3.8 Test file Number [173]

3.8.1 Mathematica

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

[B]   time = 0.388247 (sec), size = 826 ,normalized size = 25.03 $$\frac{-9a(a^2+3b^2)\cosh (c+dx)+a^3\cosh (3(c+dx))-a{b}^2\cosh (3(c+dx))-2ab\text{RootSum}[a-b+3a{\mathrm{\#}\text{1}}^2+3b{\mathrm{\#}\text{1}}^2+3a{\mathrm{\#}\text{1}}^4-3b{\mathrm{\#}\text{1}}^4+a{\mathrm{\#}\text{1}}^6+b{\mathrm{\#}\text{1}}^6\mathrm{\&},\frac{3a^{2}c+3abc+3b^{2}c+3a^{2}dx+3abdx+3b^{2}dx+6a^2\log (-\cosh (\frac{1}{2}(c+dx))-\sinh (\frac{1}{2}(c+dx))+\cosh (\frac{1}{2}(c+dx))\mathrm{\#}\text{1}-\sinh (\frac{1}{2}(c+dx))\mathrm{\#}\text{1})+6ab\log (-\cosh (\frac{1}{2}(c+dx))-\sinh (\frac{1}{2}(c+dx))+\cosh (\frac{1}{2}(c+dx))\mathrm{\#}\text{1}-\sinh (\frac{1}{2}(c+dx))\mathrm{\#}\text{1})+6b^2\log (-\cosh (\frac{1}{2}(c+dx))-\sinh (\frac{1}{2}(c+dx))+\cosh (\frac{1}{2}(c+dx))\mathrm{\#}\text{1}-\sinh (\frac{1}{2}(c+dx))\mathrm{\#}\text{1})+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 (-\cosh (\frac{1}{2}(c+dx))-\sinh (\frac{1}{2}(c+dx))+\cosh (\frac{1}{2}(c+dx))\mathrm{\#}\text{1}-\sinh (\frac{1}{2}(c+dx))\mathrm{\#}\text{1}){\mathrm{\#}\text{1}}^2-4b^2\log (-\cosh (\frac{1}{2}(c+dx))-\sinh (\frac{1}{2}(c+dx))+\cosh (\frac{1}{2}(c+dx))\mathrm{\#}\text{1}-\sinh (\frac{1}{2}(c+dx))\mathrm{\#}\text{1}){\mathrm{\#}\text{1}}^2+3a^{2}c{\mathrm{\#}\text{1}}^4-3abc{\mathrm{\#}\text{1}}^4+3b^{2}c{\mathrm{\#}\text{1}}^4+3a^{2}dx{\mathrm{\#}\text{1}}^4-3abdx{\mathrm{\#}\text{1}}^4+3b^{2}dx{\mathrm{\#}\text{1}}^4+6a^2\log (-\cosh (\frac{1}{2}(c+dx))-\sinh (\frac{1}{2}(c+dx))+\cosh (\frac{1}{2}(c+dx))\mathrm{\#}\text{1}-\sinh (\frac{1}{2}(c+dx))\mathrm{\#}\text{1}){\mathrm{\#}\text{1}}^4-6ab\log (-\cosh (\frac{1}{2}(c+dx))-\sinh (\frac{1}{2}(c+dx))+\cosh (\frac{1}{2}(c+dx))\mathrm{\#}\text{1}-\sinh (\frac{1}{2}(c+dx))\mathrm{\#}\text{1}){\mathrm{\#}\text{1}}^4+6b^2\log (-\cosh (\frac{1}{2}(c+dx))-\sinh (\frac{1}{2}(c+dx))+\cosh (\frac{1}{2}(c+dx))\mathrm{\#}\text{1}-\sinh (\frac{1}{2}(c+dx))\mathrm{\#}\text{1}){\mathrm{\#}\text{1}}^4}{a\mathrm{\#}\text{1}+b\mathrm{\#}\text{1}+2a{\mathrm{\#}\text{1}}^3-2b{\mathrm{\#}\text{1}}^3+a{\mathrm{\#}\text{1}}^5+b{\mathrm{\#}\text{1}}^5}\mathrm{\&}]+27a^{2}b\sinh (c+dx)+9b^3\sinh (c+dx)-a^{2}b\sinh (3(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 = 0.164957 (sec), size = 409 ,normalized size = 13.19 $$\frac{6a\cosh (c+dx)+b\text{RootSum}[a-b+3a{\mathrm{\#}\text{1}}^2+3b{\mathrm{\#}\text{1}}^2+3a{\mathrm{\#}\text{1}}^4-3b{\mathrm{\#}\text{1}}^4+a{\mathrm{\#}\text{1}}^6+b{\mathrm{\#}\text{1}}^6\mathrm{\&},\frac{2ac+bc+2adx+bdx+4a\log (-\cosh (\frac{1}{2}(c+dx))-\sinh (\frac{1}{2}(c+dx))+\cosh (\frac{1}{2}(c+dx))\mathrm{\#}\text{1}-\sinh (\frac{1}{2}(c+dx))\mathrm{\#}\text{1})+2b\log (-\cosh (\frac{1}{2}(c+dx))-\sinh (\frac{1}{2}(c+dx))+\cosh (\frac{1}{2}(c+dx))\mathrm{\#}\text{1}-\sinh (\frac{1}{2}(c+dx))\mathrm{\#}\text{1})+2ac{\mathrm{\#}\text{1}}^4-bc{\mathrm{\#}\text{1}}^4+2adx{\mathrm{\#}\text{1}}^4-bdx{\mathrm{\#}\text{1}}^4+4a\log (-\cosh (\frac{1}{2}(c+dx))-\sinh (\frac{1}{2}(c+dx))+\cosh (\frac{1}{2}(c+dx))\mathrm{\#}\text{1}-\sinh (\frac{1}{2}(c+dx))\mathrm{\#}\text{1}){\mathrm{\#}\text{1}}^4-2b\log (-\cosh (\frac{1}{2}(c+dx))-\sinh (\frac{1}{2}(c+dx))+\cosh (\frac{1}{2}(c+dx))\mathrm{\#}\text{1}-\sinh (\frac{1}{2}(c+dx))\mathrm{\#}\text{1}){\mathrm{\#}\text{1}}^4}{a\mathrm{\#}\text{1}+b\mathrm{\#}\text{1}+2a{\mathrm{\#}\text{1}}^3-2b{\mathrm{\#}\text{1}}^3+a{\mathrm{\#}\text{1}}^5+b{\mathrm{\#}\text{1}}^5}\mathrm{\&}]-6b\sinh (c+dx)}{6(a-b)(a+b)d}$$

[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 = 0.124779 (sec), size = 319 ,normalized size = 10.29 $$\frac{6\log (\tanh (\frac{1}{2}(c+dx)))-b\text{RootSum}[a-b+3a{\mathrm{\#}\text{1}}^2+3b{\mathrm{\#}\text{1}}^2+3a{\mathrm{\#}\text{1}}^4-3b{\mathrm{\#}\text{1}}^4+a{\mathrm{\#}\text{1}}^6+b{\mathrm{\#}\text{1}}^6\mathrm{\&},\frac{c+dx+2\log (-\cosh (\frac{1}{2}(c+dx))-\sinh (\frac{1}{2}(c+dx))+\cosh (\frac{1}{2}(c+dx))\mathrm{\#}\text{1}-\sinh (\frac{1}{2}(c+dx))\mathrm{\#}\text{1})-2c{\mathrm{\#}\text{1}}^2-2dx{\mathrm{\#}\text{1}}^2-4\log (-\cosh (\frac{1}{2}(c+dx))-\sinh (\frac{1}{2}(c+dx))+\cosh (\frac{1}{2}(c+dx))\mathrm{\#}\text{1}-\sinh (\frac{1}{2}(c+dx))\mathrm{\#}\text{1}){\mathrm{\#}\text{1}}^2+c{\mathrm{\#}\text{1}}^4+dx{\mathrm{\#}\text{1}}^4+2\log (-\cosh (\frac{1}{2}(c+dx))-\sinh (\frac{1}{2}(c+dx))+\cosh (\frac{1}{2}(c+dx))\mathrm{\#}\text{1}-\sinh (\frac{1}{2}(c+dx))\mathrm{\#}\text{1}){\mathrm{\#}\text{1}}^4}{a\mathrm{\#}\text{1}+b\mathrm{\#}\text{1}+2a{\mathrm{\#}\text{1}}^3-2b{\mathrm{\#}\text{1}}^3+a{\mathrm{\#}\text{1}}^5+b{\mathrm{\#}\text{1}}^5}\mathrm{\&}]}{6ad}$$

[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 = 0.279736 (sec), size = 201 ,normalized size = 6.09 $$-\frac{16b\text{RootSum}[a-b+3a{\mathrm{\#}\text{1}}^2+3b{\mathrm{\#}\text{1}}^2+3a{\mathrm{\#}\text{1}}^4-3b{\mathrm{\#}\text{1}}^4+a{\mathrm{\#}\text{1}}^6+b{\mathrm{\#}\text{1}}^6\mathrm{\&},\frac{c\mathrm{\#}\text{1}+dx\mathrm{\#}\text{1}+2\log (-\cosh (\frac{1}{2}(c+dx))-\sinh (\frac{1}{2}(c+dx))+\cosh (\frac{1}{2}(c+dx))\mathrm{\#}\text{1}-\sinh (\frac{1}{2}(c+dx))\mathrm{\#}\text{1})\mathrm{\#}\text{1}}{a+b+2a{\mathrm{\#}\text{1}}^2-2b{\mathrm{\#}\text{1}}^2+a{\mathrm{\#}\text{1}}^4+b{\mathrm{\#}\text{1}}^4}\mathrm{\&}]+3({\text{csch}}^2(\frac{1}{2}(c+dx))+4\log (\tanh (\frac{1}{2}(c+dx)))+{\text{sech}}^2(\frac{1}{2}(c+dx)))}{24ad}$$

[In]

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

[Out]

-1/24*(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
))/(a*d)

3.8.2 Maple

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

[B]   time = 7.433 (sec), size = 289 ,normalized size = 8.76

method	result	size
derivativedivides	$\frac{-\frac{ab(\sum _{\mathit{\text{\_R}}=\mathrm{RootOf}(a{\mathit{\text{\_Z}}}^6+3a{\mathit{\text{\_Z}}}^4+8b{\mathit{\text{\_Z}}}^3+3a{\mathit{\text{\_Z}}}^2+a)}\frac{((2a^2+b^2){\mathit{\text{\_R}}}^4-6ab{\mathit{\text{\_R}}}^3+2(4a^2+5b^2){\mathit{\text{\_R}}}^2-6ab\mathit{\text{\_R}}+2a^2+b^2)\ln (\tanh (\frac{dx}{2}+\frac{c}{2})-\mathit{\text{\_R}})}{{\mathit{\text{\_R}}}^{5}a+2{\mathit{\text{\_R}}}^{3}a+4{\mathit{\text{\_R}}}^{2}b+\mathit{\text{\_R}}a})}{3{(a+b)}^2{(a-b)}^2}-\frac{8}{(16a-16b){(\tanh (\frac{dx}{2}+\frac{c}{2})+1)}^2}+\frac{16}{3{(\tanh (\frac{dx}{2}+\frac{c}{2})+1)}^3(16a-16b)}-\frac{2b+a}{2{(a-b)}^2(\tanh (\frac{dx}{2}+\frac{c}{2})+1)}-\frac{16}{3{(\tanh (\frac{dx}{2}+\frac{c}{2})-1)}^3(16a+16b)}-\frac{8}{(16a+16b){(\tanh (\frac{dx}{2}+\frac{c}{2})-1)}^2}-\frac{2b-a}{2{(a+b)}^2(\tanh (\frac{dx}{2}+\frac{c}{2})-1)}}{d}$	$289$
default	$\frac{-\frac{ab(\sum _{\mathit{\text{\_R}}=\mathrm{RootOf}(a{\mathit{\text{\_Z}}}^6+3a{\mathit{\text{\_Z}}}^4+8b{\mathit{\text{\_Z}}}^3+3a{\mathit{\text{\_Z}}}^2+a)}\frac{((2a^2+b^2){\mathit{\text{\_R}}}^4-6ab{\mathit{\text{\_R}}}^3+2(4a^2+5b^2){\mathit{\text{\_R}}}^2-6ab\mathit{\text{\_R}}+2a^2+b^2)\ln (\tanh (\frac{dx}{2}+\frac{c}{2})-\mathit{\text{\_R}})}{{\mathit{\text{\_R}}}^{5}a+2{\mathit{\text{\_R}}}^{3}a+4{\mathit{\text{\_R}}}^{2}b+\mathit{\text{\_R}}a})}{3{(a+b)}^2{(a-b)}^2}-\frac{8}{(16a-16b){(\tanh (\frac{dx}{2}+\frac{c}{2})+1)}^2}+\frac{16}{3{(\tanh (\frac{dx}{2}+\frac{c}{2})+1)}^3(16a-16b)}-\frac{2b+a}{2{(a-b)}^2(\tanh (\frac{dx}{2}+\frac{c}{2})+1)}-\frac{16}{3{(\tanh (\frac{dx}{2}+\frac{c}{2})-1)}^3(16a+16b)}-\frac{8}{(16a+16b){(\tanh (\frac{dx}{2}+\frac{c}{2})-1)}^2}-\frac{2b-a}{2{(a+b)}^2(\tanh (\frac{dx}{2}+\frac{c}{2})-1)}}{d}$	$289$
risch	$\text{Expression too large to display}$	$2825$

