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

[B]   time = 0.483965 (sec), size = 826 ,normalized size = 25.81 \[ \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 = 0.226157 (sec), size = 409 ,normalized size = 13.63 \[ \frac{b \text{RootSum}\left [\text{$\#$1}^6 a+3 \text{$\#$1}^4 a+3 \text{$\#$1}^2 a+\text{$\#$1}^6 b-3 \text{$\#$1}^4 b+3 \text{$\#$1}^2 b+a-b\& ,\frac{4 \text{$\#$1}^4 a \log \left (-\text{$\#$1} \sinh \left (\frac{1}{2} (c+d x)\right )+\text{$\#$1} \cosh \left (\frac{1}{2} (c+d x)\right )-\sinh \left (\frac{1}{2} (c+d x)\right )-\cosh \left (\frac{1}{2} (c+d x)\right )\right )+2 \text{$\#$1}^4 a c+2 \text{$\#$1}^4 a d x-2 \text{$\#$1}^4 b \log \left (-\text{$\#$1} \sinh \left (\frac{1}{2} (c+d x)\right )+\text{$\#$1} \cosh \left (\frac{1}{2} (c+d x)\right )-\sinh \left (\frac{1}{2} (c+d x)\right )-\cosh \left (\frac{1}{2} (c+d x)\right )\right )-\text{$\#$1}^4 b c-\text{$\#$1}^4 b d x+4 a \log \left (-\text{$\#$1} \sinh \left (\frac{1}{2} (c+d x)\right )+\text{$\#$1} \cosh \left (\frac{1}{2} (c+d x)\right )-\sinh \left (\frac{1}{2} (c+d x)\right )-\cosh \left (\frac{1}{2} (c+d x)\right )\right )+2 b \log \left (-\text{$\#$1} \sinh \left (\frac{1}{2} (c+d x)\right )+\text{$\#$1} \cosh \left (\frac{1}{2} (c+d x)\right )-\sinh \left (\frac{1}{2} (c+d x)\right )-\cosh \left (\frac{1}{2} (c+d x)\right )\right )+2 a c+2 a d x+b c+b d x}{\text{$\#$1}^5 a+2 \text{$\#$1}^3 a+\text{$\#$1}^5 b-2 \text{$\#$1}^3 b+\text{$\#$1} a+\text{$\#$1} b}\& \right ]+6 a \cosh (c+d x)-6 b \sinh (c+d x)}{6 d (a-b) (a+b)} \]

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

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

[In]

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

[Out]

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

[B]   time = 0.11 (sec), size = 346 ,normalized size = 10.81 \[ -8\,{\frac{1}{d \left ( 16\,a-16\,b \right ) \left ( \tanh \left ( 1/2\,dx+c/2 \right ) +1 \right ) ^{2}}}+{\frac{16}{3\,d \left ( 16\,a-16\,b \right ) } \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) ^{-3}}-{\frac{a}{2\,d \left ( a-b \right ) ^{2}} \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) ^{-1}}-{\frac{b}{d \left ( a-b \right ) ^{2}} \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) ^{-1}}-{\frac{16}{3\,d \left ( 16\,a+16\,b \right ) } \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -1 \right ) ^{-3}}-8\,{\frac{1}{d \left ( 16\,a+16\,b \right ) \left ( \tanh \left ( 1/2\,dx+c/2 \right ) -1 \right ) ^{2}}}+{\frac{a}{2\,d \left ( a+b \right ) ^{2}} \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -1 \right ) ^{-1}}-{\frac{b}{d \left ( a+b \right ) ^{2}} \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -1 \right ) ^{-1}}-{\frac{ab}{3\,d \left ( a+b \right ) ^{2} \left ( a-b \right ) ^{2}}\sum _{{\it \_R}={\it RootOf} \left ( a{{\it \_Z}}^{6}+3\,a{{\it \_Z}}^{4}+8\,b{{\it \_Z}}^{3}+3\,a{{\it \_Z}}^{2}+a \right ) }{\frac{ \left ( 2\,{a}^{2}+{b}^{2} \right ){{\it \_R}}^{4}-6\,{{\it \_R}}^{3}ab+2\, \left ( 4\,{a}^{2}+5\,{b}^{2} \right ){{\it \_R}}^{2}-6\,ab{\it \_R}+2\,{a}^{2}+{b}^{2}}{{{\it \_R}}^{5}a+2\,{{\it \_R}}^{3}a+4\,{{\it \_R}}^{2}b+{\it \_R}\,a}\ln \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -{\it \_R} \right ) }} \]

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

[B]   time = 0.108 (sec), size = 164 ,normalized size = 5.47 \[ -4\,{\frac{1}{d \left ( 4\,a+4\,b \right ) \left ( \tanh \left ( 1/2\,dx+c/2 \right ) -1 \right ) }}+{\frac{b}{3\,d \left ( a-b \right ) \left ( a+b \right ) }\sum _{{\it \_R}={\it RootOf} \left ( a{{\it \_Z}}^{6}+3\,a{{\it \_Z}}^{4}+8\,b{{\it \_Z}}^{3}+3\,a{{\it \_Z}}^{2}+a \right ) }{\frac{{{\it \_R}}^{4}a-2\,{{\it \_R}}^{3}b+6\,{{\it \_R}}^{2}a-2\,{\it \_R}\,b+a}{{{\it \_R}}^{5}a+2\,{{\it \_R}}^{3}a+4\,{{\it \_R}}^{2}b+{\it \_R}\,a}\ln \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -{\it \_R} \right ) }}+4\,{\frac{1}{d \left ( 4\,a-4\,b \right ) \left ( \tanh \left ( 1/2\,dx+c/2 \right ) +1 \right ) }} \]

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

[B]   time = 0.105 (sec), size = 98 ,normalized size = 3.27 \[{\frac{1}{da}\ln \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) }-{\frac{4\,b}{3\,da}\sum _{{\it \_R}={\it RootOf} \left ( a{{\it \_Z}}^{6}+3\,a{{\it \_Z}}^{4}+8\,b{{\it \_Z}}^{3}+3\,a{{\it \_Z}}^{2}+a \right ) }{\frac{{{\it \_R}}^{2}}{{{\it \_R}}^{5}a+2\,{{\it \_R}}^{3}a+4\,{{\it \_R}}^{2}b+{\it \_R}\,a}\ln \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -{\it \_R} \right ) }} \]

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

[B]   time = 0.125 (sec), size = 144 ,normalized size = 4.5 \[{\frac{1}{8\,da} \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}}-{\frac{1}{8\,da} \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-2}}-{\frac{1}{2\,da}\ln \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) }-{\frac{b}{3\,da}\sum _{{\it \_R}={\it RootOf} \left ( a{{\it \_Z}}^{6}+3\,a{{\it \_Z}}^{4}+8\,b{{\it \_Z}}^{3}+3\,a{{\it \_Z}}^{2}+a \right ) }{\frac{{{\it \_R}}^{4}-2\,{{\it \_R}}^{2}+1}{{{\it \_R}}^{5}a+2\,{{\it \_R}}^{3}a+4\,{{\it \_R}}^{2}b+{\it \_R}\,a}\ln \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -{\it \_R} \right ) }} \]

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

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

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

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

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

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

[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)