Optimal. Leaf size=1263 \[ \text{result too large to display} \]
[Out]
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Rubi [A] time = 2.58476, antiderivative size = 1263, normalized size of antiderivative = 1., number of steps used = 104, number of rules used = 30, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 1.875, Rules used = {6099, 2457, 2476, 2455, 263, 325, 207, 206, 2470, 12, 260, 6688, 5988, 5932, 2447, 6742, 203, 30, 2557, 5992, 5912, 5920, 2402, 2315, 4928, 4848, 2391, 4856, 4924, 4868} \[ \text{result too large to display} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 6099
Rule 2457
Rule 2476
Rule 2455
Rule 263
Rule 325
Rule 207
Rule 206
Rule 2470
Rule 12
Rule 260
Rule 6688
Rule 5988
Rule 5932
Rule 2447
Rule 6742
Rule 203
Rule 30
Rule 2557
Rule 5992
Rule 5912
Rule 5920
Rule 2402
Rule 2315
Rule 4928
Rule 4848
Rule 2391
Rule 4856
Rule 4924
Rule 4868
Rubi steps
\begin{align*} \int \frac{\left (a+b \tanh ^{-1}\left (\frac{c}{x^2}\right )\right )^2}{x^4} \, dx &=\int \left (\frac{\left (2 a-b \log \left (1-\frac{c}{x^2}\right )\right )^2}{4 x^4}-\frac{b \left (-2 a+b \log \left (1-\frac{c}{x^2}\right )\right ) \log \left (1+\frac{c}{x^2}\right )}{2 x^4}+\frac{b^2 \log ^2\left (1+\frac{c}{x^2}\right )}{4 x^4}\right ) \, dx\\ &=\frac{1}{4} \int \frac{\left (2 a-b \log \left (1-\frac{c}{x^2}\right )\right )^2}{x^4} \, dx-\frac{1}{2} b \int \frac{\left (-2 a+b \log \left (1-\frac{c}{x^2}\right )\right ) \log \left (1+\frac{c}{x^2}\right )}{x^4} \, dx+\frac{1}{4} b^2 \int \frac{\log ^2\left (1+\frac{c}{x^2}\right )}{x^4} \, dx\\ &=-\frac{\left (2 a-b \log \left (1-\frac{c}{x^2}\right )\right )^2}{12 x^3}-\frac{b^2 \log ^2\left (1+\frac{c}{x^2}\right )}{12 x^3}-\frac{1}{2} b \int \left (-\frac{2 a \log \left (1+\frac{c}{x^2}\right )}{x^4}+\frac{b \log \left (1-\frac{c}{x^2}\right ) \log \left (1+\frac{c}{x^2}\right )}{x^4}\right ) \, dx-\frac{1}{3} (b c) \int \frac{2 a-b \log \left (1-\frac{c}{x^2}\right )}{\left (1-\frac{c}{x^2}\right ) x^6} \, dx-\frac{1}{3} \left (b^2 c\right ) \int \frac{\log \left (1+\frac{c}{x^2}\right )}{\left (1+\frac{c}{x^2}\right ) x^6} \, dx\\ &=-\frac{\left (2 a-b \log \left (1-\frac{c}{x^2}\right )\right )^2}{12 x^3}-\frac{b^2 \log ^2\left (1+\frac{c}{x^2}\right )}{12 x^3}+(a b) \int \frac{\log \left (1+\frac{c}{x^2}\right )}{x^4} \, dx-\frac{1}{2} b^2 \int \frac{\log \left (1-\frac{c}{x^2}\right ) \log \left (1+\frac{c}{x^2}\right )}{x^4} \, dx-\frac{1}{3} (b c) \int \left (-\frac{2 a-b \log \left (1-\frac{c}{x^2}\right )}{c x^4}-\frac{2 a-b \log \left (1-\frac{c}{x^2}\right )}{c^2 x^2}-\frac{2 a-b \log \left (1-\frac{c}{x^2}\right )}{c^2 \left (c-x^2\right )}\right ) \, dx-\frac{1}{3} \left (b^2 c\right ) \int \left (\frac{\log \left (1+\frac{c}{x^2}\right )}{c x^4}-\frac{\log \left (1+\frac{c}{x^2}\right )}{c^2 x^2}+\frac{\log \left (1+\frac{c}{x^2}\right )}{c^2 \left (c+x^2\right )}\right ) \, dx\\ &=-\frac{\left (2 a-b \log \left (1-\frac{c}{x^2}\right )\right )^2}{12 x^3}-\frac{a b \log \left (1+\frac{c}{x^2}\right )}{3 x^3}+\frac{b^2 \log \left (1-\frac{c}{x^2}\right ) \log \left (1+\frac{c}{x^2}\right )}{6 x^3}-\frac{b^2 \log ^2\left (1+\frac{c}{x^2}\right )}{12 x^3}+\frac{1}{3} b \int \frac{2 a-b \log \left (1-\frac{c}{x^2}\right )}{x^4} \, dx-\frac{1}{3} b^2 \int \frac{\log \left (1+\frac{c}{x^2}\right )}{x^4} \, dx+\frac{1}{2} b^2 \int \frac{2 c \log \left (1-\frac{c}{x^2}\right )}{3 x^4 \left (c+x^2\right )} \, dx+\frac{1}{2} b^2 \int \frac{2 c \log \left (1+\frac{c}{x^2}\right )}{3 x^4 \left (c-x^2\right )} \, dx+\frac{b \int \frac{2 a-b \log \left (1-\frac{c}{x^2}\right )}{x^2} \, dx}{3 c}+\frac{b \int \frac{2 a-b \log \left (1-\frac{c}{x^2}\right )}{c-x^2} \, dx}{3 c}+\frac{b^2 \int \frac{\log \left (1+\frac{c}{x^2}\right )}{x^2} \, dx}{3 c}-\frac{b^2 \int \frac{\log \left (1+\frac{c}{x^2}\right )}{c+x^2} \, dx}{3 c}-\frac{1}{3} (2 a b c) \int \frac{1}{\left (1+\frac{c}{x^2}\right ) x^6} \, dx\\ &=-\frac{b \left (2 a-b \log \left (1-\frac{c}{x^2}\right )\right )}{9 x^3}-\frac{b \left (2 a-b \log \left (1-\frac{c}{x^2}\right )\right )}{3 c x}+\frac{b \tanh ^{-1}\left (\frac{x}{\sqrt{c}}\right ) \left (2 a-b \log \left (1-\frac{c}{x^2}\right )\right )}{3 c^{3/2}}-\frac{\left (2 a-b \log \left (1-\frac{c}{x^2}\right )\right )^2}{12 x^3}-\frac{a b \log \left (1+\frac{c}{x^2}\right )}{3 x^3}+\frac{b^2 \log \left (1+\frac{c}{x^2}\right )}{9 x^3}-\frac{b^2 \log \left (1+\frac{c}{x^2}\right )}{3 c x}-\frac{b^2 \tan ^{-1}\left (\frac{x}{\sqrt{c}}\right ) \log \left (1+\frac{c}{x^2}\right )}{3 c^{3/2}}+\frac{b^2 \log \left (1-\frac{c}{x^2}\right ) \log \left (1+\frac{c}{x^2}\right )}{6 x^3}-\frac{b^2 \log ^2\left (1+\frac{c}{x^2}\right )}{12 x^3}-\frac{1}{3} \left (2 b^2\right ) \int \frac{1}{\left (1-\frac{c}{x^2}\right ) x^4} \, dx-\frac{1}{3} \left (2 b^2\right ) \int \frac{1}{\left (1+\frac{c}{x^2}\right ) x^4} \, dx-\frac{1}{3} \left (2 b^2\right ) \int \frac{\tan ^{-1}\left (\frac{x}{\sqrt{c}}\right )}{\sqrt{c} \left (1+\frac{c}{x^2}\right ) x^3} \, dx+\frac{1}{3} \left (2 b^2\right ) \int \frac{\tanh ^{-1}\left (\frac{x}{\sqrt{c}}\right )}{\sqrt{c} \left (1-\frac{c}{x^2}\right ) x^3} \, dx-\frac{1}{3} (2 a b c) \int \frac{1}{x^4 \left (c+x^2\right )} \, dx-\frac{1}{9} \left (2 b^2 c\right ) \int \frac{1}{\left (1-\frac{c}{x^2}\right ) x^6} \, dx+\frac{1}{9} \left (2 b^2 c\right ) \int \frac{1}{\left (1+\frac{c}{x^2}\right ) x^6} \, dx+\frac{1}{3} \left (b^2 c\right ) \int \frac{\log \left (1-\frac{c}{x^2}\right )}{x^4 \left (c+x^2\right )} \, dx+\frac{1}{3} \left (b^2 c\right ) \int \frac{\log \left (1+\frac{c}{x^2}\right )}{x^4 \left (c-x^2\right )} \, dx\\ &=\frac{2 a b}{9 x^3}-\frac{b \left (2 a-b \log \left (1-\frac{c}{x^2}\right )\right )}{9 x^3}-\frac{b \left (2 a-b \log \left (1-\frac{c}{x^2}\right )\right )}{3 c x}+\frac{b \tanh ^{-1}\left (\frac{x}{\sqrt{c}}\right ) \left (2 a-b \log \left (1-\frac{c}{x^2}\right )\right )}{3 c^{3/2}}-\frac{\left (2 a-b \log \left (1-\frac{c}{x^2}\right )\right )^2}{12 x^3}-\frac{a b \log \left (1+\frac{c}{x^2}\right )}{3 x^3}+\frac{b^2 \log \left (1+\frac{c}{x^2}\right )}{9 x^3}-\frac{b^2 \log \left (1+\frac{c}{x^2}\right )}{3 c x}-\frac{b^2 \tan ^{-1}\left (\frac{x}{\sqrt{c}}\right ) \log \left (1+\frac{c}{x^2}\right )}{3 c^{3/2}}+\frac{b^2 \log \left (1-\frac{c}{x^2}\right ) \log \left (1+\frac{c}{x^2}\right )}{6 x^3}-\frac{b^2 \log ^2\left (1+\frac{c}{x^2}\right )}{12 x^3}+\frac{1}{3} (2 a b) \int \frac{1}{x^2 \left (c+x^2\right )} \, dx-\frac{1}{3} \left (2 b^2\right ) \int \frac{1}{x^2 \left (-c+x^2\right )} \, dx-\frac{1}{3} \left (2 b^2\right ) \int \frac{1}{x^2 \left (c+x^2\right )} \, dx-\frac{\left (2 b^2\right ) \int \frac{\tan ^{-1}\left (\frac{x}{\sqrt{c}}\right )}{\left (1+\frac{c}{x^2}\right ) x^3} \, dx}{3 \sqrt{c}}+\frac{\left (2 b^2\right ) \int \frac{\tanh ^{-1}\left (\frac{x}{\sqrt{c}}\right )}{\left (1-\frac{c}{x^2}\right ) x^3} \, dx}{3 \sqrt{c}}-\frac{1}{9} \left (2 b^2 c\right ) \int \frac{1}{x^4 \left (-c+x^2\right )} \, dx+\frac{1}{9} \left (2 b^2 c\right ) \int \frac{1}{x^4 \left (c+x^2\right )} \, dx+\frac{1}{3} \left (b^2 c\right ) \int \left (\frac{\log \left (1-\frac{c}{x^2}\right )}{c x^4}-\frac{\log \left (1-\frac{c}{x^2}\right )}{c^2 x^2}+\frac{\log \left (1-\frac{c}{x^2}\right )}{c^2 \left (c+x^2\right )}\right ) \, dx+\frac{1}{3} \left (b^2 c\right ) \int \left (\frac{\log \left (1+\frac{c}{x^2}\right )}{c x^4}+\frac{\log \left (1+\frac{c}{x^2}\right )}{c^2 x^2}+\frac{\log \left (1+\frac{c}{x^2}\right )}{c^2 \left (c-x^2\right )}\right ) \, dx\\ &=\frac{2 a b}{9 x^3}-\frac{4 b^2}{27 x^3}-\frac{2 a b}{3 c x}-\frac{b \left (2 a-b \log \left (1-\frac{c}{x^2}\right )\right )}{9 x^3}-\frac{b \left (2 a-b \log \left (1-\frac{c}{x^2}\right )\right )}{3 c x}+\frac{b \tanh ^{-1}\left (\frac{x}{\sqrt{c}}\right ) \left (2 a-b \log \left (1-\frac{c}{x^2}\right )\right )}{3 c^{3/2}}-\frac{\left (2 a-b \log \left (1-\frac{c}{x^2}\right )\right )^2}{12 x^3}-\frac{a b \log \left (1+\frac{c}{x^2}\right )}{3 x^3}+\frac{b^2 \log \left (1+\frac{c}{x^2}\right )}{9 x^3}-\frac{b^2 \log \left (1+\frac{c}{x^2}\right )}{3 c x}-\frac{b^2 \tan ^{-1}\left (\frac{x}{\sqrt{c}}\right ) \log \left (1+\frac{c}{x^2}\right )}{3 c^{3/2}}+\frac{b^2 \log \left (1-\frac{c}{x^2}\right ) \log \left (1+\frac{c}{x^2}\right )}{6 x^3}-\frac{b^2 \log ^2\left (1+\frac{c}{x^2}\right )}{12 x^3}-\frac{1}{9} \left (2 b^2\right ) \int \frac{1}{x^2 \left (-c+x^2\right )} \, dx-\frac{1}{9} \left (2 b^2\right ) \int \frac{1}{x^2 \left (c+x^2\right )} \, dx+\frac{1}{3} b^2 \int \frac{\log \left (1-\frac{c}{x^2}\right )}{x^4} \, dx+\frac{1}{3} b^2 \int \frac{\log \left (1+\frac{c}{x^2}\right )}{x^4} \, dx-\frac{(2 a b) \int \frac{1}{c+x^2} \, dx}{3 c}-\frac{b^2 \int \frac{\log \left (1-\frac{c}{x^2}\right )}{x^2} \, dx}{3 c}+\frac{b^2 \int \frac{\log \left (1-\frac{c}{x^2}\right )}{c+x^2} \, dx}{3 c}+\frac{b^2 \int \frac{\log \left (1+\frac{c}{x^2}\right )}{x^2} \, dx}{3 c}+\frac{b^2 \int \frac{\log \left (1+\frac{c}{x^2}\right )}{c-x^2} \, dx}{3 c}-\frac{\left (2 b^2\right ) \int \frac{1}{-c+x^2} \, dx}{3 c}+\frac{\left (2 b^2\right ) \int \frac{1}{c+x^2} \, dx}{3 c}-\frac{\left (2 b^2\right ) \int \frac{\tan ^{-1}\left (\frac{x}{\sqrt{c}}\right )}{x \left (c+x^2\right )} \, dx}{3 \sqrt{c}}+\frac{\left (2 b^2\right ) \int \frac{\tanh ^{-1}\left (\frac{x}{\sqrt{c}}\right )}{x \left (-c+x^2\right )} \, dx}{3 \sqrt{c}}\\ &=\frac{2 a b}{9 x^3}-\frac{4 b^2}{27 x^3}-\frac{2 a b}{3 c x}-\frac{2 a b \tan ^{-1}\left (\frac{x}{\sqrt{c}}\right )}{3 c^{3/2}}+\frac{2 b^2 \tan ^{-1}\left (\frac{x}{\sqrt{c}}\right )}{3 c^{3/2}}+\frac{i b^2 \tan ^{-1}\left (\frac{x}{\sqrt{c}}\right )^2}{3 c^{3/2}}+\frac{2 b^2 \tanh ^{-1}\left (\frac{x}{\sqrt{c}}\right )}{3 c^{3/2}}-\frac{b^2 \tanh ^{-1}\left (\frac{x}{\sqrt{c}}\right )^2}{3 c^{3/2}}-\frac{b^2 \log \left (1-\frac{c}{x^2}\right )}{9 x^3}+\frac{b^2 \log \left (1-\frac{c}{x^2}\right )}{3 c x}+\frac{b^2 \tan ^{-1}\left (\frac{x}{\sqrt{c}}\right ) \log \left (1-\frac{c}{x^2}\right )}{3 c^{3/2}}-\frac{b \left (2 a-b \log \left (1-\frac{c}{x^2}\right )\right )}{9 x^3}-\frac{b \left (2 a-b \log \left (1-\frac{c}{x^2}\right )\right )}{3 c x}+\frac{b \tanh ^{-1}\left (\frac{x}{\sqrt{c}}\right ) \left (2 a-b \log \left (1-\frac{c}{x^2}\right )\right )}{3 c^{3/2}}-\frac{\left (2 a-b \log \left (1-\frac{c}{x^2}\right )\right )^2}{12 x^3}-\frac{a b \log \left (1+\frac{c}{x^2}\right )}{3 x^3}-\frac{2 b^2 \log \left (1+\frac{c}{x^2}\right )}{3 c x}-\frac{b^2 \tan ^{-1}\left (\frac{x}{\sqrt{c}}\right ) \log \left (1+\frac{c}{x^2}\right )}{3 c^{3/2}}+\frac{b^2 \tanh ^{-1}\left (\frac{x}{\sqrt{c}}\right ) \log \left (1+\frac{c}{x^2}\right )}{3 c^{3/2}}+\frac{b^2 \log \left (1-\frac{c}{x^2}\right ) \log \left (1+\frac{c}{x^2}\right )}{6 x^3}-\frac{b^2 \log ^2\left (1+\frac{c}{x^2}\right )}{12 x^3}-\frac{1}{3} \left (2 b^2\right ) \int \frac{1}{\left (1-\frac{c}{x^2}\right ) x^4} \, dx-\frac{1}{3} \left (2 b^2\right ) \int \frac{1}{\left (1+\frac{c}{x^2}\right ) x^4} \, dx-\frac{1}{3} \left (2 b^2\right ) \int \frac{\tan ^{-1}\left (\frac{x}{\sqrt{c}}\right )}{\sqrt{c} \left (1-\frac{c}{x^2}\right ) x^3} \, dx+\frac{1}{3} \left (2 b^2\right ) \int \frac{\tanh ^{-1}\left (\frac{x}{\sqrt{c}}\right )}{\sqrt{c} \left (1+\frac{c}{x^2}\right ) x^3} \, dx-\frac{\left (2 i b^2\right ) \int \frac{\tan ^{-1}\left (\frac{x}{\sqrt{c}}\right )}{x \left (i+\frac{x}{\sqrt{c}}\right )} \, dx}{3 c^{3/2}}-\frac{\left (2 b^2\right ) \int \frac{\tanh ^{-1}\left (\frac{x}{\sqrt{c}}\right )}{x \left (1+\frac{x}{\sqrt{c}}\right )} \, dx}{3 c^{3/2}}-\frac{\left (2 b^2\right ) \int \frac{1}{-c+x^2} \, dx}{9 c}+\frac{\left (2 b^2\right ) \int \frac{1}{c+x^2} \, dx}{9 c}+\frac{1}{9} \left (2 b^2 c\right ) \int \frac{1}{\left (1-\frac{c}{x^2}\right ) x^6} \, dx-\frac{1}{9} \left (2 b^2 c\right ) \int \frac{1}{\left (1+\frac{c}{x^2}\right ) x^6} \, dx\\ &=\frac{2 a b}{9 x^3}-\frac{4 b^2}{27 x^3}-\frac{2 a b}{3 c x}-\frac{2 a b \tan ^{-1}\left (\frac{x}{\sqrt{c}}\right )}{3 c^{3/2}}+\frac{8 b^2 \tan ^{-1}\left (\frac{x}{\sqrt{c}}\right )}{9 c^{3/2}}+\frac{i b^2 \tan ^{-1}\left (\frac{x}{\sqrt{c}}\right )^2}{3 c^{3/2}}+\frac{8 b^2 \tanh ^{-1}\left (\frac{x}{\sqrt{c}}\right )}{9 c^{3/2}}-\frac{b^2 \tanh ^{-1}\left (\frac{x}{\sqrt{c}}\right )^2}{3 c^{3/2}}-\frac{2 b^2 \tan ^{-1}\left (\frac{x}{\sqrt{c}}\right ) \log \left (2-\frac{2 \sqrt{c}}{\sqrt{c}-i x}\right )}{3 c^{3/2}}-\frac{b^2 \log \left (1-\frac{c}{x^2}\right )}{9 x^3}+\frac{b^2 \log \left (1-\frac{c}{x^2}\right )}{3 c x}+\frac{b^2 \tan ^{-1}\left (\frac{x}{\sqrt{c}}\right ) \log \left (1-\frac{c}{x^2}\right )}{3 c^{3/2}}-\frac{b \left (2 a-b \log \left (1-\frac{c}{x^2}\right )\right )}{9 x^3}-\frac{b \left (2 a-b \log \left (1-\frac{c}{x^2}\right )\right )}{3 c x}+\frac{b \tanh ^{-1}\left (\frac{x}{\sqrt{c}}\right ) \left (2 a-b \log \left (1-\frac{c}{x^2}\right )\right )}{3 c^{3/2}}-\frac{\left (2 a-b \log \left (1-\frac{c}{x^2}\right )\right )^2}{12 x^3}-\frac{a b \log \left (1+\frac{c}{x^2}\right )}{3 x^3}-\frac{2 b^2 \log \left (1+\frac{c}{x^2}\right )}{3 c x}-\frac{b^2 \tan ^{-1}\left (\frac{x}{\sqrt{c}}\right ) \log \left (1+\frac{c}{x^2}\right )}{3 c^{3/2}}+\frac{b^2 \tanh ^{-1}\left (\frac{x}{\sqrt{c}}\right ) \log \left (1+\frac{c}{x^2}\right )}{3 c^{3/2}}+\frac{b^2 \log \left (1-\frac{c}{x^2}\right ) \log \left (1+\frac{c}{x^2}\right )}{6 x^3}-\frac{b^2 \log ^2\left (1+\frac{c}{x^2}\right )}{12 x^3}-\frac{2 b^2 \tanh ^{-1}\left (\frac{x}{\sqrt{c}}\right ) \log \left (2-\frac{2 \sqrt{c}}{\sqrt{c}+x}\right )}{3 c^{3/2}}-\frac{1}{3} \left (2 b^2\right ) \int \frac{1}{x^2 \left (-c+x^2\right )} \, dx-\frac{1}{3} \left (2 b^2\right ) \int \frac{1}{x^2 \left (c+x^2\right )} \, dx+\frac{\left (2 b^2\right ) \int \frac{\log \left (2-\frac{2}{1-\frac{i x}{\sqrt{c}}}\right )}{1+\frac{x^2}{c}} \, dx}{3 c^2}+\frac{\left (2 b^2\right ) \int \frac{\log \left (2-\frac{2}{1+\frac{x}{\sqrt{c}}}\right )}{1-\frac{x^2}{c}} \, dx}{3 c^2}-\frac{\left (2 b^2\right ) \int \frac{\tan ^{-1}\left (\frac{x}{\sqrt{c}}\right )}{\left (1-\frac{c}{x^2}\right ) x^3} \, dx}{3 \sqrt{c}}+\frac{\left (2 b^2\right ) \int \frac{\tanh ^{-1}\left (\frac{x}{\sqrt{c}}\right )}{\left (1+\frac{c}{x^2}\right ) x^3} \, dx}{3 \sqrt{c}}+\frac{1}{9} \left (2 b^2 c\right ) \int \frac{1}{x^4 \left (-c+x^2\right )} \, dx-\frac{1}{9} \left (2 b^2 c\right ) \int \frac{1}{x^4 \left (c+x^2\right )} \, dx\\ &=\frac{2 a b}{9 x^3}-\frac{2 a b}{3 c x}-\frac{2 a b \tan ^{-1}\left (\frac{x}{\sqrt{c}}\right )}{3 c^{3/2}}+\frac{8 b^2 \tan ^{-1}\left (\frac{x}{\sqrt{c}}\right )}{9 c^{3/2}}+\frac{i b^2 \tan ^{-1}\left (\frac{x}{\sqrt{c}}\right )^2}{3 c^{3/2}}+\frac{8 b^2 \tanh ^{-1}\left (\frac{x}{\sqrt{c}}\right )}{9 c^{3/2}}-\frac{b^2 \tanh ^{-1}\left (\frac{x}{\sqrt{c}}\right )^2}{3 c^{3/2}}-\frac{2 b^2 \tan ^{-1}\left (\frac{x}{\sqrt{c}}\right ) \log \left (2-\frac{2 \sqrt{c}}{\sqrt{c}-i x}\right )}{3 c^{3/2}}-\frac{b^2 \log \left (1-\frac{c}{x^2}\right )}{9 x^3}+\frac{b^2 \log \left (1-\frac{c}{x^2}\right )}{3 c x}+\frac{b^2 \tan ^{-1}\left (\frac{x}{\sqrt{c}}\right ) \log \left (1-\frac{c}{x^2}\right )}{3 c^{3/2}}-\frac{b \left (2 a-b \log \left (1-\frac{c}{x^2}\right )\right )}{9 x^3}-\frac{b \left (2 a-b \log \left (1-\frac{c}{x^2}\right )\right )}{3 c x}+\frac{b \tanh ^{-1}\left (\frac{x}{\sqrt{c}}\right ) \left (2 a-b \log \left (1-\frac{c}{x^2}\right )\right )}{3 c^{3/2}}-\frac{\left (2 a-b \log \left (1-\frac{c}{x^2}\right )\right )^2}{12 x^3}-\frac{a b \log \left (1+\frac{c}{x^2}\right )}{3 x^3}-\frac{2 b^2 \log \left (1+\frac{c}{x^2}\right )}{3 c x}-\frac{b^2 \tan ^{-1}\left (\frac{x}{\sqrt{c}}\right ) \log \left (1+\frac{c}{x^2}\right )}{3 c^{3/2}}+\frac{b^2 \tanh ^{-1}\left (\frac{x}{\sqrt{c}}\right ) \log \left (1+\frac{c}{x^2}\right )}{3 c^{3/2}}+\frac{b^2 \log \left (1-\frac{c}{x^2}\right ) \log \left (1+\frac{c}{x^2}\right )}{6 x^3}-\frac{b^2 \log ^2\left (1+\frac{c}{x^2}\right )}{12 x^3}-\frac{2 b^2 \tanh ^{-1}\left (\frac{x}{\sqrt{c}}\right ) \log \left (2-\frac{2 \sqrt{c}}{\sqrt{c}+x}\right )}{3 c^{3/2}}+\frac{i b^2 \text{Li}_2\left (-1+\frac{2 \sqrt{c}}{\sqrt{c}-i x}\right )}{3 c^{3/2}}+\frac{b^2 \text{Li}_2\left (-1+\frac{2 \sqrt{c}}{\sqrt{c}+x}\right )}{3 c^{3/2}}+\frac{1}{9} \left (2 b^2\right ) \int \frac{1}{x^2 \left (-c+x^2\right )} \, dx+\frac{1}{9} \left (2 b^2\right ) \int \frac{1}{x^2 \left (c+x^2\right )} \, dx-\frac{\left (2 b^2\right ) \int \frac{1}{-c+x^2} \, dx}{3 c}+\frac{\left (2 b^2\right ) \int \frac{1}{c+x^2} \, dx}{3 c}-\frac{\left (2 b^2\right ) \int \frac{\tan ^{-1}\left (\frac{x}{\sqrt{c}}\right )}{x \left (-c+x^2\right )} \, dx}{3 \sqrt{c}}+\frac{\left (2 b^2\right ) \int \frac{\tanh ^{-1}\left (\frac{x}{\sqrt{c}}\right )}{x \left (c+x^2\right )} \, dx}{3 \sqrt{c}}\\ &=\frac{2 a b}{9 x^3}-\frac{2 a b}{3 c x}-\frac{2 a b \tan ^{-1}\left (\frac{x}{\sqrt{c}}\right )}{3 c^{3/2}}+\frac{14 b^2 \tan ^{-1}\left (\frac{x}{\sqrt{c}}\right )}{9 c^{3/2}}+\frac{i b^2 \tan ^{-1}\left (\frac{x}{\sqrt{c}}\right )^2}{3 c^{3/2}}+\frac{14 b^2 \tanh ^{-1}\left (\frac{x}{\sqrt{c}}\right )}{9 c^{3/2}}-\frac{b^2 \tanh ^{-1}\left (\frac{x}{\sqrt{c}}\right )^2}{3 c^{3/2}}-\frac{2 b^2 \tan ^{-1}\left (\frac{x}{\sqrt{c}}\right ) \log \left (2-\frac{2 \sqrt{c}}{\sqrt{c}-i x}\right )}{3 c^{3/2}}-\frac{b^2 \log \left (1-\frac{c}{x^2}\right )}{9 x^3}+\frac{b^2 \log \left (1-\frac{c}{x^2}\right )}{3 c x}+\frac{b^2 \tan ^{-1}\left (\frac{x}{\sqrt{c}}\right ) \log \left (1-\frac{c}{x^2}\right )}{3 c^{3/2}}-\frac{b \left (2 a-b \log \left (1-\frac{c}{x^2}\right )\right )}{9 x^3}-\frac{b \left (2 a-b \log \left (1-\frac{c}{x^2}\right )\right )}{3 c x}+\frac{b \tanh ^{-1}\left (\frac{x}{\sqrt{c}}\right ) \left (2 a-b \log \left (1-\frac{c}{x^2}\right )\right )}{3 c^{3/2}}-\frac{\left (2 a-b \log \left (1-\frac{c}{x^2}\right )\right )^2}{12 x^3}-\frac{a b \log \left (1+\frac{c}{x^2}\right )}{3 x^3}-\frac{2 b^2 \log \left (1+\frac{c}{x^2}\right )}{3 c x}-\frac{b^2 \tan ^{-1}\left (\frac{x}{\sqrt{c}}\right ) \log \left (1+\frac{c}{x^2}\right )}{3 c^{3/2}}+\frac{b^2 \tanh ^{-1}\left (\frac{x}{\sqrt{c}}\right ) \log \left (1+\frac{c}{x^2}\right )}{3 c^{3/2}}+\frac{b^2 \log \left (1-\frac{c}{x^2}\right ) \log \left (1+\frac{c}{x^2}\right )}{6 x^3}-\frac{b^2 \log ^2\left (1+\frac{c}{x^2}\right )}{12 x^3}-\frac{2 b^2 \tanh ^{-1}\left (\frac{x}{\sqrt{c}}\right ) \log \left (2-\frac{2 \sqrt{c}}{\sqrt{c}+x}\right )}{3 c^{3/2}}+\frac{i b^2 \text{Li}_2\left (-1+\frac{2 \sqrt{c}}{\sqrt{c}-i x}\right )}{3 c^{3/2}}+\frac{b^2 \text{Li}_2\left (-1+\frac{2 \sqrt{c}}{\sqrt{c}+x}\right )}{3 c^{3/2}}+\frac{\left (2 b^2\right ) \int \frac{1}{-c+x^2} \, dx}{9 c}-\frac{\left (2 b^2\right ) \int \frac{1}{c+x^2} \, dx}{9 c}-\frac{\left (2 b^2\right ) \int \left (-\frac{\tan ^{-1}\left (\frac{x}{\sqrt{c}}\right )}{c x}-\frac{x \tan ^{-1}\left (\frac{x}{\sqrt{c}}\right )}{c \left (c-x^2\right )}\right ) \, dx}{3 \sqrt{c}}+\frac{\left (2 b^2\right ) \int \left (\frac{\tanh ^{-1}\left (\frac{x}{\sqrt{c}}\right )}{c x}-\frac{x \tanh ^{-1}\left (\frac{x}{\sqrt{c}}\right )}{c \left (c+x^2\right )}\right ) \, dx}{3 \sqrt{c}}\\ &=\frac{2 a b}{9 x^3}-\frac{2 a b}{3 c x}-\frac{2 a b \tan ^{-1}\left (\frac{x}{\sqrt{c}}\right )}{3 c^{3/2}}+\frac{4 b^2 \tan ^{-1}\left (\frac{x}{\sqrt{c}}\right )}{3 c^{3/2}}+\frac{i b^2 \tan ^{-1}\left (\frac{x}{\sqrt{c}}\right )^2}{3 c^{3/2}}+\frac{4 b^2 \tanh ^{-1}\left (\frac{x}{\sqrt{c}}\right )}{3 c^{3/2}}-\frac{b^2 \tanh ^{-1}\left (\frac{x}{\sqrt{c}}\right )^2}{3 c^{3/2}}-\frac{2 b^2 \tan ^{-1}\left (\frac{x}{\sqrt{c}}\right ) \log \left (2-\frac{2 \sqrt{c}}{\sqrt{c}-i x}\right )}{3 c^{3/2}}-\frac{b^2 \log \left (1-\frac{c}{x^2}\right )}{9 x^3}+\frac{b^2 \log \left (1-\frac{c}{x^2}\right )}{3 c x}+\frac{b^2 \tan ^{-1}\left (\frac{x}{\sqrt{c}}\right ) \log \left (1-\frac{c}{x^2}\right )}{3 c^{3/2}}-\frac{b \left (2 a-b \log \left (1-\frac{c}{x^2}\right )\right )}{9 x^3}-\frac{b \left (2 a-b \log \left (1-\frac{c}{x^2}\right )\right )}{3 c x}+\frac{b \tanh ^{-1}\left (\frac{x}{\sqrt{c}}\right ) \left (2 a-b \log \left (1-\frac{c}{x^2}\right )\right )}{3 c^{3/2}}-\frac{\left (2 a-b \log \left (1-\frac{c}{x^2}\right )\right )^2}{12 x^3}-\frac{a b \log \left (1+\frac{c}{x^2}\right )}{3 x^3}-\frac{2 b^2 \log \left (1+\frac{c}{x^2}\right )}{3 c x}-\frac{b^2 \tan ^{-1}\left (\frac{x}{\sqrt{c}}\right ) \log \left (1+\frac{c}{x^2}\right )}{3 c^{3/2}}+\frac{b^2 \tanh ^{-1}\left (\frac{x}{\sqrt{c}}\right ) \log \left (1+\frac{c}{x^2}\right )}{3 c^{3/2}}+\frac{b^2 \log \left (1-\frac{c}{x^2}\right ) \log \left (1+\frac{c}{x^2}\right )}{6 x^3}-\frac{b^2 \log ^2\left (1+\frac{c}{x^2}\right )}{12 x^3}-\frac{2 b^2 \tanh ^{-1}\left (\frac{x}{\sqrt{c}}\right ) \log \left (2-\frac{2 \sqrt{c}}{\sqrt{c}+x}\right )}{3 c^{3/2}}+\frac{i b^2 \text{Li}_2\left (-1+\frac{2 \sqrt{c}}{\sqrt{c}-i x}\right )}{3 c^{3/2}}+\frac{b^2 \text{Li}_2\left (-1+\frac{2 \sqrt{c}}{\sqrt{c}+x}\right )}{3 c^{3/2}}+\frac{\left (2 b^2\right ) \int \frac{\tan ^{-1}\left (\frac{x}{\sqrt{c}}\right )}{x} \, dx}{3 c^{3/2}}+\frac{\left (2 b^2\right ) \int \frac{x \tan ^{-1}\left (\frac{x}{\sqrt{c}}\right )}{c-x^2} \, dx}{3 c^{3/2}}+\frac{\left (2 b^2\right ) \int \frac{\tanh ^{-1}\left (\frac{x}{\sqrt{c}}\right )}{x} \, dx}{3 c^{3/2}}-\frac{\left (2 b^2\right ) \int \frac{x \tanh ^{-1}\left (\frac{x}{\sqrt{c}}\right )}{c+x^2} \, dx}{3 c^{3/2}}\\ &=\frac{2 a b}{9 x^3}-\frac{2 a b}{3 c x}-\frac{2 a b \tan ^{-1}\left (\frac{x}{\sqrt{c}}\right )}{3 c^{3/2}}+\frac{4 b^2 \tan ^{-1}\left (\frac{x}{\sqrt{c}}\right )}{3 c^{3/2}}+\frac{i b^2 \tan ^{-1}\left (\frac{x}{\sqrt{c}}\right )^2}{3 c^{3/2}}+\frac{4 b^2 \tanh ^{-1}\left (\frac{x}{\sqrt{c}}\right )}{3 c^{3/2}}-\frac{b^2 \tanh ^{-1}\left (\frac{x}{\sqrt{c}}\right )^2}{3 c^{3/2}}-\frac{2 b^2 \tan ^{-1}\left (\frac{x}{\sqrt{c}}\right ) \log \left (2-\frac{2 \sqrt{c}}{\sqrt{c}-i x}\right )}{3 c^{3/2}}-\frac{b^2 \log \left (1-\frac{c}{x^2}\right )}{9 x^3}+\frac{b^2 \log \left (1-\frac{c}{x^2}\right )}{3 c x}+\frac{b^2 \tan ^{-1}\left (\frac{x}{\sqrt{c}}\right ) \log \left (1-\frac{c}{x^2}\right )}{3 c^{3/2}}-\frac{b \left (2 a-b \log \left (1-\frac{c}{x^2}\right )\right )}{9 x^3}-\frac{b \left (2 a-b \log \left (1-\frac{c}{x^2}\right )\right )}{3 c x}+\frac{b \tanh ^{-1}\left (\frac{x}{\sqrt{c}}\right ) \left (2 a-b \log \left (1-\frac{c}{x^2}\right )\right )}{3 c^{3/2}}-\frac{\left (2 a-b \log \left (1-\frac{c}{x^2}\right )\right )^2}{12 x^3}-\frac{a b \log \left (1+\frac{c}{x^2}\right )}{3 x^3}-\frac{2 b^2 \log \left (1+\frac{c}{x^2}\right )}{3 c x}-\frac{b^2 \tan ^{-1}\left (\frac{x}{\sqrt{c}}\right ) \log \left (1+\frac{c}{x^2}\right )}{3 c^{3/2}}+\frac{b^2 \tanh ^{-1}\left (\frac{x}{\sqrt{c}}\right ) \log \left (1+\frac{c}{x^2}\right )}{3 c^{3/2}}+\frac{b^2 \log \left (1-\frac{c}{x^2}\right ) \log \left (1+\frac{c}{x^2}\right )}{6 x^3}-\frac{b^2 \log ^2\left (1+\frac{c}{x^2}\right )}{12 x^3}-\frac{2 b^2 \tanh ^{-1}\left (\frac{x}{\sqrt{c}}\right ) \log \left (2-\frac{2 \sqrt{c}}{\sqrt{c}+x}\right )}{3 c^{3/2}}+\frac{i b^2 \text{Li}_2\left (-1+\frac{2 \sqrt{c}}{\sqrt{c}-i x}\right )}{3 c^{3/2}}-\frac{b^2 \text{Li}_2\left (-\frac{x}{\sqrt{c}}\right )}{3 c^{3/2}}+\frac{b^2 \text{Li}_2\left (\frac{x}{\sqrt{c}}\right )}{3 c^{3/2}}+\frac{b^2 \text{Li}_2\left (-1+\frac{2 \sqrt{c}}{\sqrt{c}+x}\right )}{3 c^{3/2}}+\frac{\left (i b^2\right ) \int \frac{\log \left (1-\frac{i x}{\sqrt{c}}\right )}{x} \, dx}{3 c^{3/2}}-\frac{\left (i b^2\right ) \int \frac{\log \left (1+\frac{i x}{\sqrt{c}}\right )}{x} \, dx}{3 c^{3/2}}+\frac{\left (2 b^2\right ) \int \left (\frac{\tan ^{-1}\left (\frac{x}{\sqrt{c}}\right )}{2 \left (\sqrt{c}-x\right )}-\frac{\tan ^{-1}\left (\frac{x}{\sqrt{c}}\right )}{2 \left (\sqrt{c}+x\right )}\right ) \, dx}{3 c^{3/2}}-\frac{\left (2 b^2\right ) \int \left (-\frac{\tanh ^{-1}\left (\frac{x}{\sqrt{c}}\right )}{2 \left (\sqrt{-c}-x\right )}+\frac{\tanh ^{-1}\left (\frac{x}{\sqrt{c}}\right )}{2 \left (\sqrt{-c}+x\right )}\right ) \, dx}{3 c^{3/2}}\\ &=\frac{2 a b}{9 x^3}-\frac{2 a b}{3 c x}-\frac{2 a b \tan ^{-1}\left (\frac{x}{\sqrt{c}}\right )}{3 c^{3/2}}+\frac{4 b^2 \tan ^{-1}\left (\frac{x}{\sqrt{c}}\right )}{3 c^{3/2}}+\frac{i b^2 \tan ^{-1}\left (\frac{x}{\sqrt{c}}\right )^2}{3 c^{3/2}}+\frac{4 b^2 \tanh ^{-1}\left (\frac{x}{\sqrt{c}}\right )}{3 c^{3/2}}-\frac{b^2 \tanh ^{-1}\left (\frac{x}{\sqrt{c}}\right )^2}{3 c^{3/2}}-\frac{2 b^2 \tan ^{-1}\left (\frac{x}{\sqrt{c}}\right ) \log \left (2-\frac{2 \sqrt{c}}{\sqrt{c}-i x}\right )}{3 c^{3/2}}-\frac{b^2 \log \left (1-\frac{c}{x^2}\right )}{9 x^3}+\frac{b^2 \log \left (1-\frac{c}{x^2}\right )}{3 c x}+\frac{b^2 \tan ^{-1}\left (\frac{x}{\sqrt{c}}\right ) \log \left (1-\frac{c}{x^2}\right )}{3 c^{3/2}}-\frac{b \left (2 a-b \log \left (1-\frac{c}{x^2}\right )\right )}{9 x^3}-\frac{b \left (2 a-b \log \left (1-\frac{c}{x^2}\right )\right )}{3 c x}+\frac{b \tanh ^{-1}\left (\frac{x}{\sqrt{c}}\right ) \left (2 a-b \log \left (1-\frac{c}{x^2}\right )\right )}{3 c^{3/2}}-\frac{\left (2 a-b \log \left (1-\frac{c}{x^2}\right )\right )^2}{12 x^3}-\frac{a b \log \left (1+\frac{c}{x^2}\right )}{3 x^3}-\frac{2 b^2 \log \left (1+\frac{c}{x^2}\right )}{3 c x}-\frac{b^2 \tan ^{-1}\left (\frac{x}{\sqrt{c}}\right ) \log \left (1+\frac{c}{x^2}\right )}{3 c^{3/2}}+\frac{b^2 \tanh ^{-1}\left (\frac{x}{\sqrt{c}}\right ) \log \left (1+\frac{c}{x^2}\right )}{3 c^{3/2}}+\frac{b^2 \log \left (1-\frac{c}{x^2}\right ) \log \left (1+\frac{c}{x^2}\right )}{6 x^3}-\frac{b^2 \log ^2\left (1+\frac{c}{x^2}\right )}{12 x^3}-\frac{2 b^2 \tanh ^{-1}\left (\frac{x}{\sqrt{c}}\right ) \log \left (2-\frac{2 \sqrt{c}}{\sqrt{c}+x}\right )}{3 c^{3/2}}+\frac{i b^2 \text{Li}_2\left (-1+\frac{2 \sqrt{c}}{\sqrt{c}-i x}\right )}{3 c^{3/2}}-\frac{b^2 \text{Li}_2\left (-\frac{x}{\sqrt{c}}\right )}{3 c^{3/2}}+\frac{i b^2 \text{Li}_2\left (-\frac{i x}{\sqrt{c}}\right )}{3 c^{3/2}}-\frac{i b^2 \text{Li}_2\left (\frac{i x}{\sqrt{c}}\right )}{3 c^{3/2}}+\frac{b^2 \text{Li}_2\left (\frac{x}{\sqrt{c}}\right )}{3 c^{3/2}}+\frac{b^2 \text{Li}_2\left (-1+\frac{2 \sqrt{c}}{\sqrt{c}+x}\right )}{3 c^{3/2}}+\frac{b^2 \int \frac{\tan ^{-1}\left (\frac{x}{\sqrt{c}}\right )}{\sqrt{c}-x} \, dx}{3 c^{3/2}}-\frac{b^2 \int \frac{\tan ^{-1}\left (\frac{x}{\sqrt{c}}\right )}{\sqrt{c}+x} \, dx}{3 c^{3/2}}+\frac{b^2 \int \frac{\tanh ^{-1}\left (\frac{x}{\sqrt{c}}\right )}{\sqrt{-c}-x} \, dx}{3 c^{3/2}}-\frac{b^2 \int \frac{\tanh ^{-1}\left (\frac{x}{\sqrt{c}}\right )}{\sqrt{-c}+x} \, dx}{3 c^{3/2}}\\ &=\frac{2 a b}{9 x^3}-\frac{2 a b}{3 c x}-\frac{2 a b \tan ^{-1}\left (\frac{x}{\sqrt{c}}\right )}{3 c^{3/2}}+\frac{4 b^2 \tan ^{-1}\left (\frac{x}{\sqrt{c}}\right )}{3 c^{3/2}}+\frac{i b^2 \tan ^{-1}\left (\frac{x}{\sqrt{c}}\right )^2}{3 c^{3/2}}+\frac{4 b^2 \tanh ^{-1}\left (\frac{x}{\sqrt{c}}\right )}{3 c^{3/2}}-\frac{b^2 \tanh ^{-1}\left (\frac{x}{\sqrt{c}}\right )^2}{3 c^{3/2}}-\frac{2 b^2 \tan ^{-1}\left (\frac{x}{\sqrt{c}}\right ) \log \left (2-\frac{2 \sqrt{c}}{\sqrt{c}-i x}\right )}{3 c^{3/2}}-\frac{b^2 \log \left (1-\frac{c}{x^2}\right )}{9 x^3}+\frac{b^2 \log \left (1-\frac{c}{x^2}\right )}{3 c x}+\frac{b^2 \tan ^{-1}\left (\frac{x}{\sqrt{c}}\right ) \log \left (1-\frac{c}{x^2}\right )}{3 c^{3/2}}-\frac{b \left (2 a-b \log \left (1-\frac{c}{x^2}\right )\right )}{9 x^3}-\frac{b \left (2 a-b \log \left (1-\frac{c}{x^2}\right )\right )}{3 c x}+\frac{b \tanh ^{-1}\left (\frac{x}{\sqrt{c}}\right ) \left (2 a-b \log \left (1-\frac{c}{x^2}\right )\right )}{3 c^{3/2}}-\frac{\left (2 a-b \log \left (1-\frac{c}{x^2}\right )\right )^2}{12 x^3}-\frac{a b \log \left (1+\frac{c}{x^2}\right )}{3 x^3}-\frac{2 b^2 \log \left (1+\frac{c}{x^2}\right )}{3 c x}-\frac{b^2 \tan ^{-1}\left (\frac{x}{\sqrt{c}}\right ) \log \left (1+\frac{c}{x^2}\right )}{3 c^{3/2}}+\frac{b^2 \tanh ^{-1}\left (\frac{x}{\sqrt{c}}\right ) \log \left (1+\frac{c}{x^2}\right )}{3 c^{3/2}}+\frac{b^2 \log \left (1-\frac{c}{x^2}\right ) \log \left (1+\frac{c}{x^2}\right )}{6 x^3}-\frac{b^2 \log ^2\left (1+\frac{c}{x^2}\right )}{12 x^3}+\frac{2 b^2 \tan ^{-1}\left (\frac{x}{\sqrt{c}}\right ) \log \left (\frac{2 \sqrt{c}}{\sqrt{c}-i x}\right )}{3 c^{3/2}}-\frac{b^2 \tan ^{-1}\left (\frac{x}{\sqrt{c}}\right ) \log \left (\frac{(1+i) \left (\sqrt{c}-x\right )}{\sqrt{c}-i x}\right )}{3 c^{3/2}}+\frac{2 b^2 \tanh ^{-1}\left (\frac{x}{\sqrt{c}}\right ) \log \left (\frac{2 \sqrt{c}}{\sqrt{c}+x}\right )}{3 c^{3/2}}-\frac{b^2 \tanh ^{-1}\left (\frac{x}{\sqrt{c}}\right ) \log \left (\frac{2 \sqrt{c} \left (\sqrt{-c}-x\right )}{\left (\sqrt{-c}-\sqrt{c}\right ) \left (\sqrt{c}+x\right )}\right )}{3 c^{3/2}}-\frac{b^2 \tanh ^{-1}\left (\frac{x}{\sqrt{c}}\right ) \log \left (\frac{2 \sqrt{c} \left (\sqrt{-c}+x\right )}{\left (\sqrt{-c}+\sqrt{c}\right ) \left (\sqrt{c}+x\right )}\right )}{3 c^{3/2}}-\frac{b^2 \tan ^{-1}\left (\frac{x}{\sqrt{c}}\right ) \log \left (\frac{(1-i) \left (\sqrt{c}+x\right )}{\sqrt{c}-i x}\right )}{3 c^{3/2}}-\frac{2 b^2 \tanh ^{-1}\left (\frac{x}{\sqrt{c}}\right ) \log \left (2-\frac{2 \sqrt{c}}{\sqrt{c}+x}\right )}{3 c^{3/2}}+\frac{i b^2 \text{Li}_2\left (-1+\frac{2 \sqrt{c}}{\sqrt{c}-i x}\right )}{3 