Integrand size = 21, antiderivative size = 29 \[ \int \frac {\sinh ^2(x) \sinh (2 x)}{\left (1-\sinh ^2(x)\right )^{3/2}} \, dx=\frac {2}{\sqrt {1-\sinh ^2(x)}}+2 \sqrt {1-\sinh ^2(x)} \]
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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} \[ \int \frac {\sinh ^2(x) \sinh (2 x)}{\left (1-\sinh ^2(x)\right )^{3/2}} \, dx=2 \sqrt {1-\sinh ^2(x)}+\frac {2}{\sqrt {1-\sinh ^2(x)}} \]
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Rule 12
Rule 45
Rule 272
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*}
Time = 0.04 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.72 \[ \int \frac {\sinh ^2(x) \sinh (2 x)}{\left (1-\sinh ^2(x)\right )^{3/2}} \, dx=\frac {5-\cosh (2 x)}{\sqrt {1-\sinh ^2(x)}} \]
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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 | \(-\frac {2 \left (\sinh ^{2}\left (x \right )\right )}{\sqrt {1-\left (\sinh ^{2}\left (x \right )\right )}}+\frac {4}{\sqrt {1-\left (\sinh ^{2}\left (x \right )\right )}}\) | \(30\) |
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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 \[ \int \frac {\sinh ^2(x) \sinh (2 x)}{\left (1-\sinh ^2(x)\right )^{3/2}} \, dx=\frac {\sqrt {2} {\left (\cosh \left (x\right )^{4} + 4 \, \cosh \left (x\right ) \sinh \left (x\right )^{3} + \sinh \left (x\right )^{4} + 2 \, {\left (3 \, \cosh \left (x\right )^{2} - 5\right )} \sinh \left (x\right )^{2} - 10 \, \cosh \left (x\right )^{2} + 4 \, {\left (\cosh \left (x\right )^{3} - 5 \, \cosh \left (x\right )\right )} \sinh \left (x\right ) + 1\right )} \sqrt {-\frac {\cosh \left (x\right )^{2} + \sinh \left (x\right )^{2} - 3}{\cosh \left (x\right )^{2} - 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2}}}}{\cosh \left (x\right )^{5} + 5 \, \cosh \left (x\right ) \sinh \left (x\right )^{4} + \sinh \left (x\right )^{5} + 2 \, {\left (5 \, \cosh \left (x\right )^{2} - 3\right )} \sinh \left (x\right )^{3} - 6 \, \cosh \left (x\right )^{3} + 2 \, {\left (5 \, \cosh \left (x\right )^{3} - 9 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{2} + {\left (5 \, \cosh \left (x\right )^{4} - 18 \, \cosh \left (x\right )^{2} + 1\right )} \sinh \left (x\right ) + \cosh \left (x\right )} \]
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\[ \int \frac {\sinh ^2(x) \sinh (2 x)}{\left (1-\sinh ^2(x)\right )^{3/2}} \, dx=\int \frac {\sinh ^{2}{\left (x \right )} \sinh {\left (2 x \right )}}{\left (- \left (\sinh {\left (x \right )} - 1\right ) \left (\sinh {\left (x \right )} + 1\right )\right )^{\frac {3}{2}}}\, dx \]
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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 \[ \int \frac {\sinh ^2(x) \sinh (2 x)}{\left (1-\sinh ^2(x)\right )^{3/2}} \, dx=-\frac {16 \, e^{\left (-x\right )}}{{\left (2 \, e^{\left (-x\right )} + e^{\left (-2 \, x\right )} - 1\right )}^{\frac {3}{2}} {\left (2 \, e^{\left (-x\right )} - e^{\left (-2 \, x\right )} + 1\right )}^{\frac {3}{2}}} + \frac {62 \, e^{\left (-3 \, x\right )}}{{\left (2 \, e^{\left (-x\right )} + e^{\left (-2 \, x\right )} - 1\right )}^{\frac {3}{2}} {\left (2 \, e^{\left (-x\right )} - e^{\left (-2 \, x\right )} + 1\right )}^{\frac {3}{2}}} - \frac {16 \, e^{\left (-5 \, x\right )}}{{\left (2 \, e^{\left (-x\right )} + e^{\left (-2 \, x\right )} - 1\right )}^{\frac {3}{2}} {\left (2 \, e^{\left (-x\right )} - e^{\left (-2 \, x\right )} + 1\right )}^{\frac {3}{2}}} + \frac {e^{\left (-7 \, x\right )}}{{\left (2 \, e^{\left (-x\right )} + e^{\left (-2 \, x\right )} - 1\right )}^{\frac {3}{2}} {\left (2 \, e^{\left (-x\right )} - e^{\left (-2 \, x\right )} + 1\right )}^{\frac {3}{2}}} + \frac {e^{x}}{{\left (2 \, e^{\left (-x\right )} + e^{\left (-2 \, x\right )} - 1\right )}^{\frac {3}{2}} {\left (2 \, e^{\left (-x\right )} - e^{\left (-2 \, x\right )} + 1\right )}^{\frac {3}{2}}} \]
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\[ \int \frac {\sinh ^2(x) \sinh (2 x)}{\left (1-\sinh ^2(x)\right )^{3/2}} \, dx=\int { \frac {\sinh \left (2 \, x\right ) \sinh \left (x\right )^{2}}{{\left (-\sinh \left (x\right )^{2} + 1\right )}^{\frac {3}{2}}} \,d x } \]
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Time = 0.53 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.62 \[ \int \frac {\sinh ^2(x) \sinh (2 x)}{\left (1-\sinh ^2(x)\right )^{3/2}} \, dx=\frac {2\,\sqrt {1-{\left (\frac {{\mathrm {e}}^{-x}}{2}-\frac {{\mathrm {e}}^x}{2}\right )}^2}\,\left ({\mathrm {e}}^{4\,x}-10\,{\mathrm {e}}^{2\,x}+1\right )}{{\mathrm {e}}^{4\,x}-6\,{\mathrm {e}}^{2\,x}+1} \]
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