Optimal. Leaf size=66 \[ -\frac {8 x}{9 \text {sech}^{\frac {3}{2}}(x)}-\frac {16}{27} i \sqrt {\cosh (x)} F\left (\left .\frac {i x}{2}\right |2\right ) \sqrt {\text {sech}(x)}+\frac {16 \sinh (x)}{27 \sqrt {\text {sech}(x)}}+\frac {2 x^2 \sinh (x)}{3 \sqrt {\text {sech}(x)}} \]
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Rubi [A]
time = 0.12, antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {4273, 4274,
3854, 3856, 2720} \begin {gather*} \frac {2 x^2 \sinh (x)}{3 \sqrt {\text {sech}(x)}}-\frac {8 x}{9 \text {sech}^{\frac {3}{2}}(x)}+\frac {16 \sinh (x)}{27 \sqrt {\text {sech}(x)}}-\frac {16}{27} i \sqrt {\cosh (x)} \sqrt {\text {sech}(x)} F\left (\left .\frac {i x}{2}\right |2\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 2720
Rule 3854
Rule 3856
Rule 4273
Rule 4274
Rubi steps
\begin {align*} \int \left (\frac {x^2}{\text {sech}^{\frac {3}{2}}(x)}-\frac {1}{3} x^2 \sqrt {\text {sech}(x)}\right ) \, dx &=-\left (\frac {1}{3} \int x^2 \sqrt {\text {sech}(x)} \, dx\right )+\int \frac {x^2}{\text {sech}^{\frac {3}{2}}(x)} \, dx\\ &=-\frac {8 x}{9 \text {sech}^{\frac {3}{2}}(x)}+\frac {2 x^2 \sinh (x)}{3 \sqrt {\text {sech}(x)}}+\frac {1}{3} \int x^2 \sqrt {\text {sech}(x)} \, dx+\frac {8}{9} \int \frac {1}{\text {sech}^{\frac {3}{2}}(x)} \, dx-\frac {1}{3} \left (\sqrt {\cosh (x)} \sqrt {\text {sech}(x)}\right ) \int \frac {x^2}{\sqrt {\cosh (x)}} \, dx\\ &=-\frac {8 x}{9 \text {sech}^{\frac {3}{2}}(x)}+\frac {16 \sinh (x)}{27 \sqrt {\text {sech}(x)}}+\frac {2 x^2 \sinh (x)}{3 \sqrt {\text {sech}(x)}}+\frac {8}{27} \int \sqrt {\text {sech}(x)} \, dx\\ &=-\frac {8 x}{9 \text {sech}^{\frac {3}{2}}(x)}+\frac {16 \sinh (x)}{27 \sqrt {\text {sech}(x)}}+\frac {2 x^2 \sinh (x)}{3 \sqrt {\text {sech}(x)}}+\frac {1}{27} \left (8 \sqrt {\cosh (x)} \sqrt {\text {sech}(x)}\right ) \int \frac {1}{\sqrt {\cosh (x)}} \, dx\\ &=-\frac {8 x}{9 \text {sech}^{\frac {3}{2}}(x)}-\frac {16}{27} i \sqrt {\cosh (x)} F\left (\left .\frac {i x}{2}\right |2\right ) \sqrt {\text {sech}(x)}+\frac {16 \sinh (x)}{27 \sqrt {\text {sech}(x)}}+\frac {2 x^2 \sinh (x)}{3 \sqrt {\text {sech}(x)}}\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 55, normalized size = 0.83 \begin {gather*} \frac {1}{27} \sqrt {\text {sech}(x)} \left (-12 x-12 x \cosh (2 x)-16 i \sqrt {\cosh (x)} F\left (\left .\frac {i x}{2}\right |2\right )+8 \sinh (2 x)+9 x^2 \sinh (2 x)\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.06, size = 0, normalized size = 0.00 \[\int \frac {x^{2}}{\mathrm {sech}\left (x \right )^{\frac {3}{2}}}-\frac {x^{2} \sqrt {\mathrm {sech}\left (x \right )}}{3}\, dx\]
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*} - \frac {\int \left (- \frac {3 x^{2}}{\operatorname {sech}^{\frac {3}{2}}{\left (x \right )}}\right )\, dx + \int x^{2} \sqrt {\operatorname {sech}{\left (x \right )}}\, dx}{3} \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 \frac {x^2\,\sqrt {\frac {1}{\mathrm {cosh}\left (x\right )}}}{3}-\frac {x^2}{{\left (\frac {1}{\mathrm {cosh}\left (x\right )}\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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