Optimal. Leaf size=274 \[ -\frac{16 i c^3 \text{PolyLog}\left (2,1-\frac{2}{1+i a x}\right )}{315 a^3}+\frac{1}{252} a^4 c^3 x^7+\frac{59 a^2 c^3 x^5}{3780}+\frac{1}{9} a^6 c^3 x^9 \tan ^{-1}(a x)^2-\frac{1}{36} a^5 c^3 x^8 \tan ^{-1}(a x)+\frac{3}{7} a^4 c^3 x^7 \tan ^{-1}(a x)^2-\frac{20}{189} a^3 c^3 x^6 \tan ^{-1}(a x)+\frac{3}{5} a^2 c^3 x^5 \tan ^{-1}(a x)^2-\frac{47 c^3 x}{3780 a^2}-\frac{16 i c^3 \tan ^{-1}(a x)^2}{315 a^3}+\frac{47 c^3 \tan ^{-1}(a x)}{3780 a^3}-\frac{32 c^3 \log \left (\frac{2}{1+i a x}\right ) \tan ^{-1}(a x)}{315 a^3}-\frac{89}{630} a c^3 x^4 \tan ^{-1}(a x)+\frac{1}{3} c^3 x^3 \tan ^{-1}(a x)^2-\frac{16 c^3 x^2 \tan ^{-1}(a x)}{315 a}+\frac{239 c^3 x^3}{11340} \]
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Rubi [A] time = 1.154, antiderivative size = 274, normalized size of antiderivative = 1., number of steps used = 68, number of rules used = 10, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.454, Rules used = {4948, 4852, 4916, 321, 203, 4920, 4854, 2402, 2315, 302} \[ -\frac{16 i c^3 \text{PolyLog}\left (2,1-\frac{2}{1+i a x}\right )}{315 a^3}+\frac{1}{252} a^4 c^3 x^7+\frac{59 a^2 c^3 x^5}{3780}+\frac{1}{9} a^6 c^3 x^9 \tan ^{-1}(a x)^2-\frac{1}{36} a^5 c^3 x^8 \tan ^{-1}(a x)+\frac{3}{7} a^4 c^3 x^7 \tan ^{-1}(a x)^2-\frac{20}{189} a^3 c^3 x^6 \tan ^{-1}(a x)+\frac{3}{5} a^2 c^3 x^5 \tan ^{-1}(a x)^2-\frac{47 c^3 x}{3780 a^2}-\frac{16 i c^3 \tan ^{-1}(a x)^2}{315 a^3}+\frac{47 c^3 \tan ^{-1}(a x)}{3780 a^3}-\frac{32 c^3 \log \left (\frac{2}{1+i a x}\right ) \tan ^{-1}(a x)}{315 a^3}-\frac{89}{630} a c^3 x^4 \tan ^{-1}(a x)+\frac{1}{3} c^3 x^3 \tan ^{-1}(a x)^2-\frac{16 c^3 x^2 \tan ^{-1}(a x)}{315 a}+\frac{239 c^3 x^3}{11340} \]
Antiderivative was successfully verified.
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Rule 4948
Rule 4852
Rule 4916
Rule 321
Rule 203
Rule 4920
Rule 4854
Rule 2402
Rule 2315
Rule 302
Rubi steps
\begin{align*} \int x^2 \left (c+a^2 c x^2\right )^3 \tan ^{-1}(a x)^2 \, dx &=\int \left (c^3 x^2 \tan ^{-1}(a x)^2+3 a^2 c^3 x^4 \tan ^{-1}(a x)^2+3 a^4 c^3 x^6 \tan ^{-1}(a x)^2+a^6 c^3 x^8 \tan ^{-1}(a x)^2\right ) \, dx\\ &=c^3 \int x^2 \tan ^{-1}(a x)^2 \, dx+\left (3 a^2 c^3\right ) \int x^4 \tan ^{-1}(a x)^2 \, dx+\left (3 a^4 c^3\right ) \int x^6 \tan ^{-1}(a x)^2 \, dx+\left (a^6 c^3\right ) \int x^8 \tan ^{-1}(a x)^2 \, dx\\ &=\frac{1}{3} c^3 x^3 \tan ^{-1}(a x)^2+\frac{3}{5} a^2 c^3 x^5 \tan ^{-1}(a x)^2+\frac{3}{7} a^4 c^3 x^7 \tan ^{-1}(a x)^2+\frac{1}{9} a^6 c^3 x^9 \tan ^{-1}(a x)^2-\frac{1}{3} \left (2 a c^3\right ) \int \frac{x^3 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx-\frac{1}{5} \left (6 a^3 c^3\right ) \int \frac{x^5 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx-\frac{1}{7} \left (6 a^5 c^3\right ) \int \frac{x^7 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx-\frac{1}{9} \left (2 a^7 c^3\right ) \int \frac{x^9 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx\\ &=\frac{1}{3} c^3 x^3 \tan ^{-1}(a x)^2+\frac{3}{5} a^2 c^3 x^5 \tan ^{-1}(a x)^2+\frac{3}{7} a^4 c^3 x^7 \tan ^{-1}(a x)^2+\frac{1}{9} a^6 c^3 x^9 \tan ^{-1}(a x)^2-\frac{\left (2 c^3\right ) \int x \tan ^{-1}(a x) \, dx}{3 a}+\frac{\left (2 c^3\right ) \int \frac{x \tan ^{-1}(a x)}{1+a^2 x^2} \, dx}{3 a}-\frac{1}{5} \left (6 a c^3\right ) \int x^3 \tan ^{-1}(a x) \, dx+\frac{1}{5} \left (6 a c^3\right ) \int \frac{x^3 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx-\frac{1}{7} \left (6 a^3 c^3\right ) \int x^5 \tan ^{-1}(a x) \, dx+\frac{1}{7} \left (6 a^3 c^3\right ) \int \frac{x^5 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx-\frac{1}{9} \left (2 a^5 c^3\right ) \int x^7 \tan ^{-1}(a x) \, dx+\frac{1}{9} \left (2 a^5 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)}{3 a}-\frac{3}{10} a c^3 x^4 \tan ^{-1}(a x)-\frac{1}{7} a^3 c^3 x^6 \tan ^{-1}(a x)-\frac{1}{36} a^5 c^3 x^8 \tan ^{-1}(a x)-\frac{i c^3 \tan ^{-1}(a x)^2}{3 a^3}+\frac{1}{3} c^3 x^3 \tan ^{-1}(a x)^2+\frac{3}{5} a^2 c^3 x^5 \tan ^{-1}(a x)^2+\frac{3}{7} a^4 c^3 x^7 \tan ^{-1}(a x)^2+\frac{1}{9} a^6 c^3 x^9 \tan ^{-1}(a x)^2+\frac{1}{3} c^3 \int \frac{x^2}{1+a^2 x^2} \, dx-\frac{\left (2 c^3\right ) \int \frac{\tan ^{-1}(a x)}{i-a x} \, dx}{3 a^2}+\frac{\left (6 c^3\right ) \int x \tan ^{-1}(a x) \, dx}{5 a}-\frac{\left (6 c^3\right ) \int \frac{x \tan ^{-1}(a x)}{1+a^2 x^2} \, dx}{5 a}+\frac{1}{7} \left (6 a c^3\right ) \int x^3 \tan ^{-1}(a x) \, dx-\frac{1}{7} \left (6 a c^3\right ) \int \frac{x^3 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx+\frac{1}{10} \left (3 a^2 c^3\right ) \int \frac{x^4}{1+a^2 x^2} \, dx+\frac{1}{9} \left (2 a^3 c^3\right ) \int x^5 \tan ^{-1}(a x) \, dx-\frac{1}{9} \left (2 a^3 c^3\right ) \int \frac{x^5 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx+\frac{1}{7} \left (a^4 c^3\right ) \int \frac{x^6}{1+a^2 x^2} \, dx+\frac{1}{36} \left (a^6 c^3\right ) \int \frac{x^8}{1+a^2 x^2} \, dx\\ &=\frac{c^3 x}{3 a^2}+\frac{4 c^3 x^2 \tan ^{-1}(a x)}{15 a}-\frac{3}{35} a c^3 x^4 \tan ^{-1}(a x)-\frac{20}{189} a^3 c^3 x^6 \tan ^{-1}(a x)-\frac{1}{36} a^5 c^3 x^8 \tan ^{-1}(a x)+\frac{4 i c^3 \tan ^{-1}(a x)^2}{15 a^3}+\frac{1}{3} c^3 x^3 \tan ^{-1}(a x)^2+\frac{3}{5} a^2 c^3 x^5 \tan ^{-1}(a