\(\int \frac {\text {sech}^3(x)}{(a+b \sinh (x))^2} \, dx\) [206]

Optimal result

Integrand size = 13, antiderivative size = 136 \[ \int \frac {\text {sech}^3(x)}{(a+b \sinh (x))^2} \, dx=\frac {\left (a^4+6 a^2 b^2-3 b^4\right ) \arctan (\sinh (x))}{2 \left (a^2+b^2\right )^3}-\frac {4 a b^3 \log (\cosh (x))}{\left (a^2+b^2\right )^3}+\frac {4 a b^3 \log (a+b \sinh (x))}{\left (a^2+b^2\right )^3}+\frac {b \left (a^2-3 b^2\right )}{2 \left (a^2+b^2\right )^2 (a+b \sinh (x))}+\frac {\text {sech}^2(x) (b+a \sinh (x))}{2 \left (a^2+b^2\right ) (a+b \sinh (x))} \]

[Out]

1/2*(a^4+6*a^2*b^2-3*b^4)*arctan(sinh(x))/(a^2+b^2)^3-4*a*b^3*ln(cosh(x))/(a^2+b^2)^3+4*a*b^3*ln(a+b*sinh(x))/
(a^2+b^2)^3+1/2*b*(a^2-3*b^2)/(a^2+b^2)^2/(a+b*sinh(x))+1/2*sech(x)^2*(b+a*sinh(x))/(a^2+b^2)/(a+b*sinh(x))

Rubi [A] (verified)

Time = 0.11 (sec) , antiderivative size = 136, 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} \[ \int \frac {\text {sech}^3(x)}{(a+b \sinh (x))^2} \, dx=\frac {b \left (a^2-3 b^2\right )}{2 \left (a^2+b^2\right )^2 (a+b \sinh (x))}+\frac {\text {sech}^2(x) (a \sinh (x)+b)}{2 \left (a^2+b^2\right ) (a+b \sinh (x))}+\frac {4 a b^3 \log (a+b \sinh (x))}{\left (a^2+b^2\right )^3}-\frac {4 a b^3 \log (\cosh (x))}{\left (a^2+b^2\right )^3}+\frac {\left (a^4+6 a^2 b^2-3 b^4\right ) \arctan (\sinh (x))}{2 \left (a^2+b^2\right )^3} \]

[In]

Int[Sech[x]^3/(a + b*Sinh[x])^2,x]

[Out]

((a^4 + 6*a^2*b^2 - 3*b^4)*ArcTan[Sinh[x]])/(2*(a^2 + b^2)^3) - (4*a*b^3*Log[Cosh[x]])/(a^2 + b^2)^3 + (4*a*b^
3*Log[a + b*Sinh[x]])/(a^2 + b^2)^3 + (b*(a^2 - 3*b^2))/(2*(a^2 + b^2)^2*(a + b*Sinh[x])) + (Sech[x]^2*(b + a*
Sinh[x]))/(2*(a^2 + b^2)*(a + b*Sinh[x]))

