Optimal. Leaf size=381 \[ \frac {1}{10} a^6 c^3 x^{10} \tan ^{-1}(a x)^3-\frac {1}{30} a^5 c^3 x^9 \tan ^{-1}(a x)^2+\frac {26 i c^3 \text {Li}_2\left (1-\frac {2}{i a x+1}\right )}{525 a^4}+\frac {3}{8} a^4 c^3 x^8 \tan ^{-1}(a x)^3+\frac {1}{120} a^4 c^3 x^8 \tan ^{-1}(a x)-\frac {c^3 \tan ^{-1}(a x)^3}{40 a^4}+\frac {26 i c^3 \tan ^{-1}(a x)^2}{525 a^4}-\frac {389 c^3 \tan ^{-1}(a x)}{12600 a^4}+\frac {52 c^3 \log \left (\frac {2}{1+i a x}\right ) \tan ^{-1}(a x)}{525 a^4}-\frac {1}{840} a^3 c^3 x^7-\frac {33}{280} a^3 c^3 x^7 \tan ^{-1}(a x)^2+\frac {389 c^3 x}{12600 a^3}+\frac {3 c^3 x \tan ^{-1}(a x)^2}{40 a^3}+\frac {1}{2} a^2 c^3 x^6 \tan ^{-1}(a x)^3+\frac {71 a^2 c^3 x^6 \tan ^{-1}(a x)}{2520}-\frac {107 c^3 x^2 \tan ^{-1}(a x)}{4200 a^2}-\frac {1}{252} a c^3 x^5-\frac {27}{200} a c^3 x^5 \tan ^{-1}(a x)^2+\frac {1}{4} c^3 x^4 \tan ^{-1}(a x)^3+\frac {53 c^3 x^4 \tan ^{-1}(a x)}{2100}-\frac {17 c^3 x^3}{9450 a}-\frac {c^3 x^3 \tan ^{-1}(a x)^2}{40 a} \]
[Out]
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Rubi [A] time = 3.73, antiderivative size = 381, normalized size of antiderivative = 1.00, number of steps used = 184, number of rules used = 12, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.546, Rules used = {4948, 4852, 4916, 321, 203, 4920, 4854, 2402, 2315, 4846, 4884, 302} \[ \frac {26 i c^3 \text {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )}{525 a^4}-\frac {1}{840} a^3 c^3 x^7+\frac {1}{10} a^6 c^3 x^{10} \tan ^{-1}(a x)^3-\frac {1}{30} a^5 c^3 x^9 \tan ^{-1}(a x)^2+\frac {3}{8} a^4 c^3 x^8 \tan ^{-1}(a x)^3+\frac {1}{120} a^4 c^3 x^8 \tan ^{-1}(a x)-\frac {33}{280} a^3 c^3 x^7 \tan ^{-1}(a x)^2+\frac {1}{2} a^2 c^3 x^6 \tan ^{-1}(a x)^3+\frac {71 a^2 c^3 x^6 \tan ^{-1}(a x)}{2520}-\frac {107 c^3 x^2 \tan ^{-1}(a x)}{4200 a^2}+\frac {389 c^3 x}{12600 a^3}+\frac {3 c^3 x \tan ^{-1}(a x)^2}{40 a^3}-\frac {c^3 \tan ^{-1}(a x)^3}{40 a^4}+\frac {26 i c^3 \tan ^{-1}(a x)^2}{525 a^4}-\frac {389 c^3 \tan ^{-1}(a x)}{12600 a^4}+\frac {52 c^3 \log \left (\frac {2}{1+i a x}\right ) \tan ^{-1}(a x)}{525 a^4}-\frac {1}{252} a c^3 x^5-\frac {17 c^3 x^3}{9450 a}-\frac {27}{200} a c^3 x^5 \tan ^{-1}(a x)^2+\frac {1}{4} c^3 x^4 \tan ^{-1}(a x)^3+\frac {53 c^3 x^4 \tan ^{-1}(a x)}{2100}-\frac {c^3 x^3 \tan ^{-1}(a x)^2}{40 a} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 203
Rule 302
Rule 321
Rule 2315
Rule 2402
Rule 4846
Rule 4852
Rule 4854
Rule 4884
Rule 4916
Rule 4920
Rule 4948
Rubi steps
\begin {align*} \int x^3 \left (c+a^2 c x^2\right )^3 \tan ^{-1}(a x)^3 \, dx &=\int \left (c^3 x^3 \tan ^{-1}(a x)^3+3 a^2 c^3 x^5 \tan ^{-1}(a x)^3+3 a^4 c^3 x^7 \tan ^{-1}(a x)^3+a^6 c^3 x^9 \tan ^{-1}(a x)^3\right ) \, dx\\ &=c^3 \int x^3 \tan ^{-1}(a x)^3 \, dx+\left (3 a^2 c^3\right ) \int x^5 \tan ^{-1}(a x)^3 \, dx+\left (3 a^4 c^3\right ) \int x^7 \tan ^{-1}(a x)^3 \, dx+\left (a^6 c^3\right ) \int x^9 \tan ^{-1}(a x)^3 \, dx\\ &=\frac {1}{4} c^3 x^4 \tan ^{-1}(a x)^3+\frac {1}{2} a^2 c^3 x^6 \tan ^{-1}(a x)^3+\frac {3}{8} a^4 c^3 x^8 \tan ^{-1}(a x)^3+\frac {1}{10} a^6 c^3 x^{10} \tan ^{-1}(a x)^3-\frac {1}{4} \left (3 a c^3\right ) \int \frac {x^4 \tan ^{-1}(a x)^2}{1+a^2 x^2} \, dx-\frac {1}{2} \left (3 a^3 c^3\right ) \int \frac {x^6 \tan ^{-1}(a x)^2}{1+a^2 x^2} \, dx-\frac {1}{8} \left (9 a^5 c^3\right ) \int \frac {x^8 \tan ^{-1}(a x)^2}{1+a^2 x^2} \, dx-\frac {1}{10} \left (3 a^7 c^3\right ) \int \frac {x^{10} \tan ^{-1}(a x)^2}{1+a^2 x^2} \, dx\\ &=\frac {1}{4} c^3 x^4 \tan ^{-1}(a x)^3+\frac {1}{2} a^2 c^3 