Optimal. Leaf size=1214 \[ \frac {1}{20} \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )^2 x^5+\frac {1}{20} b^2 \log ^2\left (\frac {c}{x^2}+1\right ) x^5+\frac {1}{5} a b \log \left (\frac {c}{x^2}+1\right ) x^5-\frac {1}{10} b^2 \log \left (1-\frac {c}{x^2}\right ) \log \left (\frac {c}{x^2}+1\right ) x^5+\frac {2}{15} a b c x^3-\frac {1}{15} b^2 c \log \left (1-\frac {c}{x^2}\right ) x^3+\frac {1}{15} b c \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right ) x^3+\frac {2}{15} b^2 c \log \left (\frac {c}{x^2}+1\right ) x^3+\frac {8}{15} b^2 c^2 x-\frac {1}{5} i b^2 c^{5/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )^2+\frac {1}{5} b^2 c^{5/2} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right )^2-\frac {4}{15} b^2 c^{5/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )+\frac {2}{5} a b c^{5/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )-\frac {4}{15} b^2 c^{5/2} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right )+\frac {2}{5} b^2 c^{5/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (2-\frac {2 \sqrt {c}}{\sqrt {c}-i x}\right )-\frac {1}{5} b^2 c^{5/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (1-\frac {c}{x^2}\right )-\frac {1}{5} b c^{5/2} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )+\frac {1}{5} b^2 c^{5/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (\frac {c}{x^2}+1\right )-\frac {1}{5} b^2 c^{5/2} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (\frac {c}{x^2}+1\right )-\frac {2}{5} b^2 c^{5/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (\frac {2 \sqrt {c}}{\sqrt {c}-i x}\right )+\frac {1}{5} b^2 c^{5/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (\frac {(1+i) \left (\sqrt {c}-x\right )}{\sqrt {c}-i x}\right )-\frac {2}{5} b^2 c^{5/2} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (\frac {2 \sqrt {c}}{x+\sqrt {c}}\right )+\frac {1}{5} b^2 c^{5/2} \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 (x+\sqrt {c}\right )}\right )+\frac {1}{5} b^2 c^{5/2} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (\frac {2 \sqrt {c} \left (x+\sqrt {-c}\right )}{\left (\sqrt {-c}+\sqrt {c}\right ) \left (x+\sqrt {c}\right )}\right )+\frac {1}{5} b^2 c^{5/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (\frac {(1-i) \left (x+\sqrt {c}\right )}{\sqrt {c}-i x}\right )+\frac {2}{5} b^2 c^{5/2} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (2-\frac {2 \sqrt {c}}{x+\sqrt {c}}\right )+\frac {1}{5} i b^2 c^{5/2} \text {Li}_2\left (1-\frac {2 \sqrt {c}}{\sqrt {c}-i x}\right )-\frac {1}{5} i b^2 c^{5/2} \text {Li}_2\left (\frac {2 \sqrt {c}}{\sqrt {c}-i x}-1\right )-\frac {1}{10} i b^2 c^{5/2} \text {Li}_2\left (1-\frac {(1+i) \left (\sqrt {c}-x\right )}{\sqrt {c}-i x}\right )+\frac {1}{5} b^2 c^{5/2} \text {Li}_2\left (-\frac {x}{\sqrt {c}}\right )-\frac {1}{5} i b^2 c^{5/2} \text {Li}_2\left (-\frac {i x}{\sqrt {c}}\right )+\frac {1}{5} i b^2 c^{5/2} \text {Li}_2\left (\frac {i x}{\sqrt {c}}\right )-\frac {1}{5} b^2 c^{5/2} \text {Li}_2\left (\frac {x}{\sqrt {c}}\right )+\frac {1}{5} b^2 c^{5/2} \text {Li}_2\left (1-\frac {2 \sqrt {c}}{x+\sqrt {c}}\right )-\frac {1}{5} b^2 c^{5/2} \text {Li}_2\left (\frac {2 \sqrt {c}}{x+\sqrt {c}}-1\right )-\frac {1}{10} b^2 c^{5/2} \text {Li}_2\left (1-\frac {2 \sqrt {c} \left (\sqrt {-c}-x\right )}{\left (\sqrt {-c}-\sqrt {c}\right ) \left (x+\sqrt {c}\right )}\right )-\frac {1}{10} b^2 c^{5/2} \text {Li}_2\left (1-\frac {2 \sqrt {c} \left (x+\sqrt {-c}\right )}{\left (\sqrt {-c}+\sqrt {c}\right ) \left (x+\sqrt {c}\right )}\right )-\frac {1}{10} i b^2 c^{5/2} \text {Li}_2\left (1-\frac {(1-i) \left (x+\sqrt {c}\right )}{\sqrt {c}-i x}\right ) \]
[Out]
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Rubi [A] time = 2.71, antiderivative size = 1214, normalized size of antiderivative = 1.00, number of steps used = 97, number of rules used = 33, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 2.063, Rules used = {6099, 2457, 2476, 2448, 263, 207, 2455, 193, 321, 2470, 12, 260, 6688, 5988, 5932, 2447, 302, 6742, 203, 30, 2557, 5992, 5912, 5920, 2402, 2315, 204, 4928, 4848, 2391, 4856, 4924, 4868} \[ \text {result too large to display} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 30
