\(\int \frac {\tanh ^3(x)}{a+b \text {csch}(x)} \, dx\) [115]

Optimal result

Integrand size = 13, antiderivative size = 113 \[ \int \frac {\tanh ^3(x)}{a+b \text {csch}(x)} \, dx=-\frac {b^3 \arctan (\sinh (x))}{\left (a^2+b^2\right )^2}-\frac {b \arctan (\sinh (x))}{2 \left (a^2+b^2\right )}+\frac {b^4 \log (a+b \text {csch}(x))}{a \left (a^2+b^2\right )^2}+\frac {\log (\sinh (x))}{a}-\frac {a \left (a^2+2 b^2\right ) \log (\tanh (x))}{\left (a^2+b^2\right )^2}-\frac {(a-b \text {csch}(x)) \tanh ^2(x)}{2 \left (a^2+b^2\right )} \]

[Out]

-b^3*arctan(sinh(x))/(a^2+b^2)^2-1/2*b*arctan(sinh(x))/(a^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)^2-1/2*(a-b*csch(x))*tanh(x)^2/(a^2+b^2)

Rubi [A] (verified)

Time = 0.12 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {3970, 908, 653, 209, 649, 266} \[ \int \frac {\tanh ^3(x)}{a+b \text {csch}(x)} \, dx=-\frac {b \arctan (\sinh (x))}{2 \left (a^2+b^2\right )}-\frac {b^3 \arctan (\sinh (x))}{\left (a^2+b^2\right )^2}-\frac {a \left (a^2+2 b^2\right ) \log (\tanh (x))}{\left (a^2+b^2\right )^2}-\frac {\tanh ^2(x) (a-b \text {csch}(x))}{2 \left (a^2+b^2\right )}+\frac {b^4 \log (a+b \text {csch}(x))}{a \left (a^2+b^2\right )^2}+\frac {\log (\sinh (x))}{a} \]

[In]

Int[Tanh[x]^3/(a + b*Csch[x]),x]

[Out]

-((b^3*ArcTan[Sinh[x]])/(a^2 + b^2)^2) - (b*ArcTan[Sinh[x]])/(2*(a^2 + b^2)) + (b^4*Log[a + b*Csch[x]])/(a*(a^
2 + b^2)^2) + Log[Sinh[x]]/a - (a*(a^2 + 2*b^2)*Log[Tanh[x]])/(a^2 + b^2)^2 - ((a - b*Csch[x])*Tanh[x]^2)/(2*(
a^2 + b^2))

Rule 209

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*ArcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 266

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

Rule 649

Int[((d_) + (e_.)*(x_))/((a_) + (c_.)*(x_)^2), x_Symbol] :> Dist[d, Int[1/(a + c*x^2), x], x] + Dist[e, Int[x/
(a + c*x^2), x], x] /; FreeQ[{a, c, d, e}, x] &&  !NiceSqrtQ[(-a)*c]

Rule 653

Int[((d_) + (e_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((a*e - c*d*x)/(2*a*c*(p + 1)))*(a + c*x
^2)^(p + 1), x] + Dist[d*((2*p + 3)/(2*a*(p + 1))), Int[(a + c*x^2)^(p + 1), x], x] /; FreeQ[{a, c, d, e}, x]
&& LtQ[p, -1] && NeQ[p, -3/2]

Rule 908

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIn
tegrand[(d + e*x)^m*(f + g*x)^n*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] &&
NeQ[c*d^2 + a*e^2, 0] && IntegerQ[p] && ((EqQ[p, 1] && IntegersQ[m, n]) || (ILtQ[m, 0] && ILtQ[n, 0]))

Rule 3970

Int[cot[(c_.) + (d_.)*(x_)]^(m_.)*(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_))^(n_), x_Symbol] :> Dist[-(-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]

