Optimal. Leaf size=120 \[ -\frac {1}{2} i \text {Li}_2\left (1-\frac {2}{1-i (a+b x)}\right )+\frac {1}{2} i \text {Li}_2\left (1-\frac {2 b x}{(i-a) (1-i (a+b x))}\right )+\log \left (\frac {2}{1-i (a+b x)}\right ) \left (-\cot ^{-1}(a+b x)\right )+\log \left (\frac {2 b x}{(-a+i) (1-i (a+b x))}\right ) \cot ^{-1}(a+b x) \]
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Rubi [A] time = 0.11, antiderivative size = 120, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, 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.
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Rule 2315
Rule 2402
Rule 2447
Rule 4857
Rule 5048
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*}
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Mathematica [B] time = 0.03, size = 251, normalized size = 2.09 \[ -\frac {1}{2} i \text {Li}_2\left (-\frac {b \left (\frac {a+b x}{b}-\frac {a}{b}\right )}{a-i}\right )+\frac {1}{2} i \text {Li}_2\left (-\frac {b \left (\frac {a+b x}{b}-\frac {a}{b}\right )}{a+i}\right )-\frac {1}{2} i \log \left (\frac {a+b x-i}{b \left (\frac {a}{b}-\frac {i}{b}\right )}\right ) \log \left (\frac {a+b x}{b}-\frac {a}{b}\right )+\frac {1}{2} i \log \left (\frac {a+b x-i}{a+b x}\right ) \log \left (\frac {a+b x}{b}-\frac {a}{b}\right )+\frac {1}{2} i \log \left (\frac {a+b x+i}{b \left (\frac {a}{b}+\frac {i}{b}\right )}\right ) \log \left (\frac {a+b x}{b}-\frac {a}{b}\right )-\frac {1}{2} i \log \left (\frac {a+b x+i}{a+b x}\right ) \log \left (\frac {a+b x}{b}-\frac {a}{b}\right ) \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.53, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\operatorname {arccot}\left (b x + a\right )}{x}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {arccot}\left (b x + a\right )}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 103, normalized size = 0.86 \[ \ln \left (b x \right ) \mathrm {arccot}\left (b x +a \right )-\frac {i \ln \left (b x \right ) \ln \left (\frac {-b x -a +i}{i-a}\right )}{2}+\frac {i \ln \left (b x \right ) \ln \left (\frac {b x +a +i}{i+a}\right )}{2}-\frac {i \dilog \left (\frac {-b x -a +i}{i-a}\right )}{2}+\frac {i \dilog \left (\frac {b x +a +i}{i+a}\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.48, size = 133, normalized size = 1.11 \[ \frac {1}{2} \, \arctan \left (\frac {b x}{a^{2} + 1}, -\frac {a b x}{a^{2} + 1}\right ) \log \left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right ) - \frac {1}{2} \, \arctan \left (b x + a\right ) \log \left (\frac {b^{2} x^{2}}{a^{2} + 1}\right ) + \operatorname {arccot}\left (b x + a\right ) \log \relax (x) + \arctan \left (\frac {b^{2} x + a b}{b}\right ) \log \relax (x) + \frac {1}{2} i \, {\rm Li}_2\left (\frac {i \, b x + i \, a + 1}{i \, a + 1}\right ) - \frac {1}{2} i \, {\rm Li}_2\left (\frac {i \, b x + i \, a - 1}{i \, a - 1}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\mathrm {acot}\left (a+b\,x\right )}{x} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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