x^6 \tan ^{-1}(a x)^3+\frac {3}{8} a^4 c^3 x^8 \tan ^{-1}(a x)^3+\frac {1}{10} a^6 c^3 x^{10} \tan ^{-1}(a x)^3-\frac {\left (3 c^3\right ) \int x^2 \tan ^{-1}(a x)^2 \, dx}{4 a}+\frac {\left (3 c^3\right ) \int \frac {x^2 \tan ^{-1}(a x)^2}{1+a^2 x^2} \, dx}{4 a}-\frac {1}{2} \left (3 a c^3\right ) \int x^4 \tan ^{-1}(a x)^2 \, dx+\frac {1}{2} \left (3 a c^3\right ) \int \frac {x^4 \tan ^{-1}(a x)^2}{1+a^2 x^2} \, dx-\frac {1}{8} \left (9 a^3 c^3\right ) \int x^6 \tan ^{-1}(a x)^2 \, dx+\frac {1}{8} \left (9 a^3 c^3\right ) \int \frac {x^6 \tan ^{-1}(a x)^2}{1+a^2 x^2} \, dx-\frac {1}{10} \left (3 a^5 c^3\right ) \int x^8 \tan ^{-1}(a x)^2 \, dx+\frac {1}{10} \left (3 a^5 c^3\right ) \int \frac {x^8 \tan ^{-1}(a x)^2}{1+a^2 x^2} \, dx\\ &=-\frac {c^3 x^3 \tan ^{-1}(a x)^2}{4 a}-\frac {3}{10} a c^3 x^5 \tan ^{-1}(a x)^2-\frac {9}{56} a^3 c^3 x^7 \tan ^{-1}(a x)^2-\frac {1}{30} a^5 c^3 x^9 \tan ^{-1}(a x)^2+\frac {1}{4} c^3 x^4 \tan ^{-1}(a x)^3+\frac {1}{2} a^2 c^3 x^6 \tan ^{-1}(a x)^3+\frac {3}{8} a^4 c^3 x^8 \tan ^{-1}(a x)^3+\frac {1}{10} a^6 c^3 x^{10} \tan ^{-1}(a x)^3+\frac {1}{2} c^3 \int \frac {x^3 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx+\frac {\left (3 c^3\right ) \int \tan ^{-1}(a x)^2 \, dx}{4 a^3}-\frac {\left (3 c^3\right ) \int \frac {\tan ^{-1}(a x)^2}{1+a^2 x^2} \, dx}{4 a^3}+\frac {\left (3 c^3\right ) \int x^2 \tan ^{-1}(a x)^2 \, dx}{2 a}-\frac {\left (3 c^3\right ) \int \frac {x^2 \tan ^{-1}(a x)^2}{1+a^2 x^2} \, dx}{2 a}+\frac {1}{8} \left (9 a c^3\right ) \int x^4 \tan ^{-1}(a x)^2 \, dx-\frac {1}{8} \left (9 a c^3\right ) \int \frac {x^4 \tan ^{-1}(a x)^2}{1+a^2 x^2} \, dx+\frac {1}{5} \left (3 a^2 c^3\right ) \int \frac {x^5 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx+\frac {1}{10} \left (3 a^3 c^3\right ) \int x^6 \tan ^{-1}(a x)^2 \, dx-\frac {1}{10} \left (3 a^3 c^3\right ) \int \frac {x^6 \tan ^{-1}(a x)^2}{1+a^2 x^2} \, dx+\frac {1}{28} \left (9 a^4 c^3\right ) \int \frac {x^7 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx+\frac {1}{15} \left (a^6 c^3\right ) \int \frac {x^9 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx\\ &=\frac {3 c^3 x \tan ^{-1}(a x)^2}{4 a^3}+\frac {c^3 x^3 \tan ^{-1}(a x)^2}{4 a}-\frac {3}{40} a c^3 x^5 \tan ^{-1}(a x)^2-\frac {33}{280} a^3 c^3 x^7 \tan ^{-1}(a x)^2-\frac {1}{30} a^5 c^3 x^9 \tan ^{-1}(a x)^2-\frac {c^3 \tan ^{-1}(a x)^3}{4 a^4}+\frac {1}{4} c^3 x^4 \tan ^{-1}(a x)^3+\frac {1}{2} a^2 c^3 x^6 \tan ^{-1}(a x)^3+\frac {3}{8} a^4 c^3 x^8 \tan ^{-1}(a x)^3+\frac {1}{10} a^6 c^3 x^{10} \tan ^{-1}(a x)^3+\frac {1}{5} \left (3 c^3\right ) \int x^3 \tan ^{-1}(a x) \, dx-\frac {1}{5} \left (3 c^3\right ) \int \frac {x^3 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx-c^3 \int \frac {x^3 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx-\frac {\left (3 c^3\right ) \int \tan ^{-1}(a x)^2 \, dx}{2 a^3}+\frac {\left (3 c^3\right ) \int \frac {\tan ^{-1}(a x)^2}{1+a^2 x^2} \, dx}{2 a^3}+\frac {c^3 \int x \tan ^{-1}(a x) \, dx}{2 a^2}-\frac {c^3 \int \frac {x \tan ^{-1}(a x)}{1+a^2 x^2} \, dx}{2 a^2}-\frac {\left (3 c^3\right ) \int \frac {x \tan ^{-1}(a x)}{1+a^2 x^2} \, dx}{2 a^2}-\frac {\left (9 c^3\right ) \int x^2 \tan ^{-1}(a x)^2 \, dx}{8 a}+\frac {\left (9 c^3\right ) \int \frac {x^2 \tan ^{-1}(a x)^2}{1+a^2 x^2} \, dx}{8 a}-\frac {1}{10} \left (3 a c^3\right ) \int x^4 \tan ^{-1}(a x)^2 \, dx+\frac {1}{10} \left (3 a c^3\right ) \int \frac {x^4 \tan ^{-1}(a x)^2}{1+a^2 x^2} \, dx+\frac {1}{28} \left (9 a^2 c^3\right ) \int