Integrand size = 23, antiderivative size = 71 \[ \int \frac {\csc (a+b x) \sin (3 a+3 b x)}{c+d x} \, dx=\frac {2 \cos \left (2 a-\frac {2 b c}{d}\right ) \operatorname {CosIntegral}\left (\frac {2 b c}{d}+2 b x\right )}{d}+\frac {\log (c+d x)}{d}-\frac {2 \sin \left (2 a-\frac {2 b c}{d}\right ) \text {Si}\left (\frac {2 b c}{d}+2 b x\right )}{d} \]
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Time = 0.33 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {4516, 3393, 3384, 3380, 3383} \[ \int \frac {\csc (a+b x) \sin (3 a+3 b x)}{c+d x} \, dx=\frac {2 \cos \left (2 a-\frac {2 b c}{d}\right ) \operatorname {CosIntegral}\left (\frac {2 b c}{d}+2 b x\right )}{d}-\frac {2 \sin \left (2 a-\frac {2 b c}{d}\right ) \text {Si}\left (\frac {2 b c}{d}+2 b x\right )}{d}+\frac {\log (c+d x)}{d} \]
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Rule 3380
Rule 3383
Rule 3384
Rule 3393
Rule 4516
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {3 \cos ^2(a+b x)}{c+d x}-\frac {\sin ^2(a+b x)}{c+d x}\right ) \, dx \\ & = 3 \int \frac {\cos ^2(a+b x)}{c+d x} \, dx-\int \frac {\sin ^2(a+b x)}{c+d x} \, dx \\ & = 3 \int \left (\frac {1}{2 (c+d x)}+\frac {\cos (2 a+2 b x)}{2 (c+d x)}\right ) \, dx-\int \left (\frac {1}{2 (c+d x)}-\frac {\cos (2 a+2 b x)}{2 (c+d x)}\right ) \, dx \\ & = \frac {\log (c+d x)}{d}+\frac {1}{2} \int \frac {\cos (2 a+2 b x)}{c+d x} \, dx+\frac {3}{2} \int \frac {\cos (2 a+2 b x)}{c+d x} \, dx \\ & = \frac {\log (c+d x)}{d}+\frac {1}{2} \cos \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 {1}{2} \left (3 \cos \left (2 a-\frac {2 b c}{d}\right )\right ) \int \frac {\cos \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 {\sin \left (\frac {2 b c}{d}+2 b x\right )}{c+d x} \, dx-\frac {1}{2} \left (3 \sin \left (2 a-\frac {2 b c}{d}\right )\right ) \int \frac {\sin \left (\frac {2 b c}{d}+2 b x\right )}{c+d x} \, dx \\ & = \frac {2 \cos \left (2 a-\frac {2 b c}{d}\right ) \operatorname {CosIntegral}\left (\frac {2 b c}{d}+2 b x\right )}{d}+\frac {\log (c+d x)}{d}-\frac {2 \sin \left (2 a-\frac {2 b c}{d}\right ) \text {Si}\left (\frac {2 b c}{d}+2 b x\right )}{d} \\ \end{align*}
Time = 0.14 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.89 \[ \int \frac {\csc (a+b x) \sin (3 a+3 b x)}{c+d x} \, dx=\frac {2 \cos \left (2 a-\frac {2 b c}{d}\right ) \operatorname {CosIntegral}\left (\frac {2 b (c+d x)}{d}\right )+\log (c+d x)-2 \sin \left (2 a-\frac {2 b c}{d}\right ) \text {Si}\left (\frac {2 b (c+d x)}{d}\right )}{d} \]
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Result contains complex when optimal does not.
Time = 0.57 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.49
method | result | size |
risch | \(\frac {\ln \left (d x +c \right )}{d}-\frac {{\mathrm e}^{-\frac {2 i \left (a d -c b \right )}{d}} \operatorname {Ei}_{1}\left (2 i b x +2 i a -\frac {2 i \left (a d -c b \right )}{d}\right )}{d}-\frac {{\mathrm e}^{\frac {2 i \left (a d -c b \right )}{d}} \operatorname {Ei}_{1}\left (-2 i b x -2 i a -\frac {2 \left (-i a d +i c b \right )}{d}\right )}{d}\) | \(106\) |
default | \(-\frac {\ln \left (d x +c \right )}{d}+\frac {2 \,\operatorname {Si}\left (2 x b +2 a +\frac {-2 a d +2 c b}{d}\right ) \sin \left (\frac {-2 a d +2 c b}{d}\right )}{d}+\frac {2 \,\operatorname {Ci}\left (2 x b +2 a +\frac {-2 a d +2 c b}{d}\right ) \cos \left (\frac {-2 a d +2 c b}{d}\right )}{d}+\frac {2 \ln \left (-a d +c b +d \left (x b +a \right )\right )}{d}\) | \(116\) |
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Time = 0.25 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.00 \[ \int \frac {\csc (a+b x) \sin (3 a+3 b x)}{c+d x} \, dx=\frac {2 \, \cos \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) \operatorname {Ci}\left (\frac {2 \, {\left (b d x + b c\right )}}{d}\right ) - 2 \, \sin \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) \operatorname {Si}\left (\frac {2 \, {\left (b d x + b c\right )}}{d}\right ) + \log \left (d x + c\right )}{d} \]
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\[ \int \frac {\csc (a+b x) \sin (3 a+3 b x)}{c+d x} \, dx=\int \frac {\sin {\left (3 a + 3 b x \right )} \csc {\left (a + b x \right )}}{c + d x}\, dx \]
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Result contains complex when optimal does not.
Time = 0.26 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.68 \[ \int \frac {\csc (a+b x) \sin (3 a+3 b x)}{c+d x} \, dx=-\frac {{\left (E_{1}\left (\frac {2 \, {\left (-i \, b d x - i \, b c\right )}}{d}\right ) + E_{1}\left (-\frac {2 \, {\left (-i \, b d x - i \, b c\right )}}{d}\right )\right )} \cos \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) - {\left (-i \, E_{1}\left (\frac {2 \, {\left (-i \, b d x - i \, b c\right )}}{d}\right ) + i \, E_{1}\left (-\frac {2 \, {\left (-i \, b d x - i \, b c\right )}}{d}\right )\right )} \sin \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) - \log \left (d x + c\right )}{d} \]
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Time = 0.29 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.44 \[ \int \frac {\csc (a+b x) \sin (3 a+3 b x)}{c+d x} \, dx=\frac {2 \, b \cos \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) \operatorname {Ci}\left (\frac {2 \, {\left (b c + {\left (b x + a\right )} d - a d\right )}}{d}\right ) + 2 \, b \sin \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) \operatorname {Si}\left (-\frac {2 \, {\left (b c + {\left (b x + a\right )} d - a d\right )}}{d}\right ) + b \log \left (b c + {\left (b x + a\right )} d - a d\right )}{b d} \]
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Timed out. \[ \int \frac {\csc (a+b x) \sin (3 a+3 b x)}{c+d x} \, dx=\int \frac {\sin \left (3\,a+3\,b\,x\right )}{\sin \left (a+b\,x\right )\,\left (c+d\,x\right )} \,d x \]
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