Integrand size = 20, antiderivative size = 79 \[ \int \frac {A+B \sinh (x)}{(a+i a \sinh (x))^{3/2}} \, dx=\frac {(i A+3 B) \text {arctanh}\left (\frac {\sqrt {a} \cosh (x)}{\sqrt {2} \sqrt {a+i a \sinh (x)}}\right )}{2 \sqrt {2} a^{3/2}}+\frac {(i A-B) \cosh (x)}{2 (a+i a \sinh (x))^{3/2}} \]
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Time = 0.05 (sec) , antiderivative size = 79, 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 = {2829, 2728, 212} \[ \int \frac {A+B \sinh (x)}{(a+i a \sinh (x))^{3/2}} \, dx=\frac {(3 B+i A) \text {arctanh}\left (\frac {\sqrt {a} \cosh (x)}{\sqrt {2} \sqrt {a+i a \sinh (x)}}\right )}{2 \sqrt {2} a^{3/2}}+\frac {(-B+i A) \cosh (x)}{2 (a+i a \sinh (x))^{3/2}} \]
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Rule 212
Rule 2728
Rule 2829
Rubi steps \begin{align*} \text {integral}& = \frac {(i A-B) \cosh (x)}{2 (a+i a \sinh (x))^{3/2}}+\frac {(A-3 i B) \int \frac {1}{\sqrt {a+i a \sinh (x)}} \, dx}{4 a} \\ & = \frac {(i A-B) \cosh (x)}{2 (a+i a \sinh (x))^{3/2}}+\frac {(i A+3 B) \text {Subst}\left (\int \frac {1}{2 a-x^2} \, dx,x,\frac {a \cosh (x)}{\sqrt {a+i a \sinh (x)}}\right )}{2 a} \\ & = \frac {(i A+3 B) \text {arctanh}\left (\frac {\sqrt {a} \cosh (x)}{\sqrt {2} \sqrt {a+i a \sinh (x)}}\right )}{2 \sqrt {2} a^{3/2}}+\frac {(i A-B) \cosh (x)}{2 (a+i a \sinh (x))^{3/2}} \\ \end{align*}
Time = 0.34 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.33 \[ \int \frac {A+B \sinh (x)}{(a+i a \sinh (x))^{3/2}} \, dx=\frac {\left (\cosh \left (\frac {x}{2}\right )+i \sinh \left (\frac {x}{2}\right )\right ) \left (i (A+i B) \cosh \left (\frac {x}{2}\right )+(A+i B) \sinh \left (\frac {x}{2}\right )+(1+i) \sqrt [4]{-1} (A-3 i B) \arctan \left (\frac {i+\tanh \left (\frac {x}{4}\right )}{\sqrt {2}}\right ) (-i+\sinh (x))\right )}{2 (a+i a \sinh (x))^{3/2}} \]
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\[\int \frac {A +B \sinh \left (x \right )}{\left (a +i a \sinh \left (x \right )\right )^{\frac {3}{2}}}d x\]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 264 vs. \(2 (54) = 108\).
Time = 0.31 (sec) , antiderivative size = 264, normalized size of antiderivative = 3.34 \[ \int \frac {A+B \sinh (x)}{(a+i a \sinh (x))^{3/2}} \, dx=\frac {\sqrt {\frac {1}{2}} {\left (a^{2} e^{\left (2 \, x\right )} - 2 i \, a^{2} e^{x} - a^{2}\right )} \sqrt {-\frac {A^{2} - 6 i \, A B - 9 \, B^{2}}{a^{3}}} \log \left (\frac {\sqrt {\frac {1}{2}} a^{2} \sqrt {-\frac {A^{2} - 6 i \, A B - 9 \, B^{2}}{a^{3}}} + \sqrt {\frac {1}{2} i \, a e^{\left (-x\right )}} {\left (i \, A + 3 \, B\right )}}{i \, A + 3 \, B}\right ) - \sqrt {\frac {1}{2}} {\left (a^{2} e^{\left (2 \, x\right )} - 2 i \, a^{2} e^{x} - a^{2}\right )} \sqrt {-\frac {A^{2} - 6 i \, A B - 9 \, B^{2}}{a^{3}}} \log \left (-\frac {\sqrt {\frac {1}{2}} a^{2} \sqrt {-\frac {A^{2} - 6 i \, A B - 9 \, B^{2}}{a^{3}}} - \sqrt {\frac {1}{2} i \, a e^{\left (-x\right )}} {\left (i \, A + 3 \, B\right )}}{i \, A + 3 \, B}\right ) - 2 \, {\left ({\left (i \, A - B\right )} e^{\left (2 \, x\right )} - {\left (A + i \, B\right )} e^{x}\right )} \sqrt {\frac {1}{2} i \, a e^{\left (-x\right )}}}{2 \, {\left (a^{2} e^{\left (2 \, x\right )} - 2 i \, a^{2} e^{x} - a^{2}\right )}} \]
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\[ \int \frac {A+B \sinh (x)}{(a+i a \sinh (x))^{3/2}} \, dx=\int \frac {A + B \sinh {\left (x \right )}}{\left (i a \left (\sinh {\left (x \right )} - i\right )\right )^{\frac {3}{2}}}\, dx \]
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\[ \int \frac {A+B \sinh (x)}{(a+i a \sinh (x))^{3/2}} \, dx=\int { \frac {B \sinh \left (x\right ) + A}{{\left (i \, a \sinh \left (x\right ) + a\right )}^{\frac {3}{2}}} \,d x } \]
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\[ \int \frac {A+B \sinh (x)}{(a+i a \sinh (x))^{3/2}} \, dx=\int { \frac {B \sinh \left (x\right ) + A}{{\left (i \, a \sinh \left (x\right ) + a\right )}^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {A+B \sinh (x)}{(a+i a \sinh (x))^{3/2}} \, dx=\int \frac {A+B\,\mathrm {sinh}\left (x\right )}{{\left (a+a\,\mathrm {sinh}\left (x\right )\,1{}\mathrm {i}\right )}^{3/2}} \,d x \]
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