Optimal. Leaf size=24 \[ -\frac {4}{9 \text {sech}^{\frac {3}{2}}(x)}+\frac {2 x \sinh (x)}{3 \sqrt {\text {sech}(x)}} \]
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Rubi [A]
time = 0.06, antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {4272, 4274}
\begin {gather*} \frac {2 x \sinh (x)}{3 \sqrt {\text {sech}(x)}}-\frac {4}{9 \text {sech}^{\frac {3}{2}}(x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 4272
Rule 4274
Rubi steps
\begin {align*} \int \left (\frac {x}{\text {sech}^{\frac {3}{2}}(x)}-\frac {1}{3} x \sqrt {\text {sech}(x)}\right ) \, dx &=-\left (\frac {1}{3} \int x \sqrt {\text {sech}(x)} \, dx\right )+\int \frac {x}{\text {sech}^{\frac {3}{2}}(x)} \, dx\\ &=-\frac {4}{9 \text {sech}^{\frac {3}{2}}(x)}+\frac {2 x \sinh (x)}{3 \sqrt {\text {sech}(x)}}+\frac {1}{3} \int x \sqrt {\text {sech}(x)} \, dx-\frac {1}{3} \left (\sqrt {\cosh (x)} \sqrt {\text {sech}(x)}\right ) \int \frac {x}{\sqrt {\cosh (x)}} \, dx\\ &=-\frac {4}{9 \text {sech}^{\frac {3}{2}}(x)}+\frac {2 x \sinh (x)}{3 \sqrt {\text {sech}(x)}}\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 17, normalized size = 0.71 \begin {gather*} \frac {2 (-2+3 x \tanh (x))}{9 \text {sech}^{\frac {3}{2}}(x)} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.07, size = 0, normalized size = 0.00 \[\int \frac {x}{\mathrm {sech}\left (x \right )^{\frac {3}{2}}}-\frac {x \sqrt {\mathrm {sech}\left (x \right )}}{3}\, dx\]
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}{\operatorname {sech}^{\frac {3}{2}}{\left (x \right )}}\right )\, dx + \int x \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 = -0.04 \begin {gather*} -\int \frac {x\,\sqrt {\frac {1}{\mathrm {cosh}\left (x\right )}}}{3}-\frac {x}{{\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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