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

#### 3.8.1 Mathematica

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

[B]   time = 0.534942 (sec), size = 826 ,normalized size = 25.03 $\frac {\cosh (3 (c+d x)) a^3+27 b \sinh (c+d x) a^2-b \sinh (3 (c+d x)) a^2-9 \left (a^2+3 b^2\right ) \cosh (c+d x) a-b^2 \cosh (3 (c+d x)) a-2 b \text {RootSum}\left [a \text {\#1}^6+b \text {\#1}^6+3 a \text {\#1}^4-3 b \text {\#1}^4+3 a \text {\#1}^2+3 b \text {\#1}^2+a-b\& ,\frac {3 a^2 c \text {\#1}^4+3 b^2 c \text {\#1}^4-3 a b c \text {\#1}^4+3 a^2 d x \text {\#1}^4+3 b^2 d x \text {\#1}^4-3 a b d x \text {\#1}^4+6 a^2 \log \left (\text {\#1} \cosh \left (\frac {1}{2} (c+d x)\right )-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {\#1}\right ) \text {\#1}^4+6 b^2 \log \left (\text {\#1} \cosh \left (\frac {1}{2} (c+d x)\right )-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {\#1}\right ) \text {\#1}^4-6 a b \log \left (\text {\#1} \cosh \left (\frac {1}{2} (c+d x)\right )-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {\#1}\right ) \text {\#1}^4+2 a^2 c \text {\#1}^2-2 b^2 c \text {\#1}^2+2 a^2 d x \text {\#1}^2-2 b^2 d x \text {\#1}^2+4 a^2 \log \left (\text {\#1} \cosh \left (\frac {1}{2} (c+d x)\right )-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {\#1}\right ) \text {\#1}^2-4 b^2 \log \left (\text {\#1} \cosh \left (\frac {1}{2} (c+d x)\right )-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {\#1}\right ) \text {\#1}^2+3 a^2 c+3 b^2 c+3 a b c+3 a^2 d x+3 b^2 d x+3 a b d x+6 a^2 \log \left (\text {\#1} \cosh \left (\frac {1}{2} (c+d x)\right )-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {\#1}\right )+6 b^2 \log \left (\text {\#1} \cosh \left (\frac {1}{2} (c+d x)\right )-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {\#1}\right )+6 a b \log \left (\text {\#1} \cosh \left (\frac {1}{2} (c+d x)\right )-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {\#1}\right )}{a \text {\#1}^5+b \text {\#1}^5+2 a \text {\#1}^3-2 b \text {\#1}^3+a \text {\#1}+b \text {\#1}}\& \right ] a+9 b^3 \sinh (c+d x)+b^3 \sinh (3 (c+d x))}{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] $\int \frac {\sinh (c+d x)}{a+b \tanh ^3(c+d x)} \, dx$

[B]   time = 0.26943 (sec), size = 409 ,normalized size = 13.19 $\frac {b \text {RootSum}\left [\text {\#1}^6 a+\text {\#1}^6 b+3 \text {\#1}^4 a-3 \text {\#1}^4 b+3 \text {\#1}^2 a+3 \text {\#1}^2 b+a-b\& ,\frac {4 \text {\#1}^4 a \log \left (-\text {\#1} \sinh \left (\frac {1}{2} (c+d x)\right )+\text {\#1} \cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )-\cosh \left (\frac {1}{2} (c+d x)\right )\right )+2 \text {\#1}^4 a c+2 \text {\#1}^4 a d x-2 \text {\#1}^4 b \log \left (-\text {\#1} \sinh \left (\frac {1}{2} (c+d x)\right )+\text {\#1} \cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )-\cosh \left (\frac {1}{2} (c+d x)\right )\right )-\text {\#1}^4 b c-\text {\#1}^4 b d x+4 a \log \left (-\text {\#1} \sinh \left (\frac {1}{2} (c+d x)\right )+\text {\#1} \cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )-\cosh \left (\frac {1}{2} (c+d x)\right )\right )+2 b \log \left (-\text {\#1} \sinh \left (\frac {1}{2} (c+d x)\right )+\text {\#1} \cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )-\cosh \left (\frac {1}{2} (c+d x)\right )\right )+2 a c+2 a d x+b c+b d x}{\text {\#1}^5 a+\text {\#1}^5 b+2 \text {\#1}^3 a-2 \text {\#1}^3 b+\text {\#1} a+\text {\#1} b}\& \right ]+6 a \cosh (c+d x)-6 b \sinh (c+d x)}{6 d (a-b) (a+b)}$

