Optimal. Leaf size=31 \[ \frac{2}{3} \cosh (x) \sqrt{\cosh (x) \coth (x)}-\frac{8}{3} \text{sech}(x) \sqrt{\cosh (x) \coth (x)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.122285, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.556, Rules used = {4397, 4398, 4400, 2598, 2589} \[ \frac{2}{3} \cosh (x) \sqrt{\cosh (x) \coth (x)}-\frac{8}{3} \text{sech}(x) \sqrt{\cosh (x) \coth (x)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 4397
Rule 4398
Rule 4400
Rule 2598
Rule 2589
Rubi steps
\begin{align*} \int (\text{csch}(x)+\sinh (x))^{3/2} \, dx &=\int (\cosh (x) \coth (x))^{3/2} \, dx\\ &=\frac{\left (i \sqrt{\cosh (x) \coth (x)}\right ) \int (-i \cosh (x) \coth (x))^{3/2} \, dx}{\sqrt{-i \cosh (x) \coth (x)}}\\ &=\frac{\left (i \sqrt{\cosh (x) \coth (x)}\right ) \int \cosh ^{\frac{3}{2}}(x) (-i \coth (x))^{3/2} \, dx}{\sqrt{\cosh (x)} \sqrt{-i \coth (x)}}\\ &=\frac{2}{3} \cosh (x) \sqrt{\cosh (x) \coth (x)}+\frac{\left (4 i \sqrt{\cosh (x) \coth (x)}\right ) \int \frac{(-i \coth (x))^{3/2}}{\sqrt{\cosh (x)}} \, dx}{3 \sqrt{\cosh (x)} \sqrt{-i \coth (x)}}\\ &=\frac{2}{3} \cosh (x) \sqrt{\cosh (x) \coth (x)}-\frac{8}{3} \sqrt{\cosh (x) \coth (x)} \text{sech}(x)\\ \end{align*}
Mathematica [A] time = 0.0497481, size = 21, normalized size = 0.68 \[ \frac{2}{3} \left (\cosh ^2(x)-4\right ) \text{sech}(x) \sqrt{\cosh (x) \coth (x)} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.079, size = 0, normalized size = 0. \begin{align*} \int \left ({\rm csch} \left (x\right )+\sinh \left ( x \right ) \right ) ^{{\frac{3}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] time = 1.84759, size = 147, normalized size = 4.74 \begin{align*} \frac{\sqrt{2} e^{\left (\frac{3}{2} \, x\right )}}{6 \,{\left (e^{\left (-x\right )} + 1\right )}^{\frac{3}{2}}{\left (-e^{\left (-x\right )} + 1\right )}^{\frac{3}{2}}} - \frac{5 \, \sqrt{2} e^{\left (-\frac{1}{2} \, x\right )}}{2 \,{\left (e^{\left (-x\right )} + 1\right )}^{\frac{3}{2}}{\left (-e^{\left (-x\right )} + 1\right )}^{\frac{3}{2}}} + \frac{5 \, \sqrt{2} e^{\left (-\frac{5}{2} \, x\right )}}{2 \,{\left (e^{\left (-x\right )} + 1\right )}^{\frac{3}{2}}{\left (-e^{\left (-x\right )} + 1\right )}^{\frac{3}{2}}} - \frac{\sqrt{2} e^{\left (-\frac{9}{2} \, x\right )}}{6 \,{\left (e^{\left (-x\right )} + 1\right )}^{\frac{3}{2}}{\left (-e^{\left (-x\right )} + 1\right )}^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 1.9067, size = 348, normalized size = 11.23 \begin{align*} \frac{\sqrt{\frac{1}{2}}{\left (\cosh \left (x\right )^{4} + 4 \, \cosh \left (x\right ) \sinh \left (x\right )^{3} + \sinh \left (x\right )^{4} + 2 \,{\left (3 \, \cosh \left (x\right )^{2} - 7\right )} \sinh \left (x\right )^{2} - 14 \, \cosh \left (x\right )^{2} + 4 \,{\left (\cosh \left (x\right )^{3} - 7 \, \cosh \left (x\right )\right )} \sinh \left (x\right ) + 1\right )}}{3 \, \sqrt{\cosh \left (x\right )^{3} + 3 \, \cosh \left (x\right ) \sinh \left (x\right )^{2} + \sinh \left (x\right )^{3} +{\left (3 \, \cosh \left (x\right )^{2} - 1\right )} \sinh \left (x\right ) - \cosh \left (x\right )}{\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (\operatorname{csch}\left (x\right ) + \sinh \left (x\right )\right )}^{\frac{3}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]