Optimal. Leaf size=284 \[ \frac {1}{3} a^4 c^2 x^3 \tan ^{-1}(a x)^3-\frac {1}{2} a^3 c^2 x^2 \tan ^{-1}(a x)^2-\frac {1}{2} a c^2 \log \left (a^2 x^2+1\right )+2 a^2 c^2 x \tan ^{-1}(a x)^3+a^2 c^2 x \tan ^{-1}(a x)+\frac {3}{2} a c^2 \text {Li}_3\left (\frac {2}{1-i a x}-1\right )+\frac {5}{2} a c^2 \text {Li}_3\left (1-\frac {2}{i a x+1}\right )-3 i a c^2 \text {Li}_2\left (\frac {2}{1-i a x}-1\right ) \tan ^{-1}(a x)+5 i a c^2 \text {Li}_2\left (1-\frac {2}{i a x+1}\right ) \tan ^{-1}(a x)+\frac {2}{3} i a c^2 \tan ^{-1}(a x)^3-\frac {1}{2} a c^2 \tan ^{-1}(a x)^2-\frac {c^2 \tan ^{-1}(a x)^3}{x}+5 a c^2 \log \left (\frac {2}{1+i a x}\right ) \tan ^{-1}(a x)^2+3 a c^2 \log \left (2-\frac {2}{1-i a x}\right ) \tan ^{-1}(a x)^2 \]
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Rubi [A] time = 0.75, antiderivative size = 284, normalized size of antiderivative = 1.00, number of steps used = 23, number of rules used = 13, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.591, Rules used = {4948, 4846, 4920, 4854, 4884, 4994, 6610, 4852, 4924, 4868, 4992, 4916, 260} \[ \frac {3}{2} a c^2 \text {PolyLog}\left (3,-1+\frac {2}{1-i a x}\right )+\frac {5}{2} a c^2 \text {PolyLog}\left (3,1-\frac {2}{1+i a x}\right )-3 i a c^2 \tan ^{-1}(a x) \text {PolyLog}\left (2,-1+\frac {2}{1-i a x}\right )+5 i a c^2 \tan ^{-1}(a x) \text {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )-\frac {1}{2} a c^2 \log \left (a^2 x^2+1\right )+\frac {1}{3} a^4 c^2 x^3 \tan ^{-1}(a x)^3-\frac {1}{2} a^3 c^2 x^2 \tan ^{-1}(a x)^2+2 a^2 c^2 x \tan ^{-1}(a x)^3+a^2 c^2 x \tan ^{-1}(a x)+\frac {2}{3} i a c^2 \tan ^{-1}(a x)^3-\frac {1}{2} a c^2 \tan ^{-1}(a x)^2-\frac {c^2 \tan ^{-1}(a x)^3}{x}+5 a c^2 \log \left (\frac {2}{1+i a x}\right ) \tan ^{-1}(a x)^2+3 a c^2 \log \left (2-\frac {2}{1-i a x}\right ) \tan ^{-1}(a x)^2 \]
Antiderivative was successfully verified.
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Rule 260
Rule 4846
Rule 4852
Rule 4854
Rule 4868
Rule 4884
Rule 4916
Rule 4920
Rule 4924
Rule 4948
Rule 4992
Rule 4994
Rule 6610
Rubi steps
\begin {align*} \int \frac {\left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)^3}{x^2} \, dx &=\int \left (2 a^2 c^2 \tan ^{-1}(a x)^3+\frac {c^2 \tan ^{-1}(a x)^3}{x^2}+a^4 c^2 x^2 \tan ^{-1}(a x)^3\right ) \, dx\\ &=c^2 \int \frac {\tan ^{-1}(a x)^3}{x^2} \, dx+\left (2 a^2 c^2\right ) \int \tan ^{-1}(a x)^3 \, dx+\left (a^4 c^2\right ) \int x^2 \tan ^{-1}(a x)^3 \, dx\\ &=-\frac {c^2 \tan ^{-1}(a x)^3}{x}+2 a^2 c^2 x \tan ^{-1}(a x)^3+\frac {1}{3} a^4 c^2 x^3 \tan ^{-1}(a x)^3+\left (3 a c^2\right ) \int \frac {\tan ^{-1}(a x)^2}{x \left (1+a^2 x^2\right )} \, dx-\left (6 a^3 c^2\right ) \int \frac {x \tan ^{-1}(a x)^2}{1+a^2 x^2} \, dx-\left (a^5 c^2\right ) \int \frac {x^3 \tan ^{-1}(a x)^2}{1+a^2 x^2} \, dx\\ &=i a c^2 \tan ^{-1}(a x)^3-\frac {c^2 \tan ^{-1}(a x)^3}{x}+2 a^2 c^2 x \tan ^{-1}(a x)^3+\frac {1}{3} a^4 c^2 x^3 \tan ^{-1}(a x)^3+\left (3 i a c^2\right ) \int \frac {\tan ^{-1}(a x)^2}{x (i+a x)} \, dx+\left (6 a^2 c^2\right ) \int \frac {\tan ^{-1}(a x)^2}{i-a x} \, dx-\left (a^3 c^2\right ) \int x \tan ^{-1}(a x)^2 \, dx+\left (a^3 c^2\right ) \int \frac {x \tan ^{-1}(a x)^2}{1+a^2 x^2} \, dx\\ &=-\frac {1}{2} a^3 c^2 x^2 \tan ^{-1}(a x)^2+\frac {2}{3} i a c^2 \tan ^{-1}(a x)^3-\frac {c^2 \tan ^{-1}(a x)^3}{x}+2 a^2 c^2 x \tan ^{-1}(a x)^3+\frac {1}{3} a^4 c^2 x^3 \tan ^{-1}(a x)^3+6 a c^2 \tan ^{-1}(a x)^2 \log \left (\frac {2}{1+i a x}\right )+3 a c^2 \tan ^{-1}(a x)^2 \log \left (2-\frac {2}{1-i a x}\right )-\left (a^2 c^2\right ) \int \frac {\tan ^{-1}(a x)^2}{i-a x} \, dx-\left (6 a^2 c^2\right ) \int \frac {\tan ^{-1}(a x) \log \left (2-\frac {2}{1-i a x}\right )}{1+a^2 x^2} \, dx-\left (12 a^2 c^2\right ) \int \frac {\tan ^{-1}(a x) \log \left (\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx+\left (a^4 c^2\right ) \int \frac {x^2 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx\\ &=-\frac {1}{2} a^3 c^2 x^2 \tan ^{-1}(a x)^2+\frac {2}{3} i a c^2 \tan ^{-1}(a x)^3-\frac {c^2 \tan ^{-1}(a x)^3}{x}+2 a^2 c^2 x \tan ^{-1}(a x)^3+\frac {1}{3} a^4 c^2 x^3 \tan ^{-1}(a x)^3+5 a c^2 \tan ^{-1}(a x)^2 \log \left (\frac {2}{1+i a x}\right )+3 a c^2 \tan ^{-1}(a x)^2 \log \left (2-\frac {2}{1-i a x}\right )-3 i a c^2 \tan ^{-1}(a x) \text {Li}_2\left (-1+\frac {2}{1-i a x}\right )+6 i a c^2 \tan ^{-1}(a x) \text {Li}_2\left (1-\frac {2}{1+i a x}\right )+\left (3 i a^2 c^2\right ) \int \frac {\text {Li}_2\left (-1+\frac {2}{1-i a x}\right )}{1+a^2 x^2} \, dx-\left (6 i a^2 c^2\right ) \int \frac {\text {Li}_2\left (1-\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx+\left (a^2 c^2\right ) \int \tan ^{-1}(a x) \, dx-\left (a^2 c^2\right ) \int \frac {\tan ^{-1}(a x)}{1+a^2 x^2} \, dx+\left (2 a^2 c^2\right ) \int \frac {\tan ^{-1}(a x) \log \left (\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx\\ &=a^2 c^2 x \tan ^{-1}(a x)-\frac {1}{2} a c^2 \tan ^{-1}(a x)^2-\frac {1}{2} a^3 c^2 x^2 \tan ^{-1}(a x)^2+\frac {2}{3} i a c^2 \tan ^{-1}(a x)^3-\frac {c^2 \tan ^{-1}(a x)^3}{x}+2 a^2 c^2 x \tan ^{-1}(a x)^3+\frac {1}{3} a^4 c^2 x^3 \tan ^{-1}(a x)^3+5 a c^2 \tan ^{-1}(a x)^2 \log \left (\frac {2}{1+i a x}\right )+3 a c^2 \tan ^{-1}(a x)^2 \log \left (2-\frac {2}{1-i a x}\right )-3 i a c^2 \tan ^{-1}(a x) \text {Li}_2\left (-1+\frac {2}{1-i a x}\right )+5 i a c^2 \tan ^{-1}(a x) \text {Li}_2\left (1-\frac {2}{1+i a x}\right )+\frac {3}{2} a c^2 \text {Li}_3\left (-1+\frac {2}{1-i a x}\right )+3 a c^2 \text {Li}_3\left (1-\frac {2}{1+i a x}\right )+\left (i a^2 c^2\right ) \int \frac {\text {Li}_2\left (1-\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx-\left (a^3 c^2\right ) \int \frac {x}{1+a^2 x^2} \, dx\\ &=a^2 c^2 x \tan ^{-1}(a x)-\frac {1}{2} a c^2 \tan ^{-1}(a x)^2-\frac {1}{2} a^3 c^2 x^2 \tan ^{-1}(a x)^2+\frac {2}{3} i a c^2 \tan ^{-1}(a x)^3-\frac {c^2 \tan ^{-1}(a x)^3}{x}+2 a^2 c^2 x \tan ^{-1}(a x)^3+\frac {1}{3} a^4 c^2 x^3 \tan ^{-1}(a x)^3+5 a c^2 \tan ^{-1}(a x)^2 \log \left (\frac {2}{1+i a x}\right )-\frac {1}{2} a c^2 \log \left (1+a^2 x^2\right )+3 a c^2 \tan ^{-1}(a x)^2 \log \left (2-\frac {2}{1-i a x}\right )-3 i a c^2 \tan ^{-1}(a x) \text {Li}_2\left (-1+\frac {2}{1-i a x}\right )+5 i a c^2 \tan ^{-1}(a x) \text {Li}_2\left (1-\frac {2}{1+i a x}\right )+\frac {3}{2} a c^2 \text {Li}_3\left (-1+\frac {2}{1-i a x}\right )+\frac {5}{2} a c^2 \text {Li}_3\left (1-\frac {2}{1+i a x}\right )\\ \end {align*}
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Mathematica [A] time = 0.40, size = 246, normalized size = 0.87 \[ \frac {c^2 \left (8 a^4 x^4 \tan ^{-1}(a x)^3-12 a^3 x^3 \tan ^{-1}(a x)^2-12 a x \log \left (a^2 x^2+1\right )+48 a^2 x^2 \tan ^{-1}(a x)^3+24 a^2 x^2 \tan ^{-1}(a x)+72 i a x \tan ^{-1}(a x) \text {Li}_2\left (e^{-2 i \tan ^{-1}(a x)}\right )-120 i a x \tan ^{-1}(a x) \text {Li}_2\left (-e^{2 i \tan ^{-1}(a x)}\right )+36 a x \text {Li}_3\left (e^{-2 i \tan ^{-1}(a x)}\right )+60 a x \text {Li}_3\left (-e^{2 i \tan ^{-1}(a x)}\right )-3 i \pi ^3 a x-16 i a x \tan ^{-1}(a x)^3-12 a x \tan ^{-1}(a x)^2-24 \tan ^{-1}(a x)^3+72 a x \tan ^{-1}(a x)^2 \log \left (1-e^{-2 i \tan ^{-1}(a x)}\right )+120 a x \tan ^{-1}(a x)^2 \log \left (1+e^{2 i \tan ^{-1}(a x)}\right )\right )}{24 x} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.73, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (a^{4} c^{2} x^{4} + 2 \, a^{2} c^{2} x^{2} + c^{2}\right )} \arctan \left (a x\right )^{3}}{x^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [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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maple [C] time = 7.84, size = 5486, normalized size = 19.32 \[ \text {output too large to display} \]
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 = -0.00 \[ \int \frac {{\mathrm {atan}\left (a\,x\right )}^3\,{\left (c\,a^2\,x^2+c\right )}^2}{x^2} \,d x \]
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 \[ c^{2} \left (\int 2 a^{2} \operatorname {atan}^{3}{\left (a x \right )}\, dx + \int \frac {\operatorname {atan}^{3}{\left (a x \right )}}{x^{2}}\, dx + \int a^{4} x^{2} \operatorname {atan}^{3}{\left (a x \right )}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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