Optimal. Leaf size=211 \[ -\frac {c \text {Li}_3\left (1-\frac {2}{i a x+1}\right )}{5 a^3}-\frac {2 i c \text {Li}_2\left (1-\frac {2}{i a x+1}\right ) \tan ^{-1}(a x)}{5 a^3}-\frac {2 i c \tan ^{-1}(a x)^3}{15 a^3}-\frac {c \tan ^{-1}(a x)^2}{20 a^3}-\frac {2 c \log \left (\frac {2}{1+i a x}\right ) \tan ^{-1}(a x)^2}{5 a^3}+\frac {1}{5} a^2 c x^5 \tan ^{-1}(a x)^3+\frac {c x \tan ^{-1}(a x)}{10 a^2}-\frac {3}{20} a c x^4 \tan ^{-1}(a x)^2+\frac {1}{3} c x^3 \tan ^{-1}(a x)^3+\frac {1}{10} c x^3 \tan ^{-1}(a x)-\frac {c x^2}{20 a}-\frac {c x^2 \tan ^{-1}(a x)^2}{5 a} \]
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Rubi [A] time = 0.88, antiderivative size = 211, normalized size of antiderivative = 1.00, number of steps used = 34, number of rules used = 12, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {4950, 4852, 4916, 4846, 260, 4884, 4920, 4854, 4994, 6610, 266, 43} \[ -\frac {c \text {PolyLog}\left (3,1-\frac {2}{1+i a x}\right )}{5 a^3}-\frac {2 i c \tan ^{-1}(a x) \text {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )}{5 a^3}+\frac {1}{5} a^2 c x^5 \tan ^{-1}(a x)^3+\frac {c x \tan ^{-1}(a x)}{10 a^2}-\frac {2 i c \tan ^{-1}(a x)^3}{15 a^3}-\frac {c \tan ^{-1}(a x)^2}{20 a^3}-\frac {2 c \log \left (\frac {2}{1+i a x}\right ) \tan ^{-1}(a x)^2}{5 a^3}-\frac {c x^2}{20 a}-\frac {3}{20} a c x^4 \tan ^{-1}(a x)^2+\frac {1}{3} c x^3 \tan ^{-1}(a x)^3+\frac {1}{10} c x^3 \tan ^{-1}(a x)-\frac {c x^2 \tan ^{-1}(a x)^2}{5 a} \]
Antiderivative was successfully verified.
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Rule 43
Rule 260
Rule 266
Rule 4846
Rule 4852
Rule 4854
Rule 4884
Rule 4916
Rule 4920
Rule 4950
Rule 4994
Rule 6610
Rubi steps
\begin {align*} \int x^2 \left (c+a^2 c x^2\right ) \tan ^{-1}(a x)^3 \, dx &=c \int x^2 \tan ^{-1}(a x)^3 \, dx+\left (a^2 c\right ) \int x^4 \tan ^{-1}(a x)^3 \, dx\\ &=\frac {1}{3} c x^3 \tan ^{-1}(a x)^3+\frac {1}{5} a^2 c x^5 \tan ^{-1}(a x)^3-(a c) \int \frac {x^3 \tan ^{-1}(a x)^2}{1+a^2 x^2} \, dx-\frac {1}{5} \left (3 a^3 c\right ) \int \frac {x^5 \tan ^{-1}(a x)^2}{1+a^2 x^2} \, dx\\ &=\frac {1}{3} c x^3 \tan ^{-1}(a x)^3+\frac {1}{5} a^2 c x^5 \tan ^{-1}(a x)^3-\frac {c \int x \tan ^{-1}(a x)^2 \, dx}{a}+\frac {c \int \frac {x \tan ^{-1}(a x)^2}{1+a^2 x^2} \, dx}{a}-\frac {1}{5} (3 a c) \int x^3 \tan ^{-1}(a x)^2 \, dx+\frac {1}{5} (3 a c) \int \frac {x^3 \tan ^{-1}(a x)^2}{1+a^2 x^2} \, dx\\ &=-\frac {c x^2 \tan ^{-1}(a x)^2}{2 a}-\frac {3}{20} a c x^4 \tan ^{-1}(a x)^2-\frac {i c \tan ^{-1}(a x)^3}{3 a^3}+\frac {1}{3} c x^3 \tan ^{-1}(a x)^3+\frac {1}{5} a^2 c x^5 \tan ^{-1}(a x)^3+c \int \frac {x^2 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx-\frac {c \int \frac {\tan ^{-1}(a x)^2}{i-a x} \, dx}{a^2}+\frac {(3 c) \int x \tan ^{-1}(a x)^2 \, dx}{5 a}-\frac {(3 c) \int \frac {x \tan ^{-1}(a x)^2}{1+a^2 x^2} \, dx}{5 a}+\frac {1}{10} \left (3 a^2 c\right ) \int \frac {x^4 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx\\ &=-\frac {c x^2 \tan ^{-1}(a x)^2}{5 a}-\frac {3}{20} a c x^4 \tan ^{-1}(a