Integrand size = 12, antiderivative size = 88 \[ \int (b \sinh (c+d x))^{3/2} \, dx=\frac {2 i b^2 \operatorname {EllipticF}\left (\frac {1}{2} \left (i c-\frac {\pi }{2}+i d x\right ),2\right ) \sqrt {i \sinh (c+d x)}}{3 d \sqrt {b \sinh (c+d x)}}+\frac {2 b \cosh (c+d x) \sqrt {b \sinh (c+d x)}}{3 d} \]
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Time = 0.03 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2715, 2721, 2720} \[ \int (b \sinh (c+d x))^{3/2} \, dx=\frac {2 b \cosh (c+d x) \sqrt {b \sinh (c+d x)}}{3 d}+\frac {2 i b^2 \sqrt {i \sinh (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} \left (i c+i d x-\frac {\pi }{2}\right ),2\right )}{3 d \sqrt {b \sinh (c+d x)}} \]
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Rule 2715
Rule 2720
Rule 2721
Rubi steps \begin{align*} \text {integral}& = \frac {2 b \cosh (c+d x) \sqrt {b \sinh (c+d x)}}{3 d}-\frac {1}{3} b^2 \int \frac {1}{\sqrt {b \sinh (c+d x)}} \, dx \\ & = \frac {2 b \cosh (c+d x) \sqrt {b \sinh (c+d x)}}{3 d}-\frac {\left (b^2 \sqrt {i \sinh (c+d x)}\right ) \int \frac {1}{\sqrt {i \sinh (c+d x)}} \, dx}{3 \sqrt {b \sinh (c+d x)}} \\ & = \frac {2 i b^2 \operatorname {EllipticF}\left (\frac {1}{2} \left (i c-\frac {\pi }{2}+i d x\right ),2\right ) \sqrt {i \sinh (c+d x)}}{3 d \sqrt {b \sinh (c+d x)}}+\frac {2 b \cosh (c+d x) \sqrt {b \sinh (c+d x)}}{3 d} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 0.10 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.00 \[ \int (b \sinh (c+d x))^{3/2} \, dx=\frac {b^2 \left (\sinh (2 (c+d x))-2 \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},\cosh (2 (c+d x))+\sinh (2 (c+d x))\right ) \sqrt {1-\cosh (2 c+2 d x)-\sinh (2 c+2 d x)}\right )}{3 d \sqrt {b \sinh (c+d x)}} \]
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Time = 0.89 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.20
method | result | size |
default | \(-\frac {b^{2} \left (i \sqrt {1-i \sinh \left (d x +c \right )}\, \sqrt {2}\, \sqrt {1+i \sinh \left (d x +c \right )}\, \sqrt {i \sinh \left (d x +c \right )}\, \operatorname {EllipticF}\left (\sqrt {1-i \sinh \left (d x +c \right )}, \frac {\sqrt {2}}{2}\right )-2 \cosh \left (d x +c \right )^{2} \sinh \left (d x +c \right )\right )}{3 \cosh \left (d x +c \right ) \sqrt {b \sinh \left (d x +c \right )}\, d}\) | \(106\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.08 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.31 \[ \int (b \sinh (c+d x))^{3/2} \, dx=-\frac {2 \, {\left (\sqrt {2} b \cosh \left (d x + c\right ) + \sqrt {2} b \sinh \left (d x + c\right )\right )} \sqrt {b} {\rm weierstrassPInverse}\left (4, 0, \cosh \left (d x + c\right ) + \sinh \left (d x + c\right )\right ) - {\left (b \cosh \left (d x + c\right )^{2} + 2 \, b \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + b \sinh \left (d x + c\right )^{2} + b\right )} \sqrt {b \sinh \left (d x + c\right )}}{3 \, {\left (d \cosh \left (d x + c\right ) + d \sinh \left (d x + c\right )\right )}} \]
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\[ \int (b \sinh (c+d x))^{3/2} \, dx=\int \left (b \sinh {\left (c + d x \right )}\right )^{\frac {3}{2}}\, dx \]
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\[ \int (b \sinh (c+d x))^{3/2} \, dx=\int { \left (b \sinh \left (d x + c\right )\right )^{\frac {3}{2}} \,d x } \]
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\[ \int (b \sinh (c+d x))^{3/2} \, dx=\int { \left (b \sinh \left (d x + c\right )\right )^{\frac {3}{2}} \,d x } \]
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Timed out. \[ \int (b \sinh (c+d x))^{3/2} \, dx=\int {\left (b\,\mathrm {sinh}\left (c+d\,x\right )\right )}^{3/2} \,d x \]
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