Optimal. Leaf size=206 \[ \frac{1}{4} \text{PolyLog}\left (2,1+\frac{2 i (-c x+i)}{(1-c) (1-i x)}\right )-\frac{1}{4} \text{PolyLog}\left (2,1+\frac{2 i (c x+i)}{(c+1) (1-i x)}\right )+\frac{\log \left (c^2 x^2+1\right )}{2 c}+x \cot ^{-1}(c x)-\frac{1}{2} i \tan ^{-1}(x) \log \left (1-\frac{i}{c x}\right )+\frac{1}{2} i \tan ^{-1}(x) \log \left (1+\frac{i}{c x}\right )+\frac{1}{2} i \tan ^{-1}(x) \log \left (-\frac{2 i (-c x+i)}{(1-c) (1-i x)}\right )-\frac{1}{2} i \tan ^{-1}(x) \log \left (-\frac{2 i (c x+i)}{(c+1) (1-i x)}\right ) \]
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Rubi [A] time = 0.579502, antiderivative size = 206, normalized size of antiderivative = 1., number of steps used = 28, number of rules used = 15, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 1., Rules used = {4917, 4847, 260, 4909, 203, 2470, 6688, 12, 4876, 4848, 2391, 4856, 2402, 2315, 2447} \[ \frac{1}{4} \text{PolyLog}\left (2,1+\frac{2 i (-c x+i)}{(1-c) (1-i x)}\right )-\frac{1}{4} \text{PolyLog}\left (2,1+\frac{2 i (c x+i)}{(c+1) (1-i x)}\right )+\frac{\log \left (c^2 x^2+1\right )}{2 c}+x \cot ^{-1}(c x)-\frac{1}{2} i \tan ^{-1}(x) \log \left (1-\frac{i}{c x}\right )+\frac{1}{2} i \tan ^{-1}(x) \log \left (1+\frac{i}{c x}\right )+\frac{1}{2} i \tan ^{-1}(x) \log \left (-\frac{2 i (-c x+i)}{(1-c) (1-i x)}\right )-\frac{1}{2} i \tan ^{-1}(x) \log \left (-\frac{2 i (c x+i)}{(c+1) (1-i x)}\right ) \]
Antiderivative was successfully verified.
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Rule 4917
Rule 4847
Rule 260
Rule 4909
Rule 203
Rule 2470
Rule 6688
Rule 12
Rule 4876
Rule 4848
Rule 2391
Rule 4856
Rule 2402
Rule 2315
Rule 2447
Rubi steps
\begin{align*} \int \frac{x^2 \cot ^{-1}(c x)}{1+x^2} \, dx &=\int \cot ^{-1}(c x) \, dx-\int \frac{\cot ^{-1}(c x)}{1+x^2} \, dx\\ &=x \cot ^{-1}(c x)-\frac{1}{2} i \int \frac{\log \left (1-\frac{i}{c x}\right )}{1+x^2} \, dx+\frac{1}{2} i \int \frac{\log \left (1+\frac{i}{c x}\right )}{1+x^2} \, dx+c \int \frac{x}{1+c^2 x^2} \, dx\\ &=x \cot ^{-1}(c x)-\frac{1}{2} i \tan ^{-1}(x) \log \left (1-\frac{i}{c x}\right )+\frac{1}{2} i \tan ^{-1}(x) \log \left (1+\frac{i}{c x}\right )+\frac{\log \left (1+c^2 x^2\right )}{2 c}-\frac{\int \frac{\tan ^{-1}(x)}{\left (1-\frac{i}{c x}\right ) x^2} \, dx}{2 c}-\frac{\int \frac{\tan ^{-1}(x)}{\left (1+\frac{i}{c x}\right ) x^2} \, dx}{2 c}\\ &=x \cot ^{-1}(c x)-\frac{1}{2} i \tan ^{-1}(x) \log \left (1-\frac{i}{c x}\right )+\frac{1}{2} i \tan ^{-1}(x) \log \left (1+\frac{i}{c x}\right )+\frac{\log \left (1+c^2 x^2\right )}{2 c}-\frac{\int \frac{c \tan ^{-1}(x)}{x (-i+c x)} \, dx}{2 c}-\frac{\int \frac{c \tan ^{-1}(x)}{x (i+c x)} \, dx}{2 c}\\ &=x \cot ^{-1}(c x)-\frac{1}{2} i \tan ^{-1}(x) \log \left (1-\frac{i}{c x}\right )+\frac{1}{2} i \tan ^{-1}(x) \log \left (1+\frac{i}{c x}\right )+\frac{\log \left (1+c^2 x^2\right )}{2 c}-\frac{1}{2} \int \frac{\tan ^{-1}(x)}{x (-i+c x)} \, dx-\frac{1}{2} \int \frac{\tan ^{-1}(x)}{x (i+c x)} \, dx\\ &=x \cot ^{-1}(c x)-\frac{1}{2} i \tan ^{-1}(x) \log \left (1-\frac{i}{c x}\right )+\frac{1}{2} i \tan ^{-1}(x) \log \left (1+\frac{i}{c x}\right )+\frac{\log \left (1+c^2 x^2\right )}{2 c}-\frac{1}{2} \int \left (\frac{i \tan ^{-1}(x)}{x}-\frac{i c \tan ^{-1}(x)}{-i+c x}\right ) \, dx-\frac{1}{2} \int \left (-\frac{i \tan ^{-1}(x)}{x}+\frac{i c \tan ^{-1}(x)}{i+c x}\right ) \, dx\\ &=x \cot ^{-1}(c x)-\frac{1}{2} i \tan ^{-1}(x) \log \left (1-\frac{i}{c x}\right )+\frac{1}{2} i \tan ^{-1}(x) \log \left (1+\frac{i}{c x}\right )+\frac{\log \left (1+c^2 x^2\right )}{2 c}+\frac{1}{2} (i c) \int \frac{\tan ^{-1}(x)}{-i+c x} \, dx-\frac{1}{2} (i c) \int \frac{\tan ^{-1}(x)}{i+c x} \, dx\\ &=x \cot ^{-1}(c x)-\frac{1}{2} i \tan ^{-1}(x) \log \left (1-\frac{i}{c x}\right )+\frac{1}{2} i \tan ^{-1}(x) \log \left (1+\frac{i}{c x}\right )+\frac{1}{2} i \tan ^{-1}(x) \log \left (-\frac{2 i (i-c x)}{(1-c) (1-i x)}\right )-\frac{1}{2} i \tan ^{-1}(x) \log \left (-\frac{2 i (i+c x)}{(1+c) (1-i x)}\right )+\frac{\log \left (1+c^2 x^2\right )}{2 c}-\frac{1}{2} i \int \frac{\log \left (\frac{2 (-i+c x)}{(-i+i c) (1-i x)}\right )}{1+x^2} \, dx+\frac{1}{2} i \int \frac{\log \left (\frac{2 (i+c x)}{(i+i c) (1-i x)}\right )}{1+x^2} \, dx\\ &=x \cot ^{-1}(c x)-\frac{1}{2} i \tan ^{-1}(x) \log \left (1-\frac{i}{c x}\right )+\frac{1}{2} i \tan ^{-1}(x) \log \left (1+\frac{i}{c x}\right )+\frac{1}{2} i \tan ^{-1}(x) \log \left (-\frac{2 i (i-c x)}{(1-c) (1-i x)}\right )-\frac{1}{2} i \tan ^{-1}(x) \log \left (-\frac{2 i (i+c x)}{(1+c) (1-i x)}\right )+\frac{\log \left (1+c^2 x^2\right )}{2 c}+\frac{1}{4} \text{Li}_2\left (1+\frac{2 i (i-c x)}{(1-c) (1-i x)}\right )-\frac{1}{4} \text{Li}_2\left (1+\frac{2 i (i+c x)}{(1+c) (1-i x)}\right )\\ \end{align*}
Mathematica [B] time = 1.58503, size = 626, normalized size = 3.04 \[ \frac{\frac{1}{4} \sqrt{-c^2} \left (i \left (\text{PolyLog}\left (2,\frac{\left (c^2+2 i \sqrt{-c^2}+1\right ) \left (\sqrt{-c^2}+c x\right )}{\left (c^2-1\right ) \left (\sqrt{-c^2}-c x\right )}\right )-\text{PolyLog}\left (2,\frac{\left (c^2-2 i \sqrt{-c^2}+1\right ) \left (\sqrt{-c^2}+c x\right )}{\left (c^2-1\right ) \left (\sqrt{-c^2}-c x\right )}\right )\right )+2 \cos ^{-1}\left (\frac{c^2+1}{c^2-1}\right ) \tanh ^{-1}\left (\frac{\sqrt{-c^2}}{c x}\right )-4 \cot ^{-1}(c x) \tanh ^{-1}\left (\frac{c x}{\sqrt{-c^2}}\right )-\log \left (-\frac{2 \left (c^2+i \sqrt{-c^2}\right ) (c x-i)}{\left (c^2-1\right ) \left (\sqrt{-c^2}-c x\right )}\right ) \left (\cos ^{-1}\left (\frac{c^2+1}{c^2-1}\right )-2 i \tanh ^{-1}\left (\frac{\sqrt{-c^2}}{c x}\right )\right )-\log \left (\frac{2 i \left (\sqrt{-c^2}+i c^2\right ) (c x+i)}{\left (c^2-1\right ) \left (\sqrt{-c^2}-c x\right )}\right ) \left (\cos ^{-1}\left (\frac{c^2+1}{c^2-1}\right )+2 i \tanh ^{-1}\left (\frac{\sqrt{-c^2}}{c x}\right )\right )+\left (-2 i \tanh ^{-1}\left (\frac{\sqrt{-c^2}}{c x}\right )+2 i \tanh ^{-1}\left (\frac{c x}{\sqrt{-c^2}}\right )+\cos ^{-1}\left (\frac{c^2+1}{c^2-1}\right )\right ) \log \left (\frac{\sqrt{2} \sqrt{-c^2} e^{-i \cot ^{-1}(c x)}}{\sqrt{c^2-1} \sqrt{\left (c^2-1\right ) \cos \left (2 \cot ^{-1}(c x)\right )-c^2-1}}\right )+\left (2 i \tanh ^{-1}\left (\frac{\sqrt{-c^2}}{c x}\right )-2 i \tanh ^{-1}\left (\frac{c