x)^2-\frac {2 i c \tan ^{-1}(a x)^3}{15 a^3}+\frac {1}{3} c x^3 \tan ^{-1}(a x)^3+\frac {1}{5} a^2 c x^5 \tan ^{-1}(a x)^3-\frac {c \tan ^{-1}(a x)^2 \log \left (\frac {2}{1+i a x}\right )}{a^3}+\frac {1}{10} (3 c) \int x^2 \tan ^{-1}(a x) \, dx-\frac {1}{10} (3 c) \int \frac {x^2 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx-\frac {1}{5} (3 c) \int \frac {x^2 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx+\frac {(3 c) \int \frac {\tan ^{-1}(a x)^2}{i-a x} \, dx}{5 a^2}+\frac {c \int \tan ^{-1}(a x) \, dx}{a^2}-\frac {c \int \frac {\tan ^{-1}(a x)}{1+a^2 x^2} \, dx}{a^2}+\frac {(2 c) \int \frac {\tan ^{-1}(a x) \log \left (\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx}{a^2}\\ &=\frac {c x \tan ^{-1}(a x)}{a^2}+\frac {1}{10} c x^3 \tan ^{-1}(a x)-\frac {c \tan ^{-1}(a x)^2}{2 a^3}-\frac {c x^2 \tan ^{-1}(a x)^2}{5 a}-\frac {3}{20} a c x^4 \tan ^{-1}(a x)^2-\frac {2 i c \tan ^{-1}(a x)^3}{15 a^3}+\frac {1}{3} c x^3 \tan ^{-1}(a x)^3+\frac {1}{5} a^2 c x^5 \tan ^{-1}(a x)^3-\frac {2 c \tan ^{-1}(a x)^2 \log \left (\frac {2}{1+i a x}\right )}{5 a^3}-\frac {i c \tan ^{-1}(a x) \text {Li}_2\left (1-\frac {2}{1+i a x}\right )}{a^3}+\frac {(i c) \int \frac {\text {Li}_2\left (1-\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx}{a^2}-\frac {(3 c) \int \tan ^{-1}(a x) \, dx}{10 a^2}+\frac {(3 c) \int \frac {\tan ^{-1}(a x)}{1+a^2 x^2} \, dx}{10 a^2}-\frac {(3 c) \int \tan ^{-1}(a x) \, dx}{5 a^2}+\frac {(3 c) \int \frac {\tan ^{-1}(a x)}{1+a^2 x^2} \, dx}{5 a^2}-\frac {(6 c) \int \frac {\tan ^{-1}(a x) \log \left (\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx}{5 a^2}-\frac {c \int \frac {x}{1+a^2 x^2} \, dx}{a}-\frac {1}{10} (a c) \int \frac {x^3}{1+a^2 x^2} \, dx\\ &=\frac {c x \tan ^{-1}(a x)}{10 a^2}+\frac {1}{10} c x^3 \tan ^{-1}(a x)-\frac {c \tan ^{-1}(a x)^2}{20 a^3}-\frac {c x^2 \tan ^{-1}(a x)^2}{5 a}-\frac {3}{20} a c x^4 \tan ^{-1}(a x)^2-\frac {2 i c \tan ^{-1}(a x)^3}{15 a^3}+\frac {1}{3} c x^3 \tan ^{-1}(a x)^3+\frac {1}{5} a^2 c x^5 \tan ^{-1}(a x)^3-\frac {2 c \tan ^{-1}(a x)^2 \log \left (\frac {2}{1+i a x}\right )}{5 a^3}-\frac {c \log \left (1+a^2 x^2\right )}{2 a^3}-\frac {2 i c \tan ^{-1}(a x) \text {Li}_2\left (1-\frac {2}{1+i a x}\right )}{5 a^3}-\frac {c \text {Li}_3\left (1-\frac {2}{1+i a x}\right )}{2 a^3}-\frac {(3 i c) \int \frac {\text {Li}_2\left (1-\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx}{5 a^2}+\frac {(3 c) \int \frac {x}{1+a^2 x^2} \, dx}{10 a}+\frac {(3 c) \int \frac {x}{1+a^2 x^2} \, dx}{5 a}-\frac {1}{20} (a c) \operatorname {Subst}\left (\int \frac {x}{1+a^2 x} \, dx,x,x^2\right )\\ &=\frac {c x \tan ^{-1}(a x)}{10 a^2}+\frac {1}{10} c x^3 \tan ^{-1}(a x)-\frac {c \tan ^{-1}(a x)^2}{20 a^3}-\frac {c x^2 \tan ^{-1}(a x)^2}{5 a}-\frac {3}{20} a c x^4 \tan ^{-1}(a x)^2-\frac {2 i c \tan ^{-1}(a x)^3}{15 a^3}+\frac {1}{3} c x^3 \tan ^{-1}(a x)^3+\frac {1}{5} a^2 c x^5 \tan ^{-1}(a x)^3-\frac {2 c \tan ^{-1}(a x)^2 \log \left (\frac {2}{1+i a x}\right )}{5 a^3}-\frac {c \log \left (1+a^2 x^2\right )}{20 a^3}-\frac {2 i c \tan ^{-1}(a x) \text {Li}_2\left (1-\frac {2}{1+i a x}\right )}{5 a^3}-\frac {c \text {Li}_3\left (1-\frac {2}{1+i a x}\right )}{5 a^3}-\frac {1}{20} (a c) \operatorname {Subst}\left (\int \left (\frac {1}{a^2}-\frac {1}{a^2 \left (1+a^2 x\right )}\right ) \, dx,x,x^2\right )\\ &=-\frac {c x^2}{20 a}+\frac {c x \tan ^{-1}(a x)}{10 a^2}+\frac {1}{10} c x^3 \tan ^{-1}(a x)-\frac {c \tan ^{-1}(a x)^2}{20 a^3}-\frac {c x^2 \tan ^{-1}(a x)^2}{5 a}-\frac {3}{20} a c x^4 \tan ^{-1}(a x)^2-\frac {2 i c \tan ^{-1}(a x)^3}{15 a^3}+\frac {1}{3} c x^3 \tan ^{-1}(a x)^3+\frac {1}{5} a^2 c x^5 \tan ^{-1}(a x)^3-\frac {2 c \tan ^{-1}(a x)^2 \log \left (\frac {2}{1+i a x}\right )}{5 a^3}-\frac {2 i c \tan ^{-1}(a x) \text {Li}_2\left (1-\frac {2}{1+i a x}\right )}{5 a^3}-\frac {c \text {Li}_3\left (1-\frac {2}{1+i a x}\right )}{5 a^3}\\ \end {align*}
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Mathematica [A] time = 0.57, size = 171, normalized size = 0.81 \[ \frac {c \left (12 a^5 x^5 \tan ^{-1}(a x)^3-9 a^4 x^4 \tan ^{-1}(a x)^2+20 a^3 x^3 \tan ^{-1}(a x)^3+6 a^3 x^3 \tan ^{-1}(a x)-3 a^2 x^2-12 a^2 x^2 \tan ^{-1}(a x)^2+24 i \tan ^{-1}(a x) \text {Li}_2\left (-e^{2 i \tan ^{-1}(a x)}\right )-12 \text {Li}_3\left (-e^{2 i \tan ^{-1}(a x)}\right )+6 a x \tan ^{-1}(a x)+8 i \tan ^{-1}(a x)^3-3 \tan ^{-1}(a x)^2-24 \tan ^{-1}(a x)^2 \log \left (1+e^{2 i \tan ^{-1}(a x)}\right )-3\right )}{60 a^3} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.55, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (a^{2} c x^{4} + c x^{2}\right )} \arctan \left (a x\right )^{3}, x\right ) \]
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 = 5.27, size = 2555, normalized size = 12.11 \[ \text {Expression too large to display} \]
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 \[ \frac {1}{120} \, {\left (3 \, a^{2} c x^{5} + 5 \, c x^{3}\right )} \arctan \left (a x\right )^{3} - \frac {1}{160} \, {\left (3 \, a^{2} c x^{5} + 5 \, c x^{3}\right )} \arctan \left (a x\right ) \log \left (a^{2} x^{2} + 1\right )^{2} + \int \frac {140 \, {\left (a^{4} c x^{6} + 2 \, a^{2} c x^{4} + c x^{2}\right )} \arctan \left (a x\right )^{3} - 4 \, {\left (3 \, a^{3} c x^{5} + 5 \, a c x^{3}\right )} \arctan \left (a x\right )^{2} + 4 \, {\left (3 \, a^{4} c x^{6} + 5 \, a^{2} c x^{4}\right )} \arctan \left (a x\right ) \log \left (a^{2} x^{2} + 1\right ) + {\left (3 \, a^{3} c x^{5} + 5 \, a c x^{3} + 15 \, {\left (a^{4} c x^{6} + 2 \, a^{2} c x^{4} + c x^{2}\right )} \arctan \left (a x\right )\right )} \log \left (a^{2} x^{2} + 1\right )^{2}}{160 \, {\left (a^{2} x^{2} + 1\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int x^2\,{\mathrm {atan}\left (a\,x\right )}^3\,\left (c\,a^2\,x^2+c\right ) \,d x \]
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 \[ c \left (\int x^{2} \operatorname {atan}^{3}{\left (a x \right )}\, dx + \int a^{2} x^{4} \operatorname {atan}^{3}{\left (a x \right )}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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