Rubi steps \begin{align*} \text {integral}& = -\left (b^4 \text {Subst}\left (\int \frac {1}{x (a+x) \left (-b^2-x^2\right )^2} \, dx,x,b \text {csch}(x)\right )\right ) \\ & = -\left (b^4 \text {Subst}\left (\int \left (\frac {1}{a b^4 x}-\frac {1}{a \left (a^2+b^2\right )^2 (a+x)}+\frac {-b^2-a x}{b^2 \left (a^2+b^2\right ) \left (b^2+x^2\right )^2}+\frac {-b^4-a \left (a^2+2 b^2\right ) x}{b^4 \left (a^2+b^2\right )^2 \left (b^2+x^2\right )}\right ) \, dx,x,b \text {csch}(x)\right )\right ) \\ & = \frac {b^4 \log (a+b \text {csch}(x))}{a \left (a^2+b^2\right )^2}+\frac {\log (\sinh (x))}{a}-\frac {\text {Subst}\left (\int \frac {-b^4-a \left (a^2+2 b^2\right ) x}{b^2+x^2} \, dx,x,b \text {csch}(x)\right )}{\left (a^2+b^2\right )^2}-\frac {b^2 \text {Subst}\left (\int \frac {-b^2-a x}{\left (b^2+x^2\right )^2} \, dx,x,b \text {csch}(x)\right )}{a^2+b^2} \\ & = \frac {b^4 \log (a+b \text {csch}(x))}{a \left (a^2+b^2\right )^2}+\frac {\log (\sinh (x))}{a}-\frac {(a-b \text {csch}(x)) \tanh ^2(x)}{2 \left (a^2+b^2\right )}+\frac {b^4 \text {Subst}\left (\int \frac {1}{b^2+x^2} \, dx,x,b \text {csch}(x)\right )}{\left (a^2+b^2\right )^2}+\frac {b^2 \text {Subst}\left (\int \frac {1}{b^2+x^2} \, dx,x,b \text {csch}(x)\right )}{2 \left (a^2+b^2\right )}+\frac {\left (a \left (a^2+2 b^2\right )\right ) \text {Subst}\left (\int \frac {x}{b^2+x^2} \, dx,x,b \text {csch}(x)\right )}{\left (a^2+b^2\right )^2} \\ & = -\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 )} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.13 (sec) , antiderivative size = 191, normalized size of antiderivative = 1.69 \[ \int \frac {\tanh ^3(x)}{a+b \text {csch}(x)} \, dx=\frac {a b \left (a^2+b^2\right ) \arctan (\sinh (x))+a^4 \log (i-\sinh (x))+i a^3 b \log (i-\sinh (x))+2 a^2 b^2 \log (i-\sinh (x))+2 i a b^3 \log (i-\sinh (x))+a^4 \log (i+\sinh (x))-i a^3 b \log (i+\sinh (x))+2 a^2 b^2 \log (i+\sinh (x))-2 i a b^3 \log (i+\sinh (x))+2 b^4 \log (b+a \sinh (x))+a^2 \left (a^2+b^2\right ) \text {sech}^2(x)+a b \left (a^2+b^2\right ) \text {sech}(x) \tanh (x)}{2 a \left (a^2+b^2\right )^2} \]

[In]

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

[Out]

(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)

Maple [A] (verified)

Time = 0.45 (sec) , antiderivative size = 182, normalized size of antiderivative = 1.61

method	result	size
default	\(-\frac {\ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{a}-\frac {\ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{a}+\frac {b^{4} \ln \left (-\tanh \left (\frac {x}{2}\right )^{2} b +2 a \tanh \left (\frac {x}{2}\right )+b \right )}{a \left (a^{2}+b^{2}\right )^{2}}+\frac {\frac {2 \left (\left (-\frac {1}{2} a^{2} b -\frac {1}{2} b^{3}\right ) \tanh \left (\frac {x}{2}\right )^{3}+\left (-a^{3}-a \,b^{2}\right ) \tanh \left (\frac {x}{2}\right )^{2}+\left (\frac {1}{2} a^{2} b +\frac {1}{2} b^{3}\right ) \tanh \left (\frac {x}{2}\right )\right )}{\left (1+\tanh \left (\frac {x}{2}\right )^{2}\right )^{2}}+\frac {\left (2 a^{3}+4 a \,b^{2}\right ) \ln \left (1+\tanh \left (\frac {x}{2}\right )^{2}\right )}{2}+\left (-a^{2} b -3 b^{3}\right ) \arctan \left (\tanh \left (\frac {x}{2}\right )\right )}{\left (a^{2}+b^{2}\right )^{2}}\)	\(182\)
risch	\(\frac {x}{a}-\frac {2 x \,a^{3}}{a^{4}+2 a^{2} b^{2}+b^{4}}-\frac {4 x a \,b^{2}}{a^{4}+2 a^{2} b^{2}+b^{4}}-\frac {2 x \,b^{4}}{a \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}+\frac {{\mathrm e}^{x} \left (b \,{\mathrm e}^{2 x}+2 a \,{\mathrm e}^{x}-b \right )}{\left (1+{\mathrm e}^{2 x}\right )^{2} \left (a^{2}+b^{2}\right )}+\frac {i \ln \left ({\mathrm e}^{x}-i\right ) a^{2} b}{2 a^{4}+4 a^{2} b^{2}+2 b^{4}}+\frac {3 i \ln \left ({\mathrm e}^{x}-i\right ) b^{3}}{2 \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}+\frac {\ln \left ({\mathrm e}^{x}-i\right ) a^{3}}{a^{4}+2 a^{2} b^{2}+b^{4}}+\frac {2 \ln \left ({\mathrm e}^{x}-i\right ) a \,b^{2}}{a^{4}+2 a^{2} b^{2}+b^{4}}-\frac {i \ln \left ({\mathrm e}^{x}+i\right ) a^{2} b}{2 \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}-\frac {3 i \ln \left ({\mathrm e}^{x}+i\right ) b^{3}}{2 \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}+\frac {\ln \left ({\mathrm e}^{x}+i\right ) a^{3}}{a^{4}+2 a^{2} b^{2}+b^{4}}+\frac {2 \ln \left ({\mathrm e}^{x}+i\right ) a \,b^{2}}{a^{4}+2 a^{2} b^{2}+b^{4}}+\frac {b^{4} \ln \left ({\mathrm e}^{2 x}+\frac {2 b \,{\mathrm e}^{x}}{a}-1\right )}{a \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}\)	\(384\)

