Integrand size = 22, antiderivative size = 381 \[ \int x^3 \left (c+a^2 c x^2\right )^3 \arctan (a x)^3 \, dx=\frac {389 c^3 x}{12600 a^3}-\frac {17 c^3 x^3}{9450 a}-\frac {1}{252} a c^3 x^5-\frac {1}{840} a^3 c^3 x^7-\frac {389 c^3 \arctan (a x)}{12600 a^4}-\frac {107 c^3 x^2 \arctan (a x)}{4200 a^2}+\frac {53 c^3 x^4 \arctan (a x)}{2100}+\frac {71 a^2 c^3 x^6 \arctan (a x)}{2520}+\frac {1}{120} a^4 c^3 x^8 \arctan (a x)+\frac {26 i c^3 \arctan (a x)^2}{525 a^4}+\frac {3 c^3 x \arctan (a x)^2}{40 a^3}-\frac {c^3 x^3 \arctan (a x)^2}{40 a}-\frac {27}{200} a c^3 x^5 \arctan (a x)^2-\frac {33}{280} a^3 c^3 x^7 \arctan (a x)^2-\frac {1}{30} a^5 c^3 x^9 \arctan (a x)^2-\frac {c^3 \arctan (a x)^3}{40 a^4}+\frac {1}{4} c^3 x^4 \arctan (a x)^3+\frac {1}{2} a^2 c^3 x^6 \arctan (a x)^3+\frac {3}{8} a^4 c^3 x^8 \arctan (a x)^3+\frac {1}{10} a^6 c^3 x^{10} \arctan (a x)^3+\frac {52 c^3 \arctan (a x) \log \left (\frac {2}{1+i a x}\right )}{525 a^4}+\frac {26 i c^3 \operatorname {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )}{525 a^4} \]
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Time = 2.67 (sec) , antiderivative size = 381, normalized size of antiderivative = 1.00, number of steps used = 184, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.545, Rules used = {5068, 4946, 5036, 327, 209, 5040, 4964, 2449, 2352, 4930, 5004, 308} \[ \int x^3 \left (c+a^2 c x^2\right )^3 \arctan (a x)^3 \, dx=\frac {1}{10} a^6 c^3 x^{10} \arctan (a x)^3-\frac {1}{30} a^5 c^3 x^9 \arctan (a x)^2+\frac {3}{8} a^4 c^3 x^8 \arctan (a x)^3+\frac {1}{120} a^4 c^3 x^8 \arctan (a x)-\frac {c^3 \arctan (a x)^3}{40 a^4}+\frac {26 i c^3 \arctan (a x)^2}{525 a^4}-\frac {389 c^3 \arctan (a x)}{12600 a^4}+\frac {52 c^3 \arctan (a x) \log \left (\frac {2}{1+i a x}\right )}{525 a^4}+\frac {26 i c^3 \operatorname {PolyLog}\left (2,1-\frac {2}{i a x+1}\right )}{525 a^4}-\frac {33}{280} a^3 c^3 x^7 \arctan (a x)^2+\frac {3 c^3 x \arctan (a x)^2}{40 a^3}-\frac {1}{840} a^3 c^3 x^7+\frac {389 c^3 x}{12600 a^3}+\frac {1}{2} a^2 c^3 x^6 \arctan (a x)^3+\frac {71 a^2 c^3 x^6 \arctan (a x)}{2520}-\frac {107 c^3 x^2 \arctan (a x)}{4200 a^2}-\frac {27}{200} a c^3 x^5 \arctan (a x)^2+\frac {1}{4} c^3 x^4 \arctan (a x)^3+\frac {53 c^3 x^4 \arctan (a x)}{2100}-\frac {c^3 x^3 \arctan (a x)^2}{40 a}-\frac {1}{252} a c^3 x^5-\frac {17 c^3 x^3}{9450 a} \]
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Rule 209
Rule 308
Rule 327
Rule 2352
Rule 2449
Rule 4930
Rule 4946
Rule 4964
Rule 5004
Rule 5036
Rule 5040
Rule 5068
