Integrand size = 10, antiderivative size = 69 \[ \int \left (a \text {sech}^3(x)\right )^{3/2} \, dx=-\frac {10}{21} i a \cosh ^{\frac {3}{2}}(x) \operatorname {EllipticF}\left (\frac {i x}{2},2\right ) \sqrt {a \text {sech}^3(x)}+\frac {10}{21} a \sqrt {a \text {sech}^3(x)} \sinh (x)+\frac {2}{7} a \text {sech}(x) \sqrt {a \text {sech}^3(x)} \tanh (x) \]
[Out]
Time = 0.03 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {4208, 3853, 3856, 2720} \[ \int \left (a \text {sech}^3(x)\right )^{3/2} \, dx=\frac {10}{21} a \sinh (x) \sqrt {a \text {sech}^3(x)}+\frac {2}{7} a \tanh (x) \text {sech}(x) \sqrt {a \text {sech}^3(x)}-\frac {10}{21} i a \cosh ^{\frac {3}{2}}(x) \operatorname {EllipticF}\left (\frac {i x}{2},2\right ) \sqrt {a \text {sech}^3(x)} \]
[In]
[Out]
Rule 2720
Rule 3853
Rule 3856
Rule 4208
Rubi steps \begin{align*} \text {integral}& = \frac {\left (a \sqrt {a \text {sech}^3(x)}\right ) \int \text {sech}^{\frac {9}{2}}(x) \, dx}{\text {sech}^{\frac {3}{2}}(x)} \\ & = \frac {2}{7} a \text {sech}(x) \sqrt {a \text {sech}^3(x)} \tanh (x)+\frac {\left (5 a \sqrt {a \text {sech}^3(x)}\right ) \int \text {sech}^{\frac {5}{2}}(x) \, dx}{7 \text {sech}^{\frac {3}{2}}(x)} \\ & = \frac {10}{21} a \sqrt {a \text {sech}^3(x)} \sinh (x)+\frac {2}{7} a \text {sech}(x) \sqrt {a \text {sech}^3(x)} \tanh (x)+\frac {\left (5 a \sqrt {a \text {sech}^3(x)}\right ) \int \sqrt {\text {sech}(x)} \, dx}{21 \text {sech}^{\frac {3}{2}}(x)} \\ & = \frac {10}{21} a \sqrt {a \text {sech}^3(x)} \sinh (x)+\frac {2}{7} a \text {sech}(x) \sqrt {a \text {sech}^3(x)} \tanh (x)+\frac {1}{21} \left (5 a \cosh ^{\frac {3}{2}}(x) \sqrt {a \text {sech}^3(x)}\right ) \int \frac {1}{\sqrt {\cosh (x)}} \, dx \\ & = -\frac {10}{21} i a \cosh ^{\frac {3}{2}}(x) \operatorname {EllipticF}\left (\frac {i x}{2},2\right ) \sqrt {a \text {sech}^3(x)}+\frac {10}{21} a \sqrt {a \text {sech}^3(x)} \sinh (x)+\frac {2}{7} a \text {sech}(x) \sqrt {a \text {sech}^3(x)} \tanh (x) \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.68 \[ \int \left (a \text {sech}^3(x)\right )^{3/2} \, dx=\frac {2}{21} a \text {sech}(x) \sqrt {a \text {sech}^3(x)} \left (-5 i \cosh ^{\frac {5}{2}}(x) \operatorname {EllipticF}\left (\frac {i x}{2},2\right )+5 \cosh (x) \sinh (x)+3 \tanh (x)\right ) \]
[In]
[Out]
\[\int \left (a \operatorname {sech}\left (x \right )^{3}\right )^{\frac {3}{2}}d x\]
[In]