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)^2 \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 ) (a+b \sinh (x))}-\frac {b \text {Subst}\left (\int \frac {a^2+3 b^2+2 a x}{(a+x)^2 \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 ) (a+b \sinh (x))}-\frac {b \text {Subst}\left (\int \left (\frac {a^2-3 b^2}{\left (a^2+b^2\right ) (a+x)^2}-\frac {8 a b^2}{\left (a^2+b^2\right )^2 (a+x)}+\frac {-a^4-6 a^2 b^2+3 b^4+8 a b^2 x}{\left (a^2+b^2\right )^2 \left (b^2+x^2\right )}\right ) \, dx,x,b \sinh (x)\right )}{2 \left (a^2+b^2\right )} \\ & = \frac {4 a b^3 \log (a+b \sinh (x))}{\left (a^2+b^2\right )^3}+\frac {b \left (a^2-3 b^2\right )}{2 \left (a^2+b^2\right )^2 (a+b \sinh (x))}+\frac {\text {sech}^2(x) (b+a \sinh (x))}{2 \left (a^2+b^2\right ) (a+b \sinh (x))}-\frac {b \text {Subst}\left (\int \frac {-a^4-6 a^2 b^2+3 b^4+8 a b^2 x}{b^2+x^2} \, dx,x,b \sinh (x)\right )}{2 \left (a^2+b^2\right )^3} \\ & = \frac {4 a b^3 \log (a+b \sinh (x))}{\left (a^2+b^2\right )^3}+\frac {b \left (a^2-3 b^2\right )}{2 \left (a^2+b^2\right )^2 (a+b \sinh (x))}+\frac {\text {sech}^2(x) (b+a \sinh (x))}{2 \left (a^2+b^2\right ) (a+b \sinh (x))}-\frac {\left (4 a b^3\right ) \text {Subst}\left (\int \frac {x}{b^2+x^2} \, dx,x,b \sinh (x)\right )}{\left (a^2+b^2\right )^3}+\frac {\left (b \left (a^4+6 a^2 b^2-3 b^4\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 )^3} \\ & = \frac {\left (a^4+6 a^2 b^2-3 b^4\right ) \arctan (\sinh (x))}{2 \left (a^2+b^2\right )^3}-\frac {4 a b^3 \log (\cosh (x))}{\left (a^2+b^2\right )^3}+\frac {4 a b^3 \log (a+b \sinh (x))}{\left (a^2+b^2\right )^3}+\frac {b \left (a^2-3 b^2\right )}{2 \left (a^2+b^2\right )^2 (a+b \sinh (x))}+\frac {\text {sech}^2(x) (b+a \sinh (x))}{2 \left (a^2+b^2\right ) (a+b \sinh (x))} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.52 (sec) , antiderivative size = 260, normalized size of antiderivative = 1.91 \[ \int \frac {\text {sech}^3(x)}{(a+b \sinh (x))^2} \, dx=-\frac {-\frac {2 \text {sech}^2(x) (b+a \sinh (x))}{a+b \sinh (x)}+\frac {b \left (\frac {2 a \left (a^2+b^2\right ) \left (\left (-a+\sqrt {-b^2}\right ) \log \left (\sqrt {-b^2}-b \sinh (x)\right )-2 \sqrt {-b^2} \log (a+b \sinh (x))+\left (a+\sqrt {-b^2}\right ) \log \left (\sqrt {-b^2}+b \sinh (x)\right )\right )}{\sqrt {-b^2}}+\left (-a^2+3 b^2\right ) \left (\left (2 a+\frac {-a^2+b^2}{\sqrt {-b^2}}\right ) \log \left (\sqrt {-b^2}-b \sinh (x)\right )-4 a \log (a+b \sinh (x))+\left (2 a+\frac {a^2-b^2}{\sqrt {-b^2}}\right ) \log \left (\sqrt {-b^2}+b \sinh (x)\right )+\frac {2 \left (a^2+b^2\right )}{a+b \sinh (x)}\right )\right )}{\left (a^2+b^2\right )^2}}{4 \left (a^2+b^2\right )} \]

[In]

Integrate[Sech[x]^3/(a + b*Sinh[x])^2,x]

[Out]

-1/4*((-2*Sech[x]^2*(b + a*Sinh[x]))/(a + b*Sinh[x]) + (b*((2*a*(a^2 + b^2)*((-a + Sqrt[-b^2])*Log[Sqrt[-b^2]
- b*Sinh[x]] - 2*Sqrt[-b^2]*Log[a + b*Sinh[x]] + (a + Sqrt[-b^2])*Log[Sqrt[-b^2] + b*Sinh[x]]))/Sqrt[-b^2] + (
-a^2 + 3*b^2)*((2*a + (-a^2 + b^2)/Sqrt[-b^2])*Log[Sqrt[-b^2] - b*Sinh[x]] - 4*a*Log[a + b*Sinh[x]] + (2*a + (
a^2 - b^2)/Sqrt[-b^2])*Log[Sqrt[-b^2] + b*Sinh[x]] + (2*(a^2 + b^2))/(a + b*Sinh[x]))))/(a^2 + b^2)^2)/(a^2 +
b^2)

Maple [A] (verified)

Time = 95.75 (sec) , antiderivative size = 211, normalized size of antiderivative = 1.55

