\(\int \frac {\text {sech}^3(x)}{a+b \text {csch}(x)} \, dx\) [99]

Optimal result

Integrand size = 13, antiderivative size = 95 \[ \int \frac {\text {sech}^3(x)}{a+b \text {csch}(x)} \, dx=-\frac {i a \log (i-\sinh (x))}{4 (a-i b)^2}+\frac {i a \log (i+\sinh (x))}{4 (a+i b)^2}-\frac {a^2 b \log (b+a \sinh (x))}{\left (a^2+b^2\right )^2}-\frac {\text {sech}^2(x) (b-a \sinh (x))}{2 \left (a^2+b^2\right )} \]

[Out]

-1/4*I*a*ln(I-sinh(x))/(a-I*b)^2+1/4*I*a*ln(I+sinh(x))/(a+I*b)^2-a^2*b*ln(b+a*sinh(x))/(a^2+b^2)^2-1/2*sech(x)
^2*(b-a*sinh(x))/(a^2+b^2)

Rubi [A] (verified)

Time = 0.16 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {3957, 2916, 12, 837, 815} \[ \int \frac {\text {sech}^3(x)}{a+b \text {csch}(x)} \, dx=-\frac {a^2 b \log (a \sinh (x)+b)}{\left (a^2+b^2\right )^2}-\frac {\text {sech}^2(x) (b-a \sinh (x))}{2 \left (a^2+b^2\right )}-\frac {i a \log (-\sinh (x)+i)}{4 (a-i b)^2}+\frac {i a \log (\sinh (x)+i)}{4 (a+i b)^2} \]

[In]

Int[Sech[x]^3/(a + b*Csch[x]),x]

[Out]

((-1/4*I)*a*Log[I - Sinh[x]])/(a - I*b)^2 + ((I/4)*a*Log[I + Sinh[x]])/(a + I*b)^2 - (a^2*b*Log[b + a*Sinh[x]]
)/(a^2 + b^2)^2 - (Sech[x]^2*(b - a*Sinh[x]))/(2*(a^2 + b^2))

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 815

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_) + (c_.)*(x_)^2), x_Symbol] :> Int[ExpandIntegrand[(
d + e*x)^m*((f + g*x)/(a + c*x^2)), x], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[c*d^2 + a*e^2, 0] && Integer
Q[m]

Rule 837

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-(d + e*x)^(
m + 1))*(f*a*c*e - a*g*c*d + c*(c*d*f + a*e*g)*x)*((a + c*x^2)^(p + 1)/(2*a*c*(p + 1)*(c*d^2 + a*e^2))), x] +
Dist[1/(2*a*c*(p + 1)*(c*d^2 + a*e^2)), Int[(d + e*x)^m*(a + c*x^2)^(p + 1)*Simp[f*(c^2*d^2*(2*p + 3) + a*c*e^
2*(m + 2*p + 3)) - a*c*d*e*g*m + c*e*(c*d*f + a*e*g)*(m + 2*p + 4)*x, x], x], x] /; FreeQ[{a, c, d, e, f, g},
x] && NeQ[c*d^2 + a*e^2, 0] && LtQ[p, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])

Rule 2916

Int[cos[(e_.) + (f_.)*(x_)]^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)
*(x_)])^(n_.), x_Symbol] :> Dist[1/(b^p*f), Subst[Int[(a + x)^m*(c + (d/b)*x)^n*(b^2 - x^2)^((p - 1)/2), x], x
, b*Sin[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && IntegerQ[(p - 1)/2] && NeQ[a^2 - b^2, 0]

Rule 3957

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_.), x_Symbol] :> Int[(g*Co
s[e + f*x])^p*((b + a*Sin[e + f*x])^m/Sin[e + f*x]^m), x] /; FreeQ[{a, b, e, f, g, p}, x] && IntegerQ[m]

