Optimal. Leaf size=206 \[ -\frac {i b^2 \text {Li}_2\left (1-\frac {2}{i c x+1}\right ) \left (a+b \tan ^{-1}(c x)\right )}{c^3}+\frac {a b^2 x}{c^2}-\frac {b \left (a+b \tan ^{-1}(c x)\right )^2}{2 c^3}-\frac {i \left (a+b \tan ^{-1}(c x)\right )^3}{3 c^3}-\frac {b \log \left (\frac {2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )^2}{c^3}+\frac {1}{3} x^3 \left (a+b \tan ^{-1}(c x)\right )^3-\frac {b x^2 \left (a+b \tan ^{-1}(c x)\right )^2}{2 c}-\frac {b^3 \text {Li}_3\left (1-\frac {2}{i c x+1}\right )}{2 c^3}+\frac {b^3 x \tan ^{-1}(c x)}{c^2}-\frac {b^3 \log \left (c^2 x^2+1\right )}{2 c^3} \]
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Rubi [A] time = 0.43, antiderivative size = 206, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 9, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.643, Rules used = {4852, 4916, 4846, 260, 4884, 4920, 4854, 4994, 6610} \[ -\frac {i b^2 \text {PolyLog}\left (2,1-\frac {2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{c^3}-\frac {b^3 \text {PolyLog}\left (3,1-\frac {2}{1+i c x}\right )}{2 c^3}+\frac {a b^2 x}{c^2}-\frac {b \left (a+b \tan ^{-1}(c x)\right )^2}{2 c^3}-\frac {i \left (a+b \tan ^{-1}(c x)\right )^3}{3 c^3}-\frac {b \log \left (\frac {2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )^2}{c^3}-\frac {b x^2 \left (a+b \tan ^{-1}(c x)\right )^2}{2 c}+\frac {1}{3} x^3 \left (a+b \tan ^{-1}(c x)\right )^3-\frac {b^3 \log \left (c^2 x^2+1\right )}{2 c^3}+\frac {b^3 x \tan ^{-1}(c x)}{c^2} \]
Antiderivative was successfully verified.
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Rule 260
Rule 4846
Rule 4852
Rule 4854
Rule 4884
Rule 4916
Rule 4920
Rule 4994
Rule 6610
Rubi steps
\begin {align*} \int x^2 \left (a+b \tan ^{-1}(c x)\right )^3 \, dx &=\frac {1}{3} x^3 \left (a+b \tan ^{-1}(c x)\right )^3-(b c) \int \frac {x^3 \left (a+b \tan ^{-1}(c x)\right )^2}{1+c^2 x^2} \, dx\\ &=\frac {1}{3} x^3 \left (a+b \tan ^{-1}(c x)\right )^3-\frac {b \int x \left (a+b \tan ^{-1}(c x)\right )^2 \, dx}{c}+\frac {b \int \frac {x \left (a+b \tan ^{-1}(c x)\right )^2}{1+c^2 x^2} \, dx}{c}\\ &=-\frac {b x^2 \left (a+b \tan ^{-1}(c x)\right )^2}{2 c}-\frac {i \left (a+b \tan ^{-1}(c x)\right )^3}{3 c^3}+\frac {1}{3} x^3 \left (a+b \tan ^{-1}(c x)\right )^3+b^2 \int \frac {x^2 \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx-\frac {b \int \frac {\left (a+b \tan ^{-1}(c x)\right )^2}{i-c x} \, dx}{c^2}\\ &=-\frac {b x^2 \left (a+b \tan ^{-1}(c x)\right )^2}{2 c}-\frac {i \left (a+b \tan ^{-1}(c x)\right )^3}{3 c^3}+\frac {1}{3} x^3 \left (a+b \tan ^{-1}(c x)\right )^3-\frac {b \left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac {2}{1+i c x}\right )}{c^3}+\frac {b^2 \int \left (a+b \tan ^{-1}(c x)\right ) \, dx}{c^2}-\frac {b^2 \int \frac {a+b \tan ^{-1}(c x)}{1+c^2 x^2} \, dx}{c^2}+\frac {\left (2 b^2\right ) \int \frac {\left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1+i c x}\right )}{1+c^2 x^2} \, dx}{c^2}\\ &=\frac {a b^2 x}{c^2}-\frac {b \left (a+b \tan ^{-1}(c x)\right )^2}{2 c^3}-\frac {b x^2 \left (a+b \tan ^{-1}(c x)\right )^2}{2 c}-\frac {i \left (a+b \tan ^{-1}(c x)\right )^3}{3 c^3}+\frac {1}{3} x^3 \left (a+b \tan ^{-1}(c x)\right )^3-\frac {b \left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac {2}{1+i c x}\right )}{c^3}-\frac {i b^2 \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1+i c x}\right )}{c^3}+\frac {\left (i b^3\right ) \int \frac {\text {Li}_2\left (1-\frac {2}{1+i c x}\right )}{1+c^2 x^2} \, dx}{c^2}+\frac {b^3 \int \tan ^{-1}(c x) \, dx}{c^2}\\ &=\frac {a b^2 x}{c^2}+\frac {b^3 x \tan ^{-1}(c x)}{c^2}-\frac {b \left (a+b \tan ^{-1}(c