\(\int \frac {\sinh ^2(x) \sinh (2 x)}{(1-\sinh ^2(x))^{3/2}} \, dx\) [594]

Optimal result

Integrand size = 21, antiderivative size = 29 \[ \int \frac {\sinh ^2(x) \sinh (2 x)}{\left (1-\sinh ^2(x)\right )^{3/2}} \, dx=\frac {2}{\sqrt {1-\sinh ^2(x)}}+2 \sqrt {1-\sinh ^2(x)} \]

[Out]

2/(1-sinh(x)^2)^(1/2)+2*(1-sinh(x)^2)^(1/2)

Rubi [A] (verified)

Time = 0.07 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {12, 272, 45} \[ \int \frac {\sinh ^2(x) \sinh (2 x)}{\left (1-\sinh ^2(x)\right )^{3/2}} \, dx=2 \sqrt {1-\sinh ^2(x)}+\frac {2}{\sqrt {1-\sinh ^2(x)}} \]

[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 45

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])

Rule 272

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]]

Rubi steps \begin{align*} \text {integral}& = i \text {Subst}\left (\int -\frac {2 i x^3}{\left (1-x^2\right )^{3/2}} \, dx,x,\sinh (x)\right ) \\ & = 2 \text {Subst}\left (\int \frac {x^3}{\left (1-x^2\right )^{3/2}} \, dx,x,\sinh (x)\right ) \\ & = \text {Subst}\left (\int \frac {x}{(1-x)^{3/2}} \, dx,x,\sinh ^2(x)\right ) \\ & = \text {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] (verified)

Time = 0.04 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.72 \[ \int \frac {\sinh ^2(x) \sinh (2 x)}{\left (1-\sinh ^2(x)\right )^{3/2}} \, dx=\frac {5-\cosh (2 x)}{\sqrt {1-\sinh ^2(x)}} \]

[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]

Maple [A] (verified)

Time = 0.24 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.03

method	result	size
derivativedivides	\(-\frac {2 \left (\sinh ^{2}\left (x \right )\right )}{\sqrt {1-\left (\sinh ^{2}\left (x \right )\right )}}+\frac {4}{\sqrt {1-\left (\sinh ^{2}\left (x \right )\right )}}\)	\(30\)
default	\(-\frac {2 \left (\sinh ^{2}\left (x \right )\right )}{\sqrt {1-\left (\sinh ^{2}\left (x \right )\right )}}+\frac {4}{\sqrt {1-\left (\sinh ^{2}\left (x \right )\right )}}\)	\(30\)

[In]

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

[Out]

-2*sinh(x)^2/(1-sinh(x)^2)^(1/2)+4/(1-sinh(x)^2)^(1/2)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 161 vs. \(2 (25) = 50\).

Time = 0.26 (sec) , antiderivative size = 161, normalized size of antiderivative = 5.55 \[ \int \frac {\sinh ^2(x) \sinh (2 x)}{\left (1-\sinh ^2(x)\right )^{3/2}} \, dx=\frac {\sqrt {2} {\left (\cosh \left (x\right )^{4} + 4 \, \cosh \left (x\right ) \sinh \left (x\right )^{3} + \sinh \left (x\right )^{4} + 2 \, {\left (3 \, \cosh \left (x\right )^{2} - 5\right )} \sinh \left (x\right )^{2} - 10 \, \cosh \left (x\right )^{2} + 4 \, {\left (\cosh \left (x\right )^{3} - 5 \, \cosh \left (x\right )\right )} \sinh \left (x\right ) + 1\right )} \sqrt {-\frac {\cosh \left (x\right )^{2} + \sinh \left (x\right )^{2} - 3}{\cosh \left (x\right )^{2} - 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2}}}}{\cosh \left (x\right )^{5} + 5 \, \cosh \left (x\right ) \sinh \left (x\right )^{4} + \sinh \left (x\right )^{5} + 2 \, {\left (5 \, \cosh \left (x\right )^{2} - 3\right )} \sinh \left (x\right )^{3} - 6 \, \cosh \left (x\right )^{3} + 2 \, {\left (5 \, \cosh \left (x\right )^{3} - 9 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{2} + {\left (5 \, \cosh \left (x\right )^{4} - 18 \, \cosh \left (x\right )^{2} + 1\right )} \sinh \left (x\right ) + \cosh \left (x\right )} \]

