Optimal. Leaf size=20 \[ \frac {2 x \sinh (x)}{\sqrt {\cosh (x)}}-4 \sqrt {\cosh (x)} \]
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Rubi [A] time = 0.05, 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 = {3315} \[ \frac {2 x \sinh (x)}{\sqrt {\cosh (x)}}-4 \sqrt {\cosh (x)} \]
Antiderivative was successfully verified.
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Rule 3315
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] time = 0.36, size = 46, normalized size = 2.30 \[ \frac {2 \sinh (x) \left (x-\frac {2 \sinh (x) \cosh (x) \sqrt {\tanh ^2\left (\frac {x}{2}\right )}}{(\cosh (x)-1)^{3/2} \sqrt {\cosh (x)+1}}\right )}{\sqrt {\cosh (x)}} \]
Warning: Unable to verify antiderivative.
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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 \[ \int x \sqrt {\cosh \relax (x)} + \frac {x}{\cosh \relax (x)^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F(-2)] time = 180.00, size = 0, normalized size = 0.00 \[ \int \frac {x}{\cosh \relax (x )^{\frac {3}{2}}}+x \left (\sqrt {\cosh }\relax (x )\right )\, dx \]
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 \[ \int x \sqrt {\cosh \relax (x)} + \frac {x}{\cosh \relax (x)^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.15, size = 39, normalized size = 1.95 \[ -\frac {2\,\sqrt {\frac {{\mathrm {e}}^{-x}}{2}+\frac {{\mathrm {e}}^x}{2}}\,\left (x+2\,{\mathrm {e}}^{2\,x}-x\,{\mathrm {e}}^{2\,x}+2\right )}{{\mathrm {e}}^{2\,x}+1} \]
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 \[ \int \frac {x \left (\cosh ^{2}{\relax (x )} + 1\right )}{\cosh ^{\frac {3}{2}}{\relax (x )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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