[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] $\int \frac {\text {csch}(c+d x)}{a+b \tanh ^3(c+d x)} \, dx$

[B]   time = 0.179177 (sec), size = 319 ,normalized size = 10.29 $\frac {6 \log \left (\tanh \left (\frac {1}{2} (c+d x)\right )\right )-b \text {RootSum}\left [\text {\#1}^6 a+\text {\#1}^6 b+3 \text {\#1}^4 a-3 \text {\#1}^4 b+3 \text {\#1}^2 a+3 \text {\#1}^2 b+a-b\& ,\frac {2 \text {\#1}^4 \log \left (-\text {\#1} \sinh \left (\frac {1}{2} (c+d x)\right )+\text {\#1} \cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )-\cosh \left (\frac {1}{2} (c+d x)\right )\right )+\text {\#1}^4 c+\text {\#1}^4 d x-4 \text {\#1}^2 \log \left (-\text {\#1} \sinh \left (\frac {1}{2} (c+d x)\right )+\text {\#1} \cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )-\cosh \left (\frac {1}{2} (c+d x)\right )\right )-2 \text {\#1}^2 c-2 \text {\#1}^2 d x+2 \log \left (-\text {\#1} \sinh \left (\frac {1}{2} (c+d x)\right )+\text {\#1} \cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )-\cosh \left (\frac {1}{2} (c+d x)\right )\right )+c+d x}{\text {\#1}^5 a+\text {\#1}^5 b+2 \text {\#1}^3 a-2 \text {\#1}^3 b+\text {\#1} a+\text {\#1} b}\& \right ]}{6 a d}$

[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] $\int \frac {\text {csch}^3(c+d x)}{a+b \tanh ^3(c+d x)} \, dx$

[B]   time = 0.386272 (sec), size = 201 ,normalized size = 6.09 $-\frac {16 b \text {RootSum}\left [\text {\#1}^6 a+\text {\#1}^6 b+3 \text {\#1}^4 a-3 \text {\#1}^4 b+3 \text {\#1}^2 a+3 \text {\#1}^2 b+a-b\& ,\frac {2 \text {\#1} \log \left (-\text {\#1} \sinh \left (\frac {1}{2} (c+d x)\right )+\text {\#1} \cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )-\cosh \left (\frac {1}{2} (c+d x)\right )\right )+\text {\#1} c+\text {\#1} d x}{\text {\#1}^4 a+\text {\#1}^4 b+2 \text {\#1}^2 a-2 \text {\#1}^2 b+a+b}\& \right ]+3 \left (\text {csch}^2\left (\frac {1}{2} (c+d x)\right )+\text {sech}^2\left (\frac {1}{2} (c+d x)\right )+4 \log \left (\tanh \left (\frac {1}{2} (c+d x)\right )\right )\right )}{24 a d}$

[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] $\int \frac {\sinh ^3(c+d x)}{a+b \tanh ^3(c+d x)} \, dx$