Rubi steps \begin{align*} \text {integral}& = i \int \frac {\text {sech}^2(x) \tanh (x)}{i b+i a \sinh (x)} \, dx \\ & = -\left (\left (i a^3\right ) \text {Subst}\left (\int \frac {x}{a (i b+x) \left (a^2-x^2\right )^2} \, dx,x,i a \sinh (x)\right )\right ) \\ & = -\left (\left (i a^2\right ) \text {Subst}\left (\int \frac {x}{(i b+x) \left (a^2-x^2\right )^2} \, dx,x,i a \sinh (x)\right )\right ) \\ & = -\frac {\text {sech}^2(x) (b-a \sinh (x))}{2 \left (a^2+b^2\right )}-\frac {i \text {Subst}\left (\int \frac {-i a^2 b+a^2 x}{(i b+x) \left (a^2-x^2\right )} \, dx,x,i a \sinh (x)\right )}{2 \left (a^2+b^2\right )} \\ & = -\frac {\text {sech}^2(x) (b-a \sinh (x))}{2 \left (a^2+b^2\right )}-\frac {i \text {Subst}\left (\int \left (\frac {a (a-i b)}{2 (a+i b) (a-x)}-\frac {2 a^2 b}{\left (a^2+b^2\right ) (b-i x)}+\frac {a (a+i b)}{2 (a-i b) (a+x)}\right ) \, dx,x,i a \sinh (x)\right )}{2 \left (a^2+b^2\right )} \\ & = -\frac {i a \log (i-\sinh (x))}{4 (a-i b)^2}+\frac {i a \log (i+\sinh (x))}{4 (a+i b)^2}-\frac {a^2 b \log (b+a \sinh (x))}{\left (a^2+b^2\right )^2}-\frac {\text {sech}^2(x) (b-a \sinh (x))}{2 \left (a^2+b^2\right )} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.25 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.34 \[ \int \frac {\text {sech}^3(x)}{a+b \text {csch}(x)} \, dx=-\frac {i \text {csch}(x) (b+a \sinh (x)) \left (\frac {a \log (i-\sinh (x))}{(a-i b)^2}-\frac {a \log (i+\sinh (x))}{(a+i b)^2}-\frac {4 i a^2 b \log (b+a \sinh (x))}{\left (a^2+b^2\right )^2}+\frac {1}{(i a+b) (i-\sinh (x))}+\frac {i}{(a+i b) (i+\sinh (x))}\right )}{4 (a+b \text {csch}(x))} \]

[In]

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

[Out]

((-1/4*I)*Csch[x]*(b + a*Sinh[x])*((a*Log[I - Sinh[x]])/(a - I*b)^2 - (a*Log[I + Sinh[x]])/(a + I*b)^2 - ((4*I
)*a^2*b*Log[b + a*Sinh[x]])/(a^2 + b^2)^2 + 1/((I*a + b)*(I - Sinh[x])) + I/((a + I*b)*(I + Sinh[x]))))/(a + b
*Csch[x])

Maple [A] (verified)

