\(\int \frac {\text {sech}^3(x)}{a+b \sinh (x)} \, dx\) [196]

Optimal result

Integrand size = 13, antiderivative size = 87 \[ \int \frac {\text {sech}^3(x)}{a+b \sinh (x)} \, dx=\frac {a \left (a^2+3 b^2\right ) \arctan (\sinh (x))}{2 \left (a^2+b^2\right )^2}-\frac {b^3 \log (\cosh (x))}{\left (a^2+b^2\right )^2}+\frac {b^3 \log (a+b \sinh (x))}{\left (a^2+b^2\right )^2}+\frac {\text {sech}^2(x) (b+a \sinh (x))}{2 \left (a^2+b^2\right )} \]

[Out]

1/2*a*(a^2+3*b^2)*arctan(sinh(x))/(a^2+b^2)^2-b^3*ln(cosh(x))/(a^2+b^2)^2+b^3*ln(a+b*sinh(x))/(a^2+b^2)^2+1/2*
sech(x)^2*(b+a*sinh(x))/(a^2+b^2)

Rubi [A] (verified)

Time = 0.09 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {2747, 755, 815, 649, 209, 266} \[ \int \frac {\text {sech}^3(x)}{a+b \sinh (x)} \, dx=\frac {a \left (a^2+3 b^2\right ) \arctan (\sinh (x))}{2 \left (a^2+b^2\right )^2}+\frac {\text {sech}^2(x) (a \sinh (x)+b)}{2 \left (a^2+b^2\right )}+\frac {b^3 \log (a+b \sinh (x))}{\left (a^2+b^2\right )^2}-\frac {b^3 \log (\cosh (x))}{\left (a^2+b^2\right )^2} \]

[In]

Int[Sech[x]^3/(a + b*Sinh[x]),x]

[Out]

(a*(a^2 + 3*b^2)*ArcTan[Sinh[x]])/(2*(a^2 + b^2)^2) - (b^3*Log[Cosh[x]])/(a^2 + b^2)^2 + (b^3*Log[a + b*Sinh[x
]])/(a^2 + b^2)^2 + (Sech[x]^2*(b + a*Sinh[x]))/(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 755

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-(d + e*x)^(m + 1))*(a*e + c*d*x)*
((a + c*x^2)^(p + 1)/(2*a*(p + 1)*(c*d^2 + a*e^2))), x] + Dist[1/(2*a*(p + 1)*(c*d^2 + a*e^2)), Int[(d + e*x)^
m*Simp[c*d^2*(2*p + 3) + a*e^2*(m + 2*p + 3) + c*e*d*(m + 2*p + 4)*x, x]*(a + c*x^2)^(p + 1), x], x] /; FreeQ[
{a, c, d, e, m}, x] && NeQ[c*d^2 + a*e^2, 0] && LtQ[p, -1] && IntQuadraticQ[a, 0, c, d, e, m, p, 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 2747

Int[cos[(e_.) + (f_.)*(x_)]^(p_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_Symbol] :> Dist[1/(b^p*f), S
ubst[Int[(a + x)^m*(b^2 - x^2)^((p - 1)/2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x] && Integer
Q[(p - 1)/2] && NeQ[a^2 - b^2, 0]