x^5 \tan ^{-1}(a x) \, dx-\frac {1}{28} \left (9 a^2 c^3\right ) \int \frac {x^5 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx-\frac {1}{20} \left (9 a^2 c^3\right ) \int \frac {x^5 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx+\frac {1}{15} \left (a^4 c^3\right ) \int x^7 \tan ^{-1}(a x) \, dx-\frac {1}{15} \left (a^4 c^3\right ) \int \frac {x^7 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx-\frac {1}{35} \left (3 a^4 c^3\right ) \int \frac {x^7 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx\\ &=\frac {c^3 x^2 \tan ^{-1}(a x)}{4 a^2}+\frac {3}{20} c^3 x^4 \tan ^{-1}(a x)+\frac {3}{56} a^2 c^3 x^6 \tan ^{-1}(a x)+\frac {1}{120} a^4 c^3 x^8 \tan ^{-1}(a x)+\frac {i c^3 \tan ^{-1}(a x)^2}{a^4}-\frac {3 c^3 x \tan ^{-1}(a x)^2}{4 a^3}-\frac {c^3 x^3 \tan ^{-1}(a x)^2}{8 a}-\frac {27}{200} a c^3 x^5 \tan ^{-1}(a x)^2-\frac {33}{280} a^3 c^3 x^7 \tan ^{-1}(a x)^2-\frac {1}{30} a^5 c^3 x^9 \tan ^{-1}(a x)^2+\frac {c^3 \tan ^{-1}(a x)^3}{4 a^4}+\frac {1}{4} c^3 x^4 \tan ^{-1}(a x)^3+\frac {1}{2} a^2 c^3 x^6 \tan ^{-1}(a x)^3+\frac {3}{8} a^4 c^3 x^8 \tan ^{-1}(a x)^3+\frac {1}{10} a^6 c^3 x^{10} \tan ^{-1}(a x)^3-\frac {1}{28} \left (9 c^3\right ) \int x^3 \tan ^{-1}(a x) \, dx+\frac {1}{28} \left (9 c^3\right ) \int \frac {x^3 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx-\frac {1}{20} \left (9 c^3\right ) \int x^3 \tan ^{-1}(a x) \, dx+\frac {1}{20} \left (9 c^3\right ) \int \frac {x^3 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx+\frac {1}{4} \left (3 c^3\right ) \int \frac {x^3 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx+\frac {c^3 \int \frac {\tan ^{-1}(a x)}{i-a x} \, dx}{2 a^3}+\frac {\left (9 c^3\right ) \int \tan ^{-1}(a x)^2 \, dx}{8 a^3}-\frac {\left (9 c^3\right ) \int \frac {\tan ^{-1}(a x)^2}{1+a^2 x^2} \, dx}{8 a^3}+\frac {\left (3 c^3\right ) \int \frac {\tan ^{-1}(a x)}{i-a x} \, dx}{2 a^3}-\frac {\left (3 c^3\right ) \int x \tan ^{-1}(a x) \, dx}{5 a^2}+\frac {\left (3 c^3\right ) \int \frac {x \tan ^{-1}(a x)}{1+a^2 x^2} \, dx}{5 a^2}-\frac {c^3 \int x \tan ^{-1}(a x) \, dx}{a^2}+\frac {c^3 \int \frac {x \tan ^{-1}(a x)}{1+a^2 x^2} \, dx}{a^2}+\frac {\left (3 c^3\right ) \int \frac {x \tan ^{-1}(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 \tan ^{-1}(a x)^2 \, dx}{10 a}-\frac {\left (3 c^3\right ) \int \frac {x^2 \tan ^{-1}(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 \tan ^{-1}(a x) \, dx+\frac {1}{15} \left (a^2 c^3\right ) \int \frac {x^5 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx-\frac {1}{35} \left (3 a^2 c^3\right ) \int x^5 \tan ^{-1}(a x) \, dx+\frac {1}{35} \left (3 a^2 c^3\right ) \int \frac {x^5 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx+\frac {1}{25} \left (3 a^2 c^3\right ) \int \frac {x^5 \tan ^{-1}(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\\ &=-\frac {c^3 x}{4 a^3}-\frac {11 c^3 x^2 \tan ^{-1}(a x)}{20 a^2}-\frac {3}{70} c^3 x^4 \tan ^{-1}(a x)+\frac {71 a^2 c^3 x^6 \tan ^{-1}(a x)}{2520}+\frac {1}{120} a^4 c^3 x^8 \tan ^{-1}(a x)-\frac {13 i c^3 \tan ^{-1}(a x)^2}{10 a^4}+\frac {3 c^3 x \tan ^{-1}(a x)^2}{8 a^3}-\frac {c^3 x^3 \tan ^{-1}(a x)^2}{40 a}-\frac {27}{200} a c^3 x^5 \tan ^{-1}(a x)^2-\frac {33}{280} a^3 c^3 x^7 \tan ^{-1}(a x)^2-\frac {1}{30} a^5 c^3 x^9 \tan ^{-1}(a x)^2-\frac {c^3 \tan ^{-1}(a x)^3}{8 a^4}+\frac {1}{4} c^3 x^4 \tan ^{-1}(a x)^3+\frac {1}{2} a^2 c^3 x^6 \tan ^{-1}(a x)^3+\frac {3}{8} a^4 c^3 x^8 \tan ^{-1}(a x)^3+\frac {1}{10} a^6 c^3 x^{10} \tan ^{-1}(a x)^3+\frac {2 c^3 \tan ^{-1}(a x) \log \left (\frac {2}{1+i a x}\right )}{a^4}+\frac {1}{15} c^3 \int x^3 \tan ^{-1}(a x) \, dx-\frac {1}{15} c^3 \int \frac {x^3 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx+\frac {1}{35} \left (3 c^3\right ) \int x^3 \tan ^{-1}(a x) \, dx-\frac {1}{35} \left (3 c^3\right ) \int \frac {x^3 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx+\frac {1}{25} \left (3 c^3\right ) \int x^3 \tan ^{-1}(a x) \, dx-\frac {1}{25} \left (3 c^3\right ) \int \frac {x^3 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx-\frac {1}{5} c^3 \int \frac {x^3 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx+\frac {c^3 \int \frac {1}{1+a^2 x^2} \, dx}{4 a^3}-\frac {\left (3 c^3\right ) \int \tan ^{-1}(a x)^2 \, dx}{10 a^3}+\frac {\left (3 c^3\right ) \int \frac {\tan ^{-1}(a x)^2}{1+a^2 x^2} \, dx}{10 a^3}-\frac {c^3 \int \frac {\log \left (\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx}{2 a^3}-\frac {\left (3 c^3\right ) \int \frac {\tan ^{-1}(a x)}{i-a x} \, dx}{5 a^3}-\frac {c^3 \int \frac {\tan ^{-1}(a x)}{i-a x} \, dx}{a^3}-\frac {\left (3 c^3\right ) \int \frac {\log \left (\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx}{2 a^3}-\frac {\left (3 c^3\right ) \int \frac {\tan ^{-1}(a x)}{i-a x} \, dx}{a^3}+\frac {\left (9 c^3\right ) \int x \tan ^{-1}(a x) \, dx}{28 a^2}-\frac {\left (9 c^3\right ) \int \frac {x \tan ^{-1}(a x)}{1+a^2 x^2} \, dx}{28 a^2}+\frac {\left (9 c^3\right ) \int x \tan ^{-1}(a x) \, dx}{20 a^2}-\frac {\left (9 c^3\right ) \int \frac {x \tan ^{-1}(a x)}{1+a^2 x^2} \, dx}{20 a^2}+\frac {\left (3 c^3\right ) \int x \tan ^{-1}(a x) \, dx}{4 a^2}-\frac {\left (3 c^3\right ) \int \frac {x \tan ^{-1}(a x)}{1+a^2 x^2} \, dx}{4 a^2}-\frac {\left (9 c^3\right ) \int \frac {x \tan ^{-1}(a x)}{1+a^2 x^2} \, dx}{4 a^2}+\frac {\left (3 c^3\right ) \int \frac {x^2}{1+a^2 x^2} \, dx}{10 a}+\frac {c^3 \int \frac {x^2}{1+a^2 x^2} \, dx}{2 a}+\frac {1}{112} \left (9 a c^3\right ) \int \frac {x^4}{1+a^2 x^2} \, dx+\frac {1}{80} \left (9 a c^3\right ) \int \frac {x^4}{1+a^2 x^2} \, dx-\frac {1}{20} \left (3 a c^3\right ) \int \left (-\frac {1}{a^4}+\frac {x^2}{a^2}+\frac {1}{a^4 \left (1+a^2 x^2\right )}\right ) \, dx+\frac {1}{90} \left (a^3 c^3\right ) \int \frac {x^6}{1+a^2 x^2} \, dx+\frac {1}{70} \left (a^3 c^3\right ) \int \frac {x^6}{1+a^2 x^2} \, dx-\frac {1}{56} \left (3 a^3 c^3\right ) \int \left (\frac {1}{a^6}-\frac {x^2}{a^4}+\frac {x^4}{a^2}-\frac {1}{a^6 \left (1+a^2 x^2\right )}\right ) \, dx-\frac {1}{120} \left (a^5 c^3\right ) \int \left (-\frac {1}{a^8}+\frac {x^2}{a^6}-\frac {x^4}{a^4}+\frac {x^6}{a^2}+\frac {1}{a^8 \left (1+a^2 x^2\right )}\right ) \, dx\\ &=\frac {55 c^3 x}{84 a^3}-\frac {11 c^3 x^3}{315 a}-\frac {19 a c^3 x^5}{2100}-\frac {1}{840} a^3 c^3 x^7+\frac {c^3 \tan ^{-1}(a x)}{4 a^4}+\frac {59 c^3 x^2 \tan ^{-1}(a x)}{280 a^2}+\frac {53 c^3 x^4 \tan ^{-1}(a x)}{2100}+\frac {71 a^2 c^3 x^6 \tan ^{-1}(a x)}{2520}+\frac {1}{120} a^4 c^3 x^8 \tan ^{-1}(a x)+\frac {41 i c^3 \tan ^{-1}(a x)^2}{70 a^4}+\frac {3 c^3 x \tan ^{-1}(a x)^2}{40 a^3}-\frac {c^3 x^3 \tan ^{-1}(a x)^2}{40 a}-\frac {27}{200} a c^3 x^5 \tan ^{-1}(a x)^2-\frac {33}{280} a^3 c^3 x^7 \tan ^{-1}(a x)^2-\frac {1}{30} a^5 c^3 x^9 \tan ^{-1}(a x)^2-\frac {c^3 \tan ^{-1}(a x)^3}{40 a^4}+\frac {1}{4} c^3 x^4 \tan ^{-1}(a x)^3+\frac {1}{2} a^2 c^3 x^6 \tan ^{-1}(a x)^3+\frac {3}{8} a^4 c^3 x^8 \tan ^{-1}(a x)^3+\frac {1}{10} a^6 c^3 x^{10} \tan ^{-1}(a x)^3-\frac {13 c^3 \tan ^{-1}(a x) \log \left (\frac {2}{1+i