Rubi steps \begin{align*} \text {integral}& = \int \left (c^3 x^3 \arctan (a x)^3+3 a^2 c^3 x^5 \arctan (a x)^3+3 a^4 c^3 x^7 \arctan (a x)^3+a^6 c^3 x^9 \arctan (a x)^3\right ) \, dx \\ & = c^3 \int x^3 \arctan (a x)^3 \, dx+\left (3 a^2 c^3\right ) \int x^5 \arctan (a x)^3 \, dx+\left (3 a^4 c^3\right ) \int x^7 \arctan (a x)^3 \, dx+\left (a^6 c^3\right ) \int x^9 \arctan (a x)^3 \, dx \\ & = \frac {1}{4} c^3 x^4 \arctan (a x)^3+\frac {1}{2} a^2 c^3 x^6 \arctan (a x)^3+\frac {3}{8} a^4 c^3 x^8 \arctan (a x)^3+\frac {1}{10} a^6 c^3 x^{10} \arctan (a x)^3-\frac {1}{4} \left (3 a c^3\right ) \int \frac {x^4 \arctan (a x)^2}{1+a^2 x^2} \, dx-\frac {1}{2} \left (3 a^3 c^3\right ) \int \frac {x^6 \arctan (a x)^2}{1+a^2 x^2} \, dx-\frac {1}{8} \left (9 a^5 c^3\right ) \int \frac {x^8 \arctan (a x)^2}{1+a^2 x^2} \, dx-\frac {1}{10} \left (3 a^7 c^3\right ) \int \frac {x^{10} \arctan (a x)^2}{1+a^2 x^2} \, dx \\ & = \frac {1}{4} c^3 x^4 \arctan (a x)^3+\frac {1}{2} a^2 c^3 x^6 \arctan (a x)^3+\frac {3}{8} a^4 c^3 x^8 \arctan (a x)^3+\frac {1}{10} a^6 c^3 x^{10} \arctan (a x)^3-\frac {\left (3 c^3\right ) \int x^2 \arctan (a x)^2 \, dx}{4 a}+\frac {\left (3 c^3\right ) \int \frac {x^2 \arctan (a x)^2}{1+a^2 x^2} \, dx}{4 a}-\frac {1}{2} \left (3 a c^3\right ) \int x^4 \arctan (a x)^2 \, dx+\frac {1}{2} \left (3 a c^3\right ) \int \frac {x^4 \arctan (a x)^2}{1+a^2 x^2} \, dx-\frac {1}{8} \left (9 a^3 c^3\right ) \int x^6 \arctan (a x)^2 \, dx+\frac {1}{8} \left (9 a^3 c^3\right ) \int \frac {x^6 \arctan (a x)^2}{1+a^2 x^2} \, dx-\frac {1}{10} \left (3 a^5 c^3\right ) \int x^8 \arctan (a x)^2 \, dx+\frac {1}{10} \left (3 a^5 c^3\right ) \int \frac {x^8 \arctan (a x)^2}{1+a^2 x^2} \, dx \\ & = -\frac {c^3 x^3 \arctan (a x)^2}{4 a}-\frac {3}{10} a c^3 x^5 \arctan (a x)^2-\frac {9}{56} a^3 c^3 x^7 \arctan (a x)^2-\frac {1}{30} a^5 c^3 x^9 \arctan (a x)^2+\frac {1}{4} c^3 x^4 \arctan (a x)^3+\frac {1}{2} a^2 c^3 x^6 \arctan (a x)^3+\frac {3}{8} a^4 c^3 x^8 \arctan (a x)^3+\frac {1}{10} a^6 c^3 x^{10} \arctan (a x)^3+\frac {1}{2} c^3 \int \frac {x^3 \arctan (a x)}{1+a^2 x^2} \, dx+\frac {\left (3 c^3\right ) \int \arctan (a x)^2 \, dx}{4 a^3}-\frac {\left (3 c^3\right ) \int \frac {\arctan (a x)^2}{1+a^2 x^2} \, dx}{4 a^3}+\frac {\left (3 c^3\right ) \int x^2 \arctan (a x)^2 \, dx}{2 a}-\frac {\left (3 c^3\right ) \int \frac {x^2 \arctan (a x)^2}{1+a^2 x^2} \, dx}{2 a}+\frac {1}{8} \left (9 a c^3\right ) \int x^4 \arctan (a x)^2 \, dx-\frac {1}{8} \left (9 a c^3\right ) \int \frac {x^4 \arctan (a x)^2}{1+a^2 x^2} \, dx+\frac {1}{5} \left (3 a^2 c^3\right ) \int \frac {x^5 \arctan (a x)}{1+a^2 x^2} \, dx+\frac {1}{10} \left (3 a^3 c^3\right ) \int x^6 \arctan (a x)^2 \, dx-\frac {1}{10} \left (3 a^3 c^3\right ) \int \frac {x^6 \arctan (a x)^2}{1+a^2 x^2} \, dx+\frac {1}{28} \left (9 a^4 c^3\right ) \int \frac {x^7 \arctan (a x)}{1+a^2 x^2} \, dx+\frac {1}{15} \left (a^6 c^3\right ) \int \frac {x^9 \arctan (a x)}{1+a^2 x^2} \, dx \\ & = \frac {3 c^3 x \arctan (a x)^2}{4 a^3}+\frac {c^3 x^3 \arctan (a x)^2}{4 a}-\frac {3}{40} a c^3 x^5 \arctan (a x)^2-\frac {33}{280} a^3 c^3 x^7 \arctan (a x)^2-\frac {1}{30} a^5 c^3 x^9 \arctan (a x)^2-\frac {c^3 \arctan (a x)^3}{4 a^4}+\frac {1}{4} c^3 x^4 \arctan (a x)^3+\frac {1}{2} a^2 c^3 x^6 \arctan (a x)^3+\frac {3}{8} a^4 c^3 x^8 \arctan (a x)^3+\frac {1}{10} a^6 c^3 x^{10} \arctan (a x)^3+\frac {1}{5} \left (3 c^3\right ) \int x^3 \arctan (a x) \, dx-\frac {1}{5} \left (3 c^3\right ) \int \frac {x^3 \arctan (a x)}{1+a^2 x^2} \, dx-c^3 \int \frac {x^3 \arctan (a x)}{1+a^2 x^2} \, dx-\frac {\left (3 c^3\right ) \int \arctan (a x)^2 \, dx}{2 a^3}+\frac {\left (3 c^3\right ) \int \frac {\arctan (a x)^2}{1+a^2 x^2} \, dx}{2 a^3}+\frac {c^3 \int x \arctan (a x) \, dx}{2 a^2}-\frac {c^3 \int \frac {x \arctan (a x)}{1+a^2 x^2} \, dx}{2 a^2}-\frac {\left (3 c^3\right ) \int \frac {x \arctan (a x)}{1+a^2 x^2} \, dx}{2 a^2}-\frac {\left (9 c^3\right ) \int x^2 \arctan (a x)^2 \, dx}{8 a}+\frac {\left (9 c^3\right ) \int \frac {x^2 \arctan (a x)^2}{1+a^2 x^2} \, dx}{8 a}-\frac {1}{10} \left (3 a c^3\right ) \int x^4 \arctan (a x)^2 \, dx+\frac {1}{10} \left (3 a c^3\right ) \int \frac {x^4 \arctan (a x)^2}{1+a^2 x^2} \, dx+\frac {1}{28} \left (9 a^2 c^3\right ) \int x^5 \arctan (a x) \, dx-\frac {1}{28} \left (9 a^2 c^3\right ) \int \frac {x^5 \arctan (a x)}{1+a^2 x^2} \, dx-\frac {1}{20} \left (9 a^2 c^3\right ) \int \frac {x^5 \arctan (a x)}{1+a^2 x^2} \, dx+\frac {1}{15} \left (a^4 c^3\right ) \int x^7 \arctan (a x) \, dx-\frac {1}{15} \left (a^4 c^3\right ) \int \frac {x^7 \arctan (a x)}{1+a^2 x^2} \, dx-\frac {1}{35} \left (3 a^4 c^3\right ) \int \frac {x^7 \arctan (a x)}{1+a^2 x^2} \, dx \\ & = \frac {c^3 x^2 \arctan (a x)}{4 a^2}+\frac {3}{20} c^3 x^4 \arctan (a x)+\frac {3}{56} a^2 c^3 x^6 \arctan (a x)+\frac {1}{120} a^4 c^3 x^8 \arctan (a x)+\frac {i c^3 \arctan (a x)^2}{a^4}-\frac {3 c^3 x \arctan (a x)^2}{4 a^3}-\frac {c^3 x^3 \arctan (a x)^2}{8 a}-\frac {27}{200} a c^3 x^5 \arctan (a x)^2-\frac {33}{280} a^3 c^3 x^7 \arctan (a x)^2-\frac {1}{30} a^5 c^3 x^9 \arctan (a x)^2+\frac {c^3 \arctan (a x)^3}{4 a^4}+\frac {1}{4} c^3 x^4 \arctan (a x)^3+\frac {1}{2} a^2 c^3 x^6 \arctan (a x)^3+\frac {3}{8} a^4 c^3 x^8 \arctan (a x)^3+\frac {1}{10} a^6 c^3 x^{10} \arctan (a x)^3-\frac {1}{28} \left (9 c^3\right ) \int x^3 \arctan (a x) \, dx+\frac {1}{28} \left (9 c^3\right ) \int \frac {x^3 \arctan (a x)}{1+a^2 x^2} \, dx-\frac {1}{20} \left (9 c^3\right ) \int x^3 \arctan (a x) \, dx+\frac {1}{20} \left (9 c^3\right ) \int \frac {x^3 \arctan (a x)}{1+a^2 x^2} \, dx+\frac {1}{4} \left (3 c^3\right ) \int \frac {x^3 \arctan (a x)}{1+a^2 x^2} \, dx+\frac {c^3 \int \frac {\arctan (a x)}{i-a x} \, dx}{2 a^3}+\frac {\left (9 c^3\right ) \int \arctan (a x)^2 \, dx}{8 a^3}-\frac {\left (9 c^3\right ) \int \frac {\arctan (a