Optimal. Leaf size=149 \[ \frac {1}{2} f \sinh \left (\frac {1}{4} (2 e+i \pi )\right ) \text {Chi}\left (\frac {f x}{2}\right ) \text {sech}\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \sqrt {a+i a \sinh (e+f x)}+\frac {1}{2} f \cosh \left (\frac {1}{4} (2 e+i \pi )\right ) \text {Shi}\left (\frac {f x}{2}\right ) \text {sech}\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \sqrt {a+i a \sinh (e+f x)}-\frac {\sqrt {a+i a \sinh (e+f x)}}{x} \]
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Rubi [A] time = 0.18, antiderivative size = 149, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {3319, 3297, 3303, 3298, 3301} \[ \frac {1}{2} f \sinh \left (\frac {1}{4} (2 e+i \pi )\right ) \text {Chi}\left (\frac {f x}{2}\right ) \text {sech}\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \sqrt {a+i a \sinh (e+f x)}+\frac {1}{2} f \cosh \left (\frac {1}{4} (2 e+i \pi )\right ) \text {Shi}\left (\frac {f x}{2}\right ) \text {sech}\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right ) \sqrt {a+i a \sinh (e+f x)}-\frac {\sqrt {a+i a \sinh (e+f x)}}{x} \]
Antiderivative was successfully verified.
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Rule 3297
Rule 3298
Rule 3301
Rule 3303
Rule 3319
Rubi steps
\begin {align*} \int \frac {\sqrt {a+i a \sinh (e+f x)}}{x^2} \, dx &=\left (\text {csch}\left (\frac {e}{2}-\frac {i \pi }{4}+\frac {f x}{2}\right ) \sqrt {a+i a \sinh (e+f x)}\right ) \int \frac {\sinh \left (\frac {e}{2}-\frac {i \pi }{4}+\frac {f x}{2}\right )}{x^2} \, dx\\ &=-\frac {\sqrt {a+i a \sinh (e+f x)}}{x}+\frac {1}{2} \left (f \text {csch}\left (\frac {e}{2}-\frac {i \pi }{4}+\frac {f x}{2}\right ) \sqrt {a+i a \sinh (e+f x)}\right ) \int \frac {\cosh \left (\frac {e}{2}-\frac {i \pi }{4}+\frac {f x}{2}\right )}{x} \, dx\\ &=-\frac {\sqrt {a+i a \sinh (e+f x)}}{x}-\frac {1}{2} \left (i f \cosh \left (\frac {1}{4} (2 e+i \pi )\right ) \text {csch}\left (\frac {e}{2}-\frac {i \pi }{4}+\frac {f x}{2}\right ) \sqrt {a+i a \sinh (e+f x)}\right ) \int \frac {\sinh \left (\frac {f x}{2}\right )}{x} \, dx-\frac {1}{2} \left (i f \text {csch}\left (\frac {e}{2}-\frac {i \pi }{4}+\frac {f x}{2}\right ) \sinh \left (\frac {1}{4} (2 e+i \pi )\right ) \sqrt {a+i a \sinh (e+f x)}\right ) \int \frac {\cosh \left (\frac {f x}{2}\right )}{x} \, dx\\ &=-\frac {\sqrt {a+i a \sinh (e+f x)}}{x}+\frac {1}{2} f \text {Chi}\left (\frac {f x}{2}\right ) \text {sech}\left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right ) \sinh \left (\frac {1}{4} (2 e+i \pi )\right ) \sqrt {a+i a \sinh (e+f x)}+\frac {1}{2} f \cosh \left (\frac {1}{4} (2 e+i \pi )\right ) \text {sech}\left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right ) \sqrt {a+i a \sinh (e+f x)} \text {Shi}\left (\frac {f x}{2}\right )\\ \end {align*}
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Mathematica [A] time = 0.26, size = 133, normalized size = 0.89 \[ \frac {\sqrt {a+i a \sinh (e+f x)} \left (f x \text {Chi}\left (\frac {f x}{2}\right ) \left (\sinh \left (\frac {e}{2}\right )+i \cosh \left (\frac {e}{2}\right )\right )+f x \left (\cosh \left (\frac {e}{2}\right )+i \sinh \left (\frac {e}{2}\right )\right ) \text {Shi}\left (\frac {f x}{2}\right )-2 \left (\cosh \left (\frac {1}{2} (e+f x)\right )+i \sinh \left (\frac {1}{2} (e+f x)\right )\right )\right )}{2 x \left (\cosh \left (\frac {1}{2} (e+f x)\right )+i \sinh \left (\frac {1}{2} (e+f x)\right )\right )} \]
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 \[ \int \frac {\sqrt {i \, a \sinh \left (f x + e\right ) + a}}{x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.06, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {a +i a \sinh \left (f x +e \right )}}{x^{2}}\, dx \]
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 \[ \int \frac {\sqrt {i \, a \sinh \left (f x + e\right ) + a}}{x^{2}}\,{d x} \]
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 \[ \int \frac {\sqrt {a+a\,\mathrm {sinh}\left (e+f\,x\right )\,1{}\mathrm {i}}}{x^2} \,d x \]
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 \[ \int \frac {\sqrt {i a \left (\sinh {\left (e + f x \right )} - i\right )}}{x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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