Optimal. Leaf size=71 \[ 3 \text{Unintegrable}\left (\frac{\csc (a+b x)}{c+d x},x\right )-\frac{4 \sin \left (a-\frac{b c}{d}\right ) \text{CosIntegral}\left (\frac{b c}{d}+b x\right )}{d}-\frac{4 \cos \left (a-\frac{b c}{d}\right ) \text{Si}\left (\frac{b c}{d}+b x\right )}{d} \]
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Rubi [A] time = 0.2104, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{\csc ^2(a+b x) \sin (3 a+3 b x)}{c+d x} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin{align*} \int \frac{\csc ^2(a+b x) \sin (3 a+3 b x)}{c+d x} \, dx &=\int \left (\frac{3 \cos (a+b x) \cot (a+b x)}{c+d x}-\frac{\sin (a+b x)}{c+d x}\right ) \, dx\\ &=3 \int \frac{\cos (a+b x) \cot (a+b x)}{c+d x} \, dx-\int \frac{\sin (a+b x)}{c+d x} \, dx\\ &=3 \int \frac{\csc (a+b x)}{c+d x} \, dx-3 \int \frac{\sin (a+b x)}{c+d x} \, dx-\cos \left (a-\frac{b c}{d}\right ) \int \frac{\sin \left (\frac{b c}{d}+b x\right )}{c+d x} \, dx-\sin \left (a-\frac{b c}{d}\right ) \int \frac{\cos \left (\frac{b c}{d}+b x\right )}{c+d x} \, dx\\ &=-\frac{\text{Ci}\left (\frac{b c}{d}+b x\right ) \sin \left (a-\frac{b c}{d}\right )}{d}-\frac{\cos \left (a-\frac{b c}{d}\right ) \text{Si}\left (\frac{b c}{d}+b x\right )}{d}+3 \int \frac{\csc (a+b x)}{c+d x} \, dx-\left (3 \cos \left (a-\frac{b c}{d}\right )\right ) \int \frac{\sin \left (\frac{b c}{d}+b x\right )}{c+d x} \, dx-\left (3 \sin \left (a-\frac{b c}{d}\right )\right ) \int \frac{\cos \left (\frac{b c}{d}+b x\right )}{c+d x} \, dx\\ &=-\frac{4 \text{Ci}\left (\frac{b c}{d}+b x\right ) \sin \left (a-\frac{b c}{d}\right )}{d}-\frac{4 \cos \left (a-\frac{b c}{d}\right ) \text{Si}\left (\frac{b c}{d}+b x\right )}{d}+3 \int \frac{\csc (a+b x)}{c+d x} \, dx\\ \end{align*}
Mathematica [A] time = 6.23243, size = 0, normalized size = 0. \[ \int \frac{\csc ^2(a+b x) \sin (3 a+3 b x)}{c+d x} \, dx \]
Verification is Not applicable to the result.
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Maple [A] time = 0.349, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( \csc \left ( bx+a \right ) \right ) ^{2}\sin \left ( 3\,bx+3\,a \right ) }{dx+c}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{{\left (2 i \, E_{1}\left (\frac{i \, b d x + i \, b c}{d}\right ) - 2 i \, E_{1}\left (-\frac{i \, b d x + i \, b c}{d}\right )\right )} \cos \left (-\frac{b c - a d}{d}\right ) + 3 \, d \int \frac{\sin \left (b x + a\right )}{{\left (d x + c\right )}{\left (\cos \left (b x + a\right )^{2} + \sin \left (b x + a\right )^{2} + 2 \, \cos \left (b x + a\right ) + 1\right )}}\,{d x} + 3 \, d \int \frac{\sin \left (b x + a\right )}{{\left (d x + c\right )}{\left (\cos \left (b x + a\right )^{2} + \sin \left (b x + a\right )^{2} - 2 \, \cos \left (b x + a\right ) + 1\right )}}\,{d x} + 2 \,{\left (E_{1}\left (\frac{i \, b d x + i \, b c}{d}\right ) + E_{1}\left (-\frac{i \, b d x + i \, b c}{d}\right )\right )} \sin \left (-\frac{b c - a d}{d}\right )}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\csc \left (b x + a\right )^{2} \sin \left (3 \, b x + 3 \, a\right )}{d x + c}, x\right ) \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 [A] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\csc \left (b x + a\right )^{2} \sin \left (3 \, b x + 3 \, a\right )}{d x + c}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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