method	result	size
default	\(\frac {2 b^{3} \left (-\frac {b \left (a^{2}+b^{2}\right ) \tanh \left (\frac {x}{2}\right )}{a \left (\tanh \left (\frac {x}{2}\right )^{2} a -2 b \tanh \left (\frac {x}{2}\right )-a \right )}+2 a \ln \left (\tanh \left (\frac {x}{2}\right )^{2} a -2 b \tanh \left (\frac {x}{2}\right )-a \right )\right )}{\left (a^{2}+b^{2}\right )^{3}}+\frac {\frac {2 \left (\left (-\frac {a^{4}}{2}+\frac {b^{4}}{2}\right ) \tanh \left (\frac {x}{2}\right )^{3}+\left (-2 a^{3} b -2 b^{3} a \right ) \tanh \left (\frac {x}{2}\right )^{2}+\left (\frac {a^{4}}{2}-\frac {b^{4}}{2}\right ) \tanh \left (\frac {x}{2}\right )\right )}{\left (1+\tanh \left (\frac {x}{2}\right )^{2}\right )^{2}}-4 b^{3} a \ln \left (1+\tanh \left (\frac {x}{2}\right )^{2}\right )+\left (a^{4}+6 a^{2} b^{2}-3 b^{4}\right ) \arctan \left (\tanh \left (\frac {x}{2}\right )\right )}{a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}}\)	\(211\)
risch	\(\frac {\left (a^{2} b \,{\mathrm e}^{4 x}-3 \,{\mathrm e}^{4 x} b^{3}+2 a^{3} {\mathrm e}^{3 x}+2 a \,b^{2} {\mathrm e}^{3 x}+6 a^{2} b \,{\mathrm e}^{2 x}-2 b^{3} {\mathrm e}^{2 x}-2 a^{3} {\mathrm e}^{x}-2 a \,b^{2} {\mathrm e}^{x}+a^{2} b -3 b^{3}\right ) {\mathrm e}^{x}}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) \left (1+{\mathrm e}^{2 x}\right )^{2} \left (b \,{\mathrm e}^{2 x}+2 \,{\mathrm e}^{x} a -b \right )}-\frac {i \ln \left ({\mathrm e}^{x}-i\right ) a^{4}}{2 \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}-\frac {3 i \ln \left ({\mathrm e}^{x}-i\right ) a^{2} b^{2}}{a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}}+\frac {3 i \ln \left ({\mathrm e}^{x}-i\right ) b^{4}}{2 \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}-\frac {4 \ln \left ({\mathrm e}^{x}-i\right ) b^{3} a}{a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}}+\frac {i \ln \left ({\mathrm e}^{x}+i\right ) a^{4}}{2 a^{6}+6 a^{4} b^{2}+6 a^{2} b^{4}+2 b^{6}}+\frac {3 i \ln \left ({\mathrm e}^{x}+i\right ) a^{2} b^{2}}{a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}}-\frac {3 i \ln \left ({\mathrm e}^{x}+i\right ) b^{4}}{2 \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}-\frac {4 \ln \left ({\mathrm e}^{x}+i\right ) b^{3} a}{a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}}+\frac {4 b^{3} a \ln \left ({\mathrm e}^{2 x}+\frac {2 a \,{\mathrm e}^{x}}{b}-1\right )}{a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}}\)	\(477\)

[In]

int(sech(x)^3/(a+b*sinh(x))^2,x,method=_RETURNVERBOSE)

[Out]

2*b^3/(a^2+b^2)^3*(-b*(a^2+b^2)/a*tanh(1/2*x)/(tanh(1/2*x)^2*a-2*b*tanh(1/2*x)-a)+2*a*ln(tanh(1/2*x)^2*a-2*b*t
anh(1/2*x)-a))+2/(a^6+3*a^4*b^2+3*a^2*b^4+b^6)*(((-1/2*a^4+1/2*b^4)*tanh(1/2*x)^3+(-2*a^3*b-2*a*b^3)*tanh(1/2*
x)^2+(1/2*a^4-1/2*b^4)*tanh(1/2*x))/(1+tanh(1/2*x)^2)^2-2*b^3*a*ln(1+tanh(1/2*x)^2)+1/2*(a^4+6*a^2*b^2-3*b^4)*
arctan(tanh(1/2*x)))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2615 vs. \(2 (130) = 260\).

Time = 0.33 (sec) , antiderivative size = 2615, normalized size of antiderivative = 19.23 \[ \int \frac {\text {sech}^3(x)}{(a+b \sinh (x))^2} \, dx=\text {Too large to display} \]

[In]

integrate(sech(x)^3/(a+b*sinh(x))^2,x, algorithm="fricas")

[Out]

