Optimal. Leaf size=20 \[ -4 \sqrt {\cosh (x)}+\frac {2 x \sinh (x)}{\sqrt {\cosh (x)}} \]
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Rubi [A]
time = 0.04, antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 1, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {3396}
\begin {gather*} \frac {2 x \sinh (x)}{\sqrt {\cosh (x)}}-4 \sqrt {\cosh (x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 3396
Rubi steps
\begin {align*} \int \left (\frac {x}{\cosh ^{\frac {3}{2}}(x)}+x \sqrt {\cosh (x)}\right ) \, dx &=\int \frac {x}{\cosh ^{\frac {3}{2}}(x)} \, dx+\int x \sqrt {\cosh (x)} \, dx\\ &=-4 \sqrt {\cosh (x)}+\frac {2 x \sinh (x)}{\sqrt {\cosh (x)}}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(46\) vs. \(2(20)=40\).
time = 0.27, size = 46, normalized size = 2.30 \begin {gather*} \frac {2 \sinh (x) \left (x-\frac {2 \cosh (x) \sinh (x) \sqrt {\tanh ^2\left (\frac {x}{2}\right )}}{(-1+\cosh (x))^{3/2} \sqrt {1+\cosh (x)}}\right )}{\sqrt {\cosh (x)}} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [F]
time = 1.20, size = 0, normalized size = 0.00 \[\int \frac {x}{\cosh \left (x \right )^{\frac {3}{2}}}+x \left (\sqrt {\cosh }\left (x \right )\right )\, dx\]
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 [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*} \int \frac {x \left (\cosh ^{2}{\left (x \right )} + 1\right )}{\cosh ^{\frac {3}{2}}{\left (x \right )}}\, dx \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 = 0.97, size = 39, normalized size = 1.95 \begin {gather*} -\frac {2\,\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 )}{{\mathrm {e}}^{2\,x}+1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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