Optimal. Leaf size=58 \[ \frac {2 i E\left (\left .\frac {1}{2} \left (i c-\frac {\pi }{2}+i d x\right )\right |2\right )}{d}+\frac {2 i \cosh (c+d x)}{d \sqrt {i \sinh (c+d x)}} \]
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Rubi [A]
time = 0.02, antiderivative size = 58, normalized size of antiderivative = 1.00, 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 = {2716, 2719}
\begin {gather*} \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)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2716
Rule 2719
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*}
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Mathematica [A]
time = 0.07, size = 50, normalized size = 0.86 \begin {gather*} \frac {2 \left (-i E\left (\left .\frac {1}{4} (-2 i c+\pi -2 i d x)\right |2\right )+\coth (c+d x) \sqrt {i \sinh (c+d x)}\right )}{d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.82, size = 159, normalized size = 2.74
method | result | size |
default | \(-\frac {i \left (2 \sqrt {1-i \sinh \left (d x +c \right )}\, \sqrt {2}\, \sqrt {1+i \sinh \left (d x +c \right )}\, \sqrt {i \sinh \left (d x +c \right )}\, \EllipticE \left (\sqrt {1-i \sinh \left (d x +c \right )}, \frac {\sqrt {2}}{2}\right )-\sqrt {1-i \sinh \left (d x +c \right )}\, \sqrt {2}\, \sqrt {1+i \sinh \left (d x +c \right )}\, \sqrt {i \sinh \left (d x +c \right )}\, \EllipticF \left (\sqrt {1-i \sinh \left (d x +c \right )}, \frac {\sqrt {2}}{2}\right )-2 \left (\cosh ^{2}\left (d x +c \right )\right )\right )}{\cosh \left (d x +c \right ) \sqrt {i \sinh \left (d x +c \right )}\, d}\) | \(159\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 0.08, size = 87, normalized size = 1.50 \begin {gather*} \frac {2 \, {\left (2 \, \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 (\sqrt {2} \sqrt {i} e^{\left (2 \, d x + 2 \, c\right )} - \sqrt {2} \sqrt {i}\right )} {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, e^{\left (d x + c\right )}\right )\right )\right )}}{d e^{\left (2 \, d x + 2 \, c\right )} - d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (i \sinh {\left (c + d x \right )}\right )^{\frac {3}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {1}{{\left (\mathrm {sinh}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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