[In]

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

[Out]

1/d*(-1/3*a*b/(a+b)^2/(a-b)^2*sum(((2*a^2+b^2)*_R^4-6*a*b*_R^3+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))-8/(16*a-1
6*b)/(tanh(1/2*d*x+1/2*c)+1)^2+16/3/(tanh(1/2*d*x+1/2*c)+1)^3/(16*a-16*b)-1/2*(2*b+a)/(a-b)^2/(tanh(1/2*d*x+1/
2*c)+1)-16/3/(tanh(1/2*d*x+1/2*c)-1)^3/(16*a+16*b)-8/(16*a+16*b)/(tanh(1/2*d*x+1/2*c)-1)^2-1/2/(a+b)^2*(2*b-a)
/(tanh(1/2*d*x+1/2*c)-1))

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

[B]   time = 5.369 (sec), size = 159 ,normalized size = 5.13 $$\text{result too large to display}$$

[In]

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

[Out]

1/d*(1/3*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/(4*a+4*b)/(tanh(1/2*d*x+1/2*c)-1)+4/(4*a-4*b)/(t
anh(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 = 4.535 (sec), size = 96 ,normalized size = 3.1

method	result	size
derivativedivides	$\frac{\frac{\ln (\tanh (\frac{dx}{2}+\frac{c}{2}))}{a}-\frac{4b(\sum _{\mathit{\text{\_R}}=\mathrm{RootOf}(a{\mathit{\text{\_Z}}}^6+3a{\mathit{\text{\_Z}}}^4+8b{\mathit{\text{\_Z}}}^3+3a{\mathit{\text{\_Z}}}^2+a)}\frac{{\mathit{\text{\_R}}}^2\ln (\tanh (\frac{dx}{2}+\frac{c}{2})-\mathit{\text{\_R}})}{{\mathit{\text{\_R}}}^{5}a+2{\mathit{\text{\_R}}}^{3}a+4{\mathit{\text{\_R}}}^{2}b+\mathit{\text{\_R}}a})}{3a}}{d}$	$96$
default	$\frac{\frac{\ln (\tanh (\frac{dx}{2}+\frac{c}{2}))}{a}-\frac{4b(\sum _{\mathit{\text{\_R}}=\mathrm{RootOf}(a{\mathit{\text{\_Z}}}^6+3a{\mathit{\text{\_Z}}}^4+8b{\mathit{\text{\_Z}}}^3+3a{\mathit{\text{\_Z}}}^2+a)}\frac{{\mathit{\text{\_R}}}^2\ln (\tanh (\frac{dx}{2}+\frac{c}{2})-\mathit{\text{\_R}})}{{\mathit{\text{\_R}}}^{5}a+2{\mathit{\text{\_R}}}^{3}a+4{\mathit{\text{\_R}}}^{2}b+\mathit{\text{\_R}}a})}{3a}}{d}$	$96$
risch	$-\frac{\ln ({\mathrm{e}}^{dx+c}+1)}{da}+2(\sum _{\mathit{\text{\_R}}=\mathrm{RootOf}((46656a^8{d}^6-46656a^6{b}^2{d}^6){\mathit{\text{\_Z}}}^6+3888a^4{b}^2{d}^4{\mathit{\text{\_Z}}}^4-108a^2{b}^2{d}^2{\mathit{\text{\_Z}}}^2+b^2)}\mathit{\text{\_R}}\ln ({\mathrm{e}}^{dx+c}+(\frac{7776d^5{a}^6}{b}-7776b{d}^5{a}^4){\mathit{\text{\_R}}}^5+(\frac{216d^3{a}^4}{b}+432a^{2}b{d}^3){\mathit{\text{\_R}}}^3+(6ad-6bd)\mathit{\text{\_R}}))+\frac{\ln ({\mathrm{e}}^{dx+c}-1)}{da}$	$168$

[In]

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

[Out]

1/d*(1/a*ln(tanh(1/2*d*x+1/2*c))-4/3/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 = 4.852 (sec), size = 136 ,normalized size = 4.12

method	result	size
derivativedivides	$\frac{\frac{{\tanh }^2(\frac{dx}{2}+\frac{c}{2})}{8a}-\frac{1}{8a\tanh {(\frac{dx}{2}+\frac{c}{2})}^2}-\frac{\ln (\tanh (\frac{dx}{2}+\frac{c}{2}))}{2a}-\frac{b(\sum _{\mathit{\text{\_R}}=\mathrm{RootOf}(a{\mathit{\text{\_Z}}}^6+3a{\mathit{\text{\_Z}}}^4+8b{\mathit{\text{\_Z}}}^3+3a{\mathit{\text{\_Z}}}^2+a)}\frac{({\mathit{\text{\_R}}}^4-2{\mathit{\text{\_R}}}^2+1)\ln (\tanh (\frac{dx}{2}+\frac{c}{2})-\mathit{\text{\_R}})}{{\mathit{\text{\_R}}}^{5}a+2{\mathit{\text{\_R}}}^{3}a+4{\mathit{\text{\_R}}}^{2}b+\mathit{\text{\_R}}a})}{3a}}{d}$	$136$
default	$\frac{\frac{{\tanh }^2(\frac{dx}{2}+\frac{c}{2})}{8a}-\frac{1}{8a\tanh {(\frac{dx}{2}+\frac{c}{2})}^2}-\frac{\ln (\tanh (\frac{dx}{2}+\frac{c}{2}))}{2a}-\frac{b(\sum _{\mathit{\text{\_R}}=\mathrm{RootOf}(a{\mathit{\text{\_Z}}}^6+3a{\mathit{\text{\_Z}}}^4+8b{\mathit{\text{\_Z}}}^3+3a{\mathit{\text{\_Z}}}^2+a)}\frac{({\mathit{\text{\_R}}}^4-2{\mathit{\text{\_R}}}^2+1)\ln (\tanh (\frac{dx}{2}+\frac{c}{2})-\mathit{\text{\_R}})}{{\mathit{\text{\_R}}}^{5}a+2{\mathit{\text{\_R}}}^{3}a+4{\mathit{\text{\_R}}}^{2}b+\mathit{\text{\_R}}a})}{3a}}{d}$	$136$
risch	$-\frac{{\mathrm{e}}^{dx+c}(1+{\mathrm{e}}^{2dx+2c})}{da{({\mathrm{e}}^{2dx+2c}-1)}^2}+\frac{\ln ({\mathrm{e}}^{dx+c}+1)}{2da}-\frac{\ln ({\mathrm{e}}^{dx+c}-1)}{2da}+8(\sum _{\mathit{\text{\_R}}=\mathrm{RootOf}(191102976d^6{\mathit{\text{\_Z}}}^6{a}^{10}+1728a^4{d}^2{\mathit{\text{\_Z}}}^2{b}^2+a^2{b}^2-b^4)}\mathit{\text{\_R}}\ln ({\mathrm{e}}^{dx+c}+\frac{15925248d^5{a}^9{\mathit{\text{\_R}}}^5}{a^{3}b+a^2{b}^2+a{b}^3+b^4}+(-\frac{13824d^3{a}^7}{a^{3}b+a^2{b}^2+a{b}^3+b^4}+\frac{13824d^3{b}^2{a}^5}{a^{3}b+a^2{b}^2+a{b}^3+b^4}){\mathit{\text{\_R}}}^3+(-\frac{24d{a}^{4}b}{a^{3}b+a^2{b}^2+a{b}^3+b^4}+\frac{96d{b}^2{a}^3}{a^{3}b+a^2{b}^2+a{b}^3+b^4}-\frac{24d{b}^3{a}^2}{a^{3}b+a^2{b}^2+a{b}^3+b^4})\mathit{\text{\_R}}))$	$329$

[In]

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

[Out]

1/d*(1/8*tanh(1/2*d*x+1/2*c)^2/a-1/8/a/tanh(1/2*d*x+1/2*c)^2-1/2/a*ln(tanh(1/2*d*x+1/2*c))-1/3/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)))

3.8.3 Fricas

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

[C]   time = 4.53383 (sec), size = 62017 ,normalized size = 1879.3 $$\text{Too large to display}$$

[In]

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

[Out]

