Optimal. Leaf size=1164 \[ \text{result too large to display} \]
[Out]
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Rubi [A] time = 1.62168, antiderivative size = 1164, normalized size of antiderivative = 1., number of steps used = 47, number of rules used = 23, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 1.438, Rules used = {5035, 2457, 203, 2470, 12, 4920, 4854, 2402, 2315, 2455, 6742, 206, 30, 2557, 205, 4928, 4856, 2447, 208, 5992, 5920, 5984, 5918} \[ \sqrt [4]{-1} \sqrt{c} \tan ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right )^2 b^2-(-1)^{3/4} \sqrt{c} \tanh ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right )^2 b^2+\frac{\log ^2\left (i c x^2+1\right ) b^2}{4 x}-2 (-1)^{3/4} \sqrt{c} \tan ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right ) \log \left (\frac{2}{1-\sqrt [4]{-1} \sqrt{c} x}\right ) b^2+2 (-1)^{3/4} \sqrt{c} \tan ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right ) \log \left (\frac{2}{\sqrt [4]{-1} \sqrt{c} x+1}\right ) b^2-(-1)^{3/4} \sqrt{c} \tan ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right ) \log \left (\frac{\sqrt{2} \left (\sqrt{c} x+\sqrt [4]{-1}\right )}{\sqrt [4]{-1} \sqrt{c} x+1}\right ) b^2+2 (-1)^{3/4} \sqrt{c} \tanh ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right ) \log \left (\frac{2}{1-(-1)^{3/4} \sqrt{c} x}\right ) b^2-2 (-1)^{3/4} \sqrt{c} \tanh ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right ) \log \left (\frac{2}{(-1)^{3/4} \sqrt{c} x+1}\right ) b^2+(-1)^{3/4} \sqrt{c} \tanh ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right ) \log \left (-\frac{\sqrt{2} \left (\sqrt{c} x+(-1)^{3/4}\right )}{(-1)^{3/4} \sqrt{c} x+1}\right ) b^2+(-1)^{3/4} \sqrt{c} \tanh ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right ) \log \left (\frac{(1+i) \left (\sqrt [4]{-1} \sqrt{c} x+1\right )}{(-1)^{3/4} \sqrt{c} x+1}\right ) b^2-(-1)^{3/4} \sqrt{c} \tan ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right ) \log \left (\frac{(1-i) \left ((-1)^{3/4} \sqrt{c} x+1\right )}{\sqrt [4]{-1} \sqrt{c} x+1}\right ) b^2-(-1)^{3/4} \sqrt{c} \tanh ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right ) \log \left (1-i c x^2\right ) b^2+(-1)^{3/4} \sqrt{c} \tan ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right ) \log \left (i c x^2+1\right ) b^2+(-1)^{3/4} \sqrt{c} \tanh ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right ) \log \left (i c x^2+1\right ) b^2-\frac{\log \left (1-i c x^2\right ) \log \left (i c x^2+1\right ) b^2}{2 x}+\sqrt [4]{-1} \sqrt{c} \text{PolyLog}\left (2,1-\frac{2}{1-\sqrt [4]{-1} \sqrt{c} x}\right ) b^2+\sqrt [4]{-1} \sqrt{c} \text{PolyLog}\left (2,1-\frac{2}{\sqrt [4]{-1} \sqrt{c} x+1}\right ) b^2-\frac{1}{2} \sqrt [4]{-1} \sqrt{c} \text{PolyLog}\left (2,1-\frac{\sqrt{2} \left (\sqrt{c} x+\sqrt [4]{-1}\right )}{\sqrt [4]{-1} \sqrt{c} x+1}\right ) b^2+(-1)^{3/4} \sqrt{c} \text{PolyLog}\left (2,1-\frac{2}{1-(-1)^{3/4} \sqrt{c} x}\right ) b^2+(-1)^{3/4} \sqrt{c} \text{PolyLog}\left (2,1-\frac{2}{(-1)^{3/4} \sqrt{c} x+1}\right ) b^2-\frac{1}{2} (-1)^{3/4} \sqrt{c} \text{PolyLog}\left (2,\frac{\sqrt{2} \left (\sqrt{c} x+(-1)^{3/4}\right )}{(-1)^{3/4} \sqrt{c} x+1}+1\right ) b^2-\frac{1}{2} (-1)^{3/4} \sqrt{c} \text{PolyLog}\left (2,1-\frac{(1+i) \left (\sqrt [4]{-1} \sqrt{c} x+1\right )}{(-1)^{3/4} \sqrt{c} x+1}\right ) b^2-\frac{1}{2} \sqrt [4]{-1} \sqrt{c} \text{PolyLog}\left (2,1-\frac{(1-i) \left ((-1)^{3/4} \sqrt{c} x+1\right )}{\sqrt [4]{-1} \sqrt{c} x+1}\right ) b^2-2 \sqrt [4]{-1} a \sqrt{c} \tanh ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right ) b-\sqrt [4]{-1} \sqrt{c} \tan ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right ) \left (2 a+i b \log \left (1-i c x^2\right )\right ) b+\frac{i a \log \left (i c x^2+1\right ) b}{x}-\frac{\left (2 a+i b \log \left (1-i c x^2\right )\right )^2}{4 x} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 5035
