Integrand size = 18, antiderivative size = 64 \[ \int \frac {a+b \sin (e+f x)}{c+d x} \, dx=\frac {a \log (c+d x)}{d}+\frac {b \operatorname {CosIntegral}\left (\frac {c f}{d}+f x\right ) \sin \left (e-\frac {c f}{d}\right )}{d}+\frac {b \cos \left (e-\frac {c f}{d}\right ) \text {Si}\left (\frac {c f}{d}+f x\right )}{d} \]
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Time = 0.08 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {3398, 3384, 3380, 3383} \[ \int \frac {a+b \sin (e+f x)}{c+d x} \, dx=\frac {a \log (c+d x)}{d}+\frac {b \operatorname {CosIntegral}\left (x f+\frac {c f}{d}\right ) \sin \left (e-\frac {c f}{d}\right )}{d}+\frac {b \cos \left (e-\frac {c f}{d}\right ) \text {Si}\left (x f+\frac {c f}{d}\right )}{d} \]
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Rule 3380
Rule 3383
Rule 3384
Rule 3398
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {a}{c+d x}+\frac {b \sin (e+f x)}{c+d x}\right ) \, dx \\ & = \frac {a \log (c+d x)}{d}+b \int \frac {\sin (e+f x)}{c+d x} \, dx \\ & = \frac {a \log (c+d x)}{d}+\left (b \cos \left (e-\frac {c f}{d}\right )\right ) \int \frac {\sin \left (\frac {c f}{d}+f x\right )}{c+d x} \, dx+\left (b \sin \left (e-\frac {c f}{d}\right )\right ) \int \frac {\cos \left (\frac {c f}{d}+f x\right )}{c+d x} \, dx \\ & = \frac {a \log (c+d x)}{d}+\frac {b \operatorname {CosIntegral}\left (\frac {c f}{d}+f x\right ) \sin \left (e-\frac {c f}{d}\right )}{d}+\frac {b \cos \left (e-\frac {c f}{d}\right ) \text {Si}\left (\frac {c f}{d}+f x\right )}{d} \\ \end{align*}
Time = 0.12 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.89 \[ \int \frac {a+b \sin (e+f x)}{c+d x} \, dx=\frac {a \log (c+d x)+b \operatorname {CosIntegral}\left (f \left (\frac {c}{d}+x\right )\right ) \sin \left (e-\frac {c f}{d}\right )+b \cos \left (e-\frac {c f}{d}\right ) \text {Si}\left (f \left (\frac {c}{d}+x\right )\right )}{d} \]
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Time = 0.17 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.36
method | result | size |
parts | \(\frac {a \ln \left (d x +c \right )}{d}+b \left (\frac {\operatorname {Si}\left (f x +e +\frac {c f -d e}{d}\right ) \cos \left (\frac {c f -d e}{d}\right )}{d}-\frac {\operatorname {Ci}\left (f x +e +\frac {c f -d e}{d}\right ) \sin \left (\frac {c f -d e}{d}\right )}{d}\right )\) | \(87\) |
derivativedivides | \(\frac {\frac {a f \ln \left (c f -d e +d \left (f x +e \right )\right )}{d}+b f \left (\frac {\operatorname {Si}\left (f x +e +\frac {c f -d e}{d}\right ) \cos \left (\frac {c f -d e}{d}\right )}{d}-\frac {\operatorname {Ci}\left (f x +e +\frac {c f -d e}{d}\right ) \sin \left (\frac {c f -d e}{d}\right )}{d}\right )}{f}\) | \(103\) |
default | \(\frac {\frac {a f \ln \left (c f -d e +d \left (f x +e \right )\right )}{d}+b f \left (\frac {\operatorname {Si}\left (f x +e +\frac {c f -d e}{d}\right ) \cos \left (\frac {c f -d e}{d}\right )}{d}-\frac {\operatorname {Ci}\left (f x +e +\frac {c f -d e}{d}\right ) \sin \left (\frac {c f -d e}{d}\right )}{d}\right )}{f}\) | \(103\) |
risch | \(\frac {a \ln \left (d x +c \right )}{d}-\frac {i b \,{\mathrm e}^{\frac {i \left (c f -d e \right )}{d}} \operatorname {Ei}_{1}\left (i f x +i e +\frac {i \left (c f -d e \right )}{d}\right )}{2 d}+\frac {i b \,{\mathrm e}^{-\frac {i \left (c f -d e \right )}{d}} \operatorname {Ei}_{1}\left (-i f x -i e -\frac {i c f -i d e}{d}\right )}{2 d}\) | \(111\) |
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Time = 0.27 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.16 \[ \int \frac {a+b \sin (e+f x)}{c+d x} \, dx=-\frac {b \operatorname {Ci}\left (\frac {d f x + c f}{d}\right ) \sin \left (-\frac {d e - c f}{d}\right ) - b \cos \left (-\frac {d e - c f}{d}\right ) \operatorname {Si}\left (\frac {d f x + c f}{d}\right ) - a \log \left (d x + c\right )}{d} \]
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\[ \int \frac {a+b \sin (e+f x)}{c+d x} \, dx=\int \frac {a + b \sin {\left (e + f x \right )}}{c + d x}\, dx \]
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Result contains complex when optimal does not.
Time = 0.24 (sec) , antiderivative size = 171, normalized size of antiderivative = 2.67 \[ \int \frac {a+b \sin (e+f x)}{c+d x} \, dx=\frac {\frac {2 \, a f \log \left (c + \frac {{\left (f x + e\right )} d}{f} - \frac {d e}{f}\right )}{d} + \frac {{\left (f {\left (-i \, E_{1}\left (\frac {i \, {\left (f x + e\right )} d - i \, d e + i \, c f}{d}\right ) + i \, E_{1}\left (-\frac {i \, {\left (f x + e\right )} d - i \, d e + i \, c f}{d}\right )\right )} \cos \left (-\frac {d e - c f}{d}\right ) + f {\left (E_{1}\left (\frac {i \, {\left (f x + e\right )} d - i \, d e + i \, c f}{d}\right ) + E_{1}\left (-\frac {i \, {\left (f x + e\right )} d - i \, d e + i \, c f}{d}\right )\right )} \sin \left (-\frac {d e - c f}{d}\right )\right )} b}{d}}{2 \, f} \]
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.33 (sec) , antiderivative size = 693, normalized size of antiderivative = 10.83 \[ \int \frac {a+b \sin (e+f x)}{c+d x} \, dx=\text {Too large to display} \]
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Timed out. \[ \int \frac {a+b \sin (e+f x)}{c+d x} \, dx=\int \frac {a+b\,\sin \left (e+f\,x\right )}{c+d\,x} \,d x \]
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