Optimal. Leaf size=58 \[ -\frac {2 x^2 \cosh (x)}{\sqrt {\sinh (x)}}+8 x \sqrt {\sinh (x)}-\frac {16 i E\left (\left .\frac {\pi }{4}-\frac {i x}{2}\right |2\right ) \sqrt {\sinh (x)}}{\sqrt {i \sinh (x)}} \]
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Rubi [A]
time = 0.08, antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {3397, 2721,
2719} \begin {gather*} -\frac {2 x^2 \cosh (x)}{\sqrt {\sinh (x)}}+8 x \sqrt {\sinh (x)}-\frac {16 i \sqrt {\sinh (x)} E\left (\left .\frac {\pi }{4}-\frac {i x}{2}\right |2\right )}{\sqrt {i \sinh (x)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2719
Rule 2721
Rule 3397
Rubi steps
\begin {align*} \int \left (\frac {x^2}{\sinh ^{\frac {3}{2}}(x)}-x^2 \sqrt {\sinh (x)}\right ) \, dx &=\int \frac {x^2}{\sinh ^{\frac {3}{2}}(x)} \, dx-\int x^2 \sqrt {\sinh (x)} \, dx\\ &=-\frac {2 x^2 \cosh (x)}{\sqrt {\sinh (x)}}+8 x \sqrt {\sinh (x)}-8 \int \sqrt {\sinh (x)} \, dx\\ &=-\frac {2 x^2 \cosh (x)}{\sqrt {\sinh (x)}}+8 x \sqrt {\sinh (x)}-\frac {\left (8 \sqrt {\sinh (x)}\right ) \int \sqrt {i \sinh (x)} \, dx}{\sqrt {i \sinh (x)}}\\ &=-\frac {2 x^2 \cosh (x)}{\sqrt {\sinh (x)}}+8 x \sqrt {\sinh (x)}-\frac {16 i E\left (\left .\frac {\pi }{4}-\frac {i x}{2}\right |2\right ) \sqrt {\sinh (x)}}{\sqrt {i \sinh (x)}}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in
optimal.
time = 0.85, size = 68, normalized size = 1.17 \begin {gather*} -\frac {2 \left (x^2 \cosh (x)-4 (-2+x) \sinh (x)-8 \sqrt {2} \, _2F_1\left (-\frac {1}{4},\frac {1}{2};\frac {3}{4};\cosh (2 x)+\sinh (2 x)\right ) (-\cosh (x)+\sinh (x)) \sqrt {-\sinh (x) (\cosh (x)+\sinh (x))}\right )}{\sqrt {\sinh (x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 1.68, size = 0, normalized size = 0.00 \[\int \frac {x^{2}}{\sinh \left (x \right )^{\frac {3}{2}}}-x^{2} \left (\sqrt {\sinh }\left (x \right )\right )\, dx\]
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^{2}}{\sinh ^{\frac {3}{2}}{\left (x \right )}}\right )\, dx - \int x^{2} \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 [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} -\int x^2\,\sqrt {\mathrm {sinh}\left (x\right )}-\frac {x^2}{{\mathrm {sinh}\left (x\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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