Optimal. Leaf size=83 \[ \frac{i b^2 \text{PolyLog}\left (2,1-\frac{2}{1+i c x}\right )}{c}+x \left (a+b \tan ^{-1}(c x)\right )^2+\frac{i \left (a+b \tan ^{-1}(c x)\right )^2}{c}+\frac{2 b \log \left (\frac{2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{c} \]
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Rubi [A] time = 0.0978475, antiderivative size = 83, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {4846, 4920, 4854, 2402, 2315} \[ \frac{i b^2 \text{PolyLog}\left (2,1-\frac{2}{1+i c x}\right )}{c}+x \left (a+b \tan ^{-1}(c x)\right )^2+\frac{i \left (a+b \tan ^{-1}(c x)\right )^2}{c}+\frac{2 b \log \left (\frac{2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{c} \]
Antiderivative was successfully verified.
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Rule 4846
Rule 4920
Rule 4854
Rule 2402
Rule 2315
Rubi steps
\begin{align*} \int \left (a+b \tan ^{-1}(c x)\right )^2 \, dx &=x \left (a+b \tan ^{-1}(c x)\right )^2-(2 b c) \int \frac{x \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx\\ &=\frac{i \left (a+b \tan ^{-1}(c x)\right )^2}{c}+x \left (a+b \tan ^{-1}(c x)\right )^2+(2 b) \int \frac{a+b \tan ^{-1}(c x)}{i-c x} \, dx\\ &=\frac{i \left (a+b \tan ^{-1}(c x)\right )^2}{c}+x \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 )}{c}-\left (2 b^2\right ) \int \frac{\log \left (\frac{2}{1+i c x}\right )}{1+c^2 x^2} \, dx\\ &=\frac{i \left (a+b \tan ^{-1}(c x)\right )^2}{c}+x \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 )}{c}+\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 )}{c}\\ &=\frac{i \left (a+b \tan ^{-1}(c x)\right )^2}{c}+x \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 )}{c}+\frac{i b^2 \text{Li}_2\left (1-\frac{2}{1+i c x}\right )}{c}\\ \end{align*}
Mathematica [A] time = 0.0755589, size = 90, normalized size = 1.08 \[ \frac{-i b^2 \text{PolyLog}\left (2,-e^{2 i \tan ^{-1}(c x)}\right )+a \left (a c x-b \log \left (c^2 x^2+1\right )\right )+2 b \tan ^{-1}(c x) \left (a c x+b \log \left (1+e^{2 i \tan ^{-1}(c x)}\right )\right )+b^2 (c x-i) \tan ^{-1}(c x)^2}{c} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.071, size = 128, normalized size = 1.5 \begin{align*} x{b}^{2} \left ( \arctan \left ( cx \right ) \right ) ^{2}-{\frac{i \left ( \arctan \left ( cx \right ) \right ) ^{2}{b}^{2}}{c}}+2\,xab\arctan \left ( cx \right ) +2\,{\frac{\arctan \left ( cx \right ){b}^{2}}{c}\ln \left ({\frac{ \left ( 1+icx \right ) ^{2}}{{c}^{2}{x}^{2}+1}}+1 \right ) }-{\frac{i{b}^{2}}{c}{\it polylog} \left ( 2,-{\frac{ \left ( 1+icx \right ) ^{2}}{{c}^{2}{x}^{2}+1}} \right ) }+{a}^{2}x-{\frac{ab\ln \left ({c}^{2}{x}^{2}+1 \right ) }{c}} \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}{16} \,{\left (4 \, x \arctan \left (c x\right )^{2} + 192 \, c^{2} \int \frac{x^{2} \arctan \left (c x\right )^{2}}{16 \,{\left (c^{2} x^{2} + 1\right )}}\,{d x} + 16 \, c^{2} \int \frac{x^{2} \log \left (c^{2} x^{2} + 1\right )^{2}}{16 \,{\left (c^{2} x^{2} + 1\right )}}\,{d x} + 64 \, c^{2} \int \frac{x^{2} \log \left (c^{2} x^{2} + 1\right )}{16 \,{\left (c^{2} x^{2} + 1\right )}}\,{d x} - x \log \left (c^{2} x^{2} + 1\right )^{2} + \frac{4 \, \arctan \left (c x\right )^{3}}{c} - 128 \, c \int \frac{x \arctan \left (c x\right )}{16 \,{\left (c^{2} x^{2} + 1\right )}}\,{d x} + 16 \, \int \frac{\log \left (c^{2} x^{2} + 1\right )^{2}}{16 \,{\left (c^{2} x^{2} + 1\right )}}\,{d x}\right )} b^{2} + a^{2} x + \frac{{\left (2 \, c x \arctan \left (c x\right ) - \log \left (c^{2} x^{2} + 1\right )\right )} a b}{c} \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} \arctan \left (c x\right )^{2} + 2 \, a b \arctan \left (c x\right ) + a^{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 \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}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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