1/24*((a^3 - a^2*b - a*b^2 + b^3)*cosh(d*x + c)^6 + 6*(a^3 - a^2*b - a*b^2 + b^3)*cosh(d*x + c)*sinh(d*x + c)^
5 + (a^3 - a^2*b - a*b^2 + b^3)*sinh(d*x + c)^6 - 9*(a^3 - 3*a^2*b + 3*a*b^2 - b^3)*cosh(d*x + c)^4 - 3*(3*a^3
- 9*a^2*b + 9*a*b^2 - 3*b^3 - 5*(a^3 - a^2*b - a*b^2 + b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + 4*(5*(a^3 - a^
2*b - a*b^2 + b^3)*cosh(d*x + c)^3 - 9*(a^3 - 3*a^2*b + 3*a*b^2 - b^3)*cosh(d*x + c))*sinh(d*x + c)^3 - 4*sqrt
(2/3)*sqrt(1/6)*((a^4 - 2*a^2*b^2 + b^4)*d*cosh(d*x + c)^3 + 3*(a^4 - 2*a^2*b^2 + b^4)*d*cosh(d*x + c)^2*sinh(
d*x + c) + 3*(a^4 - 2*a^2*b^2 + b^4)*d*cosh(d*x + c)*sinh(d*x + c)^2 + (a^4 - 2*a^2*b^2 + b^4)*d*sinh(d*x + c)
^3)*sqrt(-(810*a^6*b^2 + 2754*a^4*b^4 + 810*a^2*b^6 - (a^10 - 5*a^8*b^2 + 10*a^6*b^4 - 10*a^4*b^6 + 5*a^2*b^8
- b^10)*((5*a^2*b^2/(a^8*d^4 - 4*a^6*b^2*d^4 + 6*a^4*b^4*d^4 - 4*a^2*b^6*d^4 + b^8*d^4) + 9*(5*a^6*b^2 + 17*a^
4*b^4 + 5*a^2*b^6)^2/(a^10*d^2 - 5*a^8*b^2*d^2 + 10*a^6*b^4*d^2 - 10*a^4*b^6*d^2 + 5*a^2*b^8*d^2 - b^10*d^2)^2
)*(-I*sqrt(3) + 1)/(-1/1458*a^2*b^2/(a^10*d^6 - 5*a^8*b^2*d^6 + 10*a^6*b^4*d^6 - 10*a^4*b^6*d^6 + 5*a^2*b^8*d^
6 - b^10*d^6) - 5/162*(5*a^6*b^2 + 17*a^4*b^4 + 5*a^2*b^6)*a^2*b^2/((a^10*d^2 - 5*a^8*b^2*d^2 + 10*a^6*b^4*d^2
- 10*a^4*b^6*d^2 + 5*a^2*b^8*d^2 - b^10*d^2)*(a^8*d^4 - 4*a^6*b^2*d^4 + 6*a^4*b^4*d^4 - 4*a^2*b^6*d^4 + b^8*d
^4)) - 1/27*(5*a^6*b^2 + 17*a^4*b^4 + 5*a^2*b^6)^3/(a^10*d^2 - 5*a^8*b^2*d^2 + 10*a^6*b^4*d^2 - 10*a^4*b^6*d^2
+ 5*a^2*b^8*d^2 - b^10*d^2)^3 + 1/1458*(a^10 - 30*a^8*b^2 - 700*a^6*b^4 - 700*a^4*b^6 - 30*a^2*b^8 + b^10)*a^
2*b^2/((a^2 - b^2)^10*d^6))^(1/3) + 81*(-1/1458*a^2*b^2/(a^10*d^6 - 5*a^8*b^2*d^6 + 10*a^6*b^4*d^6 - 10*a^4*b^
6*d^6 + 5*a^2*b^8*d^6 - b^10*d^6) - 5/162*(5*a^6*b^2 + 17*a^4*b^4 + 5*a^2*b^6)*a^2*b^2/((a^10*d^2 - 5*a^8*b^2*
d^2 + 10*a^6*b^4*d^2 - 10*a^4*b^6*d^2 + 5*a^2*b^8*d^2 - b^10*d^2)*(a^8*d^4 - 4*a^6*b^2*d^4 + 6*a^4*b^4*d^4 - 4
*a^2*b^6*d^4 + b^8*d^4)) - 1/27*(5*a^6*b^2 + 17*a^4*b^4 + 5*a^2*b^6)^3/(a^10*d^2 - 5*a^8*b^2*d^2 + 10*a^6*b^4*
d^2 - 10*a^4*b^6*d^2 + 5*a^2*b^8*d^2 - b^10*d^2)^3 + 1/1458*(a^10 - 30*a^8*b^2 - 700*a^6*b^4 - 700*a^4*b^6 - 3
0*a^2*b^8 + b^10)*a^2*b^2/((a^2 - b^2)^10*d^6))^(1/3)*(I*sqrt(3) + 1) + 54*(5*a^6*b^2 + 17*a^4*b^4 + 5*a^2*b^6
)/(a^10*d^2 - 5*a^8*b^2*d^2 + 10*a^6*b^4*d^2 - 10*a^4*b^6*d^2 + 5*a^2*b^8*d^2 - b^10*d^2))*d^2 + 3*sqrt(1/3)*(
a^10 - 5*a^8*b^2 + 10*a^6*b^4 - 10*a^4*b^6 + 5*a^2*b^8 - b^10)*d^2*sqrt((6480*a^14*b^2 + 179820*a^12*b^4 + 158
4360*a^10*b^6 + 2835972*a^8*b^8 + 1584360*a^6*b^10 + 179820*a^4*b^12 + 6480*a^2*b^14 - (a^20 - 10*a^18*b^2 + 4
5*a^16*b^4 - 120*a^14*b^6 + 210*a^12*b^8 - 252*a^10*b^10 + 210*a^8*b^12 - 120*a^6*b^14 + 45*a^4*b^16 - 10*a^2*
b^18 + b^20)*((5*a^2*b^2/(a^8*d^4 - 4*a^6*b^2*d^4 + 6*a^4*b^4*d^4 - 4*a^2*b^6*d^4 + b^8*d^4) + 9*(5*a^6*b^2 +
17*a^4*b^4 + 5*a^2*b^6)^2/(a^10*d^2 - 5*a^8*b^2*d^2 + 10*a^6*b^4*d^2 - 10*a^4*b^6*d^2 + 5*a^2*b^8*d^2 - b^10*d
^2)^2)*(-I*sqrt(3) + 1)/(-1/1458*a^2*b^2/(a^10*d^6 - 5*a^8*b^2*d^6 + 10*a^6*b^4*d^6 - 10*a^4*b^6*d^6 + 5*a^2*b
^8*d^6 - b^10*d^6) - 5/162*(5*a^6*b^2 + 17*a^4*b^4 + 5*a^2*b^6)*a^2*b^2/((a^10*d^2 - 5*a^8*b^2*d^2 + 10*a^6*b^
4*d^2 - 10*a^4*b^6*d^2 + 5*a^2*b^8*d^2 - b^10*d^2)*(a^8*d^4 - 4*a^6*b^2*d^4 + 6*a^4*b^4*d^4 - 4*a^2*b^6*d^4 +
b^8*d^4)) - 1/27*(5*a^6*b^2 + 17*a^4*b^4 + 5*a^2*b^6)^3/(a^10*d^2 - 5*a^8*b^2*d^2 + 10*a^6*b^4*d^2 - 10*a^4*b^
6*d^2 + 5*a^2*b^8*d^2 - b^10*d^2)^3 + 1/1458*(a^10 - 30*a^8*b^2 - 700*a^6*b^4 - 700*a^4*b^6 - 30*a^2*b^8 + b^1
0)*a^2*b^2/((a^2 - b^2)^10*d^6))^(1/3) + 81*(-1/1458*a^2*b^2/(a^10*d^6 - 5*a^8*b^2*d^6 + 10*a^6*b^4*d^6 - 10*a
^4*b^6*d^6 + 5*a^2*b^8*d^6 - b^10*d^6) - 5/162*(5*a^6*b^2 + 17*a^4*b^4 + 5*a^2*b^6)*a^2*b^2/((a^10*d^2 - 5*a^8
*b^2*d^2 + 10*a^6*b^4*d^2 - 10*a^4*b^6*d^2 + 5*a^2*b^8*d^2 - b^10*d^2)*(a^8*d^4 - 4*a^6*b^2*d^4 + 6*a^4*b^4*d^
4 - 4*a^2*b^6*d^4 + b^8*d^4)) - 1/27*(5*a^6*b^2 + 17*a^4*b^4 + 5*a^2*b^6)^3/(a^10*d^2 - 5*a^8*b^2*d^2 + 10*a^6
*b^4*d^2 - 10*a^4*b^6*d^2 + 5*a^2*b^8*d^2 - b^10*d^2)^3 + 1/1458*(a^10 - 30*a^8*b^2 - 700*a^6*b^4 - 700*a^4*b^
6 - 30*a^2*b^8 + b^10)*a^2*b^2/((a^2 - b^2)^10*d^6))^(1/3)*(I*sqrt(3) + 1) + 54*(5*a^6*b^2 + 17*a^4*b^4 + 5*a^
2*b^6)/(a^10*d^2 - 5*a^8*b^2*d^2 + 10*a^6*b^4*d^2 - 10*a^4*b^6*d^2 + 5*a^2*b^8*d^2 - b^10*d^2))^2*d^4 + 108*(5
*a^16*b^2 - 8*a^14*b^4 - 30*a^12*b^6 + 95*a^10*b^8 - 95*a^8*b^10 + 30*a^6*b^12 + 8*a^4*b^14 - 5*a^2*b^16)*((5*
a^2*b^2/(a^8*d^4 - 4*a^6*b^2*d^4 + 6*a^4*b^4*d^4 - 4*a^2*b^6*d^4 + b^8*d^4) + 9*(5*a^6*b^2 + 17*a^4*b^4 + 5*a^
2*b^6)^2/(a^10*d^2 - 5*a^8*b^2*d^2 + 10*a^6*b^4*d^2 - 10*a^4*b^6*d^2 + 5*a^2*b^8*d^2 - b^10*d^2)^2)*(-I*sqrt(3
) + 1)/(-1/1458*a^2*b^2/(a^10*d^6 - 5*a^8*b^2*d^6 + 10*a^6*b^4*d^6 - 10*a^4*b^6*d^6 + 5*a^2*b^8*d^6 - b^10*d^6
) - 5/162*(5*a^6*b^2 + 17*a^4*b^4 + 5*a^2*b^6)*a^2*b^2/((a^10*d^2 - 5*a^8*b^2*d^2 + 10*a^6*b^4*d^2 - 10*a^4*b^
6*d^2 + 5*a^2*b^8*d^2 - b^10*d^2)*(a^8*d^4 - 4*a^6*b^2*d^4 + 6*a^4*b^4*d^4 - 4*a^2*b^6*d^4 + b^8*d^4)) - 1/27*
(5*a^6*b^2 + 17*a^4*b^4 + 5*a^2*b^6)^3/(a^10*d^2 - 5*a^8*b^2*d^2 + 10*a^6*b^4*d^2 - 10*a^4*b^6*d^2 + 5*a^2*b^8
*d^2 - b^10*d^2)^3 + 1/1458*(a^10 - 30*a^8*b^2 ...

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

[C]   time = 1.98982 (sec), size = 40923 ,normalized size = 1320.1 $$\text{Too large to display}$$

[In]

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

[Out]

