Optimal. Leaf size=211 \[ \frac{b^2 d \text{PolyLog}\left (2,1-\frac{2}{1+i c x}\right )}{3 c^2}+\frac{5 d \left (a+b \tan ^{-1}(c x)\right )^2}{6 c^2}-\frac{2 i b d \log \left (\frac{2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{3 c^2}+\frac{1}{3} i c d x^3 \left (a+b \tan ^{-1}(c x)\right )^2+\frac{1}{2} d x^2 \left (a+b \tan ^{-1}(c x)\right )^2-\frac{1}{3} i b d x^2 \left (a+b \tan ^{-1}(c x)\right )-\frac{a b d x}{c}+\frac{b^2 d \log \left (c^2 x^2+1\right )}{2 c^2}-\frac{i b^2 d \tan ^{-1}(c x)}{3 c^2}+\frac{i b^2 d x}{3 c}-\frac{b^2 d x \tan ^{-1}(c x)}{c} \]
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Rubi [A] time = 0.358016, antiderivative size = 211, normalized size of antiderivative = 1., number of steps used = 17, number of rules used = 12, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.571, Rules used = {4876, 4852, 4916, 4846, 260, 4884, 321, 203, 4920, 4854, 2402, 2315} \[ \frac{b^2 d \text{PolyLog}\left (2,1-\frac{2}{1+i c x}\right )}{3 c^2}+\frac{5 d \left (a+b \tan ^{-1}(c x)\right )^2}{6 c^2}-\frac{2 i b d \log \left (\frac{2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{3 c^2}+\frac{1}{3} i c d x^3 \left (a+b \tan ^{-1}(c x)\right )^2+\frac{1}{2} d x^2 \left (a+b \tan ^{-1}(c x)\right )^2-\frac{1}{3} i b d x^2 \left (a+b \tan ^{-1}(c x)\right )-\frac{a b d x}{c}+\frac{b^2 d \log \left (c^2 x^2+1\right )}{2 c^2}-\frac{i b^2 d \tan ^{-1}(c x)}{3 c^2}+\frac{i b^2 d x}{3 c}-\frac{b^2 d x \tan ^{-1}(c x)}{c} \]
Antiderivative was successfully verified.
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Rule 4876
Rule 4852
Rule 4916
Rule 4846
Rule 260
Rule 4884
Rule 321
Rule 203
Rule 4920
Rule 4854
Rule 2402
Rule 2315
Rubi steps
\begin{align*} \int x (d+i c d x) \left (a+b \tan ^{-1}(c x)\right )^2 \, dx &=\int \left (d x \left (a+b \tan ^{-1}(c x)\right )^2+i c d x^2 \left (a+b \tan ^{-1}(c x)\right )^2\right ) \, dx\\ &=d \int x \left (a+b \tan ^{-1}(c x)\right )^2 \, dx+(i c d) \int x^2 \left (a+b \tan ^{-1}(c x)\right )^2 \, dx\\ &=\frac{1}{2} d x^2 \left (a+b \tan ^{-1}(c x)\right )^2+\frac{1}{3} i c d x^3 \left (a+b \tan ^{-1}(c x)\right )^2-(b c d) \int \frac{x^2 \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx-\frac{1}{3} \left (2 i b c^2 d\right ) \int \frac{x^3 \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx\\ &=\frac{1}{2} d x^2 \left (a+b \tan ^{-1}(c x)\right )^2+\frac{1}{3} i c d x^3 \left (a+b \tan ^{-1}(c x)\right )^2-\frac{1}{3} (2 i b d) \int x \left (a+b \tan ^{-1}(c x)\right ) \, dx+\frac{1}{3} (2 i b d) \int \frac{x \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx-\frac{(b d) \int \left (a+b \tan ^{-1}(c x)\right ) \, dx}{c}+\frac{(b d) \int \frac{a+b \tan ^{-1}(c x)}{1+c^2 x^2} \, dx}{c}\\ &=-\frac{a b d x}{c}-\frac{1}{3} i b d x^2 \left (a+b \tan ^{-1}(c x)\right )+\frac{5 d \left (a+b \tan ^{-1}(c x)\right )^2}{6 c^2}+\frac{1}{2} d x^2 \left (a+b \tan ^{-1}(c x)\right )^2+\frac{1}{3} i c d x^3 \left (a+b \tan ^{-1}(c x)\right )^2-\frac{(2 i b d) \int \frac{a+b \tan ^{-1}(c x)}{i-c x} \, dx}{3 c}-\frac{\left (b^2 d\right ) \int \tan ^{-1}(c x) \, dx}{c}+\frac{1}{3} \left (i b^2 c d\right ) \int \frac{x^2}{1+c^2 x^2} \, dx\\ &=-\frac{a b d x}{c}+\frac{i b^2 d x}{3 c}-\frac{b^2 d x \tan ^{-1}(c x)}{c}-\frac{1}{3} i b d x^2 \left (a+b \tan ^{-1}(c x)\right )+\frac{5 d \left (a+b \tan ^{-1}(c x)\right )^2}{6 c^2}+\frac{1}{2} d x^2 \left (a+b \tan ^{-1}(c x)\right )^2+\frac{1}{3} i c d x^3 \left (a+b \tan ^{-1}(c x)\right )^2-\frac{2 i b d \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac{2}{1+i c x}\right )}{3 c^2}+\left (b^2 d\right ) \int \frac{x}{1+c^2 x^2} \, dx-\frac{\left (i b^2 d\right ) \int \frac{1}{1+c^2 x^2} \, dx}{3 c}+\frac{\left (2 i b^2 d\right ) \int \frac{\log \left (\frac{2}{1+i c x}\right )}{1+c^2 x^2} \, dx}{3 c}\\ &=-\frac{a b d x}{c}+\frac{i b^2 d x}{3 c}-\frac{i b^2 d \tan ^{-1}(c x)}{3 c^2}-\frac{b^2 d x \tan ^{-1}(c x)}{c}-\frac{1}{3} i b d x^2 \left (a+b \tan ^{-1}(c x)\right )+\frac{5 d \left (a+b \tan ^{-1}(c x)\right )^2}{6 c^2}+\frac{1}{2} d x^2 \left (a+b \tan ^{-1}(c x)\right )^2+\frac{1}{3} i c d x^3 \left (a+b \tan ^{-1}(c x)\right )^2-\frac{2 i b d \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac{2}{1+i c x}\right )}{3 c^2}+\frac{b^2 d \log \left (1+c^2 x^2\right )}{2 c^2}+\frac{\left (2 b^2 d\right ) \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1+i c x}\right )}{3 c^2}\\ &=-\frac{a b d x}{c}+\frac{i b^2 d x}{3 c}-\frac{i b^2 d \tan ^{-1}(c x)}{3 c^2}-\frac{b^2 d x \tan ^{-1}(c x)}{c}-\frac{1}{3} i b d x^2 \left (a+b \tan ^{-1}(c x)\right )+\frac{5 d \left (a+b \tan ^{-1}(c x)\right )^2}{6 c^2}+\frac{1}{2} d x^2 \left (a+b \tan ^{-1}(c x)\right )^2+\frac{1}{3} i c d x^3 \left (a+b \tan ^{-1}(c x)\right )^2-\frac{2 i b d \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac{2}{1+i c x}\right )}{3 c^2}+\frac{b^2 d \log \left (1+c^2 x^2\right )}{2 c^2}+\frac{b^2 d \text{Li}_2\left (1-\frac{2}{1+i c x}\right )}{3 c^2}\\ \end{align*}
Mathematica [A] time = 0.496672, size = 208, normalized size = 0.99 \[ \frac{d \left (-2 b^2 \text{PolyLog}\left (2,-e^{2 i \tan ^{-1}(c x)}\right )+2 i a^2 c^3 x^3+3 a^2 c^2 x^2-2 i a b c^2 x^2+2 i a b \log \left (c^2 x^2+1\right )+2 b \tan ^{-1}(c x) \left (a \left (2 i c^3 x^3+3 c^2 x^2+3\right )-i b \left (c^2 x^2-3 i c x+1\right )-2 i b \log \left (1+e^{2 i \tan ^{-1}(c x)}\right )\right )-6 a b c x+3 b^2 \log \left (c^2 x^2+1\right )+b^2 \left (2 i c^3 x^3+3 c^2 x^2+1\right ) \tan ^{-1}(c x)^2+2 i b^2 c x\right )}{6 c^2} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.093, size = 416, normalized size = 2. \begin{align*}{\frac{{\frac{i}{3}}dab\ln \left ({c}^{2}{x}^{2}+1 \right ) }{{c}^{2}}}+{\frac{d{b}^{2}\ln \left ({c}^{2}{x}^{2}+1 \right ) }{2\,{c}^{2}}}+{\frac{2\,i}{3}}cdab\arctan \left ( cx \right ){x}^{3}-{\frac{i}{3}}dab{x}^{2}+{\frac{i}{3}}cd{a}^{2}{x}^{3}-{\frac{i}{3}}d{b}^{2}\arctan \left ( cx \right ){x}^{2}+{\frac{dab\arctan \left ( cx \right ) }{{c}^{2}}}+{\frac{d{b}^{2}\ln \left ( cx+i \right ) \ln \left ({c}^{2}{x}^{2}+1 \right ) }{6\,{c}^{2}}}-{\frac{d{b}^{2}\ln \left ( cx+i \right ) \ln \left ({\frac{i}{2}} \left ( cx-i \right ) \right ) }{6\,{c}^{2}}}-{\frac{d{b}^{2}\ln \left ( cx-i \right ) \ln \left ({c}^{2}{x}^{2}+1 \right ) }{6\,{c}^{2}}}+{\frac{d{b}^{2}\ln \left ( cx-i \right ) \ln \left ( -{\frac{i}{2}} \left ( cx+i \right ) \right ) }{6\,{c}^{2}}}+dab\arctan \left ( cx \right ){x}^{2}+{\frac{{\frac{i}{3}}d{b}^{2}\arctan \left ( cx \right ) \ln \left ({c}^{2}{x}^{2}+1 \right ) }{{c}^{2}}}+{\frac{d{a}^{2}{x}^{2}}{2}}+{\frac{d{b}^{2} \left ( \arctan \left ( cx \right ) \right ) ^{2}}{2\,{c}^{2}}}+{\frac{d{b}^{2} \left ( \ln \left ( cx-i \right ) \right ) ^{2}}{12\,{c}^{2}}}+{\frac{d{b}^{2} \left ( \arctan \left ( cx \right ) \right ) ^{2}{x}^{2}}{2}}+{\frac{d{b}^{2}{\it dilog} \left ( -{\frac{i}{2}} \left ( cx+i \right ) \right ) }{6\,{c}^{2}}}-{\frac{d{b}^{2} \left ( \ln \left ( cx+i \right ) \right ) ^{2}}{12\,{c}^{2}}}-{\frac{d{b}^{2}{\it dilog} \left ({\frac{i}{2}} \left ( cx-i \right ) \right ) }{6\,{c}^{2}}}+{\frac{{\frac{i}{3}}d{b}^{2}x}{c}}-{\frac{{\frac{i}{3}}d{b}^{2}\arctan \left ( cx \right ) }{{c}^{2}}}-{\frac{dabx}{c}}-{\frac{d{b}^{2}x\arctan \left ( cx \right ) }{c}}+{\frac{i}{3}}cd{b}^{2} \left ( \arctan \left ( cx \right ) \right ) ^{2}{x}^{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} i \, a^{2} c d x^{3} + \frac{1}{2} \, b^{2} d x^{2} \arctan \left (c x\right )^{2} + \frac{1}{3} i \,{\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 c d + \frac{1}{48} i \,{\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} c d + \frac{1}{2} \, a^{2} d x^{2} +{\left (x^{2} \arctan \left (c x\right ) - c{\left (\frac{x}{c^{2}} - \frac{\arctan \left (c x\right )}{c^{3}}\right )}\right )} a b d - \frac{1}{2} \,{\left (2 \, c{\left (\frac{x}{c^{2}} - \frac{\arctan \left (c x\right )}{c^{3}}\right )} \arctan \left (c x\right ) + \frac{\arctan \left (c x\right )^{2} - \log \left (c^{2} x^{2} + 1\right )}{c^{2}}\right )} b^{2} d \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*} \frac{1}{24} \,{\left (-2 i \, b^{2} c d x^{3} - 3 \, b^{2} d x^{2}\right )} \log \left (-\frac{c x + i}{c x - i}\right )^{2} +{\rm integral}\left (\frac{6 i \, a^{2} c^{3} d x^{4} + 6 \, a^{2} c^{2} d x^{3} + 6 i \, a^{2} c d x^{2} + 6 \, a^{2} d x -{\left (6 \, a b c^{3} d x^{4} -{\left (6 i \, a b + 2 \, b^{2}\right )} c^{2} d x^{3} + 3 \,{\left (2 \, a b + i \, b^{2}\right )} c d x^{2} - 6 i \, a b d x\right )} \log \left (-\frac{c x + i}{c x - i}\right )}{6 \,{\left (c^{2} x^{2} + 1\right )}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: AttributeError} \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 (i \, c d x + d\right )}{\left (b \arctan \left (c x\right ) + a\right )}^{2} x\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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