Optimal. Leaf size=24 \[ -\frac {4}{3 \sqrt {\sinh (x)}}-\frac {2 x \cosh (x)}{3 \sinh ^{\frac {3}{2}}(x)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.06, 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} \[ -\frac {4}{3 \sqrt {\sinh (x)}}-\frac {2 x \cosh (x)}{3 \sinh ^{\frac {3}{2}}(x)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3315
Rubi steps
\begin {align*} \int \left (\frac {x}{\sinh ^{\frac {5}{2}}(x)}+\frac {x}{3 \sqrt {\sinh (x)}}\right ) \, dx &=\frac {1}{3} \int \frac {x}{\sqrt {\sinh (x)}} \, dx+\int \frac {x}{\sinh ^{\frac {5}{2}}(x)} \, dx\\ &=-\frac {2 x \cosh (x)}{3 \sinh ^{\frac {3}{2}}(x)}-\frac {4}{3 \sqrt {\sinh (x)}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.07, size = 22, normalized size = 0.92 \[ \frac {1}{6} \sqrt {\sinh (x)} (-8 \text {csch}(x)-4 x \coth (x) \text {csch}(x)) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.55, size = 108, normalized size = 4.50 \[ -\frac {4 \, {\left ({\left (x + 2\right )} \cosh \relax (x)^{3} + 3 \, {\left (x + 2\right )} \cosh \relax (x) \sinh \relax (x)^{2} + {\left (x + 2\right )} \sinh \relax (x)^{3} + {\left (x - 2\right )} \cosh \relax (x) + {\left (3 \, {\left (x + 2\right )} \cosh \relax (x)^{2} + x - 2\right )} \sinh \relax (x)\right )} \sqrt {\sinh \relax (x)}}{3 \, {\left (\cosh \relax (x)^{4} + 4 \, \cosh \relax (x) \sinh \relax (x)^{3} + \sinh \relax (x)^{4} + 2 \, {\left (3 \, \cosh \relax (x)^{2} - 1\right )} \sinh \relax (x)^{2} - 2 \, \cosh \relax (x)^{2} + 4 \, {\left (\cosh \relax (x)^{3} - \cosh \relax (x)\right )} \sinh \relax (x) + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x}{3 \, \sqrt {\sinh \relax (x)}} + \frac {x}{\sinh \relax (x)^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.12, size = 0, normalized size = 0.00 \[ \int \frac {x}{\sinh \relax (x )^{\frac {5}{2}}}+\frac {x}{3 \sqrt {\sinh \relax (x )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x}{3 \, \sqrt {\sinh \relax (x)}} + \frac {x}{\sinh \relax (x)^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.15, size = 40, normalized size = 1.67 \[ -\frac {4\,{\mathrm {e}}^x\,\sqrt {\frac {{\mathrm {e}}^x}{2}-\frac {{\mathrm {e}}^{-x}}{2}}\,\left (x+2\,{\mathrm {e}}^{2\,x}+x\,{\mathrm {e}}^{2\,x}-2\right )}{3\,{\left ({\mathrm {e}}^{2\,x}-1\right )}^2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {3 x}{\sinh ^{\frac {5}{2}}{\relax (x )}}\, dx + \int \frac {x}{\sqrt {\sinh {\relax (x )}}}\, dx}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________