-1/6*(sqrt(2/3)*sqrt(1/2)*((a^2 - b^2)*d*cosh(d*x + c) + (a^2 - b^2)*d*sinh(d*x + c))*sqrt(-(108*a^2*b^2 + 54*
b^4 - (a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*((b^2/(a^6*d^4 - 3*a^4*b^2*d^4 + 3*a^2*b^4*d^4 - b^6*d^4) + 3*(2*a^2
*b^2 + b^4)^2/(a^6*d^2 - 3*a^4*b^2*d^2 + 3*a^2*b^4*d^2 - b^6*d^2)^2)*(-I*sqrt(3) + 1)/(-1/1458*b^2/(a^8*d^6 -
3*a^6*b^2*d^6 + 3*a^4*b^4*d^6 - a^2*b^6*d^6) - 1/54*(2*a^2*b^2 + b^4)*b^2/((a^6*d^4 - 3*a^4*b^2*d^4 + 3*a^2*b^
4*d^4 - b^6*d^4)*(a^6*d^2 - 3*a^4*b^2*d^2 + 3*a^2*b^4*d^2 - b^6*d^2)) - 1/27*(2*a^2*b^2 + b^4)^3/(a^6*d^2 - 3*
a^4*b^2*d^2 + 3*a^2*b^4*d^2 - b^6*d^2)^3 - 1/1458*(a^6 - 3*a^4*b^2 - 24*a^2*b^4 - b^6)*b^2/((a^2 - b^2)^6*a^2*
d^6))^(1/3) + 27*(-1/1458*b^2/(a^8*d^6 - 3*a^6*b^2*d^6 + 3*a^4*b^4*d^6 - a^2*b^6*d^6) - 1/54*(2*a^2*b^2 + b^4)
*b^2/((a^6*d^4 - 3*a^4*b^2*d^4 + 3*a^2*b^4*d^4 - b^6*d^4)*(a^6*d^2 - 3*a^4*b^2*d^2 + 3*a^2*b^4*d^2 - b^6*d^2))
- 1/27*(2*a^2*b^2 + b^4)^3/(a^6*d^2 - 3*a^4*b^2*d^2 + 3*a^2*b^4*d^2 - b^6*d^2)^3 - 1/1458*(a^6 - 3*a^4*b^2 -
24*a^2*b^4 - b^6)*b^2/((a^2 - b^2)^6*a^2*d^6))^(1/3)*(I*sqrt(3) + 1) + 18*(2*a^2*b^2 + b^4)/(a^6*d^2 - 3*a^4*b
^2*d^2 + 3*a^2*b^4*d^2 - b^6*d^2))*d^2 + 3*sqrt(1/3)*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*d^2*sqrt((432*a^6*b^2
+ 2592*a^4*b^4 + 5184*a^2*b^6 + 540*b^8 - (a^12 - 6*a^10*b^2 + 15*a^8*b^4 - 20*a^6*b^6 + 15*a^4*b^8 - 6*a^2*b
^10 + b^12)*((b^2/(a^6*d^4 - 3*a^4*b^2*d^4 + 3*a^2*b^4*d^4 - b^6*d^4) + 3*(2*a^2*b^2 + b^4)^2/(a^6*d^2 - 3*a^4
*b^2*d^2 + 3*a^2*b^4*d^2 - b^6*d^2)^2)*(-I*sqrt(3) + 1)/(-1/1458*b^2/(a^8*d^6 - 3*a^6*b^2*d^6 + 3*a^4*b^4*d^6
- a^2*b^6*d^6) - 1/54*(2*a^2*b^2 + b^4)*b^2/((a^6*d^4 - 3*a^4*b^2*d^4 + 3*a^2*b^4*d^4 - b^6*d^4)*(a^6*d^2 - 3*
a^4*b^2*d^2 + 3*a^2*b^4*d^2 - b^6*d^2)) - 1/27*(2*a^2*b^2 + b^4)^3/(a^6*d^2 - 3*a^4*b^2*d^2 + 3*a^2*b^4*d^2 -
b^6*d^2)^3 - 1/1458*(a^6 - 3*a^4*b^2 - 24*a^2*b^4 - b^6)*b^2/((a^2 - b^2)^6*a^2*d^6))^(1/3) + 27*(-1/1458*b^2/
(a^8*d^6 - 3*a^6*b^2*d^6 + 3*a^4*b^4*d^6 - a^2*b^6*d^6) - 1/54*(2*a^2*b^2 + b^4)*b^2/((a^6*d^4 - 3*a^4*b^2*d^4
+ 3*a^2*b^4*d^4 - b^6*d^4)*(a^6*d^2 - 3*a^4*b^2*d^2 + 3*a^2*b^4*d^2 - b^6*d^2)) - 1/27*(2*a^2*b^2 + b^4)^3/(a
^6*d^2 - 3*a^4*b^2*d^2 + 3*a^2*b^4*d^2 - b^6*d^2)^3 - 1/1458*(a^6 - 3*a^4*b^2 - 24*a^2*b^4 - b^6)*b^2/((a^2 -
b^2)^6*a^2*d^6))^(1/3)*(I*sqrt(3) + 1) + 18*(2*a^2*b^2 + b^4)/(a^6*d^2 - 3*a^4*b^2*d^2 + 3*a^2*b^4*d^2 - b^6*d
^2))^2*d^4 + 36*(2*a^8*b^2 - 5*a^6*b^4 + 3*a^4*b^6 + a^2*b^8 - b^10)*((b^2/(a^6*d^4 - 3*a^4*b^2*d^4 + 3*a^2*b^
4*d^4 - b^6*d^4) + 3*(2*a^2*b^2 + b^4)^2/(a^6*d^2 - 3*a^4*b^2*d^2 + 3*a^2*b^4*d^2 - b^6*d^2)^2)*(-I*sqrt(3) +
1)/(-1/1458*b^2/(a^8*d^6 - 3*a^6*b^2*d^6 + 3*a^4*b^4*d^6 - a^2*b^6*d^6) - 1/54*(2*a^2*b^2 + b^4)*b^2/((a^6*d^4
- 3*a^4*b^2*d^4 + 3*a^2*b^4*d^4 - b^6*d^4)*(a^6*d^2 - 3*a^4*b^2*d^2 + 3*a^2*b^4*d^2 - b^6*d^2)) - 1/27*(2*a^2
*b^2 + b^4)^3/(a^6*d^2 - 3*a^4*b^2*d^2 + 3*a^2*b^4*d^2 - b^6*d^2)^3 - 1/1458*(a^6 - 3*a^4*b^2 - 24*a^2*b^4 - b
^6)*b^2/((a^2 - b^2)^6*a^2*d^6))^(1/3) + 27*(-1/1458*b^2/(a^8*d^6 - 3*a^6*b^2*d^6 + 3*a^4*b^4*d^6 - a^2*b^6*d^
6) - 1/54*(2*a^2*b^2 + b^4)*b^2/((a^6*d^4 - 3*a^4*b^2*d^4 + 3*a^2*b^4*d^4 - b^6*d^4)*(a^6*d^2 - 3*a^4*b^2*d^2
+ 3*a^2*b^4*d^2 - b^6*d^2)) - 1/27*(2*a^2*b^2 + b^4)^3/(a^6*d^2 - 3*a^4*b^2*d^2 + 3*a^2*b^4*d^2 - b^6*d^2)^3 -
1/1458*(a^6 - 3*a^4*b^2 - 24*a^2*b^4 - b^6)*b^2/((a^2 - b^2)^6*a^2*d^6))^(1/3)*(I*sqrt(3) + 1) + 18*(2*a^2*b^
2 + b^4)/(a^6*d^2 - 3*a^4*b^2*d^2 + 3*a^2*b^4*d^2 - b^6*d^2))*d^2)/((a^12 - 6*a^10*b^2 + 15*a^8*b^4 - 20*a^6*b
^6 + 15*a^4*b^8 - 6*a^2*b^10 + b^12)*d^4)))/((a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*d^2))*log(1/36*sqrt(2/3)*sqrt
(1/2)*((4*a^12 + 3*a^11*b + a^10*b^2 - 3*a^9*b^3 - 26*a^8*b^4 - 9*a^7*b^5 + 32*a^6*b^6 + 15*a^5*b^7 - 10*a^4*b
^8 - 6*a^3*b^9 - a^2*b^10)*((b^2/(a^6*d^4 - 3*a^4*b^2*d^4 + 3*a^2*b^4*d^4 - b^6*d^4) + 3*(2*a^2*b^2 + b^4)^2/(
a^6*d^2 - 3*a^4*b^2*d^2 + 3*a^2*b^4*d^2 - b^6*d^2)^2)*(-I*sqrt(3) + 1)/(-1/1458*b^2/(a^8*d^6 - 3*a^6*b^2*d^6 +
3*a^4*b^4*d^6 - a^2*b^6*d^6) - 1/54*(2*a^2*b^2 + b^4)*b^2/((a^6*d^4 - 3*a^4*b^2*d^4 + 3*a^2*b^4*d^4 - b^6*d^4
)*(a^6*d^2 - 3*a^4*b^2*d^2 + 3*a^2*b^4*d^2 - b^6*d^2)) - 1/27*(2*a^2*b^2 + b^4)^3/(a^6*d^2 - 3*a^4*b^2*d^2 + 3
*a^2*b^4*d^2 - b^6*d^2)^3 - 1/1458*(a^6 - 3*a^4*b^2 - 24*a^2*b^4 - b^6)*b^2/((a^2 - b^2)^6*a^2*d^6))^(1/3) + 2
7*(-1/1458*b^2/(a^8*d^6 - 3*a^6*b^2*d^6 + 3*a^4*b^4*d^6 - a^2*b^6*d^6) - 1/54*(2*a^2*b^2 + b^4)*b^2/((a^6*d^4
- 3*a^4*b^2*d^4 + 3*a^2*b^4*d^4 - b^6*d^4)*(a^6*d^2 - 3*a^4*b^2*d^2 + 3*a^2*b^4*d^2 - b^6*d^2)) - 1/27*(2*a^2*
b^2 + b^4)^3/(a^6*d^2 - 3*a^4*b^2*d^2 + 3*a^2*b^4*d^2 - b^6*d^2)^3 - 1/1458*(a^6 - 3*a^4*b^2 - 24*a^2*b^4 - b^
6)*b^2/((a^2 - b^2)^6*a^2*d^6))^(1/3)*(I*sqrt(3) + 1) + 18*(2*a^2*b^2 + b^4)/(a^6*d^2 - 3*a^4*b^2*d^2 + 3*a^2*
b^4*d^2 - b^6*d^2))^2*d^5 - 6*(a^10 + a^9*b + 71*a^8*b^2 + 50*a^7*b^3 + 267*a^6*b^4 + 141*a^5*b^5 + 140*a^4*b^
6 + 50*a^3*b^7 + 7*a^2*b^8 + a*b^9)*((b^2/(a^6*d^4 - 3*a^4*b^2*d^4 + 3*a^2*b^4*d^4 - b^6*d^4) + 3*(2*a^2*b^2 +
b^4)^2/(a^6*d^2 - 3*a^4*b^2*d^2 + 3*a^2*b^4*d^2 - b^6*d^2)^2)*(-I*sqrt(3) + 1)/(-1/1458*b^2/(a^8*d^6 - 3*a^6*
b^2*d^6 + 3*a^4*b^4*d^6 - a^2*b^6*d^6) - 1/54*(...

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

[C]   time = 2.03441 (sec), size = 20085 ,normalized size = 647.9 $$\text{Too large to display}$$

[In]

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

[Out]

-1/6*(sqrt(2/3)*sqrt(1/6)*a*d*sqrt(((a^4 - a^2*b^2)*((b^4/(a^4*d^2 - a^2*b^2*d^2)^2 + b^2/(a^6*d^4 - a^4*b^2*d
^4))*(-I*sqrt(3) + 1)/(-1/729*b^6/(a^4*d^2 - a^2*b^2*d^2)^3 - 1/486*b^4/((a^6*d^4 - a^4*b^2*d^4)*(a^4*d^2 - a^
2*b^2*d^2)) - 1/1458*b^2/(a^8*d^6 - a^6*b^2*d^6) + 1/1458*b^2/((a^2 - b^2)^2*a^4*d^6))^(1/3) + 81*(-1/729*b^6/
(a^4*d^2 - a^2*b^2*d^2)^3 - 1/486*b^4/((a^6*d^4 - a^4*b^2*d^4)*(a^4*d^2 - a^2*b^2*d^2)) - 1/1458*b^2/(a^8*d^6
- a^6*b^2*d^6) + 1/1458*b^2/((a^2 - b^2)^2*a^4*d^6))^(1/3)*(I*sqrt(3) + 1) + 18*b^2/(a^4*d^2 - a^2*b^2*d^2))*d
^2 + 3*sqrt(1/3)*(a^4 - a^2*b^2)*d^2*sqrt(-((a^8 - 2*a^6*b^2 + a^4*b^4)*((b^4/(a^4*d^2 - a^2*b^2*d^2)^2 + b^2/
(a^6*d^4 - a^4*b^2*d^4))*(-I*sqrt(3) + 1)/(-1/729*b^6/(a^4*d^2 - a^2*b^2*d^2)^3 - 1/486*b^4/((a^6*d^4 - a^4*b^
2*d^4)*(a^4*d^2 - a^2*b^2*d^2)) - 1/1458*b^2/(a^8*d^6 - a^6*b^2*d^6) + 1/1458*b^2/((a^2 - b^2)^2*a^4*d^6))^(1/
3) + 81*(-1/729*b^6/(a^4*d^2 - a^2*b^2*d^2)^3 - 1/486*b^4/((a^6*d^4 - a^4*b^2*d^4)*(a^4*d^2 - a^2*b^2*d^2)) -
1/1458*b^2/(a^8*d^6 - a^6*b^2*d^6) + 1/1458*b^2/((a^2 - b^2)^2*a^4*d^6))^(1/3)*(I*sqrt(3) + 1) + 18*b^2/(a^4*d
^2 - a^2*b^2*d^2))^2*d^4 - 1296*a^2*b^2 + 324*b^4 - 36*(a^4*b^2 - a^2*b^4)*((b^4/(a^4*d^2 - a^2*b^2*d^2)^2 + b
^2/(a^6*d^4 - a^4*b^2*d^4))*(-I*sqrt(3) + 1)/(-1/729*b^6/(a^4*d^2 - a^2*b^2*d^2)^3 - 1/486*b^4/((a^6*d^4 - a^4
*b^2*d^4)*(a^4*d^2 - a^2*b^2*d^2)) - 1/1458*b^2/(a^8*d^6 - a^6*b^2*d^6) + 1/1458*b^2/((a^2 - b^2)^2*a^4*d^6))^
(1/3) + 81*(-1/729*b^6/(a^4*d^2 - a^2*b^2*d^2)^3 - 1/486*b^4/((a^6*d^4 - a^4*b^2*d^4)*(a^4*d^2 - a^2*b^2*d^2))
- 1/1458*b^2/(a^8*d^6 - a^6*b^2*d^6) + 1/1458*b^2/((a^2 - b^2)^2*a^4*d^6))^(1/3)*(I*sqrt(3) + 1) + 18*b^2/(a^
4*d^2 - a^2*b^2*d^2))*d^2)/((a^8 - 2*a^6*b^2 + a^4*b^4)*d^4)) - 54*b^2)/((a^4 - a^2*b^2)*d^2))*log(1/324*sqrt(
2/3)*sqrt(1/6)*((a^6 - a^4*b^2)*((b^4/(a^4*d^2 - a^2*b^2*d^2)^2 + b^2/(a^6*d^4 - a^4*b^2*d^4))*(-I*sqrt(3) + 1
)/(-1/729*b^6/(a^4*d^2 - a^2*b^2*d^2)^3 - 1/486*b^4/((a^6*d^4 - a^4*b^2*d^4)*(a^4*d^2 - a^2*b^2*d^2)) - 1/1458
*b^2/(a^8*d^6 - a^6*b^2*d^6) + 1/1458*b^2/((a^2 - b^2)^2*a^4*d^6))^(1/3) + 81*(-1/729*b^6/(a^4*d^2 - a^2*b^2*d
^2)^3 - 1/486*b^4/((a^6*d^4 - a^4*b^2*d^4)*(a^4*d^2 - a^2*b^2*d^2)) - 1/1458*b^2/(a^8*d^6 - a^6*b^2*d^6) + 1/1
458*b^2/((a^2 - b^2)^2*a^4*d^6))^(1/3)*(I*sqrt(3) + 1) + 18*b^2/(a^4*d^2 - a^2*b^2*d^2))^2*d^5 - 18*(a^4 + 2*a
^2*b^2)*((b^4/(a^4*d^2 - a^2*b^2*d^2)^2 + b^2/(a^6*d^4 - a^4*b^2*d^4))*(-I*sqrt(3) + 1)/(-1/729*b^6/(a^4*d^2 -
a^2*b^2*d^2)^3 - 1/486*b^4/((a^6*d^4 - a^4*b^2*d^4)*(a^4*d^2 - a^2*b^2*d^2)) - 1/1458*b^2/(a^8*d^6 - a^6*b^2*
d^6) + 1/1458*b^2/((a^2 - b^2)^2*a^4*d^6))^(1/3) + 81*(-1/729*b^6/(a^4*d^2 - a^2*b^2*d^2)^3 - 1/486*b^4/((a^6*
d^4 - a^4*b^2*d^4)*(a^4*d^2 - a^2*b^2*d^2)) - 1/1458*b^2/(a^8*d^6 - a^6*b^2*d^6) + 1/1458*b^2/((a^2 - b^2)^2*a
^4*d^6))^(1/3)*(I*sqrt(3) + 1) + 18*b^2/(a^4*d^2 - a^2*b^2*d^2))*d^3 - 324*(2*a*b + b^2)*d - 3*sqrt(1/3)*((a^6
- a^4*b^2)*((b^4/(a^4*d^2 - a^2*b^2*d^2)^2 + b^2/(a^6*d^4 - a^4*b^2*d^4))*(-I*sqrt(3) + 1)/(-1/729*b^6/(a^4*d
^2 - a^2*b^2*d^2)^3 - 1/486*b^4/((a^6*d^4 - a^4*b^2*d^4)*(a^4*d^2 - a^2*b^2*d^2)) - 1/1458*b^2/(a^8*d^6 - a^6*
b^2*d^6) + 1/1458*b^2/((a^2 - b^2)^2*a^4*d^6))^(1/3) + 81*(-1/729*b^6/(a^4*d^2 - a^2*b^2*d^2)^3 - 1/486*b^4/((
a^6*d^4 - a^4*b^2*d^4)*(a^4*d^2 - a^2*b^2*d^2)) - 1/1458*b^2/(a^8*d^6 - a^6*b^2*d^6) + 1/1458*b^2/((a^2 - b^2)
^2*a^4*d^6))^(1/3)*(I*sqrt(3) + 1) + 18*b^2/(a^4*d^2 - a^2*b^2*d^2))*d^5 + 18*(a^4 - a^2*b^2)*d^3)*sqrt(-((a^8
- 2*a^6*b^2 + a^4*b^4)*((b^4/(a^4*d^2 - a^2*b^2*d^2)^2 + b^2/(a^6*d^4 - a^4*b^2*d^4))*(-I*sqrt(3) + 1)/(-1/72
9*b^6/(a^4*d^2 - a^2*b^2*d^2)^3 - 1/486*b^4/((a^6*d^4 - a^4*b^2*d^4)*(a^4*d^2 - a^2*b^2*d^2)) - 1/1458*b^2/(a^
8*d^6 - a^6*b^2*d^6) + 1/1458*b^2/((a^2 - b^2)^2*a^4*d^6))^(1/3) + 81*(-1/729*b^6/(a^4*d^2 - a^2*b^2*d^2)^3 -
1/486*b^4/((a^6*d^4 - a^4*b^2*d^4)*(a^4*d^2 - a^2*b^2*d^2)) - 1/1458*b^2/(a^8*d^6 - a^6*b^2*d^6) + 1/1458*b^2/
((a^2 - b^2)^2*a^4*d^6))^(1/3)*(I*sqrt(3) + 1) + 18*b^2/(a^4*d^2 - a^2*b^2*d^2))^2*d^4 - 1296*a^2*b^2 + 324*b^
4 - 36*(a^4*b^2 - a^2*b^4)*((b^4/(a^4*d^2 - a^2*b^2*d^2)^2 + b^2/(a^6*d^4 - a^4*b^2*d^4))*(-I*sqrt(3) + 1)/(-1
/729*b^6/(a^4*d^2 - a^2*b^2*d^2)^3 - 1/486*b^4/((a^6*d^4 - a^4*b^2*d^4)*(a^4*d^2 - a^2*b^2*d^2)) - 1/1458*b^2/
(a^8*d^6 - a^6*b^2*d^6) + 1/1458*b^2/((a^2 - b^2)^2*a^4*d^6))^(1/3) + 81*(-1/729*b^6/(a^4*d^2 - a^2*b^2*d^2)^3
- 1/486*b^4/((a^6*d^4 - a^4*b^2*d^4)*(a^4*d^2 - a^2*b^2*d^2)) - 1/1458*b^2/(a^8*d^6 - a^6*b^2*d^6) + 1/1458*b
^2/((a^2 - b^2)^2*a^4*d^6))^(1/3)*(I*sqrt(3) + 1) + 18*b^2/(a^4*d^2 - a^2*b^2*d^2))*d^2)/((a^8 - 2*a^6*b^2 + a
^4*b^4)*d^4)))*sqrt(((a^4 - a^2*b^2)*((b^4/(a^4*d^2 - a^2*b^2*d^2)^2 + b^2/(a^6*d^4 - a^4*b^2*d^4))*(-I*sqrt(3
) + 1)/(-1/729*b^6/(a^4*d^2 - a^2*b^2*d^2)^3 - 1/486*b^4/((a^6*d^4 - a^4*b^2*d^4)*(a^4*d^2 - a^2*b^2*d^2)) - 1
/1458*b^2/(a^8*d^6 - a^6*b^2*d^6) + 1/1458*b^2/((a^2 - b^2)^2*a^4*d^6))^(1/3) + 81*(-1/729*b^6/(a^4*d^2 - a^2*
b^2*d^2)^3 - 1/486*b^4/((a^6*d^4 - a^4*b^2*d^4)*(a^4*d^2 - a^2*b^2*d^2)) - 1/1458*b^2/(a^8*d^6 - a^6*b^2*d^6)
+ 1/1458*b^2/((a^2 - b^2)^2*a^4*d^6))^(1/3)*(I*...

