Optimal. Leaf size=287 \[ \frac {1}{6} a^6 c^3 x^6 \tan ^{-1}(a x)^2-\frac {1}{15} a^5 c^3 x^5 \tan ^{-1}(a x)+\frac {1}{60} a^4 c^3 x^4+\frac {3}{4} a^4 c^3 x^4 \tan ^{-1}(a x)^2-\frac {7}{18} a^3 c^3 x^3 \tan ^{-1}(a x)+\frac {29}{180} a^2 c^3 x^2+\frac {34}{45} c^3 \log \left (a^2 x^2+1\right )+\frac {3}{2} a^2 c^3 x^2 \tan ^{-1}(a x)^2-\frac {1}{2} c^3 \text {Li}_3\left (1-\frac {2}{i a x+1}\right )+\frac {1}{2} c^3 \text {Li}_3\left (\frac {2}{i a x+1}-1\right )-i c^3 \text {Li}_2\left (1-\frac {2}{i a x+1}\right ) \tan ^{-1}(a x)+i c^3 \text {Li}_2\left (\frac {2}{i a x+1}-1\right ) \tan ^{-1}(a x)-\frac {11}{6} a c^3 x \tan ^{-1}(a x)+\frac {11}{12} c^3 \tan ^{-1}(a x)^2+2 c^3 \tan ^{-1}(a x)^2 \tanh ^{-1}\left (1-\frac {2}{1+i a x}\right ) \]
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Rubi [A] time = 0.74, antiderivative size = 287, normalized size of antiderivative = 1.00, number of steps used = 38, number of rules used = 12, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.546, Rules used = {4948, 4850, 4988, 4884, 4994, 6610, 4852, 4916, 4846, 260, 266, 43} \[ -\frac {1}{2} c^3 \text {PolyLog}\left (3,1-\frac {2}{1+i a x}\right )+\frac {1}{2} c^3 \text {PolyLog}\left (3,-1+\frac {2}{1+i a x}\right )-i c^3 \tan ^{-1}(a x) \text {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )+i c^3 \tan ^{-1}(a x) \text {PolyLog}\left (2,-1+\frac {2}{1+i a x}\right )+\frac {1}{60} a^4 c^3 x^4+\frac {29}{180} a^2 c^3 x^2+\frac {34}{45} c^3 \log \left (a^2 x^2+1\right )+\frac {1}{6} a^6 c^3 x^6 \tan ^{-1}(a x)^2-\frac {1}{15} a^5 c^3 x^5 \tan ^{-1}(a x)+\frac {3}{4} a^4 c^3 x^4 \tan ^{-1}(a x)^2-\frac {7}{18} a^3 c^3 x^3 \tan ^{-1}(a x)+\frac {3}{2} a^2 c^3 x^2 \tan ^{-1}(a x)^2-\frac {11}{6} a c^3 x \tan ^{-1}(a x)+\frac {11}{12} c^3 \tan ^{-1}(a x)^2+2 c^3 \tan ^{-1}(a x)^2 \tanh ^{-1}\left (1-\frac {2}{1+i a x}\right ) \]
Antiderivative was successfully verified.