Time = 2.24 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.51

method	result	size
default	\(\frac {\frac {2 \left (\left (-\frac {1}{2} a^{3}-\frac {1}{2} a \,b^{2}\right ) \tanh \left (\frac {x}{2}\right )^{3}+\left (a^{2} b +b^{3}\right ) \tanh \left (\frac {x}{2}\right )^{2}+\left (\frac {1}{2} a^{3}+\frac {1}{2} a \,b^{2}\right ) \tanh \left (\frac {x}{2}\right )\right )}{\left (1+\tanh \left (\frac {x}{2}\right )^{2}\right )^{2}}+a \left (a b \ln \left (1+\tanh \left (\frac {x}{2}\right )^{2}\right )+\left (a^{2}-b^{2}\right ) \arctan \left (\tanh \left (\frac {x}{2}\right )\right )\right )}{\left (a^{2}+b^{2}\right )^{2}}-\frac {a^{2} b \ln \left (-\tanh \left (\frac {x}{2}\right )^{2} b +2 a \tanh \left (\frac {x}{2}\right )+b \right )}{\left (a^{2}+b^{2}\right )^{2}}\)	\(143\)
risch	\(\frac {{\mathrm e}^{x} \left (a \,{\mathrm e}^{2 x}-2 \,{\mathrm e}^{x} b -a \right )}{\left (1+{\mathrm e}^{2 x}\right )^{2} \left (a^{2}+b^{2}\right )}-\frac {i a^{3} \ln \left ({\mathrm e}^{x}-i\right )}{2 \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}+\frac {i a \ln \left ({\mathrm e}^{x}-i\right ) b^{2}}{2 a^{4}+4 a^{2} b^{2}+2 b^{4}}+\frac {a^{2} \ln \left ({\mathrm e}^{x}-i\right ) b}{a^{4}+2 a^{2} b^{2}+b^{4}}+\frac {i a^{3} \ln \left ({\mathrm e}^{x}+i\right )}{2 a^{4}+4 a^{2} b^{2}+2 b^{4}}-\frac {i a \ln \left ({\mathrm e}^{x}+i\right ) b^{2}}{2 \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}+\frac {a^{2} \ln \left ({\mathrm e}^{x}+i\right ) b}{a^{4}+2 a^{2} b^{2}+b^{4}}-\frac {a^{2} b \ln \left ({\mathrm e}^{2 x}+\frac {2 b \,{\mathrm e}^{x}}{a}-1\right )}{a^{4}+2 a^{2} b^{2}+b^{4}}\)	\(249\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 675 vs. \(2 (78) = 156\).

Time = 0.27 (sec) , antiderivative size = 675, normalized size of antiderivative = 7.11 \[ \int \frac {\text {sech}^3(x)}{a+b \text {csch}(x)} \, dx=\frac {{\left (a^{3} + a b^{2}\right )} \cosh \left (x\right )^{3} + {\left (a^{3} + a b^{2}\right )} \sinh \left (x\right )^{3} - 2 \, {\left (a^{2} b + b^{3}\right )} \cosh \left (x\right )^{2} - {\left (2 \, a^{2} b + 2 \, b^{3} - 3 \, {\left (a^{3} + a b^{2}\right )} \cosh \left (x\right )\right )} \sinh \left (x\right )^{2} + {\left ({\left (a^{3} - a b^{2}\right )} \cosh \left (x\right )^{4} + 4 \, {\left (a^{3} - a b^{2}\right )} \cosh \left (x\right ) \sinh \left (x\right )^{3} + {\left (a^{3} - a b^{2}\right )} \sinh \left (x\right )^{4} + a^{3} - a b^{2} + 2 \, {\left (a^{3} - a b^{2}\right )} \cosh \left (x\right )^{2} + 2 \, {\left (a^{3} - a b^{2} + 3 \, {\left (a^{3} - a b^{2}\right )} \cosh \left (x\right )^{2}\right )} \sinh \left (x\right )^{2} + 4 \, {\left ({\left (a^{3} - a b^{2}\right )} \cosh \left (x\right )^{3} + {\left (a^{3} - a b^{2}\right )} \cosh \left (x\right )\right )} \sinh \left (x\right )\right )} \arctan \left (\cosh \left (x\right ) + \sinh \left (x\right )\right ) - {\left (a^{3} + a b^{2}\right )} \cosh \left (x\right ) - {\left (a^{2} b \cosh \left (x\right )^{4} + 4 \, a^{2} b \cosh \left (x\right ) \sinh \left (x\right )^{3} + a^{2} b \sinh \left (x\right )^{4} + 2 \, a^{2} b \cosh \left (x\right )^{2} + a^{2} b + 2 \, {\left (3 \, a^{2} b \cosh \left (x\right )^{2} + a^{2} b\right )} \sinh \left (x\right )^{2} + 4 \, {\left (a^{2} b \cosh \left (x\right )^{3} + a^{2} b \cosh \left (x\right )\right )} \sinh \left (x\right )\right )} \log \left (\frac {2 \, {\left (a \sinh \left (x\right ) + b\right )}}{\cosh \left (x\right ) - \sinh \left (x\right )}\right ) + {\left (a^{2} b \cosh \left (x\right )^{4} + 4 \, a^{2} b \cosh \left (x\right ) \sinh \left (x\right )^{3} + a^{2} b \sinh \left (x\right )^{4} + 2 \, a^{2} b \cosh \left (x\right )^{2} + a^{2} b + 2 \, {\left (3 \, a^{2} b \cosh \left (x\right )^{2} + a^{2} b\right )} \sinh \left (x\right )^{2} + 4 \, {\left (a^{2} b \cosh \left (x\right )^{3} + a^{2} b \cosh \left (x\right )\right )} \sinh \left (x\right )\right )} \log \left (\frac {2 \, \cosh \left (x\right )}{\cosh \left (x\right ) - \sinh \left (x\right )}\right ) - {\left (a^{3} + a b^{2} - 3 \, {\left (a^{3} + a b^{2}\right )} \cosh \left (x\right )^{2} + 4 \, {\left (a^{2} b + b^{3}\right )} \cosh \left (x\right )\right )} \sinh \left (x\right )}{{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} \cosh \left (x\right )^{4} + 4 \, {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} \cosh \left (x\right ) \sinh \left (x\right )^{3} + {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} \sinh \left (x\right )^{4} + a^{4} + 2 \, a^{2} b^{2} + b^{4} + 2 \, {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} \cosh \left (x\right )^{2} + 2 \, {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4} + 3 \, {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} \cosh \left (x\right )^{2}\right )} \sinh \left (x\right )^{2} + 4 \, {\left ({\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} \cosh \left (x\right )^{3} + {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} \cosh \left (x\right )\right )} \sinh \left (x\right )} \]