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

[C]   time = 5.91416 (sec), size = 24389 ,normalized size = 739.06 $$\text{Too large to display}$$

[In]

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

[Out]

1/32*(24*sqrt(1/2)*sqrt(1/3)*(1/12)^(3/4)*(1/27)^(1/4)*(a^11*d^7*e^(4*d*x + 4*c) - 2*a^11*d^7*e^(2*d*x + 2*c)
+ a^11*d^7)*((1/2)^(1/3)*(I*sqrt(3) + 1)*((a^2 + b^2)*b^2/(a^10*d^6) - (a^2*b^2 - b^4)/(a^10*d^6))^(1/3) - 2*(
1/2)^(2/3)*b^2*(-I*sqrt(3) + 1)/(a^6*d^4*((a^2 + b^2)*b^2/(a^10*d^6) - (a^2*b^2 - b^4)/(a^10*d^6))^(1/3)))*sqr
t((4*((1/2)^(1/3)*(I*sqrt(3) + 1)*((a^2 + b^2)*b^2/(a^10*d^6) - (a^2*b^2 - b^4)/(a^10*d^6))^(1/3) - 2*(1/2)^(2
/3)*b^2*(-I*sqrt(3) + 1)/(a^6*d^4*((a^2 + b^2)*b^2/(a^10*d^6) - (a^2*b^2 - b^4)/(a^10*d^6))^(1/3)))^2*a^8*b^2*
d^4 + 16*a^4*b^2 + 16*a^2*b^4 + 16*b^6 - 8*(a^6*b^2 - a^4*b^4)*((1/2)^(1/3)*(I*sqrt(3) + 1)*((a^2 + b^2)*b^2/(
a^10*d^6) - (a^2*b^2 - b^4)/(a^10*d^6))^(1/3) - 2*(1/2)^(2/3)*b^2*(-I*sqrt(3) + 1)/(a^6*d^4*((a^2 + b^2)*b^2/(
a^10*d^6) - (a^2*b^2 - b^4)/(a^10*d^6))^(1/3)))*d^2 - (8*((1/2)^(1/3)*(I*sqrt(3) + 1)*((a^2 + b^2)*b^2/(a^10*d
^6) - (a^2*b^2 - b^4)/(a^10*d^6))^(1/3) - 2*(1/2)^(2/3)*b^2*(-I*sqrt(3) + 1)/(a^6*d^4*((a^2 + b^2)*b^2/(a^10*d
^6) - (a^2*b^2 - b^4)/(a^10*d^6))^(1/3)))*a^8*b^2*d^4 + (a^12 - a^10*b^2)*((1/2)^(1/3)*(I*sqrt(3) + 1)*((a^2 +
b^2)*b^2/(a^10*d^6) - (a^2*b^2 - b^4)/(a^10*d^6))^(1/3) - 2*(1/2)^(2/3)*b^2*(-I*sqrt(3) + 1)/(a^6*d^4*((a^2 +
b^2)*b^2/(a^10*d^6) - (a^2*b^2 - b^4)/(a^10*d^6))^(1/3)))^2*d^6 - 4*(a^6*b^2 - a^4*b^4)*d^2)*sqrt((((1/2)^(1/
3)*(I*sqrt(3) + 1)*((a^2 + b^2)*b^2/(a^10*d^6) - (a^2*b^2 - b^4)/(a^10*d^6))^(1/3) - 2*(1/2)^(2/3)*b^2*(-I*sqr
t(3) + 1)/(a^6*d^4*((a^2 + b^2)*b^2/(a^10*d^6) - (a^2*b^2 - b^4)/(a^10*d^6))^(1/3)))^2*a^6*d^4 + 12*b^2)/(a^6*
d^4)))/(a^4*b^2 + 2*a^2*b^4 + b^6))*sqrt((((1/2)^(1/3)*(I*sqrt(3) + 1)*((a^2 + b^2)*b^2/(a^10*d^6) - (a^2*b^2
- b^4)/(a^10*d^6))^(1/3) - 2*(1/2)^(2/3)*b^2*(-I*sqrt(3) + 1)/(a^6*d^4*((a^2 + b^2)*b^2/(a^10*d^6) - (a^2*b^2
- b^4)/(a^10*d^6))^(1/3)))^2*a^6*d^4 + 16*b^2)/(a^6*d^4))*((((1/2)^(1/3)*(I*sqrt(3) + 1)*((a^2 + b^2)*b^2/(a^1
0*d^6) - (a^2*b^2 - b^4)/(a^10*d^6))^(1/3) - 2*(1/2)^(2/3)*b^2*(-I*sqrt(3) + 1)/(a^6*d^4*((a^2 + b^2)*b^2/(a^1
0*d^6) - (a^2*b^2 - b^4)/(a^10*d^6))^(1/3)))^2*a^6*d^4 + 12*b^2)/(a^6*d^4))^(3/4)*arctan(1/128*(27*sqrt(1/2)*s
qrt(1/3)*(1/12)^(3/4)*(1/27)^(3/4)*(sqrt(1/3)*((a^19 - a^18*b)*((1/2)^(1/3)*(I*sqrt(3) + 1)*((a^2 + b^2)*b^2/(
a^10*d^6) - (a^2*b^2 - b^4)/(a^10*d^6))^(1/3) - 2*(1/2)^(2/3)*b^2*(-I*sqrt(3) + 1)/(a^6*d^4*((a^2 + b^2)*b^2/(
a^10*d^6) - (a^2*b^2 - b^4)/(a^10*d^6))^(1/3)))^2*d^11 + 2*(a^16*b + a^15*b^2)*((1/2)^(1/3)*(I*sqrt(3) + 1)*((
a^2 + b^2)*b^2/(a^10*d^6) - (a^2*b^2 - b^4)/(a^10*d^6))^(1/3) - 2*(1/2)^(2/3)*b^2*(-I*sqrt(3) + 1)/(a^6*d^4*((
a^2 + b^2)*b^2/(a^10*d^6) - (a^2*b^2 - b^4)/(a^10*d^6))^(1/3)))*d^9)*sqrt((((1/2)^(1/3)*(I*sqrt(3) + 1)*((a^2
+ b^2)*b^2/(a^10*d^6) - (a^2*b^2 - b^4)/(a^10*d^6))^(1/3) - 2*(1/2)^(2/3)*b^2*(-I*sqrt(3) + 1)/(a^6*d^4*((a^2
+ b^2)*b^2/(a^10*d^6) - (a^2*b^2 - b^4)/(a^10*d^6))^(1/3)))^2*a^6*d^4 + 16*b^2)/(a^6*d^4))*sqrt((((1/2)^(1/3)*
(I*sqrt(3) + 1)*((a^2 + b^2)*b^2/(a^10*d^6) - (a^2*b^2 - b^4)/(a^10*d^6))^(1/3) - 2*(1/2)^(2/3)*b^2*(-I*sqrt(3
) + 1)/(a^6*d^4*((a^2 + b^2)*b^2/(a^10*d^6) - (a^2*b^2 - b^4)/(a^10*d^6))^(1/3)))^2*a^6*d^4 + 12*b^2)/(a^6*d^4
)) + 4*sqrt(1/3)*((a^16*b + a^15*b^2)*((1/2)^(1/3)*(I*sqrt(3) + 1)*((a^2 + b^2)*b^2/(a^10*d^6) - (a^2*b^2 - b^
4)/(a^10*d^6))^(1/3) - 2*(1/2)^(2/3)*b^2*(-I*sqrt(3) + 1)/(a^6*d^4*((a^2 + b^2)*b^2/(a^10*d^6) - (a^2*b^2 - b^
4)/(a^10*d^6))^(1/3)))^2*d^9 - 2*(a^11*b^2 - a^10*b^3 - a^9*b^4 + a^8*b^5)*d^5)*sqrt((((1/2)^(1/3)*(I*sqrt(3)
+ 1)*((a^2 + b^2)*b^2/(a^10*d^6) - (a^2*b^2 - b^4)/(a^10*d^6))^(1/3) - 2*(1/2)^(2/3)*b^2*(-I*sqrt(3) + 1)/(a^6
*d^4*((a^2 + b^2)*b^2/(a^10*d^6) - (a^2*b^2 - b^4)/(a^10*d^6))^(1/3)))^2*a^6*d^4 + 16*b^2)/(a^6*d^4)))*sqrt((4
*((1/2)^(1/3)*(I*sqrt(3) + 1)*((a^2 + b^2)*b^2/(a^10*d^6) - (a^2*b^2 - b^4)/(a^10*d^6))^(1/3) - 2*(1/2)^(2/3)*
b^2*(-I*sqrt(3) + 1)/(a^6*d^4*((a^2 + b^2)*b^2/(a^10*d^6) - (a^2*b^2 - b^4)/(a^10*d^6))^(1/3)))^2*a^8*b^2*d^4
+ 16*a^4*b^2 + 16*a^2*b^4 + 16*b^6 - 8*(a^6*b^2 - a^4*b^4)*((1/2)^(1/3)*(I*sqrt(3) + 1)*((a^2 + b^2)*b^2/(a^10
*d^6) - (a^2*b^2 - b^4)/(a^10*d^6))^(1/3) - 2*(1/2)^(2/3)*b^2*(-I*sqrt(3) + 1)/(a^6*d^4*((a^2 + b^2)*b^2/(a^10
*d^6) - (a^2*b^2 - b^4)/(a^10*d^6))^(1/3)))*d^2 - (8*((1/2)^(1/3)*(I*sqrt(3) + 1)*((a^2 + b^2)*b^2/(a^10*d^6)
- (a^2*b^2 - b^4)/(a^10*d^6))^(1/3) - 2*(1/2)^(2/3)*b^2*(-I*sqrt(3) + 1)/(a^6*d^4*((a^2 + b^2)*b^2/(a^10*d^6)
- (a^2*b^2 - b^4)/(a^10*d^6))^(1/3)))*a^8*b^2*d^4 + (a^12 - a^10*b^2)*((1/2)^(1/3)*(I*sqrt(3) + 1)*((a^2 + b^2
)*b^2/(a^10*d^6) - (a^2*b^2 - b^4)/(a^10*d^6))^(1/3) - 2*(1/2)^(2/3)*b^2*(-I*sqrt(3) + 1)/(a^6*d^4*((a^2 + b^2
)*b^2/(a^10*d^6) - (a^2*b^2 - b^4)/(a^10*d^6))^(1/3)))^2*d^6 - 4*(a^6*b^2 - a^4*b^4)*d^2)*sqrt((((1/2)^(1/3)*(
I*sqrt(3) + 1)*((a^2 + b^2)*b^2/(a^10*d^6) - (a^2*b^2 - b^4)/(a^10*d^6))^(1/3) - 2*(1/2)^(2/3)*b^2*(-I*sqrt(3)
+ 1)/(a^6*d^4*((a^2 + b^2)*b^2/(a^10*d^6) - (a^2*b^2 - b^4)/(a^10*d^6))^(1/3)))^2*a^6*d^4 + 12*b^2)/(a^6*d^4)
))/(a^4*b^2 + 2*a^2*b^4 + b^6))*sqrt((3*sqrt(1/2)*sqrt(1/3)*(1/12)^(1/4)*(1/27)^(1/4)*(4*(a^13*b - a^9*b^5)*((
1/2)^(1/3)*(I*sqrt(3) + 1)*((a^2 + b^2)*b^2/(a^...