[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))

Sympy [F]

\[ \int \frac {\sinh ^2(x) \sinh (2 x)}{\left (1-\sinh ^2(x)\right )^{3/2}} \, dx=\int \frac {\sinh ^{2}{\left (x \right )} \sinh {\left (2 x \right )}}{\left (- \left (\sinh {\left (x \right )} - 1\right ) \left (\sinh {\left (x \right )} + 1\right )\right )^{\frac {3}{2}}}\, dx \]

[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)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 177 vs. \(2 (25) = 50\).

Time = 0.33 (sec) , antiderivative size = 177, normalized size of antiderivative = 6.10 \[ \int \frac {\sinh ^2(x) \sinh (2 x)}{\left (1-\sinh ^2(x)\right )^{3/2}} \, dx=-\frac {16 \, e^{\left (-x\right )}}{{\left (2 \, e^{\left (-x\right )} + e^{\left (-2 \, x\right )} - 1\right )}^{\frac {3}{2}} {\left (2 \, e^{\left (-x\right )} - e^{\left (-2 \, x\right )} + 1\right )}^{\frac {3}{2}}} + \frac {62 \, e^{\left (-3 \, x\right )}}{{\left (2 \, e^{\left (-x\right )} + e^{\left (-2 \, x\right )} - 1\right )}^{\frac {3}{2}} {\left (2 \, e^{\left (-x\right )} - e^{\left (-2 \, x\right )} + 1\right )}^{\frac {3}{2}}} - \frac {16 \, e^{\left (-5 \, x\right )}}{{\left (2 \, e^{\left (-x\right )} + e^{\left (-2 \, x\right )} - 1\right )}^{\frac {3}{2}} {\left (2 \, e^{\left (-x\right )} - e^{\left (-2 \, x\right )} + 1\right )}^{\frac {3}{2}}} + \frac {e^{\left (-7 \, x\right )}}{{\left (2 \, e^{\left (-x\right )} + e^{\left (-2 \, x\right )} - 1\right )}^{\frac {3}{2}} {\left (2 \, e^{\left (-x\right )} - e^{\left (-2 \, x\right )} + 1\right )}^{\frac {3}{2}}} + \frac {e^{x}}{{\left (2 \, e^{\left (-x\right )} + e^{\left (-2 \, x\right )} - 1\right )}^{\frac {3}{2}} {\left (2 \, e^{\left (-x\right )} - e^{\left (-2 \, x\right )} + 1\right )}^{\frac {3}{2}}} \]

[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))

Giac [F]

\[ \int \frac {\sinh ^2(x) \sinh (2 x)}{\left (1-\sinh ^2(x)\right )^{3/2}} \, dx=\int { \frac {\sinh \left (2 \, x\right ) \sinh \left (x\right )^{2}}{{\left (-\sinh \left (x\right )^{2} + 1\right )}^{\frac {3}{2}}} \,d x } \]

[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)

Mupad [B] (verification not implemented)

Time = 0.53 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.62 \[ \int \frac {\sinh ^2(x) \sinh (2 x)}{\left (1-\sinh ^2(x)\right )^{3/2}} \, dx=\frac {2\,\sqrt {1-{\left (\frac {{\mathrm {e}}^{-x}}{2}-\frac {{\mathrm {e}}^x}{2}\right )}^2}\,\left ({\mathrm {e}}^{4\,x}-10\,{\mathrm {e}}^{2\,x}+1\right )}{{\mathrm {e}}^{4\,x}-6\,{\mathrm {e}}^{2\,x}+1} \]

[In]

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

[Out]

(2*(1 - (exp(-x)/2 - exp(x)/2)^2)^(1/2)*(exp(4*x) - 10*exp(2*x) + 1))/(exp(4*x) - 6*exp(2*x) + 1)