Optimal. Leaf size=112 \[ \frac {64 a^3 (5 B+7 i A) \cosh (x)}{105 \sqrt {a+i a \sinh (x)}}+\frac {16}{105} a^2 (5 B+7 i A) \cosh (x) \sqrt {a+i a \sinh (x)}+\frac {2}{35} a (5 B+7 i A) \cosh (x) (a+i a \sinh (x))^{3/2}+\frac {2}{7} B \cosh (x) (a+i a \sinh (x))^{5/2} \]
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Rubi [A] time = 0.10, antiderivative size = 112, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {2751, 2647, 2646} \[ \frac {64 a^3 (5 B+7 i A) \cosh (x)}{105 \sqrt {a+i a \sinh (x)}}+\frac {16}{105} a^2 (5 B+7 i A) \cosh (x) \sqrt {a+i a \sinh (x)}+\frac {2}{35} a (5 B+7 i A) \cosh (x) (a+i a \sinh (x))^{3/2}+\frac {2}{7} B \cosh (x) (a+i a \sinh (x))^{5/2} \]
Antiderivative was successfully verified.
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Rule 2646
Rule 2647
Rule 2751
Rubi steps
\begin {align*} \int (a+i a \sinh (x))^{5/2} (A+B \sinh (x)) \, dx &=\frac {2}{7} B \cosh (x) (a+i a \sinh (x))^{5/2}+\frac {1}{7} (7 A-5 i B) \int (a+i a \sinh (x))^{5/2} \, dx\\ &=\frac {2}{35} a (7 i A+5 B) \cosh (x) (a+i a \sinh (x))^{3/2}+\frac {2}{7} B \cosh (x) (a+i a \sinh (x))^{5/2}+\frac {1}{35} (8 a (7 A-5 i B)) \int (a+i a \sinh (x))^{3/2} \, dx\\ &=\frac {16}{105} a^2 (7 i A+5 B) \cosh (x) \sqrt {a+i a \sinh (x)}+\frac {2}{35} a (7 i A+5 B) \cosh (x) (a+i a \sinh (x))^{3/2}+\frac {2}{7} B \cosh (x) (a+i a \sinh (x))^{5/2}+\frac {1}{105} \left (32 a^2 (7 A-5 i B)\right ) \int \sqrt {a+i a \sinh (x)} \, dx\\ &=\frac {64 a^3 (7 i A+5 B) \cosh (x)}{105 \sqrt {a+i a \sinh (x)}}+\frac {16}{105} a^2 (7 i A+5 B) \cosh (x) \sqrt {a+i a \sinh (x)}+\frac {2}{35} a (7 i A+5 B) \cosh (x) (a+i a \sinh (x))^{3/2}+\frac {2}{7} B \cosh (x) (a+i a \sinh (x))^{5/2}\\ \end {align*}
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Mathematica [A] time = 0.36, size = 100, normalized size = 0.89 \[ \frac {a^2 \sqrt {a+i a \sinh (x)} \left (\cosh \left (\frac {x}{2}\right )-i \sinh \left (\frac {x}{2}\right )\right ) ((-392 A+505 i B) \sinh (x)+(-120 B-42 i A) \cosh (2 x)+1246 i A-15 i B \sinh (3 x)+1040 B)}{210 \left (\cosh \left (\frac {x}{2}\right )+i \sinh \left (\frac {x}{2}\right )\right )} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.48, size = 125, normalized size = 1.12 \[ -\frac {1}{420} \, {\left (15 \, B a^{2} e^{\left (7 \, x\right )} + {\left (42 \, A - 105 i \, B\right )} a^{2} e^{\left (6 \, x\right )} + 35 \, {\left (-10 i \, A - 11 \, B\right )} a^{2} e^{\left (5 \, x\right )} - {\left (2100 \, A - 1575 i \, B\right )} a^{2} e^{\left (4 \, x\right )} + 525 \, {\left (-4 i \, A - 3 \, B\right )} a^{2} e^{\left (3 \, x\right )} - {\left (350 \, A - 385 i \, B\right )} a^{2} e^{\left (2 \, x\right )} + 21 \, {\left (2 i \, A + 5 \, B\right )} a^{2} e^{x} - 15 i \, B a^{2}\right )} \sqrt {\frac {1}{2} i \, a e^{\left (-x\right )}} e^{\left (-3 \, x\right )} \]
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 \[ \int {\left (B \sinh \relax (x) + A\right )} {\left (i \, a \sinh \relax (x) + a\right )}^{\frac {5}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.16, size = 0, normalized size = 0.00 \[ \int \left (a +i a \sinh \relax (x )\right )^{\frac {5}{2}} \left (A +B \sinh \relax (x )\right )\, dx \]
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 \[ \int {\left (B \sinh \relax (x) + A\right )} {\left (i \, a \sinh \relax (x) + a\right )}^{\frac {5}{2}}\,{d x} \]
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 \[ \int \left (A+B\,\mathrm {sinh}\relax (x)\right )\,{\left (a+a\,\mathrm {sinh}\relax (x)\,1{}\mathrm {i}\right )}^{5/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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