Integrand size = 21, antiderivative size = 70 \[ \int \frac {a+i a \sinh (e+f x)}{c+d x} \, dx=\frac {a \log (c+d x)}{d}+\frac {i a \text {Chi}\left (\frac {c f}{d}+f x\right ) \sinh \left (e-\frac {c f}{d}\right )}{d}+\frac {i a \cosh \left (e-\frac {c f}{d}\right ) \text {Shi}\left (\frac {c f}{d}+f x\right )}{d} \]
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Time = 0.12 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {3398, 3384, 3379, 3382} \[ \int \frac {a+i a \sinh (e+f x)}{c+d x} \, dx=\frac {i a \text {Chi}\left (x f+\frac {c f}{d}\right ) \sinh \left (e-\frac {c f}{d}\right )}{d}+\frac {i a \cosh \left (e-\frac {c f}{d}\right ) \text {Shi}\left (x f+\frac {c f}{d}\right )}{d}+\frac {a \log (c+d x)}{d} \]
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Rule 3379
Rule 3382
Rule 3384
Rule 3398
Rubi steps \begin{align*} \text {integral}& = \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*}
Time = 0.46 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.86 \[ \int \frac {a+i a \sinh (e+f x)}{c+d x} \, dx=\frac {a \left (\log (c+d x)+i \text {Chi}\left (f \left (\frac {c}{d}+x\right )\right ) \sinh \left (e-\frac {c f}{d}\right )+i \cosh \left (e-\frac {c f}{d}\right ) \text {Shi}\left (f \left (\frac {c}{d}+x\right )\right )\right )}{d} \]
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Time = 0.98 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.37
method | result | size |
risch | \(\frac {a \ln \left (d x +c \right )}{d}+\frac {i a \,{\mathrm e}^{\frac {c f -d e}{d}} \operatorname {Ei}_{1}\left (f x +e +\frac {c f -d e}{d}\right )}{2 d}-\frac {i a \,{\mathrm e}^{-\frac {c f -d e}{d}} \operatorname {Ei}_{1}\left (-f x -e -\frac {c f -d e}{d}\right )}{2 d}\) | \(96\) |
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Time = 0.27 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.13 \[ \int \frac {a+i a \sinh (e+f x)}{c+d x} \, dx=\frac {i \, a {\rm Ei}\left (\frac {d f x + c f}{d}\right ) e^{\left (\frac {d e - c f}{d}\right )} - i \, a {\rm Ei}\left (-\frac {d f x + c f}{d}\right ) e^{\left (-\frac {d e - c f}{d}\right )} + 2 \, a \log \left (\frac {d x + c}{d}\right )}{2 \, d} \]
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\[ \int \frac {a+i a \sinh (e+f x)}{c+d x} \, dx=i a \left (\int \left (- \frac {i}{c + d x}\right )\, dx + \int \frac {\sinh {\left (e + f x \right )}}{c + d x}\, dx\right ) \]
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Time = 0.24 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.01 \[ \int \frac {a+i a \sinh (e+f x)}{c+d x} \, dx=\frac {1}{2} i \, a {\left (\frac {e^{\left (-e + \frac {c f}{d}\right )} E_{1}\left (\frac {{\left (d x + c\right )} f}{d}\right )}{d} - \frac {e^{\left (e - \frac {c f}{d}\right )} E_{1}\left (-\frac {{\left (d x + c\right )} f}{d}\right )}{d}\right )} + \frac {a \log \left (d x + c\right )}{d} \]
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Time = 0.27 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.99 \[ \int \frac {a+i a \sinh (e+f x)}{c+d x} \, dx=-\frac {-i \, a {\rm Ei}\left (\frac {d f x + c f}{d}\right ) e^{\left (e - \frac {c f}{d}\right )} + i \, a {\rm Ei}\left (-\frac {d f x + c f}{d}\right ) e^{\left (-e + \frac {c f}{d}\right )} - 2 \, a \log \left (d x + c\right )}{2 \, d} \]
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Timed out. \[ \int \frac {a+i a \sinh (e+f x)}{c+d x} \, dx=\int \frac {a+a\,\mathrm {sinh}\left (e+f\,x\right )\,1{}\mathrm {i}}{c+d\,x} \,d x \]
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