x)\right )^2}{2 c^3}-\frac {b x^2 \left (a+b \tan ^{-1}(c x)\right )^2}{2 c}-\frac {i \left (a+b \tan ^{-1}(c x)\right )^3}{3 c^3}+\frac {1}{3} x^3 \left (a+b \tan ^{-1}(c x)\right )^3-\frac {b \left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac {2}{1+i c x}\right )}{c^3}-\frac {i b^2 \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1+i c x}\right )}{c^3}-\frac {b^3 \text {Li}_3\left (1-\frac {2}{1+i c x}\right )}{2 c^3}-\frac {b^3 \int \frac {x}{1+c^2 x^2} \, dx}{c}\\ &=\frac {a b^2 x}{c^2}+\frac {b^3 x \tan ^{-1}(c x)}{c^2}-\frac {b \left (a+b \tan ^{-1}(c x)\right )^2}{2 c^3}-\frac {b x^2 \left (a+b \tan ^{-1}(c x)\right )^2}{2 c}-\frac {i \left (a+b \tan ^{-1}(c x)\right )^3}{3 c^3}+\frac {1}{3} x^3 \left (a+b \tan ^{-1}(c x)\right )^3-\frac {b \left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac {2}{1+i c x}\right )}{c^3}-\frac {b^3 \log \left (1+c^2 x^2\right )}{2 c^3}-\frac {i b^2 \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1+i c x}\right )}{c^3}-\frac {b^3 \text {Li}_3\left (1-\frac {2}{1+i c x}\right )}{2 c^3}\\ \end {align*}
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Mathematica [A] time = 0.62, size = 269, normalized size = 1.31 \[ \frac {2 a^3 c^3 x^3+6 a^2 b c^3 x^3 \tan ^{-1}(c x)-3 a^2 b c^2 x^2+3 a^2 b \log \left (c^2 x^2+1\right )+6 a b^2 \left (\left (c^3 x^3+i\right ) \tan ^{-1}(c x)^2-\tan ^{-1}(c x) \left (c^2 x^2+2 \log \left (1+e^{2 i \tan ^{-1}(c x)}\right )+1\right )+i \text {Li}_2\left (-e^{2 i \tan ^{-1}(c x)}\right )+c x\right )+b^3 \left (2 c^3 x^3 \tan ^{-1}(c x)^3-3 \log \left (c^2 x^2+1\right )-3 c^2 x^2 \tan ^{-1}(c x)^2+6 i \tan ^{-1}(c x) \text {Li}_2\left (-e^{2 i \tan ^{-1}(c x)}\right )-3 \text {Li}_3\left (-e^{2 i \tan ^{-1}(c x)}\right )+2 i \tan ^{-1}(c x)^3-3 \tan ^{-1}(c x)^2+6 c x \tan ^{-1}(c x)-6 \tan ^{-1}(c x)^2 \log \left (1+e^{2 i \tan ^{-1}(c x)}\right )\right )}{6 c^3} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.42, size = 0, normalized size = 0.00 \[ {\rm integral}\left (b^{3} x^{2} \arctan \left (c x\right )^{3} + 3 \, a b^{2} x^{2} \arctan \left (c x\right )^{2} + 3 \, a^{2} b x^{2} \arctan \left (c x\right ) + a^{3} x^{2}, x\right ) \]
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 [C] time = 2.53, size = 2020, normalized size = 9.81 \[ \text {result too large to display} \]
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 \[ \frac {1}{24} \, b^{3} x^{3} \arctan \left (c x\right )^{3} - \frac {1}{32} \, b^{3} x^{3} \arctan \left (c x\right ) \log \left (c^{2} x^{2} + 1\right )^{2} + \frac {1}{3} \, a^{3} x^{3} + \frac {1}{2} \, {\left (2 \, x^{3} \arctan \left (c x\right ) - c {\left (\frac {x^{2}}{c^{2}} - \frac {\log \left (c^{2} x^{2} + 1\right )}{c^{4}}\right )}\right )} a^{2} b + \int \frac {4 \, b^{3} c^{2} x^{4} \arctan \left (c x\right ) \log \left (c^{2} x^{2} + 1\right ) + 28 \, {\left (b^{3} c^{2} x^{4} + b^{3} x^{2}\right )} \arctan \left (c x\right )^{3} + 4 \, {\left (24 \, a b^{2} c^{2} x^{4} - b^{3} c x^{3} + 24 \, a b^{2} x^{2}\right )} \arctan \left (c x\right )^{2} + {\left (b^{3} c x^{3} + 3 \, {\left (b^{3} c^{2} x^{4} + b^{3} x^{2}\right )} \arctan \left (c x\right )\right )} \log \left (c^{2} x^{2} + 1\right )^{2}}{32 \, {\left (c^{2} x^{2} + 1\right )}}\,{d x} \]
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 \[ \int x^2\,{\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )}^3 \,d x \]
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 \[ \int x^{2} \left (a + b \operatorname {atan}{\left (c x \right )}\right )^{3}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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