x)^2}{1+a^2 x^2} \, dx}{8 a^3}+\frac {\left (3 c^3\right ) \int \frac {\arctan (a x)}{i-a x} \, dx}{2 a^3}-\frac {\left (3 c^3\right ) \int x \arctan (a x) \, dx}{5 a^2}+\frac {\left (3 c^3\right ) \int \frac {x \arctan (a x)}{1+a^2 x^2} \, dx}{5 a^2}-\frac {c^3 \int x \arctan (a x) \, dx}{a^2}+\frac {c^3 \int \frac {x \arctan (a x)}{1+a^2 x^2} \, dx}{a^2}+\frac {\left (3 c^3\right ) \int \frac {x \arctan (a x)}{1+a^2 x^2} \, dx}{a^2}-\frac {c^3 \int \frac {x^2}{1+a^2 x^2} \, dx}{4 a}+\frac {\left (3 c^3\right ) \int x^2 \arctan (a x)^2 \, dx}{10 a}-\frac {\left (3 c^3\right ) \int \frac {x^2 \arctan (a x)^2}{1+a^2 x^2} \, dx}{10 a}-\frac {1}{20} \left (3 a c^3\right ) \int \frac {x^4}{1+a^2 x^2} \, dx-\frac {1}{15} \left (a^2 c^3\right ) \int x^5 \arctan (a x) \, dx+\frac {1}{15} \left (a^2 c^3\right ) \int \frac {x^5 \arctan (a x)}{1+a^2 x^2} \, dx-\frac {1}{35} \left (3 a^2 c^3\right ) \int x^5 \arctan (a x) \, dx+\frac {1}{35} \left (3 a^2 c^3\right ) \int \frac {x^5 \arctan (a x)}{1+a^2 x^2} \, dx+\frac {1}{25} \left (3 a^2 c^3\right ) \int \frac {x^5 \arctan (a x)}{1+a^2 x^2} \, dx-\frac {1}{56} \left (3 a^3 c^3\right ) \int \frac {x^6}{1+a^2 x^2} \, dx-\frac {1}{120} \left (a^5 c^3\right ) \int \frac {x^8}{1+a^2 x^2} \, dx \\ & = \text {Too large to display} \\ \end{align*}
Time = 1.70 (sec) , antiderivative size = 191, normalized size of antiderivative = 0.50 \[ \int x^3 \left (c+a^2 c x^2\right )^3 \arctan (a x)^3 \, dx=\frac {c^3 \left (-a x \left (-1167+68 a^2 x^2+150 a^4 x^4+45 a^6 x^6\right )-9 \left (208 i-315 a x+105 a^3 x^3+567 a^5 x^5+495 a^7 x^7+140 a^9 x^9\right ) \arctan (a x)^2+945 \left (1+a^2 x^2\right )^4 \left (-1+4 a^2 x^2\right ) \arctan (a x)^3+3 \arctan (a x) \left (-389-321 a^2 x^2+318 a^4 x^4+355 a^6 x^6+105 a^8 x^8+1248 \log \left (1+e^{2 i \arctan (a x)}\right )\right )-1872 i \operatorname {PolyLog}\left (2,-e^{2 i \arctan (a x)}\right )\right )}{37800 a^4} \]
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Time = 4.95 (sec) , antiderivative size = 383, normalized size of antiderivative = 1.01
method | result | size |
derivativedivides | \(\frac {\frac {c^{3} \arctan \left (a x \right )^{3} a^{10} x^{10}}{10}+\frac {3 c^{3} \arctan \left (a x \right )^{3} a^{8} x^{8}}{8}+\frac {a^{6} c^{3} x^{6} \arctan \left (a x \right )^{3}}{2}+\frac {a^{4} c^{3} x^{4} \arctan \left (a x \right )^{3}}{4}-\frac {c^{3} \arctan \left (a x \right )^{3}}{40}-\frac {3 c^{3} \left (\frac {4 \arctan \left (a x \right )^{2} a^{9} x^{9}}{9}+\frac {11 \arctan \left (a x \right )^{2} a^{7} x^{7}}{7}+\frac {9 a^{5} \arctan \left (a x \right )^{2} x^{5}}{5}+\frac {a^{3} \arctan \left (a x \right )^{2} x^{3}}{3}-a \arctan \left (a x \right )^{2} x -\frac {\arctan \left (a x \right ) a^{8} x^{8}}{9}-\frac {71 a^{6} \arctan \left (a x \right ) x^{6}}{189}-\frac {106 \arctan \left (a x \right ) a^{4} x^{4}}{315}+\frac {107 a^{2} \arctan \left (a x \right ) x^{2}}{315}+\frac {208 \arctan \left (a x \right ) \ln \left (a^{2} x^{2}+1\right )}{315}+\frac {a^{7} x^{7}}{63}+\frac {10 a^{5} x^{5}}{189}+\frac {68 a^{3} x^{3}}{2835}-\frac {389 a x}{945}+\frac {389 \arctan \left (a x \right )}{945}+\frac {104 i \left (\ln \left (a x -i\right ) \ln \left (a^{2} x^{2}+1\right )-\operatorname {dilog}\left (-\frac {i \left (a x +i\right )}{2}\right )-\ln \left (a x -i\right ) \ln \left (-\frac {i \left (a x +i\right )}{2}\right )-\frac {\ln \left (a x -i\right )^{2}}{2}\right )}{315}-\frac {104 i \left (\ln \left (a x +i\right ) \ln \left (a^{2} x^{2}+1\right )-\operatorname {dilog}\left (\frac {i \left (a x -i\right )}{2}\right )-\ln \left (a x +i\right ) \ln \left (\frac {i \left (a x -i\right )}{2}\right )-\frac {\ln \left (a x +i\right )^{2}}{2}\right )}{315}\right )}{40}}{a^{4}}\) | \(383\) |
default | \(\frac {\frac {c^{3} \arctan \left (a x \right )^{3} a^{10} x^{10}}{10}+\frac {3 c^{3} \arctan \left (a x \right )^{3} a^{8} x^{8}}{8}+\frac {a^{6} c^{3} x^{6} \arctan \left (a x \right )^{3}}{2}+\frac {a^{4} c^{3} x^{4} \arctan \left (a x \right )^{3}}{4}-\frac {c^{3} \arctan \left (a x \right )^{3}}{40}-\frac {3 c^{3} \left (\frac {4 \arctan \left (a x \right )^{2} a^{9} x^{9}}{9}+\frac {11 \arctan \left (a x \right )^{2} a^{7} x^{7}}{7}+\frac {9 a^{5} \arctan \left (a x \right )^{2} x^{5}}{5}+\frac {a^{3} \arctan \left (a x \right )^{2} x^{3}}{3}-a \arctan \left (a x \right )^{2} x -\frac {\arctan \left (a x \right ) a^{8} x^{8}}{9}-\frac {71 a^{6} \arctan \left (a x \right ) x^{6}}{189}-\frac {106 \arctan \left (a x \right ) a^{4} x^{4}}{315}+\frac {107 a^{2} \arctan \left (a x \right ) x^{2}}{315}+\frac {208 \arctan \left (a x \right ) \ln \left (a^{2} x^{2}+1\right )}{315}+\frac {a^{7} x^{7}}{63}+\frac {10 a^{5} x^{5}}{189}+\frac {68 a^{3} x^{3}}{2835}-\frac {389 a x}{945}+\frac {389 \arctan \left (a x \right )}{945}+\frac {104 i \left (\ln \left (a x -i\right ) \ln \left (a^{2} x^{2}+1\right )-\operatorname {dilog}\left (-\frac {i \left (a x +i\right )}{2}\right )-\ln \left (a x -i\right ) \ln \left (-\frac {i \left (a x +i\right )}{2}\right )-\frac {\ln \left (a x -i\right )^{2}}{2}\right )}{315}-\frac {104 i \left (\ln \left (a x +i\right ) \ln \left (a^{2} x^{2}+1\right )-\operatorname {dilog}\left (\frac {i \left (a x -i\right )}{2}\right )-\ln \left (a x +i\right ) \ln \left (\frac {i \left (a x -i\right )}{2}\right )-\frac {\ln \left (a x +i\right )^{2}}{2}\right )}{315}\right )}{40}}{a^{4}}\) | \(383\) |
