Optimal. Leaf size=45 \[ \frac{i \text{PolyLog}\left (2,\frac{i}{a+b x}\right )}{2 b}-\frac{i \text{PolyLog}\left (2,-\frac{i}{a+b x}\right )}{2 b} \]
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Rubi [A] time = 0.0414755, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {5044, 4849, 2391} \[ \frac{i \text{PolyLog}\left (2,\frac{i}{a+b x}\right )}{2 b}-\frac{i \text{PolyLog}\left (2,-\frac{i}{a+b x}\right )}{2 b} \]
Antiderivative was successfully verified.
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Rule 5044
Rule 4849
Rule 2391
Rubi steps
\begin{align*} \int \frac{\cot ^{-1}(a+b x)}{a+b x} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\cot ^{-1}(x)}{x} \, dx,x,a+b x\right )}{b}\\ &=\frac{i \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{i}{x}\right )}{x} \, dx,x,a+b x\right )}{2 b}-\frac{i \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{i}{x}\right )}{x} \, dx,x,a+b x\right )}{2 b}\\ &=-\frac{i \text{Li}_2\left (-\frac{i}{a+b x}\right )}{2 b}+\frac{i \text{Li}_2\left (\frac{i}{a+b x}\right )}{2 b}\\ \end{align*}
Mathematica [A] time = 0.0068087, size = 38, normalized size = 0.84 \[ -\frac{i \left (\text{PolyLog}\left (2,-\frac{i}{a+b x}\right )-\text{PolyLog}\left (2,\frac{i}{a+b x}\right )\right )}{2 b} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.052, size = 98, normalized size = 2.2 \begin{align*}{\frac{\ln \left ( bx+a \right ){\rm arccot} \left (bx+a\right )}{b}}-{\frac{{\frac{i}{2}}\ln \left ( bx+a \right ) \ln \left ( 1+i \left ( bx+a \right ) \right ) }{b}}+{\frac{{\frac{i}{2}}\ln \left ( bx+a \right ) \ln \left ( 1-i \left ( bx+a \right ) \right ) }{b}}-{\frac{{\frac{i}{2}}{\it dilog} \left ( 1+i \left ( bx+a \right ) \right ) }{b}}+{\frac{{\frac{i}{2}}{\it dilog} \left ( 1-i \left ( bx+a \right ) \right ) }{b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.63219, size = 151, normalized size = 3.36 \begin{align*} \frac{\operatorname{arccot}\left (b x + a\right ) \log \left (b x + a\right )}{b} + \frac{\arctan \left (\frac{b^{2} x + a b}{b}\right ) \log \left (b x + a\right )}{b} + \frac{\arctan \left (b x + a, 0\right ) \log \left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right ) - 2 \, \arctan \left (b x + a\right ) \log \left ({\left | b x + a \right |}\right ) + i \,{\rm Li}_2\left (i \, b x + i \, a + 1\right ) - i \,{\rm Li}_2\left (-i \, b x - i \, a + 1\right )}{2 \, b} \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 )}{b x + a}, 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 )}}{a + b 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 )}{b x + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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