Rule 193
Rule 203
Rule 204
Rule 207
Rule 260
Rule 263
Rule 302
Rule 321
Rule 2315
Rule 2391
Rule 2402
Rule 2447
Rule 2448
Rule 2455
Rule 2457
Rule 2470
Rule 2476
Rule 2557
Rule 4848
Rule 4856
Rule 4868
Rule 4924
Rule 4928
Rule 5912
Rule 5920
Rule 5932
Rule 5988
Rule 5992
Rule 6099
Rule 6688
Rule 6742
Rubi steps
\begin {align*} \int x^4 \left (a+b \tanh ^{-1}\left (\frac {c}{x^2}\right )\right )^2 \, dx &=\int \left (\frac {1}{4} x^4 \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )^2-\frac {1}{2} b x^4 \left (-2 a+b \log \left (1-\frac {c}{x^2}\right )\right ) \log \left (1+\frac {c}{x^2}\right )+\frac {1}{4} b^2 x^4 \log ^2\left (1+\frac {c}{x^2}\right )\right ) \, dx\\ &=\frac {1}{4} \int x^4 \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )^2 \, dx-\frac {1}{2} b \int x^4 \left (-2 a+b \log \left (1-\frac {c}{x^2}\right )\right ) \log \left (1+\frac {c}{x^2}\right ) \, dx+\frac {1}{4} b^2 \int x^4 \log ^2\left (1+\frac {c}{x^2}\right ) \, dx\\ &=\frac {1}{20} x^5 \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )^2+\frac {1}{20} b^2 x^5 \log ^2\left (1+\frac {c}{x^2}\right )-\frac {1}{2} b \int \left (-2 a x^4 \log \left (1+\frac {c}{x^2}\right )+b x^4 \log \left (1-\frac {c}{x^2}\right ) \log \left (1+\frac {c}{x^2}\right )\right ) \, dx+\frac {1}{5} (b c) \int \frac {x^2 \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )}{1-\frac {c}{x^2}} \, dx+\frac {1}{5} \left (b^2 c\right ) \int \frac {x^2 \log \left (1+\frac {c}{x^2}\right )}{1+\frac {c}{x^2}} \, dx\\ &=\frac {1}{20} x^5 \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )^2+\frac {1}{20} b^2 x^5 \log ^2\left (1+\frac {c}{x^2}\right )+(a b) \int x^4 \log \left (1+\frac {c}{x^2}\right ) \, dx-\frac {1}{2} b^2 \int x^4 \log \left (1-\frac {c}{x^2}\right ) \log \left (1+\frac {c}{x^2}\right ) \, dx+\frac {1}{5} (b c) \int \left (c \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )+x^2 \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )+\frac {c^2 \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )}{-c+x^2}\right ) \, dx+\frac {1}{5} \left (b^2 c\right ) \int \left (-c \log \left (1+\frac {c}{x^2}\right )+x^2 \log \left (1+\frac {c}{x^2}\right )+\frac {c^2 \log \left (1+\frac {c}{x^2}\right )}{c+x^2}\right ) \, dx\\ &=\frac {1}{20} x^5 \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )^2+\frac {1}{5} a b x^5 \log \left (1+\frac {c}{x^2}\right )-\frac {1}{10} b^2 x^5 \log \left (1-\frac {c}{x^2}\right ) \log \left (1+\frac {c}{x^2}\right )+\frac {1}{20} b^2 x^5 \log ^2\left (1+\frac {c}{x^2}\right )+\frac {1}{2} b^2 \int \frac {2 c x^4 \log \left (1-\frac {c}{x^2}\right )}{5 \left (-c-x^2\right )} \, dx+\frac {1}{2} b^2 \int \frac {2 c x^4 \log \left (1+\frac {c}{x^2}\right )}{5 \left (-c+x^2\right )} \, dx+\frac {1}{5} (b c) \int x^2 \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right ) \, dx+\frac {1}{5} (2 a b c) \int \frac {x^2}{1+\frac {c}{x^2}} \, dx+\frac {1}{5} \left (b^2 c\right ) \int x^2 \log \left (1+\frac {c}{x^2}\right ) \, dx+\frac {1}{5} \left (b c^2\right ) \int \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right ) \, dx-\frac {1}{5} \left (b^2 c^2\right ) \int \log \left (1+\frac {c}{x^2}\right ) \, dx+\frac {1}{5} \left (b c^3\right ) \int \frac {2 a-b \log \left (1-\frac {c}{x^2}\right )}{-c+x^2} \, dx+\frac {1}{5} \left (b^2 c^3\right ) \int \frac {\log \left (1+\frac {c}{x^2}\right )}{c+x^2} \, dx\\ &=\frac {2}{5} a b c^2 x+\frac {1}{15} b c x^3 \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )-\frac {1}{5} b c^{5/2} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )+\frac {1}{20} x^5 \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )^2-\frac {1}{5} b^2 c^2 x \log \left (1+\frac {c}{x^2}\right )+\frac {1}{15} b^2 c x^3 \log \left (1+\frac {c}{x^2}\right )+\frac {1}{5} a b x^5 \log \left (1+\frac {c}{x^2}\right )+\frac {1}{5} b^2 c^{5/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (1+\frac {c}{x^2}\right )-\frac {1}{10} b^2 x^5 \log \left (1-\frac {c}{x^2}\right ) \log \left (1+\frac {c}{x^2}\right )+\frac {1}{20} b^2 