Optimal. Leaf size=138 \[ -\frac{i b^2 \text{PolyLog}\left (2,1-\frac{2}{1+i c x}\right )}{3 c^3}-\frac{i \left (a+b \tan ^{-1}(c x)\right )^2}{3 c^3}-\frac{2 b \log \left (\frac{2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{3 c^3}+\frac{1}{3} x^3 \left (a+b \tan ^{-1}(c x)\right )^2-\frac{b x^2 \left (a+b \tan ^{-1}(c x)\right )}{3 c}+\frac{b^2 x}{3 c^2}-\frac{b^2 \tan ^{-1}(c x)}{3 c^3} \]
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Rubi [A] time = 0.193661, antiderivative size = 138, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.571, Rules used = {4852, 4916, 321, 203, 4920, 4854, 2402, 2315} \[ -\frac{i b^2 \text{PolyLog}\left (2,1-\frac{2}{1+i c x}\right )}{3 c^3}-\frac{i \left (a+b \tan ^{-1}(c x)\right )^2}{3 c^3}-\frac{2 b \log \left (\frac{2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{3 c^3}+\frac{1}{3} x^3 \left (a+b \tan ^{-1}(c x)\right )^2-\frac{b x^2 \left (a+b \tan ^{-1}(c x)\right )}{3 c}+\frac{b^2 x}{3 c^2}-\frac{b^2 \tan ^{-1}(c x)}{3 c^3} \]
Antiderivative was successfully verified.
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Rule 4852
Rule 4916
Rule 321
Rule 203
Rule 4920
Rule 4854
Rule 2402
Rule 2315
Rubi steps
\begin{align*} \int x^2 \left (a+b \tan ^{-1}(c x)\right )^2 \, dx &=\frac{1}{3} x^3 \left (a+b \tan ^{-1}(c x)\right )^2-\frac{1}{3} (2 b c) \int \frac{x^3 \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx\\ &=\frac{1}{3} x^3 \left (a+b \tan ^{-1}(c x)\right )^2-\frac{(2 b) \int x \left (a+b \tan ^{-1}(c x)\right ) \, dx}{3 c}+\frac{(2 b) \int \frac{x \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx}{3 c}\\ &=-\frac{b x^2 \left (a+b \tan ^{-1}(c x)\right )}{3 c}-\frac{i \left (a+b \tan ^{-1}(c x)\right )^2}{3 c^3}+\frac{1}{3} x^3 \left (a+b \tan ^{-1}(c x)\right )^2+\frac{1}{3} b^2 \int \frac{x^2}{1+c^2 x^2} \, dx-\frac{(2 b) \int \frac{a+b \tan ^{-1}(c x)}{i-c x} \, dx}{3 c^2}\\ &=\frac{b^2 x}{3 c^2}-\frac{b x^2 \left (a+b \tan ^{-1}(c x)\right )}{3 c}-\frac{i \left (a+b \tan ^{-1}(c x)\right )^2}{3 c^3}+\frac{1}{3} x^3 \left (a+b \tan ^{-1}(c x)\right )^2-\frac{2 b \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac{2}{1+i c x}\right )}{3 c^3}-\frac{b^2 \int \frac{1}{1+c^2 x^2} \, dx}{3 c^2}+\frac{\left (2 b^2\right ) \int \frac{\log \left (\frac{2}{1+i c x}\right )}{1+c^2 x^2} \, dx}{3 c^2}\\ &=\frac{b^2 x}{3 c^2}-\frac{b^2 \tan ^{-1}(c x)}{3 c^3}-\frac{b x^2 \left (a+b \tan ^{-1}(c x)\right )}{3 c}-\frac{i \left (a+b \tan ^{-1}(c x)\right )^2}{3 c^3}+\frac{1}{3} x^3 \left (a+b \tan ^{-1}(c x)\right )^2-\frac{2 b \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac{2}{1+i c x}\right )}{3 c^3}-\frac{\left (2 i b^2\right ) \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1+i c x}\right )}{3 c^3}\\ &=\frac{b^2 x}{3 c^2}-\frac{b^2 \tan ^{-1}(c x)}{3 c^3}-\frac{b x^2 \left (a+b \tan ^{-1}(c x)\right )}{3 c}-\frac{i \left (a+b \tan ^{-1}(c x)\right )^2}{3 c^3}+\frac{1}{3} x^3 \left (a+b \tan ^{-1}(c x)\right )^2-\frac{2 b \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac{2}{1+i c x}\right )}{3 c^3}-\frac{i b^2 \text{Li}_2\left (1-\frac{2}{1+i c x}\right )}{3 c^3}\\ \end{align*}