x)^2+\frac{3}{7} a^4 c^3 x^7 \tan ^{-1}(a x)^2+\frac{1}{9} a^6 c^3 x^9 \tan ^{-1}(a x)^2-\frac{2 c^3 \tan ^{-1}(a x) \log \left (\frac{2}{1+i a x}\right )}{3 a^3}-\frac{1}{5} \left (3 c^3\right ) \int \frac{x^2}{1+a^2 x^2} \, dx-\frac{c^3 \int \frac{1}{1+a^2 x^2} \, dx}{3 a^2}+\frac{\left (2 c^3\right ) \int \frac{\log \left (\frac{2}{1+i a x}\right )}{1+a^2 x^2} \, dx}{3 a^2}+\frac{\left (6 c^3\right ) \int \frac{\tan ^{-1}(a x)}{i-a x} \, dx}{5 a^2}-\frac{\left (6 c^3\right ) \int x \tan ^{-1}(a x) \, dx}{7 a}+\frac{\left (6 c^3\right ) \int \frac{x \tan ^{-1}(a x)}{1+a^2 x^2} \, dx}{7 a}-\frac{1}{9} \left (2 a c^3\right ) \int x^3 \tan ^{-1}(a x) \, dx+\frac{1}{9} \left (2 a c^3\right ) \int \frac{x^3 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx-\frac{1}{14} \left (3 a^2 c^3\right ) \int \frac{x^4}{1+a^2 x^2} \, dx+\frac{1}{10} \left (3 a^2 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}{27} \left (a^4 c^3\right ) \int \frac{x^6}{1+a^2 x^2} \, dx+\frac{1}{7} \left (a^4 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}{36} \left (a^6 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{569 c^3 x}{1260 a^2}+\frac{233 c^3 x^3}{3780}+\frac{29 a^2 c^3 x^5}{1260}+\frac{1}{252} a^4 c^3 x^7-\frac{c^3 \tan ^{-1}(a x)}{3 a^3}-\frac{17 c^3 x^2 \tan ^{-1}(a x)}{105 a}-\frac{89}{630} a c^3 x^4 \tan ^{-1}(a x)-\frac{20}{189} a^3 c^3 x^6 \tan ^{-1}(a x)-\frac{1}{36} a^5 c^3 x^8 \tan ^{-1}(a x)-\frac{17 i c^3 \tan ^{-1}(a x)^2}{105 a^3}+\frac{1}{3} c^3 x^3 \tan ^{-1}(a x)^2+\frac{3}{5} a^2 c^3 x^5 \tan ^{-1}(a x)^2+\frac{3}{7} a^4 c^3 x^7 \tan ^{-1}(a x)^2+\frac{1}{9} a^6 c^3 x^9 \tan ^{-1}(a x)^2+\frac{8 c^3 \tan ^{-1}(a x) \log \left (\frac{2}{1+i a x}\right )}{15 a^3}+\frac{1}{7} \left (3 c^3\right ) \int \frac{x^2}{1+a^2 x^2} \, dx-\frac{\left (2 i c^3\right ) \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1+i a x}\right )}{3 a^3}+\frac{c^3 \int \frac{1}{1+a^2 x^2} \, dx}{36 a^2}-\frac{c^3 \int \frac{1}{1+a^2 x^2} \, dx}{7 a^2}+\frac{\left (3 c^3\right ) \int \frac{1}{1+a^2 x^2} \, dx}{10 a^2}+\frac{\left (3 c^3\right ) \int \frac{1}{1+a^2 x^2} \, dx}{5 a^2}-\frac{\left (6 c^3\right ) \int \frac{\tan ^{-1}(a x)}{i-a x} \, dx}{7 a^2}-\frac{\left (6 c^3\right ) \int \frac{\log \left (\frac{2}{1+i a x}\right )}{1+a^2 x^2} \, dx}{5 a^2}+\frac{\left (2 c^3\right ) \int x \tan ^{-1}(a x) \, dx}{9 a}-\frac{\left (2 c^3\right ) \int \frac{x \tan ^{-1}(a x)}{1+a^2 x^2} \, dx}{9 a}+\frac{1}{18} \left (a^2 c^3\right ) \int \frac{x^4}{1+a^2 x^2} \, dx-\frac{1}{14} \left (3 a^2 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}{27} \left (a^4 