-((a^4*b - 2*a^2*b^3 - 3*b^5)*cosh(x)^5 + (a^4*b - 2*a^2*b^3 - 3*b^5)*sinh(x)^5 + 2*(a^5 + 2*a^3*b^2 + a*b^4)*
cosh(x)^4 + (2*a^5 + 4*a^3*b^2 + 2*a*b^4 + 5*(a^4*b - 2*a^2*b^3 - 3*b^5)*cosh(x))*sinh(x)^4 + 2*(3*a^4*b + 2*a
^2*b^3 - b^5)*cosh(x)^3 + 2*(3*a^4*b + 2*a^2*b^3 - b^5 + 5*(a^4*b - 2*a^2*b^3 - 3*b^5)*cosh(x)^2 + 4*(a^5 + 2*
a^3*b^2 + a*b^4)*cosh(x))*sinh(x)^3 - 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 - 5*(
a^4*b - 2*a^2*b^3 - 3*b^5)*cosh(x)^3 - 6*(a^5 + 2*a^3*b^2 + a*b^4)*cosh(x)^2 - 3*(3*a^4*b + 2*a^2*b^3 - b^5)*c
osh(x))*sinh(x)^2 + ((a^4*b + 6*a^2*b^3 - 3*b^5)*cosh(x)^6 + (a^4*b + 6*a^2*b^3 - 3*b^5)*sinh(x)^6 + 2*(a^5 +
6*a^3*b^2 - 3*a*b^4)*cosh(x)^5 + 2*(a^5 + 6*a^3*b^2 - 3*a*b^4 + 3*(a^4*b + 6*a^2*b^3 - 3*b^5)*cosh(x))*sinh(x)
^5 - a^4*b - 6*a^2*b^3 + 3*b^5 + (a^4*b + 6*a^2*b^3 - 3*b^5)*cosh(x)^4 + (a^4*b + 6*a^2*b^3 - 3*b^5 + 15*(a^4*
b + 6*a^2*b^3 - 3*b^5)*cosh(x)^2 + 10*(a^5 + 6*a^3*b^2 - 3*a*b^4)*cosh(x))*sinh(x)^4 + 4*(a^5 + 6*a^3*b^2 - 3*
a*b^4)*cosh(x)^3 + 4*(a^5 + 6*a^3*b^2 - 3*a*b^4 + 5*(a^4*b + 6*a^2*b^3 - 3*b^5)*cosh(x)^3 + 5*(a^5 + 6*a^3*b^2
- 3*a*b^4)*cosh(x)^2 + (a^4*b + 6*a^2*b^3 - 3*b^5)*cosh(x))*sinh(x)^3 - (a^4*b + 6*a^2*b^3 - 3*b^5)*cosh(x)^2
- (a^4*b + 6*a^2*b^3 - 3*b^5 - 15*(a^4*b + 6*a^2*b^3 - 3*b^5)*cosh(x)^4 - 20*(a^5 + 6*a^3*b^2 - 3*a*b^4)*cosh
(x)^3 - 6*(a^4*b + 6*a^2*b^3 - 3*b^5)*cosh(x)^2 - 12*(a^5 + 6*a^3*b^2 - 3*a*b^4)*cosh(x))*sinh(x)^2 + 2*(a^5 +
6*a^3*b^2 - 3*a*b^4)*cosh(x) + 2*(3*(a^4*b + 6*a^2*b^3 - 3*b^5)*cosh(x)^5 + a^5 + 6*a^3*b^2 - 3*a*b^4 + 5*(a^
5 + 6*a^3*b^2 - 3*a*b^4)*cosh(x)^4 + 2*(a^4*b + 6*a^2*b^3 - 3*b^5)*cosh(x)^3 + 6*(a^5 + 6*a^3*b^2 - 3*a*b^4)*c
osh(x)^2 - (a^4*b + 6*a^2*b^3 - 3*b^5)*cosh(x))*sinh(x))*arctan(cosh(x) + sinh(x)) + (a^4*b - 2*a^2*b^3 - 3*b^
5)*cosh(x) + 4*(a*b^4*cosh(x)^6 + a*b^4*sinh(x)^6 + 2*a^2*b^3*cosh(x)^5 + a*b^4*cosh(x)^4 + 4*a^2*b^3*cosh(x)^
3 - a*b^4*cosh(x)^2 + 2*a^2*b^3*cosh(x) + 2*(3*a*b^4*cosh(x) + a^2*b^3)*sinh(x)^5 - a*b^4 + (15*a*b^4*cosh(x)^
2 + 10*a^2*b^3*cosh(x) + a*b^4)*sinh(x)^4 + 4*(5*a*b^4*cosh(x)^3 + 5*a^2*b^3*cosh(x)^2 + a*b^4*cosh(x) + a^2*b
^3)*sinh(x)^3 + (15*a*b^4*cosh(x)^4 + 20*a^2*b^3*cosh(x)^3 + 6*a*b^4*cosh(x)^2 + 12*a^2*b^3*cosh(x) - a*b^4)*s
inh(x)^2 + 2*(3*a*b^4*cosh(x)^5 + 5*a^2*b^3*cosh(x)^4 + 2*a*b^4*cosh(x)^3 + 6*a^2*b^3*cosh(x)^2 - a*b^4*cosh(x
