Optimal. Leaf size=172 \[ \frac{6 i d (c+d x) \text{PolyLog}\left (2,-e^{i (a+b x)}\right )}{b^2}-\frac{6 i d (c+d x) \text{PolyLog}\left (2,e^{i (a+b x)}\right )}{b^2}-\frac{6 d^2 \text{PolyLog}\left (3,-e^{i (a+b x)}\right )}{b^3}+\frac{6 d^2 \text{PolyLog}\left (3,e^{i (a+b x)}\right )}{b^3}-\frac{8 d (c+d x) \sin (a+b x)}{b^2}-\frac{8 d^2 \cos (a+b x)}{b^3}+\frac{4 (c+d x)^2 \cos (a+b x)}{b}-\frac{6 (c+d x)^2 \tanh ^{-1}\left (e^{i (a+b x)}\right )}{b} \]
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Rubi [A] time = 0.228793, antiderivative size = 172, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 8, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.32, Rules used = {4431, 4408, 3296, 2638, 4183, 2531, 2282, 6589} \[ \frac{6 i d (c+d x) \text{PolyLog}\left (2,-e^{i (a+b x)}\right )}{b^2}-\frac{6 i d (c+d x) \text{PolyLog}\left (2,e^{i (a+b x)}\right )}{b^2}-\frac{6 d^2 \text{PolyLog}\left (3,-e^{i (a+b x)}\right )}{b^3}+\frac{6 d^2 \text{PolyLog}\left (3,e^{i (a+b x)}\right )}{b^3}-\frac{8 d (c+d x) \sin (a+b x)}{b^2}-\frac{8 d^2 \cos (a+b x)}{b^3}+\frac{4 (c+d x)^2 \cos (a+b x)}{b}-\frac{6 (c+d x)^2 \tanh ^{-1}\left (e^{i (a+b x)}\right )}{b} \]
Antiderivative was successfully verified.
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Rule 4431
Rule 4408
Rule 3296
Rule 2638
Rule 4183
Rule 2531
Rule 2282
Rule 6589
Rubi steps
\begin{align*} \int (c+d x)^2 \csc ^2(a+b x) \sin (3 a+3 b x) \, dx &=\int \left (3 (c+d x)^2 \cos (a+b x) \cot (a+b x)-(c+d x)^2 \sin (a+b x)\right ) \, dx\\ &=3 \int (c+d x)^2 \cos (a+b x) \cot (a+b x) \, dx-\int (c+d x)^2 \sin (a+b x) \, dx\\ &=\frac{(c+d x)^2 \cos (a+b x)}{b}+3 \int (c+d x)^2 \csc (a+b x) \, dx-3 \int (c+d x)^2 \sin (a+b x) \, dx-\frac{(2 d) \int (c+d x) \cos (a+b x) \, dx}{b}\\ &=-\frac{6 (c+d x)^2 \tanh ^{-1}\left (e^{i (a+b x)}\right )}{b}+\frac{4 (c+d x)^2 \cos (a+b x)}{b}-\frac{2 d (c+d x) \sin (a+b x)}{b^2}-\frac{(6 d) \int (c+d x) \cos (a+b x) \, dx}{b}-\frac{(6 d) \int (c+d x) \log \left (1-e^{i (a+b x)}\right ) \, dx}{b}+\frac{(6 d) \int (c+d x) \log \left (1+e^{i (a+b x)}\right ) \, dx}{b}+\frac{\left (2 d^2\right ) \int \sin (a+b x) \, dx}{b^2}\\ &=-\frac{6 (c+d x)^2 \tanh ^{-1}\left (e^{i (a+b x)}\right )}{b}-\frac{2 d^2 \cos (a+b x)}{b^3}+\frac{4 (c+d x)^2 \cos (a+b x)}{b}+\frac{6 i d (c+d x) \text{Li}_2\left (-e^{i (a+b x)}\right )}{b^2}-\frac{6 i d (c+d x) \text{Li}_2\left (e^{i (a+b x)}\right )}{b^2}-\frac{8 d (c+d x) \sin (a+b x)}{b^2}-\frac{\left (6 i d^2\right ) \int \text{Li}_2\left (-e^{i (a+b x)}\right ) \, dx}{b^2}+\frac{\left (6 i d^2\right ) \int \text{Li}_2\left (e^{i (a+b x)}\right ) \, dx}{b^2}+\frac{\left (6 d^2\right ) \int \sin (a+b x) \, dx}{b^2}\\ &=-\frac{6 (c+d x)^2 \tanh ^{-1}\left (e^{i (a+b x)}\right )}{b}-\frac{8 d^2 \cos (a+b x)}{b^3}+\frac{4 (c+d x)^2 \cos (a+b x)}{b}+\frac{6 i d (c+d x) \text{Li}_2\left (-e^{i (a+b x)}\right )}{b^2}-\frac{6 i d (c+d x) \text{Li}_2\left (e^{i (a+b x)}\right )}{b^2}-\frac{8 d (c+d x) \sin (a+b x)}{b^2}-\frac{\left (6 d^2\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(-x)}{x} \, dx,x,e^{i (a+b x)}\right )}{b^3}+\frac{\left (6 d^2\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(x)}{x} \, dx,x,e^{i (a+b x)}\right )}{b^3}\\ &=-\frac{6 (c+d x)^2 \tanh ^{-1}\left (e^{i (a+b x)}\right )}{b}-\frac{8 d^2 \cos (a+b x)}{b^3}+\frac{4 (c+d x)^2 \cos (a+b x)}{b}+\frac{6 i d (c+d x) \text{Li}_2\left (-e^{i (a+b x)}\right )}{b^2}-\frac{6 i d (c+d x) \text{Li}_2\left (e^{i (a+b x)}\right )}{b^2}-\frac{6 d^2 \text{Li}_3\left (-e^{i (a+b x)}\right )}{b^3}+\frac{6 d^2 \text{Li}_3\left (e^{i (a+b x)}\right )}{b^3}-\frac{8 d (c+d x) \sin (a+b x)}{b^2}\\ \end{align*}
Mathematica [A] time = 1.15642, size = 223, normalized size = 1.3 \[ \frac{6 i b d (c+d x) \text{PolyLog}\left (2,-e^{i (a+b x)}\right )-6 i b d (c+d x) \text{PolyLog}\left (2,e^{i (a+b x)}\right )-6 d^2 \text{PolyLog}\left (3,-e^{i (a+b x)}\right )+6 d^2 \text{PolyLog}\left (3,e^{i (a+b x)}\right )+4 \cos (b x) \left (\cos (a) \left (b^2 (c+d x)^2-2 d^2\right )-2 b d \sin (a) (c+d x)\right )-4 \sin (b x) \left (\sin (a) \left (b^2 (c+d x)^2-2 d^2\right )+2 b d \cos (a) (c+d x)\right )+3 b^2 (c+d x)^2 \log \left (1-e^{i (a+b x)}\right )-3 b^2 (c+d x)^2 \log \left (1+e^{i (a+b x)}\right )}{b^3} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.259, size = 481, normalized size = 2.8 \begin{align*} 2\,{\frac{ \left ({d}^{2}{x}^{2}{b}^{2}+2\,{b}^{2}cdx+{b}^{2}{c}^{2}+2\,ib{d}^{2}x-2\,{d}^{2}+2\,ibcd \right ){{\rm e}^{i \left ( bx+a \right ) }}}{{b}^{3}}}+2\,{\frac{ \left ({d}^{2}{x}^{2}{b}^{2}+2\,{b}^{2}cdx+{b}^{2}{c}^{2}-2\,ib{d}^{2}x-2\,{d}^{2}-2\,ibcd \right ){{\rm e}^{-i \left ( bx+a \right ) }}}{{b}^{3}}}-6\,{\frac{{d}^{2}{a}^{2}{\it Artanh} \left ({{\rm e}^{i \left ( bx+a \right ) }} \right ) }{{b}^{3}}}-6\,{\frac{{d}^{2}{\it polylog} \left ( 3,-{{\rm e}^{i \left ( bx+a \right ) }} \right ) }{{b}^{3}}}+6\,{\frac{cd\ln \left ( 1-{{\rm e}^{i \left ( bx+a \right ) }} \right ) x}{b}}+6\,{\frac{cd\ln \left ( 1-{{\rm e}^{i \left ( bx+a \right ) }} \right ) a}{{b}^{2}}}-6\,{\frac{cd\ln \left ({{\rm e}^{i \left ( bx+a \right ) }}+1 \right ) x}{b}}-6\,{\frac{cd\ln \left ({{\rm e}^{i \left ( bx+a \right ) }}+1 \right ) a}{{b}^{2}}}+12\,{\frac{cda{\it