3.8 Test file Number [173]
3.8.1 Mathematica
Integral number [74] \[ \int \frac {\sinh ^3(c+d x)}{a+b \tanh ^3(c+d x)} \, dx \]
[B] time = 0.388247 (sec), size = 826 ,normalized size = 25.03 \[ \frac {-9 a \left (a^2+3 b^2\right ) \cosh (c+d x)+a^3 \cosh (3 (c+d x))-a b^2 \cosh (3 (c+d x))-2 a b \text {RootSum}\left [a-b+3 a \text {$\#$1}^2+3 b \text {$\#$1}^2+3 a \text {$\#$1}^4-3 b \text {$\#$1}^4+a \text {$\#$1}^6+b \text {$\#$1}^6\& ,\frac {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 \left (-\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 ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right )+6 a b \log \left (-\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 ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right )+6 b^2 \log \left (-\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 ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right )+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 (-\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 ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right ) \text {$\#$1}^2-4 b^2 \log \left (-\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 ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right ) \text {$\#$1}^2+3 a^2 c \text {$\#$1}^4-3 a b c \text {$\#$1}^4+3 b^2 c \text {$\#$1}^4+3 a^2 d x \text {$\#$1}^4-3 a b d x \text {$\#$1}^4+3 b^2 d x \text {$\#$1}^4+6 a^2 \log \left (-\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 ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right ) \text {$\#$1}^4-6 a b \log \left (-\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 ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right ) \text {$\#$1}^4+6 b^2 \log \left (-\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 ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right ) \text {$\#$1}^4}{a \text {$\#$1}+b \text {$\#$1}+2 a \text {$\#$1}^3-2 b \text {$\#$1}^3+a \text {$\#$1}^5+b \text {$\#$1}^5}\& \right ]+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} \]
[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 = 0.164957 (sec), size = 409 ,normalized size = 13.19 \[ \frac {6 a \cosh (c+d x)+b \text {RootSum}\left [a-b+3 a \text {$\#$1}^2+3 b \text {$\#$1}^2+3 a \text {$\#$1}^4-3 b \text {$\#$1}^4+a \text {$\#$1}^6+b \text {$\#$1}^6\& ,\frac {2 a c+b c+2 a d x+b d x+4 a \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right )+2 b \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right )+2 a c \text {$\#$1}^4-b c \text {$\#$1}^4+2 a d x \text {$\#$1}^4-b d x \text {$\#$1}^4+4 a \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right ) \text {$\#$1}^4-2 b \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right ) \text {$\#$1}^4}{a \text {$\#$1}+b \text {$\#$1}+2 a \text {$\#$1}^3-2 b \text {$\#$1}^3+a \text {$\#$1}^5+b \text {$\#$1}^5}\& \right ]-6 b \sinh (c+d x)}{6 (a-b) (a+b) d} \]
[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] \[ \int \frac {\text {csch}(c+d x)}{a+b \tanh ^3(c+d x)} \, dx \]
[B] time = 0.124779 (sec), size = 319 ,normalized size = 10.29 \[ \frac {6 \log \left (\tanh \left (\frac {1}{2} (c+d x)\right )\right )-b \text {RootSum}\left [a-b+3 a \text {$\#$1}^2+3 b \text {$\#$1}^2+3 a \text {$\#$1}^4-3 b \text {$\#$1}^4+a \text {$\#$1}^6+b \text {$\#$1}^6\& ,\frac {c+d x+2 \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right )-2 c \text {$\#$1}^2-2 d x \text {$\#$1}^2-4 \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right ) \text {$\#$1}^2+c \text {$\#$1}^4+d x \text {$\#$1}^4+2 \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right ) \text {$\#$1}^4}{a \text {$\#$1}+b \text {$\#$1}+2 a \text {$\#$1}^3-2 b \text {$\#$1}^3+a \text {$\#$1}^5+b \text {$\#$1}^5}\& \right ]}{6 a d} \]