parts | \(\frac {a^{6} c^{3} x^{10} \arctan \left (a x \right )^{3}}{10}+\frac {3 a^{4} c^{3} x^{8} \arctan \left (a x \right )^{3}}{8}+\frac {a^{2} c^{3} x^{6} \arctan \left (a x \right )^{3}}{2}+\frac {c^{3} x^{4} \arctan \left (a x \right )^{3}}{4}-\frac {3 c^{3} \left (\frac {4 a^{5} \arctan \left (a x \right )^{2} x^{9}}{9}+\frac {11 a^{3} \arctan \left (a x \right )^{2} x^{7}}{7}+\frac {9 a \arctan \left (a x \right )^{2} x^{5}}{5}+\frac {\arctan \left (a x \right )^{2} x^{3}}{3 a}-\frac {\arctan \left (a x \right )^{2} x}{a^{3}}+\frac {\arctan \left (a x \right )^{3}}{a^{4}}-\frac {2 \left (\frac {35 \arctan \left (a x \right ) a^{8} x^{8}}{2}+\frac {355 a^{6} \arctan \left (a x \right ) x^{6}}{6}+53 \arctan \left (a x \right ) a^{4} x^{4}-\frac {107 a^{2} \arctan \left (a x \right ) x^{2}}{2}-104 \arctan \left (a x \right ) \ln \left (a^{2} x^{2}+1\right )-\frac {5 a^{7} x^{7}}{2}-\frac {25 a^{5} x^{5}}{3}-\frac {34 a^{3} x^{3}}{9}+\frac {389 a x}{6}-\frac {389 \arctan \left (a x \right )}{6}-52 i \left (\ln \left (a x -i\right ) \ln \left (a^{2} x^{2}+1\right )-\operatorname {dilog}\left (-\frac {i \left (a x +i\right )}{2}\right )-\ln \left (a x -i\right ) \ln \left (-\frac {i \left (a x +i\right )}{2}\right )-\frac {\ln \left (a x -i\right )^{2}}{2}\right )+52 i \left (\ln \left (a x +i\right ) \ln \left (a^{2} x^{2}+1\right )-\operatorname {dilog}\left (\frac {i \left (a x -i\right )}{2}\right )-\ln \left (a x +i\right ) \ln \left (\frac {i \left (a x -i\right )}{2}\right )-\frac {\ln \left (a x +i\right )^{2}}{2}\right )+105 \arctan \left (a x \right )^{3}\right )}{315 a^{4}}\right )}{40}\) | \(389\) |
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\[ \int x^3 \left (c+a^2 c x^2\right )^3 \arctan (a x)^3 \, dx=\int { {\left (a^{2} c x^{2} + c\right )}^{3} x^{3} \arctan \left (a x\right )^{3} \,d x } \]
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\[ \int x^3 \left (c+a^2 c x^2\right )^3 \arctan (a x)^3 \, dx=c^{3} \left (\int x^{3} \operatorname {atan}^{3}{\left (a x \right )}\, dx + \int 3 a^{2} x^{5} \operatorname {atan}^{3}{\left (a x \right )}\, dx + \int 3 a^{4} x^{7} \operatorname {atan}^{3}{\left (a x \right )}\, dx + \int a^{6} x^{9} \operatorname {atan}^{3}{\left (a x \right )}\, dx\right ) \]
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\[ \int x^3 \left (c+a^2 c x^2\right )^3 \arctan (a x)^3 \, dx=\int { {\left (a^{2} c x^{2} + c\right )}^{3} x^{3} \arctan \left (a x\right )^{3} \,d x } \]
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\[ \int x^3 \left (c+a^2 c x^2\right )^3 \arctan (a x)^3 \, dx=\int { {\left (a^{2} c x^{2} + c\right )}^{3} x^{3} \arctan \left (a x\right )^{3} \,d x } \]
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Timed out. \[ \int x^3 \left (c+a^2 c x^2\right )^3 \arctan (a x)^3 \, dx=\int x^3\,{\mathrm {atan}\left (a\,x\right )}^3\,{\left (c\,a^2\,x^2+c\right )}^3 \,d x \]
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