) + a^2*b^3)*sinh(x))*log(2*(b*sinh(x) + a)/(cosh(x) - sinh(x))) - 4*(a*b^4*cosh(x)^6 + a*b^4*sinh(x)^6 + 2*a^
2*b^3*cosh(x)^5 + a*b^4*cosh(x)^4 + 4*a^2*b^3*cosh(x)^3 - a*b^4*cosh(x)^2 + 2*a^2*b^3*cosh(x) + 2*(3*a*b^4*cos
h(x) + a^2*b^3)*sinh(x)^5 - a*b^4 + (15*a*b^4*cosh(x)^2 + 10*a^2*b^3*cosh(x) + a*b^4)*sinh(x)^4 + 4*(5*a*b^4*c
osh(x)^3 + 5*a^2*b^3*cosh(x)^2 + a*b^4*cosh(x) + a^2*b^3)*sinh(x)^3 + (15*a*b^4*cosh(x)^4 + 20*a^2*b^3*cosh(x)
^3 + 6*a*b^4*cosh(x)^2 + 12*a^2*b^3*cosh(x) - a*b^4)*sinh(x)^2 + 2*(3*a*b^4*cosh(x)^5 + 5*a^2*b^3*cosh(x)^4 +
2*a*b^4*cosh(x)^3 + 6*a^2*b^3*cosh(x)^2 - a*b^4*cosh(x) + a^2*b^3)*sinh(x))*log(2*cosh(x)/(cosh(x) - sinh(x)))
+ (a^4*b - 2*a^2*b^3 - 3*b^5 + 5*(a^4*b - 2*a^2*b^3 - 3*b^5)*cosh(x)^4 + 8*(a^5 + 2*a^3*b^2 + a*b^4)*cosh(x)^
3 + 6*(3*a^4*b + 2*a^2*b^3 - b^5)*cosh(x)^2 - 4*(a^5 + 2*a^3*b^2 + a*b^4)*cosh(x))*sinh(x))/(a^6*b + 3*a^4*b^3
+ 3*a^2*b^5 + b^7 - (a^6*b + 3*a^4*b^3 + 3*a^2*b^5 + b^7)*cosh(x)^6 - (a^6*b + 3*a^4*b^3 + 3*a^2*b^5 + b^7)*s
inh(x)^6 - 2*(a^7 + 3*a^5*b^2 + 3*a^3*b^4 + a*b^6)*cosh(x)^5 - 2*(a^7 + 3*a^5*b^2 + 3*a^3*b^4 + a*b^6 + 3*(a^6
*b + 3*a^4*b^3 + 3*a^2*b^5 + b^7)*cosh(x))*sinh(x)^5 - (a^6*b + 3*a^4*b^3 + 3*a^2*b^5 + b^7)*cosh(x)^4 - (a^6*
b + 3*a^4*b^3 + 3*a^2*b^5 + b^7 + 15*(a^6*b + 3*a^4*b^3 + 3*a^2*b^5 + b^7)*cosh(x)^2 + 10*(a^7 + 3*a^5*b^2 + 3
*a^3*b^4 + a*b^6)*cosh(x))*sinh(x)^4 - 4*(a^7 + 3*a^5*b^2 + 3*a^3*b^4 + a*b^6)*cosh(x)^3 - 4*(a^7 + 3*a^5*b^2
+ 3*a^3*b^4 + a*b^6 + 5*(a^6*b + 3*a^4*b^3 + 3*a^2*b^5 + b^7)*cosh(x)^3 + 5*(a^7 + 3*a^5*b^2 + 3*a^3*b^4 + a*b
^6)*cosh(x)^2 + (a^6*b + 3*a^4*b^3 + 3*a^2*b^5 + b^7)*cosh(x))*sinh(x)^3 + (a^6*b + 3*a^4*b^3 + 3*a^2*b^5 + b^
7)*cosh(x)^2 + (a^6*b + 3*a^4*b^3 + 3*a^2*b^5 + b^7 - 15*(a^6*b + 3*a^4*b^3 + 3*a^2*b^5 + b^7)*cosh(x)^4 - 20*
(a^7 + 3*a^5*b^2 + 3*a^3*b^4 + a*b^6)*cosh(x)^3 - 6*(a^6*b + 3*a^4*b^3 + 3*a^2*b^5 + b^7)*cosh(x)^2 - 12*(a^7
+ 3*a^5*b^2 + 3*a^3*b^4 + a*b^6)*cosh(x))*sinh(x)^2 - 2*(a^7 + 3*a^5*b^2 + 3*a^3*b^4 + a*b^6)*cosh(x) - 2*(a^7
+ 3*a^5*b^2 + 3*a^3*b^4 + a*b^6 + 3*(a^6*b + 3*a^4*b^3 + 3*a^2*b^5 + b^7)*cosh(x)^5 + 5*(a^7 + 3*a^5*b^2 + 3*
a^3*b^4 + a*b^6)*cosh(x)^4 + 2*(a^6*b + 3*a^4*b^3 + 3*a^2*b^5 + b^7)*cosh(x)^3 + 6*(a^7 + 3*a^5*b^2 + 3*a^3*b^
4 + a*b^6)*cosh(x)^2 - (a^6*b + 3*a^4*b^3 + 3*a^2*b^5 + b^7)*cosh(x))*sinh(x))

