Integrand size = 14, antiderivative size = 57 \[ \int \frac {\csc (x) \sin (3 x)}{c+d x} \, dx=\frac {2 \cos \left (\frac {2 c}{d}\right ) \operatorname {CosIntegral}\left (\frac {2 c}{d}+2 x\right )}{d}+\frac {\log (c+d x)}{d}+\frac {2 \sin \left (\frac {2 c}{d}\right ) \text {Si}\left (\frac {2 c}{d}+2 x\right )}{d} \]
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Time = 0.35 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {4516, 3393, 3384, 3380, 3383} \[ \int \frac {\csc (x) \sin (3 x)}{c+d x} \, dx=\frac {2 \cos \left (\frac {2 c}{d}\right ) \operatorname {CosIntegral}\left (\frac {2 c}{d}+2 x\right )}{d}+\frac {2 \sin \left (\frac {2 c}{d}\right ) \text {Si}\left (\frac {2 c}{d}+2 x\right )}{d}+\frac {\log (c+d x)}{d} \]
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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(x)}{c+d x}-\frac {\sin ^2(x)}{c+d x}\right ) \, dx \\ & = 3 \int \frac {\cos ^2(x)}{c+d x} \, dx-\int \frac {\sin ^2(x)}{c+d x} \, dx \\ & = 3 \int \left (\frac {1}{2 (c+d x)}+\frac {\cos (2 x)}{2 (c+d x)}\right ) \, dx-\int \left (\frac {1}{2 (c+d x)}-\frac {\cos (2 x)}{2 (c+d x)}\right ) \, dx \\ & = \frac {\log (c+d x)}{d}+\frac {1}{2} \int \frac {\cos (2 x)}{c+d x} \, dx+\frac {3}{2} \int \frac {\cos (2 x)}{c+d x} \, dx \\ & = \frac {\log (c+d x)}{d}+\frac {1}{2} \cos \left (\frac {2 c}{d}\right ) \int \frac {\cos \left (\frac {2 c}{d}+2 x\right )}{c+d x} \, dx+\frac {1}{2} \left (3 \cos \left (\frac {2 c}{d}\right )\right ) \int \frac {\cos \left (\frac {2 c}{d}+2 x\right )}{c+d x} \, dx+\frac {1}{2} \sin \left (\frac {2 c}{d}\right ) \int \frac {\sin \left (\frac {2 c}{d}+2 x\right )}{c+d x} \, dx+\frac {1}{2} \left (3 \sin \left (\frac {2 c}{d}\right )\right ) \int \frac {\sin \left (\frac {2 c}{d}+2 x\right )}{c+d x} \, dx \\ & = \frac {2 \cos \left (\frac {2 c}{d}\right ) \operatorname {CosIntegral}\left (\frac {2 c}{d}+2 x\right )}{d}+\frac {\log (c+d x)}{d}+\frac {2 \sin \left (\frac {2 c}{d}\right ) \text {Si}\left (\frac {2 c}{d}+2 x\right )}{d} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.86 \[ \int \frac {\csc (x) \sin (3 x)}{c+d x} \, dx=\frac {2 \cos \left (\frac {2 c}{d}\right ) \operatorname {CosIntegral}\left (2 \left (\frac {c}{d}+x\right )\right )+\log (c+d x)+2 \sin \left (\frac {2 c}{d}\right ) \text {Si}\left (2 \left (\frac {c}{d}+x\right )\right )}{d} \]
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Time = 0.68 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.02
method | result | size |
default | \(\frac {2 \,\operatorname {Ci}\left (\frac {2 c}{d}+2 x \right ) \cos \left (\frac {2 c}{d}\right )}{d}+\frac {\ln \left (d x +c \right )}{d}+\frac {2 \,\operatorname {Si}\left (\frac {2 c}{d}+2 x \right ) \sin \left (\frac {2 c}{d}\right )}{d}\) | \(58\) |
risch | \(\frac {\ln \left (d x +c \right )}{d}-\frac {{\mathrm e}^{\frac {2 i c}{d}} \operatorname {Ei}_{1}\left (2 i x +\frac {2 i c}{d}\right )}{d}-\frac {{\mathrm e}^{-\frac {2 i c}{d}} \operatorname {Ei}_{1}\left (-2 i x -\frac {2 i c}{d}\right )}{d}\) | \(66\) |
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Time = 0.25 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.89 \[ \int \frac {\csc (x) \sin (3 x)}{c+d x} \, dx=\frac {2 \, \cos \left (\frac {2 \, c}{d}\right ) \operatorname {Ci}\left (\frac {2 \, {\left (d x + c\right )}}{d}\right ) + 2 \, \sin \left (\frac {2 \, c}{d}\right ) \operatorname {Si}\left (\frac {2 \, {\left (d x + c\right )}}{d}\right ) + \log \left (d x + c\right )}{d} \]
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\[ \int \frac {\csc (x) \sin (3 x)}{c+d x} \, dx=\int \frac {\sin {\left (3 x \right )} \csc {\left (x \right )}}{c + d x}\, dx \]
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Result contains complex when optimal does not.
Time = 0.25 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.70 \[ \int \frac {\csc (x) \sin (3 x)}{c+d x} \, dx=-\frac {{\left (E_{1}\left (\frac {2 \, {\left (-i \, d x - i \, c\right )}}{d}\right ) + E_{1}\left (-\frac {2 \, {\left (-i \, d x - i \, c\right )}}{d}\right )\right )} \cos \left (\frac {2 \, c}{d}\right ) - {\left (i \, E_{1}\left (\frac {2 \, {\left (-i \, d x - i \, c\right )}}{d}\right ) - i \, E_{1}\left (-\frac {2 \, {\left (-i \, d x - i \, c\right )}}{d}\right )\right )} \sin \left (\frac {2 \, c}{d}\right ) - \log \left (d x + c\right )}{d} \]
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Time = 0.26 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.89 \[ \int \frac {\csc (x) \sin (3 x)}{c+d x} \, dx=\frac {2 \, \cos \left (\frac {2 \, c}{d}\right ) \operatorname {Ci}\left (\frac {2 \, {\left (d x + c\right )}}{d}\right ) + 2 \, \sin \left (\frac {2 \, c}{d}\right ) \operatorname {Si}\left (\frac {2 \, {\left (d x + c\right )}}{d}\right ) + \log \left (d x + c\right )}{d} \]
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Timed out. \[ \int \frac {\csc (x) \sin (3 x)}{c+d x} \, dx=\int \frac {\sin \left (3\,x\right )}{\sin \left (x\right )\,\left (c+d\,x\right )} \,d x \]
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