Integrand size = 13, antiderivative size = 113 \[ \int \frac {\tanh ^3(x)}{a+b \text {csch}(x)} \, dx=-\frac {b^3 \arctan (\sinh (x))}{\left (a^2+b^2\right )^2}-\frac {b \arctan (\sinh (x))}{2 \left (a^2+b^2\right )}+\frac {b^4 \log (a+b \text {csch}(x))}{a \left (a^2+b^2\right )^2}+\frac {\log (\sinh (x))}{a}-\frac {a \left (a^2+2 b^2\right ) \log (\tanh (x))}{\left (a^2+b^2\right )^2}-\frac {(a-b \text {csch}(x)) \tanh ^2(x)}{2 \left (a^2+b^2\right )} \] Output:
-b^3*arctan(sinh(x))/(a^2+b^2)^2-b*arctan(sinh(x))/(2*a^2+2*b^2)+b^4*ln(a+ b*csch(x))/a/(a^2+b^2)^2+ln(sinh(x))/a-a*(a^2+2*b^2)*ln(tanh(x))/(a^2+b^2) ^2-(a-b*csch(x))*tanh(x)^2/(2*a^2+2*b^2)
Result contains complex when optimal does not.
Time = 0.13 (sec) , antiderivative size = 191, normalized size of antiderivative = 1.69 \[ \int \frac {\tanh ^3(x)}{a+b \text {csch}(x)} \, dx=\frac {a b \left (a^2+b^2\right ) \arctan (\sinh (x))+a^4 \log (i-\sinh (x))+i a^3 b \log (i-\sinh (x))+2 a^2 b^2 \log (i-\sinh (x))+2 i a b^3 \log (i-\sinh (x))+a^4 \log (i+\sinh (x))-i a^3 b \log (i+\sinh (x))+2 a^2 b^2 \log (i+\sinh (x))-2 i a b^3 \log (i+\sinh (x))+2 b^4 \log (b+a \sinh (x))+a^2 \left (a^2+b^2\right ) \text {sech}^2(x)+a b \left (a^2+b^2\right ) \text {sech}(x) \tanh (x)}{2 a \left (a^2+b^2\right )^2} \] Input:
Integrate[Tanh[x]^3/(a + b*Csch[x]),x]
Output:
(a*b*(a^2 + b^2)*ArcTan[Sinh[x]] + a^4*Log[I - Sinh[x]] + I*a^3*b*Log[I - Sinh[x]] + 2*a^2*b^2*Log[I - Sinh[x]] + (2*I)*a*b^3*Log[I - Sinh[x]] + a^4 *Log[I + Sinh[x]] - I*a^3*b*Log[I + Sinh[x]] + 2*a^2*b^2*Log[I + Sinh[x]] - (2*I)*a*b^3*Log[I + Sinh[x]] + 2*b^4*Log[b + a*Sinh[x]] + a^2*(a^2 + b^2 )*Sech[x]^2 + a*b*(a^2 + b^2)*Sech[x]*Tanh[x])/(2*a*(a^2 + b^2)^2)
Time = 0.48 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.34, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {3042, 26, 4373, 615, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {\tanh ^3(x)}{a+b \text {csch}(x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {i}{\cot (i x)^3 (a+i b \csc (i x))}dx\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle i \int \frac {1}{\cot (i x)^3 (a+i b \csc (i x))}dx\) |
| \(\Big \downarrow \) 4373 |
| \(\displaystyle -b^4 \int \frac {\sinh (x)}{b (a+b \text {csch}(x)) \left (\text {csch}^2(x) b^2+b^2\right )^2}d(b \text {csch}(x))\) |