x^5 \log ^2\left (1+\frac {c}{x^2}\right )+\frac {1}{5} (2 a b c) \int \frac {x^4}{c+x^2} \, dx+\frac {1}{5} \left (b^2 c\right ) \int \frac {x^4 \log \left (1-\frac {c}{x^2}\right )}{-c-x^2} \, dx+\frac {1}{5} \left (b^2 c\right ) \int \frac {x^4 \log \left (1+\frac {c}{x^2}\right )}{-c+x^2} \, dx+\frac {1}{15} \left (2 b^2 c^2\right ) \int \frac {1}{1-\frac {c}{x^2}} \, dx+\frac {1}{15} \left (2 b^2 c^2\right ) \int \frac {1}{1+\frac {c}{x^2}} \, dx-\frac {1}{5} \left (b^2 c^2\right ) \int \log \left (1-\frac {c}{x^2}\right ) \, dx-\frac {1}{5} \left (2 b^2 c^3\right ) \int \frac {1}{\left (1+\frac {c}{x^2}\right ) x^2} \, dx+\frac {1}{5} \left (2 b^2 c^4\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}{5} \left (2 b^2 c^4\right ) \int \frac {\tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right )}{\sqrt {c} \left (1-\frac {c}{x^2}\right ) x^3} \, dx\\ &=\frac {2}{5} a b c^2 x-\frac {1}{5} b^2 c^2 x \log \left (1-\frac {c}{x^2}\right )+\frac {1}{15} b c x^3 \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )-\frac {1}{5} b c^{5/2} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )+\frac {1}{20} x^5 \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )^2-\frac {1}{5} b^2 c^2 x \log \left (1+\frac {c}{x^2}\right )+\frac {1}{15} b^2 c x^3 \log \left (1+\frac {c}{x^2}\right )+\frac {1}{5} a b x^5 \log \left (1+\frac {c}{x^2}\right )+\frac {1}{5} b^2 c^{5/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (1+\frac {c}{x^2}\right )-\frac {1}{10} b^2 x^5 \log \left (1-\frac {c}{x^2}\right ) \log \left (1+\frac {c}{x^2}\right )+\frac {1}{20} b^2 x^5 \log ^2\left (1+\frac {c}{x^2}\right )+\frac {1}{5} (2 a b c) \int \left (-c+x^2+\frac {c^2}{c+x^2}\right ) \, dx+\frac {1}{5} \left (b^2 c\right ) \int \left (c \log \left (1-\frac {c}{x^2}\right )-x^2 \log \left (1-\frac {c}{x^2}\right )+\frac {c^2 \log \left (1-\frac {c}{x^2}\right )}{-c-x^2}\right ) \, dx+\frac {1}{5} \left (b^2 c\right ) \int \left (c \log \left (1+\frac {c}{x^2}\right )+x^2 \log \left (1+\frac {c}{x^2}\right )+\frac {c^2 \log \left (1+\frac {c}{x^2}\right )}{-c+x^2}\right ) \, dx+\frac {1}{15} \left (2 b^2 c^2\right ) \int \frac {x^2}{-c+x^2} \, dx+\frac {1}{15} \left (2 b^2 c^2\right ) \int \frac {x^2}{c+x^2} \, dx+\frac {1}{5} \left (2 b^2 c^3\right ) \int \frac {1}{\left (1-\frac {c}{x^2}\right ) x^2} \, dx-\frac {1}{5} \left (2 b^2 c^3\right ) \int \frac {1}{c+x^2} \, dx+\frac {1}{5} \left (2 b^2 c^{7/2}\right ) \int \frac {\tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )}{\left (1+\frac {c}{x^2}\right ) x^3} \, dx-\frac {1}{5} \left (2 b^2 c^{7/2}\right ) \int \frac {\tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right )}{\left (1-\frac {c}{x^2}\right ) x^3} \, dx\\ &=\frac {4}{15} b^2 c^2 x+\frac {2}{15} a b c x^3-\frac {2}{5} b^2 c^{5/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )-\frac {1}{5} b^2 c^2 x \log \left (1-\frac {c}{x^2}\right )+\frac {1}{15} b c x^3 \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )-\frac {1}{5} b c^{5/2} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )+\frac {1}{20} x^5 \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )^2-\frac {1}{5} b^2 c^2 x \log \left (1+\frac {c}{x^2}\right )+\frac {1}{15} b^2 c x^3 \log \left (1+\frac {c}{x^2}\right )+\frac {1}{5} a b x^5 \log \left (1+\frac {c}{x^2}\right )+\frac {1}{5} b^2 c^{5/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (1+\frac {c}{x^2}\right )-\frac {1}{10} b^2 x^5 \log \left (1-\frac {c}{x^2}\right ) \log \left (1+\frac {c}{x^2}\right )+\frac {1}{20} b^2 x^5 \log ^2\left (1+\frac {c}{x^2}\right )-\frac {1}{5} \left (b^2 c\right ) \int x^2 \log \left (1-\frac {c}{x^2}\right ) \, dx+\frac {1}{5} \left (b^2 c\right ) \int x^2 \log \left (1+\frac {c}{x^2}\right ) \, dx+\frac {1}{5} \left (b^2 c^2\right ) \int \log \left (1-\frac {c}{x^2}\right ) \, dx+\frac {1}{5} \left (b^2 c^2\right ) \int \log \left (1+\frac {c}{x^2}\right ) \, dx+\frac {1}{5} \left (2 a b c^3\right ) \int \frac {1}{c+x^2} \, dx+\frac {1}{15} \left (2 b^2 c^3\right ) \int \frac {1}{-c+x^2} \, dx-\frac {1}{15} \left (2 b^2 c^3\right ) \int \frac {1}{c+x^2} \, dx+\frac {1}{5} \left (b^2 c^3\right ) \int \frac {\log \left (1-\frac {c}{x^2}\right )}{-c-x^2} \, dx+\frac {1}{5} \left (b^2 c^3\right ) \int \frac {\log \left (1+\frac {c}{x^2}\right )}{-c+x^2} \, dx+\frac {1}{5} \left (2 b^2 c^3\right ) \int \frac {1}{-c+x^2} \, dx+\frac {1}{5} \left (2 b^2 c^{7/2}\right ) \int \frac {\tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )}{x \left (c+x^2\right )} \, dx-\frac {1}{5} \left (2 b^2 c^{7/2}\right ) \int \frac {\tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right )}{x \left (-c+x^2\right )} \, dx\\ &=\frac {4}{15} b^2 c^2 x+\frac {2}{15} a b c x^3+\frac {2}{5} a b c^{5/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )-\frac {8}{15} b^2 c^{5/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )-\frac {1}{5} i b^2 c^{5/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )^2-\frac {8}{15} b^2 c^{5/2} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right )+\frac {1}{5} b^2 c^{5/2} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right )^2-\frac {1}{15} b^2 c x^3 \log \left (1-\frac {c}{x^2}\right )-\frac {1}{5} b^2 c^{5/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (1-\frac {c}{x^2}\right )+\frac {1}{15} b c x^3 \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )-\frac {1}{5} b c^{5/2} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )+\frac {1}{20} x^5 \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )^2+\frac {2}{15} b^2 c x^3 \log \left (1+\frac {c}{x^2}\right )+\frac {1}{5} a b x^5 \log \left (1+\frac {c}{x^2}\right )+\frac {1}{5} b^2 c^{5/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (1+\frac {c}{x^2}\right )-\frac {1}{5} b^2 c^{5/2} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (1+\frac {c}{x^2}\right )-\frac {1}{10} b^2 x^5 \log \left (1-\frac {c}{x^2}\right ) \log \left (1+\frac {c}{x^2}\right )+\frac {1}{20} b^2 x^5 \log ^2\left (1+\frac {c}{x^2}\right )+\frac {1}{15} \left (2 b^2 c^2\right ) \int \frac {1}{1-\frac {c}{x^2}} \, dx+\frac {1}{15} \left (2 b^2 c^2\right ) \int \frac {1}{1+\frac {c}{x^2}} \, dx+\frac {1}{5} \left (2 i b^2 c^{5/2}\right ) \int \frac {\tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )}{x \left (i+\frac {x}{\sqrt {c}}\right )} \, dx+\frac {1}{5} \left (2 b^2 c^{5/2}\right ) \int \frac {\tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right )}{x \left (1+\frac {x}{\sqrt {c}}\right )} \, dx-\frac {1}{5} \left (2 b^2 c^3\right ) \int \frac {1}{\left (1-\frac {c}{x^2}\right ) x^2} \, dx+\frac {1}{5} \left (2 b^2 c^3\right ) \int \frac {1}{\left (1+\frac {c}{x^2}\right ) x^2} \, dx+\frac {1}{5} \left (2 b^2 c^4\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}{5} \left (2 b^2 c^4\right ) \int \frac {\tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right )}{\sqrt {c} \left (1+\frac {c}{x^2}\right ) x^3} \, dx\\ &=\frac {4}{15} b^2 c^2 x+\frac {2}{15} a b c x^3+\frac {2}{5} a b c^{5/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )-\frac {8}{15} b^2 c^{5/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )-\frac {1}{5} i b^2 c^{5/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )^2-\frac {8}{15} b^2 c^{5/2} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right )+\frac {1}{5} b^2 c^{5/2} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right )^2+\frac {2}{5} b^2 c^{5/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (2-\frac {2 \sqrt {c}}{\sqrt {c}-i x}\right )-\frac {1}{15} b^2 c x^3 \log \left (1-\frac {c}{x^2}\right )-\frac {1}{5} b^2 c^{5/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (1-\frac {c}{x^2}\right )+\frac {1}{15} b c x^3 \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )-\frac {1}{5} b c^{5/2} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )+\frac {1}{20} x^5 \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )^2+\frac {2}{15} b^2 c x^3 \log \left (1+\frac {c}{x^2}\right )+\frac {1}{5} a b x^5 \log \left (1+\frac {c}{x^2}\right )+\frac {1}{5} b^2 c^{5/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (1+\frac {c}{x^2}\right )-\frac {1}{5} b^2 c^{5/2} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (1+\frac {c}{x^2}\right )-\frac {1}{10} b^2 x^5 \log \left (1-\frac {c}{x^2}\right ) \log \left (1+\frac {c}{x^2}\right )+\frac {1}{20} b^2 x^5 \log ^2\left (1+\frac {c}{x^2}\right )+\frac {2}{5} b^2 c^{5/2} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (2-\frac {2 \sqrt {c}}{\sqrt {c}+x}\right )+\frac {1}{15} \left (2 b^2 c^2\right ) \int \frac {x^2}{-c+x^2} \, dx+\frac {1}{15} \left (2 b^2 c^2\right ) \int \frac {x^2}{c+x^2} \, dx-\frac {1}{5} \left (2 b^2 c^2\right ) \int \frac {\log \left (2-\frac {2}{1-\frac {i x}{\sqrt {c}}}\right )}{1+\frac {x^2}{c}} \, dx-\frac {1}{5} \left (2 b^2 c^2\right ) \int \frac {\log \left (2-\frac {2}{1+\frac {x}{\sqrt {c}}}\right )}{1-\frac {x^2}{c}} \, dx-\frac {1}{5} \left (2 b^2 c^3\right ) \int \frac {1}{-c+x^2} \, dx+\frac {1}{5} \left (2 b^2 c^3\right ) \int \frac {1}{c+x^2} \, dx+\frac {1}{5} \left (2 b^2 c^{7/2}\right ) \int \frac {\tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )}{\left (1-\frac {c}{x^2}\right ) x^3} \, dx-\frac {1}{5} \left (2 b^2 c^{7/2}\right ) \int \frac {\tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right )}{\left (1+\frac {c}{x^2}\right ) x^3} \, dx\\ &=\frac {8}{15} b^2 c^2 x+\frac {2}{15} a b c x^3+\frac {2}{5} a b c^{5/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )-\frac {2}{15} b^2 c^{5/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )-\frac {1}{5} i b^2 c^{5/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )^2-\frac {2}{15} b^2 c^{5/2} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right )+\frac {1}{5} b^2 c^{5/2} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right )^2+\frac {2}{5} b^2 c^{5/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (2-\frac {2 \sqrt {c}}{\sqrt {c}-i x}\right )-\frac {1}{15} b^2 c x^3 \log \left (1-\frac {c}{x^2}\right )-\frac {1}{5} b^2 c^{5/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (1-\frac {c}{x^2}\right )+\frac {1}{15} b c x^3 \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )-\frac {1}{5} b c^{5/2} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )+\frac {1}{20} x^5 \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )^2+\frac {2}{15} b^2 c x^3 \log \left (1+\frac {c}{x^2}\right )+\frac {1}{5} a b x^5 \log \left (1+\frac {c}{x^2}\right )+\frac {1}{5} b^2 c^{5/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (1+\frac {c}{x^2}\right )-\frac {1}{5} b^2 c^{5/2} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (1+\frac {c}{x^2}\right )-\frac {1}{10} b^2 x^5 \log \left (1-\frac {c}{x^2}\right ) \log \left (1+\frac {c}{x^2}\right )+\frac {1}{20} b^2 x^5 \log ^2\left (1+\frac {c}{x^2}\right )+\frac {2}{5} b^2 c^{5/2} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (2-\frac {2 \sqrt {c}}{\sqrt {c}+x}\right )-\frac {1}{5} i b^2 c^{5/2} \text {Li}_2\left (-1+\frac {2 \sqrt {c}}{\sqrt {c}-i x}\right )-\frac {1}{5} b^2 c^{5/2} \text {Li}_2\left (-1+\frac {2 \sqrt {c}}{\sqrt {c}+x}\right )+\frac {1}{15} \left (2 b^2 c^3\right ) \int \frac {1}{-c+x^2} \, dx-\frac {1}{15} \left (2 b^2 c^3\right ) \int \frac {1}{c+x^2} \, dx+\frac {1}{5} \left (2 b^2 c^{7/2}\right ) \int \frac {\tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )}{x \left (-c+x^2\right )} \, dx-\frac {1}{5} \left (2 b^2 c^{7/2}\right ) \int \frac {\tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right )}{x \left (c+x^2\right )} \, dx\\ &=\frac {8}{15} b^2 c^2 x+\frac {2}{15} a b c x^3+\frac {2}{5} a b c^{5/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )-\frac {4}{15} b^2 c^{5/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )-\frac {1}{5} i b^2 c^{5/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )^2-\frac {4}{15} b^2 c^{5/2} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right )+\frac {1}{5} b^2 c^{5/2} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right )^2+\frac {2}{5} b^2 c^{5/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (2-\frac {2 \sqrt {c}}{\sqrt {c}-i x}\right )-\frac {1}{15} b^2 c x^3 \log \left (1-\frac {c}{x^2}\right )-\frac {1}{5} b^2 c^{5/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (1-\frac {c}{x^2}\right )+\frac {1}{15} b c x^3 \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )-\frac {1}{5} b