Optimal. Leaf size=313 \[ \frac{2 i c^2 \text{PolyLog}\left (2,1-\frac{2}{1+i a x}\right )}{21 a^4}+\frac{1}{8} a^4 c^2 x^8 \tan ^{-1}(a x)^3-\frac{3}{56} a^3 c^2 x^7 \tan ^{-1}(a x)^2+\frac{1}{3} a^2 c^2 x^6 \tan ^{-1}(a x)^3+\frac{1}{56} a^2 c^2 x^6 \tan ^{-1}(a x)-\frac{5 c^2 x^2 \tan ^{-1}(a x)}{168 a^2}+\frac{c^2 x}{21 a^3}+\frac{c^2 x \tan ^{-1}(a x)^2}{8 a^3}-\frac{c^2 \tan ^{-1}(a x)^3}{24 a^4}+\frac{2 i c^2 \tan ^{-1}(a x)^2}{21 a^4}-\frac{c^2 \tan ^{-1}(a x)}{21 a^4}+\frac{4 c^2 \log \left (\frac{2}{1+i a x}\right ) \tan ^{-1}(a x)}{21 a^4}-\frac{1}{280} a c^2 x^5-\frac{c^2 x^3}{168 a}-\frac{1}{8} a c^2 x^5 \tan ^{-1}(a x)^2+\frac{1}{4} c^2 x^4 \tan ^{-1}(a x)^3+\frac{1}{28} c^2 x^4 \tan ^{-1}(a x)-\frac{c^2 x^3 \tan ^{-1}(a x)^2}{24 a} \]
[Out]
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Rubi [A] time = 2.28292, antiderivative size = 313, normalized size of antiderivative = 1., number of steps used = 106, 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} \[ \frac{2 i c^2 \text{PolyLog}\left (2,1-\frac{2}{1+i a x}\right )}{21 a^4}+\frac{1}{8} a^4 c^2 x^8 \tan ^{-1}(a x)^3-\frac{3}{56} a^3 c^2 x^7 \tan ^{-1}(a x)^2+\frac{1}{3} a^2 c^2 x^6 \tan ^{-1}(a x)^3+\frac{1}{56} a^2 c^2 x^6 \tan ^{-1}(a x)-\frac{5 c^2 x^2 \tan ^{-1}(a x)}{168 a^2}+\frac{c^2 x}{21 a^3}+\frac{c^2 x \tan ^{-1}(a x)^2}{8 a^3}-\frac{c^2 \tan ^{-1}(a x)^3}{24 a^4}+\frac{2 i c^2 \tan ^{-1}(a x)^2}{21 a^4}-\frac{c^2 \tan ^{-1}(a x)}{21 a^4}+\frac{4 c^2 \log \left (\frac{2}{1+i a x}\right ) \tan ^{-1}(a x)}{21 a^4}-\frac{1}{280} a c^2 x^5-\frac{c^2 x^3}{168 a}-\frac{1}{8} a c^2 x^5 \tan ^{-1}(a x)^2+\frac{1}{4} c^2 x^4 \tan ^{-1}(a x)^3+\frac{1}{28} c^2 x^4 \tan ^{-1}(a x)-\frac{c^2 x^3 \tan ^{-1}(a x)^2}{24 a} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 4948
Rule 4852
Rule 4916
Rule 321
Rule 203
Rule 4920
Rule 4854
Rule 2402
Rule 2315
Rule 4846
Rule 4884
Rule 302
Rubi steps
\begin{align*} \int x^3 \left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)^3 \, dx &=\int \left (c^2 x^3 \tan ^{-1}(a x)^3+2 a^2 c^2 x^5 \tan ^{-1}(a x)^3+a^4 c^2 x^7 \tan ^{-1}(a x)^3\right ) \, dx\\ &=c^2 \int x^3 \tan ^{-1}(a x)^3 \, dx+\left (2 a^2 c^2\right ) \int x^5 \tan ^{-1}(a x)^3 \, dx+\left (a^4 c^2\right ) \int x^7 \tan ^{-1}(a x)^3 \, dx\\ &=\frac{1}{4} c^2 x^4 \tan ^{-1}(a x)^3+\frac{1}{3} a^2 c^2 x^6 \tan ^{-1}(a x)^3+\frac{1}{8} a^4 c^2 x^8 \tan ^{-1}(a x)^3-\frac{1}{4} \left (3 a c^2\right ) \int \frac{x^4 \tan ^{-1}(a x)^2}{1+a^2 x^2} \, dx-\left (a^3 c^2\right ) \int \frac{x^6 \tan ^{-1}(a x)^2}{1+a^2 x^2} \, dx-\frac{1}{8} \left (3 a^5 c^2\right ) \int \frac{x^8 \tan ^{-1}(a x)^2}{1+a^2 x^2} \, dx\\ &=\frac{1}{4} c^2 x^4 \tan ^{-1}(a x)^3+\frac{1}{3} a^2 c^2 x^6 \tan ^{-1}(a x)^3+\frac{1}{8} a^4 c^2 x^8 \tan ^{-1}(a x)^3-\frac{\left (3 c^2\right ) \int x^2 \tan ^{-1}(a x)^2 \, dx}{4 a}+\frac{\left (3 c^2\right ) \int \frac{x^2 \tan ^{-1}(a x)^2}{1+a^2 x^2} \, dx}{4 a}-\left (a c^2\right ) \int x^4 \tan ^{-1}(a x)^2 \, dx+\left (a c^2\right ) \int \frac{x^4 \tan ^{-1}(a x)^2}{1+a^2 x^2} \, dx-\frac{1}{8} \left (3 a^3 c^2\right ) \int x^6 \tan ^{-1}(a x)^2 \, dx+\frac{1}{8} \left (3 a^3 c^2\right ) \int \frac{x^6 \tan ^{-1}(a x)^2}{1+a^2 x^2} \, dx\\ &=-\frac{c^2 x^3 \tan ^{-1}(a x)^2}{4 a}-\frac{1}{5} a c^2 x^5 \tan ^{-1}(a x)^2-\frac{3}{56} a^3 c^2 x^7 \tan ^{-1}(a x)^2+\frac{1}{4} c^2 x^4 \tan ^{-1}(a x)^3+\frac{1}{3} a^2 c^2 x^6 \tan ^{-1}(a x)^3+\frac{1}{8} a^4 c^2 x^8 \tan ^{-1}(a x)^3+\frac{1}{2} c^2 \int \frac{x^3 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx+\frac{\left (3 c^2\right ) \int \tan ^{-1}(a x)^2 \, dx}{4 a^3}-\frac{\left (3 c^2\right ) \int \frac{\tan ^{-1}(a x)^2}{1+a^2 x^2} \, dx}{4 a^3}+\frac{c^2 \int x^2 \tan ^{-1}(a x)^2 \, dx}{a}-\frac{c^2 \int \frac{x^2 \tan ^{-1}(a x)^2}{1+a^2 x^2} \, dx}{a}+\frac{1}{8} \left (3 a c^2\right ) \int x^4 \tan ^{-1}(a x)^2 \, dx-\frac{1}{8} \left (3 a c^2\right ) \int \frac{x^4 \tan ^{-1}(a x)^2}{1+a^2 x^2} \, dx+\frac{1}{5} \left (2 a^2 c^2\right ) \int \frac{x^5 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx+\frac{1}{28} \left (3 a^4 c^2\right ) \int \frac{x^7 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx\\ &=\frac{3 c^2 x \tan ^{-1}(a x)^2}{4 a^3}+\frac{c^2 x^3 \tan ^{-1}(a x)^2}{12 a}-\frac{1}{8} a c^2 x^5 \tan ^{-1}(a x)^2-\frac{3}{56} a^3 c^2 x^7 \tan ^{-1}(a x)^2-\frac{c^2 \tan ^{-1}(a x)^3}{4 a^4}+\frac{1}{4} c^2 x^4 \tan ^{-1}(a x)^3+\frac{1}{3} a^2 c^2 x^6 \tan ^{-1}(a x)^3+\frac{1}{8} a^4 c^2 x^8 \tan ^{-1}(a x)^3+\frac{1}{5} \left (2 c^2\right ) \int x^3 \tan ^{-1}(a x) \, dx-\frac{1}{5} \left (2 c^2\right ) \int \frac{x^3 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx-\frac{1}{3} \left (2 c^2\right ) \int \frac{x^3 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx-\frac{c^2 \int \tan ^{-1}(a x)^2 \, dx}{a^3}+\frac{c^2 \int \frac{\tan ^{-1}(a x)^2}{1+a^2 x^2} \, dx}{a^3}+\frac{c^2 \int x \tan ^{-1}(a x) \, dx}{2 a^2}-\frac{c^2 \int \frac{x \tan ^{-1}(a x)}{1+a^2 x^2} \, dx}{2 a^2}-\frac{\left (3 c^2\right ) \int \frac{x \tan ^{-1}(a x)}{1+a^2 x^2} \, dx}{2 a^2}-\frac{\left (3 c^2\right ) \int x^2 \tan ^{-1}(a x)^2 \, dx}{8 a}+\frac{\left (3 c^2\right ) \int \frac{x^2 \tan ^{-1}(a x)^2}{1+a^2 x^2} \, dx}{8 a}+\frac{1}{28} \left (3 a^2 c^2\right ) \int x^5 \tan ^{-1}(a x) \, dx-\frac{1}{28} \left (3 a^2 c^2\right ) \int \frac{x^5 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx-\frac{1}{20} \left (3 a^2 c^2\right ) \int \frac{x^5 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx\\ &=\frac{c^2 x^2 \tan ^{-1}(a x)}{4 a^2}+\frac{1}{10} c^2 x^4 \tan ^{-1}(a x)+\frac{1}{56} a^2 c^2 x^6 \tan ^{-1}(a x)+\frac{i c^2 \tan ^{-1}(a x)^2}{a^4}-\frac{c^2 x \tan ^{-1}(a x)^2}{4 a^3}-\frac{c^2 x^3 \tan ^{-1}(a x)^2}{24 a}-\frac{1}{8} a c^2 x^5 \tan ^{-1}(a x)^2-\frac{3}{56} a^3 c^2 x^7 \tan ^{-1}(a x)^2+\frac{c^2 \tan ^{-1}(a x)^3}{12 a^4}+\frac{1}{4} c^2 x^4 \tan ^{-1}(a x)^3+\frac{1}{3} a^2 c^2 x^6 \tan ^{-1}(a x)^3+\frac{1}{8} a^4 c^2 x^8 \tan ^{-1}(a x)^3-\frac{1}{28} \left (3 c^2\right ) \int x^3 \tan ^{-1}(a x) \, dx+\frac{1}{28} \left (3 c^2\right ) \int \frac{x^3 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx-\frac{1}{20} \left (3 c^2\right ) \int x^3 \tan ^{-1}(a x) \, dx+\frac{1}{20} \left (3 c^2\right ) \int \frac{x^3 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx+\frac{1}{4} c^2 \int \frac{x^3 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx+\frac{\left (3 c^2\right ) \int \tan ^{-1}(a x)^2 \, dx}{8 a^3}-\frac{\left (3 c^2\right ) \int \frac{\tan ^{-1}(a x)^2}{1+a^2 x^2} \, dx}{8 a^3}+\frac{c^2 \int \frac{\tan ^{-1}(a x)}{i-a x} \, dx}{2 a^3}+\frac{\left (3 c^2\right ) \int \frac{\tan ^{-1}(a x)}{i-a x} \, dx}{2 a^3}-\frac{\left (2 c^2\right ) \int x \tan ^{-1}(a x) \, dx}{5 a^2}+\frac{\left (2 c^2\right ) \int \frac{x \tan ^{-1}(a x)}{1+a^2 x^2} \, dx}{5 a^2}-\frac{\left (2 c^2\right ) \int x \tan ^{-1}(a x) \, dx}{3 a^2}+\frac{\left (2 c^2\right ) \int \frac{x \tan ^{-1}(a x)}{1+a^2 x^2} \, dx}{3 a^2}+\frac{\left (2 c^2\right ) \int \frac{x \tan ^{-1}(a x)}{1+a^2 x^2} \, dx}{a^2}-\frac{c^2 \int \frac{x^2}{1+a^2 x^2} \, dx}{4 a}-\frac{1}{10} \left (a c^2\right ) \int \frac{x^4}{1+a^2 x^2} \, dx-\frac{1}{56} \left (a^3 c^2\right ) \int \frac{x^6}{1+a^2 x^2} \, dx\\ &=-\frac{c^2 x}{4 a^3}-\frac{17 c^2 x^2 \tan ^{-1}(a x)}{60 a^2}+\frac{1}{28} c^2 x^4 \tan ^{-1}(a x)+\frac{1}{56} a^2 c^2 x^6 \tan ^{-1}(a x)-\frac{8 i c^2 \tan ^{-1}(a x)^2}{15 a^4}+\frac{c^2 x \tan ^{-1}(a x)^2}{8 a^3}-\frac{c^2 x^3 \tan ^{-1}(a x)^2}{24 a}-\frac{1}{8} a c^2 x^5 \tan ^{-1}(a