| \(\Big \downarrow \) 615 |
| \(\displaystyle -b^4 \int \left (\frac {-b^2-a \text {csch}(x) b}{b^2 \left (a^2+b^2\right ) \left (\text {csch}^2(x) b^2+b^2\right )^2}+\frac {\sinh (x)}{a b^5}-\frac {1}{a \left (a^2+b^2\right )^2 (a+b \text {csch}(x))}+\frac {-b^4-a \left (a^2+2 b^2\right ) \text {csch}(x) b}{b^4 \left (a^2+b^2\right )^2 \left (\text {csch}^2(x) b^2+b^2\right )}\right )d(b \text {csch}(x))\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -b^4 \left (-\frac {\arctan (\text {csch}(x))}{b \left (a^2+b^2\right )^2}-\frac {\arctan (\text {csch}(x))}{2 b^3 \left (a^2+b^2\right )}+\frac {a-b \text {csch}(x)}{2 b^2 \left (a^2+b^2\right ) \left (b^2 \text {csch}^2(x)+b^2\right )}-\frac {\log (a+b \text {csch}(x))}{a \left (a^2+b^2\right )^2}-\frac {a \left (a^2+2 b^2\right ) \log \left (b^2 \text {csch}^2(x)+b^2\right )}{2 b^4 \left (a^2+b^2\right )^2}+\frac {\log (b \text {csch}(x))}{a b^4}\right )\) |
Input:
Int[Tanh[x]^3/(a + b*Csch[x]),x]
Output:
-(b^4*(-(ArcTan[Csch[x]]/(b*(a^2 + b^2)^2)) - ArcTan[Csch[x]]/(2*b^3*(a^2 + b^2)) + (a - b*Csch[x])/(2*b^2*(a^2 + b^2)*(b^2 + b^2*Csch[x]^2)) + Log[ b*Csch[x]]/(a*b^4) - Log[a + b*Csch[x]]/(a*(a^2 + b^2)^2) - (a*(a^2 + 2*b^ 2)*Log[b^2 + b^2*Csch[x]^2])/(2*b^4*(a^2 + b^2)^2)))
Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a]) I nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
Int[((e_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Int[ExpandIntegrand[(e*x)^m*(c + d*x)^n*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && ILtQ[p, 0]
Int[cot[(c_.) + (d_.)*(x_)]^(m_.)*(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_))^(n _), x_Symbol] :> Simp[-(-1)^((m - 1)/2)/(d*b^(m - 1)) Subst[Int[(b^2 - x^ 2)^((m - 1)/2)*((a + x)^n/x), x], x, b*Csc[c + d*x]], x] /; FreeQ[{a, b, c, d, n}, x] && IntegerQ[(m - 1)/2] && NeQ[a^2 - b^2, 0]
Time = 0.44 (sec) , antiderivative size = 182, normalized size of antiderivative = 1.61
| method | result | size |
