Integrand size = 20, antiderivative size = 81 \[ \int (a+i a \sinh (x))^{3/2} (A+B \sinh (x)) \, dx=\frac {8 a^2 (5 i A+3 B) \cosh (x)}{15 \sqrt {a+i a \sinh (x)}}+\frac {2}{15} a (5 i A+3 B) \cosh (x) \sqrt {a+i a \sinh (x)}+\frac {2}{5} B \cosh (x) (a+i a \sinh (x))^{3/2} \]
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Time = 0.06 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {2830, 2726, 2725} \[ \int (a+i a \sinh (x))^{3/2} (A+B \sinh (x)) \, dx=\frac {8 a^2 (3 B+5 i A) \cosh (x)}{15 \sqrt {a+i a \sinh (x)}}+\frac {2}{15} a (3 B+5 i A) \cosh (x) \sqrt {a+i a \sinh (x)}+\frac {2}{5} B \cosh (x) (a+i a \sinh (x))^{3/2} \]
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Rule 2725
Rule 2726
Rule 2830
Rubi steps \begin{align*} \text {integral}& = \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*}
Time = 1.73 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.02 \[ \int (a+i a \sinh (x))^{3/2} (A+B \sinh (x)) \, dx=-\frac {a \left (\cosh \left (\frac {x}{2}\right )-i \sinh \left (\frac {x}{2}\right )\right ) \sqrt {a+i a \sinh (x)} (-50 i A-39 B+3 B \cosh (2 x)+2 (5 A-9 i B) \sinh (x))}{15 \left (\cosh \left (\frac {x}{2}\right )+i \sinh \left (\frac {x}{2}\right )\right )} \]
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\[\int \left (a +i a \sinh \left (x \right )\right )^{\frac {3}{2}} \left (A +B \sinh \left (x \right )\right )d x\]
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none
Time = 0.31 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.01 \[ \int (a+i a \sinh (x))^{3/2} (A+B \sinh (x)) \, dx=\frac {1}{30} \, {\left (3 i \, B a e^{\left (5 \, x\right )} - 5 \, {\left (-2 i \, A - 3 \, B\right )} a e^{\left (4 \, x\right )} + 30 \, {\left (3 \, A - 2 i \, B\right )} a e^{\left (3 \, x\right )} - 30 \, {\left (-3 i \, A - 2 \, B\right )} a e^{\left (2 \, x\right )} + 5 \, {\left (2 \, A - 3 i \, B\right )} a e^{x} - 3 \, B a\right )} \sqrt {\frac {1}{2} i \, a e^{\left (-x\right )}} e^{\left (-2 \, x\right )} \]
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\[ \int (a+i a \sinh (x))^{3/2} (A+B \sinh (x)) \, dx=\int \left (i a \left (\sinh {\left (x \right )} - i\right )\right )^{\frac {3}{2}} \left (A + B \sinh {\left (x \right )}\right )\, dx \]
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\[ \int (a+i a \sinh (x))^{3/2} (A+B \sinh (x)) \, dx=\int { {\left (B \sinh \left (x\right ) + A\right )} {\left (i \, a \sinh \left (x\right ) + a\right )}^{\frac {3}{2}} \,d x } \]
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\[ \int (a+i a \sinh (x))^{3/2} (A+B \sinh (x)) \, dx=\int { {\left (B \sinh \left (x\right ) + A\right )} {\left (i \, a \sinh \left (x\right ) + a\right )}^{\frac {3}{2}} \,d x } \]
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Timed out. \[ \int (a+i a \sinh (x))^{3/2} (A+B \sinh (x)) \, dx=\int \left (A+B\,\mathrm {sinh}\left (x\right )\right )\,{\left (a+a\,\mathrm {sinh}\left (x\right )\,1{}\mathrm {i}\right )}^{3/2} \,d x \]
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