3.8.4 Giac

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

[C]   time = 1.52966 (sec), size = 303 ,normalized size = 9.18 $$-\frac{\frac{(9a{e}^{(2dx+2c)}+9b{e}^{(2dx+2c)}-a+b)e^{(-3dx-3c)}}{a^2-2ab+b^2}-\frac{a^2{e}^{(3dx+3c)}+2ab{e}^{(3dx+3c)}+b^2{e}^{(3dx+3c)}-9a^2{e}^{(dx+c)}+9b^2{e}^{(dx+c)}}{a^3+3a^{2}b+3a{b}^2+b^3}}{24d}-\frac{\frac{6(a^{3}b+a^2{b}^2+a{b}^3)(dx+c)}{a-b}-\frac{(a^{3}b+a^2{b}^2+a{b}^3)\log \left(|a{e}^{(6dx+6c)}+b{e}^{(6dx+6c)}+3a{e}^{(4dx+4c)}-3b{e}^{(4dx+4c)}+3a{e}^{(2dx+2c)}+3b{e}^{(2dx+2c)}+a-b|\right)}{a-b}}{(a^4-2a^2{b}^2+b^4)d^2}$$

[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 - 3*c)/(a^2 - 2*a*b + b^2) - (a^2*e^(3*d*
x + 3*c) + 2*a*b*e^(3*d*x + 3*c) + b^2*e^(3*d*x + 3*c) - 9*a^2*e^(d*x + c) + 9*b^2*e^(d*x + c))/(a^3 + 3*a^2*b
+ 3*a*b^2 + b^3))/d - (6*(a^3*b + a^2*b^2 + a*b^3)*(d*x + c)/(a - b) - (a^3*b + a^2*b^2 + a*b^3)*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 - b))/((a^4 - 2*a^2*b^2 + b^4)*d^2)

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

[C]   time = 1.05927 (sec), size = 169 ,normalized size = 5.45 $$\frac{\frac{e^{(dx+c)}}{a+b}+\frac{e^{(-dx-c)}}{a-b}}{2d}+\frac{\frac{6(2ab+b^2)(dx+c)}{a-b}-\frac{(2ab+b^2)\log \left(|a{e}^{(6dx+6c)}+b{e}^{(6dx+6c)}+3a{e}^{(4dx+4c)}-3b{e}^{(4dx+4c)}+3a{e}^{(2dx+2c)}+3b{e}^{(2dx+2c)}+a-b|\right)}{a-b}}{3(a^2-b^2)d^2}$$

[In]

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

[Out]

1/2*(e^(d*x + c)/(a + b) + e^(-d*x - c)/(a - b))/d + 1/3*(6*(2*a*b + b^2)*(d*x + c)/(a - b) - (2*a*b + b^2)*lo
g(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 - b))/((a^2 - b^2)*d^2)

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

[C]   time = 0.847952 (sec), size = 146 ,normalized size = 4.71 $$-\frac{\frac{\log (e^{(dx+c)}+1)}{a}-\frac{\log \left(|e^{(dx+c)}-1|\right)}{a}}{d}-\frac{\frac{6(dx+c)b}{a-b}-\frac{b\log \left(|a{e}^{(6dx+6c)}+b{e}^{(6dx+6c)}+3a{e}^{(4dx+4c)}-3b{e}^{(4dx+4c)}+3a{e}^{(2dx+2c)}+3b{e}^{(2dx+2c)}+a-b|\right)}{a-b}}{3a{d}^2}$$

[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*(d*x + c)*b/(a - b) - b*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 - b))/(a*d^2)

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

[C]   time = 0.651096 (sec), size = 68 ,normalized size = 2.06 $$\frac{\frac{\log (e^{(dx+c)}+1)}{a}-\frac{\log \left(|e^{(dx+c)}-1|\right)}{a}-\frac{2(e^{(3dx+3c)}+e^{(dx+c)})}{a{(e^{(2dx+2c)}-1)}^2}}{2d}$$

[In]

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

[Out]

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

3.8.5 Mupad

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

[B]   time = 87.9877 (sec), size = 2500 ,normalized size = 80.65 $$\text{Too large to display}$$

[In]

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

[Out]

exp(- c - d*x)/(2*(a*d - b*d)) + symsum(log((81920*a^2*b^5*exp(d*x)*exp(root(2187*a^6*b^2*d^6*z^6 - 2187*a^4*b
^4*d^6*z^6 + 729*a^2*b^6*d^6*z^6 - 729*a^8*d^6*z^6 - 1458*a^4*b^2*d^4*z^4 - 729*a^2*b^4*d^4*z^4 + 81*a^2*b^2*d
^2*z^2 - b^2, z, k)) + 221184*root(2187*a^6*b^2*d^6*z^6 - 2187*a^4*b^4*d^6*z^6 + 729*a^2*b^6*d^6*z^6 - 729*a^8
*d^6*z^6 - 1458*a^4*b^2*d^4*z^4 - 729*a^2*b^4*d^4*z^4 + 81*a^2*b^2*d^2*z^2 - b^2, z, k)^3*a^2*b^8*d^3 - 353894
4*root(2187*a^6*b^2*d^6*z^6 - 2187*a^4*b^4*d^6*z^6 + 729*a^2*b^6*d^6*z^6 - 729*a^8*d^6*z^6 - 1458*a^4*b^2*d^4*
z^4 - 729*a^2*b^4*d^4*z^4 + 81*a^2*b^2*d^2*z^2 - b^2, z, k)^3*a^3*b^7*d^3 + 1990656*root(2187*a^6*b^2*d^6*z^6
- 2187*a^4*b^4*d^6*z^6 + 729*a^2*b^6*d^6*z^6 - 729*a^8*d^6*z^6 - 1458*a^4*b^2*d^4*z^4 - 729*a^2*b^4*d^4*z^4 +
81*a^2*b^2*d^2*z^2 - b^2, z, k)^3*a^4*b^6*d^3 + 3538944*root(2187*a^6*b^2*d^6*z^6 - 2187*a^4*b^4*d^6*z^6 + 729
*a^2*b^6*d^6*z^6 - 729*a^8*d^6*z^6 - 1458*a^4*b^2*d^4*z^4 - 729*a^2*b^4*d^4*z^4 + 81*a^2*b^2*d^2*z^2 - b^2, z,
k)^3*a^5*b^5*d^3 - 2211840*root(2187*a^6*b^2*d^6*z^6 - 2187*a^4*b^4*d^6*z^6 + 729*a^2*b^6*d^6*z^6 - 729*a^8*d
^6*z^6 - 1458*a^4*b^2*d^4*z^4 - 729*a^2*b^4*d^4*z^4 + 81*a^2*b^2*d^2*z^2 - b^2, z, k)^3*a^6*b^4*d^3 + 7962624*
root(2187*a^6*b^2*d^6*z^6 - 2187*a^4*b^4*d^6*z^6 + 729*a^2*b^6*d^6*z^6 - 729*a^8*d^6*z^6 - 1458*a^4*b^2*d^4*z^
4 - 729*a^2*b^4*d^4*z^4 + 81*a^2*b^2*d^2*z^2 - b^2, z, k)^5*a^3*b^9*d^5 + 15925248*root(2187*a^6*b^2*d^6*z^6 -
2187*a^4*b^4*d^6*z^6 + 729*a^2*b^6*d^6*z^6 - 729*a^8*d^6*z^6 - 1458*a^4*b^2*d^4*z^4 - 729*a^2*b^4*d^4*z^4 + 8
1*a^2*b^2*d^2*z^2 - b^2, z, k)^5*a^4*b^8*d^5 - 7962624*root(2187*a^6*b^2*d^6*z^6 - 2187*a^4*b^4*d^6*z^6 + 729*
a^2*b^6*d^6*z^6 - 729*a^8*d^6*z^6 - 1458*a^4*b^2*d^4*z^4 - 729*a^2*b^4*d^4*z^4 + 81*a^2*b^2*d^2*z^2 - b^2, z,
k)^5*a^5*b^7*d^5 - 31850496*root(2187*a^6*b^2*d^6*z^6 - 2187*a^4*b^4*d^6*z^6 + 729*a^2*b^6*d^6*z^6 - 729*a^8*d
^6*z^6 - 1458*a^4*b^2*d^4*z^4 - 729*a^2*b^4*d^4*z^4 + 81*a^2*b^2*d^2*z^2 - b^2, z, k)^5*a^6*b^6*d^5 - 7962624*
root(2187*a^6*b^2*d^6*z^6 - 2187*a^4*b^4*d^6*z^6 + 729*a^2*b^6*d^6*z^6 - 729*a^8*d^6*z^6 - 1458*a^4*b^2*d^4*z^
4 - 729*a^2*b^4*d^4*z^4 + 81*a^2*b^2*d^2*z^2 - b^2, z, k)^5*a^7*b^5*d^5 + 15925248*root(2187*a^6*b^2*d^6*z^6 -
2187*a^4*b^4*d^6*z^6 + 729*a^2*b^6*d^6*z^6 - 729*a^8*d^6*z^6 - 1458*a^4*b^2*d^4*z^4 - 729*a^2*b^4*d^4*z^4 + 8
1*a^2*b^2*d^2*z^2 - b^2, z, k)^5*a^8*b^4*d^5 + 7962624*root(2187*a^6*b^2*d^6*z^6 - 2187*a^4*b^4*d^6*z^6 + 729*
a^2*b^6*d^6*z^6 - 729*a^8*d^6*z^6 - 1458*a^4*b^2*d^4*z^4 - 729*a^2*b^4*d^4*z^4 + 81*a^2*b^2*d^2*z^2 - b^2, z,
k)^5*a^9*b^3*d^5 + 98304*root(2187*a^6*b^2*d^6*z^6 - 2187*a^4*b^4*d^6*z^6 + 729*a^2*b^6*d^6*z^6 - 729*a^8*d^6*
z^6 - 1458*a^4*b^2*d^4*z^4 - 729*a^2*b^4*d^4*z^4 + 81*a^2*b^2*d^2*z^2 - b^2, z, k)*a^2*b^6*d - 98304*root(2187
*a^6*b^2*d^6*z^6 - 2187*a^4*b^4*d^6*z^6 + 729*a^2*b^6*d^6*z^6 - 729*a^8*d^6*z^6 - 1458*a^4*b^2*d^4*z^4 - 729*a
^2*b^4*d^4*z^4 + 81*a^2*b^2*d^2*z^2 - b^2, z, k)*a^3*b^5*d + 24576*root(2187*a^6*b^2*d^6*z^6 - 2187*a^4*b^4*d^
6*z^6 + 729*a^2*b^6*d^6*z^6 - 729*a^8*d^6*z^6 - 1458*a^4*b^2*d^4*z^4 - 729*a^2*b^4*d^4*z^4 + 81*a^2*b^2*d^2*z^
2 - b^2, z, k)*a^4*b^4*d + 8192*a*b^6*exp(d*x)*exp(root(2187*a^6*b^2*d^6*z^6 - 2187*a^4*b^4*d^6*z^6 + 729*a^2*
b^6*d^6*z^6 - 729*a^8*d^6*z^6 - 1458*a^4*b^2*d^4*z^4 - 729*a^2*b^4*d^4*z^4 + 81*a^2*b^2*d^2*z^2 - b^2, z, k))
+ 368640*root(2187*a^6*b^2*d^6*z^6 - 2187*a^4*b^4*d^6*z^6 + 729*a^2*b^6*d^6*z^6 - 729*a^8*d^6*z^6 - 1458*a^4*b
^2*d^4*z^4 - 729*a^2*b^4*d^4*z^4 + 81*a^2*b^2*d^2*z^2 - b^2, z, k)^2*a^2*b^7*d^2*exp(d*x)*exp(root(2187*a^6*b^
2*d^6*z^6 - 2187*a^4*b^4*d^6*z^6 + 729*a^2*b^6*d^6*z^6 - 729*a^8*d^6*z^6 - 1458*a^4*b^2*d^4*z^4 - 729*a^2*b^4*
d^4*z^4 + 81*a^2*b^2*d^2*z^2 - b^2, z, k)) - 2285568*root(2187*a^6*b^2*d^6*z^6 - 2187*a^4*b^4*d^6*z^6 + 729*a^
2*b^6*d^6*z^6 - 729*a^8*d^6*z^6 - 1458*a^4*b^2*d^4*z^4 - 729*a^2*b^4*d^4*z^4 + 81*a^2*b^2*d^2*z^2 - b^2, z, k)
^2*a^3*b^6*d^2*exp(d*x)*exp(root(2187*a^6*b^2*d^6*z^6 - 2187*a^4*b^4*d^6*z^6 + 729*a^2*b^6*d^6*z^6 - 729*a^8*d
^6*z^6 - 1458*a^4*b^2*d^4*z^4 - 729*a^2*b^4*d^4*z^4 + 81*a^2*b^2*d^2*z^2 - b^2, z, k)) - 5013504*root(2187*a^6
*b^2*d^6*z^6 - 2187*a^4*b^4*d^6*z^6 + 729*a^2*b^6*d^6*z^6 - 729*a^8*d^6*z^6 - 1458*a^4*b^2*d^4*z^4 - 729*a^2*b
^4*d^4*z^4 + 81*a^2*b^2*d^2*z^2 - b^2, z, k)^2*a^4*b^5*d^2*exp(d*x)*exp(root(2187*a^6*b^2*d^6*z^6 - 2187*a^4*b
^4*d^6*z^6 + 729*a^2*b^6*d^6*z^6 - 729*a^8*d^6*z^6 - 1458*a^4*b^2*d^4*z^4 - 729*a^2*b^4*d^4*z^4 + 81*a^2*b^2*d
^2*z^2 - b^2, z, k)) - 368640*root(2187*a^6*b^2*d^6*z^6 - 2187*a^4*b^4*d^6*z^6 + 729*a^2*b^6*d^6*z^6 - 729*a^8
*d^6*z^6 - 1458*a^4*b^2*d^4*z^4 - 729*a^2*b^4*d^4*z^4 + 81*a^2*b^2*d^2*z^2 - b^2, z, k)^2*a^5*b^4*d^2*exp(d*x)
*exp(root(2187*a^6*b^2*d^6*z^6 - 2187*a^4*b^4*d^6*z^6 + 729*a^2*b^6*d^6*z^6 - 729*a^8*d^6*z^6 - 1458*a^4*b^2*d
^4*z^4 - 729*a^2*b^4*d^4*z^4 + 81*a^2*b^2*d^2*z^2 - b^2, z, k)) + 8626176*root(2187*a^6*b^2*d^6*z^6 - 2187*a^4
*b^4*d^6*z^6 + 729*a^2*b^6*d^6*z^6 - 729*a^8*d^6*z^6 - 1458*a^4*b^2*d^4*z^4 - 729*a^2*b^4*d^4*z^4 + 81*a^2*b^2
*d^2*z^2 - b^2, z, k)^4*a^3*b^8*d^4*exp(d*x)*ex...

