Optimal. Leaf size=48 \[ \frac{2 a (B+3 i A) \cosh (x)}{3 \sqrt{a+i a \sinh (x)}}+\frac{2}{3} B \cosh (x) \sqrt{a+i a \sinh (x)} \]
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Rubi [A] time = 0.0538788, antiderivative size = 48, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {2751, 2646} \[ \frac{2 a (B+3 i A) \cosh (x)}{3 \sqrt{a+i a \sinh (x)}}+\frac{2}{3} B \cosh (x) \sqrt{a+i a \sinh (x)} \]
Antiderivative was successfully verified.
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Rule 2751
Rule 2646
Rubi steps
\begin{align*} \int \sqrt{a+i a \sinh (x)} (A+B \sinh (x)) \, dx &=\frac{2}{3} B \cosh (x) \sqrt{a+i a \sinh (x)}+\frac{1}{3} (3 A-i B) \int \sqrt{a+i a \sinh (x)} \, dx\\ &=\frac{2 a (3 i A+B) \cosh (x)}{3 \sqrt{a+i a \sinh (x)}}+\frac{2}{3} B \cosh (x) \sqrt{a+i a \sinh (x)}\\ \end{align*}
Mathematica [A] time = 0.0739076, size = 66, normalized size = 1.38 \[ \frac{2 \sqrt{a+i a \sinh (x)} \left (\sinh \left (\frac{x}{2}\right )+i \cosh \left (\frac{x}{2}\right )\right ) (3 A+B \sinh (x)-2 i B)}{3 \left (\cosh \left (\frac{x}{2}\right )+i \sinh \left (\frac{x}{2}\right )\right )} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.108, size = 0, normalized size = 0. \begin{align*} \int \sqrt{a+ia\sinh \left ( x \right ) } \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 )} \sqrt{i \, a \sinh \left (x\right ) + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.72814, size = 189, normalized size = 3.94 \begin{align*} \frac{\sqrt{\frac{1}{2}}{\left (B e^{\left (3 \, x\right )} + 3 \,{\left (2 \, A - i \, B\right )} e^{\left (2 \, x\right )} +{\left (6 i \, A + 3 \, B\right )} e^{x} - i \, B\right )} \sqrt{i \, a e^{\left (2 \, x\right )} + 2 \, a e^{x} - i \, a} e^{\left (-\frac{1}{2} \, x\right )}}{3 \,{\left (e^{\left (2 \, x\right )} - i \, e^{x}\right )}} \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 \sqrt{a \left (i \sinh{\left (x \right )} + 1\right )} \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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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (B \sinh \left (x\right ) + A\right )} \sqrt{i \, a \sinh \left (x\right ) + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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