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 = 0.05, 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 = {3315} \[ \frac {4}{3 \sqrt {\cosh (x)}}+\frac {2 x \sinh (x)}{3 \cosh ^{\frac {3}{2}}(x)} \]
Antiderivative was successfully verified.
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Rule 3315
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 = 0.08, size = 16, normalized size = 0.67 \[ \frac {2 (x \tanh (x)+2)}{3 \sqrt {\cosh (x)}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.50, size = 109, normalized size = 4.54 \[ \frac {4 \, {\left ({\left (x + 2\right )} \cosh \relax (x)^{3} + 3 \, {\left (x + 2\right )} \cosh \relax (x) \sinh \relax (x)^{2} + {\left (x + 2\right )} \sinh \relax (x)^{3} - {\left (x - 2\right )} \cosh \relax (x) + {\left (3 \, {\left (x + 2\right )} \cosh \relax (x)^{2} - x + 2\right )} \sinh \relax (x)\right )} \sqrt {\cosh \relax (x)}}{3 \, {\left (\cosh \relax (x)^{4} + 4 \, \cosh \relax (x) \sinh \relax (x)^{3} + \sinh \relax (x)^{4} + 2 \, {\left (3 \, \cosh \relax (x)^{2} + 1\right )} \sinh \relax (x)^{2} + 2 \, \cosh \relax (x)^{2} + 4 \, {\left (\cosh \relax (x)^{3} + \cosh \relax (x)\right )} \sinh \relax (x) + 1\right )}} \]
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 \[ \int -\frac {x}{3 \, \sqrt {\cosh \relax (x)}} + \frac {x}{\cosh \relax (x)^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x}{\cosh \relax (x )^{\frac {5}{2}}}-\frac {x}{3 \sqrt {\cosh \relax (x )}}\, dx \]
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 \[ \int -\frac {x}{3 \, \sqrt {\cosh \relax (x)}} + \frac {x}{\cosh \relax (x)^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.97, size = 42, normalized size = 1.75 \[ \frac {4\,{\mathrm {e}}^x\,\sqrt {\frac {{\mathrm {e}}^{-x}}{2}+\frac {{\mathrm {e}}^x}{2}}\,\left (2\,{\mathrm {e}}^{2\,x}-x+x\,{\mathrm {e}}^{2\,x}+2\right )}{3\,{\left ({\mathrm {e}}^{2\,x}+1\right )}^2} \]
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 \[ - \frac {\int \left (- \frac {3 x}{\cosh ^{\frac {5}{2}}{\relax (x )}}\right )\, dx + \int \frac {x}{\sqrt {\cosh {\relax (x )}}}\, dx}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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