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

[B]   time = 16.5034 (sec), size = 2500 ,normalized size = 80.65 $$\text{Too large to display}$$

[In]

int(1/(sinh(c + d*x)*(a + b*tanh(c + d*x)^3)),x)

[Out]

symsum(log(-(1409286144*b^6*exp(d*x)*exp(root(729*a^6*b^2*d^6*z^6 - 729*a^8*d^6*z^6 - 243*a^4*b^2*d^4*z^4 + 27
*a^2*b^2*d^2*z^2 - b^2, z, k)) + 134217728*root(729*a^6*b^2*d^6*z^6 - 729*a^8*d^6*z^6 - 243*a^4*b^2*d^4*z^4 +
27*a^2*b^2*d^2*z^2 - b^2, z, k)*b^7*d + 1879048192*root(729*a^6*b^2*d^6*z^6 - 729*a^8*d^6*z^6 - 243*a^4*b^2*d^
4*z^4 + 27*a^2*b^2*d^2*z^2 - b^2, z, k)*a*b^6*d - 2818572288*root(729*a^6*b^2*d^6*z^6 - 729*a^8*d^6*z^6 - 243*
a^4*b^2*d^4*z^4 + 27*a^2*b^2*d^2*z^2 - b^2, z, k)^3*a^2*b^7*d^3 - 40869298176*root(729*a^6*b^2*d^6*z^6 - 729*a
^8*d^6*z^6 - 243*a^4*b^2*d^4*z^4 + 27*a^2*b^2*d^2*z^2 - b^2, z, k)^3*a^3*b^6*d^3 + 28185722880*root(729*a^6*b^
2*d^6*z^6 - 729*a^8*d^6*z^6 - 243*a^4*b^2*d^4*z^4 + 27*a^2*b^2*d^2*z^2 - b^2, z, k)^3*a^4*b^5*d^3 + 1550214758
4*root(729*a^6*b^2*d^6*z^6 - 729*a^8*d^6*z^6 - 243*a^4*b^2*d^4*z^4 + 27*a^2*b^2*d^2*z^2 - b^2, z, k)^3*a^5*b^4
*d^3 + 18119393280*root(729*a^6*b^2*d^6*z^6 - 729*a^8*d^6*z^6 - 243*a^4*b^2*d^4*z^4 + 27*a^2*b^2*d^2*z^2 - b^2
, z, k)^5*a^4*b^7*d^5 + 235552112640*root(729*a^6*b^2*d^6*z^6 - 729*a^8*d^6*z^6 - 243*a^4*b^2*d^4*z^4 + 27*a^2
*b^2*d^2*z^2 - b^2, z, k)^5*a^5*b^6*d^5 + 14495514624*root(729*a^6*b^2*d^6*z^6 - 729*a^8*d^6*z^6 - 243*a^4*b^2
*d^4*z^4 + 27*a^2*b^2*d^2*z^2 - b^2, z, k)^5*a^6*b^5*d^5 - 219244658688*root(729*a^6*b^2*d^6*z^6 - 729*a^8*d^6
*z^6 - 243*a^4*b^2*d^4*z^4 + 27*a^2*b^2*d^2*z^2 - b^2, z, k)^5*a^7*b^4*d^5 - 48922361856*root(729*a^6*b^2*d^6*
z^6 - 729*a^8*d^6*z^6 - 243*a^4*b^2*d^4*z^4 + 27*a^2*b^2*d^2*z^2 - b^2, z, k)^5*a^8*b^3*d^5 - 32614907904*root
(729*a^6*b^2*d^6*z^6 - 729*a^8*d^6*z^6 - 243*a^4*b^2*d^4*z^4 + 27*a^2*b^2*d^2*z^2 - b^2, z, k)^7*a^6*b^7*d^7 -
179381993472*root(729*a^6*b^2*d^6*z^6 - 729*a^8*d^6*z^6 - 243*a^4*b^2*d^4*z^4 + 27*a^2*b^2*d^2*z^2 - b^2, z,
k)^7*a^7*b^6*d^7 - 16307453952*root(729*a^6*b^2*d^6*z^6 - 729*a^8*d^6*z^6 - 243*a^4*b^2*d^4*z^4 + 27*a^2*b^2*d
^2*z^2 - b^2, z, k)^7*a^8*b^5*d^7 + 179381993472*root(729*a^6*b^2*d^6*z^6 - 729*a^8*d^6*z^6 - 243*a^4*b^2*d^4*
z^4 + 27*a^2*b^2*d^2*z^2 - b^2, z, k)^7*a^9*b^4*d^7 + 48922361856*root(729*a^6*b^2*d^6*z^6 - 729*a^8*d^6*z^6 -
243*a^4*b^2*d^4*z^4 + 27*a^2*b^2*d^2*z^2 - b^2, z, k)^7*a^10*b^3*d^7 - 1912602624*root(729*a^6*b^2*d^6*z^6 -
729*a^8*d^6*z^6 - 243*a^4*b^2*d^4*z^4 + 27*a^2*b^2*d^2*z^2 - b^2, z, k)*a^2*b^5*d - 100663296*root(729*a^6*b^2
*d^6*z^6 - 729*a^8*d^6*z^6 - 243*a^4*b^2*d^4*z^4 + 27*a^2*b^2*d^2*z^2 - b^2, z, k)*a^3*b^4*d + 738197504*a*b^5
*exp(d*x)*exp(root(729*a^6*b^2*d^6*z^6 - 729*a^8*d^6*z^6 - 243*a^4*b^2*d^4*z^4 + 27*a^2*b^2*d^2*z^2 - b^2, z,
k)) + 268435456*root(729*a^6*b^2*d^6*z^6 - 729*a^8*d^6*z^6 - 243*a^4*b^2*d^4*z^4 + 27*a^2*b^2*d^2*z^2 - b^2, z
, k)^2*a*b^7*d^2*exp(d*x)*exp(root(729*a^6*b^2*d^6*z^6 - 729*a^8*d^6*z^6 - 243*a^4*b^2*d^4*z^4 + 27*a^2*b^2*d^
2*z^2 - b^2, z, k)) - 29158801408*root(729*a^6*b^2*d^6*z^6 - 729*a^8*d^6*z^6 - 243*a^4*b^2*d^4*z^4 + 27*a^2*b^
2*d^2*z^2 - b^2, z, k)^2*a^2*b^6*d^2*exp(d*x)*exp(root(729*a^6*b^2*d^6*z^6 - 729*a^8*d^6*z^6 - 243*a^4*b^2*d^4
*z^4 + 27*a^2*b^2*d^2*z^2 - b^2, z, k)) - 29125246976*root(729*a^6*b^2*d^6*z^6 - 729*a^8*d^6*z^6 - 243*a^4*b^2
*d^4*z^4 + 27*a^2*b^2*d^2*z^2 - b^2, z, k)^2*a^3*b^5*d^2*exp(d*x)*exp(root(729*a^6*b^2*d^6*z^6 - 729*a^8*d^6*z
^6 - 243*a^4*b^2*d^4*z^4 + 27*a^2*b^2*d^2*z^2 - b^2, z, k)) - 2113929216*root(729*a^6*b^2*d^6*z^6 - 729*a^8*d^
6*z^6 - 243*a^4*b^2*d^4*z^4 + 27*a^2*b^2*d^2*z^2 - b^2, z, k)^2*a^4*b^4*d^2*exp(d*x)*exp(root(729*a^6*b^2*d^6*
z^6 - 729*a^8*d^6*z^6 - 243*a^4*b^2*d^4*z^4 + 27*a^2*b^2*d^2*z^2 - b^2, z, k)) - 4831838208*root(729*a^6*b^2*d
^6*z^6 - 729*a^8*d^6*z^6 - 243*a^4*b^2*d^4*z^4 + 27*a^2*b^2*d^2*z^2 - b^2, z, k)^4*a^3*b^7*d^4*exp(d*x)*exp(ro
ot(729*a^6*b^2*d^6*z^6 - 729*a^8*d^6*z^6 - 243*a^4*b^2*d^4*z^4 + 27*a^2*b^2*d^2*z^2 - b^2, z, k)) + 1654904586
24*root(729*a^6*b^2*d^6*z^6 - 729*a^8*d^6*z^6 - 243*a^4*b^2*d^4*z^4 + 27*a^2*b^2*d^2*z^2 - b^2, z, k)^4*a^4*b^
6*d^4*exp(d*x)*exp(root(729*a^6*b^2*d^6*z^6 - 729*a^8*d^6*z^6 - 243*a^4*b^2*d^4*z^4 + 27*a^2*b^2*d^2*z^2 - b^2
, z, k)) + 283870494720*root(729*a^6*b^2*d^6*z^6 - 729*a^8*d^6*z^6 - 243*a^4*b^2*d^4*z^4 + 27*a^2*b^2*d^2*z^2
- b^2, z, k)^4*a^5*b^5*d^4*exp(d*x)*exp(root(729*a^6*b^2*d^6*z^6 - 729*a^8*d^6*z^6 - 243*a^4*b^2*d^4*z^4 + 27*
a^2*b^2*d^2*z^2 - b^2, z, k)) + 132573560832*root(729*a^6*b^2*d^6*z^6 - 729*a^8*d^6*z^6 - 243*a^4*b^2*d^4*z^4
+ 27*a^2*b^2*d^2*z^2 - b^2, z, k)^4*a^6*b^4*d^4*exp(d*x)*exp(root(729*a^6*b^2*d^6*z^6 - 729*a^8*d^6*z^6 - 243*
a^4*b^2*d^4*z^4 + 27*a^2*b^2*d^2*z^2 - b^2, z, k)) + 2717908992*root(729*a^6*b^2*d^6*z^6 - 729*a^8*d^6*z^6 - 2
43*a^4*b^2*d^4*z^4 + 27*a^2*b^2*d^2*z^2 - b^2, z, k)^4*a^7*b^3*d^4*exp(d*x)*exp(root(729*a^6*b^2*d^6*z^6 - 729
*a^8*d^6*z^6 - 243*a^4*b^2*d^4*z^4 + 27*a^2*b^2*d^2*z^2 - b^2, z, k)) + 21743271936*root(729*a^6*b^2*d^6*z^6 -
729*a^8*d^6*z^6 - 243*a^4*b^2*d^4*z^4 + 27*a^2*b^2*d^2*z^2 - b^2, z, k)^6*a^5*b^7*d^6*exp(d*x)*exp(root(729*a
^6*b^2*d^6*z^6 - 729*a^8*d^6*z^6 - 243*a^4*b^2*d^4*z^4 + 27*a^2*b^2*d^2*z^2 - b^2, z, k)) - 154920812544*root(
729*a^6*b^2*d^6*z^6 - 729*a^8*d^6*z^6 - 243*a^4...

