Optimal. Leaf size=58 \[ -\frac {2 x^2 \cosh (x)}{\sqrt {\sinh (x)}}+8 x \sqrt {\sinh (x)}-\frac {16 i \sqrt {\sinh (x)} E\left (\left .\frac {\pi }{4}-\frac {i x}{2}\right |2\right )}{\sqrt {i \sinh (x)}} \]
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Rubi [A] time = 0.12, 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 = {3316, 2640, 2639} \[ -\frac {2 x^2 \cosh (x)}{\sqrt {\sinh (x)}}+8 x \sqrt {\sinh (x)}-\frac {16 i \sqrt {\sinh (x)} E\left (\left .\frac {\pi }{4}-\frac {i x}{2}\right |2\right )}{\sqrt {i \sinh (x)}} \]
Antiderivative was successfully verified.
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Rule 2639
Rule 2640
Rule 3316
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] time = 1.23, size = 68, normalized size = 1.17 \[ -\frac {2 \left (-8 \sqrt {2} (\sinh (x)-\cosh (x)) \sqrt {-\sinh (x) (\sinh (x)+\cosh (x))} \, _2F_1\left (-\frac {1}{4},\frac {1}{2};\frac {3}{4};\cosh (2 x)+\sinh (2 x)\right )+x^2 \cosh (x)-4 (x-2) \sinh (x)\right )}{\sqrt {\sinh (x)}} \]
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 \[ \int -x^{2} \sqrt {\sinh \relax (x)} + \frac {x^{2}}{\sinh \relax (x)^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {x^{2}}{\sinh \relax (x )^{\frac {3}{2}}}-x^{2} \left (\sqrt {\sinh }\relax (x )\right )\, dx \]
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 \[ \int -x^{2} \sqrt {\sinh \relax (x)} + \frac {x^{2}}{\sinh \relax (x)^{\frac {3}{2}}}\,{d x} \]
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 \[ -\int x^2\,\sqrt {\mathrm {sinh}\relax (x)}-\frac {x^2}{{\mathrm {sinh}\relax (x)}^{3/2}} \,d x \]
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 \[ - \int \left (- \frac {x^{2}}{\sinh ^{\frac {3}{2}}{\relax (x )}}\right )\, dx - \int x^{2} \sqrt {\sinh {\relax (x )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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