Optimal. Leaf size=184 \[ \frac{i d (c+d x) \text{PolyLog}\left (2,-e^{2 i (a+b x)}\right )}{b^2}-\frac{d^2 \text{PolyLog}\left (3,-e^{2 i (a+b x)}\right )}{2 b^3}-\frac{d (c+d x) \sin (a+b x) \cos (a+b x)}{2 b^2}+\frac{d^2 \sin ^2(a+b x)}{4 b^3}-\frac{(c+d x)^2 \log \left (1+e^{2 i (a+b x)}\right )}{b}-\frac{(c+d x)^2 \sin ^2(a+b x)}{2 b}+\frac{c d x}{2 b}+\frac{d^2 x^2}{4 b}+\frac{i (c+d x)^3}{3 d} \]
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Rubi [A] time = 0.228075, antiderivative size = 184, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364, Rules used = {4407, 4404, 3310, 3719, 2190, 2531, 2282, 6589} \[ \frac{i d (c+d x) \text{PolyLog}\left (2,-e^{2 i (a+b x)}\right )}{b^2}-\frac{d^2 \text{PolyLog}\left (3,-e^{2 i (a+b x)}\right )}{2 b^3}-\frac{d (c+d x) \sin (a+b x) \cos (a+b x)}{2 b^2}+\frac{d^2 \sin ^2(a+b x)}{4 b^3}-\frac{(c+d x)^2 \log \left (1+e^{2 i (a+b x)}\right )}{b}-\frac{(c+d x)^2 \sin ^2(a+b x)}{2 b}+\frac{c d x}{2 b}+\frac{d^2 x^2}{4 b}+\frac{i (c+d x)^3}{3 d} \]
Antiderivative was successfully verified.
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Rule 4407
Rule 4404
Rule 3310
Rule 3719
Rule 2190
Rule 2531
Rule 2282
Rule 6589
Rubi steps
\begin{align*} \int (c+d x)^2 \sin ^2(a+b x) \tan (a+b x) \, dx &=-\int (c+d x)^2 \cos (a+b x) \sin (a+b x) \, dx+\int (c+d x)^2 \tan (a+b x) \, dx\\ &=\frac{i (c+d x)^3}{3 d}-\frac{(c+d x)^2 \sin ^2(a+b x)}{2 b}-2 i \int \frac{e^{2 i (a+b x)} (c+d x)^2}{1+e^{2 i (a+b x)}} \, dx+\frac{d \int (c+d x) \sin ^2(a+b x) \, dx}{b}\\ &=\frac{i (c+d x)^3}{3 d}-\frac{(c+d x)^2 \log \left (1+e^{2 i (a+b x)}\right )}{b}-\frac{d (c+d x) \cos (a+b x) \sin (a+b x)}{2 b^2}+\frac{d^2 \sin ^2(a+b x)}{4 b^3}-\frac{(c+d x)^2 \sin ^2(a+b x)}{2 b}+\frac{d \int (c+d x) \, dx}{2 b}+\frac{(2 d) \int (c+d x) \log \left (1+e^{2 i (a+b x)}\right ) \, dx}{b}\\ &=\frac{c d x}{2 b}+\frac{d^2 x^2}{4 b}+\frac{i (c+d x)^3}{3 d}-\frac{(c+d x)^2 \log \left (1+e^{2 i (a+b x)}\right )}{b}+\frac{i d (c+d x) \text{Li}_2\left (-e^{2 i (a+b x)}\right )}{b^2}-\frac{d (c+d x) \cos (a+b x) \sin (a+b x)}{2 b^2}+\frac{d^2 \sin ^2(a+b x)}{4 b^3}-\frac{(c+d x)^2 \sin ^2(a+b x)}{2 b}-\frac{\left (i d^2\right ) \int \text{Li}_2\left (-e^{2 i (a+b x)}\right ) \, dx}{b^2}\\ &=\frac{c d x}{2 b}+\frac{d^2 x^2}{4 b}+\frac{i (c+d x)^3}{3 d}-\frac{(c+d x)^2 \log \left (1+e^{2 i (a+b x)}\right )}{b}+\frac{i d (c+d x) \text{Li}_2\left (-e^{2 i (a+b x)}\right )}{b^2}-\frac{d (c+d x) \cos (a+b x) \sin (a+b x)}{2 b^2}+\frac{d^2 \sin ^2(a+b x)}{4 b^3}-\frac{(c+d x)^2 \sin ^2(a+b x)}{2 b}-\frac{d^2 \operatorname{Subst}\left (\int \frac{\text{Li}_2(-x)}{x} \, dx,x,e^{2 i (a+b x)}\right )}{2 b^3}\\ &=\frac{c d x}{2 b}+\frac{d^2 x^2}{4 b}+\frac{i (c+d x)^3}{3 d}-\frac{(c+d x)^2 \log \left (1+e^{2 i (a+b x)}\right )}{b}+\frac{i d (c+d x) \text{Li}_2\left (-e^{2 i (a+b x)}\right )}{b^2}-\frac{d^2 \text{Li}_3\left (-e^{2 i (a+b x)}\right )}{2 b^3}-\frac{d (c+d x) \cos (a+b x) \sin (a+b x)}{2 b^2}+\frac{d^2 \sin ^2(a+b x)}{4 b^3}-\frac{(c+d x)^2 \sin ^2(a+b x)}{2 b}\\ \end{align*}
