Optimal. Leaf size=121 \[ -\frac{26 i \text{EllipticF}\left (\frac{i x}{2},2\right )}{77 a^2 \cosh ^{\frac{3}{2}}(x) \sqrt{a \text{sech}^3(x)}}+\frac{26 \tanh (x)}{77 a^2 \sqrt{a \text{sech}^3(x)}}+\frac{2 \sinh (x) \cosh ^5(x)}{15 a^2 \sqrt{a \text{sech}^3(x)}}+\frac{26 \sinh (x) \cosh ^3(x)}{165 a^2 \sqrt{a \text{sech}^3(x)}}+\frac{78 \sinh (x) \cosh (x)}{385 a^2 \sqrt{a \text{sech}^3(x)}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0642417, antiderivative size = 121, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {4123, 3769, 3771, 2641} \[ \frac{26 \tanh (x)}{77 a^2 \sqrt{a \text{sech}^3(x)}}+\frac{2 \sinh (x) \cosh ^5(x)}{15 a^2 \sqrt{a \text{sech}^3(x)}}+\frac{26 \sinh (x) \cosh ^3(x)}{165 a^2 \sqrt{a \text{sech}^3(x)}}-\frac{26 i F\left (\left .\frac{i x}{2}\right |2\right )}{77 a^2 \cosh ^{\frac{3}{2}}(x) \sqrt{a \text{sech}^3(x)}}+\frac{78 \sinh (x) \cosh (x)}{385 a^2 \sqrt{a \text{sech}^3(x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 4123
Rule 3769
Rule 3771
Rule 2641
Rubi steps
\begin{align*} \int \frac{1}{\left (a \text{sech}^3(x)\right )^{5/2}} \, dx &=\frac{\text{sech}^{\frac{3}{2}}(x) \int \frac{1}{\text{sech}^{\frac{15}{2}}(x)} \, dx}{a^2 \sqrt{a \text{sech}^3(x)}}\\ &=\frac{2 \cosh ^5(x) \sinh (x)}{15 a^2 \sqrt{a \text{sech}^3(x)}}+\frac{\left (13 \text{sech}^{\frac{3}{2}}(x)\right ) \int \frac{1}{\text{sech}^{\frac{11}{2}}(x)} \, dx}{15 a^2 \sqrt{a \text{sech}^3(x)}}\\ &=\frac{26 \cosh ^3(x) \sinh (x)}{165 a^2 \sqrt{a \text{sech}^3(x)}}+\frac{2 \cosh ^5(x) \sinh (x)}{15 a^2 \sqrt{a \text{sech}^3(x)}}+\frac{\left (39 \text{sech}^{\frac{3}{2}}(x)\right ) \int \frac{1}{\text{sech}^{\frac{7}{2}}(x)} \, dx}{55 a^2 \sqrt{a \text{sech}^3(x)}}\\ &=\frac{78 \cosh (x) \sinh (x)}{385 a^2 \sqrt{a \text{sech}^3(x)}}+\frac{26 \cosh ^3(x) \sinh (x)}{165 a^2 \sqrt{a \text{sech}^3(x)}}+\frac{2 \cosh ^5(x) \sinh (x)}{15 a^2 \sqrt{a \text{sech}^3(x)}}+\frac{\left (39 \text{sech}^{\frac{3}{2}}(x)\right ) \int \frac{1}{\text{sech}^{\frac{3}{2}}(x)} \, dx}{77 a^2 \sqrt{a \text{sech}^3(x)}}\\ &=\frac{78 \cosh (x) \sinh (x)}{385 a^2 \sqrt{a \text{sech}^3(x)}}+\frac{26 \cosh ^3(x) \sinh (x)}{165 a^2 \sqrt{a \text{sech}^3(x)}}+\frac{2 \cosh ^5(x) \sinh (x)}{15 a^2 \sqrt{a \text{sech}^3(x)}}+\frac{26 \tanh (x)}{77 a^2 \sqrt{a \text{sech}^3(x)}}+\frac{\left (13 \text{sech}^{\frac{3}{2}}(x)\right ) \int \sqrt{\text{sech}(x)} \, dx}{77 a^2 \sqrt{a \text{sech}^3(x)}}\\ &=\frac{78 \cosh (x) \sinh (x)}{385 a^2 \sqrt{a \text{sech}^3(x)}}+\frac{26 \cosh ^3(x) \sinh (x)}{165 a^2 \sqrt{a \text{sech}^3(x)}}+\frac{2 \cosh ^5(x) \sinh (x)}{15 a^2 \sqrt{a \text{sech}^3(x)}}+\frac{26 \tanh (x)}{77 a^2 \sqrt{a \text{sech}^3(x)}}+\frac{13 \int \frac{1}{\sqrt{\cosh (x)}} \, dx}{77 a^2 \cosh ^{\frac{3}{2}}(x) \sqrt{a \text{sech}^3(x)}}\\ &=-\frac{26 i F\left (\left .\frac{i x}{2}\right |2\right )}{77 a^2 \cosh ^{\frac{3}{2}}(x) \sqrt{a \text{sech}^3(x)}}+\frac{78 \cosh (x) \sinh (x)}{385 a^2 \sqrt{a \text{sech}^3(x)}}+\frac{26 \cosh ^3(x) \sinh (x)}{165 a^2 \sqrt{a \text{sech}^3(x)}}+\frac{2 \cosh ^5(x) \sinh (x)}{15 a^2 \sqrt{a \text{sech}^3(x)}}+\frac{26 \tanh (x)}{77 a^2 \sqrt{a \text{sech}^3(x)}}\\ \end{align*}
Mathematica [A] time = 0.0907472, size = 63, normalized size = 0.52 \[ \frac{\cosh (x) \sqrt{a \text{sech}^3(x)} \left (-24960 i \sqrt{\cosh (x)} \text{EllipticF}\left (\frac{i x}{2},2\right )+19122 \sinh (2 x)+4406 \sinh (4 x)+826 \sinh (6 x)+77 \sinh (8 x)\right )}{73920 a^3} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.053, size = 0, normalized size = 0. \begin{align*} \int \left ( a \left ({\rm sech} \left (x\right ) \right ) ^{3} \right ) ^{-{\frac{5}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (a \operatorname{sech}\left (x\right )^{3}\right )^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{a \operatorname{sech}\left (x\right )^{3}}}{a^{3} \operatorname{sech}\left (x\right )^{9}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (a \operatorname{sech}^{3}{\left (x \right )}\right )^{\frac{5}{2}}}\, dx \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 \frac{1}{\left (a \operatorname{sech}\left (x\right )^{3}\right )^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]