Optimal. Leaf size=24 \[ \frac {4}{3 \sqrt {\cosh (x)}}+\frac {2 x \sinh (x)}{3 \cosh ^{\frac {3}{2}}(x)} \]
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Rubi [A]
time = 0.03, antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {3396}
\begin {gather*} \frac {4}{3 \sqrt {\cosh (x)}}+\frac {2 x \sinh (x)}{3 \cosh ^{\frac {3}{2}}(x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 3396
Rubi steps
\begin {align*} \int \left (\frac {x}{\cosh ^{\frac {5}{2}}(x)}-\frac {x}{3 \sqrt {\cosh (x)}}\right ) \, dx &=-\left (\frac {1}{3} \int \frac {x}{\sqrt {\cosh (x)}} \, dx\right )+\int \frac {x}{\cosh ^{\frac {5}{2}}(x)} \, dx\\ &=\frac {4}{3 \sqrt {\cosh (x)}}+\frac {2 x \sinh (x)}{3 \cosh ^{\frac {3}{2}}(x)}\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 16, normalized size = 0.67 \begin {gather*} \frac {2 (2+x \tanh (x))}{3 \sqrt {\cosh (x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 1.26, size = 0, normalized size = 0.00 \[\int \frac {x}{\cosh \left (x \right )^{\frac {5}{2}}}-\frac {x}{3 \sqrt {\cosh \left (x \right )}}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 109 vs.
\(2 (16) = 32\).
time = 0.38, size = 109, normalized size = 4.54 \begin {gather*} \frac {4 \, {\left ({\left (x + 2\right )} \cosh \left (x\right )^{3} + 3 \, {\left (x + 2\right )} \cosh \left (x\right ) \sinh \left (x\right )^{2} + {\left (x + 2\right )} \sinh \left (x\right )^{3} - {\left (x - 2\right )} \cosh \left (x\right ) + {\left (3 \, {\left (x + 2\right )} \cosh \left (x\right )^{2} - x + 2\right )} \sinh \left (x\right )\right )} \sqrt {\cosh \left (x\right )}}{3 \, {\left (\cosh \left (x\right )^{4} + 4 \, \cosh \left (x\right ) \sinh \left (x\right )^{3} + \sinh \left (x\right )^{4} + 2 \, {\left (3 \, \cosh \left (x\right )^{2} + 1\right )} \sinh \left (x\right )^{2} + 2 \, \cosh \left (x\right )^{2} + 4 \, {\left (\cosh \left (x\right )^{3} + \cosh \left (x\right )\right )} \sinh \left (x\right ) + 1\right )}} \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}{\cosh ^{\frac {5}{2}}{\left (x \right )}}\right )\, dx + \int \frac {x}{\sqrt {\cosh {\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 [B]
time = 0.94, size = 42, normalized size = 1.75 \begin {gather*} \frac {4\,{\mathrm {e}}^x\,\sqrt {\frac {{\mathrm {e}}^{-x}}{2}+\frac {{\mathrm {e}}^x}{2}}\,\left (2\,{\mathrm {e}}^{2\,x}-x+x\,{\mathrm {e}}^{2\,x}+2\right )}{3\,{\left ({\mathrm {e}}^{2\,x}+1\right )}^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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