a x}\right )}{5 a^4}+\frac {\left (i c^3\right ) \operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+i a x}\right )}{2 a^4}+\frac {\left (3 i c^3\right ) \operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+i a x}\right )}{2 a^4}-\frac {c^3 \int \frac {1}{1+a^2 x^2} \, dx}{120 a^3}+\frac {\left (3 c^3\right ) \int \frac {1}{1+a^2 x^2} \, dx}{56 a^3}-\frac {\left (3 c^3\right ) \int \frac {1}{1+a^2 x^2} \, dx}{20 a^3}-\frac {\left (3 c^3\right ) \int \frac {1}{1+a^2 x^2} \, dx}{10 a^3}+\frac {\left (9 c^3\right ) \int \frac {\tan ^{-1}(a x)}{i-a x} \, dx}{28 a^3}+\frac {\left (9 c^3\right ) \int \frac {\tan ^{-1}(a x)}{i-a x} \, dx}{20 a^3}-\frac {c^3 \int \frac {1}{1+a^2 x^2} \, dx}{2 a^3}+\frac {\left (3 c^3\right ) \int \frac {\log \left (\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx}{5 a^3}+\frac {\left (3 c^3\right ) \int \frac {\tan ^{-1}(a x)}{i-a x} \, dx}{4 a^3}+\frac {c^3 \int \frac {\log \left (\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx}{a^3}+\frac {\left (9 c^3\right ) \int \frac {\tan ^{-1}(a x)}{i-a x} \, dx}{4 a^3}+\frac {\left (3 c^3\right ) \int \frac {\log \left (\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx}{a^3}-\frac {c^3 \int x \tan ^{-1}(a x) \, dx}{15 a^2}+\frac {c^3 \int \frac {x \tan ^{-1}(a x)}{1+a^2 x^2} \, dx}{15 a^2}-\frac {\left (3 c^3\right ) \int x \tan ^{-1}(a x) \, dx}{35 a^2}+\frac {\left (3 c^3\right ) \int \frac {x \tan ^{-1}(a x)}{1+a^2 x^2} \, dx}{35 a^2}-\frac {\left (3 c^3\right ) \int x \tan ^{-1}(a x) \, dx}{25 a^2}+\frac {\left (3 c^3\right ) \int \frac {x \tan ^{-1}(a x)}{1+a^2 x^2} \, dx}{25 a^2}-\frac {c^3 \int x \tan ^{-1}(a x) \, dx}{5 a^2}+\frac {c^3 \int \frac {x \tan ^{-1}(a x)}{1+a^2 x^2} \, dx}{5 a^2}+\frac {\left (3 c^3\right ) \int \frac {x \tan ^{-1}(a x)}{1+a^2 x^2} \, dx}{5 a^2}-\frac {\left (9 c^3\right ) \int \frac {x^2}{1+a^2 x^2} \, dx}{56 a}-\frac {\left (9 c^3\right ) \int \frac {x^2}{1+a^2 x^2} \, dx}{40 a}-\frac {\left (3 c^3\right ) \int \frac {x^2}{1+a^2 x^2} \, dx}{8 a}-\frac {1}{60} \left (a c^3\right ) \int \frac {x^4}{1+a^2 x^2} \, dx-\frac {1}{140} \left (3 a c^3\right ) \int \frac {x^4}{1+a^2 x^2} \, dx-\frac {1}{100} \left (3 a c^3\right ) \int \frac {x^4}{1+a^2 x^2} \, dx+\frac {1}{112} \left (9 a c^3\right ) \int \left (-\frac {1}{a^4}+\frac {x^2}{a^2}+\frac {1}{a^4 \left (1+a^2 x^2\right )}\right ) \, dx+\frac {1}{80} \left (9 a c^3\right ) \int \left (-\frac {1}{a^4}+\frac {x^2}{a^2}+\frac {1}{a^4 \left (1+a^2 x^2\right )}\right ) \, dx+\frac {1}{90} \left (a^3 c^3\right ) \int \left (\frac {1}{a^6}-\frac {x^2}{a^4}+\frac {x^4}{a^2}-\frac {1}{a^6 \left (1+a^2 x^2\right )}\right ) \, dx+\frac {1}{70} \left (a^3 c^3\right ) \int \left (\frac {1}{a^6}-\frac {x^2}{a^4}+\frac {x^4}{a^2}-\frac {1}{a^6 \left (1+a^2 x^2\right )}\right ) \, dx\\ &=-\frac {689 c^3 x}{2520 a^3}+\frac {79 c^3 x^3}{3780 a}-\frac {1}{252} a c^3 x^5-\frac {1}{840} a^3 c^3 x^7-\frac {55 c^3 \tan ^{-1}(a x)}{84 a^4}-\frac {107 c^3 x^2 \tan ^{-1}(a x)}{4200 a^2}+\frac {53 c^3 x^4 \tan ^{-1}(a x)}{2100}+\frac {71 a^2 c^3 x^6 \tan ^{-1}(a x)}{2520}+\frac {1}{120} a^4 c^3 x^8 \tan ^{-1}(a x)+\frac {26 i c^3 \tan ^{-1}(a x)^2}{525 a^4}+\frac {3 c^3 x \tan ^{-1}(a x)^2}{40 a^3}-\frac {c^3 x^3 \tan ^{-1}(a x)^2}{40 a}-\frac {27}{200} a c^3 x^5 \tan ^{-1}(a x)^2-\frac {33}{280} a^3 c^3 x^7 \tan ^{-1}(a x)^2-\frac {1}{30} a^5 c^3 x^9 \tan ^{-1}(a x)^2-\frac {c^3 \tan ^{-1}(a x)^3}{40 a^4}+\frac {1}{4} c^3 x^4 \tan ^{-1}(a x)^3+\frac {1}{2} a^2 c^3 x^6 \tan ^{-1}(a x)^3+\frac {3}{8} a^4 c^3 x^8 \tan ^{-1}(a x)^3+\frac {1}{10} a^6 c^3 x^{10} \tan ^{-1}(a x)^3+\frac {41 c^3 \tan ^{-1}(a x) \log \left (\frac {2}{1+i a x}\right )}{35 a^4}+\frac {i c^3 \text {Li}_2\left (1-\frac {2}{1+i a x}\right )}{a^4}-\frac {\left (3 i c^3\right ) \operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+i a x}\right )}{5 a^4}-\frac {\left (i c^3\right ) \operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+i a x}\right )}{a^4}-\frac {\left (3 i c^3\right ) \operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+i a x}\right )}{a^4}-\frac {c^3 \int \frac {1}{1+a^2 x^2} \, dx}{90 a^3}-\frac {c^3 \int \frac {1}{1+a^2 x^2} \, dx}{70 a^3}-\frac {c^3 \int \frac {\tan ^{-1}(a x)}{i-a x} \, dx}{15 a^3}+\frac {\left (9 c^3\right ) \int \frac {1}{1+a^2 x^2} \, dx}{112 a^3}-\frac {\left (3 c^3\right ) \int \frac {\tan ^{-1}(a x)}{i-a x} \, dx}{35 a^3}+\frac {\left (9 c^3\right ) \int \frac {1}{1+a^2 x^2} \, dx}{80 a^3}-\frac {\left (3 c^3\right ) \int \frac {\tan ^{-1}(a x)}{i-a x} \, dx}{25 a^3}+\frac {\left (9 c^3\right ) \int \frac {1}{1+a^2 x^2} \, dx}{56 a^3}-\frac {c^3 \int \frac {\tan ^{-1}(a x)}{i-a x} \, dx}{5 a^3}+\frac {\left (9 c^3\right ) \int \frac {1}{1+a^2 x^2} \, dx}{40 a^3}-\frac {\left (9 c^3\right ) \int \frac {\log \left (\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx}{28 a^3}+\frac {\left (3 c^3\right ) \int \frac {1}{1+a^2 x^2} \, dx}{8 a^3}-\frac {\left (9 c^3\right ) \int \frac {\log \left (\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx}{20 a^3}-\frac {\left (3 c^3\right ) \int \frac {\tan ^{-1}(a x)}{i-a x} \, dx}{5 a^3}-\frac {\left (3 c^3\right ) \int \frac {\log \left (\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx}{4 a^3}-\frac {\left (9 c^3\right ) \int \frac {\log \left (\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx}{4 a^3}+\frac {c^3 \int \frac {x^2}{1+a^2 x^2} \, dx}{30 a}+\frac {\left (3 c^3\right ) \int \frac {x^2}{1+a^2 x^2} \, dx}{70 a}+\frac {\left (3 c^3\right ) \int \frac {x^2}{1+a^2 x^2} \, dx}{50 a}+\frac {c^3 \int \frac {x^2}{1+a^2 x^2} \, dx}{10 a}-\frac {1}{60} \left (a c^3\right ) \int \left (-\frac {1}{a^4}+\frac {x^2}{a^2}+\frac {1}{a^4 \left (1+a^2 x^2\right )}\right ) \, dx-\frac {1}{140} \left (3 a c^3\right ) \int \left (-\frac {1}{a^4}+\frac {x^2}{a^2}+\frac {1}{a^4 \left (1+a^2 x^2\right )}\right ) \, dx-\frac {1}{100} \left (3 a c^3\right ) \int \left (-\frac {1}{a^4}+\frac {x^2}{a^2}+\frac {1}{a^4 \left (1+a^2 x^2\right )}\right ) \, 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 {689 c^3 \tan ^{-1}(a x)}{2520 a^4}-\frac {107 c^3 x^2 \tan ^{-1}(a x)}{4200 a^2}+\frac {53 c^3 x^4 \tan ^{-1}(a x)}{2100}+\frac {71 a^2 c^3 x^6 \tan ^{-1}(a x)}{2520}+\frac {1}{120} a^4 c^3 x^8 \tan ^{-1}(a x)+\frac {26 i c^3 \tan ^{-1}(a x)^2}{525 a^4}+\frac {3 c^3 x \tan ^{-1}(a x)^2}{40 a^3}-\frac {c^3 x^3 \tan ^{-1}(a x)^2}{40 a}-\frac {27}{200} a c^3 x^5 \tan ^{-1}(a x)^2-\frac {33}{280} a^3 c^3 x^7 \tan ^{-1}(a x)^2-\frac {1}{30} a^5 c^3 x^9 \tan ^{-1}(a x)^2-\frac {c^3 \tan ^{-1}(a x)^3}{40 a^4}+\frac {1}{4} c^3 x^4 \tan ^{-1}(a x)^3+\frac {1}{2} a^2 c^3 x^6 \tan ^{-1}(a x)^3+\frac {3}{8} a^4 c^3 x^8 \tan ^{-1}(a x)^3+\frac {1}{10} a^6 c^3 x^{10} \tan ^{-1}(a x)^3+\frac {52 c^3 \tan ^{-1}(a x) \log \left (\frac {2}{1+i a x}\right )}{525 a^4}-\frac {13 i c^3 \text {Li}_2\left (1-\frac {2}{1+i a x}\right )}{10 a^4}+\frac {\left (9 i c^3\right ) \operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+i a x}\right )}{28 a^4}+\frac {\left (9 i c^3\right ) \operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+i a x}\right )}{20 a^4}+\frac {\left (3 i c^3\right ) \operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+i a x}\right )}{4 a^4}+\frac {\left (9 i c^3\right ) \operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+i a x}\right )}{4 a^4}-\frac {c^3 \int \frac {1}{1+a^2 x^2} \, dx}{60 a^3}-\frac {\left (3 c^3\right ) \int \frac {1}{1+a^2 x^2} \, dx}{140 a^3}-\frac {\left (3 c^3\right ) \int \frac {1}{1+a^2 x^2} \, dx}{100 a^3}-\frac {c^3 \int \frac {1}{1+a^2 x^2} \, dx}{30 a^3}-\frac {\left (3 c^3\right ) \int \frac {1}{1+a^2 x^2} \, dx}{70 a^3}-\frac {\left (3 c^3\right ) \int \frac {1}{1+a^2 x^2} \, dx}{50 a^3}+\frac {c^3 \int \frac {\log \left (\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx}{15 a^3}+\frac {\left (3 c^3\right ) \int \frac {\log \left (\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx}{35 a^3}-\frac {c^3 \int \frac {1}{1+a^2 x^2} \, dx}{10 a^3}+\frac {\left (3 c^3\right ) \int \frac {\log \left (\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx}{25 a^3}+\frac {c^3 \int \frac {\log \left (\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx}{5 a^3}+\frac {\left (3 c^3\right ) \int \frac {\log \left (\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx}{5 a^3}\\ &=\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 \tan ^{-1}(a x)}{12600 a^4}-\frac {107 c^3 x^2 \tan ^{-1}(a x)}{4200 a^2}+\frac {53 c^3 x^4 \tan ^{-1}(a x)}{2100}+\frac {71 a^2 c^3 x^6 \tan ^{-1}(a x)}{2520}+\frac {1}{120} a^4 c^3 x^8 \tan ^{-1}(a x)+\frac {26 i c^3 \tan ^{-1}(a x)^2}{525 a^4}+\frac {3 c^3 x \tan ^{-1}(a x)^2}{40 a^3}-\frac {c^3 x^3 \tan ^{-1}(a x)^2}{40 a}-\frac {27}{200} a c^3 x^5 \tan ^{-1}(a x)^2-\frac {33}{280} a^3 c^3 x^7 \tan ^{-1}(a x)^2-\frac {1}{30} a^5 c^3 x^9 \tan ^{-1}(a x)^2-\frac {c^3 \tan ^{-1}(a x)^3}{40 a^4}+\frac {1}{4} c^3 x^4 \tan ^{-1}(a x)^3+\frac {1}{2} a^2 c^3 x^6 \tan ^{-1}(a x)^3+\frac {3}{8} a^4 c^3 x^8 \tan ^{-1}(a x)^3+\frac {1}{10} a^6 c^3 x^{10} \tan ^{-1}(a x)^3+\frac {52 c^3 \tan ^{-1}(a x) \log \left (\frac {2}{1+i a x}\right )}{525 a^4}+\frac {41 i c^3 \text {Li}_2\left (1-\frac {2}{1+i a x}\right )}{70 a^4}-\frac {\left (i c^3\right ) \operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+i a x}\right )}{15 a^4}-\frac {\left (3 i c^3\right ) \operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+i a x}\right )}{35 a^4}-\frac {\left (3 i c^3\right ) \operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+i a x}\right )}{25 a^4}-\frac {\left (i c^3\right ) \operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+i a x}\right )}{5 a^4}-\frac {\left (3 i c^3\right ) \operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+i a x}\right )}{5 a^4}\\ &=\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 \tan ^{-1}(a x)}{12600 a^4}-\frac {107 c^3 x^2 \tan ^{-1}(a x)}{4200 a^2}+\frac {53 c^3 x^4 \tan ^{-1}(a x)}{2100}+\frac {71 a^2 c^3 x^6 \tan ^{-1}(a x)}{2520}+\frac {1}{120} a^4 c^3 x^8 \tan ^{-1}(a x)+\frac {26 i c^3 \tan ^{-1}(a x)^2}{525 a^4}+\frac {3 c^3 x \tan ^{-1}(a x)^2}{40 a^3}-\frac {c^3 x^3 \tan ^{-1}(a x)^2}{40 a}-\frac {27}{200} a c^3 x^5 \tan ^{-1}(a x)^2-\frac {33}{280} a^3 c^3 x^7 \tan ^{-1}(a x)^2-\frac {1}{30} a^5 c^3 x^9 \tan ^{-1}(a x)^2-\frac {c^3 \tan ^{-1}(a x)^3}{40 a^4}+\frac {1}{4} c^3 x^4 \tan ^{-1}(a x)^3+\frac {1}{2} a^2 c^3 x^6 \tan ^{-1}(a x)^3+\frac {3}{8} a^4 c^3 x^8 \tan ^{-1}(a x)^3+\frac {1}{10} a^6 c^3 x^{10} \tan ^{-1}(a x)^3+\frac {52 c^3 \tan ^{-1}(a x) \log \left (\frac {2}{1+i a x}\right )}{525 a^4}+\frac {26 i c^3 \text {Li}_2\left (1-\frac {2}{1+i a x}\right )}{525 a^4}\\ \end {align*}
