Optimal. Leaf size=274 \[ \frac {1}{4} \log \left (\frac {b (i-x)}{a+i (1+b)}\right ) \log (1-i a-i b x)-\frac {1}{4} \log \left (-\frac {b (i+x)}{a+i (1-b)}\right ) \log (1-i a-i b x)-\frac {1}{4} \log \left (\frac {b (i-x)}{a-i (1-b)}\right ) \log (1+i a+i b x)+\frac {1}{4} \log \left (-\frac {b (i+x)}{a-i (1+b)}\right ) \log (1+i a+i b x)-\frac {1}{4} \text {PolyLog}\left (2,-\frac {i-a-b x}{a-i (1-b)}\right )+\frac {1}{4} \text {PolyLog}\left (2,-\frac {i-a-b x}{a-i (1+b)}\right )-\frac {1}{4} \text {PolyLog}\left (2,\frac {i+a+b x}{i+a-i b}\right )+\frac {1}{4} \text {PolyLog}\left (2,\frac {i+a+b x}{a+i (1+b)}\right ) \]
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Rubi [A]
time = 0.22, antiderivative size = 274, normalized size of antiderivative = 1.00, number of steps
used = 17, number of rules used = 5, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {5159, 2456,
2441, 2440, 2438} \begin {gather*} -\frac {1}{4} \text {Li}_2\left (-\frac {-a-b x+i}{a-i (1-b)}\right )+\frac {1}{4} \text {Li}_2\left (-\frac {-a-b x+i}{a-i (b+1)}\right )-\frac {1}{4} \text {Li}_2\left (\frac {a+b x+i}{a-i b+i}\right )+\frac {1}{4} \text {Li}_2\left (\frac {a+b x+i}{a+i (b+1)}\right )+\frac {1}{4} \log \left (\frac {b (-x+i)}{a+i (b+1)}\right ) \log (-i a-i b x+1)-\frac {1}{4} \log \left (-\frac {b (x+i)}{a+i (1-b)}\right ) \log (-i a-i b x+1)-\frac {1}{4} \log \left (\frac {b (-x+i)}{a-i (1-b)}\right ) \log (i a+i b x+1)+\frac {1}{4} \log \left (-\frac {b (x+i)}{a-i (b+1)}\right ) \log (i a+i b x+1) \end {gather*}
Antiderivative was successfully verified.
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Rule 2438
Rule 2440
Rule 2441
Rule 2456
Rule 5159
Rubi steps
\begin {align*} \int \frac {\tan ^{-1}(a+b x)}{1+x^2} \, dx &=\frac {1}{2} i \int \frac {\log (1-i a-i b x)}{1+x^2} \, dx-\frac {1}{2} i \int \frac {\log (1+i a+i b x)}{1+x^2} \, dx\\ &=\frac {1}{2} i \int \left (\frac {i \log (1-i a-i b x)}{2 (i-x)}+\frac {i \log (1-i a-i b x)}{2 (i+x)}\right ) \, dx-\frac {1}{2} i \int \left (\frac {i \log (1+i a+i b x)}{2 (i-x)}+\frac {i \log (1+i a+i b x)}{2 (i+x)}\right ) \, dx\\ &=-\left (\frac {1}{4} \int \frac {\log (1-i a-i b x)}{i-x} \, dx\right )-\frac {1}{4} \int \frac {\log (1-i a-i b x)}{i+x} \, dx+\frac {1}{4} \int \frac {\log (1+i a+i b x)}{i-x} \, dx+\frac {1}{4} \int \frac {\log (1+i a+i b x)}{i+x} \, dx\\ &=\frac {1}{4} \log \left (\frac {b (i-x)}{a+i (1+b)}\right ) \log (1-i a-i b x)-\frac {1}{4} \log \left (-\frac {b (i+x)}{a+i (1-b)}\right ) \log (1-i a-i b x)-\frac {1}{4} \log \left (\frac {b (i-x)}{a-i (1-b)}\right ) \log (1+i a+i b x)+\frac {1}{4} \log \left (-\frac {b (i+x)}{a-i (1+b)}\right ) \log (1+i a+i b x)+\frac {1}{4} (i b) \int \frac {\log \left (\frac {i b (i-x)}{1+i a-b}\right )}{1+i a+i b x} \, dx+\frac {1}{4} (i b) \int \frac {\log \left (-\frac {i b (i-x)}{1-i a+b}\right )}{1-i a-i b x} \, dx-\frac {1}{4} (i b) \int \frac {\log \left (\frac {i b (i+x)}{-1-i a-b}\right )}{1+i a+i b x} \, dx-\frac {1}{4} (i b) \int \frac {\log \left (-\frac {i b (i+x)}{-1+i a+b}\right )}{1-i a-i b x} \, dx\\ &=\frac {1}{4} \log \left (\frac {b (i-x)}{a+i (1+b)}\right ) \log (1-i a-i b x)-\frac {1}{4} \log \left (-\frac {b (i+x)}{a+i (1-b)}\right ) \log (1-i a-i b x)-\frac {1}{4} \log \left (\frac {b (i-x)}{a-i (1-b)}\right ) \log (1+i a+i b x)+\frac {1}{4} \log \left (-\frac {b (i+x)}{a-i (1+b)}\right ) \log (1+i a+i b x)-\frac {1}{4} \text {Subst}\left (\int \frac {\log \left (1+\frac {x}{-1-i a-b}\right )}{x} \, dx,x,1+i a+i b x\right )+\frac {1}{4} \text {Subst}\left (\int \frac {\log \left (1-\frac {x}{1+i a-b}\right )}{x} \, dx,x,1+i a+i b x\right )-\frac {1}{4} \text {Subst}\left (\int \frac {\log \left (1-\frac {x}{1-i a+b}\right )}{x} \, dx,x,1-i a-i b x\right )+\frac {1}{4} \text {Subst}\left (\int \frac {\log \left (1+\frac {x}{-1+i a+b}\right )}{x} \, dx,x,1-i a-i b x\right )\\ &=\frac {1}{4} \log \left (\frac {b (i-x)}{a+i (1+b)}\right ) \log (1-i a-i b x)-\frac {1}{4} \log \left (-\frac {b (i+x)}{a+i (1-b)}\right ) \log (1-i a-i b x)-\frac {1}{4} \log \left (\frac {b (i-x)}{a-i (1-b)}\right ) \log (1+i a+i b x)+\frac {1}{4} \log \left (-\frac {b (i+x)}{a-i (1+b)}\right ) \log (1+i a+i b x)-\frac {1}{4} \text {Li}_2\left (-\frac {i-a-b x}{a-i (1-b)}\right )+\frac {1}{4} \text {Li}_2\left (-\frac {i-a-b x}{a-i (1+b)}\right )-\frac {1}{4} \text {Li}_2\left (\frac {i+a+b x}{i+a-i b}\right )+\frac {1}{4} \text {Li}_2\left (\frac {i+a+b x}{a+i (1+b)}\right )\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 283, normalized size = 1.03 \begin {gather*} \frac {1}{4} \log \left (\frac {b (i-x)}{a+i (1+b)}\right ) \log (1-i a-i b x)-\frac {1}{4} \log \left (-\frac {b (i+x)}{a+i (1-b)}\right ) \log (1-i a-i b x)-\frac {1}{4} \log \left (\frac {b (i-x)}{a-i (1-b)}\right ) \log (1+i a+i b x)+\frac {1}{4} \log \left (-\frac {b (i+x)}{a-i (1+b)}\right ) \log (1+i a+i b x)-\frac {1}{4} \text {PolyLog}\left (2,\frac {1-i a-i b x}{1-i a-b}\right )+\frac {1}{4} \text {PolyLog}\left (2,\frac {1-i a-i b x}{1-i a+b}\right )-\frac {1}{4} \text {PolyLog}\left (2,\frac {1+i a+i b x}{1+i a-b}\right )+\frac {1}{4} \text {PolyLog}\left (2,\frac {1+i a+i b x}{1+i a+b}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 500 vs. \(2 (225 ) = 450\).
time = 0.31, size = 501, normalized size = 1.83 Too large to display
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.51, size = 328, normalized size = 1.20 \begin {gather*} \frac {1}{8} \, b {\left (\frac {8 \, \arctan \left (x\right ) \arctan \left (\frac {b^{2} x + a b}{b}\right )}{b} - \frac {4 \, \arctan \left (x\right ) \arctan \left (\frac {a b + {\left (b^{2} + b\right )} x}{a^{2} + b^{2} + 2 \, b + 1}, \frac {a b x + a^{2} + b + 1}{a^{2} + b^{2} + 2 \, b + 1}\right ) - 4 \, \arctan \left (x\right ) \arctan \left (\frac {a b + {\left (b^{2} - b\right )} x}{a^{2} + b^{2} - 2 \, b + 1}, \frac {a b x + a^{2} - b + 1}{a^{2} + b^{2} - 2 \, b + 1}\right ) + \log \left (x^{2} + 1\right ) \log \left (\frac {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}{a^{2} + b^{2} + 2 \, b + 1}\right ) - \log \left (x^{2} + 1\right ) \log \left (\frac {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}{a^{2} + b^{2} - 2 \, b + 1}\right ) + 2 \, {\rm Li}_2\left (-\frac {i \, b x - b}{i \, a + b + 1}\right ) - 2 \, {\rm Li}_2\left (-\frac {i \, b x - b}{i \, a + b - 1}\right ) + 2 \, {\rm Li}_2\left (\frac {i \, b x + b}{-i \, a + b + 1}\right ) - 2 \, {\rm Li}_2\left (\frac {i \, b x + b}{-i \, a + b - 1}\right )}{b}\right )} + \arctan \left (b x + a\right ) \arctan \left (x\right ) - \arctan \left (x\right ) \arctan \left (\frac {b^{2} x + a b}{b}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {\operatorname {atan}{\left (a + b x \right )}}{x^{2} + 1}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {\mathrm {atan}\left (a+b\,x\right )}{x^2+1} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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