Sympy [F]

\[ \int \frac {\text {sech}^3(x)}{(a+b \sinh (x))^2} \, dx=\int \frac {\operatorname {sech}^{3}{\left (x \right )}}{\left (a + b \sinh {\left (x \right )}\right )^{2}}\, dx \]

[In]

integrate(sech(x)**3/(a+b*sinh(x))**2,x)

[Out]

Integral(sech(x)**3/(a + b*sinh(x))**2, x)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 375 vs. \(2 (130) = 260\).

Time = 0.30 (sec) , antiderivative size = 375, normalized size of antiderivative = 2.76 \[ \int \frac {\text {sech}^3(x)}{(a+b \sinh (x))^2} \, dx=\frac {4 \, a b^{3} \log \left (-2 \, a e^{\left (-x\right )} + b e^{\left (-2 \, x\right )} - b\right )}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} - \frac {4 \, a b^{3} \log \left (e^{\left (-2 \, x\right )} + 1\right )}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} - \frac {{\left (a^{4} + 6 \, a^{2} b^{2} - 3 \, b^{4}\right )} \arctan \left (e^{\left (-x\right )}\right )}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} + \frac {{\left (a^{2} b - 3 \, b^{3}\right )} e^{\left (-x\right )} + 2 \, {\left (a^{3} + a b^{2}\right )} e^{\left (-2 \, x\right )} + 2 \, {\left (3 \, a^{2} b - b^{3}\right )} e^{\left (-3 \, x\right )} - 2 \, {\left (a^{3} + a b^{2}\right )} e^{\left (-4 \, x\right )} + {\left (a^{2} b - 3 \, b^{3}\right )} e^{\left (-5 \, x\right )}}{a^{4} b + 2 \, a^{2} b^{3} + b^{5} + 2 \, {\left (a^{5} + 2 \, a^{3} b^{2} + a b^{4}\right )} e^{\left (-x\right )} + {\left (a^{4} b + 2 \, a^{2} b^{3} + b^{5}\right )} e^{\left (-2 \, x\right )} + 4 \, {\left (a^{5} + 2 \, a^{3} b^{2} + a b^{4}\right )} e^{\left (-3 \, x\right )} - {\left (a^{4} b + 2 \, a^{2} b^{3} + b^{5}\right )} e^{\left (-4 \, x\right )} + 2 \, {\left (a^{5} + 2 \, a^{3} b^{2} + a b^{4}\right )} e^{\left (-5 \, x\right )} - {\left (a^{4} b + 2 \, a^{2} b^{3} + b^{5}\right )} e^{\left (-6 \, x\right )}} \]

[In]

integrate(sech(x)^3/(a+b*sinh(x))^2,x, algorithm="maxima")

[Out]

