Optimal. Leaf size=444 \[ -\frac {4 a^2 \cosh ^4\left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right ) \sqrt {a+i a \sinh (c+d x)}}{x}-\frac {5}{8} a^2 d \text {Chi}\left (\frac {5 d x}{2}\right ) \text {sech}\left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right ) \sinh \left (\frac {5 c}{2}+\frac {i \pi }{4}\right ) \sqrt {a+i a \sinh (c+d x)}-\frac {15}{8} a^2 d \text {Chi}\left (\frac {3 d x}{2}\right ) \text {sech}\left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right ) \sinh \left (\frac {1}{4} (6 c-i \pi )\right ) \sqrt {a+i a \sinh (c+d x)}+\frac {5}{4} a^2 d \text {Chi}\left (\frac {d x}{2}\right ) \text {sech}\left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right ) \sinh \left (\frac {1}{4} (2 c+i \pi )\right ) \sqrt {a+i a \sinh (c+d x)}+\frac {5}{4} a^2 d \cosh \left (\frac {1}{4} (2 c+i \pi )\right ) \text {sech}\left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right ) \sqrt {a+i a \sinh (c+d x)} \text {Shi}\left (\frac {d x}{2}\right )-\frac {15}{8} a^2 d \cosh \left (\frac {1}{4} (6 c-i \pi )\right ) \text {sech}\left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right ) \sqrt {a+i a \sinh (c+d x)} \text {Shi}\left (\frac {3 d x}{2}\right )-\frac {5}{8} a^2 d \cosh \left (\frac {5 c}{2}+\frac {i \pi }{4}\right ) \text {sech}\left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right ) \sqrt {a+i a \sinh (c+d x)} \text {Shi}\left (\frac {5 d x}{2}\right ) \]
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Rubi [A]
time = 0.30, antiderivative size = 444, normalized size of antiderivative = 1.00, number of steps
used = 12, number of rules used = 5, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {3400, 3394,
3384, 3379, 3382} \begin {gather*} -\frac {5}{8} a^2 d \sinh \left (\frac {5 c}{2}+\frac {i \pi }{4}\right ) \text {Chi}\left (\frac {5 d x}{2}\right ) \text {sech}\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \sqrt {a+i a \sinh (c+d x)}-\frac {15}{8} a^2 d \sinh \left (\frac {1}{4} (6 c-i \pi )\right ) \text {Chi}\left (\frac {3 d x}{2}\right ) \text {sech}\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \sqrt {a+i a \sinh (c+d x)}+\frac {5}{4} a^2 d \sinh \left (\frac {1}{4} (2 c+i \pi )\right ) \text {Chi}\left (\frac {d x}{2}\right ) \text {sech}\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \sqrt {a+i a \sinh (c+d x)}+\frac {5}{4} a^2 d \cosh \left (\frac {1}{4} (2 c+i \pi )\right ) \text {Shi}\left (\frac {d x}{2}\right ) \text {sech}\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \sqrt {a+i a \sinh (c+d x)}-\frac {15}{8} a^2 d \cosh \left (\frac {1}{4} (6 c-i \pi )\right ) \text {Shi}\left (\frac {3 d x}{2}\right ) \text {sech}\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \sqrt {a+i a \sinh (c+d x)}-\frac {5}{8} a^2 d \cosh \left (\frac {5 c}{2}+\frac {i \pi }{4}\right ) \text {Shi}\left (\frac {5 d x}{2}\right ) \text {sech}\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \sqrt {a+i a \sinh (c+d x)}-\frac {4 a^2 \cosh ^4\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \sqrt {a+i a \sinh (c+d x)}}{x} \end {gather*}
Antiderivative was successfully verified.