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{583 c^3 x}{3780 a^2}+\frac{29 c^3 x^3}{11340}+\frac{59 a^2 c^3 x^5}{3780}+\frac{1}{252} a^4 c^3 x^7+\frac{569 c^3 \tan ^{-1}(a x)}{1260 a^3}-\frac{16 c^3 x^2 \tan ^{-1}(a x)}{315 a}-\frac{89}{630} a c^3 x^4 \tan ^{-1}(a x)-\frac{20}{189} a^3 c^3 x^6 \tan ^{-1}(a x)-\frac{1}{36} a^5 c^3 x^8 \tan ^{-1}(a x)-\frac{16 i c^3 \tan ^{-1}(a x)^2}{315 a^3}+\frac{1}{3} c^3 x^3 \tan ^{-1}(a x)^2+\frac{3}{5} a^2 c^3 x^5 \tan ^{-1}(a x)^2+\frac{3}{7} a^4 c^3 x^7 \tan ^{-1}(a x)^2+\frac{1}{9} a^6 c^3 x^9 \tan ^{-1}(a x)^2-\frac{34 c^3 \tan ^{-1}(a x) \log \left (\frac{2}{1+i a x}\right )}{105 a^3}-\frac{i c^3 \text{Li}_2\left (1-\frac{2}{1+i a x}\right )}{3 a^3}-\frac{1}{9} c^3 \int \frac{x^2}{1+a^2 x^2} \, dx+\frac{\left (6 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^3}+\frac{c^3 \int \frac{1}{1+a^2 x^2} \, dx}{27 a^2}-\frac{\left (3 c^3\right ) \int \frac{1}{1+a^2 x^2} \, dx}{14 a^2}+\frac{\left (2 c^3\right ) \int \frac{\tan ^{-1}(a x)}{i-a x} \, dx}{9 a^2}-\frac{\left (3 c^3\right ) \int \frac{1}{1+a^2 x^2} \, dx}{7 a^2}+\frac{\left (6 c^3\right ) \int \frac{\log \left (\frac{2}{1+i a x}\right )}{1+a^2 x^2} \, dx}{7 a^2}+\frac{1}{18} \left (a^2 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{47 c^3 x}{3780 a^2}+\frac{239 c^3 x^3}{11340}+\frac{59 a^2 c^3 x^5}{3780}+\frac{1}{252} a^4 c^3 x^7-\frac{583 c^3 \tan ^{-1}(a x)}{3780 a^3}-\frac{16 c^3 x^2 \tan ^{-1}(a x)}{315 a}-\frac{89}{630} a c^3 x^4 \tan ^{-1}(a x)-\frac{20}{189} a^3 c^3 x^6 \tan ^{-1}(a x)-\frac{1}{36} a^5 c^3 x^8 \tan ^{-1}(a x)-\frac{16 i c^3 \tan ^{-1}(a x)^2}{315 a^3}+\frac{1}{3} c^3 x^3 \tan ^{-1}(a x)^2+\frac{3}{5} a^2 c^3 x^5 \tan ^{-1}(a x)^2+\frac{3}{7} a^4 c^3 x^7 \tan ^{-1}(a x)^2+\frac{1}{9} a^6 c^3 x^9 \tan ^{-1}(a x)^2-\frac{32 c^3 \tan ^{-1}(a x) \log \left (\frac{2}{1+i a x}\right )}{315 a^3}+\frac{4 i c^3 \text{Li}_2\left (1-\frac{2}{1+i a x}\right )}{15 a^3}-\frac{\left (6 i c^3\right ) \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1+i a x}\right )}{7 a^3}+\frac{c^3 \int \frac{1}{1+a^2 x^2} \, dx}{18 a^2}+\frac{c^3 \int \frac{1}{1+a^2 x^2} \, dx}{9 a^2}-\frac{\left (2 c^3\right ) \int \frac{\log \left (\frac{2}{1+i a x}\right )}{1+a^2 x^2} \, dx}{9 a^2}\\ &=-\frac{47 c^3 x}{3780 a^2}+\frac{239 c^3 x^3}{11340}+\frac{59 a^2 c^3 x^5}{3780}+\frac{1}{252} a^4 c^3 x^7+\frac{47 c^3 \tan ^{-1}(a x)}{3780 a^3}-\frac{16 c^3 x^2 \tan ^{-1}(a x)}{315 a}-\frac{89}{630} a c^3 x^4 \tan ^{-1}(a x)-\frac{20}{189} a^3 c^3 x^6 \tan ^{-1}(a x)-\frac{1}{36} a^5 c^3 x^8 \tan ^{-1}(a x)-\frac{16 i c^3 \tan ^{-1}(a x)^2}{315 a^3}+\frac{1}{3} c^3 