Rubi steps \begin{align*} \text {integral}& = b^3 \text {Subst}\left (\int \frac {1}{(a+x) \left (-b^2-x^2\right )^2} \, dx,x,b \sinh (x)\right ) \\ & = \frac {\text {sech}^2(x) (b+a \sinh (x))}{2 \left (a^2+b^2\right )}-\frac {b \text {Subst}\left (\int \frac {a^2+2 b^2+a x}{(a+x) \left (-b^2-x^2\right )} \, dx,x,b \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 {b \text {Subst}\left (\int \left (-\frac {2 b^2}{\left (a^2+b^2\right ) (a+x)}+\frac {-a^3-3 a b^2+2 b^2 x}{\left (a^2+b^2\right ) \left (b^2+x^2\right )}\right ) \, dx,x,b \sinh (x)\right )}{2 \left (a^2+b^2\right )} \\ & = \frac {b^3 \log (a+b \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 )}-\frac {b \text {Subst}\left (\int \frac {-a^3-3 a b^2+2 b^2 x}{b^2+x^2} \, dx,x,b \sinh (x)\right )}{2 \left (a^2+b^2\right )^2} \\ & = \frac {b^3 \log (a+b \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 )}-\frac {b^3 \text {Subst}\left (\int \frac {x}{b^2+x^2} \, dx,x,b \sinh (x)\right )}{\left (a^2+b^2\right )^2}+\frac {\left (a b \left (a^2+3 b^2\right )\right ) \text {Subst}\left (\int \frac {1}{b^2+x^2} \, dx,x,b \sinh (x)\right )}{2 \left (a^2+b^2\right )^2} \\ & = \frac {a \left (a^2+3 b^2\right ) \arctan (\sinh (x))}{2 \left (a^2+b^2\right )^2}-\frac {b^3 \log (\cosh (x))}{\left (a^2+b^2\right )^2}+\frac {b^3 \log (a+b \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 = 0.14 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.89 \[ \int \frac {\text {sech}^3(x)}{a+b \sinh (x)} \, dx=\frac {2 a \left (a^2+3 b^2\right ) \arctan \left (\tanh \left (\frac {x}{2}\right )\right )+2 b^3 (-\log (\cosh (x))+\log (a+b \sinh (x)))+b \left (a^2+b^2\right ) \text {sech}^2(x)+a \left (a^2+b^2\right ) \text {sech}(x) \tanh (x)}{2 \left (a^2+b^2\right )^2} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 17.27 (sec) , antiderivative size = 161, normalized size of antiderivative = 1.85

method	result	size
default	\(\frac {b^{3} \ln \left (\tanh \left (\frac {x}{2}\right )^{2} a -2 b \tanh \left (\frac {x}{2}\right )-a \right )}{a^{4}+2 a^{2} b^{2}+b^{4}}+\frac {\frac {2 \left (\left (-\frac {1}{2} a^{3}-\frac {1}{2} a \,b^{2}\right ) \tanh \left (\frac {x}{2}\right )^{3}+\left (-a^{2} b -b^{3}\right ) \tanh \left (\frac {x}{2}\right )^{2}+\left (\frac {1}{2} a^{3}+\frac {1}{2} a \,b^{2}\right ) \tanh \left (\frac {x}{2}\right )\right )}{\left (1+\tanh \left (\frac {x}{2}\right )^{2}\right )^{2}}-b^{3} \ln \left (1+\tanh \left (\frac {x}{2}\right )^{2}\right )+\left (a^{3}+3 a \,b^{2}\right ) \arctan \left (\tanh \left (\frac {x}{2}\right )\right )}{a^{4}+2 a^{2} b^{2}+b^{4}}\)	\(161\)
risch	\(\frac {{\mathrm e}^{x} \left ({\mathrm e}^{2 x} a +2 \,{\mathrm e}^{x} b -a \right )}{\left (1+{\mathrm e}^{2 x}\right )^{2} \left (a^{2}+b^{2}\right )}+\frac {i \ln \left ({\mathrm e}^{x}+i\right ) a^{3}}{2 a^{4}+4 a^{2} b^{2}+2 b^{4}}+\frac {3 i \ln \left ({\mathrm e}^{x}+i\right ) a \,b^{2}}{2 \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}-\frac {\ln \left ({\mathrm e}^{x}+i\right ) b^{3}}{a^{4}+2 a^{2} b^{2}+b^{4}}-\frac {i \ln \left ({\mathrm e}^{x}-i\right ) a^{3}}{2 \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}-\frac {3 i \ln \left ({\mathrm e}^{x}-i\right ) a \,b^{2}}{2 \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}-\frac {\ln \left ({\mathrm e}^{x}-i\right ) b^{3}}{a^{4}+2 a^{2} b^{2}+b^{4}}+\frac {b^{3} \ln \left ({\mathrm e}^{2 x}+\frac {2 a \,{\mathrm e}^{x}}{b}-1\right )}{a^{4}+2 a^{2} b^{2}+b^{4}}\)	\(247\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 652 vs. \(2 (83) = 166\).

