Optimal. Leaf size=29 \[ 2 \sqrt {1-\sinh ^2(x)}+\frac {2}{\sqrt {1-\sinh ^2(x)}} \]
________________________________________________________________________________________
Rubi [A] time = 0.11, antiderivative size = 29, normalized size of antiderivative = 1.00, 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]
[Out]
Rule 12
Rule 43
Rule 266
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 = 0.06, size = 21, normalized size = 0.72 \[ \frac {5-\cosh (2 x)}{\sqrt {1-\sinh ^2(x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sinh ^2(x) \sinh (2 x)}{\left (1-\sinh ^2(x)\right )^{3/2}} \, dx \]
Verification is Not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 1.34, size = 161, normalized size = 5.55 \[ \frac {\sqrt {2} {\left (\cosh \relax (x)^{4} + 4 \, \cosh \relax (x) \sinh \relax (x)^{3} + \sinh \relax (x)^{4} + 2 \, {\left (3 \, \cosh \relax (x)^{2} - 5\right )} \sinh \relax (x)^{2} - 10 \, \cosh \relax (x)^{2} + 4 \, {\left (\cosh \relax (x)^{3} - 5 \, \cosh \relax (x)\right )} \sinh \relax (x) + 1\right )} \sqrt {-\frac {\cosh \relax (x)^{2} + \sinh \relax (x)^{2} - 3}{\cosh \relax (x)^{2} - 2 \, \cosh \relax (x) \sinh \relax (x) + \sinh \relax (x)^{2}}}}{\cosh \relax (x)^{5} + 5 \, \cosh \relax (x) \sinh \relax (x)^{4} + \sinh \relax (x)^{5} + 2 \, {\left (5 \, \cosh \relax (x)^{2} - 3\right )} \sinh \relax (x)^{3} - 6 \, \cosh \relax (x)^{3} + 2 \, {\left (5 \, \cosh \relax (x)^{3} - 9 \, \cosh \relax (x)\right )} \sinh \relax (x)^{2} + {\left (5 \, \cosh \relax (x)^{4} - 18 \, \cosh \relax (x)^{2} + 1\right )} \sinh \relax (x) + \cosh \relax (x)} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sinh \left (2 \, x\right ) \sinh \relax (x)^{2}}{{\left (-\sinh \relax (x)^{2} + 1\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [C] time = 0.13, size = 28, normalized size = 0.97
method | result | size |
default | \(\mathit {`\,int/indef0`\,}\left (-\frac {2 \left (\sinh ^{3}\relax (x )\right )}{\left (\sinh ^{2}\relax (x )-1\right ) \sqrt {1-\left (\sinh ^{2}\relax (x )\right )}}, \sinh \relax (x )\right )\) | \(28\) |
meijerg | error in int/gbinthm/express: improper op or subscript selector\ | N/A |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [B] time = 1.21, size = 177, normalized size = 6.10 \[ -\frac {16 \, e^{\left (-x\right )}}{{\left (2 \, e^{\left (-x\right )} + e^{\left (-2 \, x\right )} - 1\right )}^{\frac {3}{2}} {\left (2 \, e^{\left (-x\right )} - e^{\left (-2 \, x\right )} + 1\right )}^{\frac {3}{2}}} + \frac {62 \, e^{\left (-3 \, x\right )}}{{\left (2 \, e^{\left (-x\right )} + e^{\left (-2 \, x\right )} - 1\right )}^{\frac {3}{2}} {\left (2 \, e^{\left (-x\right )} - e^{\left (-2 \, x\right )} + 1\right )}^{\frac {3}{2}}} - \frac {16 \, e^{\left (-5 \, x\right )}}{{\left (2 \, e^{\left (-x\right )} + e^{\left (-2 \, x\right )} - 1\right )}^{\frac {3}{2}} {\left (2 \, e^{\left (-x\right )} - e^{\left (-2 \, x\right )} + 1\right )}^{\frac {3}{2}}} + \frac {e^{\left (-7 \, x\right )}}{{\left (2 \, e^{\left (-x\right )} + e^{\left (-2 \, x\right )} - 1\right )}^{\frac {3}{2}} {\left (2 \, e^{\left (-x\right )} - e^{\left (-2 \, x\right )} + 1\right )}^{\frac {3}{2}}} + \frac {e^{x}}{{\left (2 \, e^{\left (-x\right )} + e^{\left (-2 \, x\right )} - 1\right )}^{\frac {3}{2}} {\left (2 \, e^{\left (-x\right )} - e^{\left (-2 \, x\right )} + 1\right )}^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.49, size = 47, normalized size = 1.62 \[ \frac {2\,\sqrt {1-{\left (\frac {{\mathrm {e}}^{-x}}{2}-\frac {{\mathrm {e}}^x}{2}\right )}^2}\,\left ({\mathrm {e}}^{4\,x}-10\,{\mathrm {e}}^{2\,x}+1\right )}{{\mathrm {e}}^{4\,x}-6\,{\mathrm {e}}^{2\,x}+1} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sinh ^{2}{\relax (x )} \sinh {\left (2 x \right )}}{\left (- \left (\sinh {\relax (x )} - 1\right ) \left (\sinh {\relax (x )} + 1\right )\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________