[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 = 0.279736 (sec), size = 201 ,normalized size = 6.09 \[ -\frac {16 b \text {RootSum}\left [a-b+3 a \text {$\#$1}^2+3 b \text {$\#$1}^2+3 a \text {$\#$1}^4-3 b \text {$\#$1}^4+a \text {$\#$1}^6+b \text {$\#$1}^6\& ,\frac {c \text {$\#$1}+d x \text {$\#$1}+2 \log \left (-\cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )+\cosh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}-\sinh \left (\frac {1}{2} (c+d x)\right ) \text {$\#$1}\right ) \text {$\#$1}}{a+b+2 a \text {$\#$1}^2-2 b \text {$\#$1}^2+a \text {$\#$1}^4+b \text {$\#$1}^4}\& \right ]+3 \left (\text {csch}^2\left (\frac {1}{2} (c+d x)\right )+4 \log \left (\tanh \left (\frac {1}{2} (c+d x)\right )\right )+\text {sech}^2\left (\frac {1}{2} (c+d x)\right )\right )}{24 a d} \]
[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 = 7.433 (sec), size = 289 ,normalized size = 8.76
| | |
| method |
result |
size |
| | |
| derivativedivides |
\(\frac {-\frac {a b \left (\munderset {\textit {\_R} =\RootOf \left (a \,\textit {\_Z}^{6}+3 a \,\textit {\_Z}^{4}+8 b \,\textit {\_Z}^{3}+3 a \,\textit {\_Z}^{2}+a \right )}{\sum }\frac {\left (\left (2 a^{2}+b^{2}\right ) \textit {\_R}^{4}-6 a b \,\textit {\_R}^{3}+2 \left (4 a^{2}+5 b^{2}\right ) \textit {\_R}^{2}-6 a b \textit {\_R} +2 a^{2}+b^{2}\right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-\textit {\_R} \right )}{\textit {\_R}^{5} a +2 \textit {\_R}^{3} a +4 \textit {\_R}^{2} b +\textit {\_R} a}\right )}{3 \left (a +b \right )^{2} \left (a -b \right )^{2}}-\frac {8}{\left (16 a -16 b \right ) \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}+\frac {16}{3 \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3} \left (16 a -16 b \right )}-\frac {2 b +a}{2 \left (a -b \right )^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}-\frac {16}{3 \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3} \left (16 a +16 b \right )}-\frac {8}{\left (16 a +16 b \right ) \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {2 b -a}{2 \left (a +b \right )^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}}{d}\) |
\(289\) |
| default |
\(\frac {-\frac {a b \left (\munderset {\textit {\_R} =\RootOf \left (a \,\textit {\_Z}^{6}+3 a \,\textit {\_Z}^{4}+8 b \,\textit {\_Z}^{3}+3 a \,\textit {\_Z}^{2}+a \right )}{\sum }\frac {\left (\left (2 a^{2}+b^{2}\right ) \textit {\_R}^{4}-6 a b \,\textit {\_R}^{3}+2 \left (4 a^{2}+5 b^{2}\right ) \textit {\_R}^{2}-6 a b \textit {\_R} +2 a^{2}+b^{2}\right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-\textit {\_R} \right )}{\textit {\_R}^{5} a +2 \textit {\_R}^{3} a +4 \textit {\_R}^{2} b +\textit {\_R} a}\right )}{3 \left (a +b \right )^{2} \left (a -b \right )^{2}}-\frac {8}{\left (16 a -16 b \right ) \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}+\frac {16}{3 \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3} \left (16 a -16 b \right )}-\frac {2 b +a}{2 \left (a -b \right )^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}-\frac {16}{3 \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3} \left (16 a +16 b \right )}-\frac {8}{\left (16 a +16 b \right ) \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {2 b -a}{2 \left (a +b \right )^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}}{d}\) |
\(289\) |
| risch |
\(\text {Expression too large to display}\) |
\(2825\) |
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[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] \[ \int \frac {\sinh (c+d x)}{a+b \tanh ^3(c+d x)} \, dx \]
[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] \[ \int \frac {\text {csch}(c+d x)}{a+b \tanh ^3(c+d x)} \, dx \]
[B] time = 4.535 (sec), size = 96 ,normalized size = 3.1
| | |
| method |
result |
size |
| | |
| derivativedivides |
\(\frac {\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}-\frac {4 b \left (\munderset {\textit {\_R} =\RootOf \left (a \,\textit {\_Z}^{6}+3 a \,\textit {\_Z}^{4}+8 b \,\textit {\_Z}^{3}+3 a \,\textit {\_Z}^{2}+a \right )}{\sum }\frac {\textit {\_R}^{2} \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-\textit {\_R} \right )}{\textit {\_R}^{5} a +2 \textit {\_R}^{3} a +4 \textit {\_R}^{2} b +\textit {\_R} a}\right )}{3 a}}{d}\) |