[B]   time = 0.467 (sec), size = 346 ,normalized size = 10.48 $-\frac {a b \left (\munderset {\textit {\_R} =\RootOf \left (a \,\textit {\_Z}^{6}+3 a \,\textit {\_Z}^{4}+8 b \,\textit {\_Z}^{3}+3 a \,\textit {\_Z}^{2}+a \right )}{\sum }\frac {\left (\left (2 a^{2}+b^{2}\right ) \textit {\_R}^{4}-6 \textit {\_R}^{3} a b +2 \left (4 a^{2}+5 b^{2}\right ) \textit {\_R}^{2}-6 a b \textit {\_R} +2 a^{2}+b^{2}\right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-\textit {\_R} \right )}{\textit {\_R}^{5} a +2 \textit {\_R}^{3} a +4 \textit {\_R}^{2} b +\textit {\_R} a}\right )}{3 d \left (a +b \right )^{2} \left (a -b \right )^{2}}-\frac {16}{3 d \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3} \left (16 a +16 b \right )}-\frac {8}{d \left (16 a +16 b \right ) \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}+\frac {a}{2 d \left (a +b \right )^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}-\frac {b}{d \left (a +b \right )^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}-\frac {8}{d \left (16 a -16 b \right ) \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}+\frac {16}{3 d \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3} \left (16 a -16 b \right )}-\frac {a}{2 d \left (a -b \right )^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}-\frac {b}{d \left (a -b \right )^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}$

[In]

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

[Out]

-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))-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-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

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

[B]   time = 0.528 (sec), size = 164 ,normalized size = 5.29 $\frac {b \left (\munderset {\textit {\_R} =\RootOf \left (a \,\textit {\_Z}^{6}+3 a \,\textit {\_Z}^{4}+8 b \,\textit {\_Z}^{3}+3 a \,\textit {\_Z}^{2}+a \right )}{\sum }\frac {\left (\textit {\_R}^{4} a -2 \textit {\_R}^{3} b +6 \textit {\_R}^{2} a -2 \textit {\_R} b +a \right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-\textit {\_R} \right )}{\textit {\_R}^{5} a +2 \textit {\_R}^{3} a +4 \textit {\_R}^{2} b +\textit {\_R} a}\right )}{3 d \left (a -b \right ) \left (a +b \right )}+\frac {4}{d \left (4 a -4 b \right ) \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}-\frac {4}{d \left (4 a +4 b \right ) \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}$

[In]

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

[Out]

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)-4/d/(4*a+4*b)/(
tanh(1/2*d*x+1/2*c)-1)

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

[B]   time = 0.619 (sec), size = 98 ,normalized size = 3.16 $-\frac {4 b \left (\munderset {\textit {\_R} =\RootOf \left (a \,\textit {\_Z}^{6}+3 a \,\textit {\_Z}^{4}+8 b \,\textit {\_Z}^{3}+3 a \,\textit {\_Z}^{2}+a \right )}{\sum }\frac {\textit {\_R}^{2} \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-\textit {\_R} \right )}{\textit {\_R}^{5} a +2 \textit {\_R}^{3} a +4 \textit {\_R}^{2} b +\textit {\_R} a}\right )}{3 d a}+\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}$

[In]

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

[Out]

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

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

[B]   time = 0.595 (sec), size = 144 ,normalized size = 4.36 $\frac {\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d a}-\frac {b \left (\munderset {\textit {\_R} =\RootOf \left (a \,\textit {\_Z}^{6}+3 a \,\textit {\_Z}^{4}+8 b \,\textit {\_Z}^{3}+3 a \,\textit {\_Z}^{2}+a \right )}{\sum }\frac {\left (\textit {\_R}^{4}-2 \textit {\_R}^{2}+1\right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-\textit {\_R} \right )}{\textit {\_R}^{5} a +2 \textit {\_R}^{3} a +4 \textit {\_R}^{2} b +\textit {\_R} a}\right )}{3 d a}-\frac {1}{8 d a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}-\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d a}$

[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)^2-1/3/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))-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))