x)^2-\frac{3}{56} a^3 c^2 x^7 \tan ^{-1}(a x)^2-\frac{c^2 \tan ^{-1}(a x)^3}{24 a^4}+\frac{1}{4} c^2 x^4 \tan ^{-1}(a x)^3+\frac{1}{3} a^2 c^2 x^6 \tan ^{-1}(a x)^3+\frac{1}{8} a^4 c^2 x^8 \tan ^{-1}(a x)^3+\frac{2 c^2 \tan ^{-1}(a x) \log \left (\frac{2}{1+i a x}\right )}{a^4}+\frac{c^2 \int \frac{1}{1+a^2 x^2} \, dx}{4 a^3}-\frac{\left (2 c^2\right ) \int \frac{\tan ^{-1}(a x)}{i-a x} \, dx}{5 a^3}-\frac{c^2 \int \frac{\log \left (\frac{2}{1+i a x}\right )}{1+a^2 x^2} \, dx}{2 a^3}-\frac{\left (2 c^2\right ) \int \frac{\tan ^{-1}(a x)}{i-a x} \, dx}{3 a^3}-\frac{\left (3 c^2\right ) \int \frac{\log \left (\frac{2}{1+i a x}\right )}{1+a^2 x^2} \, dx}{2 a^3}-\frac{\left (2 c^2\right ) \int \frac{\tan ^{-1}(a x)}{i-a x} \, dx}{a^3}+\frac{\left (3 c^2\right ) \int x \tan ^{-1}(a x) \, dx}{28 a^2}-\frac{\left (3 c^2\right ) \int \frac{x \tan ^{-1}(a x)}{1+a^2 x^2} \, dx}{28 a^2}+\frac{\left (3 c^2\right ) \int x \tan ^{-1}(a x) \, dx}{20 a^2}-\frac{\left (3 c^2\right ) \int \frac{x \tan ^{-1}(a x)}{1+a^2 x^2} \, dx}{20 a^2}+\frac{c^2 \int x \tan ^{-1}(a x) \, dx}{4 a^2}-\frac{c^2 \int \frac{x \tan ^{-1}(a x)}{1+a^2 x^2} \, dx}{4 a^2}-\frac{\left (3 c^2\right ) \int \frac{x \tan ^{-1}(a x)}{1+a^2 x^2} \, dx}{4 a^2}+\frac{c^2 \int \frac{x^2}{1+a^2 x^2} \, dx}{5 a}+\frac{c^2 \int \frac{x^2}{1+a^2 x^2} \, dx}{3 a}+\frac{1}{112} \left (3 a c^2\right ) \int \frac{x^4}{1+a^2 x^2} \, dx+\frac{1}{80} \left (3 a c^2\right ) \int \frac{x^4}{1+a^2 x^2} \, dx-\frac{1}{10} \left (a c^2\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}{56} \left (a^3 c^2\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{307 c^2 x}{840 a^3}-\frac{23 c^2 x^3}{840 a}-\frac{1}{280} a c^2 x^5+\frac{c^2 \tan ^{-1}(a x)}{4 a^4}-\frac{5 c^2 x^2 \tan ^{-1}(a x)}{168 a^2}+\frac{1}{28} c^2 x^4 \tan ^{-1}(a x)+\frac{1}{56} a^2 c^2 x^6 \tan ^{-1}(a x)+\frac{2 i c^2 \tan ^{-1}(a x)^2}{21 a^4}+\frac{c^2 x \tan ^{-1}(a x)^2}{8 a^3}-\frac{c^2 x^3 \tan ^{-1}(a x)^2}{24 a}-\frac{1}{8} a c^2 x^5 \tan ^{-1}(a x)^2-\frac{3}{56} a^3 c^2 x^7 \tan ^{-1}(a x)^2-\frac{c^2 \tan ^{-1}(a x)^3}{24 a^4}+\frac{1}{4} c^2 x^4 \tan ^{-1}(a x)^3+\frac{1}{3} a^2 c^2 x^6 \tan ^{-1}(a x)^3+\frac{1}{8} a^4 c^2 x^8 \tan ^{-1}(a x)^3-\frac{16 c^2 \tan ^{-1}(a x) \log \left (\frac{2}{1+i a x}\right )}{15 a^4}+\frac{\left (i c^2\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^2\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^2 \int \frac{1}{1+a^2 x^2} \, dx}{56 a^3}-\frac{c^2 \int \frac{1}{1+a^2 x^2} \, dx}{10 a^3}+\frac{\left (3 c^2\right ) \int \frac{\tan ^{-1}(a x)}{i-a x} \, dx}{28 a^3}+\frac{\left (3 c^2\right ) \int \frac{\tan ^{-1}(a x)}{i-a x} \, dx}{20 a^3}-\frac{c^2 \int \frac{1}{1+a^2 x^2} \, dx}{5 a^3}+\frac{c^2 \int \frac{\tan ^{-1}(a x)}{i-a x} \, dx}{4 a^3}-\frac{c^2 \int \frac{1}{1+a^2 x^2} \, dx}{3 a^3}+\frac{\left (2 c^2\right ) \int \frac{\log \left (\frac{2}{1+i a x}\right )}{1+a^2 x^2} \, dx}{5 a^3}+\frac{\left (2 c^2\right ) \int \frac{\log \left (\frac{2}{1+i a x}\right )}{1+a^2 x^2} \, dx}{3 a^3}+\frac{\left (3 c^2\right ) \int \frac{\tan ^{-1}(a x)}{i-a x} \, dx}{4 a^3}+\frac{\left (2 c^2\right ) \int \frac{\log \left (\frac{2}{1+i a x}\right )}{1+a^2 x^2} \, dx}{a^3}-\frac{\left (3 c^2\right ) \int \frac{x^2}{1+a^2 x^2} \, dx}{56 a}-\frac{\left (3 c^2\right ) \int \frac{x^2}{1+a^2 x^2} \, dx}{40 a}-\frac{c^2 \int \frac{x^2}{1+a^2 x^2} \, dx}{8 a}+\frac{1}{112} \left (3 a c^2\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 (3 a c^2\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{c^2 x}{21 a^3}-\frac{c^2 x^3}{168 a}-\frac{1}{280} a c^2 x^5-\frac{307 c^2 \tan ^{-1}(a x)}{840 a^4}-\frac{5 c^2 x^2 \tan ^{-1}(a x)}{168 a^2}+\frac{1}{28} c^2 x^4 \tan ^{-1}(a x)+\frac{1}{56} a^2 c^2 x^6 \tan ^{-1}(a x)+\frac{2 i c^2 \tan ^{-1}(a x)^2}{21 a^4}+\frac{c^2 x \tan ^{-1}(a x)^2}{8 a^3}-\frac{c^2 x^3 \tan ^{-1}(a x)^2}{24 a}-\frac{1}{8} a c^2 x^5 \tan ^{-1}(a x)^2-\frac{3}{56} a^3 c^2 x^7 \tan ^{-1}(a x)^2-\frac{c^2 \tan ^{-1}(a x)^3}{24 a^4}+\frac{1}{4} c^2 x^4 \tan ^{-1}(a x)^3+\frac{1}{3} a^2 c^2 x^6 \tan ^{-1}(a x)^3+\frac{1}{8} a^4 c^2 x^8 \tan ^{-1}(a x)^3+\frac{4 c^2 \tan ^{-1}(a x) \log \left (\frac{2}{1+i a x}\right )}{21 a^4}+\frac{i c^2 \text{Li}_2\left (1-\frac{2}{1+i a x}\right )}{a^4}-\frac{\left (2 i c^2\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 (2 i c^2\right ) \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1+i a x}\right )}{3 a^4}-\frac{\left (2 i c^2\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 c^2\right ) \int \frac{1}{1+a^2 x^2} \, dx}{112 a^3}+\frac{\left (3 c^2\right ) \int \frac{1}{1+a^2 x^2} \, dx}{80 a^3}+\frac{\left (3 c^2\right ) \int \frac{1}{1+a^2 x^2} \, dx}{56 a^3}+\frac{\left (3 c^2\right ) \int \frac{1}{1+a^2 x^2} \, dx}{40 a^3}-\frac{\left (3 c^2\right ) \int \frac{\log \left (\frac{2}{1+i a x}\right )}{1+a^2 x^2} \, dx}{28 a^3}+\frac{c^2 \int \frac{1}{1+a^2 x^2} \, dx}{8 a^3}-\frac{\left (3 c^2\right ) \int \frac{\log \left (\frac{2}{1+i a x}\right )}{1+a^2 x^2} \, dx}{20 a^3}-\frac{c^2 \int \frac{\log \left (\frac{2}{1+i a x}\right )}{1+a^2 x^2} \, dx}{4 a^3}-\frac{\left (3 c^2\right ) \int \frac{\log \left (\frac{2}{1+i a x}\right )}{1+a^2 x^2} \, dx}{4 a^3}\\ &=\frac{c^2 x}{21 a^3}-\frac{c^2 x^3}{168 a}-\frac{1}{280} a c^2 x^5-\frac{c^2 \tan ^{-1}(a x)}{21 a^4}-\frac{5 c^2 x^2 \tan ^{-1}(a x)}{168 a^2}+\frac{1}{28} c^2 x^4 \tan ^{-1}(a x)+\frac{1}{56} a^2 c^2 x^6 \tan ^{-1}(a x)+\frac{2 i c^2 \tan ^{-1}(a x)^2}{21 a^4}+\frac{c^2 x \tan ^{-1}(a x)^2}{8 a^3}-\frac{c^2 x^3 \tan ^{-1}(a x)^2}{24 a}-\frac{1}{8} a c^2 x^5 \tan ^{-1}(a x)^2-\frac{3}{56} a^3 c^2 x^7 \tan ^{-1}(a x)^2-\frac{c^2 \tan ^{-1}(a x)^3}{24 a^4}+\frac{1}{4} c^2 x^4 \tan ^{-1}(a x)^3+\frac{1}{3} a^2 c^2 x^6 \tan ^{-1}(a x)^3+\frac{1}{8} a^4 c^2 x^8 \tan ^{-1}(a x)^3+\frac{4 c^2 \tan ^{-1}(a x) \log \left (\frac{2}{1+i a x}\right )}{21 a^4}-\frac{8 i c^2 \text{Li}_2\left (1-\frac{2}{1+i a x}\right )}{15 a^4}+\frac{\left (3 i c^2\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 (3 i c^2\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 (i c^2\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 (3 i c^2\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^2 x}{21 a^3}-\frac{c^2 x^3}{168 a}-\frac{1}{280} a c^2 x^5-\frac{c^2 \tan ^{-1}(a x)}{21 a^4}-\frac{5 c^2 x^2 \tan ^{-1}(a x)}{168 a^2}+\frac{1}{28} c^2 x^4 \tan ^{-1}(a x)+\frac{1}{56} a^2 c^2 x^6 \tan ^{-1}(a x)+\frac{2 i c^2 \tan ^{-1}(a x)^2}{21 a^4}+\frac{c^2 x \tan ^{-1}(a x)^2}{8 a^3}-\frac{c^2 x^3 \tan ^{-1}(a x)^2}{24 a}-\frac{1}{8} a c^2 x^5 \tan ^{-1}(a x)^2-\frac{3}{56} a^3 c^2 x^7 \tan ^{-1}(a x)^2-\frac{c^2 \tan ^{-1}(a x)^3}{24 a^4}+\frac{1}{4} c^2 x^4 \tan ^{-1}(a x)^3+\frac{1}{3} a^2 c^2 x^6 \tan ^{-1}(a x)^3+\frac{1}{8} a^4 c^2 x^8 \tan ^{-1}(a x)^3+\frac{4 c^2 \tan ^{-1}(a x) \log \left (\frac{2}{1+i a x}\right )}{21 a^4}+\frac{2 i c^2 \text{Li}_2\left (1-\frac{2}{1+i a x}\right )}{21 a^4}\\ \end{align*}
Mathematica [A] time = 1.22391, size = 165, normalized size = 0.53 \[ \frac{c^2 \left (-80 i \text{PolyLog}\left (2,-e^{2 i \tan ^{-1}(a x)}\right )-3 a^5 x^5-5 a^3 x^3+35 \left (a^2 x^2+1\right )^3 \left (3 a^2 x^2-1\right ) \tan ^{-1}(a x)^3-5 \left (9 a^7 x^7+21 a^5 x^5+7 a^3 x^3-21 a x+16 i\right ) \tan ^{-1}(a x)^2+5 \tan ^{-1}(a x) \left (3 a^6 x^6+6 a^4 x^4-5 a^2 x^2+32 \log \left (1+e^{2 i \tan ^{-1}(a x)}\right )-8\right )+40 a x\right )}{840 a^4} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.099, size = 411, normalized size = 1.3 \begin{align*}{\frac{{a}^{4}{c}^{2}{x}^{8} \left ( \arctan \left ( ax \right ) \right ) ^{3}}{8}}+{\frac{{a}^{2}{c}^{2}{x}^{6} \left ( \arctan \left ( ax \right ) \right ) ^{3}}{3}}+{\frac{{c}^{2}{x}^{4} \left ( \arctan \left ( ax \right ) \right ) ^{3}}{4}}-{\frac{3\,{a}^{3}{c}^{2}{x}^{7} \left ( \arctan \left ( ax \right ) \right ) ^{2}}{56}}-{\frac{a{c}^{2}{x}^{5} \left ( \arctan \left ( ax \right ) \right ) ^{2}}{8}}-{\frac{{c}^{2}{x}^{3} \left ( \arctan \left ( ax \right ) \right ) ^{2}}{24\,a}}+{\frac{{c}^{2}x \left ( \arctan \left ( ax \right ) \right ) ^{2}}{8\,{a}^{3}}}-{\frac{{c}^{2} \left ( \arctan \left ( ax \right ) \right ) ^{3}}{24\,{a}^{4}}}+{\frac{{a}^{2}{c}^{2}{x}^{6}\arctan \left ( ax \right ) }{56}}+{\frac{{c}^{2}{x}^{4}\arctan \left ( ax \right ) }{28}}-{\frac{5\,{c}^{2}{x}^{2}\arctan \left ( ax \right ) }{168\,{a}^{2}}}-{\frac{2\,{c}^{2}\arctan \left ( ax \right ) \ln \left ({a}^{2}{x}^{2}+1 \right ) }{21\,{a}^{4}}}-{\frac{a{c}^{2}{x}^{5}}{280}}-{\frac{{c}^{2}{x}^{3}}{168\,a}}+{\frac{{c}^{2}x}{21\,{a}^{3}}}-{\frac{{c}^{2}\arctan \left ( ax \right ) }{21\,{a}^{4}}}-{\frac{{\frac{i}{21}}{c}^{2}\ln \left ( ax-i \right ) \ln \left ({a}^{2}{x}^{2}+1 \right ) }{{a}^{4}}}-{\frac{{\frac{i}{21}}{c}^{2}\ln \left ( ax+i \right ) \ln \left ({\frac{i}{2}} \left ( ax-i \right ) \right ) }{{a}^{4}}}-{\frac{{\frac{i}{21}}{c}^{2}{\it dilog} \left ({\frac{i}{2}} \left ( ax-i \right ) \right ) }{{a}^{4}}}+{\frac{{\frac{i}{21}}{c}^{2}{\it dilog} \left ( -{\frac{i}{2}} \left ( ax+i \right ) \right ) }{{a}^{4}}}-{\frac{{\frac{i}{42}}{c}^{2} \left ( \ln \left ( ax+i \right ) \right ) ^{2}}{{a}^{4}}}+{\frac{{\frac{i}{42}}{c}^{2} \left ( \ln \left ( ax-i \right ) \right ) ^{2}}{{a}^{4}}}+{\frac{{\frac{i}{21}}{c}^{2}\ln \left ( ax+i \right ) \ln \left ({a}^{2}{x}^{2}+1 \right ) }{{a}^{4}}}+{\frac{{\frac{i}{21}}{c}^{2}\ln \left ( ax-i \right ) \ln \left ( -{\frac{i}{2}} \left ( ax+i \right ) \right ) }{{a}^{4}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (a^{4} c^{2} x^{7} + 2 \, a^{2} c^{2} x^{5} + c^{2} x^{3}\right )} \arctan \left (a x\right )^{3}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} c^{2} \left (\int x^{3} \operatorname{atan}^{3}{\left (a x \right )}\, dx + \int 2 a^{2} x^{5} \operatorname{atan}^{3}{\left (a x \right )}\, dx + \int a^{4} x^{7} \operatorname{atan}^{3}{\left (a x \right )}\, dx\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (a^{2} c x^{2} + c\right )}^{2} x^{3} \arctan \left (a x\right )^{3}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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