Mathematica [B] time = 6.4593, size = 518, normalized size = 2.82 \[ -\frac{c d \csc (a) \sec (a) \left (b^2 x^2 e^{-i \tan ^{-1}(\cot (a))}-\frac{\cot (a) \left (i \text{PolyLog}\left (2,e^{2 i \left (b x-\tan ^{-1}(\cot (a))\right )}\right )+i b x \left (-2 \tan ^{-1}(\cot (a))-\pi \right )-2 \left (b x-\tan ^{-1}(\cot (a))\right ) \log \left (1-e^{2 i \left (b x-\tan ^{-1}(\cot (a))\right )}\right )-2 \tan ^{-1}(\cot (a)) \log \left (\sin \left (b x-\tan ^{-1}(\cot (a))\right )\right )-\pi \log \left (1+e^{-2 i b x}\right )+\pi \log (\cos (b x))\right )}{\sqrt{\cot ^2(a)+1}}\right )}{b^2 \sqrt{\csc ^2(a) \left (\sin ^2(a)+\cos ^2(a)\right )}}-\frac{i e^{-i a} d^2 \sec (a) \left (6 \left (1+e^{2 i a}\right ) b x \text{PolyLog}\left (2,-e^{-2 i (a+b x)}\right )-3 i \left (1+e^{2 i a}\right ) \text{PolyLog}\left (3,-e^{-2 i (a+b x)}\right )+2 b^2 x^2 \left (2 b x-3 i \left (1+e^{2 i a}\right ) \log \left (1+e^{-2 i (a+b x)}\right )\right )\right )}{12 b^3}+\frac{\cos (2 b x) \left (2 b^2 c^2 \cos (2 a)+4 b^2 c d x \cos (2 a)+2 b^2 d^2 x^2 \cos (2 a)-2 b c d \sin (2 a)-2 b d^2 x \sin (2 a)-d^2 \cos (2 a)\right )}{8 b^3}-\frac{\sin (2 b x) \left (2 b^2 c^2 \sin (2 a)+4 b^2 c d x \sin (2 a)+2 b^2 d^2 x^2 \sin (2 a)+2 b c d \cos (2 a)+2 b d^2 x \cos (2 a)-d^2 \sin (2 a)\right )}{8 b^3}-\frac{c^2 \sec (a) (b x \sin (a)+\cos (a) \log (\cos (a) \cos (b x)-\sin (a) \sin (b x)))}{b \left (\sin ^2(a)+\cos ^2(a)\right )}+\frac{1}{3} x \tan (a) \left (3 c^2+3 c d x+d^2 x^2\right ) \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.353, size = 379, normalized size = 2.1 \begin{align*}{\frac{2\,i{a}^{2}cd}{{b}^{2}}}+icd{x}^{2}-{\frac{{\frac{4\,i}{3}}{a}^{3}{d}^{2}}{{b}^{3}}}+{\frac{ \left ( 2\,{d}^{2}{x}^{2}{b}^{2}+2\,ib{d}^{2}x+4\,{b}^{2}cdx+2\,ibcd+2\,{b}^{2}{c}^{2}-{d}^{2} \right ){{\rm e}^{2\,i \left ( bx+a \right ) }}}{16\,{b}^{3}}}+{\frac{ \left ( 2\,{d}^{2}{x}^{2}{b}^{2}-2\,ib{d}^{2}x+4\,{b}^{2}cdx-2\,ibcd+2\,{b}^{2}{c}^{2}-{d}^{2} \right ){{\rm e}^{-2\,i \left ( bx+a \right ) }}}{16\,{b}^{3}}}-{\frac{{c}^{2}\ln \left ({{\rm e}^{2\,i \left ( bx+a \right ) }}+1 \right ) }{b}}+2\,{\frac{{c}^{2}\ln \left ({{\rm e}^{i \left ( bx+a \right ) }} \right ) }{b}}+2\,{\frac{{a}^{2}{d}^{2}\ln \left ({{\rm e}^{i \left ( bx+a \right ) }} \right ) }{{b}^{3}}}+{\frac{4\,iacdx}{b}}+{\frac{idc{\it polylog} \left ( 2,-{{\rm e}^{2\,i \left ( bx+a \right ) }} \right ) }{{b}^{2}}}-i{c}^{2}x+{\frac{i{d}^{2}{\it polylog} \left ( 2,-{{\rm e}^{2\,i \left ( bx+a \right ) }} \right ) x}{{b}^{2}}}-{\frac{{d}^{2}\ln \left ({{\rm e}^{2\,i \left ( bx+a \right ) }}+1 \right ){x}^{2}}{b}}-{\frac{2\,i{a}^{2}{d}^{2}x}{{b}^{2}}}-{\frac{{d}^{2}{\it polylog} \left ( 3,-{{\rm e}^{2\,i \left ( bx+a \right ) }} \right ) }{2\,{b}^{3}}}-4\,{\frac{cda\ln \left ({{\rm e}^{i \left ( bx+a \right ) }} \right ) }{{b}^{2}}}+{\frac{i}{3}}{d}^{2}{x}^{3}-2\,{\frac{cd\ln \left ({{\rm e}^{2\,i \left ( bx+a \right ) }}+1 \right ) x}{b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.83483, size = 512, normalized size = 2.78 \begin{align*} -\frac{12 \,{\left (\sin \left (b x + a\right )^{2} + \log \left (\sin \left (b x + a\right )^{2} - 1\right )\right )} c^{2} - \frac{24 \,{\left (\sin \left (b x + a\right )^{2} + \log \left (\sin \left (b x + a\right )^{2} - 1\right )\right )} a c d}{b} + \frac{12 \,{\left (\sin \left (b x + a\right )^{2} + \log \left (\sin \left (b x + a\right )^{2} - 1\right )\right )} a^{2} d^{2}}{b^{2}} + \frac{-8 i \,{\left (b x + a\right )}^{3} d^{2} +{\left (-24 i \, b c d + 24 i \, a d^{2}\right )}{\left (b x + a\right )}^{2} + 12 \, d^{2}{\rm Li}_{3}(-e^{\left (2 i \, b x + 2 i \, a\right )}) +{\left (24 i \,{\left (b x + a\right )}^{2} d^{2} +{\left (48 i \, b c d - 48 i \, a d^{2}\right )}{\left (b x + a\right )}\right )} \arctan \left (\sin \left (2 \, b x + 2 \, a\right ), \cos \left (2 \, b x + 2 \, a\right ) + 1\right ) - 3 \,{\left (2 \,{\left (b x + a\right )}^{2} d^{2} + 4 \,{\left (b c d - a d^{2}\right )}{\left (b x + a\right )} - d^{2}\right )} \cos \left (2 \, b x + 2 \, a\right ) +{\left (-24 i \, b c d - 24 i \,{\left (b x + a\right )} d^{2} + 24 i \, a d^{2}\right )}{\rm Li}_2\left (-e^{\left (2 i \, b x + 2 i \, a\right )}\right ) + 12 \,{\left ({\left (b x + a\right )}^{2} d^{2} + 2 \,{\left (b c d - a d^{2}\right )}{\left (b x + a\right )}\right )} \log \left (\cos \left (2 \, b x + 2 \, a\right )^{2} + \sin \left (2 \, b x + 2 \, a\right )^{2} + 2 \, \cos \left (2 \, b x + 2 \, a\right ) + 1\right ) + 6 \,{\left (b c d +{\left (b x + a\right )} d^{2} - a d^{2}\right )} \sin \left (2 \, b x + 2 \, a\right )}{b^{2}}}{24 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] time = 0.706088, size = 1746, normalized size = 9.49 \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} \sec \left (b x + a\right ) \sin \left (b x + a\right )^{3}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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