Integrand size = 20, antiderivative size = 65 \[ \int \frac {\cos (a+b x) \sin (a+b x)}{c+d x} \, dx=\frac {\operatorname {CosIntegral}\left (\frac {2 b c}{d}+2 b x\right ) \sin \left (2 a-\frac {2 b c}{d}\right )}{2 d}+\frac {\cos \left (2 a-\frac {2 b c}{d}\right ) \text {Si}\left (\frac {2 b c}{d}+2 b x\right )}{2 d} \]
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Time = 0.15 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {4491, 12, 3384, 3380, 3383} \[ \int \frac {\cos (a+b x) \sin (a+b x)}{c+d x} \, dx=\frac {\sin \left (2 a-\frac {2 b c}{d}\right ) \operatorname {CosIntegral}\left (\frac {2 b c}{d}+2 b x\right )}{2 d}+\frac {\cos \left (2 a-\frac {2 b c}{d}\right ) \text {Si}\left (\frac {2 b c}{d}+2 b x\right )}{2 d} \]
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Rule 12
Rule 3380
Rule 3383
Rule 3384
Rule 4491
Rubi steps \begin{align*} \text {integral}& = \int \frac {\sin (2 a+2 b x)}{2 (c+d x)} \, dx \\ & = \frac {1}{2} \int \frac {\sin (2 a+2 b x)}{c+d x} \, dx \\ & = \frac {1}{2} \cos \left (2 a-\frac {2 b c}{d}\right ) \int \frac {\sin \left (\frac {2 b c}{d}+2 b x\right )}{c+d x} \, dx+\frac {1}{2} \sin \left (2 a-\frac {2 b c}{d}\right ) \int \frac {\cos \left (\frac {2 b c}{d}+2 b x\right )}{c+d x} \, dx \\ & = \frac {\operatorname {CosIntegral}\left (\frac {2 b c}{d}+2 b x\right ) \sin \left (2 a-\frac {2 b c}{d}\right )}{2 d}+\frac {\cos \left (2 a-\frac {2 b c}{d}\right ) \text {Si}\left (\frac {2 b c}{d}+2 b x\right )}{2 d} \\ \end{align*}
Time = 0.11 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.92 \[ \int \frac {\cos (a+b x) \sin (a+b x)}{c+d x} \, dx=\frac {\operatorname {CosIntegral}\left (\frac {2 b c}{d}+2 b x\right ) \sin \left (2 a-\frac {2 b c}{d}\right )+\cos \left (2 a-\frac {2 b c}{d}\right ) \text {Si}\left (\frac {2 b c}{d}+2 b x\right )}{2 d} \]
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Time = 0.37 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.29
method | result | size |
derivativedivides | \(-\frac {\operatorname {Si}\left (-2 x b -2 a -\frac {2 \left (-a d +c b \right )}{d}\right ) \cos \left (\frac {-2 a d +2 c b}{d}\right )}{2 d}-\frac {\operatorname {Ci}\left (2 x b +2 a +\frac {-2 a d +2 c b}{d}\right ) \sin \left (\frac {-2 a d +2 c b}{d}\right )}{2 d}\) | \(84\) |
default | \(-\frac {\operatorname {Si}\left (-2 x b -2 a -\frac {2 \left (-a d +c b \right )}{d}\right ) \cos \left (\frac {-2 a d +2 c b}{d}\right )}{2 d}-\frac {\operatorname {Ci}\left (2 x b +2 a +\frac {-2 a d +2 c b}{d}\right ) \sin \left (\frac {-2 a d +2 c b}{d}\right )}{2 d}\) | \(84\) |
risch | \(-\frac {i {\mathrm e}^{-\frac {2 i \left (a d -c b \right )}{d}} \operatorname {Ei}_{1}\left (2 i b x +2 i a -\frac {2 i \left (a d -c b \right )}{d}\right )}{4 d}+\frac {i {\mathrm e}^{\frac {2 i \left (a d -c b \right )}{d}} \operatorname {Ei}_{1}\left (-2 i b x -2 i a -\frac {2 \left (-i a d +i c b \right )}{d}\right )}{4 d}\) | \(98\) |
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Time = 0.24 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.98 \[ \int \frac {\cos (a+b x) \sin (a+b x)}{c+d x} \, dx=\frac {\operatorname {Ci}\left (\frac {2 \, {\left (b d x + b c\right )}}{d}\right ) \sin \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) + \cos \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) \operatorname {Si}\left (\frac {2 \, {\left (b d x + b c\right )}}{d}\right )}{2 \, d} \]
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\[ \int \frac {\cos (a+b x) \sin (a+b x)}{c+d x} \, dx=\int \frac {\sin {\left (a + b x \right )} \cos {\left (a + b x \right )}}{c + d x}\, dx \]
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Result contains complex when optimal does not.
