3.594 \(\int \frac{\sinh ^2(x) \sinh (2 x)}{(1-\sinh ^2(x))^{3/2}} \, dx\)

Optimal. Leaf size=29 \[ 2 \sqrt{1-\sinh ^2(x)}+\frac{2}{\sqrt{1-\sinh ^2(x)}} \]

[Out]

2/Sqrt[1 - Sinh[x]^2] + 2*Sqrt[1 - Sinh[x]^2]

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Rubi [A]  time = 0.107906, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {12, 266, 43} \[ 2 \sqrt{1-\sinh ^2(x)}+\frac{2}{\sqrt{1-\sinh ^2(x)}} \]

Antiderivative was successfully verified.

[In]

Int[(Sinh[x]^2*Sinh[2*x])/(1 - Sinh[x]^2)^(3/2),x]

[Out]

2/Sqrt[1 - Sinh[x]^2] + 2*Sqrt[1 - Sinh[x]^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 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{\sinh ^2(x) \sinh (2 x)}{\left (1-\sinh ^2(x)\right )^{3/2}} \, dx &=i \operatorname{Subst}\left (\int -\frac{2 i x^3}{\left (1-x^2\right )^{3/2}} \, dx,x,\sinh (x)\right )\\ &=2 \operatorname{Subst}\left (\int \frac{x^3}{\left (1-x^2\right )^{3/2}} \, dx,x,\sinh (x)\right )\\ &=\operatorname{Subst}\left (\int \frac{x}{(1-x)^{3/2}} \, dx,x,\sinh ^2(x)\right )\\ &=\operatorname{Subst}\left (\int \left (\frac{1}{(1-x)^{3/2}}-\frac{1}{\sqrt{1-x}}\right ) \, dx,x,\sinh ^2(x)\right )\\ &=\frac{2}{\sqrt{1-\sinh ^2(x)}}+2 \sqrt{1-\sinh ^2(x)}\\ \end{align*}

Mathematica [A]  time = 0.0539855, size = 21, normalized size = 0.72 \[ \frac{5-\cosh (2 x)}{\sqrt{1-\sinh ^2(x)}} \]

Antiderivative was successfully verified.

[In]

Integrate[(Sinh[x]^2*Sinh[2*x])/(1 - Sinh[x]^2)^(3/2),x]

[Out]

(5 - Cosh[2*x])/Sqrt[1 - Sinh[x]^2]

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Maple [C]  time = 0.044, size = 28, normalized size = 1. \begin{align*} \mbox{{\tt ` int/indef0`}} \left ( -2\,{\frac{ \left ( \sinh \left ( x \right ) \right ) ^{3}}{ \left ( -1+ \left ( \sinh \left ( x \right ) \right ) ^{2} \right ) \sqrt{1- \left ( \sinh \left ( x \right ) \right ) ^{2}}}},\sinh \left ( x \right ) \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sinh(x)^2*sinh(2*x)/(1-sinh(x)^2)^(3/2),x)

[Out]

`int/indef0`(-2*sinh(x)^3/(-1+sinh(x)^2)/(1-sinh(x)^2)^(1/2),sinh(x))

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Maxima [B]  time = 1.65, size = 239, normalized size = 8.24 \begin{align*} -\frac{16 \, e^{\left (-x\right )}}{{\left (2 \, e^{\left (-x\right )} + e^{\left (-2 \, x\right )} - 1\right )}^{\frac{3}{2}}{\left (2 \, e^{\left (-x\right )} - e^{\left (-2 \, x\right )} + 1\right )}^{\frac{3}{2}}} + \frac{62 \, e^{\left (-3 \, x\right )}}{{\left (2 \, e^{\left (-x\right )} + e^{\left (-2 \, x\right )} - 1\right )}^{\frac{3}{2}}{\left (2 \, e^{\left (-x\right )} - e^{\left (-2 \, x\right )} + 1\right )}^{\frac{3}{2}}} - \frac{16 \, e^{\left (-5 \, x\right )}}{{\left (2 \, e^{\left (-x\right )} + e^{\left (-2 \, x\right )} - 1\right )}^{\frac{3}{2}}{\left (2 \, e^{\left (-x\right )} - e^{\left (-2 \, x\right )} + 1\right )}^{\frac{3}{2}}} + \frac{e^{\left (-7 \, x\right )}}{{\left (2 \, e^{\left (-x\right )} + e^{\left (-2 \, x\right )} - 1\right )}^{\frac{3}{2}}{\left (2 \, e^{\left (-x\right )} - e^{\left (-2 \, x\right )} + 1\right )}^{\frac{3}{2}}} + \frac{e^{x}}{{\left (2 \, e^{\left (-x\right )} + e^{\left (-2 \, x\right )} - 1\right )}^{\frac{3}{2}}{\left (2 \, e^{\left (-x\right )} - e^{\left (-2 \, x\right )} + 1\right )}^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(x)^2*sinh(2*x)/(1-sinh(x)^2)^(3/2),x, algorithm="maxima")

