Optimal. Leaf size=70 \[ \frac {a \log (c+d x)}{d}+\frac {i a \text {Chi}\left (\frac {c f}{d}+f x\right ) \sinh \left (e-\frac {c f}{d}\right )}{d}+\frac {i a \cosh \left (e-\frac {c f}{d}\right ) \text {Shi}\left (\frac {c f}{d}+f x\right )}{d} \]
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Rubi [A]
time = 0.12, antiderivative size = 70, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {3398, 3384,
3379, 3382} \begin {gather*} \frac {i a \text {Chi}\left (x f+\frac {c f}{d}\right ) \sinh \left (e-\frac {c f}{d}\right )}{d}+\frac {i a \cosh \left (e-\frac {c f}{d}\right ) \text {Shi}\left (x f+\frac {c f}{d}\right )}{d}+\frac {a \log (c+d x)}{d} \end {gather*}
Antiderivative was successfully verified.
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Rule 3379
Rule 3382
Rule 3384
Rule 3398
Rubi steps
\begin {align*} \int \frac {a+i a \sinh (e+f x)}{c+d x} \, dx &=\int \left (\frac {a}{c+d x}+\frac {i a \sinh (e+f x)}{c+d x}\right ) \, dx\\ &=\frac {a \log (c+d x)}{d}+(i a) \int \frac {\sinh (e+f x)}{c+d x} \, dx\\ &=\frac {a \log (c+d x)}{d}+\left (i a \cosh \left (e-\frac {c f}{d}\right )\right ) \int \frac {\sinh \left (\frac {c f}{d}+f x\right )}{c+d x} \, dx+\left (i a \sinh \left (e-\frac {c f}{d}\right )\right ) \int \frac {\cosh \left (\frac {c f}{d}+f x\right )}{c+d x} \, dx\\ &=\frac {a \log (c+d x)}{d}+\frac {i a \text {Chi}\left (\frac {c f}{d}+f x\right ) \sinh \left (e-\frac {c f}{d}\right )}{d}+\frac {i a \cosh \left (e-\frac {c f}{d}\right ) \text {Shi}\left (\frac {c f}{d}+f x\right )}{d}\\ \end {align*}
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Mathematica [A]
time = 0.17, size = 60, normalized size = 0.86 \begin {gather*} \frac {a \left (\log (c+d x)+i \text {Chi}\left (f \left (\frac {c}{d}+x\right )\right ) \sinh \left (e-\frac {c f}{d}\right )+i \cosh \left (e-\frac {c f}{d}\right ) \text {Shi}\left (f \left (\frac {c}{d}+x\right )\right )\right )}{d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.54, size = 96, normalized size = 1.37
method | result | size |
risch | \(\frac {a \ln \left (d x +c \right )}{d}+\frac {i a \,{\mathrm e}^{\frac {c f -d e}{d}} \expIntegral \left (1, f x +e +\frac {c f -d e}{d}\right )}{2 d}-\frac {i a \,{\mathrm e}^{-\frac {c f -d e}{d}} \expIntegral \left (1, -f x -e -\frac {c f -d e}{d}\right )}{2 d}\) | \(96\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.32, size = 73, normalized size = 1.04 \begin {gather*} \frac {1}{2} i \, a {\left (\frac {e^{\left (\frac {c f}{d} - e\right )} E_{1}\left (\frac {{\left (d x + c\right )} f}{d}\right )}{d} - \frac {e^{\left (-\frac {c f}{d} + e\right )} E_{1}\left (-\frac {{\left (d x + c\right )} f}{d}\right )}{d}\right )} + \frac {a \log \left (d x + c\right )}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 81, normalized size = 1.16 \begin {gather*} \frac {-i \, a {\rm Ei}\left (-\frac {d f x + c f}{d}\right ) e^{\left (\frac {c f - d e}{d}\right )} + i \, a {\rm Ei}\left (\frac {d f x + c f}{d}\right ) e^{\left (-\frac {c f - d e}{d}\right )} + 2 \, a \log \left (\frac {d x + c}{d}\right )}{2 \, d} \end {gather*}
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 \begin {gather*} i a \left (\int \left (- \frac {i}{c + d x}\right )\, dx + \int \frac {\sinh {\left (e + f x \right )}}{c + d x}\, dx\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.42, size = 69, normalized size = 0.99 \begin {gather*} -\frac {-i \, a {\rm Ei}\left (\frac {d f x + c f}{d}\right ) e^{\left (e - \frac {c f}{d}\right )} + i \, a {\rm Ei}\left (-\frac {d f x + c f}{d}\right ) e^{\left (-e + \frac {c f}{d}\right )} - 2 \, a \log \left (d x + c\right )}{2 \, d} \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 = -0.01 \begin {gather*} \int \frac {a+a\,\mathrm {sinh}\left (e+f\,x\right )\,1{}\mathrm {i}}{c+d\,x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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