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Rule 43
Rule 260
Rule 266
Rule 4846
Rule 4850
Rule 4852
Rule 4884
Rule 4916
Rule 4948
Rule 4988
Rule 4994
Rule 6610
Rubi steps
\begin {align*} \int \frac {\left (c+a^2 c x^2\right )^3 \tan ^{-1}(a x)^2}{x} \, dx &=\int \left (\frac {c^3 \tan ^{-1}(a x)^2}{x}+3 a^2 c^3 x \tan ^{-1}(a x)^2+3 a^4 c^3 x^3 \tan ^{-1}(a x)^2+a^6 c^3 x^5 \tan ^{-1}(a x)^2\right ) \, dx\\ &=c^3 \int \frac {\tan ^{-1}(a x)^2}{x} \, dx+\left (3 a^2 c^3\right ) \int x \tan ^{-1}(a x)^2 \, dx+\left (3 a^4 c^3\right ) \int x^3 \tan ^{-1}(a x)^2 \, dx+\left (a^6 c^3\right ) \int x^5 \tan ^{-1}(a x)^2 \, dx\\ &=\frac {3}{2} a^2 c^3 x^2 \tan ^{-1}(a x)^2+\frac {3}{4} a^4 c^3 x^4 \tan ^{-1}(a x)^2+\frac {1}{6} a^6 c^3 x^6 \tan ^{-1}(a x)^2+2 c^3 \tan ^{-1}(a x)^2 \tanh ^{-1}\left (1-\frac {2}{1+i a x}\right )-\left (4 a c^3\right ) \int \frac {\tan ^{-1}(a x) \tanh ^{-1}\left (1-\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx-\left (3 a^3 c^3\right ) \int \frac {x^2 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx-\frac {1}{2} \left (3 a^5 c^3\right ) \int \frac {x^4 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx-\frac {1}{3} \left (a^7 c^3\right ) \int \frac {x^6 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx\\ &=\frac {3}{2} a^2 c^3 x^2 \tan ^{-1}(a x)^2+\frac {3}{4} a^4 c^3 x^4 \tan ^{-1}(a x)^2+\frac {1}{6} a^6 c^3 x^6 \tan ^{-1}(a x)^2+2 c^3 \tan ^{-1}(a x)^2 \tanh ^{-1}\left (1-\frac {2}{1+i a x}\right )+\left (2 a c^3\right ) \int \frac {\tan ^{-1}(a x) \log \left (\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx-\left (2 a c^3\right ) \int \frac {\tan ^{-1}(a x) \log \left (2-\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx-\left (3 a c^3\right ) \int \tan ^{-1}(a x) \, dx+\left (3 a c^3\right ) \int \frac {\tan ^{-1}(a x)}{1+a^2 x^2} \, dx-\frac {1}{2} \left (3 a^3 c^3\right ) \int x^2 \tan ^{-1}(a x) \, dx+\frac {1}{2} \left (3 a^3 c^3\right ) \int \frac {x^2 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx-\frac {1}{3} \left (a^5 c^3\right ) \int x^4 \tan ^{-1}(a x) \, dx+\frac {1}{3} \left (a^5 c^3\right ) \int \frac {x^4 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx\\ &=-3 a c^3 x \tan ^{-1}(a x)-\frac {1}{2} a^3 c^3 x^3 \tan ^{-1}(a x)-\frac {1}{15} a^5 c^3 x^5 \tan ^{-1}(a x)+\frac {3}{2} c^3 \tan ^{-1}(a x)^2+\frac {3}{2} a^2 c^3 x^2 \tan ^{-1}(a x)^2+\frac {3}{4} a^4 c^3 x^4 \tan ^{-1}(a x)^2+\frac {1}{6} a^6 c^3 x^6 \tan ^{-1}(a x)^2+2 c^3 \tan ^{-1}(a x)^2 \tanh ^{-1}\left (1-\frac {2}{1+i a x}\right )-i c^3 \tan ^{-1}(a x) \text {Li}_2\left (1-\frac {2}{1+i a x}\right )+i c^3 \tan ^{-1}(a x) \text {Li}_2\left (-1+\frac {2}{1+i a x}\right )+\left (i a c^3\right ) \int \frac {\text {Li}_2\left (1-\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx-\left (i a c^3\right ) \int \frac {\text {Li}_2\left (-1+\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx+\frac {1}{2} \left (3 a c^3\right ) \int \tan ^{-1}(a x) \, dx-\frac {1}{2} \left (3 a c^3\right ) \int \frac {\tan ^{-1}(a x)}{1+a^2 x^2} \, dx+\left (3 a^2 c^3\right ) \int \frac {x}{1+a^2 x^2} \, dx+\frac {1}{3} \left (a^3 c^3\right ) \int x^2 \tan ^{-1}(a x) \, dx-\frac {1}{3} \left (a^3 c^3\right ) \int \frac {x^2 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx+\frac {1}{2} \left (a^4 c^3\right ) \int \frac {x^3}{1+a^2 x^2} \, dx+\frac {1}{15} \left (a^6 c^3\right ) \int \frac {x^5}{1+a^2 x^2} \, dx\\ &=-\frac {3}{2} a c^3 x \tan ^{-1}(a x)-\frac {7}{18} a^3 c^3 x^3 \tan ^{-1}(a x)-\frac {1}{15} a^5 c^3 x^5 \tan ^{-1}(a x)+\frac {3}{4} c^3 \tan ^{-1}(a x)^2+\frac {3}{2} a^2 c^3 x^2 \tan ^{-1}(a x)^2+\frac {3}{4} a^4 c^3 x^4 \tan ^{-1}(a x)^2+\frac {1}{6} a^6 c^3 x^6 \tan ^{-1}(a x)^2+2 c^3 \tan ^{-1}(a x)^2 \tanh ^{-1}\left (1-\frac {2}{1+i a x}\right )+\frac {3}{2} c^3 \log \left (1+a^2 x^2\right )-i c^3 \tan ^{-1}(a x) \text {Li}_2\left (1-\frac {2}{1+i a x}\right )+i c^3 \tan ^{-1}(a x) \text {Li}_2\left (-1+\frac {2}{1+i a x}\right )-\frac {1}{2} c^3 \text {Li}_3\left (1-\frac {2}{1+i a x}\right )+\frac {1}{2} c^3 \text {Li}_3\left (-1+\frac {2}{1+i a x}\right )-\frac {1}{3} \left (a c^3\right ) \int \tan ^{-1}(a x) \, dx+\frac {1}{3} \left (a c^3\right ) \int \frac {\tan ^{-1}(a x)}{1+a^2 x^2} \, dx-\frac {1}{2} \left (3 a^2 c^3\right ) \int \frac {x}{1+a^2 x^2} \, dx-\frac {1}{9} \left (a^4 c^3\right ) \int \frac {x^3}{1+a^2 x^2} \, dx+\frac {1}{4} \left (a^4 c^3\right ) \operatorname {Subst}\left (\int \frac {x}{1+a^2 x} \, dx,x,x^2\right )+\frac {1}{30} \left (a^6 c^3\right ) \operatorname {Subst}\left (\int \frac {x^2}{1+a^2 x} \, dx,x,x^2\right )\\ &=-\frac {11}{6} a c^3 x \tan ^{-1}(a x)-\frac {7}{18} a^3 c^3 x^3 \tan ^{-1}(a x)-\frac {1}{15} a^5 c^3 x^5 \tan ^{-1}(a x)+\frac {11}{12} c^3 \tan ^{-1}(a x)^2+\frac {3}{2} a^2 c^3 x^2 \tan ^{-1}(a x)^2+\frac {3}{4} a^4 c^3 x^4 \tan ^{-1}(a x)^2+\frac {1}{6} a^6 c^3 x^6 \tan ^{-1}(a x)^2+2 c^3 \tan ^{-1}(a x)^2 \tanh ^{-1}\left (1-\frac {2}{1+i a x}\right )+\frac {3}{4} c^3 \log \left (1+a^2 x^2\right )-i c^3 \tan ^{-1}(a x) \text {Li}_2\left (1-\frac {2}{1+i a x}\right )+i c^3 \tan ^{-1}(a x) \text {Li}_2\left (-1+\frac {2}{1+i a x}\right )-\frac {1}{2} c^3 \text {Li}_3\left (1-\frac {2}{1+i a x}\right )+\frac {1}{2} c^3 \text {Li}_3\left (-1+\frac {2}{1+i a x}\right )+\frac {1}{3} \left (a^2 c^3\right ) \int \frac {x}{1+a^2 x^2} \, dx-\frac {1}{18} \left (a^4 c^3\right ) \operatorname {Subst}\left (\int \frac {x}{1+a^2 x} \, dx,x,x^2\right )+\frac {1}{4} \left (a^4 c^3\right ) \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 {1}{30} \left (a^6 c^3\right ) \operatorname {Subst}\left (\int \left (-\frac {1}{a^4}+\frac {x}{a^2}+\frac {1}{a^4 \left (1+a^2 x\right )}\right ) \, dx,x,x^2\right )\\ &=\frac {13}{60} a^2 c^3 x^2+\frac {1}{60} a^4 c^3 x^4-\frac {11}{6} a c^3 x \tan ^{-1}(a x)-\frac {7}{18} a^3 c^3 x^3 \tan ^{-1}(a x)-\frac {1}{15} a^5 c^3 x^5 \tan ^{-1}(a x)+\frac {11}{12} c^3 \tan ^{-1}(a x)^2+\frac {3}{2} a^2 c^3 x^2 \tan ^{-1}(a x)^2+\frac {3}{4} a^4 c^3 x^4 \tan ^{-1}(a x)^2+\frac {1}{6} a^6 c^3 x^6 \tan ^{-1}(a x)^2+2 c^3 \tan ^{-1}(a x)^2 \tanh ^{-1}\left (1-\frac {2}{1+i a x}\right )+\frac {7}{10} c^3 \log \left (1+a^2 x^2\right )-i c^3 \tan ^{-1}(a x) \text {Li}_2\left (1-\frac {2}{1+i a x}\right )+i c^3 \tan ^{-1}(a x) \text {Li}_2\left (-1+\frac {2}{1+i a x}\right )-\frac {1}{2} c^3 \text {Li}_3\left (1-\frac {2}{1+i a x}\right )+\frac {1}{2} c^3 \text {Li}_3\left (-1+\frac {2}{1+i a x}\right )-\frac {1}{18} \left (a^4 c^3\right ) \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 {29}{180} a^2 c^3 x^2+\frac {1}{60} a^4 c^3 x^4-\frac {11}{6} a c^3 x \tan ^{-1}(a x)-\frac {7}{18} a^3 c^3 x^3 \tan ^{-1}(a x)-\frac {1}{15} a^5 c^3 x^5 \tan ^{-1}(a x)+\frac {11}{12} c^3 \tan ^{-1}(a x)^2+\frac {3}{2} a^2 c^3 x^2 \tan ^{-1}(a x)^2+\frac {3}{4} a^4 c^3 x^4 \tan ^{-1}(a x)^2+\frac {1}{6} a^6 c^3 x^6 \tan ^{-1}(a x)^2+2 c^3 \tan ^{-1}(a x)^2 \tanh ^{-1}\left (1-\frac {2}{1+i a x}\right )+\frac {34}{45} c^3 \log \left (1+a^2 x^2\right )-i c^3 \tan ^{-1}(a x) \text {Li}_2\left (1-\frac {2}{1+i a x}\right )+i c^3 \tan ^{-1}(a x) \text {Li}_2\left (-1+\frac {2}{1+i a x}\right )-\frac {1}{2} c^3 \text {Li}_3\left (1-\frac {2}{1+i a x}\right )+\frac {1}{2} c^3 \text {Li}_3\left (-1+\frac {2}{1+i a x}\right )\\ \end {align*}
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Mathematica [A] time = 0.57, size = 252, normalized size = 0.88 \[ \frac {1}{360} c^3 \left (60 a^6 x^6 \tan ^{-1}(a x)^2-24 a^5 x^5 \tan ^{-1}(a x)+6 a^4 x^4+270 a^4 x^4 \tan ^{-1}(a x)^2-140 a^3 x^3 \tan ^{-1}(a x)+58 a^2 x^2+272 \log \left (a^2 x^2+1\right )+540 a^2 x^2 \tan ^{-1}(a x)^2+360 i \tan ^{-1}(a x) \text {Li}_2\left (e^{-2 i \tan ^{-1}(a x)}\right )+360 i \tan ^{-1}(a x) \text {Li}_2\left (-e^{2 i \tan ^{-1}(a x)}\right )+180 \text {Li}_3\left (e^{-2 i \tan ^{-1}(a x)}\right )-180 \text {Li}_3\left (-e^{2 i \tan ^{-1}(a x)}\right )-660 a x \tan ^{-1}(a x)+240 i \tan ^{-1}(a x)^3+330 \tan ^{-1}(a x)^2+360 \tan ^{-1}(a x)^2 \log \left (1-e^{-2 i \tan ^{-1}(a x)}\right )-360 \tan ^{-1}(a x)^2 \log \left (1+e^{2 i \tan ^{-1}(a x)}\right )-15 i \pi ^3+52\right ) \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.42, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (a^{6} c^{3} x^{6} + 3 \, a^{4} c^{3} x^{4} + 3 \, a^{2} c^{3} x^{2} + c^{3}\right )} \arctan \left (a x\right )^{2}}{x}, x\right ) \]
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 = 11.00, size = 1217, normalized size = 4.24 \[ \text {result too large to display} \]