#### 3.8.3 Giac

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

[B]   time = 3.16393 (sec), size = 344 ,normalized size = 10.42 $-\frac {\frac {{\left (9 \, a e^{\left (2 \, d x + 2 \, c\right )} + 9 \, b e^{\left (2 \, d x + 2 \, c\right )} - a + b\right )} e^{\left (-3 \, d x\right )}}{a^{2} e^{\left (3 \, c\right )} - 2 \, a b e^{\left (3 \, c\right )} + b^{2} e^{\left (3 \, c\right )}} - \frac {a^{2} e^{\left (3 \, d x + 30 \, c\right )} + 2 \, a b e^{\left (3 \, d x + 30 \, c\right )} + b^{2} e^{\left (3 \, d x + 30 \, c\right )} - 9 \, a^{2} e^{\left (d x + 28 \, c\right )} + 9 \, b^{2} e^{\left (d x + 28 \, c\right )}}{a^{3} e^{\left (27 \, c\right )} + 3 \, a^{2} b e^{\left (27 \, c\right )} + 3 \, a b^{2} e^{\left (27 \, c\right )} + b^{3} e^{\left (27 \, c\right )}}}{24 \, d} - \frac {\frac {6 \, {\left (a^{3} b e^{c} + a^{2} b^{2} e^{c} + a b^{3} e^{c}\right )} d x}{a - b} - \frac {{\left (a^{3} b e^{c} + a^{2} b^{2} e^{c} + a b^{3} e^{c}\right )} \log \left ({\left | a e^{\left (6 \, d x + 6 \, c\right )} + b e^{\left (6 \, d x + 6 \, c\right )} + 3 \, a e^{\left (4 \, d x + 4 \, c\right )} - 3 \, b e^{\left (4 \, d x + 4 \, c\right )} + 3 \, a e^{\left (2 \, d x + 2 \, c\right )} + 3 \, b e^{\left (2 \, d x + 2 \, c\right )} + a - b \right |}\right )}{a - b}}{{\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} d^{2}}$

[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)/(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 - b) - (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 - b)
)/((a^4 - 2*a^2*b^2 + b^4)*d^2)

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

[B]   time = 1.84122 (sec), size = 189 ,normalized size = 6.1 $\frac {\frac {e^{\left (d x + 8 \, c\right )}}{a e^{\left (7 \, c\right )} + b e^{\left (7 \, c\right )}} + \frac {e^{\left (-d x\right )}}{a e^{c} - b e^{c}}}{2 \, d} + \frac {\frac {6 \, {\left (2 \, a b e^{c} + b^{2} e^{c}\right )} d x}{a - b} - \frac {{\left (2 \, a b e^{c} + b^{2} e^{c}\right )} \log \left ({\left | a e^{\left (6 \, d x + 6 \, c\right )} + b e^{\left (6 \, d x + 6 \, c\right )} + 3 \, a e^{\left (4 \, d x + 4 \, c\right )} - 3 \, b e^{\left (4 \, d x + 4 \, c\right )} + 3 \, a e^{\left (2 \, d x + 2 \, c\right )} + 3 \, b e^{\left (2 \, d x + 2 \, c\right )} + a - b \right |}\right )}{a - b}}{3 \, {\left (a^{2} - b^{2}\right )} d^{2}}$

[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
- b) - (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 - b))/((a^2 - b^2)*d^2)

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

[B]   time = 1.39841 (sec), size = 147 ,normalized size = 4.74 $-\frac {\frac {\log \left (e^{\left (d x + c\right )} + 1\right )}{a} - \frac {\log \left ({\left | e^{\left (d x + c\right )} - 1 \right |}\right )}{a}}{d} - \frac {\frac {6 \, b d x e^{c}}{a - b} - \frac {b e^{c} \log \left ({\left | a e^{\left (6 \, d x + 6 \, c\right )} + b e^{\left (6 \, d x + 6 \, c\right )} + 3 \, a e^{\left (4 \, d x + 4 \, c\right )} - 3 \, b e^{\left (4 \, d x + 4 \, c\right )} + 3 \, a e^{\left (2 \, d x + 2 \, c\right )} + 3 \, b e^{\left (2 \, d x + 2 \, c\right )} + a - b \right |}\right )}{a - b}}{3 \, a d^{2}}$

[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*b*d*x*e^c/(a - b) - 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 - b))/(a*d^2)

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

[B]   time = 0.788413 (sec), size = 68 ,normalized size = 2.06 $\frac {\frac {\log \left (e^{\left (d x + c\right )} + 1\right )}{a} - \frac {\log \left ({\left | e^{\left (d x + c\right )} - 1 \right |}\right )}{a} - \frac {2 \, {\left (e^{\left (3 \, d x + 3 \, c\right )} + e^{\left (d x + c\right )}\right )}}{a {\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )}^{2}}}{2 \, d}$

[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