Mathematica [A] time = 0.280131, size = 131, normalized size = 0.95 \[ \frac{i b^2 \text{PolyLog}\left (2,-e^{2 i \tan ^{-1}(c x)}\right )+a^2 c^3 x^3-a b c^2 x^2+a b \log \left (c^2 x^2+1\right )-b \tan ^{-1}(c x) \left (-2 a c^3 x^3+b c^2 x^2+2 b \log \left (1+e^{2 i \tan ^{-1}(c x)}\right )+b\right )+b^2 \left (c^3 x^3+i\right ) \tan ^{-1}(c x)^2+b^2 c x}{3 c^3} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.014, size = 298, normalized size = 2.2 \begin{align*}{\frac{{x}^{3}{a}^{2}}{3}}+{\frac{{b}^{2}{x}^{3} \left ( \arctan \left ( cx \right ) \right ) ^{2}}{3}}-{\frac{{b}^{2}\arctan \left ( cx \right ){x}^{2}}{3\,c}}+{\frac{{b}^{2}\arctan \left ( cx \right ) \ln \left ({c}^{2}{x}^{2}+1 \right ) }{3\,{c}^{3}}}+{\frac{{b}^{2}x}{3\,{c}^{2}}}-{\frac{{b}^{2}\arctan \left ( cx \right ) }{3\,{c}^{3}}}+{\frac{{\frac{i}{6}}{b}^{2}\ln \left ( cx+i \right ) \ln \left ({\frac{i}{2}} \left ( cx-i \right ) \right ) }{{c}^{3}}}-{\frac{{\frac{i}{6}}{b}^{2}\ln \left ( cx-i \right ) \ln \left ( -{\frac{i}{2}} \left ( cx+i \right ) \right ) }{{c}^{3}}}+{\frac{{\frac{i}{6}}{b}^{2}\ln \left ({c}^{2}{x}^{2}+1 \right ) \ln \left ( cx-i \right ) }{{c}^{3}}}-{\frac{{\frac{i}{6}}{b}^{2}\ln \left ({c}^{2}{x}^{2}+1 \right ) \ln \left ( cx+i \right ) }{{c}^{3}}}+{\frac{{\frac{i}{12}}{b}^{2} \left ( \ln \left ( cx+i \right ) \right ) ^{2}}{{c}^{3}}}+{\frac{{\frac{i}{6}}{b}^{2}{\it dilog} \left ({\frac{i}{2}} \left ( cx-i \right ) \right ) }{{c}^{3}}}-{\frac{{\frac{i}{6}}{b}^{2}{\it dilog} \left ( -{\frac{i}{2}} \left ( cx+i \right ) \right ) }{{c}^{3}}}-{\frac{{\frac{i}{12}}{b}^{2} \left ( \ln \left ( cx-i \right ) \right ) ^{2}}{{c}^{3}}}+{\frac{2\,ab{x}^{3}\arctan \left ( cx \right ) }{3}}-{\frac{ab{x}^{2}}{3\,c}}+{\frac{ab\ln \left ({c}^{2}{x}^{2}+1 \right ) }{3\,{c}^{3}}} \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{1}{3} \, a^{2} x^{3} + \frac{1}{3} \,{\left (2 \, x^{3} \arctan \left (c x\right ) - c{\left (\frac{x^{2}}{c^{2}} - \frac{\log \left (c^{2} x^{2} + 1\right )}{c^{4}}\right )}\right )} a b + \frac{1}{48} \,{\left (4 \, x^{3} \arctan \left (c x\right )^{2} - x^{3} \log \left (c^{2} x^{2} + 1\right )^{2} + 48 \, \int \frac{4 \, c^{2} x^{4} \log \left (c^{2} x^{2} + 1\right ) - 8 \, c x^{3} \arctan \left (c x\right ) + 36 \,{\left (c^{2} x^{4} + x^{2}\right )} \arctan \left (c x\right )^{2} + 3 \,{\left (c^{2} x^{4} + x^{2}\right )} \log \left (c^{2} x^{2} + 1\right )^{2}}{48 \,{\left (c^{2} x^{2} + 1\right )}}\,{d x}\right )} b^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (b^{2} x^{2} \arctan \left (c x\right )^{2} + 2 \, a b x^{2} \arctan \left (c x\right ) + a^{2} x^{2}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{2} \left (a + b \operatorname{atan}{\left (c x \right )}\right )^{2}\, dx \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 (b \arctan \left (c x\right ) + a\right )}^{2} x^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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