Optimal. Leaf size=66 \[ \frac {2 x^2 \sinh (x)}{3 \sqrt {\text {sech}(x)}}-\frac {8 x}{9 \text {sech}^{\frac {3}{2}}(x)}+\frac {16 \sinh (x)}{27 \sqrt {\text {sech}(x)}}-\frac {16}{27} i \sqrt {\cosh (x)} \sqrt {\text {sech}(x)} F\left (\left .\frac {i x}{2}\right |2\right ) \]
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Rubi [A] time = 0.16, 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 = {4188, 4189, 3769, 3771, 2641} \[ \frac {2 x^2 \sinh (x)}{3 \sqrt {\text {sech}(x)}}-\frac {8 x}{9 \text {sech}^{\frac {3}{2}}(x)}+\frac {16 \sinh (x)}{27 \sqrt {\text {sech}(x)}}-\frac {16}{27} i \sqrt {\cosh (x)} \sqrt {\text {sech}(x)} F\left (\left .\frac {i x}{2}\right |2\right ) \]
Antiderivative was successfully verified.
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Rule 2641
Rule 3769
Rule 3771
Rule 4188
Rule 4189
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 = 0.10, size = 55, normalized size = 0.83 \[ \frac {1}{27} \sqrt {\text {sech}(x)} \left (9 x^2 \sinh (2 x)-12 x+8 \sinh (2 x)-12 x \cosh (2 x)-16 i \sqrt {\cosh (x)} F\left (\left .\frac {i x}{2}\right |2\right )\right ) \]
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 \[ \int -\frac {1}{3} \, x^{2} \sqrt {\operatorname {sech}\relax (x)} + \frac {x^{2}}{\operatorname {sech}\relax (x)^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.01, size = 0, normalized size = 0.00 \[ \int \frac {x^{2}}{\mathrm {sech}\relax (x )^{\frac {3}{2}}}-\frac {x^{2} \sqrt {\mathrm {sech}\relax (x )}}{3}\, dx \]
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 \[ \int -\frac {1}{3} \, x^{2} \sqrt {\operatorname {sech}\relax (x)} + \frac {x^{2}}{\operatorname {sech}\relax (x)^{\frac {3}{2}}}\,{d x} \]
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 \[ -\int \frac {x^2\,\sqrt {\frac {1}{\mathrm {cosh}\relax (x)}}}{3}-\frac {x^2}{{\left (\frac {1}{\mathrm {cosh}\relax (x)}\right )}^{3/2}} \,d x \]
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 \[ - \frac {\int \left (- \frac {3 x^{2}}{\operatorname {sech}^{\frac {3}{2}}{\relax (x )}}\right )\, dx + \int x^{2} \sqrt {\operatorname {sech}{\relax (x )}}\, dx}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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