[Out]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.09 (sec) , antiderivative size = 391, normalized size of antiderivative = 5.67 \[ \int \left (a \text {sech}^3(x)\right )^{3/2} \, dx=\frac {2 \, {\left (5 \, \sqrt {2} {\left (a \cosh \left (x\right )^{6} + 6 \, a \cosh \left (x\right ) \sinh \left (x\right )^{5} + a \sinh \left (x\right )^{6} + 3 \, a \cosh \left (x\right )^{4} + 3 \, {\left (5 \, a \cosh \left (x\right )^{2} + a\right )} \sinh \left (x\right )^{4} + 4 \, {\left (5 \, a \cosh \left (x\right )^{3} + 3 \, a \cosh \left (x\right )\right )} \sinh \left (x\right )^{3} + 3 \, a \cosh \left (x\right )^{2} + 3 \, {\left (5 \, a \cosh \left (x\right )^{4} + 6 \, a \cosh \left (x\right )^{2} + a\right )} \sinh \left (x\right )^{2} + 6 \, {\left (a \cosh \left (x\right )^{5} + 2 \, a \cosh \left (x\right )^{3} + a \cosh \left (x\right )\right )} \sinh \left (x\right ) + a\right )} \sqrt {a} {\rm weierstrassPInverse}\left (-4, 0, \cosh \left (x\right ) + \sinh \left (x\right )\right ) + \sqrt {2} {\left (5 \, a \cosh \left (x\right )^{6} + 30 \, a \cosh \left (x\right ) \sinh \left (x\right )^{5} + 5 \, a \sinh \left (x\right )^{6} + 17 \, a \cosh \left (x\right )^{4} + {\left (75 \, a \cosh \left (x\right )^{2} + 17 \, a\right )} \sinh \left (x\right )^{4} + 4 \, {\left (25 \, a \cosh \left (x\right )^{3} + 17 \, a \cosh \left (x\right )\right )} \sinh \left (x\right )^{3} - 17 \, a \cosh \left (x\right )^{2} + {\left (75 \, a \cosh \left (x\right )^{4} + 102 \, a \cosh \left (x\right )^{2} - 17 \, a\right )} \sinh \left (x\right )^{2} + 2 \, {\left (15 \, a \cosh \left (x\right )^{5} + 34 \, a \cosh \left (x\right )^{3} - 17 \, a \cosh \left (x\right )\right )} \sinh \left (x\right ) - 5 \, a\right )} \sqrt {\frac {a \cosh \left (x\right ) + a \sinh \left (x\right )}{\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2} + 1}}\right )}}{21 \, {\left (\cosh \left (x\right )^{6} + 6 \, \cosh \left (x\right ) \sinh \left (x\right )^{5} + \sinh \left (x\right )^{6} + 3 \, {\left (5 \, \cosh \left (x\right )^{2} + 1\right )} \sinh \left (x\right )^{4} + 3 \, \cosh \left (x\right )^{4} + 4 \, {\left (5 \, \cosh \left (x\right )^{3} + 3 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{3} + 3 \, {\left (5 \, \cosh \left (x\right )^{4} + 6 \, \cosh \left (x\right )^{2} + 1\right )} \sinh \left (x\right )^{2} + 3 \, \cosh \left (x\right )^{2} + 6 \, {\left (\cosh \left (x\right )^{5} + 2 \, \cosh \left (x\right )^{3} + \cosh \left (x\right )\right )} \sinh \left (x\right ) + 1\right )}} \]
[In]
[Out]
\[ \int \left (a \text {sech}^3(x)\right )^{3/2} \, dx=\int \left (a \operatorname {sech}^{3}{\left (x \right )}\right )^{\frac {3}{2}}\, dx \]
[In]
[Out]
\[ \int \left (a \text {sech}^3(x)\right )^{3/2} \, dx=\int { \left (a \operatorname {sech}\left (x\right )^{3}\right )^{\frac {3}{2}} \,d x } \]
[In]
[Out]
\[ \int \left (a \text {sech}^3(x)\right )^{3/2} \, dx=\int { \left (a \operatorname {sech}\left (x\right )^{3}\right )^{\frac {3}{2}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \left (a \text {sech}^3(x)\right )^{3/2} \, dx=\int {\left (\frac {a}{{\mathrm {cosh}\left (x\right )}^3}\right )}^{3/2} \,d x \]
[In]
[Out]