Optimal. Leaf size=98 \[ \frac {i \text {Li}_2\left (1-\frac {2}{i a x+1}\right )}{a^3 c}-\frac {\tan ^{-1}(a x)^3}{3 a^3 c}+\frac {i \tan ^{-1}(a x)^2}{a^3 c}+\frac {2 \log \left (\frac {2}{1+i a x}\right ) \tan ^{-1}(a x)}{a^3 c}+\frac {x \tan ^{-1}(a x)^2}{a^2 c} \]
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Rubi [A] time = 0.17, antiderivative size = 98, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {4916, 4846, 4920, 4854, 2402, 2315, 4884} \[ \frac {i \text {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )}{a^3 c}-\frac {\tan ^{-1}(a x)^3}{3 a^3 c}+\frac {x \tan ^{-1}(a x)^2}{a^2 c}+\frac {i \tan ^{-1}(a x)^2}{a^3 c}+\frac {2 \log \left (\frac {2}{1+i a x}\right ) \tan ^{-1}(a x)}{a^3 c} \]
Antiderivative was successfully verified.
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Rule 2315
Rule 2402
Rule 4846
Rule 4854
Rule 4884
Rule 4916
Rule 4920
Rubi steps
\begin {align*} \int \frac {x^2 \tan ^{-1}(a x)^2}{c+a^2 c x^2} \, dx &=-\frac {\int \frac {\tan ^{-1}(a x)^2}{c+a^2 c x^2} \, dx}{a^2}+\frac {\int \tan ^{-1}(a x)^2 \, dx}{a^2 c}\\ &=\frac {x \tan ^{-1}(a x)^2}{a^2 c}-\frac {\tan ^{-1}(a x)^3}{3 a^3 c}-\frac {2 \int \frac {x \tan ^{-1}(a x)}{1+a^2 x^2} \, dx}{a c}\\ &=\frac {i \tan ^{-1}(a x)^2}{a^3 c}+\frac {x \tan ^{-1}(a x)^2}{a^2 c}-\frac {\tan ^{-1}(a x)^3}{3 a^3 c}+\frac {2 \int \frac {\tan ^{-1}(a x)}{i-a x} \, dx}{a^2 c}\\ &=\frac {i \tan ^{-1}(a x)^2}{a^3 c}+\frac {x \tan ^{-1}(a x)^2}{a^2 c}-\frac {\tan ^{-1}(a x)^3}{3 a^3 c}+\frac {2 \tan ^{-1}(a x) \log \left (\frac {2}{1+i a x}\right )}{a^3 c}-\frac {2 \int \frac {\log \left (\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx}{a^2 c}\\ &=\frac {i \tan ^{-1}(a x)^2}{a^3 c}+\frac {x \tan ^{-1}(a x)^2}{a^2 c}-\frac {\tan ^{-1}(a x)^3}{3 a^3 c}+\frac {2 \tan ^{-1}(a x) \log \left (\frac {2}{1+i a x}\right )}{a^3 c}+\frac {(2 i) \operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+i a x}\right )}{a^3 c}\\ &=\frac {i \tan ^{-1}(a x)^2}{a^3 c}+\frac {x \tan ^{-1}(a x)^2}{a^2 c}-\frac {\tan ^{-1}(a x)^3}{3 a^3 c}+\frac {2 \tan ^{-1}(a x) \log \left (\frac {2}{1+i a x}\right )}{a^3 c}+\frac {i \text {Li}_2\left (1-\frac {2}{1+i a x}\right )}{a^3 c}\\ \end {align*}
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Mathematica [A] time = 0.18, size = 69, normalized size = 0.70 \[ \frac {-i \text {Li}_2\left (-e^{2 i \tan ^{-1}(a x)}\right )-\frac {1}{3} \tan ^{-1}(a x) \left (\tan ^{-1}(a x)^2+(-3 a x+3 i) \tan ^{-1}(a x)-6 \log \left (1+e^{2 i \tan ^{-1}(a x)}\right )\right )}{a^3 c} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.76, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {x^{2} \arctan \left (a x\right )^{2}}{a^{2} c x^{2} + c}, x\right ) \]
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 [B] time = 0.09, size = 230, normalized size = 2.35 \[ \frac {x \arctan \left (a x \right )^{2}}{a^{2} c}-\frac {\arctan \left (a x \right )^{3}}{3 a^{3} c}-\frac {\arctan \left (a x \right ) \ln \left (a^{2} x^{2}+1\right )}{a^{3} c}-\frac {i \ln \left (a x -i\right ) \ln \left (a^{2} x^{2}+1\right )}{2 a^{3} c}+\frac {i \ln \left (a x -i\right )^{2}}{4 a^{3} c}+\frac {i \dilog \left (-\frac {i \left (a x +i\right )}{2}\right )}{2 a^{3} c}+\frac {i \ln \left (a x -i\right ) \ln \left (-\frac {i \left (a x +i\right )}{2}\right )}{2 a^{3} c}+\frac {i \ln \left (a x +i\right ) \ln \left (a^{2} x^{2}+1\right )}{2 a^{3} c}-\frac {i \ln \left (a x +i\right )^{2}}{4 a^{3} c}-\frac {i \dilog \left (\frac {i \left (a x -i\right )}{2}\right )}{2 a^{3} c}-\frac {i \ln \left (a x +i\right ) \ln \left (\frac {i \left (a x -i\right )}{2}\right )}{2 a^{3} c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
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 \[ \int \frac {x^2\,{\mathrm {atan}\left (a\,x\right )}^2}{c\,a^2\,x^2+c} \,d x \]
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 \[ \frac {\int \frac {x^{2} \operatorname {atan}^{2}{\left (a x \right )}}{a^{2} x^{2} + 1}\, dx}{c} \]
Verification of antiderivative is not currently implemented for this CAS.
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