Optimal. Leaf size=1393 \[ \text{result too large to display} \]
[Out]
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Rubi [A] time = 2.60206, antiderivative size = 1393, normalized size of antiderivative = 1., number of steps used = 86, number of rules used = 27, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 1.687, Rules used = {5035, 2457, 2476, 2448, 321, 203, 2455, 302, 2470, 12, 4920, 4854, 2402, 2315, 6742, 206, 30, 2557, 205, 4928, 4856, 2447, 208, 5992, 5920, 5984, 5918} \[ \text{result too large to display} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 5035
Rule 2457
Rule 2476
Rule 2448
Rule 321
Rule 203
Rule 2455
Rule 302
Rule 2470
Rule 12
Rule 4920
Rule 4854
Rule 2402
Rule 2315
Rule 6742
Rule 206
Rule 30
Rule 2557
Rule 205
Rule 4928
Rule 4856
Rule 2447
Rule 208
Rule 5992
Rule 5920
Rule 5984
Rule 5918
Rubi steps
\begin{align*} \int x^2 \left (a+b \tan ^{-1}\left (c x^2\right )\right )^2 \, dx &=\int \left (\frac{1}{4} x^2 \left (2 a+i b \log \left (1-i c x^2\right )\right )^2+\frac{1}{2} b x^2 \left (-2 i a+b \log \left (1-i c x^2\right )\right ) \log \left (1+i c x^2\right )-\frac{1}{4} b^2 x^2 \log ^2\left (1+i c x^2\right )\right ) \, dx\\ &=\frac{1}{4} \int x^2 \left (2 a+i b \log \left (1-i c x^2\right )\right )^2 \, dx+\frac{1}{2} b \int x^2 \left (-2 i a+b \log \left (1-i c x^2\right )\right ) \log \left (1+i c x^2\right ) \, dx-\frac{1}{4} b^2 \int x^2 \log ^2\left (1+i c x^2\right ) \, dx\\ &=\frac{1}{12} x^3 \left (2 a+i b \log \left (1-i c x^2\right )\right )^2-\frac{1}{12} b^2 x^3 \log ^2\left (1+i c x^2\right )+\frac{1}{2} b \int \left (-2 i a x^2 \log \left (1+i c x^2\right )+b x^2 \log \left (1-i c x^2\right ) \log \left (1+i c x^2\right )\right ) \, dx-\frac{1}{3} (b c) \int \frac{x^4 \left (2 a+i b \log \left (1-i c x^2\right )\right )}{1-i c x^2} \, dx+\frac{1}{3} \left (i b^2 c\right ) \int \frac{x^4 \log \left (1+i c x^2\right )}{1+i c x^2} \, dx\\ &=\frac{1}{12} x^3 \left (2 a+i b \log \left (1-i c x^2\right )\right )^2-\frac{1}{12} b^2 x^3 \log ^2\left (1+i c x^2\right )-(i a b) \int x^2 \log \left (1+i c x^2\right ) \, dx+\frac{1}{2} b^2 \int x^2 \log \left (1-i c x^2\right ) \log \left (1+i c x^2\right ) \, dx-\frac{1}{3} (b c) \int \left (\frac{2 a+i b \log \left (1-i c x^2\right )}{c^2}+\frac{i x^2 \left (2 a+i b \log \left (1-i c x^2\right )\right )}{c}-\frac{2 a+i b \log \left (1-i c x^2\right )}{c^2 \left (1-i c x^2\right )}\right ) \, dx+\frac{1}{3} \left (i b^2 c\right ) \int \left (\frac{\log \left (1+i c x^2\right )}{c^2}-\frac{i x^2 \log \left (1+i c x^2\right )}{c}-\frac{\log \left (1+i c x^2\right )}{c^2 \left (1+i c x^2\right )}\right ) \, dx\\ &=\frac{1}{12} x^3 \left (2 a+i b \log \left (1-i c x^2\right )\right )^2-\frac{1}{3} i a b x^3 \log \left (1+i c x^2\right )+\frac{1}{6} b^2 x^3 \log \left (1-i c x^2\right ) \log \left (1+i c x^2\right )-\frac{1}{12} b^2 x^3 \log ^2\left (1+i c x^2\right )-\frac{1}{3} (i b) \int x^2 \left (2 a+i b \log \left (1-i c x^2\right )\right ) \, dx+\frac{1}{3} b^2 \int x^2 \log \left (1+i c x^2\right ) \, dx-\frac{1}{2} b^2 \int \frac{2 c x^4 \log \left (1-i c x^2\right )}{-3 i+3 c x^2} \, dx-\frac{1}{2} b^2 \int \frac{2 c x^4 \log \left (1+i c x^2\right )}{3 i+3 c x^2} \, dx-\frac{b \int \left (2 a+i b \log \left (1-i c x^2\right )\right ) \, dx}{3 c}+\frac{b \int \frac{2 a+i b \log \left (1-i c x^2\right )}{1-i c x^2} \, dx}{3 c}+\frac{\left (i b^2\right ) \int \log \left (1+i c x^2\right ) \, dx}{3 c}-\frac{\left (i b^2\right ) \int \frac{\log \left (1+i c x^2\right )}{1+i c x^2} \, dx}{3 c}-\frac{1}{3} (2 a b c) \int \frac{x^4}{1+i c x^2} \, dx\\ &=-\frac{2 a b x}{3 c}-\frac{1}{9} i b x^3 \left (2 a+i b \log \left (1-i c x^2\right )\right )-\frac{\sqrt [4]{-1} b \tan ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right ) \left (2 a+i b \log \left (1-i c x^2\right )\right )}{3 c^{3/2}}+\frac{1}{12} x^3 \left (2 a+i b \log \left (1-i c x^2\right )\right )^2+\frac{i b^2 x \log \left (1+i c x^2\right )}{3 c}-\frac{1}{3} i a b x^3 \log \left (1+i c x^2\right )+\frac{1}{9} b^2 x^3 \log \left (1+i c x^2\right )+\frac{(-1)^{3/4} b^2 \tanh ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right ) \log \left (1+i c x^2\right )}{3 c^{3/2}}+\frac{1}{6} b^2 