Optimal. Leaf size=95 \[ \frac{3 i d \text{PolyLog}\left (2,-e^{i (a+b x)}\right )}{b^2}-\frac{3 i d \text{PolyLog}\left (2,e^{i (a+b x)}\right )}{b^2}-\frac{4 d \sin (a+b x)}{b^2}+\frac{4 (c+d x) \cos (a+b x)}{b}-\frac{6 (c+d x) \tanh ^{-1}\left (e^{i (a+b x)}\right )}{b} \]
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Rubi [A] time = 0.108855, antiderivative size = 95, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 7, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.304, Rules used = {4431, 4408, 3296, 2637, 4183, 2279, 2391} \[ \frac{3 i d \text{PolyLog}\left (2,-e^{i (a+b x)}\right )}{b^2}-\frac{3 i d \text{PolyLog}\left (2,e^{i (a+b x)}\right )}{b^2}-\frac{4 d \sin (a+b x)}{b^2}+\frac{4 (c+d x) \cos (a+b x)}{b}-\frac{6 (c+d x) \tanh ^{-1}\left (e^{i (a+b x)}\right )}{b} \]
Antiderivative was successfully verified.
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Rule 4431
Rule 4408
Rule 3296
Rule 2637
Rule 4183
Rule 2279
Rule 2391
Rubi steps
\begin{align*} \int (c+d x) \csc ^2(a+b x) \sin (3 a+3 b x) \, dx &=\int (3 (c+d x) \cos (a+b x) \cot (a+b x)-(c+d x) \sin (a+b x)) \, dx\\ &=3 \int (c+d x) \cos (a+b x) \cot (a+b x) \, dx-\int (c+d x) \sin (a+b x) \, dx\\ &=\frac{(c+d x) \cos (a+b x)}{b}+3 \int (c+d x) \csc (a+b x) \, dx-3 \int (c+d x) \sin (a+b x) \, dx-\frac{d \int \cos (a+b x) \, dx}{b}\\ &=-\frac{6 (c+d x) \tanh ^{-1}\left (e^{i (a+b x)}\right )}{b}+\frac{4 (c+d x) \cos (a+b x)}{b}-\frac{d \sin (a+b x)}{b^2}-\frac{(3 d) \int \cos (a+b x) \, dx}{b}-\frac{(3 d) \int \log \left (1-e^{i (a+b x)}\right ) \, dx}{b}+\frac{(3 d) \int \log \left (1+e^{i (a+b x)}\right ) \, dx}{b}\\ &=-\frac{6 (c+d x) \tanh ^{-1}\left (e^{i (a+b x)}\right )}{b}+\frac{4 (c+d x) \cos (a+b x)}{b}-\frac{4 d \sin (a+b x)}{b^2}+\frac{(3 i d) \operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{i (a+b x)}\right )}{b^2}-\frac{(3 i d) \operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,e^{i (a+b x)}\right )}{b^2}\\ &=-\frac{6 (c+d x) \tanh ^{-1}\left (e^{i (a+b x)}\right )}{b}+\frac{4 (c+d x) \cos (a+b x)}{b}+\frac{3 i d \text{Li}_2\left (-e^{i (a+b x)}\right )}{b^2}-\frac{3 i d \text{Li}_2\left (e^{i (a+b x)}\right )}{b^2}-\frac{4 d \sin (a+b x)}{b^2}\\ \end{align*}
Mathematica [A] time = 0.341125, size = 171, normalized size = 1.8 \[ \frac{3 d \left (i \left (\text{PolyLog}\left (2,-e^{i (a+b x)}\right )-\text{PolyLog}\left (2,e^{i (a+b x)}\right )\right )+(a+b x) \left (\log \left (1-e^{i (a+b x)}\right )-\log \left (1+e^{i (a+b x)}\right )\right )\right )}{b^2}-\frac{4 d \sin (a+b x)}{b^2}-\frac{3 a d \log \left (\tan \left (\frac{1}{2} (a+b x)\right )\right )}{b^2}+\frac{4 c \cos (a+b x)}{b}+\frac{3 c \log \left (\sin \left (\frac{1}{2} (a+b x)\right )\right )}{b}-\frac{3 c \log \left (\cos \left (\frac{1}{2} (a+b x)\right )\right )}{b}+\frac{4 d x \cos (a+b x)}{b} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.249, size = 205, normalized size = 2.2 \begin{align*} 2\,{\frac{ \left ( dxb+bc+id \right ){{\rm e}^{i \left ( bx+a \right ) }}}{{b}^{2}}}+2\,{\frac{ \left ( dxb+bc-id \right ){{\rm e}^{-i \left ( bx+a \right ) }}}{{b}^{2}}}-6\,{\frac{c{\it Artanh} \left ({{\rm e}^{i \left ( bx+a \right ) }} \right ) }{b}}+3\,{\frac{d\ln \left ( 1-{{\rm e}^{i \left ( bx+a \right ) }} \right ) x}{b}}+3\,{\frac{d\ln \left ( 1-{{\rm e}^{i \left ( bx+a \right ) }} \right ) a}{{b}^{2}}}-{\frac{3\,id{\it polylog} \left ( 2,{{\rm e}^{i \left ( bx+a \right ) }} \right ) }{{b}^{2}}}-3\,{\frac{d\ln \left ({{\rm e}^{i \left ( bx+a \right ) }}+1 \right ) x}{b}}-3\,{\frac{d\ln \left ({{\rm e}^{i \left ( bx+a \right ) }}+1 \right ) a}{{b}^{2}}}+{\frac{3\,id{\it polylog} \left ( 2,-{{\rm e}^{i \left ( bx+a \right ) }} \right ) }{{b}^{2}}}+6\,{\frac{ad{\it Artanh} \left ({{\rm e}^{i \left ( bx+a \right ) }} \right ) }{{b}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{c{\left (8 \, \cos \left (b x + a\right ) - 3 \, \log \left (\cos \left (b x\right )^{2} + 2 \, \cos \left (b x\right ) \cos \left (a\right ) + \cos \left (a\right )^{2} + \sin \left (b x\right )^{2} - 2 \, \sin \left (b x\right ) \sin \left (a\right ) + \sin \left (a\right )^{2}\right ) + 3 \, \log \left (\cos \left (b x\right )^{2} - 2 \, \cos \left (b x\right ) \cos \left (a\right ) + \cos \left (a\right )^{2} + \sin \left (b x\right )^{2} + 2 \, \sin \left (b x\right ) \sin \left (a\right ) + \sin \left (a\right )^{2}\right )\right )}}{2 \, b} + \frac{-\frac{1}{2} \,{\left (6 i \, b x \arctan \left (\sin \left (b x + a\right ), \cos \left (b x + a\right ) + 1\right ) + 6 i \, b x \arctan \left (\sin \left (b x + a\right ), -\cos \left (b x + a\right ) + 1\right ) - 8 \, b x \cos \left (b x + a\right ) + 3 \, b x \log \left (\cos \left (b x + a\right )^{2} + \sin \left (b x + a\right )^{2} + 2 \, \cos \left (b x + a\right ) + 1\right ) - 3 \, b x \log \left (\cos \left (b x + a\right )^{2} + \sin \left (b x + a\right )^{2} - 2 \, \cos \left (b x + a\right ) + 1\right ) - 6 i \,{\rm Li}_2\left (-e^{\left (i \, b x + i \, a\right )}\right ) + 6 i \,{\rm Li}_2\left (e^{\left (i \, b x + i \, a\right )}\right ) + 8 \, \sin \left (b x + a\right )\right )} d}{b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 0.58448, size = 814, normalized size = 8.57 \begin{align*} \frac{8 \,{\left (b d x + b c\right )} \cos \left (b x + a\right ) - 3 i \, d{\rm Li}_2\left (\cos \left (b x + a\right ) + i \, \sin \left (b x + a\right )\right ) + 3 i \, d{\rm Li}_2\left (\cos \left (b x + a\right ) - i \, \sin \left (b x + a\right )\right ) - 3 i \, d{\rm Li}_2\left (-\cos \left (b x + a\right ) + i \, \sin \left (b x + a\right )\right ) + 3 i \, d{\rm Li}_2\left (-\cos \left (b x + a\right ) - i \, \sin \left (b x + a\right )\right ) - 3 \,{\left (b d x + b c\right )} \log \left (\cos \left (b x + a\right ) + i \, \sin \left (b x + a\right ) + 1\right ) - 3 \,{\left (b d x + b c\right )} \log \left (\cos \left (b x + a\right ) - i \, \sin \left (b x + a\right ) + 1\right ) + 3 \,{\left (b c - a d\right )} \log \left (-\frac{1}{2} \, \cos \left (b x + a\right ) + \frac{1}{2} i \, \sin \left (b x + a\right ) + \frac{1}{2}\right ) + 3 \,{\left (b c - a d\right )} \log \left (-\frac{1}{2} \, \cos \left (b x + a\right ) - \frac{1}{2} i \, \sin \left (b x + a\right ) + \frac{1}{2}\right ) + 3 \,{\left (b d x + a d\right )} \log \left (-\cos \left (b x + a\right ) + i \, \sin \left (b x + a\right ) + 1\right ) + 3 \,{\left (b d x + a d\right )} \log \left (-\cos \left (b x + a\right ) - i \, \sin \left (b x + a\right ) + 1\right ) - 8 \, d \sin \left (b x + a\right )}{2 \, b^{2}} \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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (d x + c\right )} \csc \left (b x + a\right )^{2} \sin \left (3 \, b x + 3 \, a\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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