Time = 0.31 (sec) , antiderivative size = 652, normalized size of antiderivative = 7.49 \[ \int \frac {\text {sech}^3(x)}{a+b \sinh (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} + 3 \, a b^{2}\right )} \cosh \left (x\right )^{4} + 4 \, {\left (a^{3} + 3 \, a b^{2}\right )} \cosh \left (x\right ) \sinh \left (x\right )^{3} + {\left (a^{3} + 3 \, a b^{2}\right )} \sinh \left (x\right )^{4} + a^{3} + 3 \, a b^{2} + 2 \, {\left (a^{3} + 3 \, a b^{2}\right )} \cosh \left (x\right )^{2} + 2 \, {\left (a^{3} + 3 \, a b^{2} + 3 \, {\left (a^{3} + 3 \, a b^{2}\right )} \cosh \left (x\right )^{2}\right )} \sinh \left (x\right )^{2} + 4 \, {\left ({\left (a^{3} + 3 \, a b^{2}\right )} \cosh \left (x\right )^{3} + {\left (a^{3} + 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 (b^{3} \cosh \left (x\right )^{4} + 4 \, b^{3} \cosh \left (x\right ) \sinh \left (x\right )^{3} + b^{3} \sinh \left (x\right )^{4} + 2 \, b^{3} \cosh \left (x\right )^{2} + b^{3} + 2 \, {\left (3 \, b^{3} \cosh \left (x\right )^{2} + b^{3}\right )} \sinh \left (x\right )^{2} + 4 \, {\left (b^{3} \cosh \left (x\right )^{3} + b^{3} \cosh \left (x\right )\right )} \sinh \left (x\right )\right )} \log \left (\frac {2 \, {\left (b \sinh \left (x\right ) + a\right )}}{\cosh \left (x\right ) - \sinh \left (x\right )}\right ) - {\left (b^{3} \cosh \left (x\right )^{4} + 4 \, b^{3} \cosh \left (x\right ) \sinh \left (x\right )^{3} + b^{3} \sinh \left (x\right )^{4} + 2 \, b^{3} \cosh \left (x\right )^{2} + b^{3} + 2 \, {\left (3 \, b^{3} \cosh \left (x\right )^{2} + b^{3}\right )} \sinh \left (x\right )^{2} + 4 \, {\left (b^{3} \cosh \left (x\right )^{3} + b^{3} \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*sinh(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 + 3*a*b^2)*cosh(x)^4 + 4*(a^3 + 3*a*b^2)*cosh(x)*sinh(x)^3 + (a^3 + 3*a*b^2)*
sinh(x)^4 + a^3 + 3*a*b^2 + 2*(a^3 + 3*a*b^2)*cosh(x)^2 + 2*(a^3 + 3*a*b^2 + 3*(a^3 + 3*a*b^2)*cosh(x)^2)*sinh
(x)^2 + 4*((a^3 + 3*a*b^2)*cosh(x)^3 + (a^3 + 3*a*b^2)*cosh(x))*sinh(x))*arctan(cosh(x) + sinh(x)) - (a^3 + a*
b^2)*cosh(x) + (b^3*cosh(x)^4 + 4*b^3*cosh(x)*sinh(x)^3 + b^3*sinh(x)^4 + 2*b^3*cosh(x)^2 + b^3 + 2*(3*b^3*cos
h(x)^2 + b^3)*sinh(x)^2 + 4*(b^3*cosh(x)^3 + b^3*cosh(x))*sinh(x))*log(2*(b*sinh(x) + a)/(cosh(x) - sinh(x)))
- (b^3*cosh(x)^4 + 4*b^3*cosh(x)*sinh(x)^3 + b^3*sinh(x)^4 + 2*b^3*cosh(x)^2 + b^3 + 2*(3*b^3*cosh(x)^2 + b^3)
*sinh(x)^2 + 4*(b^3*cosh(x)^3 + b^3*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]

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

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.83 \[ \int \frac {\text {sech}^3(x)}{a+b \sinh (x)} \, dx=\frac {b^{3} \log \left (-2 \, a e^{\left (-x\right )} + b e^{\left (-2 \, x\right )} - b\right )}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} - \frac {b^{3} \log \left (e^{\left (-2 \, x\right )} + 1\right )}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} - \frac {{\left (a^{3} + 3 \, a b^{2}\right )} \arctan \left (e^{\left (-x\right )}\right )}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} + \frac {a e^{\left (-x\right )} + 2 \, b e^{\left (-2 \, x\right )} - a e^{\left (-3 \, x\right )}}{a^{2} + b^{2} + 2 \, {\left (a^{2} + b^{2}\right )} e^{\left (-2 \, x\right )} + {\left (a^{2} + b^{2}\right )} e^{\left (-4 \, x\right )}} \]

[In]

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

[Out]

b^3*log(-2*a*e^(-x) + b*e^(-2*x) - b)/(a^4 + 2*a^2*b^2 + b^4) - b^3*log(e^(-2*x) + 1)/(a^4 + 2*a^2*b^2 + b^4)
- (a^3 + 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. 214 vs. \(2 (83) = 166\).