x^3 \log \left (1-i c x^2\right ) \log \left (1+i c x^2\right )-\frac{1}{12} b^2 x^3 \log ^2\left (1+i c x^2\right )+\frac{1}{3} \left (2 b^2\right ) \int \frac{x^2}{1+i c x^2} \, dx+\frac{1}{3} \left (2 b^2\right ) \int \frac{\sqrt [4]{-1} x \tan ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right )}{\sqrt{c} \left (1-i c x^2\right )} \, dx+\frac{1}{3} \left (2 b^2\right ) \int \frac{\sqrt [4]{-1} x \tanh ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right )}{\sqrt{c} \left (1+i c x^2\right )} \, dx-\frac{\left (i b^2\right ) \int \log \left (1-i c x^2\right ) \, dx}{3 c}-\frac{1}{3} (2 a b c) \int \left (\frac{1}{c^2}-\frac{i x^2}{c}-\frac{1}{c^2 \left (1+i c x^2\right )}\right ) \, dx+\frac{1}{9} \left (2 i b^2 c\right ) \int \frac{x^4}{1-i c x^2} \, dx-\frac{1}{9} \left (2 i b^2 c\right ) \int \frac{x^4}{1+i c x^2} \, dx-\left (b^2 c\right ) \int \frac{x^4 \log \left (1-i c x^2\right )}{-3 i+3 c x^2} \, dx-\left (b^2 c\right ) \int \frac{x^4 \log \left (1+i c x^2\right )}{3 i+3 c x^2} \, dx\\ &=-\frac{4 a b x}{3 c}-\frac{2 i b^2 x}{3 c}+\frac{2}{9} i a b x^3-\frac{i b^2 x \log \left (1-i c x^2\right )}{3 c}-\frac{1}{9} i b x^3 \left (2 a+i b \log \left (1-i c x^2\right )\right )-\frac{\sqrt [4]{-1} b \tan ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right ) \left (2 a+i b \log \left (1-i c x^2\right )\right )}{3 c^{3/2}}+\frac{1}{12} x^3 \left (2 a+i b \log \left (1-i c x^2\right )\right )^2+\frac{i b^2 x \log \left (1+i c x^2\right )}{3 c}-\frac{1}{3} i a b x^3 \log \left (1+i c x^2\right )+\frac{1}{9} b^2 x^3 \log \left (1+i c x^2\right )+\frac{(-1)^{3/4} b^2 \tanh ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right ) \log \left (1+i c x^2\right )}{3 c^{3/2}}+\frac{1}{6} b^2 x^3 \log \left (1-i c x^2\right ) \log \left (1+i c x^2\right )-\frac{1}{12} b^2 x^3 \log ^2\left (1+i c x^2\right )+\frac{1}{3} \left (2 b^2\right ) \int \frac{x^2}{1-i c x^2} \, dx+\frac{(2 a b) \int \frac{1}{1+i c x^2} \, dx}{3 c}+\frac{\left (2 i b^2\right ) \int \frac{1}{1+i c x^2} \, dx}{3 c}+\frac{\left (2 \sqrt [4]{-1} b^2\right ) \int \frac{x \tan ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right )}{1-i c x^2} \, dx}{3 \sqrt{c}}+\frac{\left (2 \sqrt [4]{-1} b^2\right ) \int \frac{x \tanh ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right )}{1+i c x^2} \, dx}{3 \sqrt{c}}+\frac{1}{9} \left (2 i b^2 c\right ) \int \left (\frac{1}{c^2}+\frac{i x^2}{c}-\frac{1}{c^2 \left (1-i c x^2\right )}\right ) \, dx-\frac{1}{9} \left (2 i b^2 c\right ) \int \left (\frac{1}{c^2}-\frac{i x^2}{c}-\frac{1}{c^2 \left (1+i c x^2\right )}\right ) \, dx-\left (b^2 c\right ) \int \left (\frac{i \log \left (1-i c x^2\right )}{3 c^2}+\frac{x^2 \log \left (1-i c x^2\right )}{3 c}-\frac{\log \left (1-i c x^2\right )}{c^2 \left (-3 i+3 c x^2\right )}\right ) \, dx-\left (b^2 c\right ) \int \left (-\frac{i \log \left (1+i c x^2\right )}{3 c^2}+\frac{x^2 \log \left (1+i c x^2\right )}{3 c}-\frac{\log \left (1+i c x^2\right )}{c^2 \left (3 i+3 c x^2\right )}\right ) \, dx\\ &=-\frac{4 a b x}{3 c}+\frac{2}{9} i a b x^3-\frac{4 b^2 x^3}{27}+\frac{\sqrt [4]{-1} b^2 \tan ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right )^2}{3 c^{3/2}}-\frac{2 \sqrt [4]{-1} a b \tanh ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right )}{3 c^{3/2}}-\frac{2 (-1)^{3/4} b^2 \tanh ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right )}{3 c^{3/2}}-\frac{(-1)^{3/4} b^2 \tanh ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right )^2}{3 c^{3/2}}-\frac{i b^2 x \log \left (1-i c x^2\right )}{3 c}-\frac{1}{9} i b x^3 \left (2 a+i b \log \left (1-i c x^2\right )\right )-\frac{\sqrt [4]{-1} b \tan ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right ) \left (2 a+i b \log \left (1-i c x^2\right )\right )}{3 c^{3/2}}+\frac{1}{12} x^3 \left (2 a+i b \log \left (1-i c x^2\right )\right )^2+\frac{i b^2 x \log \left (1+i c x^2\right )}{3 c}-\frac{1}{3} i a b x^3 \log \left (1+i c x^2\right )+\frac{1}{9} b^2 x^3 \log \left (1+i c x^2\right )+\frac{(-1)^{3/4} b^2 \tanh ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right ) \log \left (1+i c x^2\right )}{3 c^{3/2}}+\frac{1}{6} b^2 x^3 \log \left (1-i c x^2\right ) \log \left (1+i c x^2\right )-\frac{1}{12} b^2 x^3 \log ^2\left (1+i c x^2\right )-\frac{1}{3} b^2 \int x^2 \log \left (1-i c x^2\right ) \, dx-\frac{1}{3} b^2 \int x^2 \log \left (1+i c x^2\right ) \, dx-\frac{\left (2 i b^2\right ) \int \frac{1}{1-i c x^2} \, dx}{9 c}+\frac{\left (2 i b^2\right ) \int \frac{1}{1+i c x^2} \, dx}{9 c}-\frac{\left (i b^2\right ) \int \log \left (1-i c x^2\right ) \, dx}{3 c}+\frac{\left (i b^2\right ) \int \log \left (1+i c x^2\right ) \, dx}{3 c}-\frac{\left (2 i b^2\right ) \int \frac{1}{1-i c x^2} \, dx}{3 c}+\frac{\left (2 i b^2\right ) \int \frac{\tan ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right )}{i-(-1)^{3/4} \sqrt{c} x} \, dx}{3 c}-\frac{\left (2 i b^2\right ) \int \frac{\tanh ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right )}{1-(-1)^{3/4} \sqrt{c} x} \, dx}{3 c}+\frac{b^2 \int \frac{\log \left (1-i c x^2\right )}{-3 i+3 c x^2} \, dx}{c}+\frac{b^2 \int \frac{\log \left (1+i c x^2\right )}{3 i+3 c x^2} \, dx}{c}\\ &=-\frac{4 a b x}{3 c}+\frac{2}{9} i a b x^3-\frac{4 b^2 x^3}{27}+\frac{8 (-1)^{3/4} b^2 \tan ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right )}{9 c^{3/2}}+\frac{\sqrt [4]{-1} b^2 \tan ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right )^2}{3 c^{3/2}}-\frac{2 \sqrt [4]{-1} a b \tanh ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right )}{3 c^{3/2}}-\frac{8 (-1)^{3/4} b^2 \tanh ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right )}{9 c^{3/2}}-\frac{(-1)^{3/4} b^2 \tanh ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right )^2}{3 c^{3/2}}-\frac{2 (-1)^{3/4} b^2 \tan ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right ) \log \left (\frac{2}{1-\sqrt [4]{-1} \sqrt{c} x}\right )}{3 c^{3/2}}+\frac{2 (-1)^{3/4} b^2 \tanh ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right ) \log \left (\frac{2}{1-(-1)^{3/4} \sqrt{c} x}\right )}{3 c^{3/2}}-\frac{2 i b^2 x \log \left (1-i c x^2\right )}{3 c}-\frac{1}{9} b^2 x^3 \log \left (1-i c x^2\right )-\frac{(-1)^{3/4} b^2 \tanh ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right ) \log \left (1-i c x^2\right )}{3 c^{3/2}}-\frac{1}{9} i b x^3 \left (2 a+i b \log \left (1-i c x^2\right )\right )-\frac{\sqrt [4]{-1} b \tan ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right ) \left (2 a+i b \log \left (1-i c x^2\right )\right )}{3 c^{3/2}}+\frac{1}{12} x^3 \left (2 a+i b \log \left (1-i c x^2\right )\right )^2+\frac{2 i b^2 x \log \left (1+i c x^2\right )}{3 c}-\frac{1}{3} i a b x^3 \log \left (1+i c x^2\right )+\frac{(-1)^{3/4} b^2 \tan ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right ) \log \left (1+i c x^2\right )}{3 c^{3/2}}+\frac{(-1)^{3/4} b^2 \tanh ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right ) \log \left (1+i c x^2\right )}{3 c^{3/2}}+\frac{1}{6} b^2 x^3 \log \left (1-i c x^2\right ) \log \left (1+i c x^2\right )-\frac{1}{12} b^2 x^3 \log ^2\left (1+i c x^2\right )-\left (2 i b^2\right ) \int \frac{(-1)^{3/4} x \tan ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right )}{3 \sqrt{c} \left (1+i c x^2\right )} \, dx+\left (2 i b^2\right ) \int -\frac{(-1)^{3/4} x \tanh ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right )}{3 \sqrt{c} \left (1-i c x^2\right )} \, dx+\frac{1}{3} \left (2 b^2\right ) \int \frac{x^2}{1-i c x^2} \, dx+\frac{1}{3} \left (2 b^2\right ) \int \frac{x^2}{1+i c x^2} \, dx-\frac{\left (2 i b^2\right ) \int \frac{\log \left (\frac{2}{1-\sqrt [4]{-1} \sqrt{c} x}\right )}{1-i c x^2} \, dx}{3 c}+\frac{\left (2 i b^2\right ) \int \frac{\log \left (\frac{2}{1-(-1)^{3/4} \sqrt{c} x}\right )}{1+i c x^2} \, dx}{3 c}-\frac{1}{9} \left (2 i b^2 c\right ) \int \frac{x^4}{1-i c x^2} \, dx+\frac{1}{9} \left (2 i b^2 c\right ) \int \frac{x^4}{1+i c x^2} \, dx\\ &=-\frac{4 a b x}{3 c}+\frac{2}{9} i a b x^3-\frac{4 b^2 x^3}{27}+\frac{8 (-1)^{3/4} b^2 \tan ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right )}{9 c^{3/2}}+\frac{\sqrt [4]{-1} b^2 \tan ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right )^2}{3 c^{3/2}}-\frac{2 \sqrt [4]{-1} a b \tanh ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right )}{3 c^{3/2}}-\frac{8 (-1)^{3/4} b^2 \tanh ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right )}{9 c^{3/2}}-\frac{(-1)^{3/4} b^2 \tanh ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right )^2}{3 c^{3/2}}-\frac{2 (-1)^{3/4} b^2 \tan ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right ) \log \left (\frac{2}{1-\sqrt [4]{-1} \sqrt{c} x}\right )}{3 c^{3/2}}+\frac{2 (-1)^{3/4} b^2 \tanh ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right ) \log \left (\frac{2}{1-(-1)^{3/4} \sqrt{c} x}\right )}{3 c^{3/2}}-\frac{2 i b^2 x \log \left (1-i c x^2\right )}{3 c}-\frac{1}{9} b^2 x^3 \log \left (1-i c x^2\right )-\frac{(-1)^{3/4} b^2 \tanh ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right ) \log \left (1-i c x^2\right )}{3 c^{3/2}}-\frac{1}{9} i b x^3 \left (2 a+i b \log \left (1-i c x^2\right )\right )-\frac{\sqrt [4]{-1} b \tan ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right ) \left (2 a+i b \log \left (1-i c x^2\right )\right )}{3 c^{3/2}}+\frac{1}{12} x^3 \left (2 a+i b \log \left (1-i c x^2\right )\right )^2+\frac{2 i b^2 x \log \left (1+i c x^2\right )}{3 c}-\frac{1}{3} i a b x^3 \log \left (1+i c x^2\right )+\frac{(-1)^{3/4} b^2 \tan ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right ) \log \left (1+i c x^2\right )}{3 c^{3/2}}+\frac{(-1)^{3/4} b^2 \tanh ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right ) \log \left (1+i c x^2\right )}{3 c^{3/2}}+\frac{1}{6} b^2 x^3 \log \left (1-i c x^2\right ) \log \left (1+i c x^2\right )-\frac{1}{12} b^2 x^3 \log ^2\left (1+i c x^2\right )+\frac{\left (2 \sqrt [4]{-1} b^2\right ) \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1-\sqrt [4]{-1} \sqrt{c} x}\right )}{3 c^{3/2}}+\frac{\left (2 (-1)^{3/4} b^2\right ) \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1-(-1)^{3/4} \sqrt{c} x}\right )}{3 c^{3/2}}-\frac{\left (2 i b^2\right ) \int \frac{1}{1-i c x^2} \, dx}{3 c}+\frac{\left (2 i b^2\right ) \int \frac{1}{1+i c x^2} \, dx}{3 c}+\frac{\left (2 \sqrt [4]{-1} b^2\right ) \int \frac{x \tan ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right )}{1+i c x^2} \, dx}{3 \sqrt{c}}+\frac{\left (2 \sqrt [4]{-1} b^2\right ) \int \frac{x \tanh ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right )}{1-i c x^2} \, dx}{3 \sqrt{c}}-\frac{1}{9} \left (2 i b^2 c\right ) \int \left (\frac{1}{c^2}+\frac{i x^2}{c}-\frac{1}{c^2 \left (1-i c x^2\right )}\right ) \, dx+\frac{1}{9} \left (2 i b^2 c\right ) \int \left (\frac{1}{c^2}-\frac{i x^2}{c}-\frac{1}{c^2 \left (1+i c x^2\right )}\right ) \, dx\\ &=-\frac{4 a b x}{3 c}+\frac{2}{9} i a b x^3+\frac{14 (-1)^{3/4} b^2 \tan ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right )}{9 c^{3/2}}+\frac{\sqrt [4]{-1} b^2 \tan ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right )^2}{3 c^{3/2}}-\frac{2 \sqrt [4]{-1} a b \tanh ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right )}{3 c^{3/2}}-\frac{14 (-1)^{3/4} b^2 \tanh ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right )}{9 c^{3/2}}-\frac{(-1)^{3/4} b^2 \tanh ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right )^2}{3 c^{3/2}}-\frac{2 (-1)^{3/4} b^2 \tan ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right ) \log \left (\frac{2}{1-\sqrt [4]{-1} \sqrt{c} x}\right )}{3 c^{3/2}}+\frac{2 (-1)^{3/4} b^2 \tanh ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right ) \log \left (\frac{2}{1-(-1)^{3/4} \sqrt{c} x}\right )}{3 c^{3/2}}-\frac{2 i b^2 x \log \left (1-i c x^2\right )}{3 c}-\frac{1}{9} b^2 x^3 \log \left (1-i c x^2\right )-\frac{(-1)^{3/4} b^2 \tanh ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right ) \log \left (1-i c x^2\right )}{3 c^{3/2}}-\frac{1}{9} i b x^3 \left (2 a+i b \log \left (1-i c x^2\right )\right )-\frac{\sqrt [4]{-1} b \tan ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right ) \left (2 a+i b \log \left (1-i c x^2\right )\right )}{3 c^{3/2}}+\frac{1}{12} x^3 \left (2 a+i b \log \left (1-i c x^2\right )\right )^2+\frac{2 i b^2 x \log \left (1+i c x^2\right )}{3 c}-\frac{1}{3} i a b x^3 \log \left (1+i c x^2\right )+\frac{(-1)^{3/4} b^2 \tan ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right ) \log \left (1+i c x^2\right )}{3 c^{3/2}}+\frac{(-1)^{3/4} b^2 \tanh ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right ) \log \left (1+i c x^2\right )}{3 c^{3/2}}+\frac{1}{6} b^2 x^3 \log \left (1-i c x^2\right ) \log \left (1+i c x^2\right )-\frac{1}{12} b^2 x^3 \log ^2\left (1+i c x^2\right )+\frac{\sqrt [4]{-1} b^2 \text{Li}_2\left (1-\frac{2}{1-\sqrt [4]{-1} \sqrt{c} x}\right )}{3 c^{3/2}}+\frac{(-1)^{3/4} b^2 \text{Li}_2\left (1-\frac{2}{1-(-1)^{3/4} \sqrt{c} x}\right )}{3 c^{3/2}}+\frac{\left (2 i b^2\right ) \int \frac{1}{1-i c x^2} \, dx}{9 c}-\frac{\left (2 i b^2\right ) \int \frac{1}{1+i c x^2} \, dx}{9 c}+\frac{\left (2 \sqrt [4]{-1} b^2\right ) \int \left (\frac{i \tan ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right )}{2 \sqrt{c} \left (\sqrt [4]{-1}-\sqrt{c} x\right )}-\frac{i \tan ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right )}{2 \sqrt{c} \left (\sqrt [4]{-1}+\sqrt{c} x\right )}\right ) \, dx}{3 \sqrt{c}}+\frac{\left (2 \sqrt [4]{-1} b^2\right ) \int \left (-\frac{i \tanh ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right )}{2 \sqrt{c} \left (-(-1)^{3/4}-\sqrt{c} x\right )}+\frac{i \tanh ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right )}{2 \sqrt{c} \left (-(-1)^{3/4}+\sqrt{c} x\right )}\right ) \, dx}{3 \sqrt{c}}\\ &=-\frac{4 a b x}{3 c}+\frac{2}{9} i a b x^3+\frac{4 (-1)^{3/4} b^2 \tan ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right )}{3 c^{3/2}}+\frac{\sqrt [4]{-1} b^2 \tan ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right )^2}{3 c^{3/2}}-\frac{2 \sqrt [4]{-1} a b \tanh ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right )}{3 c^{3/2}}-\frac{4 (-1)^{3/4} b^2 \tanh ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right )}{3 c^{3/2}}-\frac{(-1)^{3/4} b^2 \tanh ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right )^2}{3 c^{3/2}}-\frac{2 (-1)^{3/4} b^2 \tan ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right ) \log \left (\frac{2}{1-\sqrt [4]{-1} \sqrt{c} x}\right )}{3 c^{3/2}}+\frac{2 (-1)^{3/4} b^2 \tanh ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right ) \log \left (\frac{2}{1-(-1)^{3/4} \sqrt{c} x}\right )}{3 c^{3/2}}-\frac{2 i b^2 x \log \left (1-i c x^2\right )}{3 c}-\frac{1}{9} b^2 x^3 \log \left (1-i c x^2\right )-\frac{(-1)^{3/4} b^2 \tanh ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right ) \log \left (1-i c x^2\right )}{3 c^{3/2}}-\frac{1}{9} i b x^3 \left (2 a+i b \log \left (1-i c x^2\right )\right )-\frac{\sqrt [4]{-1} b \tan ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right ) \left (2 a+i b \log \left (1-i c x^2\right )\right )}{3 c^{3/2}}+\frac{1}{12} x^3 \left (2 a+i b \log \left (1-i c x^2\right )\right )^2+\frac{2 i b^2 x \log \left (1+i c x^2\right )}{3 c}-\frac{1}{3} i a b x^3 \log \left (1+i c x^2\right )+\frac{(-1)^{3/4} b^2 \tan ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right ) \log \left (1+i c x^2\right )}{3 c^{3/2}}+\frac{(-1)^{3/4} b^2 \tanh ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right ) \log \left (1+i c x^2\right )}{3 c^{3/2}}+\frac{1}{6} b^2 x^3 \log \left (1-i c x^2\right ) \log \left (1+i c x^2\right )-\frac{1}{12} b^2 x^3 \log ^2\left (1+i c x^2\right )+\frac{\sqrt [4]{-1} b^2 \text{Li}_2\left (1-\frac{2}{1-\sqrt [4]{-1} \sqrt{c} x}\right )}{3 c^{3/2}}+\frac{(-1)^{3/4} b^2 \text{Li}_2\left (1-\frac{2}{1-(-1)^{3/4} \sqrt{c} x}\right )}{3 c^{3/2}}+\frac{\left ((-1)^{3/4} b^2\right ) \int \frac{\tan ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right )}{\sqrt [4]{-1}-\sqrt{c} x} \, dx}{3 c}-\frac{\left ((-1)^{3/4} b^2\right ) \int \frac{\tan ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right )}{\sqrt [4]{-1}+\sqrt{c} x} \, dx}{3 c}-\frac{\left ((-1)^{3/4} b^2\right ) \int \frac{\tanh ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right )}{-(-1)^{3/4}-\sqrt{c} x} \, dx}{3 c}+\frac{\left ((-1)^{3/4} b^2\right ) \int \frac{\tanh ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right )}{-(-1)^{3/4}+\sqrt{c} x} \, dx}{3 c}\\ &=-\frac{4 a b x}{3 c}+\frac{2}{9} i a b x^3+\frac{4 (-1)^{3/4} b^2 \tan ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right )}{3 c^{3/2}}+\frac{\sqrt [4]{-1} b^2 \tan ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right )^2}{3 c^{3/2}}-\frac{2 \sqrt [4]{-1} a b \tanh ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right )}{3 c^{3/2}}-\frac{4 (-1)^{3/4} b^2 \tanh ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right )}{3 c^{3/2}}-\frac{(-1)^{3/4} b^2 \tanh ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right )^2}{3 c^{3/2}}-\frac{2 (-1)^{3/4} b^2 \tan ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right ) \log \left (\frac{2}{1-\sqrt [4]{-1} \sqrt{c} x}\right )}{3 c^{3/2}}+\frac{2 (-1)^{3/4} b^2 \tan ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right ) \log \left (\frac{2}{1+\sqrt [4]{-1} \sqrt{c} x}\right )}{3 c^{3/2}}-\frac{(-1)^{3/4} b^2 \tan ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right ) \log \left (\frac{\sqrt{2} \left (\sqrt [4]{-1}+\sqrt{c} x\right )}{1+\sqrt [4]{-1} \sqrt{c} x}\right )}{3 c^{3/2}}+\frac{2 (-1)^{3/4} b^2 \tanh ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right ) \log \left (\frac{2}{1-(-1)^{3/4} \sqrt{c} x}\right )}{3 c^{3/2}}-\frac{2 (-1)^{3/4} b^2 \tanh ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right ) \log \left (\frac{2}{1+(-1)^{3/4} \sqrt{c} x}\right )}{3 c^{3/2}}+\frac{(-1)^{3/4} b^2 \tanh ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right ) \log \left (-\frac{\sqrt{2} \left ((-1)^{3/4}+\sqrt{c} x\right )}{1+(-1)^{3/4} \sqrt{c} x}\right )}{3 c^{3/2}}+\frac{(-1)^{3/4} b^2 \tanh ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right ) \log \left (\frac{\sqrt [4]{-1} \sqrt{2} \left (1+\sqrt [4]{-1} \sqrt{c} x\right )}{1+(-1)^{3/4} \sqrt{c} x}\right )}{3 c^{3/2}}-\frac{(-1)^{3/4} b^2 \tan ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right ) \log \left (\frac{(1-i) \left (1+(-1)^{3/4} \sqrt{c} x\right )}{1+\sqrt [4]{-1} \sqrt{c} x}\right )}{3 c^{3/2}}-\frac{2 i b^2 x \log \left (1-i c x^2\right )}{3 c}-\frac{1}{9} b^2 x^3 \log \left (1-i c x^2\right )-\frac{(-1)^{3/4} b^2 \tanh ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right ) \log \left (1-i c x^2\right )}{3 c^{3/2}}-\frac{1}{9} i b x^3 \left (2 a+i b \log \left (1-i c x^2\right )\right )-\frac{\sqrt [4]{-1} b \tan ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right ) \left (2 a+i b \log \left (1-i c x^2\right )\right )}{3 c^{3/2}}+\frac{1}{12} x^3 \left (2 a+i b \log \left (1-i c x^2\right )\right )^2+\frac{2 i b^2 x \log \left (1+i c x^2\right )}{3 c}-\frac{1}{3} i a b x^3 \log \left (1+i c x^2\right )+\frac{(-1)^{3/4} b^2 \tan ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right ) \log \left (1+i c x^2\right )}{3 c^{3/2}}+\frac{(-1)^{3/4} b^2 \tanh ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right ) \log \left (1+i c x^2\right )}{3 c^{3/2}}+\frac{1}{6} b^2 x^3 \log \left (1-i c x^2\right ) \log \left (1+i c x^2\right )-\frac{1}{12} b^2 x^3 \log ^2\left (1+i c x^2\right )+\frac{\sqrt [4]{-1} b^2 \text{Li}_2\left (1-\frac{2}{1-\sqrt [4]{-1} \sqrt{c} x}\right )}{3 c^{3/2}}+\frac{(-1)^{3/4} b^2 \text{Li}_2\left (1-\frac{2}{1-(-1)^{3/4} \sqrt{c} x}\right )}{3 c^{3/2}}+2 \frac{\left (i b^2\right ) \int \frac{\log \left (\frac{2}{1+\sqrt [4]{-1} \sqrt{c} x}\right )}{1-i c x^2} \, dx}{3 c}-\frac{\left (i b^2\right ) \int \frac{\log \left (-\frac{(1-i) (-1)^{3/4} \left (\sqrt [4]{-1}-\sqrt{c} x\right )}{1+\sqrt [4]{-1} \sqrt{c} x}\right )}{1-i c x^2} \, dx}{3 c}-\frac{\left (i b^2\right ) \int \frac{\log \left (-\frac{(1+i) (-1)^{3/4} \left (\sqrt [4]{-1}+\sqrt{c} x\right )}{1+\sqrt [4]{-1} \sqrt{c} x}\right )}{1-i c x^2} \, dx}{3 c}-2 \frac{\left (i b^2\right ) \int \frac{\log \left (\frac{2}{1+(-1)^{3/4} \sqrt{c} x}\right )}{1+i c x^2} \, dx}{3 c}+\frac{\left (i b^2\right ) \int \frac{\log \left (-\frac{(1+i) (-1)^{3/4} \left (-(-1)^{3/4}-\sqrt{c} x\right )}{1+(-1)^{3/4} \sqrt{c} x}\right )}{1+i c x^2} \, dx}{3 c}+\frac{\left (i b^2\right ) \int \frac{\log \left (\frac{(1-i) (-1)^{3/4} \left (-(-1)^{3/4}+\sqrt{c} x\right )}{1+(-1)^{3/4} \sqrt{c} x}\right )}{1+i c x^2} \, dx}{3 c}\\ &=-\frac{4 a b x}{3 c}+\frac{2}{9} i a b x^3+\frac{4 (-1)^{3/4} b^2 \tan ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right )}{3 c^{3/2}}+\frac{\sqrt [4]{-1} b^2 \tan ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right )^2}{3 c^{3/2}}-\frac{2 \sqrt [4]{-1} a b \tanh ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right )}{3 c^{3/2}}-\frac{4 (-1)^{3/4} b^2 \tanh ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right )}{3 c^{3/2}}-\frac{(-1)^{3/4} b^2 \tanh ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right )^2}{3 c^{3/2}}-\frac{2 (-1)^{3/4} b^2 \tan ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right ) \log \left (\frac{2}{1-\sqrt [4]{-1} \sqrt{c} x}\right )}{3 c^{3/2}}+\frac{2 (-1)^{3/4} b^2 \tan ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right ) \log \left (\frac{2}{1+\sqrt [4]{-1} \sqrt{c} x}\right )}{3 c^{3/2}}-\frac{(-1)^{3/4} b^2 \tan ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right ) \log \left (\frac{\sqrt{2} \left (\sqrt [4]{-1}+\sqrt{c} x\right )}{1+\sqrt [4]{-1} \sqrt{c} x}\right )}{3 c^{3/2}}+\frac{2 (-1)^{3/4} b^2 \tanh ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right ) \log \left (\frac{2}{1-(-1)^{3/4} \sqrt{c} x}\right )}{3 c^{3/2}}-\frac{2 (-1)^{3/4} b^2 \tanh ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right ) \log \left (\frac{2}{1+(-1)^{3/4} \sqrt{c} x}\right )}{3 c^{3/2}}+\frac{(-1)^{3/4} b^2 \tanh ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right ) \log \left (-\frac{\sqrt{2} \left ((-1)^{3/4}+\sqrt{c} x\right )}{1+(-1)^{3/4} \sqrt{c} x}\right )}{3 c^{3/2}}+\frac{(-1)^{3/4} b^2 \tanh ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right ) \log \left (\frac{\sqrt [4]{-1} \sqrt{2} \left (1+\sqrt [4]{-1} \sqrt{c} x\right )}{1+(-1)^{3/4} \sqrt{c} x}\right )}{3 c^{3/2}}-\frac{(-1)^{3/4} b^2 \tan ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right ) \log \left (\frac{(1-i) \left (1+(-1)^{3/4} \sqrt{c} x\right )}{1+\sqrt [4]{-1} \sqrt{c} x}\right )}{3 c^{3/2}}-\frac{2 i b^2 x \log \left (1-i c x^2\right )}{3 c}-\frac{1}{9} b^2 x^3 \log \left (1-i c x^2\right )-\frac{(-1)^{3/4} b^2 \tanh ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right ) \log \left (1-i c x^2\right )}{3 c^{3/2}}-\frac{1}{9} i b x^3 \left (2 a+i b \log \left (1-i c x^2\right )\right )-\frac{\sqrt [4]{-1} b \tan ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right ) \left (2 a+i b \log \left (1-i c x^2\right )\right )}{3 c^{3/2}}+\frac{1}{12} x^3 \left (2 a+i b \log \left (1-i c x^2\right )\right )^2+\frac{2 i b^2 x \log \left (1+i c x^2\right )}{3 c}-\frac{1}{3} i a b x^3 \log \left (1+i c x^2\right )+\frac{(-1)^{3/4} b^2 \tan ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right ) \log \left (1+i c x^2\right )}{3 c^{3/2}}+\frac{(-1)^{3/4} b^2 \tanh ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right ) \log \left (1+i c x^2\right )}{3 c^{3/2}}+\frac{1}{6} b^2 x^3 \log \left (1-i c x^2\right ) \log \left (1+i c x^2\right )-\frac{1}{12} b^2 x^3 \log ^2\left (1+i c x^2\right )+\frac{\sqrt [4]{-1} b^2 \text{Li}_2\left (1-\frac{2}{1-\sqrt [4]{-1} \sqrt{c} x}\right )}{3 c^{3/2}}-\frac{\sqrt [4]{-1} b^2 \text{Li}_2\left (1-\frac{\sqrt{2} \left (\sqrt [4]{-1}+\sqrt{c} x\right )}{1+\sqrt [4]{-1} \sqrt{c} x}\right )}{6 c^{3/2}}+\frac{(-1)^{3/4} b^2 \text{Li}_2\left (1-\frac{2}{1-(-1)^{3/4} \sqrt{c} x}\right )}{3 c^{3/2}}-\frac{(-1)^{3/4} b^2 \text{Li}_2\left (1+\frac{\sqrt{2} \left ((-1)^{3/4}+\sqrt{c} x\right )}{1+(-1)^{3/4} \sqrt{c} x}\right )}{6 c^{3/2}}-\frac{(-1)^{3/4} b^2 \text{Li}_2\left (1-\frac{\sqrt [4]{-1} \sqrt{2} \left (1+\sqrt [4]{-1} \sqrt{c} x\right )}{1+(-1)^{3/4} \sqrt{c} x}\right )}{6 c^{3/2}}-\frac{\sqrt [4]{-1} b^2 \text{Li}_2\left (1-\frac{(1-i) \left (1+(-1)^{3/4} \sqrt{c} x\right )}{1+\sqrt [4]{-1} \sqrt{c} x}\right )}{6 c^{3/2}}+2 \frac{\left (\sqrt [4]{-1} b^2\right ) \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1+\sqrt [4]{-1} \sqrt{c} x}\right )}{3 c^{3/2}}+2 \frac{\left ((-1)^{3/4} b^2\right ) \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1+(-1)^{3/4} \sqrt{c} x}\right )}{3 c^{3/2}}\\ &=-\frac{4 a b x}{3 c}+\frac{2}{9} i a b x^3+\frac{4 (-1)^{3/4} b^2 \tan ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right )}{3 c^{3/2}}+\frac{\sqrt [4]{-1} b^2 \tan ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right )^2}{3 c^{3/2}}-\frac{2 \sqrt [4]{-1} a b \tanh ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right )}{3 c^{3/2}}-\frac{4 (-1)^{3/4} b^2 \tanh ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right )}{3 c^{3/2}}-\frac{(-1)^{3/4} b^2 \tanh ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right )^2}{3 c^{3/2}}-\frac{2 (-1)^{3/4} b^2 \tan ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right ) \log \left (\frac{2}{1-\sqrt [4]{-1} \sqrt{c} x}\right )}{3 c^{3/2}}+\frac{2 (-1)^{3/4} b^2 \tan ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right ) \log \left (\frac{2}{1+\sqrt [4]{-1} \sqrt{c} x}\right )}{3 c^{3/2}}-\frac{(-1)^{3/4} b^2 \tan ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right ) \log \left (\frac{\sqrt{2} \left (\sqrt [4]{-1}+\sqrt{c} x\right )}{1+\sqrt [4]{-1} \sqrt{c} x}\right )}{3 c^{3/2}}+\frac{2 (-1)^{3/4} b^2 \tanh ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right ) \log \left (\frac{2}{1-(-1)^{3/4} \sqrt{c} x}\right )}{3 c^{3/2}}-\frac{2 (-1)^{3/4} b^2 \tanh ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right ) \log \left (\frac{2}{1+(-1)^{3/4} \sqrt{c} x}\right )}{3 c^{3/2}}+\frac{(-1)^{3/4} b^2 \tanh ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right ) \log \left (-\frac{\sqrt{2} \left ((-1)^{3/4}+\sqrt{c} x\right )}{1+(-1)^{3/4} \sqrt{c} x}\right )}{3 c^{3/2}}+\frac{(-1)^{3/4} b^2 \tanh ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right ) \log \left (\frac{\sqrt [4]{-1} \sqrt{2} \left (1+\sqrt [4]{-1} \sqrt{c} x\right )}{1+(-1)^{3/4} \sqrt{c} x}\right )}{3 c^{3/2}}-\frac{(-1)^{3/4} b^2 \tan ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right ) \log \left (\frac{(1-i) \left (1+(-1)^{3/4} \sqrt{c} x\right )}{1+\sqrt [4]{-1} \sqrt{c} x}\right )}{3 c^{3/2}}-\frac{2 i b^2 x \log \left (1-i c x^2\right )}{3 c}-\frac{1}{9} b^2 x^3 \log \left (1-i c x^2\right )-\frac{(-1)^{3/4} b^2 \tanh ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right ) \log \left (1-i c x^2\right )}{3 c^{3/2}}-\frac{1}{9} i b x^3 \left (2 a+i b \log \left (1-i c x^2\right )\right )-\frac{\sqrt [4]{-1} b \tan ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right ) \left (2 a+i b \log \left (1-i c x^2\right )\right )}{3 c^{3/2}}+\frac{1}{12} x^3 \left (2 a+i b \log \left (1-i c x^2\right )\right )^2+\frac{2 i b^2 x \log \left (1+i c x^2\right )}{3 c}-\frac{1}{3} i a b x^3 \log \left (1+i c x^2\right )+\frac{(-1)^{3/4} b^2 \tan ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right ) \log \left (1+i c x^2\right )}{3 c^{3/2}}+\frac{(-1)^{3/4} b^2 \tanh ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right ) \log \left (1+i c x^2\right )}{3 c^{3/2}}+\frac{1}{6} b^2 x^3 \log \left (1-i c x^2\right ) \log \left (1+i c x^2\right )-\frac{1}{12} b^2 x^3 \log ^2\left (1+i c x^2\right )+\frac{\sqrt [4]{-1} b^2 \text{Li}_2\left (1-\frac{2}{1-\sqrt [4]{-1} \sqrt{c} x}\right )}{3 c^{3/2}}+\frac{\sqrt [4]{-1} b^2 \text{Li}_2\left (1-\frac{2}{1+\sqrt [4]{-1} \sqrt{c} x}\right )}{3 c^{3/2}}-\frac{\sqrt [4]{-1} b^2 \text{Li}_2\left (1-\frac{\sqrt{2} \left (\sqrt [4]{-1}+\sqrt{c} x\right )}{1+\sqrt [4]{-1} \sqrt{c} x}\right )}{6 c^{3/2}}+\frac{(-1)^{3/4} b^2 \text{Li}_2\left (1-\frac{2}{1-(-1)^{3/4} \sqrt{c} x}\right )}{3 c^{3/2}}+\frac{(-1)^{3/4} b^2 \text{Li}_2\left (1-\frac{2}{1+(-1)^{3/4} \sqrt{c} x}\right )}{3 c^{3/2}}-\frac{(-1)^{3/4} b^2 \text{Li}_2\left (1+\frac{\sqrt{2} \left ((-1)^{3/4}+\sqrt{c} x\right )}{1+(-1)^{3/4} \sqrt{c} x}\right )}{6 c^{3/2}}-\frac{(-1)^{3/4} b^2 \text{Li}_2\left (1-\frac{\sqrt [4]{-1} \sqrt{2} \left (1+\sqrt [4]{-1} \sqrt{c} x\right )}{1+(-1)^{3/4} \sqrt{c} x}\right )}{6 c^{3/2}}-\frac{\sqrt [4]{-1} b^2 \text{Li}_2\left (1-\frac{(1-i) \left (1+(-1)^{3/4} \sqrt{c} x\right )}{1+\sqrt [4]{-1} \sqrt{c} x}\right )}{6 c^{3/2}}\\ \end{align*}
Mathematica [F] time = 4.84914, size = 0, normalized size = 0. \[ \int x^2 \left (a+b \tan ^{-1}\left (c x^2\right )\right )^2 \, dx \]
Verification is Not applicable to the result.
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Maple [F] time = 0.207, size = 0, normalized size = 0. \begin{align*} \int{x}^{2} \left ( a+b\arctan \left ( c{x}^{2} \right ) \right ) ^{2}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \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 (b^{2} x^{2} \arctan \left (c x^{2}\right )^{2} + 2 \, a b x^{2} \arctan \left (c x^{2}\right ) + a^{2} x^{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*} \int x^{2} \left (a + b \operatorname{atan}{\left (c x^{2} \right )}\right )^{2}\, dx \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 (b \arctan \left (c x^{2}\right ) + a\right )}^{2} x^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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