Optimal. Leaf size=20 \[ 4 \sqrt {\sinh (x)}-\frac {2 x \cosh (x)}{\sqrt {\sinh (x)}} \]
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Rubi [A] time = 0.06, 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 = {3315} \[ 4 \sqrt {\sinh (x)}-\frac {2 x \cosh (x)}{\sqrt {\sinh (x)}} \]
Antiderivative was successfully verified.
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Rule 3315
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 = 0.12, size = 17, normalized size = 0.85 \[ \frac {4 \sinh (x)-2 x \cosh (x)}{\sqrt {\sinh (x)}} \]
Antiderivative was successfully verified.
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fricas [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
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 \[ \int -x \sqrt {\sinh \relax (x)} + \frac {x}{\sinh \relax (x)^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F(-2)] time = 180.00, size = 0, normalized size = 0.00 \[ \int \frac {x}{\sinh \relax (x )^{\frac {3}{2}}}-x \left (\sqrt {\sinh }\relax (x )\right )\, dx \]
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 \[ \int -x \sqrt {\sinh \relax (x)} + \frac {x}{\sinh \relax (x)^{\frac {3}{2}}}\,{d x} \]
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 \[ -\frac {2\,\sqrt {\frac {{\mathrm {e}}^x}{2}-\frac {{\mathrm {e}}^{-x}}{2}}\,\left (x-2\,{\mathrm {e}}^{2\,x}+x\,{\mathrm {e}}^{2\,x}+2\right )}{{\mathrm {e}}^{2\,x}-1} \]
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 \[ - \int \left (- \frac {x}{\sinh ^{\frac {3}{2}}{\relax (x )}}\right )\, dx - \int x \sqrt {\sinh {\relax (x )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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