Time = 0.28 (sec) , antiderivative size = 214, normalized size of antiderivative = 2.46 \[ \int \frac {\text {sech}^3(x)}{a+b \sinh (x)} \, dx=\frac {b^{4} \log \left ({\left | -b {\left (e^{\left (-x\right )} - e^{x}\right )} + 2 \, a \right |}\right )}{a^{4} b + 2 \, a^{2} b^{3} + b^{5}} - \frac {b^{3} \log \left ({\left (e^{\left (-x\right )} - e^{x}\right )}^{2} + 4\right )}{2 \, {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )}} + \frac {{\left (\pi + 2 \, \arctan \left (\frac {1}{2} \, {\left (e^{\left (2 \, x\right )} - 1\right )} e^{\left (-x\right )}\right )\right )} {\left (a^{3} + 3 \, a b^{2}\right )}}{4 \, {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )}} + \frac {b^{3} {\left (e^{\left (-x\right )} - e^{x}\right )}^{2} - 2 \, a^{3} {\left (e^{\left (-x\right )} - e^{x}\right )} - 2 \, a b^{2} {\left (e^{\left (-x\right )} - e^{x}\right )} + 4 \, a^{2} b + 8 \, b^{3}}{2 \, {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} {\left ({\left (e^{\left (-x\right )} - e^{x}\right )}^{2} + 4\right )}} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 3.04 (sec) , antiderivative size = 291, normalized size of antiderivative = 3.34 \[ \int \frac {\text {sech}^3(x)}{a+b \sinh (x)} \, dx=\frac {\frac {2\,\left (a^2\,b+b^3\right )}{{\left (a^2+b^2\right )}^2}+\frac {{\mathrm {e}}^x\,\left (a^3+a\,b^2\right )}{{\left (a^2+b^2\right )}^2}}{{\mathrm {e}}^{2\,x}+1}-\frac {\frac {2\,b}{a^2+b^2}+\frac {2\,a\,{\mathrm {e}}^x}{a^2+b^2}}{2\,{\mathrm {e}}^{2\,x}+{\mathrm {e}}^{4\,x}+1}-\frac {\ln \left (1+{\mathrm {e}}^x\,1{}\mathrm {i}\right )\,\left (a+b\,2{}\mathrm {i}\right )}{2\,\left (-a^2\,1{}\mathrm {i}+2\,a\,b+b^2\,1{}\mathrm {i}\right )}+\frac {b^3\,\ln \left (16\,b^7\,{\mathrm {e}}^{2\,x}-a^6\,b-16\,b^7-9\,a^2\,b^5-6\,a^4\,b^3+2\,a^7\,{\mathrm {e}}^x+9\,a^2\,b^5\,{\mathrm {e}}^{2\,x}+6\,a^4\,b^3\,{\mathrm {e}}^{2\,x}+32\,a\,b^6\,{\mathrm {e}}^x+a^6\,b\,{\mathrm {e}}^{2\,x}+18\,a^3\,b^4\,{\mathrm {e}}^x+12\,a^5\,b^2\,{\mathrm {e}}^x\right )}{a^4+2\,a^2\,b^2+b^4}-\frac {\ln \left ({\mathrm {e}}^x+1{}\mathrm {i}\right )\,\left (2\,b+a\,1{}\mathrm {i}\right )}{2\,\left (-a^2+a\,b\,2{}\mathrm {i}+b^2\right )} \]

[In]

int(1/(cosh(x)^3*(a + b*sinh(x))),x)

[Out]

((2*(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) - ((2*b)/(a^2 + b^2) +
(2*a*exp(x))/(a^2 + b^2))/(2*exp(2*x) + exp(4*x) + 1) - (log(exp(x)*1i + 1)*(a + b*2i))/(2*(2*a*b - a^2*1i +
b^2*1i)) + (b^3*log(16*b^7*exp(2*x) - a^6*b - 16*b^7 - 9*a^2*b^5 - 6*a^4*b^3 + 2*a^7*exp(x) + 9*a^2*b^5*exp(2*
x) + 6*a^4*b^3*exp(2*x) + 32*a*b^6*exp(x) + a^6*b*exp(2*x) + 18*a^3*b^4*exp(x) + 12*a^5*b^2*exp(x)))/(a^4 + b^
4 + 2*a^2*b^2) - (log(exp(x) + 1i)*(a*1i + 2*b))/(2*(a*b*2i - a^2 + b^2))