Optimal. Leaf size=102 \[ -\frac {\text {Li}_3\left (1-\frac {2}{i a x+1}\right )}{2 a^2 c}-\frac {i \text {Li}_2\left (1-\frac {2}{i a x+1}\right ) \tan ^{-1}(a x)}{a^2 c}-\frac {i \tan ^{-1}(a x)^3}{3 a^2 c}-\frac {\log \left (\frac {2}{1+i a x}\right ) \tan ^{-1}(a x)^2}{a^2 c} \]
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Rubi [A] time = 0.15, antiderivative size = 102, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 5, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {4920, 4854, 4884, 4994, 6610} \[ -\frac {\text {PolyLog}\left (3,1-\frac {2}{1+i a x}\right )}{2 a^2 c}-\frac {i \tan ^{-1}(a x) \text {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )}{a^2 c}-\frac {i \tan ^{-1}(a x)^3}{3 a^2 c}-\frac {\log \left (\frac {2}{1+i a x}\right ) \tan ^{-1}(a x)^2}{a^2 c} \]
Antiderivative was successfully verified.
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Rule 4854
Rule 4884
Rule 4920
Rule 4994
Rule 6610
Rubi steps
\begin {align*} \int \frac {x \tan ^{-1}(a x)^2}{c+a^2 c x^2} \, dx &=-\frac {i \tan ^{-1}(a x)^3}{3 a^2 c}-\frac {\int \frac {\tan ^{-1}(a x)^2}{i-a x} \, dx}{a c}\\ &=-\frac {i \tan ^{-1}(a x)^3}{3 a^2 c}-\frac {\tan ^{-1}(a x)^2 \log \left (\frac {2}{1+i a x}\right )}{a^2 c}+\frac {2 \int \frac {\tan ^{-1}(a x) \log \left (\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx}{a c}\\ &=-\frac {i \tan ^{-1}(a x)^3}{3 a^2 c}-\frac {\tan ^{-1}(a x)^2 \log \left (\frac {2}{1+i a x}\right )}{a^2 c}-\frac {i \tan ^{-1}(a x) \text {Li}_2\left (1-\frac {2}{1+i a x}\right )}{a^2 c}+\frac {i \int \frac {\text {Li}_2\left (1-\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx}{a c}\\ &=-\frac {i \tan ^{-1}(a x)^3}{3 a^2 c}-\frac {\tan ^{-1}(a x)^2 \log \left (\frac {2}{1+i a x}\right )}{a^2 c}-\frac {i \tan ^{-1}(a x) \text {Li}_2\left (1-\frac {2}{1+i a x}\right )}{a^2 c}-\frac {\text {Li}_3\left (1-\frac {2}{1+i a x}\right )}{2 a^2 c}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 110, normalized size = 1.08 \[ -\frac {\text {Li}_3\left (\frac {a x+i}{a x-i}\right )}{2 a^2 c}-\frac {i \text {Li}_2\left (\frac {a x+i}{a x-i}\right ) \tan ^{-1}(a x)}{a^2 c}-\frac {i \tan ^{-1}(a x)^3}{3 a^2 c}-\frac {\log \left (\frac {2 i}{-a x+i}\right ) \tan ^{-1}(a x)^2}{a^2 c} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.53, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {x \arctan \left (a x\right )^{2}}{a^{2} c x^{2} + c}, x\right ) \]
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 [C] time = 0.58, size = 897, normalized size = 8.79 \[ \frac {\arctan \left (a x \right )^{2} \ln \left (a^{2} x^{2}+1\right )}{2 a^{2} c}-\frac {\arctan \left (a x \right )^{2} \ln \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )}{a^{2} c}+\frac {i \mathrm {csgn}\left (\frac {i \left (i a x +1\right )^{2}}{\left (a^{2} x^{2}+1\right ) \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )^{2}}\right )^{3} \arctan \left (a x \right )^{2} \pi }{4 a^{2} c}-\frac {i \mathrm {csgn}\left (\frac {i \left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )^{2} \mathrm {csgn}\left (\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right ) \arctan \left (a x \right )^{2} \pi }{2 a^{2} c}-\frac {i \mathrm {csgn}\left (i \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )\right )^{2} \mathrm {csgn}\left (i \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )^{2}\right ) \arctan \left (a x \right )^{2} \pi }{4 a^{2} c}-\frac {i \mathrm {csgn}\left (\frac {i \left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right ) \mathrm {csgn}\left (\frac {i \left (i a x +1\right )^{2}}{\left (a^{2} x^{2}+1\right ) \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )^{2}}\right )^{2} \arctan \left (a x \right )^{2} \pi }{4 a^{2} c}+\frac {i \mathrm {csgn}\left (\frac {i \left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right ) \mathrm {csgn}\left (\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )^{2} \arctan \left (a x \right )^{2} \pi }{4 a^{2} c}+\frac {i \arctan \left (a x \right ) \polylog \left (2, -\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )}{a^{2} c}-\frac {i \mathrm {csgn}\left (\frac {i}{\left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )^{2}}\right ) \mathrm {csgn}\left (\frac {i \left (i a x +1\right )^{2}}{\left (a^{2} x^{2}+1\right ) \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )^{2}}\right )^{2} \arctan \left (a x \right )^{2} \pi }{4 a^{2} c}+\frac {i \mathrm {csgn}\left (\frac {i \left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )^{3} \arctan \left (a x \right )^{2} \pi }{4 a^{2} c}-\frac {i \mathrm {csgn}\left (i \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )^{2}\right )^{3} \arctan \left (a x \right )^{2} \pi }{4 a^{2} c}+\frac {i \arctan \left (a x \right )^{3}}{3 a^{2} c}-\frac {\ln \relax (2) \arctan \left (a x \right )^{2}}{a^{2} c}+\frac {i \mathrm {csgn}\left (\frac {i}{\left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )^{2}}\right ) \mathrm {csgn}\left (\frac {i \left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right ) \mathrm {csgn}\left (\frac {i \left (i a x +1\right )^{2}}{\left (a^{2} x^{2}+1\right ) \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )^{2}}\right ) \arctan \left (a x \right )^{2} \pi }{4 a^{2} c}-\frac {\polylog \left (3, -\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}\right )}{2 a^{2} c}+\frac {i \mathrm {csgn}\left (i \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )\right ) \mathrm {csgn}\left (i \left (\frac {\left (i a x +1\right )^{2}}{a^{2} x^{2}+1}+1\right )^{2}\right )^{2} \arctan \left (a x \right )^{2} \pi }{2 a^{2} 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 \[ \int \frac {x \arctan \left (a x\right )^{2}}{a^{2} c x^{2} + c}\,{d x} \]
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 \[ \int \frac {x\,{\mathrm {atan}\left (a\,x\right )}^2}{c\,a^2\,x^2+c} \,d x \]
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 \[ \frac {\int \frac {x \operatorname {atan}^{2}{\left (a x \right )}}{a^{2} x^{2} + 1}\, dx}{c} \]
Verification of antiderivative is not currently implemented for this CAS.
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