x^3 \tan ^{-1}(a x)^2+\frac{3}{5} a^2 c^3 x^5 \tan ^{-1}(a x)^2+\frac{3}{7} a^4 c^3 x^7 \tan ^{-1}(a x)^2+\frac{1}{9} a^6 c^3 x^9 \tan ^{-1}(a x)^2-\frac{32 c^3 \tan ^{-1}(a x) \log \left (\frac{2}{1+i a x}\right )}{315 a^3}-\frac{17 i c^3 \text{Li}_2\left (1-\frac{2}{1+i a x}\right )}{105 a^3}+\frac{\left (2 i c^3\right ) \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1+i a x}\right )}{9 a^3}\\ &=-\frac{47 c^3 x}{3780 a^2}+\frac{239 c^3 x^3}{11340}+\frac{59 a^2 c^3 x^5}{3780}+\frac{1}{252} a^4 c^3 x^7+\frac{47 c^3 \tan ^{-1}(a x)}{3780 a^3}-\frac{16 c^3 x^2 \tan ^{-1}(a x)}{315 a}-\frac{89}{630} a c^3 x^4 \tan ^{-1}(a x)-\frac{20}{189} a^3 c^3 x^6 \tan ^{-1}(a x)-\frac{1}{36} a^5 c^3 x^8 \tan ^{-1}(a x)-\frac{16 i c^3 \tan ^{-1}(a x)^2}{315 a^3}+\frac{1}{3} c^3 x^3 \tan ^{-1}(a x)^2+\frac{3}{5} a^2 c^3 x^5 \tan ^{-1}(a x)^2+\frac{3}{7} a^4 c^3 x^7 \tan ^{-1}(a x)^2+\frac{1}{9} a^6 c^3 x^9 \tan ^{-1}(a x)^2-\frac{32 c^3 \tan ^{-1}(a x) \log \left (\frac{2}{1+i a x}\right )}{315 a^3}-\frac{16 i c^3 \text{Li}_2\left (1-\frac{2}{1+i a x}\right )}{315 a^3}\\ \end{align*}
Mathematica [A] time = 2.22103, size = 157, normalized size = 0.57 \[ \frac{c^3 \left (576 i \text{PolyLog}\left (2,-e^{2 i \tan ^{-1}(a x)}\right )+a x \left (45 a^6 x^6+177 a^4 x^4+239 a^2 x^2-141\right )+36 \left (35 a^9 x^9+135 a^7 x^7+189 a^5 x^5+105 a^3 x^3+16 i\right ) \tan ^{-1}(a x)^2-3 \tan ^{-1}(a x) \left (105 a^8 x^8+400 a^6 x^6+534 a^4 x^4+192 a^2 x^2+384 \log \left (1+e^{2 i \tan ^{-1}(a x)}\right )-47\right )\right )}{11340 a^3} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.089, size = 376, normalized size = 1.4 \begin{align*}{\frac{{a}^{6}{c}^{3}{x}^{9} \left ( \arctan \left ( ax \right ) \right ) ^{2}}{9}}+{\frac{3\,{a}^{4}{c}^{3}{x}^{7} \left ( \arctan \left ( ax \right ) \right ) ^{2}}{7}}+{\frac{3\,{a}^{2}{c}^{3}{x}^{5} \left ( \arctan \left ( ax \right ) \right ) ^{2}}{5}}+{\frac{{c}^{3}{x}^{3} \left ( \arctan \left ( ax \right ) \right ) ^{2}}{3}}-{\frac{{a}^{5}{c}^{3}{x}^{8}\arctan \left ( ax \right ) }{36}}-{\frac{20\,{a}^{3}{c}^{3}{x}^{6}\arctan \left ( ax \right ) }{189}}-{\frac{89\,a{c}^{3}{x}^{4}\arctan \left ( ax \right ) }{630}}-{\frac{16\,{c}^{3}{x}^{2}\arctan \left ( ax \right ) }{315\,a}}+{\frac{16\,{c}^{3}\arctan \left ( ax \right ) \ln \left ({a}^{2}{x}^{2}+1 \right ) }{315\,{a}^{3}}}+{\frac{{a}^{4}{c}^{3}{x}^{7}}{252}}+{\frac{59\,{a}^{2}{c}^{3}{x}^{5}}{3780}}+{\frac{239\,{c}^{3}{x}^{3}}{11340}}-{\frac{47\,{c}^{3}x}{3780\,{a}^{2}}}+{\frac{47\,{c}^{3}\arctan \left ( ax \right ) }{3780\,{a}^{3}}}-{\frac{{\frac{8\,i}{315}}{c}^{3}\ln \left ( ax-i \right ) \ln \left ( -{\frac{i}{2}} \left ( ax+i \right ) \right ) }{{a}^{3}}}+{\frac{{\frac{8\,i}{315}}{c}^{3}\ln \left ( ax-i \right ) \ln \left ({a}^{2}{x}^{2}+1 \right ) }{{a}^{3}}}-{\frac{{\frac{8\,i}{315}}{c}^{3}\ln \left ({a}^{2}{x}^{2}+1 \right ) \ln \left ( ax+i \right ) }{{a}^{3}}}-{\frac{{\frac{8\,i}{315}}{c}^{3}{\it dilog} \left ( -{\frac{i}{2}} \left ( ax+i \right ) \right ) }{{a}^{3}}}-{\frac{{\frac{4\,i}{315}}{c}^{3} \left ( \ln \left ( ax-i \right ) \right ) ^{2}}{{a}^{3}}}+{\frac{{\frac{8\,i}{315}}{c}^{3}{\it dilog} \left ({\frac{i}{2}} \left ( ax-i \right ) \right ) }{{a}^{3}}}+{\frac{{\frac{4\,i}{315}}{c}^{3} \left ( \ln \left ( ax+i \right ) \right ) ^{2}}{{a}^{3}}}+{\frac{{\frac{8\,i}{315}}{c}^{3}\ln \left ( ax+i \right ) \ln \left ({\frac{i}{2}} \left ( ax-i \right ) \right ) }{{a}^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{1260} \,{\left (35 \, a^{6} c^{3} x^{9} + 135 \, a^{4} c^{3} x^{7} + 189 \, a^{2} c^{3} x^{5} + 105 \, c^{3} x^{3}\right )} \arctan \left (a x\right )^{2} - \frac{1}{5040} \,{\left (35 \, a^{6} c^{3} x^{9} + 135 \, a^{4} c^{3} x^{7} + 189 \, a^{2} c^{3} x^{5} + 105 \, c^{3} x^{3}\right )} \log \left (a^{2} x^{2} + 1\right )^{2} + \int \frac{3780 \,{\left (a^{8} c^{3} x^{10} + 4 \, a^{6} c^{3} x^{8} + 6 \, a^{4} c^{3} x^{6} + 4 \, a^{2} c^{3} x^{4} + c^{3} x^{2}\right )} \arctan \left (a x\right )^{2} + 315 \,{\left (a^{8} c^{3} x^{10} + 4 \, a^{6} c^{3} x^{8} + 6 \, a^{4} c^{3} x^{6} + 4 \, a^{2} c^{3} x^{4} + c^{3} x^{2}\right )} \log \left (a^{2} x^{2} + 1\right )^{2} - 8 \,{\left (35 \, a^{7} c^{3} x^{9} + 135 \, a^{5} c^{3} x^{7} + 189 \, a^{3} c^{3} x^{5} + 105 \, a c^{3} x^{3}\right )} \arctan \left (a x\right ) + 4 \,{\left (35 \, a^{8} c^{3} x^{10} + 135 \, a^{6} c^{3} x^{8} + 189 \, a^{4} c^{3} x^{6} + 105 \, a^{2} c^{3} x^{4}\right )} \log \left (a^{2} x^{2} + 1\right )}{5040 \,{\left (a^{2} x^{2} + 1\right )}}\,{d x} \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^{6} c^{3} x^{8} + 3 \, a^{4} c^{3} x^{6} + 3 \, a^{2} c^{3} x^{4} + c^{3} x^{2}\right )} \arctan \left (a x\right )^{2}, 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^{3} \left (\int x^{2} \operatorname{atan}^{2}{\left (a x \right )}\, dx + \int 3 a^{2} x^{4} \operatorname{atan}^{2}{\left (a x \right )}\, dx + \int 3 a^{4} x^{6} \operatorname{atan}^{2}{\left (a x \right )}\, dx + \int a^{6} x^{8} \operatorname{atan}^{2}{\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 )}^{3} x^{2} \arctan \left (a x\right )^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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