Artanh} \left ({{\rm e}^{i \left ( bx+a \right ) }} \right ) }{{b}^{2}}}-{\frac{6\,i{d}^{2}{\it polylog} \left ( 2,{{\rm e}^{i \left ( bx+a \right ) }} \right ) x}{{b}^{2}}}+{\frac{6\,i{d}^{2}{\it polylog} \left ( 2,-{{\rm e}^{i \left ( bx+a \right ) }} \right ) x}{{b}^{2}}}+3\,{\frac{{d}^{2}\ln \left ( 1-{{\rm e}^{i \left ( bx+a \right ) }} \right ){x}^{2}}{b}}-3\,{\frac{{d}^{2}\ln \left ( 1-{{\rm e}^{i \left ( bx+a \right ) }} \right ){a}^{2}}{{b}^{3}}}+{\frac{6\,icd{\it polylog} \left ( 2,-{{\rm e}^{i \left ( bx+a \right ) }} \right ) }{{b}^{2}}}-3\,{\frac{{d}^{2}\ln \left ({{\rm e}^{i \left ( bx+a \right ) }}+1 \right ){x}^{2}}{b}}+3\,{\frac{{d}^{2}\ln \left ({{\rm e}^{i \left ( bx+a \right ) }}+1 \right ){a}^{2}}{{b}^{3}}}-{\frac{6\,icd{\it polylog} \left ( 2,{{\rm e}^{i \left ( bx+a \right ) }} \right ) }{{b}^{2}}}+6\,{\frac{{d}^{2}{\it polylog} \left ( 3,{{\rm e}^{i \left ( bx+a \right ) }} \right ) }{{b}^{3}}}-6\,{\frac{{c}^{2}{\it Artanh} \left ({{\rm e}^{i \left ( bx+a \right ) }} \right ) }{b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.56023, size = 552, normalized size = 3.21 \begin{align*} \frac{c^{2}{\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{12 \, d^{2}{\rm Li}_{3}(-e^{\left (i \, b x + i \, a\right )}) - 12 \, d^{2}{\rm Li}_{3}(e^{\left (i \, b x + i \, a\right )}) +{\left (6 i \, b^{2} d^{2} x^{2} + 12 i \, b^{2} c d x\right )} \arctan \left (\sin \left (b x + a\right ), \cos \left (b x + a\right ) + 1\right ) +{\left (6 i \, b^{2} d^{2} x^{2} + 12 i \, b^{2} c d x\right )} \arctan \left (\sin \left (b x + a\right ), -\cos \left (b x + a\right ) + 1\right ) - 8 \,{\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x - 2 \, d^{2}\right )} \cos \left (b x + a\right ) +{\left (-12 i \, b d^{2} x - 12 i \, b c d\right )}{\rm Li}_2\left (-e^{\left (i \, b x + i \, a\right )}\right ) +{\left (12 i \, b d^{2} x + 12 i \, b c d\right )}{\rm Li}_2\left (e^{\left (i \, b x + i \, a\right )}\right ) + 3 \,{\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x\right )} \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 \,{\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x\right )} \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 ) + 16 \,{\left (b d^{2} x + b c d\right )} \sin \left (b x + a\right )}{2 \, b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] time = 0.637698, size = 1480, normalized size = 8.6 \begin{align*} \text{result too large to display} \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 )}^{2} \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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