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

Optimal. Leaf size=136 \[ \frac{b \left (a^2-3 b^2\right )}{2 \left (a^2+b^2\right )^2 (a+b \sinh (x))}+\frac{4 a b^3 \log (a+b \sinh (x))}{\left (a^2+b^2\right )^3}+\frac{\left (6 a^2 b^2+a^4-3 b^4\right ) \tan ^{-1}(\sinh (x))}{2 \left (a^2+b^2\right )^3}-\frac{4 a b^3 \log (\cosh (x))}{\left (a^2+b^2\right )^3}+\frac{\text{sech}^2(x) (a \sinh (x)+b)}{2 \left (a^2+b^2\right ) (a+b \sinh (x))} \]

[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]))

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Rubi [A]  time = 0.164562, antiderivative size = 136, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.462, Rules used = {2668, 741, 801, 635, 203, 260} \[ \frac{b \left (a^2-3 b^2\right )}{2 \left (a^2+b^2\right )^2 (a+b \sinh (x))}+\frac{4 a b^3 \log (a+b \sinh (x))}{\left (a^2+b^2\right )^3}+\frac{\left (6 a^2 b^2+a^4-3 b^4\right ) \tan ^{-1}(\sinh (x))}{2 \left (a^2+b^2\right )^3}-\frac{4 a b^3 \log (\cosh (x))}{\left (a^2+b^2\right )^3}+\frac{\text{sech}^2(x) (a \sinh (x)+b)}{2 \left (a^2+b^2\right ) (a+b \sinh (x))} \]

Antiderivative was successfully verified.

[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 2668

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]

Rule 741

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 801

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 635

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 203

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

Rule 260

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]

Rubi steps

\begin{align*} \int \frac{\text{sech}^3(x)}{(a+b \sinh (x))^2} \, dx &=b^3 \operatorname{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 \operatorname{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 \operatorname{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 \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \tan ^{-1}(\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]  time = 2.44154, size = 260, normalized size = 1.91 \[ -\frac{\frac{b \left (\frac{2 a \left (a^2+b^2\right ) \left (\left (\sqrt{-b^2}-a\right ) \log \left (\sqrt{-b^2}-b \sinh (x)\right )-2 \sqrt{-b^2} \log (a+b \sinh (x))+\left (a+\sqrt{-b^2}\right ) \log \left (\sqrt{-b^2}+b \sinh (x)\right )\right )}{\sqrt{-b^2}}+\left (3 b^2-a^2\right ) \left (\frac{2 \left (a^2+b^2\right )}{a+b \sinh (x)}+\left (\frac{b^2-a^2}{\sqrt{-b^2}}+2 a\right ) \log \left (\sqrt{-b^2}-b \sinh (x)\right )+\left (\frac{a^2-b^2}{\sqrt{-b^2}}+2 a\right ) \log \left (\sqrt{-b^2}+b \sinh (x)\right )-4 a \log (a+b \sinh (x))\right )\right )}{\left (a^2+b^2\right )^2}-\frac{2 \text{sech}^2(x) (a \sinh (x)+b)}{a+b \sinh (x)}}{4 \left (a^2+b^2\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((-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)/(4*(a^2 + b
^2))

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Maple [B]  time = 0.069, size = 548, normalized size = 4. \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 1.87928, size = 506, normalized size = 3.72 \begin{align*} \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 )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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Fricas [B]  time = 2.82989, size = 6197, normalized size = 45.57 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{sech}^{3}{\left (x \right )}}{\left (a + b \sinh{\left (x \right )}\right )^{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B]  time = 1.17612, size = 398, normalized size = 2.93 \begin{align*} \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 )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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))