[In]

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

[Out]

-1/a*ln(tanh(1/2*x)+1)-1/a*ln(tanh(1/2*x)-1)+b^4/a/(a^2+b^2)^2*ln(-tanh(1/2*x)^2*b+2*a*tanh(1/2*x)+b)+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))/(1+tanh
(1/2*x)^2)^2+1/4*(2*a^3+4*a*b^2)*ln(1+tanh(1/2*x)^2)+1/2*(-a^2*b-3*b^3)*arctan(tanh(1/2*x)))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 965 vs. \(2 (110) = 220\).

Time = 0.30 (sec) , antiderivative size = 965, normalized size of antiderivative = 8.54 \[ \int \frac {\tanh ^3(x)}{a+b \text {csch}(x)} \, dx=\text {Too large to display} \]

[In]

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

[Out]

-((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)*co
sh(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)*c
osh(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)*sinh(x)^2 + 4*((a^3*b + 3*a*b^3)*cosh(x)^3 + (a^3*b + 3*a*b^3)*cosh(x
))*sinh(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))*s
inh(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))*si
nh(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)*cosh(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 + 2*a^3*b^2 + a*b^4)*cosh(x)*sinh(x)^3 + (a^5 + 2*a^3*b^2 + a*b
^4)*sinh(x)^4 + 2*(a^5 + 2*a^3*b^2 + a*b^4)*cosh(x)^2 + 2*(a^5 + 2*a^3*b^2 + a*b^4 + 3*(a^5 + 2*a^3*b^2 + a*b^
4)*cosh(x)^2)*sinh(x)^2 + 4*((a^5 + 2*a^3*b^2 + a*b^4)*cosh(x)^3 + (a^5 + 2*a^3*b^2 + a*b^4)*cosh(x))*sinh(x))

Sympy [F]

\[ \int \frac {\tanh ^3(x)}{a+b \text {csch}(x)} \, dx=\int \frac {\tanh ^{3}{\left (x \right )}}{a + b \operatorname {csch}{\left (x \right )}}\, dx \]

[In]

integrate(tanh(x)**3/(a+b*csch(x)),x)

[Out]

Integral(tanh(x)**3/(a + b*csch(x)), x)

Maxima [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.52 \[ \int \frac {\tanh ^3(x)}{a+b \text {csch}(x)} \, dx=\frac {b^{4} \log \left (-2 \, b e^{\left (-x\right )} + a e^{\left (-2 \, x\right )} - a\right )}{a^{5} + 2 \, a^{3} b^{2} + a b^{4}} + \frac {{\left (a^{2} b + 3 \, b^{3}\right )} \arctan \left (e^{\left (-x\right )}\right )}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} + \frac {{\left (a^{3} + 2 \, a b^{2}\right )} \log \left (e^{\left (-2 \, x\right )} + 1\right )}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} + \frac {b e^{\left (-x\right )} + 2 \, a e^{\left (-2 \, x\right )} - b e^{\left (-3 \, x\right )}}{a^{2} + b^{2} + 2 \, {\left (a^{2} + b^{2}\right )} e^{\left (-2 \, x\right )} + {\left (a^{2} + b^{2}\right )} e^{\left (-4 \, x\right )}} + \frac {x}{a} \]

[In]

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

[Out]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 234 vs. \(2 (110) = 220\).

