Optimal. Leaf size=189 \[ -\frac {1}{2} i \text {Li}_2(-i x) \left (-\log \left (x^2+1\right )+\log (1-i x)+\log (1+i x)\right )+\frac {1}{2} i \text {Li}_2(i x) \left (-\log \left (x^2+1\right )+\log (1-i x)+\log (1+i x)\right )-i \text {Li}_3(1-i x)+i \text {Li}_3(i x+1)+i \text {Li}_2(1-i x) \log (1-i x)-i \text {Li}_2(i x+1) \log (1+i x)+\frac {1}{2} i \log (i x) \log ^2(1-i x)-\frac {1}{2} i \log ^2(1+i x) \log (-i x) \]
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Rubi [A] time = 0.18, antiderivative size = 189, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 7, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.583, Rules used = {4848, 2391, 5011, 2396, 2433, 2374, 6589} \[ -\frac {1}{2} i \left (-\log \left (x^2+1\right )+\log (1-i x)+\log (1+i x)\right ) \text {PolyLog}(2,-i x)+\frac {1}{2} i \left (-\log \left (x^2+1\right )+\log (1-i x)+\log (1+i x)\right ) \text {PolyLog}(2,i x)-i \text {PolyLog}(3,1-i x)+i \text {PolyLog}(3,1+i x)+i \log (1-i x) \text {PolyLog}(2,1-i x)-i \log (1+i x) \text {PolyLog}(2,1+i x)+\frac {1}{2} i \log (i x) \log ^2(1-i x)-\frac {1}{2} i \log ^2(1+i x) \log (-i x) \]
Antiderivative was successfully verified.
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Rule 2374
Rule 2391
Rule 2396
Rule 2433
Rule 4848
Rule 5011
Rule 6589
Rubi steps
\begin {align*} \int \frac {\tan ^{-1}(x) \log \left (1+x^2\right )}{x} \, dx &=\frac {1}{2} i \int \frac {\log ^2(1-i x)}{x} \, dx-\frac {1}{2} i \int \frac {\log ^2(1+i x)}{x} \, dx+\left (-\log (1-i x)-\log (1+i x)+\log \left (1+x^2\right )\right ) \int \frac {\tan ^{-1}(x)}{x} \, dx\\ &=-\frac {1}{2} i \log ^2(1+i x) \log (-i x)+\frac {1}{2} i \log ^2(1-i x) \log (i x)+\frac {1}{2} \left (i \left (\log (1-i x)+\log (1+i x)-\log \left (1+x^2\right )\right )\right ) \int \frac {\log (1+i x)}{x} \, dx+\frac {1}{2} \left (i \left (-\log (1-i x)-\log (1+i x)+\log \left (1+x^2\right )\right )\right ) \int \frac {\log (1-i x)}{x} \, dx-\int \frac {\log (1+i x) \log (-i x)}{1+i x} \, dx-\int \frac {\log (1-i x) \log (i x)}{1-i x} \, dx\\ &=-\frac {1}{2} i \log ^2(1+i x) \log (-i x)+\frac {1}{2} i \log ^2(1-i x) \log (i x)-\frac {1}{2} i \left (\log (1-i x)+\log (1+i x)-\log \left (1+x^2\right )\right ) \text {Li}_2(-i x)+\frac {1}{2} i \left (\log (1-i x)+\log (1+i x)-\log \left (1+x^2\right )\right ) \text {Li}_2(i x)+i \operatorname {Subst}\left (\int \frac {\log (-i (i-i x)) \log (x)}{x} \, dx,x,1+i x\right )-i \operatorname {Subst}\left (\int \frac {\log (i (-i+i x)) \log (x)}{x} \, dx,x,1-i x\right )\\ &=-\frac {1}{2} i \log ^2(1+i x) \log (-i x)+\frac {1}{2} i \log ^2(1-i x) \log (i x)+i \log (1-i x) \text {Li}_2(1-i x)-i \log (1+i x) \text {Li}_2(1+i x)-\frac {1}{2} i \left (\log (1-i x)+\log (1+i x)-\log \left (1+x^2\right )\right ) \text {Li}_2(-i x)+\frac {1}{2} i \left (\log (1-i x)+\log (1+i x)-\log \left (1+x^2\right )\right ) \text {Li}_2(i x)-i \operatorname {Subst}\left (\int \frac {\text {Li}_2(x)}{x} \, dx,x,1-i x\right )+i \operatorname {Subst}\left (\int \frac {\text {Li}_2(x)}{x} \, dx,x,1+i x\right )\\ &=-\frac {1}{2} i \log ^2(1+i x) \log (-i x)+\frac {1}{2} i \log ^2(1-i x) \log (i x)+i \log (1-i x) \text {Li}_2(1-i x)-i \log (1+i x) \text {Li}_2(1+i x)-\frac {1}{2} i \left (\log (1-i x)+\log (1+i x)-\log \left (1+x^2\right )\right ) \text {Li}_2(-i x)+\frac {1}{2} i \left (\log (1-i x)+\log (1+i x)-\log \left (1+x^2\right )\right ) \text {Li}_2(i x)-i \text {Li}_3(1-i x)+i \text {Li}_3(1+i x)\\ \end {align*}
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Mathematica [F] time = 0.78, size = 0, normalized size = 0.00 \[ \int \frac {\tan ^{-1}(x) \log \left (1+x^2\right )}{x} \, dx \]
Verification is Not applicable to the result.
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fricas [F] time = 0.40, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\arctan \relax (x) \log \left (x^{2} + 1\right )}{x}, x\right ) \]
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 \[ \int \frac {\arctan \relax (x) \log \left (x^{2} + 1\right )}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 2.80, size = 5237, normalized size = 27.71 \[ \text {output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\arctan \relax (x) \log \left (x^{2} + 1\right )}{x}\,{d x} \]
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 \[ \int \frac {\ln \left (x^2+1\right )\,\mathrm {atan}\relax (x)}{x} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\log {\left (x^{2} + 1 \right )} \operatorname {atan}{\relax (x )}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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