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Rule 3379
Rule 3382
Rule 3384
Rule 3394
Rule 3400
Rubi steps
\begin {align*} \int \frac {(a+i a \sinh (c+d x))^{5/2}}{x^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 \frac {\sinh ^5\left (\frac {c}{2}-\frac {i \pi }{4}+\frac {d x}{2}\right )}{x^2} \, dx\\ &=-\frac {4 a^2 \cosh ^4\left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right ) \sqrt {a+i a \sinh (c+d x)}}{x}+\left (10 a^2 d \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 \left (\frac {\cosh \left (\frac {1}{4} (2 c-i \pi )+\frac {d x}{2}\right )}{8 x}+\frac {3 \cosh \left (\frac {1}{4} (6 c+i \pi )+\frac {3 d x}{2}\right )}{16 x}-\frac {\cosh \left (\frac {1}{4} (10 c-i \pi )+\frac {5 d x}{2}\right )}{16 x}\right ) \, dx\\ &=-\frac {4 a^2 \cosh ^4\left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right ) \sqrt {a+i a \sinh (c+d x)}}{x}-\frac {1}{8} \left (5 a^2 d \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 \frac {\cosh \left (\frac {1}{4} (10 c-i \pi )+\frac {5 d x}{2}\right )}{x} \, dx+\frac {1}{4} \left (5 a^2 d \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 \frac {\cosh \left (\frac {1}{4} (2 c-i \pi )+\frac {d x}{2}\right )}{x} \, dx+\frac {1}{8} \left (15 a^2 d \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 \frac {\cosh \left (\frac {1}{4} (6 c+i \pi )+\frac {3 d x}{2}\right )}{x} \, dx\\ &=-\frac {4 a^2 \cosh ^4\left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right ) \sqrt {a+i a \sinh (c+d x)}}{x}+\frac {1}{8} \left (5 i a^2 d \cosh \left (\frac {5 c}{2}+\frac {i \pi }{4}\right ) \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 \frac {\sinh \left (\frac {5 d x}{2}\right )}{x} \, dx+\frac {1}{8} \left (15 i a^2 d \cosh \left (\frac {1}{4} (6 c-i \pi )\right ) \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 \frac {\sinh \left (\frac {3 d x}{2}\right )}{x} \, dx-\frac {1}{4} \left (5 i a^2 d \cosh \left (\frac {1}{4} (2 c+i \pi )\right ) \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 \frac {\sinh \left (\frac {d x}{2}\right )}{x} \, dx+\frac {1}{8} \left (5 i a^2 d \text {csch}\left (\frac {c}{2}-\frac {i \pi }{4}+\frac {d x}{2}\right ) \sinh \left (\frac {5 c}{2}+\frac {i \pi }{4}\right ) \sqrt {a+i a \sinh (c+d x)}\right ) \int \frac {\cosh \left (\frac {5 d x}{2}\right )}{x} \, dx+\frac {1}{8} \left (15 i a^2 d \text {csch}\left (\frac {c}{2}-\frac {i \pi }{4}+\frac {d x}{2}\right ) \sinh \left (\frac {1}{4} (6 c-i \pi )\right ) \sqrt {a+i a \sinh (c+d x)}\right ) \int \frac {\cosh \left (\frac {3 d x}{2}\right )}{x} \, dx-\frac {1}{4} \left (5 i a^2 d \text {csch}\left (\frac {c}{2}-\frac {i \pi }{4}+\frac {d x}{2}\right ) \sinh \left (\frac {1}{4} (2 c+i \pi )\right ) \sqrt {a+i a \sinh (c+d x)}\right ) \int \frac {\cosh \left (\frac {d x}{2}\right )}{x} \, dx\\ &=-\frac {4 a^2 \cosh ^4\left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right ) \sqrt {a+i a \sinh (c+d x)}}{x}-\frac {5}{8} a^2 d \text {Chi}\left (\frac {5 d x}{2}\right ) \text {sech}\left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right ) \sinh \left (\frac {5 c}{2}+\frac {i \pi }{4}\right ) \sqrt {a+i a \sinh (c+d x)}-\frac {15}{8} a^2 