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

[B]   time = 26.9207 (sec), size = 2500 ,normalized size = 75.76 $$\text{Too large to display}$$

[In]

int(1/(sinh(c + d*x)^3*(a + b*tanh(c + d*x)^3)),x)

[Out]

exp(c + d*x)/(a*d - a*d*exp(2*c + 2*d*x)) - (2*exp(c + d*x))/(a*d - 2*a*d*exp(2*c + 2*d*x) + a*d*exp(4*c + 4*d
*x)) + symsum(log((570425344*a^4*b^6*exp(d*x)*exp(root(729*a^10*d^6*z^6 + 27*a^4*b^2*d^2*z^2 + a^2*b^2 - b^4,
z, k)) - 33554432*root(729*a^10*d^6*z^6 + 27*a^4*b^2*d^2*z^2 + a^2*b^2 - b^4, z, k)*a*b^10*d - 553648128*a^2*b
^8*exp(d*x)*exp(root(729*a^10*d^6*z^6 + 27*a^4*b^2*d^2*z^2 + a^2*b^2 - b^4, z, k)) - 167772160*a^3*b^7*exp(d*x
)*exp(root(729*a^10*d^6*z^6 + 27*a^4*b^2*d^2*z^2 + a^2*b^2 - b^4, z, k)) - 16777216*b^10*exp(d*x)*exp(root(729
*a^10*d^6*z^6 + 27*a^4*b^2*d^2*z^2 + a^2*b^2 - b^4, z, k)) + 192937984*a^5*b^5*exp(d*x)*exp(root(729*a^10*d^6*
z^6 + 27*a^4*b^2*d^2*z^2 + a^2*b^2 - b^4, z, k)) + 2617245696*root(729*a^10*d^6*z^6 + 27*a^4*b^2*d^2*z^2 + a^2
*b^2 - b^4, z, k)^3*a^5*b^8*d^3 - 150994944*root(729*a^10*d^6*z^6 + 27*a^4*b^2*d^2*z^2 + a^2*b^2 - b^4, z, k)^
3*a^6*b^7*d^3 - 1384120320*root(729*a^10*d^6*z^6 + 27*a^4*b^2*d^2*z^2 + a^2*b^2 - b^4, z, k)^3*a^7*b^6*d^3 + 2
415919104*root(729*a^10*d^6*z^6 + 27*a^4*b^2*d^2*z^2 + a^2*b^2 - b^4, z, k)^3*a^8*b^5*d^3 - 3498049536*root(72
9*a^10*d^6*z^6 + 27*a^4*b^2*d^2*z^2 + a^2*b^2 - b^4, z, k)^3*a^9*b^4*d^3 + 5435817984*root(729*a^10*d^6*z^6 +
27*a^4*b^2*d^2*z^2 + a^2*b^2 - b^4, z, k)^5*a^8*b^7*d^5 + 679477248*root(729*a^10*d^6*z^6 + 27*a^4*b^2*d^2*z^2
+ a^2*b^2 - b^4, z, k)^5*a^9*b^6*d^5 - 70665633792*root(729*a^10*d^6*z^6 + 27*a^4*b^2*d^2*z^2 + a^2*b^2 - b^4
, z, k)^5*a^10*b^5*d^5 + 52319748096*root(729*a^10*d^6*z^6 + 27*a^4*b^2*d^2*z^2 + a^2*b^2 - b^4, z, k)^5*a^11*
b^4*d^5 + 12230590464*root(729*a^10*d^6*z^6 + 27*a^4*b^2*d^2*z^2 + a^2*b^2 - b^4, z, k)^5*a^12*b^3*d^5 + 32614
907904*root(729*a^10*d^6*z^6 + 27*a^4*b^2*d^2*z^2 + a^2*b^2 - b^4, z, k)^7*a^11*b^6*d^7 + 146767085568*root(72
9*a^10*d^6*z^6 + 27*a^4*b^2*d^2*z^2 + a^2*b^2 - b^4, z, k)^7*a^12*b^5*d^7 - 130459631616*root(729*a^10*d^6*z^6
+ 27*a^4*b^2*d^2*z^2 + a^2*b^2 - b^4, z, k)^7*a^13*b^4*d^7 - 48922361856*root(729*a^10*d^6*z^6 + 27*a^4*b^2*d
^2*z^2 + a^2*b^2 - b^4, z, k)^7*a^14*b^3*d^7 + 67108864*root(729*a^10*d^6*z^6 + 27*a^4*b^2*d^2*z^2 + a^2*b^2 -
b^4, z, k)*a^2*b^9*d - 427819008*root(729*a^10*d^6*z^6 + 27*a^4*b^2*d^2*z^2 + a^2*b^2 - b^4, z, k)*a^3*b^8*d
- 822083584*root(729*a^10*d^6*z^6 + 27*a^4*b^2*d^2*z^2 + a^2*b^2 - b^4, z, k)*a^4*b^7*d + 436207616*root(729*a
^10*d^6*z^6 + 27*a^4*b^2*d^2*z^2 + a^2*b^2 - b^4, z, k)*a^5*b^6*d + 754974720*root(729*a^10*d^6*z^6 + 27*a^4*b
^2*d^2*z^2 + a^2*b^2 - b^4, z, k)*a^6*b^5*d + 25165824*root(729*a^10*d^6*z^6 + 27*a^4*b^2*d^2*z^2 + a^2*b^2 -
b^4, z, k)*a^7*b^4*d - 25165824*a*b^9*exp(d*x)*exp(root(729*a^10*d^6*z^6 + 27*a^4*b^2*d^2*z^2 + a^2*b^2 - b^4,
z, k)) + 234881024*root(729*a^10*d^6*z^6 + 27*a^4*b^2*d^2*z^2 + a^2*b^2 - b^4, z, k)^2*a^3*b^9*d^2*exp(d*x)*e
xp(root(729*a^10*d^6*z^6 + 27*a^4*b^2*d^2*z^2 + a^2*b^2 - b^4, z, k)) + 2592079872*root(729*a^10*d^6*z^6 + 27*
a^4*b^2*d^2*z^2 + a^2*b^2 - b^4, z, k)^2*a^4*b^8*d^2*exp(d*x)*exp(root(729*a^10*d^6*z^6 + 27*a^4*b^2*d^2*z^2 +
a^2*b^2 - b^4, z, k)) - 2860515328*root(729*a^10*d^6*z^6 + 27*a^4*b^2*d^2*z^2 + a^2*b^2 - b^4, z, k)^2*a^5*b^
7*d^2*exp(d*x)*exp(root(729*a^10*d^6*z^6 + 27*a^4*b^2*d^2*z^2 + a^2*b^2 - b^4, z, k)) + 2919235584*root(729*a^
10*d^6*z^6 + 27*a^4*b^2*d^2*z^2 + a^2*b^2 - b^4, z, k)^2*a^6*b^6*d^2*exp(d*x)*exp(root(729*a^10*d^6*z^6 + 27*a
^4*b^2*d^2*z^2 + a^2*b^2 - b^4, z, k)) - 2357198848*root(729*a^10*d^6*z^6 + 27*a^4*b^2*d^2*z^2 + a^2*b^2 - b^4
, z, k)^2*a^7*b^5*d^2*exp(d*x)*exp(root(729*a^10*d^6*z^6 + 27*a^4*b^2*d^2*z^2 + a^2*b^2 - b^4, z, k)) - 528482
304*root(729*a^10*d^6*z^6 + 27*a^4*b^2*d^2*z^2 + a^2*b^2 - b^4, z, k)^2*a^8*b^4*d^2*exp(d*x)*exp(root(729*a^10
*d^6*z^6 + 27*a^4*b^2*d^2*z^2 + a^2*b^2 - b^4, z, k)) + 301989888*root(729*a^10*d^6*z^6 + 27*a^4*b^2*d^2*z^2 +
a^2*b^2 - b^4, z, k)^4*a^6*b^8*d^4*exp(d*x)*exp(root(729*a^10*d^6*z^6 + 27*a^4*b^2*d^2*z^2 + a^2*b^2 - b^4, z
, k)) + 9965666304*root(729*a^10*d^6*z^6 + 27*a^4*b^2*d^2*z^2 + a^2*b^2 - b^4, z, k)^4*a^7*b^7*d^4*exp(d*x)*ex
p(root(729*a^10*d^6*z^6 + 27*a^4*b^2*d^2*z^2 + a^2*b^2 - b^4, z, k)) - 33671872512*root(729*a^10*d^6*z^6 + 27*
a^4*b^2*d^2*z^2 + a^2*b^2 - b^4, z, k)^4*a^8*b^6*d^4*exp(d*x)*exp(root(729*a^10*d^6*z^6 + 27*a^4*b^2*d^2*z^2 +
a^2*b^2 - b^4, z, k)) - 6568280064*root(729*a^10*d^6*z^6 + 27*a^4*b^2*d^2*z^2 + a^2*b^2 - b^4, z, k)^4*a^9*b^
5*d^4*exp(d*x)*exp(root(729*a^10*d^6*z^6 + 27*a^4*b^2*d^2*z^2 + a^2*b^2 - b^4, z, k)) + 29293019136*root(729*a
^10*d^6*z^6 + 27*a^4*b^2*d^2*z^2 + a^2*b^2 - b^4, z, k)^4*a^10*b^4*d^4*exp(d*x)*exp(root(729*a^10*d^6*z^6 + 27
*a^4*b^2*d^2*z^2 + a^2*b^2 - b^4, z, k)) + 679477248*root(729*a^10*d^6*z^6 + 27*a^4*b^2*d^2*z^2 + a^2*b^2 - b^
4, z, k)^4*a^11*b^3*d^4*exp(d*x)*exp(root(729*a^10*d^6*z^6 + 27*a^4*b^2*d^2*z^2 + a^2*b^2 - b^4, z, k)) + 7202
4588288*root(729*a^10*d^6*z^6 + 27*a^4*b^2*d^2*z^2 + a^2*b^2 - b^4, z, k)^6*a^10*b^6*d^6*exp(d*x)*exp(root(729
*a^10*d^6*z^6 + 27*a^4*b^2*d^2*z^2 + a^2*b^2 - b^4, z, k)) + 27179089920*root(729*a^10*d^6*z^6 + 27*a^4*b^2*d^
2*z^2 + a^2*b^2 - b^4, z, k)^6*a^11*b^5*d^6*exp...