Optimal. Leaf size=156 \[ -\frac {2 i c \text {Li}_2\left (1-\frac {2}{i a x+1}\right )}{15 a^3}-\frac {2 i c \tan ^{-1}(a x)^2}{15 a^3}-\frac {c \tan ^{-1}(a x)}{30 a^3}-\frac {4 c \log \left (\frac {2}{1+i a x}\right ) \tan ^{-1}(a x)}{15 a^3}+\frac {1}{5} a^2 c x^5 \tan ^{-1}(a x)^2+\frac {c x}{30 a^2}-\frac {1}{10} a c x^4 \tan ^{-1}(a x)+\frac {1}{3} c x^3 \tan ^{-1}(a x)^2-\frac {2 c x^2 \tan ^{-1}(a x)}{15 a}+\frac {c x^3}{30} \]
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Rubi [A] time = 0.41, antiderivative size = 156, normalized size of antiderivative = 1.00, number of steps used = 24, number of rules used = 10, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {4950, 4852, 4916, 321, 203, 4920, 4854, 2402, 2315, 302} \[ -\frac {2 i c \text {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )}{15 a^3}+\frac {1}{5} a^2 c x^5 \tan ^{-1}(a x)^2+\frac {c x}{30 a^2}-\frac {2 i c \tan ^{-1}(a x)^2}{15 a^3}-\frac {c \tan ^{-1}(a x)}{30 a^3}-\frac {4 c \log \left (\frac {2}{1+i a x}\right ) \tan ^{-1}(a x)}{15 a^3}-\frac {1}{10} a c x^4 \tan ^{-1}(a x)+\frac {1}{3} c x^3 \tan ^{-1}(a x)^2-\frac {2 c x^2 \tan ^{-1}(a x)}{15 a}+\frac {c x^3}{30} \]
Antiderivative was successfully verified.
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Rule 203
Rule 302
Rule 321
Rule 2315
Rule 2402
Rule 4852
Rule 4854
Rule 4916
Rule 4920
Rule 4950
Rubi steps
\begin {align*} \int x^2 \left (c+a^2 c x^2\right ) \tan ^{-1}(a x)^2 \, dx &=c \int x^2 \tan ^{-1}(a x)^2 \, dx+\left (a^2 c\right ) \int x^4 \tan ^{-1}(a x)^2 \, dx\\ &=\frac {1}{3} c x^3 \tan ^{-1}(a x)^2+\frac {1}{5} a^2 c x^5 \tan ^{-1}(a x)^2-\frac {1}{3} (2 a c) \int \frac {x^3 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx-\frac {1}{5} \left (2 a^3 c\right ) \int \frac {x^5 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx\\ &=\frac {1}{3} c x^3 \tan ^{-1}(a x)^2+\frac {1}{5} a^2 c x^5 \tan ^{-1}(a x)^2-\frac {(2 c) \int x \tan ^{-1}(a x) \, dx}{3 a}+\frac {(2 c) \int \frac {x \tan ^{-1}(a x)}{1+a^2 x^2} \, dx}{3 a}-\frac {1}{5} (2 a c) \int x^3 \tan ^{-1}(a x) \, dx+\frac {1}{5} (2 a c) \int \frac {x^3 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx\\ &=-\frac {c x^2 \tan ^{-1}(a x)}{3 a}-\frac {1}{10} a c x^4 \tan ^{-1}(a x)-\frac {i c \tan ^{-1}(a x)^2}{3 a^3}+\frac {1}{3} c x^3 \tan ^{-1}(a x)^2+\frac {1}{5} a^2 c x^5 \tan ^{-1}(a x)^2+\frac {1}{3} c \int \frac {x^2}{1+a^2 x^2} \, dx-\frac {(2 c) \int \frac {\tan ^{-1}(a x)}{i-a x} \, dx}{3 a^2}+\frac {(2 c) \int x \tan ^{-1}(a x) \, dx}{5 a}-\frac {(2 c) \int \frac {x \tan ^{-1}(a x)}{1+a^2 x^2} \, dx}{5 a}+\frac {1}{10} \left (a^2 c\right ) \int \frac {x^4}{1+a^2 x^2} \, dx\\ &=\frac {c x}{3 a^2}-\frac {2 c x^2 \tan ^{-1}(a x)}{15 a}-\frac {1}{10} a c x^4 \tan ^{-1}(a x)-\frac {2 i c \tan ^{-1}(a x)^2}{15 a^3}+\frac {1}{3} c x^3 \tan ^{-1}(a x)^2+\frac {1}{5} a^2 c x^5 \tan ^{-1}(a x)^2-\frac {2 c \tan ^{-1}(a x) \log \left (\frac {2}{1+i a x}\right )}{3 a^3}-\frac {1}{5} c \int \frac {x^2}{1+a^2 x^2} \, dx-\frac {c \int \frac {1}{1+a^2 x^2} \, dx}{3 a^2}+\frac {(2 c) \int \frac {\tan ^{-1}(a x)}{i-a x} \, dx}{5 a^2}+\frac {(2 c) \int \frac {\log \left (\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx}{3 a^2}+\frac {1}{10} \left (a^2 c\right ) \int \left (-\frac {1}{a^4}+\frac {x^2}{a^2}+\frac {1}{a^4 \left (1+a^2 x^2\right )}\right ) \, dx\\ &=\frac {c x}{30 a^2}+\frac {c x^3}{30}-\frac {c \tan ^{-1}(a x)}{3 a^3}-\frac {2 c x^2 \tan ^{-1}(a x)}{15 a}-\frac {1}{10} a c x^4 \tan ^{-1}(a x)-\frac {2 i c \tan ^{-1}(a x)^2}{15 a^3}+\frac {1}{3} c x^3 \tan ^{-1}(a x)^2+\frac {1}{5} a^2 c x^5 \tan ^{-1}(a x)^2-\frac {4 c \tan ^{-1}(a x) \log \left (\frac {2}{1+i a x}\right )}{15 a^3}-\frac {(2 i c) \operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+i a x}\right )}{3 a^3}+\frac {c \int \frac {1}{1+a^2 x^2} \, dx}{10 a^2}+\frac {c \int \frac {1}{1+a^2 x^2} \, dx}{5 a^2}-\frac {(2 c) \int \frac {\log \left (\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx}{5 a^2}\\ &=\frac {c x}{30 a^2}+\frac {c x^3}{30}-\frac {c \tan ^{-1}(a x)}{30 a^3}-\frac {2 c x^2 \tan ^{-1}(a x)}{15 a}-\frac {1}{10} a c x^4 \tan ^{-1}(a x)-\frac {2 i c \tan ^{-1}(a x)^2}{15 a^3}+\frac {1}{3} c x^3 \tan ^{-1}(a x)^2+\frac {1}{5} a^2 c x^5 \tan ^{-1}(a x)^2-\frac {4 c \tan ^{-1}(a x) \log \left (\frac {2}{1+i a x}\right )}{15 a^3}-\frac {i c \text {Li}_2\left (1-\frac {2}{1+i a x}\right )}{3 a^3}+\frac {(2 i c) \operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+i a x}\right )}{5 a^3}\\ &=\frac {c x}{30 a^2}+\frac {c x^3}{30}-\frac {c \tan ^{-1}(a x)}{30 a^3}-\frac {2 c x^2 \tan ^{-1}(a x)}{15 a}-\frac {1}{10} a c x^4 \tan ^{-1}(a x)-\frac {2 i c \tan ^{-1}(a x)^2}{15 a^3}+\frac {1}{3} c x^3 \tan ^{-1}(a x)^2+\frac {1}{5} a^2 c x^5 \tan ^{-1}(a x)^2-\frac {4 c \tan ^{-1}(a x) \log \left (\frac {2}{1+i a x}\right )}{15 a^3}-\frac {2 i c \text {Li}_2\left (1-\frac {2}{1+i a x}\right )}{15 a^3}\\ \end {align*}