\(96\) |
| default |
\(\frac {\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a}-\frac {4 b \left (\munderset {\textit {\_R} =\RootOf \left (a \,\textit {\_Z}^{6}+3 a \,\textit {\_Z}^{4}+8 b \,\textit {\_Z}^{3}+3 a \,\textit {\_Z}^{2}+a \right )}{\sum }\frac {\textit {\_R}^{2} \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-\textit {\_R} \right )}{\textit {\_R}^{5} a +2 \textit {\_R}^{3} a +4 \textit {\_R}^{2} b +\textit {\_R} a}\right )}{3 a}}{d}\) |
\(96\) |
| risch |
\(-\frac {\ln \left ({\mathrm e}^{d x +c}+1\right )}{d a}+2 \left (\munderset {\textit {\_R} =\RootOf \left (\left (46656 a^{8} d^{6}-46656 a^{6} b^{2} d^{6}\right ) \textit {\_Z}^{6}+3888 a^{4} b^{2} d^{4} \textit {\_Z}^{4}-108 a^{2} b^{2} d^{2} \textit {\_Z}^{2}+b^{2}\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{d x +c}+\left (\frac {7776 d^{5} a^{6}}{b}-7776 b \,d^{5} a^{4}\right ) \textit {\_R}^{5}+\left (\frac {216 d^{3} a^{4}}{b}+432 a^{2} b \,d^{3}\right ) \textit {\_R}^{3}+\left (6 a d -6 b d \right ) \textit {\_R} \right )\right )+\frac {\ln \left ({\mathrm e}^{d x +c}-1\right )}{d a}\) |
\(168\) |
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[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] \[ \int \frac {\text {csch}^3(c+d x)}{a+b \tanh ^3(c+d x)} \, dx \]
[B] time = 4.852 (sec), size = 136 ,normalized size = 4.12
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| method |
result |
size |
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| derivativedivides |
\(\frac {\frac {\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}{8 a}-\frac {1}{8 a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}-\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a}-\frac {b \left (\munderset {\textit {\_R} =\RootOf \left (a \,\textit {\_Z}^{6}+3 a \,\textit {\_Z}^{4}+8 b \,\textit {\_Z}^{3}+3 a \,\textit {\_Z}^{2}+a \right )}{\sum }\frac {\left (\textit {\_R}^{4}-2 \textit {\_R}^{2}+1\right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-\textit {\_R} \right )}{\textit {\_R}^{5} a +2 \textit {\_R}^{3} a +4 \textit {\_R}^{2} b +\textit {\_R} a}\right )}{3 a}}{d}\) |
\(136\) |
| default |
\(\frac {\frac {\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}{8 a}-\frac {1}{8 a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}-\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a}-\frac {b \left (\munderset {\textit {\_R} =\RootOf \left (a \,\textit {\_Z}^{6}+3 a \,\textit {\_Z}^{4}+8 b \,\textit {\_Z}^{3}+3 a \,\textit {\_Z}^{2}+a \right )}{\sum }\frac {\left (\textit {\_R}^{4}-2 \textit {\_R}^{2}+1\right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-\textit {\_R} \right )}{\textit {\_R}^{5} a +2 \textit {\_R}^{3} a +4 \textit {\_R}^{2} b +\textit {\_R} a}\right )}{3 a}}{d}\) |
\(136\) |
| risch |
\(-\frac {{\mathrm e}^{d x +c} \left (1+{\mathrm e}^{2 d x +2 c}\right )}{d a \left ({\mathrm e}^{2 d x +2 c}-1\right )^{2}}+\frac {\ln \left ({\mathrm e}^{d x +c}+1\right )}{2 d a}-\frac {\ln \left ({\mathrm e}^{d x +c}-1\right )}{2 d a}+8 \left (\munderset {\textit {\_R} =\RootOf \left (191102976 d^{6} \textit {\_Z}^{6} a^{10}+1728 a^{4} d^{2} \textit {\_Z}^{2} b^{2}+a^{2} b^{2}-b^{4}\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{d x +c}+\frac {15925248 d^{5} a^{9} \textit {\_R}^{5}}{a^{3} b +a^{2} b^{2}+a \,b^{3}+b^{4}}+\left (-\frac {13824 d^{3} a^{7}}{a^{3} b +a^{2} b^{2}+a \,b^{3}+b^{4}}+\frac {13824 d^{3} b^{2} a^{5}}{a^{3} b +a^{2} b^{2}+a \,b^{3}+b^{4}}\right ) \textit {\_R}^{3}+\left (-\frac {24 d \,a^{4} b}{a^{3} b +a^{2} b^{2}+a \,b^{3}+b^{4}}+\frac {96 d \,b^{2} a^{3}}{a^{3} b +a^{2} b^{2}+a \,b^{3}+b^{4}}-\frac {24 d \,b^{3} a^{2}}{a^{3} b +a^{2} b^{2}+a \,b^{3}+b^{4}}\right ) \textit {\_R} \right )\right )\) |
\(329\) |
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[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] \[ \int \frac {\sinh ^3(c+d x)}{a+b \tanh ^3(c+d x)} \, dx \]