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Mathematica [A] time = 2.18, size = 191, normalized size = 0.50 \[ \frac {c^3 \left (945 \left (4 a^2 x^2-1\right ) \left (a^2 x^2+1\right )^4 \tan ^{-1}(a x)^3-a x \left (45 a^6 x^6+150 a^4 x^4+68 a^2 x^2-1167\right )-9 \left (140 a^9 x^9+495 a^7 x^7+567 a^5 x^5+105 a^3 x^3-315 a x+208 i\right ) \tan ^{-1}(a x)^2+3 \tan ^{-1}(a x) \left (105 a^8 x^8+355 a^6 x^6+318 a^4 x^4-321 a^2 x^2+1248 \log \left (1+e^{2 i \tan ^{-1}(a x)}\right )-389\right )-1872 i \text {Li}_2\left (-e^{2 i \tan ^{-1}(a x)}\right )\right )}{37800 a^4} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.58, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (a^{6} c^{3} x^{9} + 3 \, a^{4} c^{3} x^{7} + 3 \, a^{2} c^{3} x^{5} + c^{3} x^{3}\right )} \arctan \left (a x\right )^{3}, x\right ) \]
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 [A] time = 0.13, size = 471, normalized size = 1.24 \[ \frac {13 i c^{3} \dilog \left (-\frac {i \left (a x +i\right )}{2}\right )}{525 a^{4}}-\frac {13 i c^{3} \dilog \left (\frac {i \left (a x -i\right )}{2}\right )}{525 a^{4}}-\frac {13 i c^{3} \ln \left (a x +i\right )^{2}}{1050 a^{4}}+\frac {13 i c^{3} \ln \left (a x -i\right )^{2}}{1050 a^{4}}-\frac {26 c^{3} \arctan \left (a x \right ) \ln \left (a^{2} x^{2}+1\right )}{525 a^{4}}+\frac {389 c^{3} x}{12600 a^{3}}-\frac {17 c^{3} x^{3}}{9450 a}-\frac {a \,c^{3} x^{5}}{252}-\frac {a^{3} c^{3} x^{7}}{840}-\frac {389 c^{3} \arctan \left (a x \right )}{12600 a^{4}}+\frac {53 c^{3} x^{4} \arctan \left (a x \right )}{2100}-\frac {c^{3} \arctan \left (a x \right )^{3}}{40 a^{4}}+\frac {c^{3} x^{4} \arctan \left (a x \right )^{3}}{4}+\frac {71 a^{2} c^{3} x^{6} \arctan \left (a x \right )}{2520}+\frac {a^{4} c^{3} x^{8} \arctan \left (a x \right )}{120}-\frac {107 c^{3} x^{2} \arctan \left (a x \right )}{4200 a^{2}}+\frac {3 c^{3} x \arctan \left (a x \right )^{2}}{40 a^{3}}-\frac {c^{3} x^{3} \arctan \left (a x \right )^{2}}{40 a}-\frac {27 a \,c^{3} x^{5} \arctan \left (a x \right )^{2}}{200}-\frac {33 a^{3} c^{3} x^{7} \arctan \left (a x \right )^{2}}{280}-\frac {a^{5} c^{3} x^{9} \arctan \left (a x \right )^{2}}{30}+\frac {a^{2} c^{3} x^{6} \arctan \left (a x \right )^{3}}{2}+\frac {3 a^{4} c^{3} x^{8} \arctan \left (a x \right )^{3}}{8}+\frac {a^{6} c^{3} x^{10} \arctan \left (a x \right )^{3}}{10}-\frac {13 i c^{3} \ln \left (a x -i\right ) \ln \left (a^{2} x^{2}+1\right )}{525 a^{4}}-\frac {13 i c^{3} \ln \left (a x +i\right ) \ln \left (\frac {i \left (a x -i\right )}{2}\right )}{525 a^{4}}+\frac {13 i c^{3} \ln \left (a x -i\right ) \ln \left (-\frac {i \left (a x +i\right )}{2}\right )}{525 a^{4}}+\frac {13 i c^{3} \ln \left (a x +i\right ) \ln \left (a^{2} x^{2}+1\right )}{525 a^{4}} \]
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 \[ \int x^3\,{\mathrm {atan}\left (a\,x\right )}^3\,{\left (c\,a^2\,x^2+c\right )}^3 \,d x \]
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 \[ c^{3} \left (\int x^{3} \operatorname {atan}^{3}{\left (a x \right )}\, dx + \int 3 a^{2} x^{5} \operatorname {atan}^{3}{\left (a x \right )}\, dx + \int 3 a^{4} x^{7} \operatorname {atan}^{3}{\left (a x \right )}\, dx + \int a^{6} x^{9} \operatorname {atan}^{3}{\left (a x \right )}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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