Optimal. Leaf size=152 \[ \frac{i \text{PolyLog}\left (2,1-\frac{2 b (c+d x)}{(1-i (a+b x)) (-a d+b c+i d)}\right )}{2 d}-\frac{i \text{PolyLog}\left (2,1-\frac{2}{1-i (a+b x)}\right )}{2 d}+\frac{\cot ^{-1}(a+b x) \log \left (\frac{2 b (c+d x)}{(1-i (a+b x)) (-a d+b c+i d)}\right )}{d}-\frac{\log \left (\frac{2}{1-i (a+b x)}\right ) \cot ^{-1}(a+b x)}{d} \]
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Rubi [A] time = 0.14161, antiderivative size = 152, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.357, Rules used = {5048, 4857, 2402, 2315, 2447} \[ \frac{i \text{PolyLog}\left (2,1-\frac{2 b (c+d x)}{(1-i (a+b x)) (-a d+b c+i d)}\right )}{2 d}-\frac{i \text{PolyLog}\left (2,1-\frac{2}{1-i (a+b x)}\right )}{2 d}+\frac{\cot ^{-1}(a+b x) \log \left (\frac{2 b (c+d x)}{(1-i (a+b x)) (-a d+b c+i d)}\right )}{d}-\frac{\log \left (\frac{2}{1-i (a+b x)}\right ) \cot ^{-1}(a+b x)}{d} \]
Antiderivative was successfully verified.
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Rule 5048
Rule 4857
Rule 2402
Rule 2315
Rule 2447
Rubi steps
\begin{align*} \int \frac{\cot ^{-1}(a+b x)}{c+d x} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\cot ^{-1}(x)}{\frac{b c-a d}{b}+\frac{d x}{b}} \, dx,x,a+b x\right )}{b}\\ &=-\frac{\cot ^{-1}(a+b x) \log \left (\frac{2}{1-i (a+b x)}\right )}{d}+\frac{\cot ^{-1}(a+b x) \log \left (\frac{2 b (c+d x)}{(b c+i d-a d) (1-i (a+b x))}\right )}{d}-\frac{\operatorname{Subst}\left (\int \frac{\log \left (\frac{2}{1-i x}\right )}{1+x^2} \, dx,x,a+b x\right )}{d}+\frac{\operatorname{Subst}\left (\int \frac{\log \left (\frac{2 \left (\frac{b c-a d}{b}+\frac{d x}{b}\right )}{\left (\frac{i d}{b}+\frac{b c-a d}{b}\right ) (1-i x)}\right )}{1+x^2} \, dx,x,a+b x\right )}{d}\\ &=-\frac{\cot ^{-1}(a+b x) \log \left (\frac{2}{1-i (a+b x)}\right )}{d}+\frac{\cot ^{-1}(a+b x) \log \left (\frac{2 b (c+d x)}{(b c+i d-a d) (1-i (a+b x))}\right )}{d}+\frac{i \text{Li}_2\left (1-\frac{2 b (c+d x)}{(b c+i d-a d) (1-i (a+b x))}\right )}{2 d}-\frac{i \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1-i (a+b x)}\right )}{d}\\ &=-\frac{\cot ^{-1}(a+b x) \log \left (\frac{2}{1-i (a+b x)}\right )}{d}+\frac{\cot ^{-1}(a+b x) \log \left (\frac{2 b (c+d x)}{(b c+i d-a d) (1-i (a+b x))}\right )}{d}-\frac{i \text{Li}_2\left (1-\frac{2}{1-i (a+b x)}\right )}{2 d}+\frac{i \text{Li}_2\left (1-\frac{2 b (c+d x)}{(b c+i d-a d) (1-i (a+b x))}\right )}{2 d}\\ \end{align*}
Mathematica [B] time = 0.0399261, size = 345, normalized size = 2.27 \[ \frac{i \text{PolyLog}\left (2,\frac{b \left (\frac{b c-a d}{b}+\frac{d (a+b x)}{b}\right )}{-a d+b c-i d}\right )}{2 d}-\frac{i \text{PolyLog}\left (2,\frac{b \left (\frac{b c-a d}{b}+\frac{d (a+b x)}{b}\right )}{-a d+b c+i d}\right )}{2 d}-\frac{i \log \left (\frac{d (a+b x-i)}{b \left (-\frac{b c-a d}{b}-\frac{i d}{b}\right )}\right ) \log \left (\frac{b c-a d}{b}+\frac{d (a+b x)}{b}\right )}{2 d}+\frac{i \log \left (\frac{a+b x-i}{a+b x}\right ) \log \left (\frac{b c-a d}{b}+\frac{d (a+b x)}{b}\right )}{2 d}+\frac{i \log \left (\frac{d (a+b x+i)}{b \left (-\frac{b c-a d}{b}+\frac{i d}{b}\right )}\right ) \log \left (\frac{b c-a d}{b}+\frac{d (a+b x)}{b}\right )}{2 d}-\frac{i \log \left (\frac{a+b x+i}{a+b x}\right ) \log \left (\frac{b c-a d}{b}+\frac{d (a+b x)}{b}\right )}{2 d} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.057, size = 198, normalized size = 1.3 \begin{align*}{\frac{\ln \left ( d \left ( bx+a \right ) -ad+cb \right ){\rm arccot} \left (bx+a\right )}{d}}-{\frac{{\frac{i}{2}}\ln \left ( d \left ( bx+a \right ) -ad+cb \right ) }{d}\ln \left ({\frac{id-d \left ( bx+a \right ) }{cb+id-ad}} \right ) }+{\frac{{\frac{i}{2}}\ln \left ( d \left ( bx+a \right ) -ad+cb \right ) }{d}\ln \left ({\frac{id+d \left ( bx+a \right ) }{id+ad-cb}} \right ) }-{\frac{{\frac{i}{2}}}{d}{\it dilog} \left ({\frac{id-d \left ( bx+a \right ) }{cb+id-ad}} \right ) }+{\frac{{\frac{i}{2}}}{d}{\it dilog} \left ({\frac{id+d \left ( bx+a \right ) }{id+ad-cb}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.89624, size = 382, normalized size = 2.51 \begin{align*} \frac{\operatorname{arccot}\left (b x + a\right ) \log \left (d x + c\right )}{d} + \frac{\arctan \left (\frac{b^{2} x + a b}{b}\right ) \log \left (d x + c\right )}{d} + \frac{\arctan \left (\frac{b d^{2} x + b c d}{b^{2} c^{2} - 2 \, a b c d +{\left (a^{2} + 1\right )} d^{2}}, \frac{b^{2} c^{2} - a b c d +{\left (b^{2} c d - a b d^{2}\right )} x}{b^{2} c^{2} - 2 \, a b c d +{\left (a^{2} + 1\right )} d^{2}}\right ) \log \left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right ) - \arctan \left (b x + a\right ) \log \left (\frac{b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2}}{b^{2} c^{2} - 2 \, a b c d +{\left (a^{2} + 1\right )} d^{2}}\right ) + i \,{\rm Li}_2\left (\frac{i \, b d x +{\left (i \, a + 1\right )} d}{-i \, b c +{\left (i \, a + 1\right )} d}\right ) - i \,{\rm Li}_2\left (\frac{i \, b d x +{\left (i \, a - 1\right )} d}{-i \, b c +{\left (i \, a - 1\right )} d}\right )}{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*}{\rm integral}\left (\frac{\operatorname{arccot}\left (b x + a\right )}{d x + c}, 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{\operatorname{acot}{\left (a + b x \right )}}{c + d x}\, 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{\operatorname{arccot}\left (b x + a\right )}{d x + c}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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