[In]

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

[Out]

((a^3 + a*b^2)*cosh(x)^3 + (a^3 + a*b^2)*sinh(x)^3 - 2*(a^2*b + b^3)*cosh(x)^2 - (2*a^2*b + 2*b^3 - 3*(a^3 + a
*b^2)*cosh(x))*sinh(x)^2 + ((a^3 - a*b^2)*cosh(x)^4 + 4*(a^3 - a*b^2)*cosh(x)*sinh(x)^3 + (a^3 - a*b^2)*sinh(x
)^4 + a^3 - a*b^2 + 2*(a^3 - a*b^2)*cosh(x)^2 + 2*(a^3 - a*b^2 + 3*(a^3 - a*b^2)*cosh(x)^2)*sinh(x)^2 + 4*((a^
3 - a*b^2)*cosh(x)^3 + (a^3 - a*b^2)*cosh(x))*sinh(x))*arctan(cosh(x) + sinh(x)) - (a^3 + a*b^2)*cosh(x) - (a^
2*b*cosh(x)^4 + 4*a^2*b*cosh(x)*sinh(x)^3 + a^2*b*sinh(x)^4 + 2*a^2*b*cosh(x)^2 + a^2*b + 2*(3*a^2*b*cosh(x)^2
+ a^2*b)*sinh(x)^2 + 4*(a^2*b*cosh(x)^3 + a^2*b*cosh(x))*sinh(x))*log(2*(a*sinh(x) + b)/(cosh(x) - sinh(x)))
+ (a^2*b*cosh(x)^4 + 4*a^2*b*cosh(x)*sinh(x)^3 + a^2*b*sinh(x)^4 + 2*a^2*b*cosh(x)^2 + a^2*b + 2*(3*a^2*b*cosh
(x)^2 + a^2*b)*sinh(x)^2 + 4*(a^2*b*cosh(x)^3 + a^2*b*cosh(x))*sinh(x))*log(2*cosh(x)/(cosh(x) - sinh(x))) - (
a^3 + a*b^2 - 3*(a^3 + a*b^2)*cosh(x)^2 + 4*(a^2*b + b^3)*cosh(x))*sinh(x))/((a^4 + 2*a^2*b^2 + b^4)*cosh(x)^4
+ 4*(a^4 + 2*a^2*b^2 + b^4)*cosh(x)*sinh(x)^3 + (a^4 + 2*a^2*b^2 + b^4)*sinh(x)^4 + a^4 + 2*a^2*b^2 + b^4 + 2
*(a^4 + 2*a^2*b^2 + b^4)*cosh(x)^2 + 2*(a^4 + 2*a^2*b^2 + b^4 + 3*(a^4 + 2*a^2*b^2 + b^4)*cosh(x)^2)*sinh(x)^2
+ 4*((a^4 + 2*a^2*b^2 + b^4)*cosh(x)^3 + (a^4 + 2*a^2*b^2 + b^4)*cosh(x))*sinh(x))

