Optimal. Leaf size=58 \[ \frac{2 i E\left (\left .\frac{1}{2} \left (i c+i d x-\frac{\pi }{2}\right )\right |2\right )}{d}+\frac{2 i \cosh (c+d x)}{d \sqrt{i \sinh (c+d x)}} \]
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Rubi [A] time = 0.0186412, antiderivative size = 58, 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 = {2636, 2639} \[ \frac{2 i E\left (\left .\frac{1}{2} \left (i c+i d x-\frac{\pi }{2}\right )\right |2\right )}{d}+\frac{2 i \cosh (c+d x)}{d \sqrt{i \sinh (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 2636
Rule 2639
Rubi steps
\begin{align*} \int \frac{1}{(i \sinh (c+d x))^{3/2}} \, dx &=\frac{2 i \cosh (c+d x)}{d \sqrt{i \sinh (c+d x)}}-\int \sqrt{i \sinh (c+d x)} \, dx\\ &=\frac{2 i E\left (\left .\frac{1}{2} \left (i c-\frac{\pi }{2}+i d x\right )\right |2\right )}{d}+\frac{2 i \cosh (c+d x)}{d \sqrt{i \sinh (c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.0938688, size = 50, normalized size = 0.86 \[ \frac{2 \left (\sqrt{i \sinh (c+d x)} \coth (c+d x)-i E\left (\left .\frac{1}{4} (-2 i c-2 i d x+\pi )\right |2\right )\right )}{d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.052, size = 159, normalized size = 2.7 \begin{align*}{\frac{-i}{d\cosh \left ( dx+c \right ) } \left ( 2\,\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 ) -\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 ) },{\frac{\sqrt{2}}{2}} \right ) -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 \frac{1}{\left (i \, \sinh \left (d x + c\right )\right )^{\frac{3}{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{4 \, \sqrt{\frac{1}{2}} \sqrt{i \, e^{\left (2 \, d x + 2 \, c\right )} - i} e^{\left (\frac{3}{2} \, d x + \frac{3}{2} \, c\right )} +{\left (d e^{\left (2 \, d x + 2 \, c\right )} - d\right )}{\rm integral}\left (-\frac{2 \, \sqrt{\frac{1}{2}} \sqrt{i \, e^{\left (2 \, d x + 2 \, c\right )} - i} e^{\left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}}{d e^{\left (2 \, d x + 2 \, c\right )} - d}, x\right )}{d e^{\left (2 \, d x + 2 \, c\right )} - d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (i \sinh{\left (c + d x \right )}\right )^{\frac{3}{2}}}\, dx \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 \frac{1}{\left (i \, \sinh \left (d x + c\right )\right )^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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