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Mathematica [A] time = 0.64, size = 104, normalized size = 0.67 \[ \frac {c \left (a^3 x^3+2 \left (3 a^5 x^5+5 a^3 x^3+2 i\right ) \tan ^{-1}(a x)^2-\tan ^{-1}(a x) \left (3 a^4 x^4+4 a^2 x^2+8 \log \left (1+e^{2 i \tan ^{-1}(a x)}\right )+1\right )+4 i \text {Li}_2\left (-e^{2 i \tan ^{-1}(a x)}\right )+a x\right )}{30 a^3} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.55, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (a^{2} c x^{4} + c x^{2}\right )} \arctan \left (a x\right )^{2}, x\right ) \]
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 = 0.12, size = 258, normalized size = 1.65 \[ \frac {a^{2} c \,x^{5} \arctan \left (a x \right )^{2}}{5}+\frac {c \,x^{3} \arctan \left (a x \right )^{2}}{3}-\frac {a c \,x^{4} \arctan \left (a x \right )}{10}-\frac {2 c \,x^{2} \arctan \left (a x \right )}{15 a}+\frac {2 c \arctan \left (a x \right ) \ln \left (a^{2} x^{2}+1\right )}{15 a^{3}}+\frac {c \,x^{3}}{30}+\frac {c x}{30 a^{2}}-\frac {c \arctan \left (a x \right )}{30 a^{3}}+\frac {i c \ln \left (a x -i\right ) \ln \left (a^{2} x^{2}+1\right )}{15 a^{3}}-\frac {i c \ln \left (a x -i\right )^{2}}{30 a^{3}}+\frac {i c \ln \left (a x +i\right )^{2}}{30 a^{3}}-\frac {i c \ln \left (a x +i\right ) \ln \left (a^{2} x^{2}+1\right )}{15 a^{3}}-\frac {i c \dilog \left (-\frac {i \left (a x +i\right )}{2}\right )}{15 a^{3}}+\frac {i c \ln \left (a x +i\right ) \ln \left (\frac {i \left (a x -i\right )}{2}\right )}{15 a^{3}}+\frac {i c \dilog \left (\frac {i \left (a x -i\right )}{2}\right )}{15 a^{3}}-\frac {i c \ln \left (a x -i\right ) \ln \left (-\frac {i \left (a x +i\right )}{2}\right )}{15 a^{3}} \]
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 \[ \frac {1}{60} \, {\left (3 \, a^{2} c x^{5} + 5 \, c x^{3}\right )} \arctan \left (a x\right )^{2} - \frac {1}{240} \, {\left (3 \, a^{2} c x^{5} + 5 \, c x^{3}\right )} \log \left (a^{2} x^{2} + 1\right )^{2} + \int \frac {180 \, {\left (a^{4} c x^{6} + 2 \, a^{2} c x^{4} + c x^{2}\right )} \arctan \left (a x\right )^{2} + 15 \, {\left (a^{4} c x^{6} + 2 \, a^{2} c x^{4} + c x^{2}\right )} \log \left (a^{2} x^{2} + 1\right )^{2} - 8 \, {\left (3 \, a^{3} c x^{5} + 5 \, a c x^{3}\right )} \arctan \left (a x\right ) + 4 \, {\left (3 \, a^{4} c x^{6} + 5 \, a^{2} c x^{4}\right )} \log \left (a^{2} x^{2} + 1\right )}{240 \, {\left (a^{2} x^{2} + 1\right )}}\,{d x} \]
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 \[ \int x^2\,{\mathrm {atan}\left (a\,x\right )}^2\,\left (c\,a^2\,x^2+c\right ) \,d x \]
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 \[ c \left (\int x^{2} \operatorname {atan}^{2}{\left (a x \right )}\, dx + \int a^{2} x^{4} \operatorname {atan}^{2}{\left (a x \right )}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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