c^{3/2}}-\frac{b^2 \text{Li}_2\left (-\frac{x}{\sqrt{c}}\right )}{3 c^{3/2}}+\frac{i b^2 \text{Li}_2\left (-\frac{i x}{\sqrt{c}}\right )}{3 c^{3/2}}-\frac{i b^2 \text{Li}_2\left (\frac{i x}{\sqrt{c}}\right )}{3 c^{3/2}}+\frac{b^2 \text{Li}_2\left (\frac{x}{\sqrt{c}}\right )}{3 c^{3/2}}+\frac{b^2 \text{Li}_2\left (-1+\frac{2 \sqrt{c}}{\sqrt{c}+x}\right )}{3 c^{3/2}}-2 \frac{b^2 \int \frac{\log \left (\frac{2}{1-\frac{i x}{\sqrt{c}}}\right )}{1+\frac{x^2}{c}} \, dx}{3 c^2}+\frac{b^2 \int \frac{\log \left (\frac{(1+i) \left (\sqrt{c}-x\right )}{\sqrt{c} \left (1-\frac{i x}{\sqrt{c}}\right )}\right )}{1+\frac{x^2}{c}} \, dx}{3 c^2}+\frac{b^2 \int \frac{\log \left (\frac{(1-i) \left (\sqrt{c}+x\right )}{\sqrt{c} \left (1-\frac{i x}{\sqrt{c}}\right )}\right )}{1+\frac{x^2}{c}} \, dx}{3 c^2}-2 \frac{b^2 \int \frac{\log \left (\frac{2}{1+\frac{x}{\sqrt{c}}}\right )}{1-\frac{x^2}{c}} \, dx}{3 c^2}+\frac{b^2 \int \frac{\log \left (\frac{2 \left (\sqrt{-c}-x\right )}{\left (-1+\frac{\sqrt{-c}}{\sqrt{c}}\right ) \sqrt{c} \left (1+\frac{x}{\sqrt{c}}\right )}\right )}{1-\frac{x^2}{c}} \, dx}{3 c^2}+\frac{b^2 \int \frac{\log \left (\frac{2 \left (\sqrt{-c}+x\right )}{\left (1+\frac{\sqrt{-c}}{\sqrt{c}}\right ) \sqrt{c} \left (1+\frac{x}{\sqrt{c}}\right )}\right )}{1-\frac{x^2}{c}} \, dx}{3 c^2}\\ &=\frac{2 a b}{9 x^3}-\frac{2 a b}{3 c x}-\frac{2 a b \tan ^{-1}\left (\frac{x}{\sqrt{c}}\right )}{3 c^{3/2}}+\frac{4 b^2 \tan ^{-1}\left (\frac{x}{\sqrt{c}}\right )}{3 c^{3/2}}+\frac{i b^2 \tan ^{-1}\left (\frac{x}{\sqrt{c}}\right )^2}{3 c^{3/2}}+\frac{4 b^2 \tanh ^{-1}\left (\frac{x}{\sqrt{c}}\right )}{3 c^{3/2}}-\frac{b^2 \tanh ^{-1}\left (\frac{x}{\sqrt{c}}\right )^2}{3 c^{3/2}}-\frac{2 b^2 \tan ^{-1}\left (\frac{x}{\sqrt{c}}\right ) \log \left (2-\frac{2 \sqrt{c}}{\sqrt{c}-i x}\right )}{3 c^{3/2}}-\frac{b^2 \log \left (1-\frac{c}{x^2}\right )}{9 x^3}+\frac{b^2 \log \left (1-\frac{c}{x^2}\right )}{3 c x}+\frac{b^2 \tan ^{-1}\left (\frac{x}{\sqrt{c}}\right ) \log \left (1-\frac{c}{x^2}\right )}{3 c^{3/2}}-\frac{b \left (2 a-b \log \left (1-\frac{c}{x^2}\right )\right )}{9 x^3}-\frac{b \left (2 a-b \log \left (1-\frac{c}{x^2}\right )\right )}{3 c x}+\frac{b \tanh ^{-1}\left (\frac{x}{\sqrt{c}}\right ) \left (2 a-b \log \left (1-\frac{c}{x^2}\right )\right )}{3 c^{3/2}}-\frac{\left (2 a-b \log \left (1-\frac{c}{x^2}\right )\right )^2}{12 x^3}-\frac{a b \log \left (1+\frac{c}{x^2}\right )}{3 x^3}-\frac{2 b^2 \log \left (1+\frac{c}{x^2}\right )}{3 c x}-\frac{b^2 \tan ^{-1}\left (\frac{x}{\sqrt{c}}\right ) \log \left (1+\frac{c}{x^2}\right )}{3 c^{3/2}}+\frac{b^2 \tanh ^{-1}\left (\frac{x}{\sqrt{c}}\right ) \log \left (1+\frac{c}{x^2}\right )}{3 c^{3/2}}+\frac{b^2 \log \left (1-\frac{c}{x^2}\right ) \log \left (1+\frac{c}{x^2}\right )}{6 x^3}-\frac{b^2 \log ^2\left (1+\frac{c}{x^2}\right )}{12 x^3}+\frac{2 b^2 \tan ^{-1}\left (\frac{x}{\sqrt{c}}\right ) \log \left (\frac{2 \sqrt{c}}{\sqrt{c}-i x}\right )}{3 c^{3/2}}-\frac{b^2 \tan ^{-1}\left (\frac{x}{\sqrt{c}}\right ) \log \left (\frac{(1+i) \left (\sqrt{c}-x\right )}{\sqrt{c}-i x}\right )}{3 c^{3/2}}+\frac{2 b^2 \tanh ^{-1}\left (\frac{x}{\sqrt{c}}\right ) \log \left (\frac{2 \sqrt{c}}{\sqrt{c}+x}\right )}{3 c^{3/2}}-\frac{b^2 \tanh ^{-1}\left (\frac{x}{\sqrt{c}}\right ) \log \left (\frac{2 \sqrt{c} \left (\sqrt{-c}-x\right )}{\left (\sqrt{-c}-\sqrt{c}\right ) \left (\sqrt{c}+x\right )}\right )}{3 c^{3/2}}-\frac{b^2 \tanh ^{-1}\left (\frac{x}{\sqrt{c}}\right ) \log \left (\frac{2 \sqrt{c} \left (\sqrt{-c}+x\right )}{\left (\sqrt{-c}+\sqrt{c}\right ) \left (\sqrt{c}+x\right )}\right )}{3 c^{3/2}}-\frac{b^2 \tan ^{-1}\left (\frac{x}{\sqrt{c}}\right ) \log \left (\frac{(1-i) \left (\sqrt{c}+x\right )}{\sqrt{c}-i x}\right )}{3 c^{3/2}}-\frac{2 b^2 \tanh ^{-1}\left (\frac{x}{\sqrt{c}}\right ) \log \left (2-\frac{2 \sqrt{c}}{\sqrt{c}+x}\right )}{3 c^{3/2}}+\frac{i b^2 \text{Li}_2\left (-1+\frac{2 \sqrt{c}}{\sqrt{c}-i x}\right )}{3 c^{3/2}}+\frac{i b^2 \text{Li}_2\left (1-\frac{(1+i) \left (\sqrt{c}-x\right )}{\sqrt{c}-i x}\right )}{6 c^{3/2}}-\frac{b^2 \text{Li}_2\left (-\frac{x}{\sqrt{c}}\right )}{3 c^{3/2}}+\frac{i b^2 \text{Li}_2\left (-\frac{i x}{\sqrt{c}}\right )}{3 c^{3/2}}-\frac{i b^2 \text{Li}_2\left (\frac{i x}{\sqrt{c}}\right )}{3 c^{3/2}}+\frac{b^2 \text{Li}_2\left (\frac{x}{\sqrt{c}}\right )}{3 c^{3/2}}+\frac{b^2 \text{Li}_2\left (-1+\frac{2 \sqrt{c}}{\sqrt{c}+x}\right )}{3 c^{3/2}}+\frac{b^2 \text{Li}_2\left (1-\frac{2 \sqrt{c} \left (\sqrt{-c}-x\right )}{\left (\sqrt{-c}-\sqrt{c}\right ) \left (\sqrt{c}+x\right )}\right )}{6 c^{3/2}}+\frac{b^2 \text{Li}_2\left (1-\frac{2 \sqrt{c} \left (\sqrt{-c}+x\right )}{\left (\sqrt{-c}+\sqrt{c}\right ) \left (\sqrt{c}+x\right )}\right )}{6 c^{3/2}}+\frac{i b^2 \text{Li}_2\left (1-\frac{(1-i) \left (\sqrt{c}+x\right )}{\sqrt{c}-i x}\right )}{6 c^{3/2}}-2 \frac{\left (i