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 = 0.08, antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, 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 2646
Rule 2647
Rule 2751
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*}
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Mathematica [A] time = 0.22, 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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fricas [A] time = 0.67, size = 80, normalized size = 0.99 \[ \frac {1}{30} \, {\left (3 i \, B a e^{\left (5 \, x\right )} - 5 \, {\left (-2 i \, A - 3 \, B\right )} a e^{\left (4 \, x\right )} + {\left (90 \, A - 60 i \, B\right )} a e^{\left (3 \, x\right )} - 30 \, {\left (-3 i \, A - 2 \, B\right )} a e^{\left (2 \, x\right )} + {\left (10 \, A - 15 i \, B\right )} a e^{x} - 3 \, B a\right )} \sqrt {\frac {1}{2} i \, a e^{\left (-x\right )}} e^{\left (-2 \, x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (B \sinh \relax (x) + A\right )} {\left (i \, a \sinh \relax (x) + a\right )}^{\frac {3}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.14, size = 0, normalized size = 0.00 \[ \int \left (a +i a \sinh \relax (x )\right )^{\frac {3}{2}} \left (A +B \sinh \relax (x )\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (B \sinh \relax (x) + A\right )} {\left (i \, a \sinh \relax (x) + a\right )}^{\frac {3}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \left (A+B\,\mathrm {sinh}\relax (x)\right )\,{\left (a+a\,\mathrm {sinh}\relax (x)\,1{}\mathrm {i}\right )}^{3/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (i a \left (\sinh {\relax (x )} - i\right )\right )^{\frac {3}{2}} \left (A + B \sinh {\relax (x )}\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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