Optimal. Leaf size=47 \[ -\frac {2 x \cosh (x)}{5 \sinh ^{\frac {5}{2}}(x)}-\frac {4}{15 \sinh ^{\frac {3}{2}}(x)}+\frac {6 x \cosh (x)}{5 \sqrt {\sinh (x)}}-\frac {12 \sqrt {\sinh (x)}}{5} \]
[Out]
________________________________________________________________________________________
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 \sinh ^{\frac {3}{2}}(x)}-\frac {12 \sqrt {\sinh (x)}}{5}-\frac {2 x \cosh (x)}{5 \sinh ^{\frac {5}{2}}(x)}+\frac {6 x \cosh (x)}{5 \sqrt {\sinh (x)}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 3396
Rubi steps
\begin {align*} \int \left (\frac {x}{\sinh ^{\frac {7}{2}}(x)}+\frac {3}{5} x \sqrt {\sinh (x)}\right ) \, dx &=\frac {3}{5} \int x \sqrt {\sinh (x)} \, dx+\int \frac {x}{\sinh ^{\frac {7}{2}}(x)} \, dx\\ &=-\frac {2 x \cosh (x)}{5 \sinh ^{\frac {5}{2}}(x)}-\frac {4}{15 \sinh ^{\frac {3}{2}}(x)}-\frac {3}{5} \int \frac {x}{\sinh ^{\frac {3}{2}}(x)} \, dx+\frac {3}{5} \int x \sqrt {\sinh (x)} \, dx\\ &=-\frac {2 x \cosh (x)}{5 \sinh ^{\frac {5}{2}}(x)}-\frac {4}{15 \sinh ^{\frac {3}{2}}(x)}+\frac {6 x \cosh (x)}{5 \sqrt {\sinh (x)}}-\frac {12 \sqrt {\sinh (x)}}{5}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.08, size = 33, normalized size = 0.70 \begin {gather*} \frac {-21 x \cosh (x)+9 x \cosh (3 x)+46 \sinh (x)-18 \sinh (3 x)}{30 \sinh ^{\frac {5}{2}}(x)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 1.81, size = 0, normalized size = 0.00 \[\int \frac {x}{\sinh \left (x \right )^{\frac {7}{2}}}+\frac {3 x \left (\sqrt {\sinh }\left (x \right )\right )}{5}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {\int \frac {5 x}{\sinh ^{\frac {7}{2}}{\left (x \right )}}\, dx + \int 3 x \sqrt {\sinh {\left (x \right )}}\, dx}{5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.29, size = 111, normalized size = 2.36 \begin {gather*} \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 {{\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 {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.
[In]
[Out]
________________________________________________________________________________________