Integrand size = 18, antiderivative size = 20 \[ \int \left (\frac {x}{\sinh ^{\frac {3}{2}}(x)}-x \sqrt {\sinh (x)}\right ) \, dx=-\frac {2 x \cosh (x)}{\sqrt {\sinh (x)}}+4 \sqrt {\sinh (x)} \]
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Time = 0.04 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {3396} \[ \int \left (\frac {x}{\sinh ^{\frac {3}{2}}(x)}-x \sqrt {\sinh (x)}\right ) \, dx=4 \sqrt {\sinh (x)}-\frac {2 x \cosh (x)}{\sqrt {\sinh (x)}} \]
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Rule 3396
Rubi steps \begin{align*} \text {integral}& = \int \frac {x}{\sinh ^{\frac {3}{2}}(x)} \, dx-\int x \sqrt {\sinh (x)} \, dx \\ & = -\frac {2 x \cosh (x)}{\sqrt {\sinh (x)}}+4 \sqrt {\sinh (x)} \\ \end{align*}
Time = 0.13 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.85 \[ \int \left (\frac {x}{\sinh ^{\frac {3}{2}}(x)}-x \sqrt {\sinh (x)}\right ) \, dx=\frac {-2 x \cosh (x)+4 \sinh (x)}{\sqrt {\sinh (x)}} \]
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\[\int \left (\frac {x}{\sinh \left (x \right )^{\frac {3}{2}}}-x \sqrt {\sinh \left (x \right )}\right )d x\]
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Exception generated. \[ \int \left (\frac {x}{\sinh ^{\frac {3}{2}}(x)}-x \sqrt {\sinh (x)}\right ) \, dx=\text {Exception raised: TypeError} \]
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\[ \int \left (\frac {x}{\sinh ^{\frac {3}{2}}(x)}-x \sqrt {\sinh (x)}\right ) \, dx=- \int \left (- \frac {x}{\sinh ^{\frac {3}{2}}{\left (x \right )}}\right )\, dx - \int x \sqrt {\sinh {\left (x \right )}}\, dx \]
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\[ \int \left (\frac {x}{\sinh ^{\frac {3}{2}}(x)}-x \sqrt {\sinh (x)}\right ) \, dx=\int { -x \sqrt {\sinh \left (x\right )} + \frac {x}{\sinh \left (x\right )^{\frac {3}{2}}} \,d x } \]
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\[ \int \left (\frac {x}{\sinh ^{\frac {3}{2}}(x)}-x \sqrt {\sinh (x)}\right ) \, dx=\int { -x \sqrt {\sinh \left (x\right )} + \frac {x}{\sinh \left (x\right )^{\frac {3}{2}}} \,d x } \]
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Time = 0.86 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.90 \[ \int \left (\frac {x}{\sinh ^{\frac {3}{2}}(x)}-x \sqrt {\sinh (x)}\right ) \, dx=-\frac {2\,\sqrt {\frac {{\mathrm {e}}^x}{2}-\frac {{\mathrm {e}}^{-x}}{2}}\,\left (x-2\,{\mathrm {e}}^{2\,x}+x\,{\mathrm {e}}^{2\,x}+2\right )}{{\mathrm {e}}^{2\,x}-1} \]
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