Integrand size = 15, antiderivative size = 37 \[ \int \frac {\sinh (x)}{\left (-9+4 \cosh ^2(x)\right )^{5/2}} \, dx=-\frac {\cosh (x)}{27 \left (-9+4 \cosh ^2(x)\right )^{3/2}}+\frac {2 \cosh (x)}{243 \sqrt {-9+4 \cosh ^2(x)}} \]
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Time = 0.03 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3269, 198, 197} \[ \int \frac {\sinh (x)}{\left (-9+4 \cosh ^2(x)\right )^{5/2}} \, dx=\frac {2 \cosh (x)}{243 \sqrt {4 \cosh ^2(x)-9}}-\frac {\cosh (x)}{27 \left (4 \cosh ^2(x)-9\right )^{3/2}} \]
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Rule 197
Rule 198
Rule 3269
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {1}{\left (-9+4 x^2\right )^{5/2}} \, dx,x,\cosh (x)\right ) \\ & = -\frac {\cosh (x)}{27 \left (-9+4 \cosh ^2(x)\right )^{3/2}}-\frac {2}{27} \text {Subst}\left (\int \frac {1}{\left (-9+4 x^2\right )^{3/2}} \, dx,x,\cosh (x)\right ) \\ & = -\frac {\cosh (x)}{27 \left (-9+4 \cosh ^2(x)\right )^{3/2}}+\frac {2 \cosh (x)}{243 \sqrt {-9+4 \cosh ^2(x)}} \\ \end{align*}
Time = 0.07 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.70 \[ \int \frac {\sinh (x)}{\left (-9+4 \cosh ^2(x)\right )^{5/2}} \, dx=\frac {\cosh (x) (-23+4 \cosh (2 x))}{243 (-7+2 \cosh (2 x))^{3/2}} \]
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Time = 0.18 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.81
method | result | size |
derivativedivides | \(-\frac {\cosh \left (x \right )}{27 {\left (-9+4 \left (\cosh ^{2}\left (x \right )\right )\right )}^{\frac {3}{2}}}+\frac {2 \cosh \left (x \right )}{243 \sqrt {-9+4 \left (\cosh ^{2}\left (x \right )\right )}}\) | \(30\) |
default | \(-\frac {\cosh \left (x \right )}{27 {\left (-9+4 \left (\cosh ^{2}\left (x \right )\right )\right )}^{\frac {3}{2}}}+\frac {2 \cosh \left (x \right )}{243 \sqrt {-9+4 \left (\cosh ^{2}\left (x \right )\right )}}\) | \(30\) |
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Leaf count of result is larger than twice the leaf count of optimal. 474 vs. \(2 (29) = 58\).
Time = 0.26 (sec) , antiderivative size = 474, normalized size of antiderivative = 12.81 \[ \int \frac {\sinh (x)}{\left (-9+4 \cosh ^2(x)\right )^{5/2}} \, dx=\frac {2 \, \cosh \left (x\right )^{8} + 16 \, \cosh \left (x\right ) \sinh \left (x\right )^{7} + 2 \, \sinh \left (x\right )^{8} + 28 \, {\left (2 \, \cosh \left (x\right )^{2} - 1\right )} \sinh \left (x\right )^{6} - 28 \, \cosh \left (x\right )^{6} + 56 \, {\left (2 \, \cosh \left (x\right )^{3} - 3 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{5} + 2 \, {\left (70 \, \cosh \left (x\right )^{4} - 210 \, \cosh \left (x\right )^{2} + 51\right )} \sinh \left (x\right )^{4} + 102 \, \cosh \left (x\right )^{4} + 8 \, {\left (14 \, \cosh \left (x\right )^{5} - 70 \, \cosh \left (x\right )^{3} + 51 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{3} + 4 \, {\left (14 \, \cosh \left (x\right )^{6} - 105 \, \cosh \left (x\right )^{4} + 153 \, \cosh \left (x\right )^{2} - 7\right )} \sinh \left (x\right )^{2} - 28 \, \cosh \left (x\right )^{2} + 8 \, {\left (2 \, \cosh \left (x\right )^{7} - 21 \, \cosh \left (x\right )^{5} + 51 \, \cosh \left (x\right )^{3} - 7 \, \cosh \left (x\right )\right )} \sinh \left (x\right ) + {\left (2 \, \cosh \left (x\right )^{6} + 12 \, \cosh \left (x\right ) \sinh \left (x\right )^{5} + 2 \, \sinh \left (x\right )^{6} + 3 \, {\left (10 \, \cosh \left (x\right )^{2} - 7\right )} \sinh \left (x\right )^{4} - 21 \, \cosh \left (x\right )^{4} + 4 \, {\left (10 \, \cosh \left (x\right )^{3} - 21 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{3} + 3 \, {\left (10 \, \cosh \left (x\right )^{4} - 42 \, \cosh \left (x\right )^{2} - 7\right )} \sinh \left (x\right )^{2} - 21 \, \cosh \left (x\right )^{2} + 6 \, {\left (2 \, \cosh \left (x\right )^{5} - 14 \, \cosh \left (x\right )^{3} - 7 \, \cosh \left (x\right )\right )} \sinh \left (x\right ) + 2\right )} \sqrt {\frac {2 \, \cosh \left (x\right )^{2} + 2 \, \sinh \left (x\right )^{2} - 7}{\cosh \left (x\right )^{2} - 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2}}} + 2}{486 \, {\left (\cosh \left (x\right )^{8} + 8 \, \cosh \left (x\right ) \sinh \left (x\right )^{7} + \sinh \left (x\right )^{8} + 14 \, {\left (2 \, \cosh \left (x\right )^{2} - 1\right )} \sinh \left (x\right )^{6} - 14 \, \cosh \left (x\right )^{6} + 28 \, {\left (2 \, \cosh \left (x\right )^{3} - 3 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{5} + {\left (70 \, \cosh \left (x\right )^{4} - 210 \, \cosh \left (x\right )^{2} + 51\right )} \sinh \left (x\right )^{4} + 51 \, \cosh \left (x\right )^{4} + 4 \, {\left (14 \, \cosh \left (x\right )^{5} - 70 \, \cosh \left (x\right )^{3} + 51 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{3} + 2 \, {\left (14 \, \cosh \left (x\right )^{6} - 105 \, \cosh \left (x\right )^{4} + 153 \, \cosh \left (x\right )^{2} - 7\right )} \sinh \left (x\right )^{2} - 14 \, \cosh \left (x\right )^{2} + 4 \, {\left (2 \, \cosh \left (x\right )^{7} - 21 \, \cosh \left (x\right )^{5} + 51 \, \cosh \left (x\right )^{3} - 7 \, \cosh \left (x\right )\right )} \sinh \left (x\right ) + 1\right )}} \]
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Timed out. \[ \int \frac {\sinh (x)}{\left (-9+4 \cosh ^2(x)\right )^{5/2}} \, dx=\text {Timed out} \]
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Leaf count of result is larger than twice the leaf count of optimal. 125 vs. \(2 (29) = 58\).
Time = 0.21 (sec) , antiderivative size = 125, normalized size of antiderivative = 3.38 \[ \int \frac {\sinh (x)}{\left (-9+4 \cosh ^2(x)\right )^{5/2}} \, dx=-\frac {1855 \, e^{\left (-2 \, x\right )} - 8485 \, e^{\left (-4 \, x\right )} + 5285 \, e^{\left (-6 \, x\right )} - 980 \, e^{\left (-8 \, x\right )} + 56 \, e^{\left (-10 \, x\right )} - 106}{12150 \, {\left (3 \, e^{\left (-x\right )} + e^{\left (-2 \, x\right )} + 1\right )}^{\frac {5}{2}} {\left (-3 \, e^{\left (-x\right )} + e^{\left (-2 \, x\right )} + 1\right )}^{\frac {5}{2}}} + \frac {980 \, e^{\left (-2 \, x\right )} - 5285 \, e^{\left (-4 \, x\right )} + 8485 \, e^{\left (-6 \, x\right )} - 1855 \, e^{\left (-8 \, x\right )} + 106 \, e^{\left (-10 \, x\right )} - 56}{12150 \, {\left (3 \, e^{\left (-x\right )} + e^{\left (-2 \, x\right )} + 1\right )}^{\frac {5}{2}} {\left (-3 \, e^{\left (-x\right )} + e^{\left (-2 \, x\right )} + 1\right )}^{\frac {5}{2}}} \]
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Time = 0.32 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.03 \[ \int \frac {\sinh (x)}{\left (-9+4 \cosh ^2(x)\right )^{5/2}} \, dx=\frac {{\left ({\left (2 \, e^{\left (2 \, x\right )} - 21\right )} e^{\left (2 \, x\right )} - 21\right )} e^{\left (2 \, x\right )} + 2}{486 \, {\left (e^{\left (4 \, x\right )} - 7 \, e^{\left (2 \, x\right )} + 1\right )}^{\frac {3}{2}}} \]
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Time = 0.14 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.54 \[ \int \frac {\sinh (x)}{\left (-9+4 \cosh ^2(x)\right )^{5/2}} \, dx=-\frac {{\mathrm {e}}^x\,\sqrt {4\,{\left (\frac {{\mathrm {e}}^{-x}}{2}+\frac {{\mathrm {e}}^x}{2}\right )}^2-9}\,\left (21\,{\mathrm {e}}^{2\,x}+21\,{\mathrm {e}}^{4\,x}-2\,{\mathrm {e}}^{6\,x}-2\right )}{486\,{\left ({\mathrm {e}}^{4\,x}-7\,{\mathrm {e}}^{2\,x}+1\right )}^2} \]
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