Optimal. Leaf size=36 \[ \frac {2 x^2 \sinh (x)}{\sqrt {\cosh (x)}}-8 x \sqrt {\cosh (x)}-16 i E\left (\left .\frac {i x}{2}\right |2\right ) \]
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Rubi [A] time = 0.09, antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {3316, 2639} \[ \frac {2 x^2 \sinh (x)}{\sqrt {\cosh (x)}}-8 x \sqrt {\cosh (x)}-16 i E\left (\left .\frac {i x}{2}\right |2\right ) \]
Antiderivative was successfully verified.
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Rule 2639
Rule 3316
Rubi steps
\begin {align*} \int \left (\frac {x^2}{\cosh ^{\frac {3}{2}}(x)}+x^2 \sqrt {\cosh (x)}\right ) \, dx &=\int \frac {x^2}{\cosh ^{\frac {3}{2}}(x)} \, dx+\int x^2 \sqrt {\cosh (x)} \, dx\\ &=-8 x \sqrt {\cosh (x)}+\frac {2 x^2 \sinh (x)}{\sqrt {\cosh (x)}}+8 \int \sqrt {\cosh (x)} \, dx\\ &=-8 x \sqrt {\cosh (x)}-16 i E\left (\left .\frac {i x}{2}\right |2\right )+\frac {2 x^2 \sinh (x)}{\sqrt {\cosh (x)}}\\ \end {align*}
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Mathematica [C] time = 0.19, size = 76, normalized size = 2.11 \[ \frac {4 \sqrt {\cosh (x)} (\sinh (x)+\cosh (x)) \left (8 \, _2F_1\left (-\frac {1}{4},\frac {1}{2};\frac {3}{4};-e^{2 x}\right ) (\sinh (x)-\cosh (x)) \sqrt {\sinh (2 x)+\cosh (2 x)+1}+x^2 \sinh (x)-4 (x-2) \cosh (x)\right )}{e^{2 x}+1} \]
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 \[ \int x^{2} \sqrt {\cosh \relax (x)} + \frac {x^{2}}{\cosh \relax (x)^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {x^{2}}{\cosh \relax (x )^{\frac {3}{2}}}+x^{2} \left (\sqrt {\cosh }\relax (x )\right )\, dx \]
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 \[ \int x^{2} \sqrt {\cosh \relax (x)} + \frac {x^{2}}{\cosh \relax (x)^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.03 \[ \int x^2\,\sqrt {\mathrm {cosh}\relax (x)}+\frac {x^2}{{\mathrm {cosh}\relax (x)}^{3/2}} \,d x \]
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 \[ \int \frac {x^{2} \left (\cosh ^{2}{\relax (x )} + 1\right )}{\cosh ^{\frac {3}{2}}{\relax (x )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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