Optimal. Leaf size=20 \[ -\frac {2 x \cosh (x)}{\sqrt {\sinh (x)}}+4 \sqrt {\sinh (x)} \]
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Rubi [A]
time = 0.04, antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {3396}
\begin {gather*} 4 \sqrt {\sinh (x)}-\frac {2 x \cosh (x)}{\sqrt {\sinh (x)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 3396
Rubi steps
\begin {align*} \int \left (\frac {x}{\sinh ^{\frac {3}{2}}(x)}-x \sqrt {\sinh (x)}\right ) \, dx &=\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*}
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Mathematica [A]
time = 0.08, size = 17, normalized size = 0.85 \begin {gather*} \frac {-2 x \cosh (x)+4 \sinh (x)}{\sqrt {\sinh (x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 1.75, size = 0, normalized size = 0.00 \[\int \frac {x}{\sinh \left (x \right )^{\frac {3}{2}}}-x \left (\sqrt {\sinh }\left (x \right )\right )\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \int \left (- \frac {x}{\sinh ^{\frac {3}{2}}{\left (x \right )}}\right )\, dx - \int x \sqrt {\sinh {\left (x \right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.17, size = 38, normalized size = 1.90 \begin {gather*} -\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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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