Optimal. Leaf size=24 \[ -\frac {2 x \cosh (x)}{3 \sinh ^{\frac {3}{2}}(x)}-\frac {4}{3 \sqrt {\sinh (x)}} \]
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Rubi [A]
time = 0.04, 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 {\sinh (x)}}-\frac {2 x \cosh (x)}{3 \sinh ^{\frac {3}{2}}(x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 3396
Rubi steps
\begin {align*} \int \left (\frac {x}{\sinh ^{\frac {5}{2}}(x)}+\frac {x}{3 \sqrt {\sinh (x)}}\right ) \, dx &=\frac {1}{3} \int \frac {x}{\sqrt {\sinh (x)}} \, dx+\int \frac {x}{\sinh ^{\frac {5}{2}}(x)} \, dx\\ &=-\frac {2 x \cosh (x)}{3 \sinh ^{\frac {3}{2}}(x)}-\frac {4}{3 \sqrt {\sinh (x)}}\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 22, normalized size = 0.92 \begin {gather*} \frac {1}{6} (-8 \text {csch}(x)-4 x \coth (x) \text {csch}(x)) \sqrt {\sinh (x)} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 1.65, size = 0, normalized size = 0.00 \[\int \frac {x}{\sinh \left (x \right )^{\frac {5}{2}}}+\frac {x}{3 \sqrt {\sinh \left (x \right )}}\, dx\]
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. 108 vs.
\(2 (16) = 32\).
time = 0.35, size = 108, normalized size = 4.50 \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 {\sinh \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 \frac {3 x}{\sinh ^{\frac {5}{2}}{\left (x \right )}}\, dx + \int \frac {x}{\sqrt {\sinh {\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 = 0.15, size = 40, normalized size = 1.67 \begin {gather*} -\frac {4\,{\mathrm {e}}^x\,\sqrt {\frac {{\mathrm {e}}^x}{2}-\frac {{\mathrm {e}}^{-x}}{2}}\,\left (x+2\,{\mathrm {e}}^{2\,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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