*c) + 3*a*e^(4*d*x + 4*c) - 3*b*e^
(4*d*x + 4*c) + 3*a*e^(2*d*x + 2*c) + 3*b*e^(2*d*x + 2*c) + a - b))/(a*d - b*d))/((a^2 - b^2)*d)

Integral number [77] $$\int \frac{\text{csch}(c+dx)}{a+b{\tanh }^3(c+dx)}dx$$

[B]   time = 1.45323 (sec), size = 207 ,normalized size = 6.9 $$-\frac{\frac{\log (e^{(dx+c)}+1)}{a}-\frac{\log \left(|e^{(dx+c)}-1|\right)}{a}}{d}-\frac{\frac{6bdx{e}^c}{ad-bd}-\frac{b{e}^c\log \left(|a{e}^{(6dx+6c)}+b{e}^{(6dx+6c)}+3a{e}^{(4dx+4c)}-3b{e}^{(4dx+4c)}+3a{e}^{(2dx+2c)}+3b{e}^{(2dx+2c)}+a-b|\right)}{ad-bd}}{3ad}$$

[In]

integrate(csch(d*x+c)/(a+b*tanh(d*x+c)^3),x, algorithm="giac")

[Out]

-(log(e^(d*x + c) + 1)/a - log(abs(e^(d*x + c) - 1))/a)/d - 1/3*(6*b*d*x*e^c/(a*d - b*d) - b*e^c*log(abs(a*e^(
6*d*x + 6*c) + b*e^(6*d*x + 6*c) + 3*a*e^(4*d*x + 4*c) - 3*b*e^(4*d*x + 4*c) + 3*a*e^(2*d*x + 2*c) + 3*b*e^(2*
d*x + 2*c) + a - b))/(a*d - b*d))/(a*d)