Optimal. Leaf size=120 \[ -\frac{1}{2} i \text{PolyLog}\left (2,1-\frac{2}{1-i (a+b x)}\right )+\frac{1}{2} i \text{PolyLog}\left (2,1-\frac{2 b x}{(-a+i) (1-i (a+b x))}\right )+\log \left (\frac{2}{1-i (a+b x)}\right ) \left (-\cot ^{-1}(a+b x)\right )+\log \left (\frac{2 b x}{(-a+i) (1-i (a+b x))}\right ) \cot ^{-1}(a+b x) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.10834, antiderivative size = 120, 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 = {5048, 4857, 2402, 2315, 2447} \[ -\frac{1}{2} i \text{PolyLog}\left (2,1-\frac{2}{1-i (a+b x)}\right )+\frac{1}{2} i \text{PolyLog}\left (2,1-\frac{2 b x}{(-a+i) (1-i (a+b x))}\right )+\log \left (\frac{2}{1-i (a+b x)}\right ) \left (-\cot ^{-1}(a+b x)\right )+\log \left (\frac{2 b x}{(-a+i) (1-i (a+b x))}\right ) \cot ^{-1}(a+b x) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 5048
Rule 4857
Rule 2402
Rule 2315
Rule 2447
Rubi steps
\begin{align*} \int \frac{\cot ^{-1}(a+b x)}{x} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\cot ^{-1}(x)}{-\frac{a}{b}+\frac{x}{b}} \, dx,x,a+b x\right )}{b}\\ &=-\cot ^{-1}(a+b x) \log \left (\frac{2}{1-i (a+b x)}\right )+\cot ^{-1}(a+b x) \log \left (\frac{2 b x}{(i-a) (1-i (a+b x))}\right )-\operatorname{Subst}\left (\int \frac{\log \left (\frac{2}{1-i x}\right )}{1+x^2} \, dx,x,a+b x\right )+\operatorname{Subst}\left (\int \frac{\log \left (\frac{2 \left (-\frac{a}{b}+\frac{x}{b}\right )}{\left (\frac{i}{b}-\frac{a}{b}\right ) (1-i x)}\right )}{1+x^2} \, dx,x,a+b x\right )\\ &=-\cot ^{-1}(a+b x) \log \left (\frac{2}{1-i (a+b x)}\right )+\cot ^{-1}(a+b x) \log \left (\frac{2 b x}{(i-a) (1-i (a+b x))}\right )+\frac{1}{2} i \text{Li}_2\left (1-\frac{2 b x}{(i-a) (1-i (a+b x))}\right )-i \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1-i (a+b x)}\right )\\ &=-\cot ^{-1}(a+b x) \log \left (\frac{2}{1-i (a+b x)}\right )+\cot ^{-1}(a+b x) \log \left (\frac{2 b x}{(i-a) (1-i (a+b x))}\right )-\frac{1}{2} i \text{Li}_2\left (1-\frac{2}{1-i (a+b x)}\right )+\frac{1}{2} i \text{Li}_2\left (1-\frac{2 b x}{(i-a) (1-i (a+b x))}\right )\\ \end{align*}
Mathematica [B] time = 0.024878, size = 251, normalized size = 2.09 \[ -\frac{1}{2} i \text{PolyLog}\left (2,-\frac{b \left (\frac{a+b x}{b}-\frac{a}{b}\right )}{a-i}\right )+\frac{1}{2} i \text{PolyLog}\left (2,-\frac{b \left (\frac{a+b x}{b}-\frac{a}{b}\right )}{a+i}\right )-\frac{1}{2} i \log \left (\frac{a+b x-i}{b \left (\frac{a}{b}-\frac{i}{b}\right )}\right ) \log \left (\frac{a+b x}{b}-\frac{a}{b}\right )+\frac{1}{2} i \log \left (\frac{a+b x-i}{a+b x}\right ) \log \left (\frac{a+b x}{b}-\frac{a}{b}\right )+\frac{1}{2} i \log \left (\frac{a+b x+i}{b \left (\frac{a}{b}+\frac{i}{b}\right )}\right ) \log \left (\frac{a+b x}{b}-\frac{a}{b}\right )-\frac{1}{2} i \log \left (\frac{a+b x+i}{a+b x}\right ) \log \left (\frac{a+b x}{b}-\frac{a}{b}\right ) \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.056, size = 103, normalized size = 0.9 \begin{align*} \ln \left ( bx \right ){\rm arccot} \left (bx+a\right )-{\frac{i}{2}}\ln \left ( bx \right ) \ln \left ({\frac{i-a-bx}{i-a}} \right ) +{\frac{i}{2}}\ln \left ( bx \right ) \ln \left ({\frac{i+a+bx}{i+a}} \right ) -{\frac{i}{2}}{\it dilog} \left ({\frac{i-a-bx}{i-a}} \right ) +{\frac{i}{2}}{\it dilog} \left ({\frac{i+a+bx}{i+a}} \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.64997, size = 180, normalized size = 1.5 \begin{align*} \frac{1}{2} \, \arctan \left (\frac{b x}{a^{2} + 1}, -\frac{a b x}{a^{2} + 1}\right ) \log \left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right ) - \frac{1}{2} \, \arctan \left (b x + a\right ) \log \left (\frac{b^{2} x^{2}}{a^{2} + 1}\right ) + \operatorname{arccot}\left (b x + a\right ) \log \left (x\right ) + \arctan \left (\frac{b^{2} x + a b}{b}\right ) \log \left (x\right ) + \frac{1}{2} i \,{\rm Li}_2\left (\frac{i \, b x + i \, a + 1}{i \, a + 1}\right ) - \frac{1}{2} i \,{\rm Li}_2\left (\frac{i \, b x + i \, a - 1}{i \, a - 1}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\operatorname{arccot}\left (b x + a\right )}{x}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{acot}{\left (a + b x \right )}}{x}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{arccot}\left (b x + a\right )}{x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]