Optimal. Leaf size=62 \[ \frac{2 i (i \sinh (c+d x))^{3/2} \cosh (c+d x)}{5 d}-\frac{6 i E\left (\left .\frac{1}{2} \left (i c+i d x-\frac{\pi }{2}\right )\right |2\right )}{5 d} \]
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Rubi [A] time = 0.0198261, antiderivative size = 62, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {2635, 2639} \[ \frac{2 i (i \sinh (c+d x))^{3/2} \cosh (c+d x)}{5 d}-\frac{6 i E\left (\left .\frac{1}{2} \left (i c+i d x-\frac{\pi }{2}\right )\right |2\right )}{5 d} \]
Antiderivative was successfully verified.
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Rule 2635
Rule 2639
Rubi steps
\begin{align*} \int (i \sinh (c+d x))^{5/2} \, dx &=\frac{2 i \cosh (c+d x) (i \sinh (c+d x))^{3/2}}{5 d}+\frac{3}{5} \int \sqrt{i \sinh (c+d x)} \, dx\\ &=-\frac{6 i E\left (\left .\frac{1}{2} \left (i c-\frac{\pi }{2}+i d x\right )\right |2\right )}{5 d}+\frac{2 i \cosh (c+d x) (i \sinh (c+d x))^{3/2}}{5 d}\\ \end{align*}
Mathematica [A] time = 0.0588187, size = 55, normalized size = 0.89 \[ \frac{6 i E\left (\left .\frac{1}{4} (-2 i c-2 i d x+\pi )\right |2\right )-\sqrt{i \sinh (c+d x)} \sinh (2 (c+d x))}{5 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.05, size = 169, normalized size = 2.7 \begin{align*}{\frac{{\frac{i}{5}}}{d\cosh \left ( dx+c \right ) } \left ( 6\,\sqrt{1-i\sinh \left ( dx+c \right ) }\sqrt{2}\sqrt{1+i\sinh \left ( dx+c \right ) }\sqrt{i\sinh \left ( dx+c \right ) }{\it EllipticE} \left ( \sqrt{1-i\sinh \left ( dx+c \right ) },1/2\,\sqrt{2} \right ) -3\,\sqrt{1-i\sinh \left ( dx+c \right ) }\sqrt{2}\sqrt{1+i\sinh \left ( dx+c \right ) }\sqrt{i\sinh \left ( dx+c \right ) }{\it EllipticF} \left ( \sqrt{1-i\sinh \left ( dx+c \right ) },1/2\,\sqrt{2} \right ) -2\, \left ( \cosh \left ( dx+c \right ) \right ) ^{4}+2\, \left ( \cosh \left ( dx+c \right ) \right ) ^{2} \right ){\frac{1}{\sqrt{i\sinh \left ( dx+c \right ) }}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (i \, \sinh \left (d x + c\right )\right )^{\frac{5}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{\sqrt{\frac{1}{2}}{\left (e^{\left (5 \, d x + 5 \, c\right )} - 2 \, e^{\left (4 \, d x + 4 \, c\right )} - 12 \, e^{\left (3 \, d x + 3 \, c\right )} - 24 \, e^{\left (2 \, d x + 2 \, c\right )} - e^{\left (d x + c\right )} + 2\right )} \sqrt{i \, e^{\left (2 \, d x + 2 \, c\right )} - i} e^{\left (-\frac{1}{2} \, d x - \frac{1}{2} \, c\right )} - 10 \,{\left (d e^{\left (3 \, d x + 3 \, c\right )} - 2 \, d e^{\left (2 \, d x + 2 \, c\right )}\right )}{\rm integral}\left (\frac{6 \, \sqrt{\frac{1}{2}}{\left (2 \, e^{\left (2 \, d x + 2 \, c\right )} + 3 \, e^{\left (d x + c\right )} - 2\right )} \sqrt{i \, e^{\left (2 \, d x + 2 \, c\right )} - i} e^{\left (-\frac{1}{2} \, d x - \frac{1}{2} \, c\right )}}{5 \,{\left (d e^{\left (4 \, d x + 4 \, c\right )} - 4 \, d e^{\left (3 \, d x + 3 \, c\right )} + 3 \, d e^{\left (2 \, d x + 2 \, c\right )} + 4 \, d e^{\left (d x + c\right )} - 4 \, d\right )}}, x\right )}{10 \,{\left (d e^{\left (3 \, d x + 3 \, c\right )} - 2 \, d e^{\left (2 \, d x + 2 \, c\right )}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (i \, \sinh \left (d x + c\right )\right )^{\frac{5}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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