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 = 0.02, antiderivative size = 62, 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 = {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*}
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Mathematica [A] time = 0.06, 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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fricas [F] time = 0.60, size = 0, normalized size = 0.00 \[ -\frac {\sqrt {\frac {1}{2}} {\left (e^{\left (5 \, d x + 5 \, c\right )} - 2 \, e^{\left (4 \, d x + 4 \, c\right )} - 12 \, e^{\left (3 \, d x + 3 \, c\right )} - 24 \, e^{\left (2 \, d x + 2 \, c\right )} - e^{\left (d x + c\right )} + 2\right )} \sqrt {i \, e^{\left (2 \, d x + 2 \, c\right )} - i} e^{\left (-\frac {1}{2} \, d x - \frac {1}{2} \, c\right )} - 10 \, {\left (d e^{\left (3 \, d x + 3 \, c\right )} - 2 \, d e^{\left (2 \, d x + 2 \, c\right )}\right )} {\rm integral}\left (\frac {6 \, \sqrt {\frac {1}{2}} {\left (2 \, e^{\left (2 \, d x + 2 \, c\right )} + 3 \, e^{\left (d x + c\right )} - 2\right )} \sqrt {i \, e^{\left (2 \, d x + 2 \, c\right )} - i} e^{\left (-\frac {1}{2} \, d x - \frac {1}{2} \, c\right )}}{5 \, {\left (d e^{\left (4 \, d x + 4 \, c\right )} - 4 \, d e^{\left (3 \, d x + 3 \, c\right )} + 3 \, d e^{\left (2 \, d x + 2 \, c\right )} + 4 \, d e^{\left (d x + c\right )} - 4 \, d\right )}}, x\right )}{10 \, {\left (d e^{\left (3 \, d x + 3 \, c\right )} - 2 \, d e^{\left (2 \, d x + 2 \, c\right )}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.07, size = 169, normalized size = 2.73 \[ \frac {i \left (6 \sqrt {1-i \sinh \left (d x +c \right )}\, \sqrt {2}\, \sqrt {1+i \sinh \left (d x +c \right )}\, \sqrt {i \sinh \left (d x +c \right )}\, \EllipticE \left (\sqrt {1-i \sinh \left (d x +c \right )}, \frac {\sqrt {2}}{2}\right )-3 \sqrt {1-i \sinh \left (d x +c \right )}\, \sqrt {2}\, \sqrt {1+i \sinh \left (d x +c \right )}\, \sqrt {i \sinh \left (d x +c \right )}\, \EllipticF \left (\sqrt {1-i \sinh \left (d x +c \right )}, \frac {\sqrt {2}}{2}\right )-2 \left (\cosh ^{4}\left (d x +c \right )\right )+2 \left (\cosh ^{2}\left (d x +c \right )\right )\right )}{5 \cosh \left (d x +c \right ) \sqrt {i \sinh \left (d x +c \right )}\, d} \]
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 \[ \int \left (i \, \sinh \left (d x + c\right )\right )^{\frac {5}{2}}\,{d x} \]
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 \[ \int {\left (\mathrm {sinh}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{5/2} \,d x \]
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 \[ \int \left (i \sinh {\left (c + d x \right )}\right )^{\frac {5}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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