4*a*b^3*log(-2*a*e^(-x) + b*e^(-2*x) - b)/(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6) - 4*a*b^3*log(e^(-2*x) + 1)/(a^6
+ 3*a^4*b^2 + 3*a^2*b^4 + b^6) - (a^4 + 6*a^2*b^2 - 3*b^4)*arctan(e^(-x))/(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)
+ ((a^2*b - 3*b^3)*e^(-x) + 2*(a^3 + a*b^2)*e^(-2*x) + 2*(3*a^2*b - b^3)*e^(-3*x) - 2*(a^3 + a*b^2)*e^(-4*x)
+ (a^2*b - 3*b^3)*e^(-5*x))/(a^4*b + 2*a^2*b^3 + b^5 + 2*(a^5 + 2*a^3*b^2 + a*b^4)*e^(-x) + (a^4*b + 2*a^2*b^3
+ b^5)*e^(-2*x) + 4*(a^5 + 2*a^3*b^2 + a*b^4)*e^(-3*x) - (a^4*b + 2*a^2*b^3 + b^5)*e^(-4*x) + 2*(a^5 + 2*a^3*
b^2 + a*b^4)*e^(-5*x) - (a^4*b + 2*a^2*b^3 + b^5)*e^(-6*x))

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 295 vs. \(2 (130) = 260\).

Time = 0.28 (sec) , antiderivative size = 295, normalized size of antiderivative = 2.17 \[ \int \frac {\text {sech}^3(x)}{(a+b \sinh (x))^2} \, dx=\frac {4 \, a b^{4} \log \left ({\left | -b {\left (e^{\left (-x\right )} - e^{x}\right )} + 2 \, a \right |}\right )}{a^{6} b + 3 \, a^{4} b^{3} + 3 \, a^{2} b^{5} + b^{7}} - \frac {2 \, a b^{3} \log \left ({\left (e^{\left (-x\right )} - e^{x}\right )}^{2} + 4\right )}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} + \frac {{\left (\pi + 2 \, \arctan \left (\frac {1}{2} \, {\left (e^{\left (2 \, x\right )} - 1\right )} e^{\left (-x\right )}\right )\right )} {\left (a^{4} + 6 \, a^{2} b^{2} - 3 \, b^{4}\right )}}{4 \, {\left (a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}\right )}} - \frac {a^{2} b {\left (e^{\left (-x\right )} - e^{x}\right )}^{2} - 3 \, b^{3} {\left (e^{\left (-x\right )} - e^{x}\right )}^{2} - 2 \, a^{3} {\left (e^{\left (-x\right )} - e^{x}\right )} - 2 \, a b^{2} {\left (e^{\left (-x\right )} - e^{x}\right )} + 8 \, a^{2} b - 8 \, b^{3}}{{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} {\left (b {\left (e^{\left (-x\right )} - e^{x}\right )}^{3} - 2 \, a {\left (e^{\left (-x\right )} - e^{x}\right )}^{2} + 4 \, b {\left (e^{\left (-x\right )} - e^{x}\right )} - 8 \, a\right )}} \]

[In]

integrate(sech(x)^3/(a+b*sinh(x))^2,x, algorithm="giac")

[Out]

4*a*b^4*log(abs(-b*(e^(-x) - e^x) + 2*a))/(a^6*b + 3*a^4*b^3 + 3*a^2*b^5 + b^7) - 2*a*b^3*log((e^(-x) - e^x)^2
+ 4)/(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6) + 1/4*(pi + 2*arctan(1/2*(e^(2*x) - 1)*e^(-x)))*(a^4 + 6*a^2*b^2 - 3
*b^4)/(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6) - (a^2*b*(e^(-x) - e^x)^2 - 3*b^3*(e^(-x) - e^x)^2 - 2*a^3*(e^(-x) -
e^x) - 2*a*b^2*(e^(-x) - e^x) + 8*a^2*b - 8*b^3)/((a^4 + 2*a^2*b^2 + b^4)*(b*(e^(-x) - e^x)^3 - 2*a*(e^(-x) -
e^x)^2 + 4*b*(e^(-x) - e^x) - 8*a))

Mupad [B] (verification not implemented)