x}{\sqrt{-c^2}}\right )+\cos ^{-1}\left (\frac{c^2+1}{c^2-1}\right )\right ) \log \left (\frac{\sqrt{2} \sqrt{-c^2} e^{i \cot ^{-1}(c x)}}{\sqrt{c^2-1} \sqrt{\left (c^2-1\right ) \cos \left (2 \cot ^{-1}(c x)\right )-c^2-1}}\right )\right )-\log \left (\frac{1}{c x \sqrt{\frac{1}{c^2 x^2}+1}}\right )+c x \cot ^{-1}(c x)}{c} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.247, size = 265, normalized size = 1.3 \begin{align*} -\arctan \left ( x \right ){\rm arccot} \left (cx\right )+x{\rm arccot} \left (cx\right )+{\frac{\ln \left ({c}^{2}{x}^{2}+1 \right ) }{2\,c}}+{\frac{{\frac{i}{2}}c\arctan \left ( x \right ) }{-1+c}\ln \left ( 1-{\frac{ \left ( 1+c \right ) \left ( 1+ix \right ) ^{2}}{ \left ({x}^{2}+1 \right ) \left ( -1+c \right ) }} \right ) }-{\frac{{\frac{i}{2}}\arctan \left ( x \right ) }{-1+c}\ln \left ( 1-{\frac{ \left ( 1+c \right ) \left ( 1+ix \right ) ^{2}}{ \left ({x}^{2}+1 \right ) \left ( -1+c \right ) }} \right ) }+{\frac{c \left ( \arctan \left ( x \right ) \right ) ^{2}}{-2+2\,c}}+{\frac{c}{-4+4\,c}{\it polylog} \left ( 2,{\frac{ \left ( 1+c \right ) \left ( 1+ix \right ) ^{2}}{ \left ({x}^{2}+1 \right ) \left ( -1+c \right ) }} \right ) }-{\frac{ \left ( \arctan \left ( x \right ) \right ) ^{2}}{-2+2\,c}}-{\frac{1}{-4+4\,c}{\it polylog} \left ( 2,{\frac{ \left ( 1+c \right ) \left ( 1+ix \right ) ^{2}}{ \left ({x}^{2}+1 \right ) \left ( -1+c \right ) }} \right ) }-{\frac{i}{2}}\arctan \left ( x \right ) \ln \left ( 1-{\frac{ \left ( -1+c \right ) \left ( 1+ix \right ) ^{2}}{ \left ({x}^{2}+1 \right ) \left ( 1+c \right ) }} \right ) -{\frac{ \left ( \arctan \left ( x \right ) \right ) ^{2}}{2}}-{\frac{1}{4}{\it polylog} \left ( 2,{\frac{ \left ( -1+c \right ) \left ( 1+ix \right ) ^{2}}{ \left ({x}^{2}+1 \right ) \left ( 1+c \right ) }} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.58836, size = 269, normalized size = 1.31 \begin{align*}{\left (x - \arctan \left (x\right )\right )} \operatorname{arccot}\left (c x\right ) - \frac{4 \, c \arctan \left (x\right ) \arctan \left (\frac{c x}{c + 1}, \frac{1}{c + 1}\right ) - 4 \, c \arctan \left (x\right ) \arctan \left (\frac{c x}{c - 1}, -\frac{1}{c - 1}\right ) + c \log \left (x^{2} + 1\right ) \log \left (\frac{c^{2} x^{2} + 1}{c^{2} + 2 \, c + 1}\right ) - c \log \left (x^{2} + 1\right ) \log \left (\frac{c^{2} x^{2} + 1}{c^{2} - 2 \, c + 1}\right ) + 2 \, c{\rm Li}_2\left (\frac{i \, c x + c}{c + 1}\right ) + 2 \, c{\rm Li}_2\left (-\frac{i \, c x - c}{c + 1}\right ) - 2 \, c{\rm Li}_2\left (\frac{i \, c x + c}{c - 1}\right ) - 2 \, c{\rm Li}_2\left (-\frac{i \, c x - c}{c - 1}\right ) - 4 \, \log \left (c^{2} x^{2} + 1\right )}{8 \, 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 (\frac{x^{2} \operatorname{arccot}\left (c x\right )}{x^{2} + 1}, 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 \frac{x^{2} \operatorname{acot}{\left (c x \right )}}{x^{2} + 1}\, 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 \frac{x^{2} \operatorname{arccot}\left (c x\right )}{x^{2} + 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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