Optimal. Leaf size=403 \[ -\frac {1}{4} i a^2 \sinh \left (\frac {5 c}{2}-\frac {i \pi }{4}\right ) \text {Chi}\left (\frac {5 d x}{2}\right ) \text {sech}\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \sqrt {a+i a \sinh (c+d x)}+\frac {5}{2} i a^2 \sinh \left (\frac {1}{4} (2 c-i \pi )\right ) \text {Chi}\left (\frac {d x}{2}\right ) \text {sech}\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \sqrt {a+i a \sinh (c+d x)}+\frac {5}{4} i a^2 \sinh \left (\frac {1}{4} (6 c+i \pi )\right ) \text {Chi}\left (\frac {3 d x}{2}\right ) \text {sech}\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \sqrt {a+i a \sinh (c+d x)}+\frac {5}{2} i a^2 \cosh \left (\frac {1}{4} (2 c-i \pi )\right ) \text {Shi}\left (\frac {d x}{2}\right ) \text {sech}\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \sqrt {a+i a \sinh (c+d x)}+\frac {5}{4} i a^2 \cosh \left (\frac {1}{4} (6 c+i \pi )\right ) \text {Shi}\left (\frac {3 d x}{2}\right ) \text {sech}\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \sqrt {a+i a \sinh (c+d x)}-\frac {1}{4} i a^2 \cosh \left (\frac {5 c}{2}-\frac {i \pi }{4}\right ) \text {Shi}\left (\frac {5 d x}{2}\right ) \text {sech}\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \sqrt {a+i a \sinh (c+d x)} \]
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Rubi [A] time = 0.42, antiderivative size = 403, 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 = {3319, 3312, 3303, 3298, 3301} \[ -\frac {1}{4} i a^2 \sinh \left (\frac {5 c}{2}-\frac {i \pi }{4}\right ) \text {Chi}\left (\frac {5 d x}{2}\right ) \text {sech}\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \sqrt {a+i a \sinh (c+d x)}+\frac {5}{2} i a^2 \sinh \left (\frac {1}{4} (2 c-i \pi )\right ) \text {Chi}\left (\frac {d x}{2}\right ) \text {sech}\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \sqrt {a+i a \sinh (c+d x)}+\frac {5}{4} i a^2 \sinh \left (\frac {1}{4} (6 c+i \pi )\right ) \text {Chi}\left (\frac {3 d x}{2}\right ) \text {sech}\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \sqrt {a+i a \sinh (c+d x)}+\frac {5}{2} i a^2 \cosh \left (\frac {1}{4} (2 c-i \pi )\right ) \text {Shi}\left (\frac {d x}{2}\right ) \text {sech}\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \sqrt {a+i a \sinh (c+d x)}+\frac {5}{4} i a^2 \cosh \left (\frac {1}{4} (6 c+i \pi )\right ) \text {Shi}\left (\frac {3 d x}{2}\right ) \text {sech}\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \sqrt {a+i a \sinh (c+d x)}-\frac {1}{4} i a^2 \cosh \left (\frac {5 c}{2}-\frac {i \pi }{4}\right ) \text {Shi}\left (\frac {5 d x}{2}\right ) \text {sech}\left (\frac {c}{2}+\frac {d x}{2}+\frac {i \pi }{4}\right ) \sqrt {a+i a \sinh (c+d x)} \]
Antiderivative was successfully verified.
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Rule 3298
Rule 3301
Rule 3303
Rule 3312
Rule 3319
Rubi steps
\begin {align*} \int \frac {(a+i a \sinh (c+d x))^{5/2}}{x} \, 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} \, dx\\ &=-\left (\left (4 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 \left (\frac {5 i \sinh \left (\frac {1}{4} (2 c-i \pi )+\frac {d x}{2}\right )}{8 x}+\frac {5 i \sinh \left (\frac {1}{4} (6 c+i \pi )+\frac {3 d x}{2}\right )}{16 x}-\frac {i \sinh \left (\frac {1}{4} (10 c-i \pi )+\frac {5 d x}{2}\right )}{16 x}\right ) \, dx\right )\\ &=-\left (\frac {1}{4} \left (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 \left (\frac {1}{4} (10 c-i \pi )+\frac {5 d x}{2}\right )}{x} \, dx\right )+\frac {1}{4} \left (5 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 \left (\frac {1}{4} (6 c+i \pi )+\frac {3 d x}{2}\right )}{x} \, dx+\frac {1}{2} \left (5 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 \left (\frac {1}{4} (2 c-i \pi )+\frac {d x}{2}\right )}{x} \, dx\\ &=-\left (\frac {1}{4} \left (a^2 \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\right )+\frac {1}{2} \left (5 a^2 \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}{4} \left (5 a^2 \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 (a^2 \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}{2} \left (5 a^2 \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 {1}{4} \left (5 a^2 \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} i a^2 \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 {5}{2} i a^2 \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} i a^2 \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}{2} i a^2 \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 {5}{4} i a^2 \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 {1}{4} i a^2 \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 = 1.37, size = 242, normalized size = 0.60 \[ \frac {a^2 (\sinh (c+d x)-i)^2 \sqrt {a+i a \sinh (c+d x)} \left (i \sinh \left (\frac {5 c}{2}\right ) \text {Chi}\left (\frac {5 d x}{2}\right )+\cosh \left (\frac {5 c}{2}\right ) \text {Chi}\left (\frac {5 d x}{2}\right )-10 \left (\cosh \left (\frac {c}{2}\right )+i \sinh \left (\frac {c}{2}\right )\right ) \text {Chi}\left (\frac {d x}{2}\right )+5 \left (\cosh \left (\frac {3 c}{2}\right )-i \sinh \left (\frac {3 c}{2}\right )\right ) \text {Chi}\left (\frac {3 d x}{2}\right )-10 \sinh \left (\frac {c}{2}\right ) \text {Shi}\left (\frac {d x}{2}\right )+5 \sinh \left (\frac {3 c}{2}\right ) \text {Shi}\left (\frac {3 d x}{2}\right )+\sinh \left (\frac {5 c}{2}\right ) \text {Shi}\left (\frac {5 d x}{2}\right )-10 i \cosh \left (\frac {c}{2}\right ) \text {Shi}\left (\frac {d x}{2}\right )-5 i \cosh \left (\frac {3 c}{2}\right ) \text {Shi}\left (\frac {3 d x}{2}\right )+i \cosh \left (\frac {5 c}{2}\right ) \text {Shi}\left (\frac {5 d x}{2}\right )\right )}{4 \left (\cosh \left (\frac {1}{2} (c+d x)\right )+i \sinh \left (\frac {1}{2} (c+d x)\right )\right )^5} \]
Antiderivative was successfully verified.
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fricas [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
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 \[ \int \frac {{\left (i \, a \sinh \left (d x + c\right ) + a\right )}^{\frac {5}{2}}}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.04, size = 0, normalized size = 0.00 \[ \int \frac {\left (a +i a \sinh \left (d x +c \right )\right )^{\frac {5}{2}}}{x}\, dx \]
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 \[ \int \frac {{\left (i \, a \sinh \left (d x + c\right ) + a\right )}^{\frac {5}{2}}}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (a+a\,\mathrm {sinh}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{5/2}}{x} \,d x \]
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 \[ \int \frac {\left (i a \left (\sinh {\left (c + d x \right )} - i\right )\right )^{\frac {5}{2}}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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