| default | \(-\frac {\ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{a}+\frac {\frac {2 \left (\left (-\frac {1}{2} a^{2} b -\frac {1}{2} b^{3}\right ) \tanh \left (\frac {x}{2}\right )^{3}+\left (-a^{3}-a \,b^{2}\right ) \tanh \left (\frac {x}{2}\right )^{2}+\left (\frac {1}{2} a^{2} b +\frac {1}{2} b^{3}\right ) \tanh \left (\frac {x}{2}\right )\right )}{\left (\tanh \left (\frac {x}{2}\right )^{2}+1\right )^{2}}+\frac {\left (2 a^{3}+4 a \,b^{2}\right ) \ln \left (\tanh \left (\frac {x}{2}\right )^{2}+1\right )}{2}+\left (-a^{2} b -3 b^{3}\right ) \arctan \left (\tanh \left (\frac {x}{2}\right )\right )}{\left (a^{2}+b^{2}\right )^{2}}+\frac {b^{4} \ln \left (-b \tanh \left (\frac {x}{2}\right )^{2}+2 a \tanh \left (\frac {x}{2}\right )+b \right )}{a \left (a^{2}+b^{2}\right )^{2}}-\frac {\ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{a}\) | \(182\) |
| risch | \(\frac {x}{a}-\frac {2 x \,a^{3}}{a^{4}+2 a^{2} b^{2}+b^{4}}-\frac {4 x a \,b^{2}}{a^{4}+2 a^{2} b^{2}+b^{4}}-\frac {2 x \,b^{4}}{a \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}+\frac {{\mathrm e}^{x} \left ({\mathrm e}^{2 x} b +2 \,{\mathrm e}^{x} a -b \right )}{\left ({\mathrm e}^{2 x}+1\right )^{2} \left (a^{2}+b^{2}\right )}+\frac {i \ln \left ({\mathrm e}^{x}-i\right ) a^{2} b}{2 a^{4}+4 a^{2} b^{2}+2 b^{4}}+\frac {3 i \ln \left ({\mathrm e}^{x}-i\right ) b^{3}}{2 \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}+\frac {\ln \left ({\mathrm e}^{x}-i\right ) a^{3}}{a^{4}+2 a^{2} b^{2}+b^{4}}+\frac {2 \ln \left ({\mathrm e}^{x}-i\right ) a \,b^{2}}{a^{4}+2 a^{2} b^{2}+b^{4}}-\frac {i \ln \left ({\mathrm e}^{x}+i\right ) a^{2} b}{2 \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}-\frac {3 i \ln \left ({\mathrm e}^{x}+i\right ) b^{3}}{2 \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}+\frac {\ln \left ({\mathrm e}^{x}+i\right ) a^{3}}{a^{4}+2 a^{2} b^{2}+b^{4}}+\frac {2 \ln \left ({\mathrm e}^{x}+i\right ) a \,b^{2}}{a^{4}+2 a^{2} b^{2}+b^{4}}+\frac {b^{4} \ln \left ({\mathrm e}^{2 x}+\frac {2 b \,{\mathrm e}^{x}}{a}-1\right )}{a \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}\) | \(384\) |
Input:
int(tanh(x)^3/(a+b*csch(x)),x,method=_RETURNVERBOSE)
Output:
-1/a*ln(tanh(1/2*x)-1)+2/(a^2+b^2)^2*(((-1/2*a^2*b-1/2*b^3)*tanh(1/2*x)^3+ (-a^3-a*b^2)*tanh(1/2*x)^2+(1/2*a^2*b+1/2*b^3)*tanh(1/2*x))/(tanh(1/2*x)^2 +1)^2+1/4*(2*a^3+4*a*b^2)*ln(tanh(1/2*x)^2+1)+1/2*(-a^2*b-3*b^3)*arctan(ta nh(1/2*x)))+b^4/a/(a^2+b^2)^2*ln(-b*tanh(1/2*x)^2+2*a*tanh(1/2*x)+b)-1/a*l n(tanh(1/2*x)+1)
Leaf count of result is larger than twice the leaf count of optimal. 965 vs. \(2 (110) = 220\).