c^{5/2} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )+\frac {1}{20} x^5 \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )^2+\frac {2}{15} b^2 c x^3 \log \left (1+\frac {c}{x^2}\right )+\frac {1}{5} a b x^5 \log \left (1+\frac {c}{x^2}\right )+\frac {1}{5} b^2 c^{5/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (1+\frac {c}{x^2}\right )-\frac {1}{5} b^2 c^{5/2} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (1+\frac {c}{x^2}\right )-\frac {1}{10} b^2 x^5 \log \left (1-\frac {c}{x^2}\right ) \log \left (1+\frac {c}{x^2}\right )+\frac {1}{20} b^2 x^5 \log ^2\left (1+\frac {c}{x^2}\right )+\frac {2}{5} b^2 c^{5/2} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (2-\frac {2 \sqrt {c}}{\sqrt {c}+x}\right )-\frac {1}{5} i b^2 c^{5/2} \text {Li}_2\left (-1+\frac {2 \sqrt {c}}{\sqrt {c}-i x}\right )-\frac {1}{5} b^2 c^{5/2} \text {Li}_2\left (-1+\frac {2 \sqrt {c}}{\sqrt {c}+x}\right )+\frac {1}{5} \left (2 b^2 c^{7/2}\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-\frac {1}{5} \left (2 b^2 c^{7/2}\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\\ &=\frac {8}{15} b^2 c^2 x+\frac {2}{15} a b c x^3+\frac {2}{5} a b c^{5/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )-\frac {4}{15} b^2 c^{5/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )-\frac {1}{5} i b^2 c^{5/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )^2-\frac {4}{15} b^2 c^{5/2} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right )+\frac {1}{5} b^2 c^{5/2} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right )^2+\frac {2}{5} b^2 c^{5/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (2-\frac {2 \sqrt {c}}{\sqrt {c}-i x}\right )-\frac {1}{15} b^2 c x^3 \log \left (1-\frac {c}{x^2}\right )-\frac {1}{5} b^2 c^{5/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (1-\frac {c}{x^2}\right )+\frac {1}{15} b c x^3 \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )-\frac {1}{5} b c^{5/2} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )+\frac {1}{20} x^5 \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )^2+\frac {2}{15} b^2 c x^3 \log \left (1+\frac {c}{x^2}\right )+\frac {1}{5} a b x^5 \log \left (1+\frac {c}{x^2}\right )+\frac {1}{5} b^2 c^{5/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (1+\frac {c}{x^2}\right )-\frac {1}{5} b^2 c^{5/2} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (1+\frac {c}{x^2}\right )-\frac {1}{10} b^2 x^5 \log \left (1-\frac {c}{x^2}\right ) \log \left (1+\frac {c}{x^2}\right )+\frac {1}{20} b^2 x^5 \log ^2\left (1+\frac {c}{x^2}\right )+\frac {2}{5} b^2 c^{5/2} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (2-\frac {2 \sqrt {c}}{\sqrt {c}+x}\right )-\frac {1}{5} i b^2 c^{5/2} \text {Li}_2\left (-1+\frac {2 \sqrt {c}}{\sqrt {c}-i x}\right )-\frac {1}{5} b^2 c^{5/2} \text {Li}_2\left (-1+\frac {2 \sqrt {c}}{\sqrt {c}+x}\right )-\frac {1}{5} \left (2 b^2 c^{5/2}\right ) \int \frac {\tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )}{x} \, dx-\frac {1}{5} \left (2 b^2 c^{5/2}\right ) \int \frac {x \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )}{c-x^2} \, dx-\frac {1}{5} \left (2 b^2 c^{5/2}\right ) \int \frac {\tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right )}{x} \, dx+\frac {1}{5} \left (2 b^2 c^{5/2}\right ) \int \frac {x \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right )}{c+x^2} \, dx\\ &=\frac {8}{15} b^2 c^2 x+\frac {2}{15} a b c x^3+\frac {2}{5} a b c^{5/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )-\frac {4}{15} b^2 c^{5/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )-\frac {1}{5} i b^2 c^{5/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )^2-\frac {4}{15} b^2 c^{5/2} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right )+\frac {1}{5} b^2 c^{5/2} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right )^2+\frac {2}{5} b^2 c^{5/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (2-\frac {2 \sqrt {c}}{\sqrt {c}-i x}\right )-\frac {1}{15} b^2 c x^3 \log \left (1-\frac {c}{x^2}\right )-\frac {1}{5} b^2 c^{5/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (1-\frac {c}{x^2}\right )+\frac {1}{15} b c x^3 \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )-\frac {1}{5} b c^{5/2} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )+\frac {1}{20} x^5 \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )^2+\frac {2}{15} b^2 c x^3 \log \left (1+\frac {c}{x^2}\right )+\frac {1}{5} a b x^5 \log \left (1+\frac {c}{x^2}\right )+\frac {1}{5} b^2 c^{5/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (1+\frac {c}{x^2}\right )-\frac {1}{5} b^2 c^{5/2} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (1+\frac {c}{x^2}\right )-\frac {1}{10} b^2 x^5 \log \left (1-\frac {c}{x^2}\right ) \log \left (1+\frac {c}{x^2}\right )+\frac {1}{20} b^2 x^5 \log ^2\left (1+\frac {c}{x^2}\right )+\frac {2}{5} b^2 c^{5/2} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (2-\frac {2 \sqrt {c}}{\sqrt {c}+x}\right )-\frac {1}{5} i b^2 c^{5/2} \text {Li}_2\left (-1+\frac {2 \sqrt {c}}{\sqrt {c}-i x}\right )+\frac {1}{5} b^2 c^{5/2} \text {Li}_2\left (-\frac {x}{\sqrt {c}}\right )-\frac {1}{5} b^2 c^{5/2} \text {Li}_2\left (\frac {x}{\sqrt {c}}\right )-\frac {1}{5} b^2 c^{5/2} \text {Li}_2\left (-1+\frac {2 \sqrt {c}}{\sqrt {c}+x}\right )-\frac {1}{5} \left (i b^2 c^{5/2}\right ) \int \frac {\log \left (1-\frac {i x}{\sqrt {c}}\right )}{x} \, dx+\frac {1}{5} \left (i b^2 c^{5/2}\right ) \int \frac {\log \left (1+\frac {i x}{\sqrt {c}}\right )}{x} \, dx-\frac {1}{5} \left (2 b^2 c^{5/2}\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+\frac {1}{5} \left (2 b^2 c^{5/2}\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\\ &=\frac {8}{15} b^2 c^2 x+\frac {2}{15} a b c x^3+\frac {2}{5} a b c^{5/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )-\frac {4}{15} b^2 c^{5/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )-\frac {1}{5} i b^2 c^{5/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )^2-\frac {4}{15} b^2 c^{5/2} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right )+\frac {1}{5} b^2 c^{5/2} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right )^2+\frac {2}{5} b^2 c^{5/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (2-\frac {2 \sqrt {c}}{\sqrt {c}-i x}\right )-\frac {1}{15} b^2 c x^3 \log \left (1-\frac {c}{x^2}\right )-\frac {1}{5} b^2 c^{5/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (1-\frac {c}{x^2}\right )+\frac {1}{15} b c x^3 \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )-\frac {1}{5} b c^{5/2} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )+\frac {1}{20} x^5 \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )^2+\frac {2}{15} b^2 c x^3 \log \left (1+\frac {c}{x^2}\right )+\frac {1}{5} a b x^5 \log \left (1+\frac {c}{x^2}\right )+\frac {1}{5} b^2 c^{5/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (1+\frac {c}{x^2}\right )-\frac {1}{5} b^2 c^{5/2} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (1+\frac {c}{x^2}\right )-\frac {1}{10} b^2 x^5 \log \left (1-\frac {c}{x^2}\right ) \log \left (1+\frac {c}{x^2}\right )+\frac {1}{20} b^2 x^5 \log ^2\left (1+\frac {c}{x^2}\right )+\frac {2}{5} b^2 c^{5/2} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (2-\frac {2 \sqrt {c}}{\sqrt {c}+x}\right )-\frac {1}{5} i b^2 c^{5/2} \text {Li}_2\left (-1+\frac {2 \sqrt {c}}{\sqrt {c}-i x}\right )+\frac {1}{5} b^2 c^{5/2} \text {Li}_2\left (-\frac {x}{\sqrt {c}}\right )-\frac {1}{5} i b^2 c^{5/2} \text {Li}_2\left (-\frac {i x}{\sqrt {c}}\right )+\frac {1}{5} i b^2 c^{5/2} \text {Li}_2\left (\frac {i x}{\sqrt {c}}\right )-\frac {1}{5} b^2 c^{5/2} \text {Li}_2\left (\frac {x}{\sqrt {c}}\right )-\frac {1}{5} b^2 c^{5/2} \text {Li}_2\left (-1+\frac {2 \sqrt {c}}{\sqrt {c}+x}\right )-\frac {1}{5} \left (b^2 c^{5/2}\right ) \int \frac {\tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )}{\sqrt {c}-x} \, dx+\frac {1}{5} \left (b^2 c^{5/2}\right ) \int \frac {\tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )}{\sqrt {c}+x} \, dx-\frac {1}{5} \left (b^2 c^{5/2}\right ) \int \frac {\tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right )}{\sqrt {-c}-x} \, dx+\frac {1}{5} \left (b^2 c^{5/2}\right ) \int \frac {\tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right )}{\sqrt {-c}+x} \, dx\\ &=\frac {8}{15} b^2 c^2 