Rule 2457
Rule 203
Rule 2470
Rule 12
Rule 4920
Rule 4854
Rule 2402
Rule 2315
Rule 2455
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 \frac{\left (a+b \tan ^{-1}\left (c x^2\right )\right )^2}{x^2} \, dx &=\int \left (\frac{\left (2 a+i b \log \left (1-i c x^2\right )\right )^2}{4 x^2}+\frac{b \left (-2 i a+b \log \left (1-i c x^2\right )\right ) \log \left (1+i c x^2\right )}{2 x^2}-\frac{b^2 \log ^2\left (1+i c x^2\right )}{4 x^2}\right ) \, dx\\ &=\frac{1}{4} \int \frac{\left (2 a+i b \log \left (1-i c x^2\right )\right )^2}{x^2} \, dx+\frac{1}{2} b \int \frac{\left (-2 i a+b \log \left (1-i c x^2\right )\right ) \log \left (1+i c x^2\right )}{x^2} \, dx-\frac{1}{4} b^2 \int \frac{\log ^2\left (1+i c x^2\right )}{x^2} \, dx\\ &=-\frac{\left (2 a+i b \log \left (1-i c x^2\right )\right )^2}{4 x}+\frac{b^2 \log ^2\left (1+i c x^2\right )}{4 x}+\frac{1}{2} b \int \left (-\frac{2 i a \log \left (1+i c x^2\right )}{x^2}+\frac{b \log \left (1-i c x^2\right ) \log \left (1+i c x^2\right )}{x^2}\right ) \, dx+(b c) \int \frac{2 a+i b \log \left (1-i c x^2\right )}{1-i c x^2} \, dx-\left (i b^2 c\right ) \int \frac{\log \left (1+i c x^2\right )}{1+i c x^2} \, dx\\ &=-\sqrt [4]{-1} b \sqrt{c} \tan ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right ) \left (2 a+i b \log \left (1-i c x^2\right )\right )-\frac{\left (2 a+i b \log \left (1-i c x^2\right )\right )^2}{4 x}+(-1)^{3/4} b^2 \sqrt{c} \tanh ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right ) \log \left (1+i c x^2\right )+\frac{b^2 \log ^2\left (1+i c x^2\right )}{4 x}-(i a b) \int \frac{\log \left (1+i c x^2\right )}{x^2} \, dx+\frac{1}{2} b^2 \int \frac{\log \left (1-i c x^2\right ) \log \left (1+i c x^2\right )}{x^2} \, dx+\left (2 b^2 c^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+\left (2 b^2 c^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\\ &=-\sqrt [4]{-1} b \sqrt{c} \tan ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right ) \left (2 a+i b \log \left (1-i c x^2\right )\right )-\frac{\left (2 a+i b \log \left (1-i c x^2\right )\right )^2}{4 x}+\frac{i a b \log \left (1+i c x^2\right )}{x}+(-1)^{3/4} b^2 \sqrt{c} \tanh ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right ) \log \left (1+i c x^2\right )-\frac{b^2 \log \left (1-i c x^2\right ) \log \left (1+i c x^2\right )}{2 x}+\frac{b^2 \log ^2\left (1+i c x^2\right )}{4 x}-\frac{1}{2} b^2 \int \frac{2 c \log \left (1-i c x^2\right )}{i-c x^2} \, dx-\frac{1}{2} b^2 \int \frac{2 c \log \left (1+i c x^2\right )}{-i-c x^2} \, dx+(2 a b c) \int \frac{1}{1+i c x^2} \, dx+\left (2 \sqrt [4]{-1} b^2 c^{3/2}\right ) \int \frac{x \tan ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right )}{1-i c x^2} \, dx+\left (2 \sqrt [4]{-1} b^2 c^{3/2}\right ) \int \frac{x \tanh ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right )}{1+i c x^2} \, dx\\ &=\sqrt [4]{-1} b^2 \sqrt{c} \tan ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right )^2-2 \sqrt [4]{-1} a b \sqrt{c} \tanh ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right )-(-1)^{3/4} b^2 \sqrt{c} \tanh ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right )^2-\sqrt [4]{-1} b \sqrt{c} \tan ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right ) \left (2 a+i b \log \left (1-i c x^2\right )\right )-\frac{\left (2 a+i b \log \left (1-i c x^2\right )\right )^2}{4 x}+\frac{i a b \log \left (1+i c x^2\right )}{x}+(-1)^{3/4} b^2 \sqrt{c} \tanh ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right ) \log \left (1+i c x^2\right )-\frac{b^2 \log \left (1-i c x^2\right ) \log \left (1+i c x^2\right )}{2 x}+\frac{b^2 \log ^2\left (1+i c x^2\right )}{4 x}+\left (2 i b^2 c\right ) \int \frac{\tan ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right )}{i-(-1)^{3/4} \sqrt{c} x} \, dx-\left (2 i b^2 c\right ) \int \frac{\tanh ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right )}{1-(-1)^{3/4} \sqrt{c} x} \, dx-\left (b^2 c\right ) \int \frac{\log \left (1-i c x^2\right )}{i-c x^2} \, dx-\left (b^2 c\right ) \int \frac{\log \left (1+i c x^2\right )}{-i-c x^2} \, dx\\ &=\sqrt [4]{-1} b^2 \sqrt{c} \tan ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right )^2-2 \sqrt [4]{-1} a b \sqrt{c} \tanh ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right )-(-1)^{3/4} b^2 \sqrt{c} \tanh ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right )^2-2 (-1)^{3/4} b^2 \sqrt{c} \tan ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right ) \log \left (\frac{2}{1-\sqrt [4]{-1} \sqrt{c} x}\right )+2 (-1)^{3/4} b^2 \sqrt{c} \tanh ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right ) \log \left (\frac{2}{1-(-1)^{3/4} \sqrt{c} x}\right )-(-1)^{3/4} b^2 \sqrt{c} \tanh ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right ) \log \left (1-i c x^2\right )-\sqrt [4]{-1} b \sqrt{c} \tan ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right ) \left (2 a+i b \log \left (1-i c x^2\right )\right )-\frac{\left (2 a+i b \log \left (1-i c x^2\right )\right )^2}{4 x}+\frac{i a b \log \left (1+i c x^2\right )}{x}+(-1)^{3/4} b^2 \sqrt{c} \tan ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right ) \log \left (1+i c x^2\right )+(-1)^{3/4} b^2 \sqrt{c} \tanh ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right ) \log \left (1+i c x^2\right )-\frac{b^2 \log \left (1-i c x^2\right ) \log \left (1+i c x^2\right )}{2 x}+\frac{b^2 \log ^2\left (1+i c x^2\right )}{4 x}-\left (2 i b^2 c\right ) \int \frac{\log \left (\frac{2}{1-\sqrt [4]{-1} \sqrt{c} x}\right )}{1-i c x^2} \, dx+\left (2 i b^2 c\right ) \int \frac{\log \left (\frac{2}{1-(-1)^{3/4} \sqrt{c} x}\right )}{1+i c x^2} \, dx-\left (2 i b^2 c^2\right ) \int \frac{(-1)^{3/4} x \tan ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right )}{\sqrt{c} \left (1+i c x^2\right )} \, dx-\left (2 i b^2 c^2\right ) \int \frac{(-1)^{3/4} x \tanh ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right )}{\sqrt{c} \left (1-i c x^2\right )} \, dx\\ &=\sqrt [4]{-1} b^2 \sqrt{c} \tan ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right )^2-2 \sqrt [4]{-1} a b \sqrt{c} \tanh ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right )-(-1)^{3/4} b^2 \sqrt{c} \tanh ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right )^2-2 (-1)^{3/4} b^2 \sqrt{c} \tan ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right ) \log \left (\frac{2}{1-\sqrt [4]{-1} \sqrt{c} x}\right )+2 (-1)^{3/4} b^2 \sqrt{c} \tanh ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right ) \log \left (\frac{2}{1-(-1)^{3/4} \sqrt{c} x}\right )-(-1)^{3/4} b^2 \sqrt{c} \tanh ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right ) \log \left (1-i c x^2\right )-\sqrt [4]{-1} b \sqrt{c} \tan ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right ) \left (2 a+i b \log \left (1-i c x^2\right )\right )-\frac{\left (2 a+i b \log \left (1-i c x^2\right )\right )^2}{4 x}+\frac{i a b \log \left (1+i c x^2\right )}{x}+(-1)^{3/4} b^2 \sqrt{c} \tan ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right ) \log \left (1+i c x^2\right )+(-1)^{3/4} b^2 \sqrt{c} \tanh ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right ) \log \left (1+i c x^2\right )-\frac{b^2 \log \left (1-i c x^2\right ) \log \left (1+i c x^2\right )}{2 x}+\frac{b^2 \log ^2\left (1+i c x^2\right )}{4 x}+\left (2 \sqrt [4]{-1} b^2 \sqrt{c}\right ) \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1-\sqrt [4]{-1} \sqrt{c} x}\right )+\left (2 (-1)^{3/4} b^2 \sqrt{c}\right ) \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1-(-1)^{3/4} \sqrt{c} x}\right )+\left (2 \sqrt [4]{-1} b^2 c^{3/2}\right ) \int \frac{x \tan ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right )}{1+i c x^2} \, dx+\left (2 \sqrt [4]{-1} b^2 c^{3/2}\right ) \int \frac{x \tanh ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right )}{1-i c x^2} \, dx\\ &=\sqrt [4]{-1} b^2 \sqrt{c} \tan ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right )^2-2 \sqrt [4]{-1} a b \sqrt{c} \tanh ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right )-(-1)^{3/4} b^2 \sqrt{c} \tanh ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right )^2-2 (-1)^{3/4} b^2 \sqrt{c} \tan ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right ) \log \left (\frac{2}{1-\sqrt [4]{-1} \sqrt{c} x}\right )+2 (-1)^{3/4} b^2 \sqrt{c} \tanh ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right ) \log \left (\frac{2}{1-(-1)^{3/4} \sqrt{c} x}\right )-(-1)^{3/4} b^2 \sqrt{c} \tanh ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right ) \log \left (1-i c x^2\right )-\sqrt [4]{-1} b \sqrt{c} \tan ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right ) \left (2 a+i b \log \left (1-i c x^2\right )\right )-\frac{\left (2 a+i b \log \left (1-i c x^2\right )\right )^2}{4 x}+\frac{i a b \log \left (1+i c x^2\right )}{x}+(-1)^{3/4} b^2 \sqrt{c} \tan ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right ) \log \left (1+i c x^2\right )+(-1)^{3/4} b^2 \sqrt{c} \tanh ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right ) \log \left (1+i c x^2\right )-\frac{b^2 \log \left (1-i c x^2\right ) \log \left (1+i c x^2\right )}{2 x}+\frac{b^2 \log ^2\left (1+i c x^2\right )}{4 x}+\sqrt [4]{-1} b^2 \sqrt{c} \text{Li}_2\left (1-\frac{2}{1-\sqrt [4]{-1} \sqrt{c} x}\right )+(-1)^{3/4} b^2 \sqrt{c} \text{Li}_2\left (1-\frac{2}{1-(-1)^{3/4} \sqrt{c} x}\right )+\left (2 \sqrt [4]{-1} b^2 c^{3/2}\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+\left (2 \sqrt [4]{-1} b^2 c^{3/2}\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\\ &=\sqrt [4]{-1} b^2 \sqrt{c} \tan ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right )^2-2 \sqrt [4]{-1} a b \sqrt{c} \tanh ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right )-(-1)^{3/4} b^2 \sqrt{c} \tanh ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right )^2-2 (-1)^{3/4} b^2 \sqrt{c} \tan ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right ) \log \left (\frac{2}{1-\sqrt [4]{-1} \sqrt{c} x}\right )+2 (-1)^{3/4} b^2 \sqrt{c} \tanh ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right ) \log \left (\frac{2}{1-(-1)^{3/4} \sqrt{c} x}\right )-(-1)^{3/4} b^2 \sqrt{c} \tanh ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right ) \log \left (1-i c x^2\right )-\sqrt [4]{-1} b \sqrt{c} \tan ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right ) \left (2 a+i b \log \left (1-i c x^2\right )\right )-\frac{\left (2 a+i b \log \left (1-i c x^2\right )\right )^2}{4 x}+\frac{i a b \log \left (1+i c x^2\right )}{x}+(-1)^{3/4} b^2 \sqrt{c} \tan ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right ) \log \left (1+i c x^2\right )+(-1)^{3/4} b^2 \sqrt{c} \tanh ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right ) \log \left (1+i c x^2\right )-\frac{b^2 \log \left (1-i c x^2\right ) \log \left (1+i c x^2\right )}{2 x}+\frac{b^2 \log ^2\left (1+i c x^2\right )}{4 x}+\sqrt [4]{-1} b^2 \sqrt{c} \text{Li}_2\left (1-\frac{2}{1-\sqrt [4]{-1} \sqrt{c} x}\right )+(-1)^{3/4} b^2 \sqrt{c} \text{Li}_2\left (1-\frac{2}{1-(-1)^{3/4} \sqrt{c} x}\right )+\left ((-1)^{3/4} b^2 c\right ) \int \frac{\tan ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right )}{\sqrt [4]{-1}-\sqrt{c} x} \, dx-\left ((-1)^{3/4} b^2 c\right ) \int \frac{\tan ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right )}{\sqrt [4]{-1}+\sqrt{c} x} \, dx-\left ((-1)^{3/4} b^2 c\right ) \int \frac{\tanh ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right )}{-(-1)^{3/4}-\sqrt{c} x} \, dx+\left ((-1)^{3/4} b^2 c\right ) \int \frac{\tanh ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right )}{-(-1)^{3/4}+\sqrt{c} x} \, dx\\ &=\sqrt [4]{-1} b^2 \sqrt{c} \tan ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right )^2-2 \sqrt [4]{-1} a b \sqrt{c} \tanh ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right )-(-1)^{3/4} b^2 \sqrt{c} \tanh ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right )^2-2 (-1)^{3/4} b^2 \sqrt{c} \tan ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right ) \log \left (\frac{2}{1-\sqrt [4]{-1} \sqrt{c} x}\right )+2 (-1)^{3/4} b^2 \sqrt{c} \tan ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right ) \log \left (\frac{2}{1+\sqrt [4]{-1} \sqrt{c} x}\right )-(-1)^{3/4} b^2 \sqrt{c} \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 )+2 (-1)^{3/4} b^2 \sqrt{c} \tanh ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right ) \log \left (\frac{2}{1-(-1)^{3/4} \sqrt{c} x}\right )-2 (-1)^{3/4} b^2 \sqrt{c} \tanh ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right ) \log \left (\frac{2}{1+(-1)^{3/4} \sqrt{c} x}\right )+(-1)^{3/4} b^2 \sqrt{c} \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 )+(-1)^{3/4} b^2 \sqrt{c} \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 )-(-1)^{3/4} b^2 \sqrt{c} \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 )-(-1)^{3/4} b^2 \sqrt{c} \tanh ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right ) \log \left (1-i c x^2\right )-\sqrt [4]{-1} b \sqrt{c} \tan ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right ) \left (2 a+i b \log \left (1-i c x^2\right )\right )-\frac{\left (2 a+i b \log \left (1-i c x^2\right )\right )^2}{4 x}+\frac{i a b \log \left (1+i c x^2\right )}{x}+(-1)^{3/4} b^2 \sqrt{c} \tan ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right ) \log \left (1+i c x^2\right )+(-1)^{3/4} b^2 \sqrt{c} \tanh ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right ) \log \left (1+i c x^2\right )-\frac{b^2 \log \left (1-i c x^2\right ) \log \left (1+i c x^2\right )}{2 x}+\frac{b^2 \log ^2\left (1+i c x^2\right )}{4 x}+\sqrt [4]{-1} b^2 \sqrt{c} \text{Li}_2\left (1-\frac{2}{1-\sqrt [4]{-1} \sqrt{c} x}\right )+(-1)^{3/4} b^2 \sqrt{c} \text{Li}_2\left (1-\frac{2}{1-(-1)^{3/4} \sqrt{c} x}\right )+2 \left (\left (i b^2 c\right ) \int \frac{\log \left (\frac{2}{1+\sqrt [4]{-1} \sqrt{c} x}\right )}{1-i c x^2} \, dx\right )-\left (i b^2 c\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-\left (i b^2 c\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-2 \left (\left (i b^2 c\right ) \int \frac{\log \left (\frac{2}{1+(-1)^{3/4} \sqrt{c} x}\right )}{1+i c x^2} \, dx\right )+\left (i b^2 c\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+\left (i b^2 c\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\\ &=\sqrt [4]{-1} b^2 \sqrt{c} \tan ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right )^2-2 \sqrt [4]{-1} a b \sqrt{c} \tanh ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right )-(-1)^{3/4} b^2 \sqrt{c} \tanh ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right )^2-2 (-1)^{3/4} b^2 \sqrt{c} \tan ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right ) \log \left (\frac{2}{1-\sqrt [4]{-1} \sqrt{c} x}\right )+2 (-1)^{3/4} b^2 \sqrt{c} \tan ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right ) \log \left (\frac{2}{1+\sqrt [4]{-1} \sqrt{c} x}\right )-(-1)^{3/4} b^2 \sqrt{c} \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 )+2 (-1)^{3/4} b^2 \sqrt{c} \tanh ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right ) \log \left (\frac{2}{1-(-1)^{3/4} \sqrt{c} x}\right )-2 (-1)^{3/4} b^2 \sqrt{c} \tanh ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right ) \log \left (\frac{2}{1+(-1)^{3/4} \sqrt{c} x}\right )+(-1)^{3/4} b^2 \sqrt{c} \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 )+(-1)^{3/4} b^2 \sqrt{c} \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 )-(-1)^{3/4} b^2 \sqrt{c} \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 )-(-1)^{3/4} b^2 \sqrt{c} \tanh ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right ) \log \left (1-i c x^2\right )-\sqrt [4]{-1} b \sqrt{c} \tan ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right ) \left (2 a+i b \log \left (1-i c x^2\right )\right )-\frac{\left (2 a+i b \log \left (1-i c x^2\right )\right )^2}{4 x}+\frac{i a b \log \left (1+i c x^2\right )}{x}+(-1)^{3/4} b^2 \sqrt{c} \tan ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right ) \log \left (1+i c x^2\right )+(-1)^{3/4} b^2 \sqrt{c} \tanh ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right ) \log \left (1+i c x^2\right )-\frac{b^2 \log \left (1-i c x^2\right ) \log \left (1+i c x^2\right )}{2 x}+\frac{b^2 \log ^2\left (1+i c x^2\right )}{4 x}+\sqrt [4]{-1} b^2 \sqrt{c} \text{Li}_2\left (1-\frac{2}{1-\sqrt [4]{-1} \sqrt{c} x}\right )-\frac{1}{2} \sqrt [4]{-1} b^2 \sqrt{c} \text{Li}_2\left (1-\frac{\sqrt{2} \left (\sqrt [4]{-1}+\sqrt{c} x\right )}{1+\sqrt [4]{-1} \sqrt{c} x}\right )+(-1)^{3/4} b^2 \sqrt{c} \text{Li}_2\left (1-\frac{2}{1-(-1)^{3/4} \sqrt{c} x}\right )-\frac{1}{2} (-1)^{3/4} b^2 \sqrt{c} \text{Li}_2\left (1+\frac{\sqrt{2} \left ((-1)^{3/4}+\sqrt{c} x\right )}{1+(-1)^{3/4} \sqrt{c} x}\right )-\frac{1}{2} (-1)^{3/4} b^2 \sqrt{c} \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 )-\frac{1}{2} \sqrt [4]{-1} b^2 \sqrt{c} \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 )+2 \left (\left (\sqrt [4]{-1} b^2 \sqrt{c}\right ) \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1+\sqrt [4]{-1} \sqrt{c} x}\right )\right )+2 \left (\left ((-1)^{3/4} b^2 \sqrt{c}\right ) \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1+(-1)^{3/4} \sqrt{c} x}\right )\right )\\ &=\sqrt [4]{-1} b^2 \sqrt{c} \tan ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right )^2-2 \sqrt [4]{-1} a b \sqrt{c} \tanh ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right )-(-1)^{3/4} b^2 \sqrt{c} \tanh ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right )^2-2 (-1)^{3/4} b^2 \sqrt{c} \tan ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right ) \log \left (\frac{2}{1-\sqrt [4]{-1} \sqrt{c} x}\right )+2 (-1)^{3/4} b^2 \sqrt{c} \tan ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right ) \log \left (\frac{2}{1+\sqrt [4]{-1} \sqrt{c} x}\right )-(-1)^{3/4} b^2 \sqrt{c} \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 )+2 (-1)^{3/4} b^2 \sqrt{c} \tanh ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right ) \log \left (\frac{2}{1-(-1)^{3/4} \sqrt{c} x}\right )-2 (-1)^{3/4} b^2 \sqrt{c} \tanh ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right ) \log \left (\frac{2}{1+(-1)^{3/4} \sqrt{c} x}\right )+(-1)^{3/4} b^2 \sqrt{c} \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 )+(-1)^{3/4} b^2 \sqrt{c} \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 )-(-1)^{3/4} b^2 \sqrt{c} \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 )-(-1)^{3/4} b^2 \sqrt{c} \tanh ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right ) \log \left (1-i c x^2\right )-\sqrt [4]{-1} b \sqrt{c} \tan ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right ) \left (2 a+i b \log \left (1-i c x^2\right )\right )-\frac{\left (2 a+i b \log \left (1-i c x^2\right )\right )^2}{4 x}+\frac{i a b \log \left (1+i c x^2\right )}{x}+(-1)^{3/4} b^2 \sqrt{c} \tan ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right ) \log \left (1+i c x^2\right )+(-1)^{3/4} b^2 \sqrt{c} \tanh ^{-1}\left ((-1)^{3/4} \sqrt{c} x\right ) \log \left (1+i c x^2\right )-\frac{b^2 \log \left (1-i c x^2\right ) \log \left (1+i c x^2\right )}{2 x}+\frac{b^2 \log ^2\left (1+i c x^2\right )}{4 x}+\sqrt [4]{-1} b^2 \sqrt{c} \text{Li}_2\left (1-\frac{2}{1-\sqrt [4]{-1} \sqrt{c} x}\right )+\sqrt [4]{-1} b^2 \sqrt{c} \text{Li}_2\left (1-\frac{2}{1+\sqrt [4]{-1} \sqrt{c} x}\right )-\frac{1}{2} \sqrt [4]{-1} b^2 \sqrt{c} \text{Li}_2\left (1-\frac{\sqrt{2} \left (\sqrt [4]{-1}+\sqrt{c} x\right )}{1+\sqrt [4]{-1} \sqrt{c} x}\right )+(-1)^{3/4} b^2 \sqrt{c} \text{Li}_2\left (1-\frac{2}{1-(-1)^{3/4} \sqrt{c} x}\right )+(-1)^{3/4} b^2 \sqrt{c} \text{Li}_2\left (1-\frac{2}{1+(-1)^{3/4} \sqrt{c} x}\right )-\frac{1}{2} (-1)^{3/4} b^2 \sqrt{c} \text{Li}_2\left (1+\frac{\sqrt{2} \left ((-1)^{3/4}+\sqrt{c} x\right )}{1+(-1)^{3/4} \sqrt{c} x}\right )-\frac{1}{2} (-1)^{3/4} b^2 \sqrt{c} \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 )-\frac{1}{2} \sqrt [4]{-1} b^2 \sqrt{c} \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 )\\ \end{align*}
Mathematica [B] time = 31.4029, size = 5434, normalized size = 4.67 \[ \text{Result too large to show} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.191, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( a+b\arctan \left ( c{x}^{2} \right ) \right ) ^{2}}{{x}^{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 (\frac{b^{2} \arctan \left (c x^{2}\right )^{2} + 2 \, a b \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 \frac{\left (a + b \operatorname{atan}{\left (c x^{2} \right )}\right )^{2}}{x^{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 \frac{{\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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