Time = 6.08 (sec) , antiderivative size = 519, normalized size of antiderivative = 3.82 \[ \int \frac {\text {sech}^3(x)}{(a+b \sinh (x))^2} \, dx=\frac {\frac {4\,\left (a^7\,b+3\,a^5\,b^3+3\,a^3\,b^5+a\,b^7\right )}{\left (a^2+b^2\right )\,{\left (a^4+2\,a^2\,b^2+b^4\right )}^2}+\frac {{\mathrm {e}}^x\,\left (a^8+2\,a^6\,b^2-2\,a^2\,b^6-b^8\right )}{\left (a^2+b^2\right )\,{\left (a^4+2\,a^2\,b^2+b^4\right )}^2}}{{\mathrm {e}}^{2\,x}+1}-\frac {\frac {4\,a\,b}{a^4+2\,a^2\,b^2+b^4}+\frac {2\,{\mathrm {e}}^x\,\left (a^2-b^2\right )}{a^4+2\,a^2\,b^2+b^4}}{2\,{\mathrm {e}}^{2\,x}+{\mathrm {e}}^{4\,x}+1}+\frac {\ln \left ({\mathrm {e}}^x+1{}\mathrm {i}\right )\,\left (a-b\,3{}\mathrm {i}\right )}{2\,\left (-a^3\,1{}\mathrm {i}-3\,a^2\,b+a\,b^2\,3{}\mathrm {i}+b^3\right )}+\frac {\ln \left (1+{\mathrm {e}}^x\,1{}\mathrm {i}\right )\,\left (-3\,b+a\,1{}\mathrm {i}\right )}{2\,\left (-a^3-a^2\,b\,3{}\mathrm {i}+3\,a\,b^2+b^3\,1{}\mathrm {i}\right )}+\frac {4\,a\,b^3\,\ln \left (9\,b^9\,{\mathrm {e}}^{2\,x}-a^8\,b-9\,b^9-220\,a^2\,b^7-30\,a^4\,b^5-12\,a^6\,b^3+2\,a^9\,{\mathrm {e}}^x+220\,a^2\,b^7\,{\mathrm {e}}^{2\,x}+30\,a^4\,b^5\,{\mathrm {e}}^{2\,x}+12\,a^6\,b^3\,{\mathrm {e}}^{2\,x}+18\,a\,b^8\,{\mathrm {e}}^x+a^8\,b\,{\mathrm {e}}^{2\,x}+440\,a^3\,b^6\,{\mathrm {e}}^x+60\,a^5\,b^4\,{\mathrm {e}}^x+24\,a^7\,b^2\,{\mathrm {e}}^x\right )}{a^6+3\,a^4\,b^2+3\,a^2\,b^4+b^6}-\frac {2\,{\mathrm {e}}^x\,\left (a^4\,b^6+2\,a^2\,b^8+b^{10}\right )}{b^2\,\left (a^2\,b+b^3\right )\,\left (a^2+b^2\right )\,\left (2\,a\,{\mathrm {e}}^x-b+b\,{\mathrm {e}}^{2\,x}\right )\,\left (a^4+2\,a^2\,b^2+b^4\right )} \]

[In]

int(1/(cosh(x)^3*(a + b*sinh(x))^2),x)

[Out]

((4*(a*b^7 + a^7*b + 3*a^3*b^5 + 3*a^5*b^3))/((a^2 + b^2)*(a^4 + b^4 + 2*a^2*b^2)^2) + (exp(x)*(a^8 - b^8 - 2*
a^2*b^6 + 2*a^6*b^2))/((a^2 + b^2)*(a^4 + b^4 + 2*a^2*b^2)^2))/(exp(2*x) + 1) - ((4*a*b)/(a^4 + b^4 + 2*a^2*b^
2) + (2*exp(x)*(a^2 - b^2))/(a^4 + b^4 + 2*a^2*b^2))/(2*exp(2*x) + exp(4*x) + 1) + (log(exp(x) + 1i)*(a - b*3i
))/(2*(a*b^2*3i - 3*a^2*b - a^3*1i + b^3)) + (log(exp(x)*1i + 1)*(a*1i - 3*b))/(2*(3*a*b^2 - a^2*b*3i - a^3 +
b^3*1i)) + (4*a*b^3*log(9*b^9*exp(2*x) - a^8*b - 9*b^9 - 220*a^2*b^7 - 30*a^4*b^5 - 12*a^6*b^3 + 2*a^9*exp(x)
+ 220*a^2*b^7*exp(2*x) + 30*a^4*b^5*exp(2*x) + 12*a^6*b^3*exp(2*x) + 18*a*b^8*exp(x) + a^8*b*exp(2*x) + 440*a^
3*b^6*exp(x) + 60*a^5*b^4*exp(x) + 24*a^7*b^2*exp(x)))/(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2) - (2*exp(x)*(b^10 +
2*a^2*b^8 + a^4*b^6))/(b^2*(a^2*b + b^3)*(a^2 + b^2)*(2*a*exp(x) - b + b*exp(2*x))*(a^4 + b^4 + 2*a^2*b^2))