Optimal. Leaf size=83 \[ x \left (a+b \tan ^{-1}(c x)\right )^2+\frac {i \left (a+b \tan ^{-1}(c x)\right )^2}{c}+\frac {2 b \log \left (\frac {2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{c}+\frac {i b^2 \text {Li}_2\left (1-\frac {2}{i c x+1}\right )}{c} \]
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Rubi [A] time = 0.10, antiderivative size = 83, 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 = {4846, 4920, 4854, 2402, 2315} \[ \frac {i b^2 \text {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{c}+x \left (a+b \tan ^{-1}(c x)\right )^2+\frac {i \left (a+b \tan ^{-1}(c x)\right )^2}{c}+\frac {2 b \log \left (\frac {2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{c} \]
Antiderivative was successfully verified.
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Rule 2315
Rule 2402
Rule 4846
Rule 4854
Rule 4920
Rubi steps
\begin {align*} \int \left (a+b \tan ^{-1}(c x)\right )^2 \, dx &=x \left (a+b \tan ^{-1}(c x)\right )^2-(2 b c) \int \frac {x \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx\\ &=\frac {i \left (a+b \tan ^{-1}(c x)\right )^2}{c}+x \left (a+b \tan ^{-1}(c x)\right )^2+(2 b) \int \frac {a+b \tan ^{-1}(c x)}{i-c x} \, dx\\ &=\frac {i \left (a+b \tan ^{-1}(c x)\right )^2}{c}+x \left (a+b \tan ^{-1}(c x)\right )^2+\frac {2 b \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1+i c x}\right )}{c}-\left (2 b^2\right ) \int \frac {\log \left (\frac {2}{1+i c x}\right )}{1+c^2 x^2} \, dx\\ &=\frac {i \left (a+b \tan ^{-1}(c x)\right )^2}{c}+x \left (a+b \tan ^{-1}(c x)\right )^2+\frac {2 b \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1+i c x}\right )}{c}+\frac {\left (2 i b^2\right ) \operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+i c x}\right )}{c}\\ &=\frac {i \left (a+b \tan ^{-1}(c x)\right )^2}{c}+x \left (a+b \tan ^{-1}(c x)\right )^2+\frac {2 b \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1+i c x}\right )}{c}+\frac {i b^2 \text {Li}_2\left (1-\frac {2}{1+i c x}\right )}{c}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 90, normalized size = 1.08 \[ \frac {a \left (a c x-b \log \left (c^2 x^2+1\right )\right )+2 b \tan ^{-1}(c x) \left (a c x+b \log \left (1+e^{2 i \tan ^{-1}(c x)}\right )\right )-i b^2 \text {Li}_2\left (-e^{2 i \tan ^{-1}(c x)}\right )+b^2 (c x-i) \tan ^{-1}(c x)^2}{c} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.43, size = 0, normalized size = 0.00 \[ {\rm integral}\left (b^{2} \arctan \left (c x\right )^{2} + 2 \, a b \arctan \left (c x\right ) + a^{2}, x\right ) \]
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 \[ \mathit {sage}_{0} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.19, size = 128, normalized size = 1.54 \[ x \,b^{2} \arctan \left (c x \right )^{2}-\frac {i \arctan \left (c x \right )^{2} b^{2}}{c}+2 x a b \arctan \left (c x \right )+\frac {2 \arctan \left (c x \right ) \ln \left (\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}+1\right ) b^{2}}{c}-\frac {i \polylog \left (2, -\frac {\left (i c x +1\right )^{2}}{c^{2} x^{2}+1}\right ) b^{2}}{c}+a^{2} x -\frac {a b \ln \left (c^{2} x^{2}+1\right )}{c} \]
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 \[ \frac {1}{16} \, {\left (4 \, x \arctan \left (c x\right )^{2} + 192 \, c^{2} \int \frac {x^{2} \arctan \left (c x\right )^{2}}{16 \, {\left (c^{2} x^{2} + 1\right )}}\,{d x} + 16 \, c^{2} \int \frac {x^{2} \log \left (c^{2} x^{2} + 1\right )^{2}}{16 \, {\left (c^{2} x^{2} + 1\right )}}\,{d x} + 64 \, c^{2} \int \frac {x^{2} \log \left (c^{2} x^{2} + 1\right )}{16 \, {\left (c^{2} x^{2} + 1\right )}}\,{d x} - x \log \left (c^{2} x^{2} + 1\right )^{2} + \frac {4 \, \arctan \left (c x\right )^{3}}{c} - 128 \, c \int \frac {x \arctan \left (c x\right )}{16 \, {\left (c^{2} x^{2} + 1\right )}}\,{d x} + 16 \, \int \frac {\log \left (c^{2} x^{2} + 1\right )^{2}}{16 \, {\left (c^{2} x^{2} + 1\right )}}\,{d x}\right )} b^{2} + a^{2} x + \frac {{\left (2 \, c x \arctan \left (c x\right ) - \log \left (c^{2} x^{2} + 1\right )\right )} a b}{c} \]
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 \[ \int {\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )}^2 \,d x \]
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 \[ \int \left (a + b \operatorname {atan}{\left (c x \right )}\right )^{2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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