Time = 0.13 (sec) , antiderivative size = 965, normalized size of antiderivative = 8.54 \[ \int \frac {\tanh ^3(x)}{a+b \text {csch}(x)} \, dx=\text {Too large to display} \] Input:
integrate(tanh(x)^3/(a+b*csch(x)),x, algorithm="fricas")
Output:
-((a^4 + 2*a^2*b^2 + b^4)*x*cosh(x)^4 + (a^4 + 2*a^2*b^2 + b^4)*x*sinh(x)^ 4 - (a^3*b + a*b^3)*cosh(x)^3 - (a^3*b + a*b^3 - 4*(a^4 + 2*a^2*b^2 + b^4) *x*cosh(x))*sinh(x)^3 - 2*(a^4 + a^2*b^2 - (a^4 + 2*a^2*b^2 + b^4)*x)*cosh (x)^2 - (2*a^4 + 2*a^2*b^2 - 6*(a^4 + 2*a^2*b^2 + b^4)*x*cosh(x)^2 - 2*(a^ 4 + 2*a^2*b^2 + b^4)*x + 3*(a^3*b + a*b^3)*cosh(x))*sinh(x)^2 + (a^4 + 2*a ^2*b^2 + b^4)*x + ((a^3*b + 3*a*b^3)*cosh(x)^4 + 4*(a^3*b + 3*a*b^3)*cosh( x)*sinh(x)^3 + (a^3*b + 3*a*b^3)*sinh(x)^4 + a^3*b + 3*a*b^3 + 2*(a^3*b + 3*a*b^3)*cosh(x)^2 + 2*(a^3*b + 3*a*b^3 + 3*(a^3*b + 3*a*b^3)*cosh(x)^2)*s inh(x)^2 + 4*((a^3*b + 3*a*b^3)*cosh(x)^3 + (a^3*b + 3*a*b^3)*cosh(x))*sin h(x))*arctan(cosh(x) + sinh(x)) + (a^3*b + a*b^3)*cosh(x) - (b^4*cosh(x)^4 + 4*b^4*cosh(x)*sinh(x)^3 + b^4*sinh(x)^4 + 2*b^4*cosh(x)^2 + b^4 + 2*(3* b^4*cosh(x)^2 + b^4)*sinh(x)^2 + 4*(b^4*cosh(x)^3 + b^4*cosh(x))*sinh(x))* log(2*(a*sinh(x) + b)/(cosh(x) - sinh(x))) - ((a^4 + 2*a^2*b^2)*cosh(x)^4 + 4*(a^4 + 2*a^2*b^2)*cosh(x)*sinh(x)^3 + (a^4 + 2*a^2*b^2)*sinh(x)^4 + a^ 4 + 2*a^2*b^2 + 2*(a^4 + 2*a^2*b^2)*cosh(x)^2 + 2*(a^4 + 2*a^2*b^2 + 3*(a^ 4 + 2*a^2*b^2)*cosh(x)^2)*sinh(x)^2 + 4*((a^4 + 2*a^2*b^2)*cosh(x)^3 + (a^ 4 + 2*a^2*b^2)*cosh(x))*sinh(x))*log(2*cosh(x)/(cosh(x) - sinh(x))) + (4*( a^4 + 2*a^2*b^2 + b^4)*x*cosh(x)^3 + a^3*b + a*b^3 - 3*(a^3*b + a*b^3)*cos h(x)^2 - 4*(a^4 + a^2*b^2 - (a^4 + 2*a^2*b^2 + b^4)*x)*cosh(x))*sinh(x))/( a^5 + 2*a^3*b^2 + a*b^4 + (a^5 + 2*a^3*b^2 + a*b^4)*cosh(x)^4 + 4*(a^5 ...
\[ \int \frac {\tanh ^3(x)}{a+b \text {csch}(x)} \, dx=\int \frac {\tanh ^{3}{\left (x \right )}}{a + b \operatorname {csch}{\left (x \right )}}\, dx \] Input:
integrate(tanh(x)**3/(a+b*csch(x)),x)
Output:
Integral(tanh(x)**3/(a + b*csch(x)), x)
Time = 0.12 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.52 \[ \int \frac {\tanh ^3(x)}{a+b \text {csch}(x)} \, dx=\frac {b^{4} \log \left (-2 \, b e^{\left (-x\right )} + a e^{\left (-2 \, x\right )} - a\right )}{a^{5} + 2 \, a^{3} b^{2} + a b^{4}} + \frac {{\left (a^{2} b + 3 \, b^{3}\right )} \arctan \left (e^{\left (-x\right )}\right )}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} + \frac {{\left (a^{3} + 2 \, a b^{2}\right )} \log \left (e^{\left (-2 \, x\right )} + 1\right )}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} + \frac {b e^{\left (-x\right )} + 2 \, a e^{\left (-2 \, x\right )} - b e^{\left (-3 \, x\right )}}{a^{2} + b^{2} + 2 \, {\left (a^{2} + b^{2}\right )} e^{\left (-2 \, x\right )} + {\left (a^{2} + b^{2}\right )} e^{\left (-4 \, x\right )}} + \frac {x}{a} \] Input:
integrate(tanh(x)^3/(a+b*csch(x)),x, algorithm="maxima")
Output:
b^4*log(-2*b*e^(-x) + a*e^(-2*x) - a)/(a^5 + 2*a^3*b^2 + a*b^4) + (a^2*b + 3*b^3)*arctan(e^(-x))/(a^4 + 2*a^2*b^2 + b^4) + (a^3 + 2*a*b^2)*log(e^(-2 *x) + 1)/(a^4 + 2*a^2*b^2 + b^4) + (b*e^(-x) + 2*a*e^(-2*x) - b*e^(-3*x))/ (a^2 + b^2 + 2*(a^2 + b^2)*e^(-2*x) + (a^2 + b^2)*e^(-4*x)) + x/a
Leaf count of result is larger than twice the leaf count of optimal. 234 vs. \(2 (110) = 220\).
