Optimal. Leaf size=71 \[ \frac{2 \cos \left (2 a-\frac{2 b c}{d}\right ) \text{CosIntegral}\left (\frac{2 b c}{d}+2 b x\right )}{d}-\frac{2 \sin \left (2 a-\frac{2 b c}{d}\right ) \text{Si}\left (\frac{2 b c}{d}+2 b x\right )}{d}+\frac{\log (c+d x)}{d} \]
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Rubi [A] time = 0.281018, antiderivative size = 71, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 5, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.217, Rules used = {4431, 3312, 3303, 3299, 3302} \[ \frac{2 \cos \left (2 a-\frac{2 b c}{d}\right ) \text{CosIntegral}\left (\frac{2 b c}{d}+2 b x\right )}{d}-\frac{2 \sin \left (2 a-\frac{2 b c}{d}\right ) \text{Si}\left (\frac{2 b c}{d}+2 b x\right )}{d}+\frac{\log (c+d x)}{d} \]
Antiderivative was successfully verified.
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Rule 4431
Rule 3312
Rule 3303
Rule 3299
Rule 3302
Rubi steps
\begin{align*} \int \frac{\csc (a+b x) \sin (3 a+3 b x)}{c+d x} \, dx &=\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 ) \text{Ci}\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*}
Mathematica [A] time = 0.15281, size = 63, normalized size = 0.89 \[ \frac{2 \cos \left (2 a-\frac{2 b c}{d}\right ) \text{CosIntegral}\left (\frac{2 b (c+d x)}{d}\right )-2 \sin \left (2 a-\frac{2 b c}{d}\right ) \text{Si}\left (\frac{2 b (c+d x)}{d}\right )+\log (c+d x)}{d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.037, size = 116, normalized size = 1.6 \begin{align*} -{\frac{\ln \left ( dx+c \right ) }{d}}+2\,{\frac{1}{d}{\it Si} \left ( 2\,bx+2\,a+2\,{\frac{-ad+bc}{d}} \right ) \sin \left ( 2\,{\frac{-ad+bc}{d}} \right ) }+2\,{\frac{1}{d}{\it Ci} \left ( 2\,bx+2\,a+2\,{\frac{-ad+bc}{d}} \right ) \cos \left ( 2\,{\frac{-ad+bc}{d}} \right ) }+2\,{\frac{\ln \left ( \left ( bx+a \right ) d-ad+bc \right ) }{d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] time = 1.2865, size = 158, normalized size = 2.23 \begin{align*} -\frac{{\left (E_{1}\left (\frac{2 i \, b d x + 2 i \, b c}{d}\right ) + E_{1}\left (-\frac{2 i \, b d x + 2 i \, b c}{d}\right )\right )} \cos \left (-\frac{2 \,{\left (b c - a d\right )}}{d}\right ) -{\left (i \, E_{1}\left (\frac{2 i \, b d x + 2 i \, b c}{d}\right ) - i \, E_{1}\left (-\frac{2 i \, b d x + 2 i \, b c}{d}\right )\right )} \sin \left (-\frac{2 \,{\left (b c - a d\right )}}{d}\right ) - \log \left (d x + c\right )}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.489241, size = 228, normalized size = 3.21 \begin{align*} \frac{{\left (\operatorname{Ci}\left (\frac{2 \,{\left (b d x + b c\right )}}{d}\right ) + \operatorname{Ci}\left (-\frac{2 \,{\left (b d x + b c\right )}}{d}\right )\right )} \cos \left (-\frac{2 \,{\left (b c - a d\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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [C] time = 1.29773, size = 1509, normalized size = 21.25 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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