Optimal. Leaf size=81 \[ \frac{8 a^2 (3 B+5 i A) \cosh (x)}{15 \sqrt{a+i a \sinh (x)}}+\frac{2}{15} a (3 B+5 i A) \cosh (x) \sqrt{a+i a \sinh (x)}+\frac{2}{5} B \cosh (x) (a+i a \sinh (x))^{3/2} \]
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Rubi [A] time = 0.0808424, antiderivative size = 81, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {2751, 2647, 2646} \[ \frac{8 a^2 (3 B+5 i A) \cosh (x)}{15 \sqrt{a+i a \sinh (x)}}+\frac{2}{15} a (3 B+5 i A) \cosh (x) \sqrt{a+i a \sinh (x)}+\frac{2}{5} B \cosh (x) (a+i a \sinh (x))^{3/2} \]
Antiderivative was successfully verified.
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Rule 2751
Rule 2647
Rule 2646
Rubi steps
\begin{align*} \int (a+i a \sinh (x))^{3/2} (A+B \sinh (x)) \, dx &=\frac{2}{5} B \cosh (x) (a+i a \sinh (x))^{3/2}+\frac{1}{5} (5 A-3 i B) \int (a+i a \sinh (x))^{3/2} \, dx\\ &=\frac{2}{15} a (5 i A+3 B) \cosh (x) \sqrt{a+i a \sinh (x)}+\frac{2}{5} B \cosh (x) (a+i a \sinh (x))^{3/2}+\frac{1}{15} (4 a (5 A-3 i B)) \int \sqrt{a+i a \sinh (x)} \, dx\\ &=\frac{8 a^2 (5 i A+3 B) \cosh (x)}{15 \sqrt{a+i a \sinh (x)}}+\frac{2}{15} a (5 i A+3 B) \cosh (x) \sqrt{a+i a \sinh (x)}+\frac{2}{5} B \cosh (x) (a+i a \sinh (x))^{3/2}\\ \end{align*}
Mathematica [A] time = 0.214721, size = 83, normalized size = 1.02 \[ -\frac{a \sqrt{a+i a \sinh (x)} \left (\cosh \left (\frac{x}{2}\right )-i \sinh \left (\frac{x}{2}\right )\right ) (2 (5 A-9 i B) \sinh (x)-50 i A+3 B \cosh (2 x)-39 B)}{15 \left (\cosh \left (\frac{x}{2}\right )+i \sinh \left (\frac{x}{2}\right )\right )} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.108, size = 0, normalized size = 0. \begin{align*} \int \left ( a+ia\sinh \left ( x \right ) \right ) ^{{\frac{3}{2}}} \left ( A+B\sinh \left ( x \right ) \right ) \, 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 (B \sinh \left (x\right ) + A\right )}{\left (i \, a \sinh \left (x\right ) + a\right )}^{\frac{3}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.76496, size = 294, normalized size = 3.63 \begin{align*} \frac{\sqrt{\frac{1}{2}}{\left (3 i \, B a e^{\left (5 \, x\right )} +{\left (10 i \, A + 15 \, B\right )} a e^{\left (4 \, x\right )} + 30 \,{\left (3 \, A - 2 i \, B\right )} a e^{\left (3 \, x\right )} +{\left (90 i \, A + 60 \, B\right )} a e^{\left (2 \, x\right )} + 5 \,{\left (2 \, A - 3 i \, B\right )} a e^{x} - 3 \, B a\right )} \sqrt{i \, a e^{\left (2 \, x\right )} + 2 \, a e^{x} - i \, a} e^{\left (-\frac{1}{2} \, x\right )}}{30 \,{\left (e^{\left (3 \, x\right )} - i \, e^{\left (2 \, x\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 (B \sinh \left (x\right ) + A\right )}{\left (i \, a \sinh \left (x\right ) + a\right )}^{\frac{3}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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