Time = 0.12 (sec) , antiderivative size = 234, normalized size of antiderivative = 2.07 \[ \int \frac {\tanh ^3(x)}{a+b \text {csch}(x)} \, dx=\frac {b^{4} \log \left ({\left | -a {\left (e^{\left (-x\right )} - e^{x}\right )} + 2 \, b \right |}\right )}{a^{5} + 2 \, a^{3} b^{2} + a b^{4}} - \frac {{\left (\pi + 2 \, \arctan \left (\frac {1}{2} \, {\left (e^{\left (2 \, x\right )} - 1\right )} e^{\left (-x\right )}\right )\right )} {\left (a^{2} b + 3 \, b^{3}\right )}}{4 \, {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )}} + \frac {{\left (a^{3} + 2 \, a b^{2}\right )} \log \left ({\left (e^{\left (-x\right )} - e^{x}\right )}^{2} + 4\right )}{2 \, {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )}} - \frac {a^{3} {\left (e^{\left (-x\right )} - e^{x}\right )}^{2} + 2 \, a b^{2} {\left (e^{\left (-x\right )} - e^{x}\right )}^{2} + 2 \, a^{2} b {\left (e^{\left (-x\right )} - e^{x}\right )} + 2 \, b^{3} {\left (e^{\left (-x\right )} - e^{x}\right )} + 4 \, a b^{2}}{2 \, {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} {\left ({\left (e^{\left (-x\right )} - e^{x}\right )}^{2} + 4\right )}} \] Input:
integrate(tanh(x)^3/(a+b*csch(x)),x, algorithm="giac")
Output:
b^4*log(abs(-a*(e^(-x) - e^x) + 2*b))/(a^5 + 2*a^3*b^2 + a*b^4) - 1/4*(pi + 2*arctan(1/2*(e^(2*x) - 1)*e^(-x)))*(a^2*b + 3*b^3)/(a^4 + 2*a^2*b^2 + b ^4) + 1/2*(a^3 + 2*a*b^2)*log((e^(-x) - e^x)^2 + 4)/(a^4 + 2*a^2*b^2 + b^4 ) - 1/2*(a^3*(e^(-x) - e^x)^2 + 2*a*b^2*(e^(-x) - e^x)^2 + 2*a^2*b*(e^(-x) - e^x) + 2*b^3*(e^(-x) - e^x) + 4*a*b^2)/((a^4 + 2*a^2*b^2 + b^4)*((e^(-x ) - e^x)^2 + 4))
Time = 4.94 (sec) , antiderivative size = 335, normalized size of antiderivative = 2.96 \[ \int \frac {\tanh ^3(x)}{a+b \text {csch}(x)} \, dx=\frac {\frac {{\mathrm {e}}^x\,\left (a^2\,b+b^3\right )}{{\left (a^2+b^2\right )}^2}+\frac {2\,\left (a^4+a^2\,b^2\right )}{a\,{\left (a^2+b^2\right )}^2}}{{\mathrm {e}}^{2\,x}+1}-\frac {\frac {2\,a}{a^2+b^2}+\frac {2\,b\,{\mathrm {e}}^x}{a^2+b^2}}{2\,{\mathrm {e}}^{2\,x}+{\mathrm {e}}^{4\,x}+1}-\frac {x}{a}+\frac {b^4\,\ln \left (4\,a^9\,{\mathrm {e}}^{2\,x}-4\,a\,b^8-4\,a^9+7\,a^3\,b^6-14\,a^5\,b^4-17\,a^7\,b^2+8\,b^9\,{\mathrm {e}}^x-7\,a^3\,b^6\,{\mathrm {e}}^{2\,x}+14\,a^5\,b^4\,{\mathrm {e}}^{2\,x}+17\,a^7\,b^2\,{\mathrm {e}}^{2\,x}+8\,a^8\,b\,{\mathrm {e}}^x+4\,a\,b^8\,{\mathrm {e}}^{2\,x}-14\,a^2\,b^7\,{\mathrm {e}}^x+28\,a^4\,b^5\,{\mathrm {e}}^x+34\,a^6\,b^3\,{\mathrm {e}}^x\right )}{a^5+2\,a^3\,b^2+a\,b^4}+\frac {\ln \left (1+{\mathrm {e}}^x\,1{}\mathrm {i}\right )\,\left (3\,b+a\,2{}\mathrm {i}\right )}{2\,\left (a^2\,1{}\mathrm {i}+2\,a\,b-b^2\,1{}\mathrm {i}\right )}+\frac {\ln \left ({\mathrm {e}}^x+1{}\mathrm {i}\right )\,\left (2\,a+b\,3{}\mathrm {i}\right )}{2\,\left (a^2+a\,b\,2{}\mathrm {i}-b^2\right )} \] Input:
int(tanh(x)^3/(a + b/sinh(x)),x)
Output:
((exp(x)*(a^2*b + b^3))/(a^2 + b^2)^2 + (2*(a^4 + a^2*b^2))/(a*(a^2 + b^2) ^2))/(exp(2*x) + 1) - ((2*a)/(a^2 + b^2) + (2*b*exp(x))/(a^2 + b^2))/(2*ex p(2*x) + exp(4*x) + 1) - x/a + (b^4*log(4*a^9*exp(2*x) - 4*a*b^8 - 4*a^9 + 7*a^3*b^6 - 14*a^5*b^4 - 17*a^7*b^2 + 8*b^9*exp(x) - 7*a^3*b^6*exp(2*x) + 14*a^5*b^4*exp(2*x) + 17*a^7*b^2*exp(2*x) + 8*a^8*b*exp(x) + 4*a*b^8*exp( 2*x) - 14*a^2*b^7*exp(x) + 28*a^4*b^5*exp(x) + 34*a^6*b^3*exp(x)))/(a*b^4 + a^5 + 2*a^3*b^2) + (log(exp(x)*1i + 1)*(a*2i + 3*b))/(2*(2*a*b + a^2*1i - b^2*1i)) + (log(exp(x) + 1i)*(2*a + b*3i))/(2*(a*b*2i + a^2 - b^2))
Time = 0.22 (sec) , antiderivative size = 515, normalized size of antiderivative = 4.56 \[ \int \frac {\tanh ^3(x)}{a+b \text {csch}(x)} \, dx=\frac {-2 e^{4 x} a^{2} b^{2} x -e^{4 x} a^{4}+e^{3 x} a^{3} b +e^{3 x} a \,b^{3}-e^{x} a^{3} b -e^{4 x} a^{4} x -e^{4 x} b^{4} x -e^{4 x} \mathit {atan} \left (e^{x}\right ) a^{3} b -\mathit {atan} \left (e^{x}\right ) a^{3} b +e^{4 x} \mathrm {log}\left (e^{2 x}+1\right ) a^{4}+e^{4 x} \mathrm {log}\left (e^{2 x} a +2 e^{x} b -a \right ) b^{4}-3 e^{4 x} \mathit {atan} \left (e^{x}\right ) a \,b^{3}-2 e^{2 x} \mathit {atan} \left (e^{x}\right ) a^{3} b -6 e^{2 x} \mathit {atan} \left (e^{x}\right ) a \,b^{3}+2 e^{4 x} \mathrm {log}\left (e^{2 x}+1\right ) a^{2} b^{2}+4 e^{2 x} \mathrm {log}\left (e^{2 x}+1\right ) a^{2} b^{2}-4 e^{2 x} a^{2} b^{2} x -2 e^{2 x} a^{4} x -2 e^{2 x} b^{4} x -e^{x} a \,b^{3}-3 \mathit {atan} \left (e^{x}\right ) a \,b^{3}+2 e^{2 x} \mathrm {log}\left (e^{2 x}+1\right ) a^{4}+2 e^{2 x} \mathrm {log}\left (e^{2 x} a +2 e^{x} b -a \right ) b^{4}+2 \,\mathrm {log}\left (e^{2 x}+1\right ) a^{2} b^{2}-2 a^{2} b^{2} x -a^{2} b^{2}-a^{4}+\mathrm {log}\left (e^{2 x}+1\right ) a^{4}+\mathrm {log}\left (e^{2 x} a +2 e^{x} b -a \right ) b^{4}-a^{4} x -b^{4} x -e^{4 x} a^{2} b^{2}}{a \left (e^{4 x} a^{4}+2 e^{4 x} a^{2} b^{2}+e^{4 x} b^{4}+2 e^{2 x} a^{4}+4 e^{2 x} a^{2} b^{2}+2 e^{2 x} b^{4}+a^{4}+2 a^{2} b^{2}+b^{4}\right )} \] Input:
int(tanh(x)^3/(a+b*csch(x)),x)
Output:
( - e**(4*x)*atan(e**x)*a**3*b - 3*e**(4*x)*atan(e**x)*a*b**3 - 2*e**(2*x) *atan(e**x)*a**3*b - 6*e**(2*x)*atan(e**x)*a*b**3 - atan(e**x)*a**3*b - 3* atan(e**x)*a*b**3 + e**(4*x)*log(e**(2*x) + 1)*a**4 + 2*e**(4*x)*log(e**(2 *x) + 1)*a**2*b**2 + e**(4*x)*log(e**(2*x)*a + 2*e**x*b - a)*b**4 - e**(4* x)*a**4*x - e**(4*x)*a**4 - 2*e**(4*x)*a**2*b**2*x - e**(4*x)*a**2*b**2 - e**(4*x)*b**4*x + e**(3*x)*a**3*b + e**(3*x)*a*b**3 + 2*e**(2*x)*log(e**(2 *x) + 1)*a**4 + 4*e**(2*x)*log(e**(2*x) + 1)*a**2*b**2 + 2*e**(2*x)*log(e* *(2*x)*a + 2*e**x*b - a)*b**4 - 2*e**(2*x)*a**4*x - 4*e**(2*x)*a**2*b**2*x - 2*e**(2*x)*b**4*x - e**x*a**3*b - e**x*a*b**3 + log(e**(2*x) + 1)*a**4 + 2*log(e**(2*x) + 1)*a**2*b**2 + log(e**(2*x)*a + 2*e**x*b - a)*b**4 - a* *4*x - a**4 - 2*a**2*b**2*x - a**2*b**2 - b**4*x)/(a*(e**(4*x)*a**4 + 2*e* *(4*x)*a**2*b**2 + e**(4*x)*b**4 + 2*e**(2*x)*a**4 + 4*e**(2*x)*a**2*b**2 + 2*e**(2*x)*b**4 + a**4 + 2*a**2*b**2 + b**4))