d \text {Chi}\left (\frac {3 d x}{2}\right ) \text {sech}\left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right ) \sinh \left (\frac {1}{4} (6 c-i \pi )\right ) \sqrt {a+i a \sinh (c+d x)}+\frac {5}{4} a^2 d \text {Chi}\left (\frac {d x}{2}\right ) \text {sech}\left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right ) \sinh \left (\frac {1}{4} (2 c+i \pi )\right ) \sqrt {a+i a \sinh (c+d x)}+\frac {5}{4} a^2 d \cosh \left (\frac {1}{4} (2 c+i \pi )\right ) \text {sech}\left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right ) \sqrt {a+i a \sinh (c+d x)} \text {Shi}\left (\frac {d x}{2}\right )-\frac {15}{8} a^2 d \cosh \left (\frac {1}{4} (6 c-i \pi )\right ) \text {sech}\left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right ) \sqrt {a+i a \sinh (c+d x)} \text {Shi}\left (\frac {3 d x}{2}\right )-\frac {5}{8} a^2 d \cosh \left (\frac {5 c}{2}+\frac {i \pi }{4}\right ) \text {sech}\left (\frac {c}{2}+\frac {i \pi }{4}+\frac {d x}{2}\right ) \sqrt {a+i a \sinh (c+d x)} \text {Shi}\left (\frac {5 d x}{2}\right )\\ \end {align*}
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Mathematica [A]
time = 1.13, size = 347, normalized size = 0.78 \begin {gather*} \frac {a^2 (-i+\sinh (c+d x))^2 \sqrt {a+i a \sinh (c+d x)} \left (20 \cosh \left (\frac {1}{2} (c+d x)\right )-10 \cosh \left (\frac {3}{2} (c+d x)\right )-2 \cosh \left (\frac {5}{2} (c+d x)\right )+5 i d x \cosh \left (\frac {5 c}{2}\right ) \text {Chi}\left (\frac {5 d x}{2}\right )-10 i d x \text {Chi}\left (\frac {d x}{2}\right ) \left (\cosh \left (\frac {c}{2}\right )-i \sinh \left (\frac {c}{2}\right )\right )+15 d x \text {Chi}\left (\frac {3 d x}{2}\right ) \left (-i \cosh \left (\frac {3 c}{2}\right )+\sinh \left (\frac {3 c}{2}\right )\right )+5 d x \text {Chi}\left (\frac {5 d x}{2}\right ) \sinh \left (\frac {5 c}{2}\right )+20 i \sinh \left (\frac {1}{2} (c+d x)\right )+10 i \sinh \left (\frac {3}{2} (c+d x)\right )-2 i \sinh \left (\frac {5}{2} (c+d x)\right )-10 d x \cosh \left (\frac {c}{2}\right ) \text {Shi}\left (\frac {d x}{2}\right )-10 i d x \sinh \left (\frac {c}{2}\right ) \text {Shi}\left (\frac {d x}{2}\right )+15 d x \cosh \left (\frac {3 c}{2}\right ) \text {Shi}\left (\frac {3 d x}{2}\right )-15 i d x \sinh \left (\frac {3 c}{2}\right ) \text {Shi}\left (\frac {3 d x}{2}\right )+5 d x \cosh \left (\frac {5 c}{2}\right ) \text {Shi}\left (\frac {5 d x}{2}\right )+5 i d x \sinh \left (\frac {5 c}{2}\right ) \text {Shi}\left (\frac {5 d x}{2}\right )\right )}{8 x \left (\cosh \left (\frac {1}{2} (c+d x)\right )+i \sinh \left (\frac {1}{2} (c+d x)\right )\right )^5} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.12, size = 0, normalized size = 0.00 \[\int \frac {\left (a +i a \sinh \left (d x +c \right )\right )^{\frac {5}{2}}}{x^{2}}\, dx\]
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(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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 \frac {{\left (a+a\,\mathrm {sinh}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{5/2}}{x^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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