b^2\right ) \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1-\frac{i x}{\sqrt{c}}}\right )}{3 c^{3/2}}-2 \frac{b^2 \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1+\frac{x}{\sqrt{c}}}\right )}{3 c^{3/2}}\\ &=\frac{2 a b}{9 x^3}-\frac{2 a b}{3 c x}-\frac{2 a b \tan ^{-1}\left (\frac{x}{\sqrt{c}}\right )}{3 c^{3/2}}+\frac{4 b^2 \tan ^{-1}\left (\frac{x}{\sqrt{c}}\right )}{3 c^{3/2}}+\frac{i b^2 \tan ^{-1}\left (\frac{x}{\sqrt{c}}\right )^2}{3 c^{3/2}}+\frac{4 b^2 \tanh ^{-1}\left (\frac{x}{\sqrt{c}}\right )}{3 c^{3/2}}-\frac{b^2 \tanh ^{-1}\left (\frac{x}{\sqrt{c}}\right )^2}{3 c^{3/2}}-\frac{2 b^2 \tan ^{-1}\left (\frac{x}{\sqrt{c}}\right ) \log \left (2-\frac{2 \sqrt{c}}{\sqrt{c}-i x}\right )}{3 c^{3/2}}-\frac{b^2 \log \left (1-\frac{c}{x^2}\right )}{9 x^3}+\frac{b^2 \log \left (1-\frac{c}{x^2}\right )}{3 c x}+\frac{b^2 \tan ^{-1}\left (\frac{x}{\sqrt{c}}\right ) \log \left (1-\frac{c}{x^2}\right )}{3 c^{3/2}}-\frac{b \left (2 a-b \log \left (1-\frac{c}{x^2}\right )\right )}{9 x^3}-\frac{b \left (2 a-b \log \left (1-\frac{c}{x^2}\right )\right )}{3 c x}+\frac{b \tanh ^{-1}\left (\frac{x}{\sqrt{c}}\right ) \left (2 a-b \log \left (1-\frac{c}{x^2}\right )\right )}{3 c^{3/2}}-\frac{\left (2 a-b \log \left (1-\frac{c}{x^2}\right )\right )^2}{12 x^3}-\frac{a b \log \left (1+\frac{c}{x^2}\right )}{3 x^3}-\frac{2 b^2 \log \left (1+\frac{c}{x^2}\right )}{3 c x}-\frac{b^2 \tan ^{-1}\left (\frac{x}{\sqrt{c}}\right ) \log \left (1+\frac{c}{x^2}\right )}{3 c^{3/2}}+\frac{b^2 \tanh ^{-1}\left (\frac{x}{\sqrt{c}}\right ) \log \left (1+\frac{c}{x^2}\right )}{3 c^{3/2}}+\frac{b^2 \log \left (1-\frac{c}{x^2}\right ) \log \left (1+\frac{c}{x^2}\right )}{6 x^3}-\frac{b^2 \log ^2\left (1+\frac{c}{x^2}\right )}{12 x^3}+\frac{2 b^2 \tan ^{-1}\left (\frac{x}{\sqrt{c}}\right ) \log \left (\frac{2 \sqrt{c}}{\sqrt{c}-i x}\right )}{3 c^{3/2}}-\frac{b^2 \tan ^{-1}\left (\frac{x}{\sqrt{c}}\right ) \log \left (\frac{(1+i) \left (\sqrt{c}-x\right )}{\sqrt{c}-i x}\right )}{3 c^{3/2}}+\frac{2 b^2 \tanh ^{-1}\left (\frac{x}{\sqrt{c}}\right ) \log \left (\frac{2 \sqrt{c}}{\sqrt{c}+x}\right )}{3 c^{3/2}}-\frac{b^2 \tanh ^{-1}\left (\frac{x}{\sqrt{c}}\right ) \log \left (\frac{2 \sqrt{c} \left (\sqrt{-c}-x\right )}{\left (\sqrt{-c}-\sqrt{c}\right ) \left (\sqrt{c}+x\right )}\right )}{3 c^{3/2}}-\frac{b^2 \tanh ^{-1}\left (\frac{x}{\sqrt{c}}\right ) \log \left (\frac{2 \sqrt{c} \left (\sqrt{-c}+x\right )}{\left (\sqrt{-c}+\sqrt{c}\right ) \left (\sqrt{c}+x\right )}\right )}{3 c^{3/2}}-\frac{b^2 \tan ^{-1}\left (\frac{x}{\sqrt{c}}\right ) \log \left (\frac{(1-i) \left (\sqrt{c}+x\right )}{\sqrt{c}-i x}\right )}{3 c^{3/2}}-\frac{2 b^2 \tanh ^{-1}\left (\frac{x}{\sqrt{c}}\right ) \log \left (2-\frac{2 \sqrt{c}}{\sqrt{c}+x}\right )}{3 c^{3/2}}-\frac{i b^2 \text{Li}_2\left (1-\frac{2 \sqrt{c}}{\sqrt{c}-i x}\right )}{3 c^{3/2}}+\frac{i b^2 \text{Li}_2\left (-1+\frac{2 \sqrt{c}}{\sqrt{c}-i x}\right )}{3 c^{3/2}}+\frac{i b^2 \text{Li}_2\left (1-\frac{(1+i) \left (\sqrt{c}-x\right )}{\sqrt{c}-i x}\right )}{6 c^{3/2}}-\frac{b^2 \text{Li}_2\left (-\frac{x}{\sqrt{c}}\right )}{3 c^{3/2}}+\frac{i b^2 \text{Li}_2\left (-\frac{i x}{\sqrt{c}}\right )}{3 c^{3/2}}-\frac{i b^2 \text{Li}_2\left (\frac{i x}{\sqrt{c}}\right )}{3 c^{3/2}}+\frac{b^2 \text{Li}_2\left (\frac{x}{\sqrt{c}}\right )}{3 c^{3/2}}-\frac{b^2 \text{Li}_2\left (1-\frac{2 \sqrt{c}}{\sqrt{c}+x}\right )}{3 c^{3/2}}+\frac{b^2 \text{Li}_2\left (-1+\frac{2 \sqrt{c}}{\sqrt{c}+x}\right )}{3 c^{3/2}}+\frac{b^2 \text{Li}_2\left (1-\frac{2 \sqrt{c} \left (\sqrt{-c}-x\right )}{\left (\sqrt{-c}-\sqrt{c}\right ) \left (\sqrt{c}+x\right )}\right )}{6 c^{3/2}}+\frac{b^2 \text{Li}_2\left (1-\frac{2 \sqrt{c} \left (\sqrt{-c}+x\right )}{\left (\sqrt{-c}+\sqrt{c}\right ) \left (\sqrt{c}+x\right )}\right )}{6 c^{3/2}}+\frac{i b^2 \text{Li}_2\left (1-\frac{(1-i) \left (\sqrt{c}+x\right )}{\sqrt{c}-i x}\right )}{6 c^{3/2}}\\ \end{align*}
Mathematica [F] time = 2.6617, size = 0, normalized size = 0. \[ \int \frac{\left (a+b \tanh ^{-1}\left (\frac{c}{x^2}\right )\right )^2}{x^4} \, dx \]
Verification is Not applicable to the result.
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Maple [F] time = 0.603, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{4}} \left ( a+b{\it Artanh} \left ({\frac{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 (\frac{b^{2} \operatorname{artanh}\left (\frac{c}{x^{2}}\right )^{2} + 2 \, a b \operatorname{artanh}\left (\frac{c}{x^{2}}\right ) + a^{2}}{x^{4}}, 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{atanh}{\left (\frac{c}{x^{2}} \right )}\right )^{2}}{x^{4}}\, 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 \operatorname{artanh}\left (\frac{c}{x^{2}}\right ) + a\right )}^{2}}{x^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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