Optimal. Leaf size=128 \[ \frac {1}{3} c x \left (a^2 x^2+1\right ) \tan ^{-1}(a x)^2-\frac {c \left (a^2 x^2+1\right ) \tan ^{-1}(a x)}{3 a}+\frac {2 i c \text {Li}_2\left (1-\frac {2}{i a x+1}\right )}{3 a}+\frac {2 i c \tan ^{-1}(a x)^2}{3 a}+\frac {2}{3} c x \tan ^{-1}(a x)^2+\frac {4 c \log \left (\frac {2}{1+i a x}\right ) \tan ^{-1}(a x)}{3 a}+\frac {c x}{3} \]
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Rubi [A] time = 0.10, antiderivative size = 128, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.412, Rules used = {4880, 4846, 4920, 4854, 2402, 2315, 8} \[ \frac {2 i c \text {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )}{3 a}+\frac {1}{3} c x \left (a^2 x^2+1\right ) \tan ^{-1}(a x)^2-\frac {c \left (a^2 x^2+1\right ) \tan ^{-1}(a x)}{3 a}+\frac {2 i c \tan ^{-1}(a x)^2}{3 a}+\frac {2}{3} c x \tan ^{-1}(a x)^2+\frac {4 c \log \left (\frac {2}{1+i a x}\right ) \tan ^{-1}(a x)}{3 a}+\frac {c x}{3} \]
Antiderivative was successfully verified.
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Rule 8
Rule 2315
Rule 2402
Rule 4846
Rule 4854
Rule 4880
Rule 4920
Rubi steps
\begin {align*} \int \left (c+a^2 c x^2\right ) \tan ^{-1}(a x)^2 \, dx &=-\frac {c \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}{3 a}+\frac {1}{3} c x \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^2+\frac {1}{3} c \int 1 \, dx+\frac {1}{3} (2 c) \int \tan ^{-1}(a x)^2 \, dx\\ &=\frac {c x}{3}-\frac {c \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}{3 a}+\frac {2}{3} c x \tan ^{-1}(a x)^2+\frac {1}{3} c x \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^2-\frac {1}{3} (4 a c) \int \frac {x \tan ^{-1}(a x)}{1+a^2 x^2} \, dx\\ &=\frac {c x}{3}-\frac {c \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}{3 a}+\frac {2 i c \tan ^{-1}(a x)^2}{3 a}+\frac {2}{3} c x \tan ^{-1}(a x)^2+\frac {1}{3} c x \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^2+\frac {1}{3} (4 c) \int \frac {\tan ^{-1}(a x)}{i-a x} \, dx\\ &=\frac {c x}{3}-\frac {c \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}{3 a}+\frac {2 i c \tan ^{-1}(a x)^2}{3 a}+\frac {2}{3} c x \tan ^{-1}(a x)^2+\frac {1}{3} c x \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^2+\frac {4 c \tan ^{-1}(a x) \log \left (\frac {2}{1+i a x}\right )}{3 a}-\frac {1}{3} (4 c) \int \frac {\log \left (\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx\\ &=\frac {c x}{3}-\frac {c \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}{3 a}+\frac {2 i c \tan ^{-1}(a x)^2}{3 a}+\frac {2}{3} c x \tan ^{-1}(a x)^2+\frac {1}{3} c x \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^2+\frac {4 c \tan ^{-1}(a x) \log \left (\frac {2}{1+i a x}\right )}{3 a}+\frac {(4 i c) \operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+i a x}\right )}{3 a}\\ &=\frac {c x}{3}-\frac {c \left (1+a^2 x^2\right ) \tan ^{-1}(a x)}{3 a}+\frac {2 i c \tan ^{-1}(a x)^2}{3 a}+\frac {2}{3} c x \tan ^{-1}(a x)^2+\frac {1}{3} c x \left (1+a^2 x^2\right ) \tan ^{-1}(a x)^2+\frac {4 c \tan ^{-1}(a x) \log \left (\frac {2}{1+i a x}\right )}{3 a}+\frac {2 i c \text {Li}_2\left (1-\frac {2}{1+i a x}\right )}{3 a}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 82, normalized size = 0.64 \[ \frac {c \left (\left (a^3 x^3+3 a x-2 i\right ) \tan ^{-1}(a x)^2-\tan ^{-1}(a x) \left (a^2 x^2-4 \log \left (1+e^{2 i \tan ^{-1}(a x)}\right )+1\right )-2 i \text {Li}_2\left (-e^{2 i \tan ^{-1}(a x)}\right )+a x\right )}{3 a} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 1.28, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (a^{2} c x^{2} + c\right )} \arctan \left (a x\right )^{2}, x\right ) \]
