Optimal. Leaf size=506 \[ -\frac {256 a^2 x \sqrt {a+i a \sinh (c+d x)}}{15 d^2}-\frac {128 a^2 x \cosh ^2\left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right ) \sqrt {a+i a \sinh (c+d x)}}{45 d^2}-\frac {32 a^2 x \cosh ^4\left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right ) \sqrt {a+i a \sinh (c+d x)}}{25 d^2}+\frac {32 a^2 x^2 \cosh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right ) \sinh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right ) \sqrt {a+i a \sinh (c+d x)}}{15 d}+\frac {8 a^2 x^2 \cosh ^3\left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right ) \sinh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right ) \sqrt {a+i a \sinh (c+d x)}}{5 d}+\frac {9536 a^2 \sqrt {a+i a \sinh (c+d x)} \tanh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{225 d^3}+\frac {64 a^2 x^2 \sqrt {a+i a \sinh (c+d x)} \tanh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{15 d}+\frac {2432 a^2 \sinh ^2\left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right ) \sqrt {a+i a \sinh (c+d x)} \tanh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{675 d^3}+\frac {64 a^2 \sinh ^4\left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right ) \sqrt {a+i a \sinh (c+d x)} \tanh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{125 d^3} \]
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Rubi [A]
time = 0.26, antiderivative size = 506, normalized size of antiderivative = 1.00, number of steps
used = 10, number of rules used = 5, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {3400, 3392,
3377, 2718, 2713} \begin {gather*} \frac {64 a^2 \sinh ^4\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \tanh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \sqrt {a+i a \sinh (c+d x)}}{125 d^3}+\frac {2432 a^2 \sinh ^2\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \tanh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \sqrt {a+i a \sinh (c+d x)}}{675 d^3}+\frac {9536 a^2 \tanh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \sqrt {a+i a \sinh (c+d x)}}{225 d^3}-\frac {256 a^2 x \sqrt {a+i a \sinh (c+d x)}}{15 d^2}-\frac {32 a^2 x \cosh ^4\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \sqrt {a+i a \sinh (c+d x)}}{25 d^2}-\frac {128 a^2 x \cosh ^2\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \sqrt {a+i a \sinh (c+d x)}}{45 d^2}+\frac {64 a^2 x^2 \tanh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \sqrt {a+i a \sinh (c+d x)}}{15 d}+\frac {8 a^2 x^2 \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \cosh ^3\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \sqrt {a+i a \sinh (c+d x)}}{5 d}+\frac {32 a^2 x^2 \sinh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \cosh \left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \sqrt {a+i a \sinh (c+d x)}}{15 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 2713
Rule 2718
Rule 3377
Rule 3392
Rule 3400
Rubi steps
\begin {align*} \int x^2 (a+i a \sinh (c+d x))^{5/2} \, dx &=\left (4 a^2 \text {csch}\left (\frac {c}{2}-\frac {i \pi }{4}+\frac {d x}{2}\right ) \sqrt {a+i a \sinh (c+d x)}\right ) \int x^2 \sinh ^5\left (\frac {c}{2}-\frac {i \pi }{4}+\frac {d x}{2}\right ) \, dx\\ &=-\frac {32 a^2 x \cosh ^4\left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right ) \sqrt {a+i a \sinh (c+d x)}}{25 d^2}+\frac {8 a^2 x^2 \cosh ^3\left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right ) \sinh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right ) \sqrt {a+i a \sinh (c+d x)}}{5 d}-\frac {1}{5} \left (16 a^2 \text {csch}\left (\frac {c}{2}-\frac {i \pi }{4}+\frac {d x}{2}\right ) \sqrt {a+i a \sinh (c+d x)}\right ) \int x^2 \sinh ^3\left (\frac {c}{2}-\frac {i \pi }{4}+\frac {d x}{2}\right ) \, dx+\frac {\left (32 a^2 \text {csch}\left (\frac {c}{2}-\frac {i \pi }{4}+\frac {d x}{2}\right ) \sqrt {a+i a \sinh (c+d x)}\right ) \int \sinh ^5\left (\frac {c}{2}-\frac {i \pi }{4}+\frac {d x}{2}\right ) \, dx}{25 d^2}\\ &=-\frac {128 a^2 x \cosh ^2\left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right ) \sqrt {a+i a \sinh (c+d x)}}{45 d^2}-\frac {32 a^2 x \cosh ^4\left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right ) \sqrt {a+i a \sinh (c+d x)}}{25 d^2}+\frac {32 a^2 x^2 \cosh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right ) \sinh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right ) \sqrt {a+i a \sinh (c+d x)}}{15 d}+\frac {8 a^2 x^2 \cosh ^3\left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right ) \sinh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right ) \sqrt {a+i a \sinh (c+d x)}}{5 d}+\frac {1}{15} \left (32 a^2 \text {csch}\left (\frac {c}{2}-\frac {i \pi }{4}+\frac {d x}{2}\right ) \sqrt {a+i a \sinh (c+d x)}\right ) \int x^2 \sinh \left (\frac {c}{2}-\frac {i \pi }{4}+\frac {d x}{2}\right ) \, dx+\frac {\left (64 a^2 \text {csch}\left (\frac {c}{2}-\frac {i \pi }{4}+\frac {d x}{2}\right ) \sqrt {a+i a \sinh (c+d x)}\right ) \text {Subst}\left (\int \left (1-2 x^2+x^4\right ) \, dx,x,\cosh \left (\frac {c}{2}-\frac {i \pi }{4}+\frac {d x}{2}\right )\right )}{25 d^3}-\frac {\left (128 a^2 \text {csch}\left (\frac {c}{2}-\frac {i \pi }{4}+\frac {d x}{2}\right ) \sqrt {a+i a \sinh (c+d x)}\right ) \int \sinh ^3\left (\frac {c}{2}-\frac {i \pi }{4}+\frac {d x}{2}\right ) \, dx}{45 d^2}\\ &=-\frac {128 a^2 x \cosh ^2\left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right ) \sqrt {a+i a \sinh (c+d x)}}{45 d^2}-\frac {32 a^2 x \cosh ^4\left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right ) \sqrt {a+i a \sinh (c+d x)}}{25 d^2}+\frac {32 a^2 x^2 \cosh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right ) \sinh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right ) \sqrt {a+i a \sinh (c+d x)}}{15 d}+\frac {8 a^2 x^2 \cosh ^3\left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right ) \sinh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right ) \sqrt {a+i a \sinh (c+d x)}}{5 d}+\frac {64 a^2 \sqrt {a+i a \sinh (c+d x)} \tanh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{25 d^3}+\frac {64 a^2 x^2 \sqrt {a+i a \sinh (c+d x)} \tanh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{15 d}+\frac {128 a^2 \sinh ^2\left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right ) \sqrt {a+i a \sinh (c+d x)} \tanh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{75 d^3}+\frac {64 a^2 \sinh ^4\left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right ) \sqrt {a+i a \sinh (c+d x)} \tanh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{125 d^3}+\frac {\left (256 a^2 \text {csch}\left (\frac {c}{2}-\frac {i \pi }{4}+\frac {d x}{2}\right ) \sqrt {a+i a \sinh (c+d x)}\right ) \text {Subst}\left (\int \left (1-x^2\right ) \, dx,x,\cosh \left (\frac {c}{2}-\frac {i \pi }{4}+\frac {d x}{2}\right )\right )}{45 d^3}-\frac {\left (128 a^2 \text {csch}\left (\frac {c}{2}-\frac {i \pi }{4}+\frac {d x}{2}\right ) \sqrt {a+i a \sinh (c+d x)}\right ) \int x \cosh \left (\frac {c}{2}-\frac {i \pi }{4}+\frac {d x}{2}\right ) \, dx}{15 d}\\ &=-\frac {256 a^2 x \sqrt {a+i a \sinh (c+d x)}}{15 d^2}-\frac {128 a^2 x \cosh ^2\left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right ) \sqrt {a+i a \sinh (c+d x)}}{45 d^2}-\frac {32 a^2 x \cosh ^4\left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right ) \sqrt {a+i a \sinh (c+d x)}}{25 d^2}+\frac {32 a^2 x^2 \cosh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right ) \sinh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right ) \sqrt {a+i a \sinh (c+d x)}}{15 d}+\frac {8 a^2 x^2 \cosh ^3\left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right ) \sinh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right ) \sqrt {a+i a \sinh (c+d x)}}{5 d}+\frac {1856 a^2 \sqrt {a+i a \sinh (c+d x)} \tanh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{225 d^3}+\frac {64 a^2 x^2 \sqrt {a+i a \sinh (c+d x)} \tanh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{15 d}+\frac {2432 a^2 \sinh ^2\left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right ) \sqrt {a+i a \sinh (c+d x)} \tanh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{675 d^3}+\frac {64 a^2 \sinh ^4\left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right ) \sqrt {a+i a \sinh (c+d x)} \tanh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{125 d^3}-\frac {\left (256 i a^2 \text {csch}\left (\frac {c}{2}-\frac {i \pi }{4}+\frac {d x}{2}\right ) \sqrt {a+i a \sinh (c+d x)}\right ) \int \cosh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right ) \, dx}{15 d^2}\\ &=-\frac {256 a^2 x \sqrt {a+i a \sinh (c+d x)}}{15 d^2}-\frac {128 a^2 x \cosh ^2\left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right ) \sqrt {a+i a \sinh (c+d x)}}{45 d^2}-\frac {32 a^2 x \cosh ^4\left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right ) \sqrt {a+i a \sinh (c+d x)}}{25 d^2}+\frac {32 a^2 x^2 \cosh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right ) \sinh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right ) \sqrt {a+i a \sinh (c+d x)}}{15 d}+\frac {8 a^2 x^2 \cosh ^3\left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right ) \sinh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right ) \sqrt {a+i a \sinh (c+d x)}}{5 d}+\frac {9536 a^2 \sqrt {a+i a \sinh (c+d x)} \tanh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{225 d^3}+\frac {64 a^2 x^2 \sqrt {a+i a \sinh (c+d x)} \tanh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{15 d}+\frac {2432 a^2 \sinh ^2\left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right ) \sqrt {a+i a \sinh (c+d x)} \tanh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{675 d^3}+\frac {64 a^2 \sinh ^4\left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right ) \sqrt {a+i a \sinh (c+d x)} \tanh \left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right )}{125 d^3}\\ \end {align*}
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Mathematica [A]
time = 0.72, size = 300, normalized size = 0.59 \begin {gather*} \frac {a^2 \sqrt {a+i a \sinh (c+d x)} \left (33750 i \left (8+4 i d x+d^2 x^2\right ) \cosh \left (\frac {1}{2} (c+d x)\right )+625 \left (8 i+12 d x+9 i d^2 x^2\right ) \cosh \left (\frac {3}{2} (c+d x)\right )-216 i \cosh \left (\frac {5}{2} (c+d x)\right )+540 d x \cosh \left (\frac {5}{2} (c+d x)\right )-675 i d^2 x^2 \cosh \left (\frac {5}{2} (c+d x)\right )+270000 \sinh \left (\frac {1}{2} (c+d x)\right )-135000 i d x \sinh \left (\frac {1}{2} (c+d x)\right )+33750 d^2 x^2 \sinh \left (\frac {1}{2} (c+d x)\right )-5000 \sinh \left (\frac {3}{2} (c+d x)\right )-7500 i d x \sinh \left (\frac {3}{2} (c+d x)\right )-5625 d^2 x^2 \sinh \left (\frac {3}{2} (c+d x)\right )-216 \sinh \left (\frac {5}{2} (c+d x)\right )+540 i d x \sinh \left (\frac {5}{2} (c+d x)\right )-675 d^2 x^2 \sinh \left (\frac {5}{2} (c+d x)\right )\right )}{6750 d^3 \left (\cosh \left (\frac {1}{2} (c+d x)\right )+i \sinh \left (\frac {1}{2} (c+d x)\right )\right )} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.12, size = 0, normalized size = 0.00 \[\int x^{2} \left (a +i a \sinh \left (d x +c \right )\right )^{\frac {5}{2}}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int x^2\,{\left (a+a\,\mathrm {sinh}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{5/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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