Sympy [F]

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

[In]

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

[Out]

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 161 vs. \(2 (78) = 156\).

Time = 0.27 (sec) , antiderivative size = 161, normalized size of antiderivative = 1.69 \[ \int \frac {\text {sech}^3(x)}{a+b \text {csch}(x)} \, dx=-\frac {a^{2} b \log \left (-2 \, b e^{\left (-x\right )} + a e^{\left (-2 \, x\right )} - a\right )}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} + \frac {a^{2} b \log \left (e^{\left (-2 \, x\right )} + 1\right )}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} - \frac {{\left (a^{3} - a b^{2}\right )} \arctan \left (e^{\left (-x\right )}\right )}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} + \frac {a e^{\left (-x\right )} - 2 \, b e^{\left (-2 \, x\right )} - a e^{\left (-3 \, x\right )}}{a^{2} + b^{2} + 2 \, {\left (a^{2} + b^{2}\right )} e^{\left (-2 \, x\right )} + {\left (a^{2} + b^{2}\right )} e^{\left (-4 \, x\right )}} \]

[In]

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

[Out]

-a^2*b*log(-2*b*e^(-x) + a*e^(-2*x) - a)/(a^4 + 2*a^2*b^2 + b^4) + a^2*b*log(e^(-2*x) + 1)/(a^4 + 2*a^2*b^2 +
b^4) - (a^3 - a*b^2)*arctan(e^(-x))/(a^4 + 2*a^2*b^2 + b^4) + (a*e^(-x) - 2*b*e^(-2*x) - a*e^(-3*x))/(a^2 + b^
2 + 2*(a^2 + b^2)*e^(-2*x) + (a^2 + b^2)*e^(-4*x))

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 218 vs. \(2 (78) = 156\).

Time = 0.29 (sec) , antiderivative size = 218, normalized size of antiderivative = 2.29 \[ \int \frac {\text {sech}^3(x)}{a+b \text {csch}(x)} \, dx=-\frac {a^{3} b \log \left ({\left | -a {\left (e^{\left (-x\right )} - e^{x}\right )} + 2 \, b \right |}\right )}{a^{5} + 2 \, a^{3} b^{2} + a b^{4}} + \frac {a^{2} b \log \left ({\left (e^{\left (-x\right )} - e^{x}\right )}^{2} + 4\right )}{2 \, {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )}} + \frac {{\left (\pi + 2 \, \arctan \left (\frac {1}{2} \, {\left (e^{\left (2 \, x\right )} - 1\right )} e^{\left (-x\right )}\right )\right )} {\left (a^{3} - a b^{2}\right )}}{4 \, {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )}} - \frac {a^{2} b {\left (e^{\left (-x\right )} - e^{x}\right )}^{2} + 2 \, a^{3} {\left (e^{\left (-x\right )} - e^{x}\right )} + 2 \, a b^{2} {\left (e^{\left (-x\right )} - e^{x}\right )} + 8 \, a^{2} b + 4 \, b^{3}}{2 \, {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} {\left ({\left (e^{\left (-x\right )} - e^{x}\right )}^{2} + 4\right )}} \]