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 [B] time = 0.10, size = 233, normalized size = 1.82 \[ \frac {a^{2} c \arctan \left (a x \right )^{2} x^{3}}{3}+c x \arctan \left (a x \right )^{2}-\frac {a c \arctan \left (a x \right ) x^{2}}{3}-\frac {2 c \arctan \left (a x \right ) \ln \left (a^{2} x^{2}+1\right )}{3 a}+\frac {c x}{3}-\frac {c \arctan \left (a x \right )}{3 a}-\frac {i c \ln \left (a x -i\right ) \ln \left (a^{2} x^{2}+1\right )}{3 a}+\frac {i c \ln \left (a x -i\right )^{2}}{6 a}+\frac {i c \dilog \left (-\frac {i \left (a x +i\right )}{2}\right )}{3 a}+\frac {i c \ln \left (a x -i\right ) \ln \left (-\frac {i \left (a x +i\right )}{2}\right )}{3 a}+\frac {i c \ln \left (a x +i\right ) \ln \left (a^{2} x^{2}+1\right )}{3 a}-\frac {i c \ln \left (a x +i\right )^{2}}{6 a}-\frac {i c \dilog \left (\frac {i \left (a x -i\right )}{2}\right )}{3 a}-\frac {i c \ln \left (a x +i\right ) \ln \left (\frac {i \left (a x -i\right )}{2}\right )}{3 a} \]
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 \[ 36 \, a^{4} c \int \frac {x^{4} \arctan \left (a x\right )^{2}}{48 \, {\left (a^{2} x^{2} + 1\right )}}\,{d x} + 3 \, a^{4} c \int \frac {x^{4} \log \left (a^{2} x^{2} + 1\right )^{2}}{48 \, {\left (a^{2} x^{2} + 1\right )}}\,{d x} + 4 \, a^{4} c \int \frac {x^{4} \log \left (a^{2} x^{2} + 1\right )}{48 \, {\left (a^{2} x^{2} + 1\right )}}\,{d x} - 8 \, a^{3} c \int \frac {x^{3} \arctan \left (a x\right )}{48 \, {\left (a^{2} x^{2} + 1\right )}}\,{d x} + 72 \, a^{2} c \int \frac {x^{2} \arctan \left (a x\right )^{2}}{48 \, {\left (a^{2} x^{2} + 1\right )}}\,{d x} + 6 \, a^{2} c \int \frac {x^{2} \log \left (a^{2} x^{2} + 1\right )^{2}}{48 \, {\left (a^{2} x^{2} + 1\right )}}\,{d x} + 12 \, a^{2} c \int \frac {x^{2} \log \left (a^{2} x^{2} + 1\right )}{48 \, {\left (a^{2} x^{2} + 1\right )}}\,{d x} + \frac {1}{12} \, {\left (a^{2} c x^{3} + 3 \, c x\right )} \arctan \left (a x\right )^{2} + \frac {c \arctan \left (a x\right )^{3}}{4 \, a} - 24 \, a c \int \frac {x \arctan \left (a x\right )}{48 \, {\left (a^{2} x^{2} + 1\right )}}\,{d x} - \frac {1}{48} \, {\left (a^{2} c x^{3} + 3 \, c x\right )} \log \left (a^{2} x^{2} + 1\right )^{2} + 3 \, c \int \frac {\log \left (a^{2} x^{2} + 1\right )^{2}}{48 \, {\left (a^{2} x^{2} + 1\right )}}\,{d x} \]
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 \[ \int {\mathrm {atan}\left (a\,x\right )}^2\,\left (c\,a^2\,x^2+c\right ) \,d x \]
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 \[ c \left (\int a^{2} x^{2} \operatorname {atan}^{2}{\left (a x \right )}\, dx + \int \operatorname {atan}^{2}{\left (a x \right )}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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