Time = 0.27 (sec) , antiderivative size = 143, normalized size of antiderivative = 2.20 \[ \int \frac {\cos (a+b x) \sin (a+b x)}{c+d x} \, dx=-\frac {b {\left (-i \, E_{1}\left (\frac {2 \, {\left (-i \, b c - i \, {\left (b x + a\right )} d + i \, a d\right )}}{d}\right ) + i \, E_{1}\left (-\frac {2 \, {\left (-i \, b c - i \, {\left (b x + a\right )} d + i \, a d\right )}}{d}\right )\right )} \cos \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) + b {\left (E_{1}\left (\frac {2 \, {\left (-i \, b c - i \, {\left (b x + a\right )} d + i \, a d\right )}}{d}\right ) + E_{1}\left (-\frac {2 \, {\left (-i \, b c - i \, {\left (b x + a\right )} d + i \, a d\right )}}{d}\right )\right )} \sin \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right )}{4 \, b d} \]
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.31 (sec) , antiderivative size = 569, normalized size of antiderivative = 8.75 \[ \int \frac {\cos (a+b x) \sin (a+b x)}{c+d x} \, dx=\frac {\Im \left ( \operatorname {Ci}\left (2 \, b x + \frac {2 \, b c}{d}\right ) \right ) \tan \left (a\right )^{2} \tan \left (\frac {b c}{d}\right )^{2} - \Im \left ( \operatorname {Ci}\left (-2 \, b x - \frac {2 \, b c}{d}\right ) \right ) \tan \left (a\right )^{2} \tan \left (\frac {b c}{d}\right )^{2} + 2 \, \operatorname {Si}\left (\frac {2 \, {\left (b d x + b c\right )}}{d}\right ) \tan \left (a\right )^{2} \tan \left (\frac {b c}{d}\right )^{2} + 2 \, \Re \left ( \operatorname {Ci}\left (2 \, b x + \frac {2 \, b c}{d}\right ) \right ) \tan \left (a\right )^{2} \tan \left (\frac {b c}{d}\right ) + 2 \, \Re \left ( \operatorname {Ci}\left (-2 \, b x - \frac {2 \, b c}{d}\right ) \right ) \tan \left (a\right )^{2} \tan \left (\frac {b c}{d}\right ) - 2 \, \Re \left ( \operatorname {Ci}\left (2 \, b x + \frac {2 \, b c}{d}\right ) \right ) \tan \left (a\right ) \tan \left (\frac {b c}{d}\right )^{2} - 2 \, \Re \left ( \operatorname {Ci}\left (-2 \, b x - \frac {2 \, b c}{d}\right ) \right ) \tan \left (a\right ) \tan \left (\frac {b c}{d}\right )^{2} - \Im \left ( \operatorname {Ci}\left (2 \, b x + \frac {2 \, b c}{d}\right ) \right ) \tan \left (a\right )^{2} + \Im \left ( \operatorname {Ci}\left (-2 \, b x - \frac {2 \, b c}{d}\right ) \right ) \tan \left (a\right )^{2} - 2 \, \operatorname {Si}\left (\frac {2 \, {\left (b d x + b c\right )}}{d}\right ) \tan \left (a\right )^{2} + 4 \, \Im \left ( \operatorname {Ci}\left (2 \, b x + \frac {2 \, b c}{d}\right ) \right ) \tan \left (a\right ) \tan \left (\frac {b c}{d}\right ) - 4 \, \Im \left ( \operatorname {Ci}\left (-2 \, b x - \frac {2 \, b c}{d}\right ) \right ) \tan \left (a\right ) \tan \left (\frac {b c}{d}\right ) + 8 \, \operatorname {Si}\left (\frac {2 \, {\left (b d x + b c\right )}}{d}\right ) \tan \left (a\right ) \tan \left (\frac {b c}{d}\right ) - \Im \left ( \operatorname {Ci}\left (2 \, b x + \frac {2 \, b c}{d}\right ) \right ) \tan \left (\frac {b c}{d}\right )^{2} + \Im \left ( \operatorname {Ci}\left (-2 \, b x - \frac {2 \, b c}{d}\right ) \right ) \tan \left (\frac {b c}{d}\right )^{2} - 2 \, \operatorname {Si}\left (\frac {2 \, {\left (b d x + b c\right )}}{d}\right ) \tan \left (\frac {b c}{d}\right )^{2} + 2 \, \Re \left ( \operatorname {Ci}\left (2 \, b x + \frac {2 \, b c}{d}\right ) \right ) \tan \left (a\right ) + 2 \, \Re \left ( \operatorname {Ci}\left (-2 \, b x - \frac {2 \, b c}{d}\right ) \right ) \tan \left (a\right ) - 2 \, \Re \left ( \operatorname {Ci}\left (2 \, b x + \frac {2 \, b c}{d}\right ) \right ) \tan \left (\frac {b c}{d}\right ) - 2 \, \Re \left ( \operatorname {Ci}\left (-2 \, b x - \frac {2 \, b c}{d}\right ) \right ) \tan \left (\frac {b c}{d}\right ) + \Im \left ( \operatorname {Ci}\left (2 \, b x + \frac {2 \, b c}{d}\right ) \right ) - \Im \left ( \operatorname {Ci}\left (-2 \, b x - \frac {2 \, b c}{d}\right ) \right ) + 2 \, \operatorname {Si}\left (\frac {2 \, {\left (b d x + b c\right )}}{d}\right )}{4 \, {\left (d \tan \left (a\right )^{2} \tan \left (\frac {b c}{d}\right )^{2} + d \tan \left (a\right )^{2} + d \tan \left (\frac {b c}{d}\right )^{2} + d\right )}} \]
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Timed out. \[ \int \frac {\cos (a+b x) \sin (a+b x)}{c+d x} \, dx=\int \frac {\cos \left (a+b\,x\right )\,\sin \left (a+b\,x\right )}{c+d\,x} \,d x \]
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