[In]

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

[Out]

-a^3*b*log(abs(-a*(e^(-x) - e^x) + 2*b))/(a^5 + 2*a^3*b^2 + a*b^4) + 1/2*a^2*b*log((e^(-x) - e^x)^2 + 4)/(a^4
+ 2*a^2*b^2 + b^4) + 1/4*(pi + 2*arctan(1/2*(e^(2*x) - 1)*e^(-x)))*(a^3 - a*b^2)/(a^4 + 2*a^2*b^2 + b^4) - 1/2
*(a^2*b*(e^(-x) - e^x)^2 + 2*a^3*(e^(-x) - e^x) + 2*a*b^2*(e^(-x) - e^x) + 8*a^2*b + 4*b^3)/((a^4 + 2*a^2*b^2
+ b^4)*((e^(-x) - e^x)^2 + 4))

Mupad [B] (verification not implemented)

Time = 3.59 (sec) , antiderivative size = 256, normalized size of antiderivative = 2.69 \[ \int \frac {\text {sech}^3(x)}{a+b \text {csch}(x)} \, dx=\frac {\frac {2\,b}{a^2+b^2}-\frac {2\,a\,{\mathrm {e}}^x}{a^2+b^2}}{2\,{\mathrm {e}}^{2\,x}+{\mathrm {e}}^{4\,x}+1}-\frac {\frac {2\,\left (a^2\,b+b^3\right )}{{\left (a^2+b^2\right )}^2}-\frac {{\mathrm {e}}^x\,\left (a^3+a\,b^2\right )}{{\left (a^2+b^2\right )}^2}}{{\mathrm {e}}^{2\,x}+1}+\frac {a\,\ln \left ({\mathrm {e}}^x+1{}\mathrm {i}\right )}{2\,\left (-a^2\,1{}\mathrm {i}+2\,a\,b+b^2\,1{}\mathrm {i}\right )}-\frac {a^2\,b\,\ln \left (a^6\,{\mathrm {e}}^{2\,x}-a^6-a^2\,b^4-14\,a^4\,b^2+a^2\,b^4\,{\mathrm {e}}^{2\,x}+14\,a^4\,b^2\,{\mathrm {e}}^{2\,x}+2\,a\,b^5\,{\mathrm {e}}^x+2\,a^5\,b\,{\mathrm {e}}^x+28\,a^3\,b^3\,{\mathrm {e}}^x\right )}{a^4+2\,a^2\,b^2+b^4}+\frac {a\,\ln \left (1+{\mathrm {e}}^x\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{2\,\left (-a^2+a\,b\,2{}\mathrm {i}+b^2\right )} \]

[In]

int(1/(cosh(x)^3*(a + b/sinh(x))),x)

[Out]

((2*b)/(a^2 + b^2) - (2*a*exp(x))/(a^2 + b^2))/(2*exp(2*x) + exp(4*x) + 1) - ((2*(a^2*b + b^3))/(a^2 + b^2)^2
- (exp(x)*(a*b^2 + a^3))/(a^2 + b^2)^2)/(exp(2*x) + 1) + (a*log(exp(x)*1i + 1)*1i)/(2*(a*b*2i - a^2 + b^2)) +
(a*log(exp(x) + 1i))/(2*(2*a*b - a^2*1i + b^2*1i)) - (a^2*b*log(a^6*exp(2*x) - a^6 - a^2*b^4 - 14*a^4*b^2 + a^
2*b^4*exp(2*x) + 14*a^4*b^2*exp(2*x) + 2*a*b^5*exp(x) + 2*a^5*b*exp(x) + 28*a^3*b^3*exp(x)))/(a^4 + b^4 + 2*a^
2*b^2)