Integrand size = 14, antiderivative size = 62 \[ \int \frac {1}{(i \sinh (c+d x))^{5/2}} \, dx=-\frac {2 i \operatorname {EllipticF}\left (\frac {1}{2} \left (i c-\frac {\pi }{2}+i d x\right ),2\right )}{3 d}+\frac {2 i \cosh (c+d x)}{3 d (i \sinh (c+d x))^{3/2}} \]
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Time = 0.02 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2716, 2720} \[ \int \frac {1}{(i \sinh (c+d x))^{5/2}} \, dx=\frac {2 i \cosh (c+d x)}{3 d (i \sinh (c+d x))^{3/2}}-\frac {2 i \operatorname {EllipticF}\left (\frac {1}{2} \left (i c+i d x-\frac {\pi }{2}\right ),2\right )}{3 d} \]
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Rule 2716
Rule 2720
Rubi steps \begin{align*} \text {integral}& = \frac {2 i \cosh (c+d x)}{3 d (i \sinh (c+d x))^{3/2}}+\frac {1}{3} \int \frac {1}{\sqrt {i \sinh (c+d x)}} \, dx \\ & = -\frac {2 i \operatorname {EllipticF}\left (\frac {1}{2} \left (i c-\frac {\pi }{2}+i d x\right ),2\right )}{3 d}+\frac {2 i \cosh (c+d x)}{3 d (i \sinh (c+d x))^{3/2}} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 0.07 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.34 \[ \int \frac {1}{(i \sinh (c+d x))^{5/2}} \, dx=\frac {2 \left (\coth (c+d x)+\sqrt {2} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},\cosh (2 (c+d x))+\sinh (2 (c+d x))\right ) \sqrt {-\left ((1+\coth (c+d x)) \sinh ^2(c+d x)\right )}\right )}{3 d \sqrt {i \sinh (c+d x)}} \]
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Time = 0.93 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.81
method | result | size |
default | \(\frac {i \left (\sqrt {1-i \sinh \left (d x +c \right )}\, \sqrt {2}\, \sqrt {1+i \sinh \left (d x +c \right )}\, \sqrt {i \sinh \left (d x +c \right )}\, \operatorname {EllipticF}\left (\sqrt {1-i \sinh \left (d x +c \right )}, \frac {\sqrt {2}}{2}\right ) \sinh \left (d x +c \right )-2 i \cosh \left (d x +c \right )^{2}\right )}{3 \sinh \left (d x +c \right ) \cosh \left (d x +c \right ) \sqrt {i \sinh \left (d x +c \right )}\, d}\) | \(112\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.09 (sec) , antiderivative size = 132, normalized size of antiderivative = 2.13 \[ \int \frac {1}{(i \sinh (c+d x))^{5/2}} \, dx=-\frac {2 \, {\left (2 \, \sqrt {\frac {1}{2}} {\left (i \, e^{\left (3 \, d x + 3 \, c\right )} + i \, e^{\left (d x + c\right )}\right )} \sqrt {i \, e^{\left (2 \, d x + 2 \, c\right )} - i} e^{\left (-\frac {1}{2} \, d x - \frac {1}{2} \, c\right )} + {\left (i \, \sqrt {2} \sqrt {i} e^{\left (4 \, d x + 4 \, c\right )} - 2 i \, \sqrt {2} \sqrt {i} e^{\left (2 \, d x + 2 \, c\right )} + i \, \sqrt {2} \sqrt {i}\right )} {\rm weierstrassPInverse}\left (4, 0, e^{\left (d x + c\right )}\right )\right )}}{3 \, {\left (d e^{\left (4 \, d x + 4 \, c\right )} - 2 \, d e^{\left (2 \, d x + 2 \, c\right )} + d\right )}} \]
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\[ \int \frac {1}{(i \sinh (c+d x))^{5/2}} \, dx=\int \frac {1}{\left (i \sinh {\left (c + d x \right )}\right )^{\frac {5}{2}}}\, dx \]
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\[ \int \frac {1}{(i \sinh (c+d x))^{5/2}} \, dx=\int { \frac {1}{\left (i \, \sinh \left (d x + c\right )\right )^{\frac {5}{2}}} \,d x } \]
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\[ \int \frac {1}{(i \sinh (c+d x))^{5/2}} \, dx=\int { \frac {1}{\left (i \, \sinh \left (d x + c\right )\right )^{\frac {5}{2}}} \,d x } \]
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Timed out. \[ \int \frac {1}{(i \sinh (c+d x))^{5/2}} \, dx=\int \frac {1}{{\left (\mathrm {sinh}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{5/2}} \,d x \]
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