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 = 0.05, 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 = {3396}
\begin {gather*} \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 {gather*}
Antiderivative was successfully verified.
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Rule 3396
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 = 0.45, size = 64, normalized size = 1.36 \begin {gather*} \frac {1}{5} \sqrt {\cosh (x)} \left (-\frac {12 \sinh ^2(x)}{\sqrt {-1+\cosh (x)} (1+\cosh (x))^{3/2} \sqrt {\tanh ^2\left (\frac {x}{2}\right )}}+6 x \tanh (x)+\text {sech}^2(x) \left (\frac {4}{3}+2 x \tanh (x)\right )\right ) \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [F]
time = 1.61, size = 0, normalized size = 0.00 \[\int \frac {x}{\cosh \left (x \right )^{\frac {7}{2}}}+\frac {3 x \left (\sqrt {\cosh }\left (x \right )\right )}{5}\, dx\]
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 \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(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \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 = 1.08, size = 110, normalized size = 2.34 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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