Optimal. Leaf size=57 \[ \frac {2 \cos \left (\frac {2 c}{d}\right ) \text {Ci}\left (\frac {2 c}{d}+2 x\right )}{d}+\frac {2 \sin \left (\frac {2 c}{d}\right ) \text {Si}\left (\frac {2 c}{d}+2 x\right )}{d}+\frac {\log (c+d x)}{d} \]
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Rubi [A] time = 0.25, antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 5, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {4431, 3312, 3303, 3299, 3302} \[ \frac {2 \cos \left (\frac {2 c}{d}\right ) \text {CosIntegral}\left (\frac {2 c}{d}+2 x\right )}{d}+\frac {2 \sin \left (\frac {2 c}{d}\right ) \text {Si}\left (\frac {2 c}{d}+2 x\right )}{d}+\frac {\log (c+d x)}{d} \]
Antiderivative was successfully verified.
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Rule 3299
Rule 3302
Rule 3303
Rule 3312
Rule 4431
Rubi steps
\begin {align*} \int \frac {\csc (x) \sin (3 x)}{c+d x} \, dx &=\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 ) \text {Ci}\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*}
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Mathematica [A] time = 0.06, size = 49, normalized size = 0.86 \[ \frac {2 \cos \left (\frac {2 c}{d}\right ) \text {Ci}\left (2 \left (\frac {c}{d}+x\right )\right )+2 \sin \left (\frac {2 c}{d}\right ) \text {Si}\left (2 \left (\frac {c}{d}+x\right )\right )+\log (c+d x)}{d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 62, normalized size = 1.09 \[ \frac {{\left (\operatorname {Ci}\left (\frac {2 \, {\left (d x + c\right )}}{d}\right ) + \operatorname {Ci}\left (-\frac {2 \, {\left (d x + c\right )}}{d}\right )\right )} \cos \left (\frac {2 \, c}{d}\right ) + 2 \, \sin \left (\frac {2 \, c}{d}\right ) \operatorname {Si}\left (\frac {2 \, {\left (d x + c\right )}}{d}\right ) + \log \left (d x + c\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 2.95, size = 51, normalized size = 0.89 \[ \frac {2 \, \cos \left (\frac {2 \, c}{d}\right ) \operatorname {Ci}\left (\frac {2 \, {\left (d x + c\right )}}{d}\right ) + 2 \, \sin \left (\frac {2 \, c}{d}\right ) \operatorname {Si}\left (\frac {2 \, {\left (d x + c\right )}}{d}\right ) + \log \left (d x + c\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 58, normalized size = 1.02 \[ \frac {2 \Ci \left (\frac {2 c}{d}+2 x \right ) \cos \left (\frac {2 c}{d}\right )}{d}+\frac {\ln \left (d x +c \right )}{d}+\frac {2 \Si \left (\frac {2 c}{d}+2 x \right ) \sin \left (\frac {2 c}{d}\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 0.38, size = 95, normalized size = 1.67 \[ -\frac {{\left (E_{1}\left (\frac {2 i \, d x + 2 i \, c}{d}\right ) + E_{1}\left (-\frac {2 i \, d x + 2 i \, c}{d}\right )\right )} \cos \left (\frac {2 \, c}{d}\right ) - {\left (-i \, E_{1}\left (\frac {2 i \, d x + 2 i \, c}{d}\right ) + i \, E_{1}\left (-\frac {2 i \, d x + 2 i \, c}{d}\right )\right )} \sin \left (\frac {2 \, c}{d}\right ) - \log \left (d x + c\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {\sin \left (3\,x\right )}{\sin \relax (x)\,\left (c+d\,x\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sin {\left (3 x \right )} \csc {\relax (x )}}{c + d x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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