Time = 0.27 (sec) , antiderivative size = 234, normalized size of antiderivative = 2.07 \[ \int \frac {\tanh ^3(x)}{a+b \text {csch}(x)} \, dx=\frac {b^{4} \log \left ({\left | -a {\left (e^{\left (-x\right )} - e^{x}\right )} + 2 \, b \right |}\right )}{a^{5} + 2 \, a^{3} b^{2} + a b^{4}} - \frac {{\left (\pi + 2 \, \arctan \left (\frac {1}{2} \, {\left (e^{\left (2 \, x\right )} - 1\right )} e^{\left (-x\right )}\right )\right )} {\left (a^{2} b + 3 \, b^{3}\right )}}{4 \, {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )}} + \frac {{\left (a^{3} + 2 \, a b^{2}\right )} \log \left ({\left (e^{\left (-x\right )} - e^{x}\right )}^{2} + 4\right )}{2 \, {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )}} - \frac {a^{3} {\left (e^{\left (-x\right )} - e^{x}\right )}^{2} + 2 \, a b^{2} {\left (e^{\left (-x\right )} - e^{x}\right )}^{2} + 2 \, a^{2} b {\left (e^{\left (-x\right )} - e^{x}\right )} + 2 \, b^{3} {\left (e^{\left (-x\right )} - e^{x}\right )} + 4 \, a b^{2}}{2 \, {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} {\left ({\left (e^{\left (-x\right )} - e^{x}\right )}^{2} + 4\right )}} \]

[In]

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

[Out]

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))

Mupad [B] (verification not implemented)

Time = 4.35 (sec) , antiderivative size = 335, normalized size of antiderivative = 2.96 \[ \int \frac {\tanh ^3(x)}{a+b \text {csch}(x)} \, dx=\frac {\frac {{\mathrm {e}}^x\,\left (a^2\,b+b^3\right )}{{\left (a^2+b^2\right )}^2}+\frac {2\,\left (a^4+a^2\,b^2\right )}{a\,{\left (a^2+b^2\right )}^2}}{{\mathrm {e}}^{2\,x}+1}-\frac {\frac {2\,a}{a^2+b^2}+\frac {2\,b\,{\mathrm {e}}^x}{a^2+b^2}}{2\,{\mathrm {e}}^{2\,x}+{\mathrm {e}}^{4\,x}+1}-\frac {x}{a}+\frac {b^4\,\ln \left (4\,a^9\,{\mathrm {e}}^{2\,x}-4\,a\,b^8-4\,a^9+7\,a^3\,b^6-14\,a^5\,b^4-17\,a^7\,b^2+8\,b^9\,{\mathrm {e}}^x-7\,a^3\,b^6\,{\mathrm {e}}^{2\,x}+14\,a^5\,b^4\,{\mathrm {e}}^{2\,x}+17\,a^7\,b^2\,{\mathrm {e}}^{2\,x}+8\,a^8\,b\,{\mathrm {e}}^x+4\,a\,b^8\,{\mathrm {e}}^{2\,x}-14\,a^2\,b^7\,{\mathrm {e}}^x+28\,a^4\,b^5\,{\mathrm {e}}^x+34\,a^6\,b^3\,{\mathrm {e}}^x\right )}{a^5+2\,a^3\,b^2+a\,b^4}+\frac {\ln \left (1+{\mathrm {e}}^x\,1{}\mathrm {i}\right )\,\left (3\,b+a\,2{}\mathrm {i}\right )}{2\,\left (a^2\,1{}\mathrm {i}+2\,a\,b-b^2\,1{}\mathrm {i}\right )}+\frac {\ln \left ({\mathrm {e}}^x+1{}\mathrm {i}\right )\,\left (2\,a+b\,3{}\mathrm {i}\right )}{2\,\left (a^2+a\,b\,2{}\mathrm {i}-b^2\right )} \]

[In]

int(tanh(x)^3/(a + b/sinh(x)),x)

[Out]

((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*exp(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))