Integrand size = 24, antiderivative size = 66 \[ \int \left (\frac {x^2}{\text {sech}^{\frac {3}{2}}(x)}-\frac {1}{3} x^2 \sqrt {\text {sech}(x)}\right ) \, dx=-\frac {8 x}{9 \text {sech}^{\frac {3}{2}}(x)}-\frac {16}{27} i \sqrt {\cosh (x)} \operatorname {EllipticF}\left (\frac {i x}{2},2\right ) \sqrt {\text {sech}(x)}+\frac {16 \sinh (x)}{27 \sqrt {\text {sech}(x)}}+\frac {2 x^2 \sinh (x)}{3 \sqrt {\text {sech}(x)}} \]
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Time = 0.12 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {4273, 4274, 3854, 3856, 2720} \[ \int \left (\frac {x^2}{\text {sech}^{\frac {3}{2}}(x)}-\frac {1}{3} x^2 \sqrt {\text {sech}(x)}\right ) \, dx=\frac {2 x^2 \sinh (x)}{3 \sqrt {\text {sech}(x)}}-\frac {8 x}{9 \text {sech}^{\frac {3}{2}}(x)}+\frac {16 \sinh (x)}{27 \sqrt {\text {sech}(x)}}-\frac {16}{27} i \sqrt {\cosh (x)} \sqrt {\text {sech}(x)} \operatorname {EllipticF}\left (\frac {i x}{2},2\right ) \]
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Rule 2720
Rule 3854
Rule 3856
Rule 4273
Rule 4274
Rubi steps \begin{align*} \text {integral}& = -\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)} \operatorname {EllipticF}\left (\frac {i x}{2},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*}
Time = 0.09 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.74 \[ \int \left (\frac {x^2}{\text {sech}^{\frac {3}{2}}(x)}-\frac {1}{3} x^2 \sqrt {\text {sech}(x)}\right ) \, dx=\frac {2}{27} \sqrt {\text {sech}(x)} \left (-8 i \sqrt {\cosh (x)} \operatorname {EllipticF}\left (\frac {i x}{2},2\right )+\cosh (x) \left (-12 x \cosh (x)+\left (8+9 x^2\right ) \sinh (x)\right )\right ) \]
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\[\int \left (\frac {x^{2}}{\operatorname {sech}\left (x \right )^{\frac {3}{2}}}-\frac {x^{2} \sqrt {\operatorname {sech}\left (x \right )}}{3}\right )d x\]
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Exception generated. \[ \int \left (\frac {x^2}{\text {sech}^{\frac {3}{2}}(x)}-\frac {1}{3} x^2 \sqrt {\text {sech}(x)}\right ) \, dx=\text {Exception raised: TypeError} \]
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\[ \int \left (\frac {x^2}{\text {sech}^{\frac {3}{2}}(x)}-\frac {1}{3} x^2 \sqrt {\text {sech}(x)}\right ) \, dx=- \frac {\int \left (- \frac {3 x^{2}}{\operatorname {sech}^{\frac {3}{2}}{\left (x \right )}}\right )\, dx + \int x^{2} \sqrt {\operatorname {sech}{\left (x \right )}}\, dx}{3} \]
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\[ \int \left (\frac {x^2}{\text {sech}^{\frac {3}{2}}(x)}-\frac {1}{3} x^2 \sqrt {\text {sech}(x)}\right ) \, dx=\int { -\frac {1}{3} \, x^{2} \sqrt {\operatorname {sech}\left (x\right )} + \frac {x^{2}}{\operatorname {sech}\left (x\right )^{\frac {3}{2}}} \,d x } \]
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\[ \int \left (\frac {x^2}{\text {sech}^{\frac {3}{2}}(x)}-\frac {1}{3} x^2 \sqrt {\text {sech}(x)}\right ) \, dx=\int { -\frac {1}{3} \, x^{2} \sqrt {\operatorname {sech}\left (x\right )} + \frac {x^{2}}{\operatorname {sech}\left (x\right )^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \left (\frac {x^2}{\text {sech}^{\frac {3}{2}}(x)}-\frac {1}{3} x^2 \sqrt {\text {sech}(x)}\right ) \, dx=-\int \frac {x^2\,\sqrt {\frac {1}{\mathrm {cosh}\left (x\right )}}}{3}-\frac {x^2}{{\left (\frac {1}{\mathrm {cosh}\left (x\right )}\right )}^{3/2}} \,d x \]
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