Optimal. Leaf size=47 \[ \frac {4}{15 \cosh ^{\frac {3}{2}}(x)}-\frac {12 \sqrt {\cosh (x)}}{5}+\frac {2 x \sinh (x)}{5 \cosh ^{\frac {5}{2}}(x)}+\frac {6 x \sinh (x)}{5 \sqrt {\cosh (x)}} \]
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Rubi [A] time = 0.07, antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {3315} \[ \frac {4}{15 \cosh ^{\frac {3}{2}}(x)}-\frac {12 \sqrt {\cosh (x)}}{5}+\frac {2 x \sinh (x)}{5 \cosh ^{\frac {5}{2}}(x)}+\frac {6 x \sinh (x)}{5 \sqrt {\cosh (x)}} \]
Antiderivative was successfully verified.
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Rule 3315
Rubi steps
\begin {align*} \int \left (\frac {x}{\cosh ^{\frac {7}{2}}(x)}+\frac {3}{5} x \sqrt {\cosh (x)}\right ) \, dx &=\frac {3}{5} \int x \sqrt {\cosh (x)} \, dx+\int \frac {x}{\cosh ^{\frac {7}{2}}(x)} \, dx\\ &=\frac {4}{15 \cosh ^{\frac {3}{2}}(x)}+\frac {2 x \sinh (x)}{5 \cosh ^{\frac {5}{2}}(x)}+\frac {3}{5} \int \frac {x}{\cosh ^{\frac {3}{2}}(x)} \, dx+\frac {3}{5} \int x \sqrt {\cosh (x)} \, dx\\ &=\frac {4}{15 \cosh ^{\frac {3}{2}}(x)}-\frac {12 \sqrt {\cosh (x)}}{5}+\frac {2 x \sinh (x)}{5 \cosh ^{\frac {5}{2}}(x)}+\frac {6 x \sinh (x)}{5 \sqrt {\cosh (x)}}\\ \end {align*}
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Mathematica [A] time = 0.61, size = 64, normalized size = 1.36 \[ \frac {1}{5} \sqrt {\cosh (x)} \left (6 x \tanh (x)+\left (2 x \tanh (x)+\frac {4}{3}\right ) \text {sech}^2(x)-\frac {12 \sinh ^2(x)}{\sqrt {\cosh (x)-1} (\cosh (x)+1)^{3/2} \sqrt {\tanh ^2\left (\frac {x}{2}\right )}}\right ) \]
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 \[ \int \frac {3}{5} \, x \sqrt {\cosh \relax (x)} + \frac {x}{\cosh \relax (x)^{\frac {7}{2}}}\,{d x} \]
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 \[ \int \frac {x}{\cosh \relax (x )^{\frac {7}{2}}}+\frac {3 x \left (\sqrt {\cosh }\relax (x )\right )}{5}\, dx \]
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 \[ \int \frac {3}{5} \, x \sqrt {\cosh \relax (x)} + \frac {x}{\cosh \relax (x)^{\frac {7}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.11, size = 110, normalized size = 2.34 \[ \frac {{\mathrm {e}}^{2\,x}\,\left (\frac {8\,x}{5}+\frac {16}{15}\right )\,\sqrt {\frac {{\mathrm {e}}^{-x}}{2}+\frac {{\mathrm {e}}^x}{2}}}{{\left ({\mathrm {e}}^{2\,x}+1\right )}^2}-\left (\frac {6\,x}{5}+\frac {12}{5}\right )\,\sqrt {\frac {{\mathrm {e}}^{-x}}{2}+\frac {{\mathrm {e}}^x}{2}}+\frac {12\,x\,{\mathrm {e}}^{2\,x}\,\sqrt {\frac {{\mathrm {e}}^{-x}}{2}+\frac {{\mathrm {e}}^x}{2}}}{5\,\left ({\mathrm {e}}^{2\,x}+1\right )}-\frac {16\,x\,{\mathrm {e}}^{2\,x}\,\sqrt {\frac {{\mathrm {e}}^{-x}}{2}+\frac {{\mathrm {e}}^x}{2}}}{5\,{\left ({\mathrm {e}}^{2\,x}+1\right )}^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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