Optimal. Leaf size=76 \[ -\frac {i \sinh \left (c-\frac {d e}{f}\right ) \text {Chi}\left (\frac {d e}{f}+d x\right )}{a f}-\frac {i \cosh \left (c-\frac {d e}{f}\right ) \text {Shi}\left (\frac {d e}{f}+d x\right )}{a f}+\frac {\log (e+f x)}{a f} \]
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Rubi [A] time = 0.20, antiderivative size = 76, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.161, Rules used = {5563, 31, 3303, 3298, 3301} \[ -\frac {i \sinh \left (c-\frac {d e}{f}\right ) \text {Chi}\left (\frac {d e}{f}+d x\right )}{a f}-\frac {i \cosh \left (c-\frac {d e}{f}\right ) \text {Shi}\left (\frac {d e}{f}+d x\right )}{a f}+\frac {\log (e+f x)}{a f} \]
Antiderivative was successfully verified.
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Rule 31
Rule 3298
Rule 3301
Rule 3303
Rule 5563
Rubi steps
\begin {align*} \int \frac {\cosh ^2(c+d x)}{(e+f x) (a+i a \sinh (c+d x))} \, dx &=-\frac {i \int \frac {\sinh (c+d x)}{e+f x} \, dx}{a}+\frac {\int \frac {1}{e+f x} \, dx}{a}\\ &=\frac {\log (e+f x)}{a f}-\frac {\left (i \cosh \left (c-\frac {d e}{f}\right )\right ) \int \frac {\sinh \left (\frac {d e}{f}+d x\right )}{e+f x} \, dx}{a}-\frac {\left (i \sinh \left (c-\frac {d e}{f}\right )\right ) \int \frac {\cosh \left (\frac {d e}{f}+d x\right )}{e+f x} \, dx}{a}\\ &=\frac {\log (e+f x)}{a f}-\frac {i \text {Chi}\left (\frac {d e}{f}+d x\right ) \sinh \left (c-\frac {d e}{f}\right )}{a f}-\frac {i \cosh \left (c-\frac {d e}{f}\right ) \text {Shi}\left (\frac {d e}{f}+d x\right )}{a f}\\ \end {align*}
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Mathematica [A] time = 0.32, size = 62, normalized size = 0.82 \[ \frac {-i \sinh \left (c-\frac {d e}{f}\right ) \text {Chi}\left (d \left (\frac {e}{f}+x\right )\right )-i \cosh \left (c-\frac {d e}{f}\right ) \text {Shi}\left (d \left (\frac {e}{f}+x\right )\right )+\log (e+f x)}{a f} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.49, size = 79, normalized size = 1.04 \[ \frac {i \, {\rm Ei}\left (-\frac {d f x + d e}{f}\right ) e^{\left (\frac {d e - c f}{f}\right )} - i \, {\rm Ei}\left (\frac {d f x + d e}{f}\right ) e^{\left (-\frac {d e - c f}{f}\right )} + 2 \, \log \left (\frac {f x + e}{f}\right )}{2 \, a f} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.23, size = 81, normalized size = 1.07 \[ -\frac {{\left (i \, {\rm Ei}\left (\frac {d f x + d e}{f}\right ) e^{\left (2 \, c - \frac {d e}{f}\right )} - i \, {\rm Ei}\left (-\frac {d f x + d e}{f}\right ) e^{\left (\frac {d e}{f}\right )} - 2 \, e^{c} \log \left (i \, f x + i \, e\right )\right )} e^{\left (-c\right )}}{2 \, a f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.18, size = 103, normalized size = 1.36 \[ \frac {\ln \left (f x +e \right )}{a f}+\frac {i {\mathrm e}^{\frac {c f -d e}{f}} \Ei \left (1, -d x -c -\frac {-c f +d e}{f}\right )}{2 a f}-\frac {i {\mathrm e}^{-\frac {c f -d e}{f}} \Ei \left (1, d x +c -\frac {c f -d e}{f}\right )}{2 a f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.46, size = 76, normalized size = 1.00 \[ -\frac {i \, e^{\left (-c + \frac {d e}{f}\right )} E_{1}\left (\frac {{\left (f x + e\right )} d}{f}\right )}{2 \, a f} + \frac {i \, e^{\left (c - \frac {d e}{f}\right )} E_{1}\left (-\frac {{\left (f x + e\right )} d}{f}\right )}{2 \, a f} + \frac {\log \left (f x + e\right )}{a f} \]
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 \[ \int \frac {{\mathrm {cosh}\left (c+d\,x\right )}^2}{\left (e+f\,x\right )\,\left (a+a\,\mathrm {sinh}\left (c+d\,x\right )\,1{}\mathrm {i}\right )} \,d x \]
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 \[ - \frac {i \int \frac {\cosh ^{2}{\left (c + d x \right )}}{e \sinh {\left (c + d x \right )} - i e + f x \sinh {\left (c + d x \right )} - i f x}\, dx}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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