[Out]

-16*e^(-x)/((2*e^(-x) + e^(-2*x) - 1)^(3/2)*(2*e^(-x) - e^(-2*x) + 1)^(3/2)) + 62*e^(-3*x)/((2*e^(-x) + e^(-2*
x) - 1)^(3/2)*(2*e^(-x) - e^(-2*x) + 1)^(3/2)) - 16*e^(-5*x)/((2*e^(-x) + e^(-2*x) - 1)^(3/2)*(2*e^(-x) - e^(-
2*x) + 1)^(3/2)) + e^(-7*x)/((2*e^(-x) + e^(-2*x) - 1)^(3/2)*(2*e^(-x) - e^(-2*x) + 1)^(3/2)) + e^x/((2*e^(-x)
+ e^(-2*x) - 1)^(3/2)*(2*e^(-x) - e^(-2*x) + 1)^(3/2))

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Fricas [B]  time = 2.16068, size = 548, normalized size = 18.9 \begin{align*} \frac{\sqrt{2}{\left (\cosh \left (x\right )^{4} + 4 \, \cosh \left (x\right ) \sinh \left (x\right )^{3} + \sinh \left (x\right )^{4} + 2 \,{\left (3 \, \cosh \left (x\right )^{2} - 5\right )} \sinh \left (x\right )^{2} - 10 \, \cosh \left (x\right )^{2} + 4 \,{\left (\cosh \left (x\right )^{3} - 5 \, \cosh \left (x\right )\right )} \sinh \left (x\right ) + 1\right )} \sqrt{-\frac{\cosh \left (x\right )^{2} + \sinh \left (x\right )^{2} - 3}{\cosh \left (x\right )^{2} - 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2}}}}{\cosh \left (x\right )^{5} + 5 \, \cosh \left (x\right ) \sinh \left (x\right )^{4} + \sinh \left (x\right )^{5} + 2 \,{\left (5 \, \cosh \left (x\right )^{2} - 3\right )} \sinh \left (x\right )^{3} - 6 \, \cosh \left (x\right )^{3} + 2 \,{\left (5 \, \cosh \left (x\right )^{3} - 9 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{2} +{\left (5 \, \cosh \left (x\right )^{4} - 18 \, \cosh \left (x\right )^{2} + 1\right )} \sinh \left (x\right ) + \cosh \left (x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(x)^2*sinh(2*x)/(1-sinh(x)^2)^(3/2),x, algorithm="fricas")

[Out]

sqrt(2)*(cosh(x)^4 + 4*cosh(x)*sinh(x)^3 + sinh(x)^4 + 2*(3*cosh(x)^2 - 5)*sinh(x)^2 - 10*cosh(x)^2 + 4*(cosh(
x)^3 - 5*cosh(x))*sinh(x) + 1)*sqrt(-(cosh(x)^2 + sinh(x)^2 - 3)/(cosh(x)^2 - 2*cosh(x)*sinh(x) + sinh(x)^2))/
(cosh(x)^5 + 5*cosh(x)*sinh(x)^4 + sinh(x)^5 + 2*(5*cosh(x)^2 - 3)*sinh(x)^3 - 6*cosh(x)^3 + 2*(5*cosh(x)^3 -
9*cosh(x))*sinh(x)^2 + (5*cosh(x)^4 - 18*cosh(x)^2 + 1)*sinh(x) + cosh(x))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sinh ^{2}{\left (x \right )} \sinh{\left (2 x \right )}}{\left (- \left (\sinh{\left (x \right )} - 1\right ) \left (\sinh{\left (x \right )} + 1\right )\right )^{\frac{3}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(x)**2*sinh(2*x)/(1-sinh(x)**2)**(3/2),x)

[Out]

Integral(sinh(x)**2*sinh(2*x)/(-(sinh(x) - 1)*(sinh(x) + 1))**(3/2), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sinh \left (2 \, x\right ) \sinh \left (x\right )^{2}}{{\left (-\sinh \left (x\right )^{2} + 1\right )}^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(x)^2*sinh(2*x)/(1-sinh(x)^2)^(3/2),x, algorithm="giac")

[Out]

integrate(sinh(2*x)*sinh(x)^2/(-sinh(x)^2 + 1)^(3/2), x)