Optimal. Leaf size=104 \[ \frac{64 i a^3 \cosh (c+d x)}{15 d \sqrt{a+i a \sinh (c+d x)}}+\frac{16 i a^2 \cosh (c+d x) \sqrt{a+i a \sinh (c+d x)}}{15 d}+\frac{2 i a \cosh (c+d x) (a+i a \sinh (c+d x))^{3/2}}{5 d} \]
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Rubi [A] time = 0.0531309, antiderivative size = 104, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {2647, 2646} \[ \frac{64 i a^3 \cosh (c+d x)}{15 d \sqrt{a+i a \sinh (c+d x)}}+\frac{16 i a^2 \cosh (c+d x) \sqrt{a+i a \sinh (c+d x)}}{15 d}+\frac{2 i a \cosh (c+d x) (a+i a \sinh (c+d x))^{3/2}}{5 d} \]
Antiderivative was successfully verified.
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Rule 2647
Rule 2646
Rubi steps
\begin{align*} \int (a+i a \sinh (c+d x))^{5/2} \, dx &=\frac{2 i a \cosh (c+d x) (a+i a \sinh (c+d x))^{3/2}}{5 d}+\frac{1}{5} (8 a) \int (a+i a \sinh (c+d x))^{3/2} \, dx\\ &=\frac{16 i a^2 \cosh (c+d x) \sqrt{a+i a \sinh (c+d x)}}{15 d}+\frac{2 i a \cosh (c+d x) (a+i a \sinh (c+d x))^{3/2}}{5 d}+\frac{1}{15} \left (32 a^2\right ) \int \sqrt{a+i a \sinh (c+d x)} \, dx\\ &=\frac{64 i a^3 \cosh (c+d x)}{15 d \sqrt{a+i a \sinh (c+d x)}}+\frac{16 i a^2 \cosh (c+d x) \sqrt{a+i a \sinh (c+d x)}}{15 d}+\frac{2 i a \cosh (c+d x) (a+i a \sinh (c+d x))^{3/2}}{5 d}\\ \end{align*}
Mathematica [A] time = 0.452801, size = 145, normalized size = 1.39 \[ \frac{a^2 (\sinh (c+d x)-i)^2 \sqrt{a+i a \sinh (c+d x)} \left (-150 \sinh \left (\frac{1}{2} (c+d x)\right )+25 \sinh \left (\frac{3}{2} (c+d x)\right )+3 \sinh \left (\frac{5}{2} (c+d x)\right )-150 i \cosh \left (\frac{1}{2} (c+d x)\right )-25 i \cosh \left (\frac{3}{2} (c+d x)\right )+3 i \cosh \left (\frac{5}{2} (c+d x)\right )\right )}{30 d \left (\cosh \left (\frac{1}{2} (c+d x)\right )+i \sinh \left (\frac{1}{2} (c+d x)\right )\right )^5} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.14, size = 0, normalized size = 0. \begin{align*} \int \left ( a+ia\sinh \left ( dx+c \right ) \right ) ^{{\frac{5}{2}}}\, dx \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 \, a \sinh \left (d x + c\right ) + a\right )}^{\frac{5}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.98379, size = 360, normalized size = 3.46 \begin{align*} -\frac{\sqrt{\frac{1}{2}}{\left (3 \, a^{2} e^{\left (5 \, d x + 5 \, c\right )} - 25 i \, a^{2} e^{\left (4 \, d x + 4 \, c\right )} - 150 \, a^{2} e^{\left (3 \, d x + 3 \, c\right )} - 150 i \, a^{2} e^{\left (2 \, d x + 2 \, c\right )} - 25 \, a^{2} e^{\left (d x + c\right )} + 3 i \, a^{2}\right )} \sqrt{i \, a e^{\left (2 \, d x + 2 \, c\right )} + 2 \, a e^{\left (d x + c\right )} - i \, a} e^{\left (-\frac{1}{2} \, d x - \frac{1}{2} \, c\right )}}{30 \, d e^{\left (3 \, d x + 3 \, c\right )} - 30 i \, d e^{\left (2 \, d x + 2 \, c\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 \, a \sinh \left (d x + c\right ) + a\right )}^{\frac{5}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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