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 \[ 36 \, a^{8} c^{3} \int \frac {x^{8} \arctan \left (a x\right )^{2}}{48 \, {\left (a^{2} x^{3} + x\right )}}\,{d x} + 3 \, a^{8} c^{3} \int \frac {x^{8} \log \left (a^{2} x^{2} + 1\right )^{2}}{48 \, {\left (a^{2} x^{3} + x\right )}}\,{d x} + 2 \, a^{8} c^{3} \int \frac {x^{8} \log \left (a^{2} x^{2} + 1\right )}{48 \, {\left (a^{2} x^{3} + x\right )}}\,{d x} - 4 \, a^{7} c^{3} \int \frac {x^{7} \arctan \left (a x\right )}{48 \, {\left (a^{2} x^{3} + x\right )}}\,{d x} + 144 \, a^{6} c^{3} \int \frac {x^{6} \arctan \left (a x\right )^{2}}{48 \, {\left (a^{2} x^{3} + x\right )}}\,{d x} + 12 \, a^{6} c^{3} \int \frac {x^{6} \log \left (a^{2} x^{2} + 1\right )^{2}}{48 \, {\left (a^{2} x^{3} + x\right )}}\,{d x} + 9 \, a^{6} c^{3} \int \frac {x^{6} \log \left (a^{2} x^{2} + 1\right )}{48 \, {\left (a^{2} x^{3} + x\right )}}\,{d x} - 18 \, a^{5} c^{3} \int \frac {x^{5} \arctan \left (a x\right )}{48 \, {\left (a^{2} x^{3} + x\right )}}\,{d x} + 216 \, a^{4} c^{3} \int \frac {x^{4} \arctan \left (a x\right )^{2}}{48 \, {\left (a^{2} x^{3} + x\right )}}\,{d x} + 18 \, a^{4} c^{3} \int \frac {x^{4} \log \left (a^{2} x^{2} + 1\right )^{2}}{48 \, {\left (a^{2} x^{3} + x\right )}}\,{d x} + 18 \, a^{4} c^{3} \int \frac {x^{4} \log \left (a^{2} x^{2} + 1\right )}{48 \, {\left (a^{2} x^{3} + x\right )}}\,{d x} - 36 \, a^{3} c^{3} \int \frac {x^{3} \arctan \left (a x\right )}{48 \, {\left (a^{2} x^{3} + x\right )}}\,{d x} + 144 \, a^{2} c^{3} \int \frac {x^{2} \arctan \left (a x\right )^{2}}{48 \, {\left (a^{2} x^{3} + x\right )}}\,{d x} + \frac {1}{24} \, c^{3} \log \left (a^{2} x^{2} + 1\right )^{3} + 36 \, c^{3} \int \frac {\arctan \left (a x\right )^{2}}{48 \, {\left (a^{2} x^{3} + x\right )}}\,{d x} + 3 \, c^{3} \int \frac {\log \left (a^{2} x^{2} + 1\right )^{2}}{48 \, {\left (a^{2} x^{3} + x\right )}}\,{d x} + \frac {1}{48} \, {\left (2 \, a^{6} c^{3} x^{6} + 9 \, a^{4} c^{3} x^{4} + 18 \, a^{2} c^{3} x^{2}\right )} \arctan \left (a x\right )^{2} - \frac {1}{192} \, {\left (2 \, a^{6} c^{3} x^{6} + 9 \, a^{4} c^{3} x^{4} + 18 \, a^{2} c^{3} x^{2}\right )} \log \left (a^{2} x^{2} + 1\right )^{2} \]
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 \[ \int \frac {{\mathrm {atan}\left (a\,x\right )}^2\,{\left (c\,a^2\,x^2+c\right )}^3}{x} \,d x \]
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 \[ c^{3} \left (\int \frac {\operatorname {atan}^{2}{\left (a x \right )}}{x}\, dx + \int 3 a^{2} x \operatorname {atan}^{2}{\left (a x \right )}\, dx + \int 3 a^{4} x^{3} \operatorname {atan}^{2}{\left (a x \right )}\, dx + \int a^{6} x^{5} \operatorname {atan}^{2}{\left (a x \right )}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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