[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] \[ \int \frac {\sinh (c+d x)}{a+b \tanh ^3(c+d x)} \, dx \]
[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] \[ \int \frac {\text {csch}(c+d x)}{a+b \tanh ^3(c+d x)} \, dx \]
[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] \[ \int \frac {\text {csch}^3(c+d x)}{a+b \tanh ^3(c+d x)} \, dx \]
[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] \[ \int \frac {\sinh ^3(c+d x)}{a+b \tanh ^3(c+d x)} \, dx \]
[C] time = 1.52966 (sec), size = 303 ,normalized size = 9.18 \[ -\frac {\frac {{\left (9 \, a e^{\left (2 \, d x + 2 \, c\right )} + 9 \, b e^{\left (2 \, d x + 2 \, c\right )} - a + b\right )} e^{\left (-3 \, d x - 3 \, c\right )}}{a^{2} - 2 \, a b + b^{2}} - \frac {a^{2} e^{\left (3 \, d x + 3 \, c\right )} + 2 \, a b e^{\left (3 \, d x + 3 \, c\right )} + b^{2} e^{\left (3 \, d x + 3 \, c\right )} - 9 \, a^{2} e^{\left (d x + c\right )} + 9 \, b^{2} e^{\left (d x + c\right )}}{a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}}}{24 \, d} - \frac {\frac {6 \, {\left (a^{3} b + a^{2} b^{2} + a b^{3}\right )} {\left (d x + c\right )}}{a - b} - \frac {{\left (a^{3} b + a^{2} b^{2} + a b^{3}\right )} \log \left ({\left | a e^{\left (6 \, d x + 6 \, c\right )} + b e^{\left (6 \, d x + 6 \, c\right )} + 3 \, a e^{\left (4 \, d x + 4 \, c\right )} - 3 \, b e^{\left (4 \, d x + 4 \, c\right )} + 3 \, a e^{\left (2 \, d x + 2 \, c\right )} + 3 \, b e^{\left (2 \, d x + 2 \, c\right )} + a - b \right |}\right )}{a - b}}{{\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} d^{2}} \]
[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 - 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] \[ \int \frac {\sinh (c+d x)}{a+b \tanh ^3(c+d x)} \, dx \]
[C] time = 1.05927 (sec), size = 169 ,normalized size = 5.45 \[ \frac {\frac {e^{\left (d x + c\right )}}{a + b} + \frac {e^{\left (-d x - c\right )}}{a - b}}{2 \, d} + \frac {\frac {6 \, {\left (2 \, a b + b^{2}\right )} {\left (d x + c\right )}}{a - b} - \frac {{\left (2 \, a b + b^{2}\right )} \log \left ({\left | a e^{\left (6 \, d x + 6 \, c\right )} + b e^{\left (6 \, d x + 6 \, c\right )} + 3 \, a e^{\left (4 \, d x + 4 \, c\right )} - 3 \, b e^{\left (4 \, d x + 4 \, c\right )} + 3 \, a e^{\left (2 \, d x + 2 \, c\right )} + 3 \, b e^{\left (2 \, d x + 2 \, c\right )} + a - b \right |}\right )}{a - b}}{3 \, {\left (a^{2} - b^{2}\right )} d^{2}} \]
[In]
integrate(sinh(d*x+c)/(a+b*tanh(d*x+c)^3),x, algorithm=""giac"")
[Out]
1/2*(e^(d*x + 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] \[ \int \frac {\text {csch}(c+d x)}{a+b \tanh ^3(c+d x)} \, dx \]
[C] time = 0.847952 (sec), size = 146 ,normalized size = 4.71 \[ -\frac {\frac {\log \left (e^{\left (d x + c\right )} + 1\right )}{a} - \frac {\log \left ({\left | e^{\left (d x + c\right )} - 1 \right |}\right )}{a}}{d} - \frac {\frac {6 \, {\left (d x + c\right )} b}{a - b} - \frac {b \log \left ({\left | a e^{\left (6 \, d x + 6 \, c\right )} + b e^{\left (6 \, d x + 6 \, c\right )} + 3 \, a e^{\left (4 \, d x + 4 \, c\right )} - 3 \, b e^{\left (4 \, d x + 4 \, c\right )} + 3 \, a e^{\left (2 \, d x + 2 \, c\right )} + 3 \, b e^{\left (2 \, d x + 2 \, c\right )} + a - b \right |}\right )}{a - b}}{3 \, a d^{2}} \]
[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*(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] \[ \int \frac {\text {csch}^3(c+d x)}{a+b \tanh ^3(c+d x)} \, dx \]
[C] time = 0.651096 (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
3.8.5 Mupad
Integral number [76] \[ \int \frac {\sinh (c+d x)}{a+b \tanh ^3(c+d x)} \, dx \]
[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] \[ \int \frac {\text {csch}(c+d x)}{a+b \tanh ^3(c+d x)} \, dx \]
[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] \[ \int \frac {\text {csch}^3(c+d x)}{a+b \tanh ^3(c+d x)} \, dx \]
[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...