Optimal. Leaf size=122 \[ -\frac{1}{2} i \text{PolyLog}\left (2,1-\frac{2}{1-i (a-x)}\right )+\frac{1}{2} i \text{PolyLog}\left (2,1+\frac{2 (a+x)}{(-2 a+i) (1-i (a-x))}\right )+\log \left (\frac{2}{1-i (a-x)}\right ) \tan ^{-1}(a-x)-\log \left (-\frac{2 (a+x)}{(-2 a+i) (1-i (a-x))}\right ) \tan ^{-1}(a-x) \]
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Rubi [A] time = 0.0914153, antiderivative size = 122, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.385, Rules used = {5047, 4856, 2402, 2315, 2447} \[ -\frac{1}{2} i \text{PolyLog}\left (2,1-\frac{2}{1-i (a-x)}\right )+\frac{1}{2} i \text{PolyLog}\left (2,1+\frac{2 (a+x)}{(-2 a+i) (1-i (a-x))}\right )+\log \left (\frac{2}{1-i (a-x)}\right ) \tan ^{-1}(a-x)-\log \left (-\frac{2 (a+x)}{(-2 a+i) (1-i (a-x))}\right ) \tan ^{-1}(a-x) \]
Antiderivative was successfully verified.
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Rule 5047
Rule 4856
Rule 2402
Rule 2315
Rule 2447
Rubi steps
\begin{align*} \int -\frac{\tan ^{-1}(a-x)}{a+x} \, dx &=\operatorname{Subst}\left (\int \frac{\tan ^{-1}(x)}{2 a-x} \, dx,x,a-x\right )\\ &=\tan ^{-1}(a-x) \log \left (\frac{2}{1-i (a-x)}\right )-\tan ^{-1}(a-x) \log \left (-\frac{2 (a+x)}{(i-2 a) (1-i (a-x))}\right )-\operatorname{Subst}\left (\int \frac{\log \left (\frac{2}{1-i x}\right )}{1+x^2} \, dx,x,a-x\right )+\operatorname{Subst}\left (\int \frac{\log \left (\frac{2 (2 a-x)}{(-i+2 a) (1-i x)}\right )}{1+x^2} \, dx,x,a-x\right )\\ &=\tan ^{-1}(a-x) \log \left (\frac{2}{1-i (a-x)}\right )-\tan ^{-1}(a-x) \log \left (-\frac{2 (a+x)}{(i-2 a) (1-i (a-x))}\right )+\frac{1}{2} i \text{Li}_2\left (1+\frac{2 (a+x)}{(i-2 a) (1-i (a-x))}\right )-i \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1-i (a-x)}\right )\\ &=\tan ^{-1}(a-x) \log \left (\frac{2}{1-i (a-x)}\right )-\tan ^{-1}(a-x) \log \left (-\frac{2 (a+x)}{(i-2 a) (1-i (a-x))}\right )-\frac{1}{2} i \text{Li}_2\left (1-\frac{2}{1-i (a-x)}\right )+\frac{1}{2} i \text{Li}_2\left (1+\frac{2 (a+x)}{(i-2 a) (1-i (a-x))}\right )\\ \end{align*}
Mathematica [A] time = 0.0304267, size = 105, normalized size = 0.86 \[ -\frac{1}{2} i \left (\text{PolyLog}\left (2,\frac{a-x+i}{2 a+i}\right )-\text{PolyLog}\left (2,\frac{-a+x+i}{-2 a+i}\right )-\log (1+i (a-x)) \log \left (\frac{a+x}{2 a-i}\right )+\log (-i a+i x+1) \log \left (\frac{a+x}{2 a+i}\right )\right ) \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.013, size = 102, normalized size = 0.8 \begin{align*} -\ln \left ( a+x \right ) \arctan \left ( a-x \right ) +{\frac{i}{2}}\ln \left ( a+x \right ) \ln \left ({\frac{a-x+i}{2\,a+i}} \right ) -{\frac{i}{2}}\ln \left ( a+x \right ) \ln \left ({\frac{-a+x+i}{i-2\,a}} \right ) +{\frac{i}{2}}{\it dilog} \left ({\frac{a-x+i}{2\,a+i}} \right ) -{\frac{i}{2}}{\it dilog} \left ({\frac{-a+x+i}{i-2\,a}} \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.6248, size = 159, normalized size = 1.3 \begin{align*} -\frac{1}{2} \, \arctan \left (\frac{a + x}{4 \, a^{2} + 1}, \frac{2 \,{\left (a^{2} + a x\right )}}{4 \, a^{2} + 1}\right ) \log \left (a^{2} - 2 \, a x + x^{2} + 1\right ) + \frac{1}{2} \, \arctan \left (-a + x\right ) \log \left (\frac{a^{2} + 2 \, a x + x^{2}}{4 \, a^{2} + 1}\right ) - \frac{1}{2} i \,{\rm Li}_2\left (-\frac{-i \, a + i \, x + 1}{2 i \, a - 1}\right ) + \frac{1}{2} i \,{\rm Li}_2\left (-\frac{-i \, a + i \, x - 1}{2 i \, a + 1}\right ) \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{\arctan \left (-a + x\right )}{a + x}, 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{atan}{\left (a - x \right )}}{a + 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{\arctan \left (a - x\right )}{a + x}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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