x+\frac {2}{15} a b c x^3+\frac {2}{5} a b c^{5/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )-\frac {4}{15} b^2 c^{5/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )-\frac {1}{5} i b^2 c^{5/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )^2-\frac {4}{15} b^2 c^{5/2} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right )+\frac {1}{5} b^2 c^{5/2} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right )^2+\frac {2}{5} b^2 c^{5/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (2-\frac {2 \sqrt {c}}{\sqrt {c}-i x}\right )-\frac {1}{15} b^2 c x^3 \log \left (1-\frac {c}{x^2}\right )-\frac {1}{5} b^2 c^{5/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (1-\frac {c}{x^2}\right )+\frac {1}{15} b c x^3 \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )-\frac {1}{5} b c^{5/2} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )+\frac {1}{20} x^5 \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )^2+\frac {2}{15} b^2 c x^3 \log \left (1+\frac {c}{x^2}\right )+\frac {1}{5} a b x^5 \log \left (1+\frac {c}{x^2}\right )+\frac {1}{5} b^2 c^{5/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (1+\frac {c}{x^2}\right )-\frac {1}{5} b^2 c^{5/2} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (1+\frac {c}{x^2}\right )-\frac {1}{10} b^2 x^5 \log \left (1-\frac {c}{x^2}\right ) \log \left (1+\frac {c}{x^2}\right )+\frac {1}{20} b^2 x^5 \log ^2\left (1+\frac {c}{x^2}\right )-\frac {2}{5} b^2 c^{5/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (\frac {2 \sqrt {c}}{\sqrt {c}-i x}\right )+\frac {1}{5} b^2 c^{5/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (\frac {(1+i) \left (\sqrt {c}-x\right )}{\sqrt {c}-i x}\right )-\frac {2}{5} b^2 c^{5/2} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (\frac {2 \sqrt {c}}{\sqrt {c}+x}\right )+\frac {1}{5} b^2 c^{5/2} \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 )+\frac {1}{5} b^2 c^{5/2} \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 )+\frac {1}{5} b^2 c^{5/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (\frac {(1-i) \left (\sqrt {c}+x\right )}{\sqrt {c}-i x}\right )+\frac {2}{5} b^2 c^{5/2} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (2-\frac {2 \sqrt {c}}{\sqrt {c}+x}\right )-\frac {1}{5} i b^2 c^{5/2} \text {Li}_2\left (-1+\frac {2 \sqrt {c}}{\sqrt {c}-i x}\right )+\frac {1}{5} b^2 c^{5/2} \text {Li}_2\left (-\frac {x}{\sqrt {c}}\right )-\frac {1}{5} i b^2 c^{5/2} \text {Li}_2\left (-\frac {i x}{\sqrt {c}}\right )+\frac {1}{5} i b^2 c^{5/2} \text {Li}_2\left (\frac {i x}{\sqrt {c}}\right )-\frac {1}{5} b^2 c^{5/2} \text {Li}_2\left (\frac {x}{\sqrt {c}}\right )-\frac {1}{5} b^2 c^{5/2} \text {Li}_2\left (-1+\frac {2 \sqrt {c}}{\sqrt {c}+x}\right )+2 \left (\frac {1}{5} \left (b^2 c^2\right ) \int \frac {\log \left (\frac {2}{1-\frac {i x}{\sqrt {c}}}\right )}{1+\frac {x^2}{c}} \, dx\right )-\frac {1}{5} \left (b^2 c^2\right ) \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-\frac {1}{5} \left (b^2 c^2\right ) \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+2 \left (\frac {1}{5} \left (b^2 c^2\right ) \int \frac {\log \left (\frac {2}{1+\frac {x}{\sqrt {c}}}\right )}{1-\frac {x^2}{c}} \, dx\right )-\frac {1}{5} \left (b^2 c^2\right ) \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-\frac {1}{5} \left (b^2 c^2\right ) \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\\ &=\frac {8}{15} b^2 c^2 x+\frac {2}{15} a b c x^3+\frac {2}{5} a b c^{5/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )-\frac {4}{15} b^2 c^{5/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )-\frac {1}{5} i b^2 c^{5/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )^2-\frac {4}{15} b^2 c^{5/2} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right )+\frac {1}{5} b^2 c^{5/2} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right )^2+\frac {2}{5} b^2 c^{5/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (2-\frac {2 \sqrt {c}}{\sqrt {c}-i x}\right )-\frac {1}{15} b^2 c x^3 \log \left (1-\frac {c}{x^2}\right )-\frac {1}{5} b^2 c^{5/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (1-\frac {c}{x^2}\right )+\frac {1}{15} b c x^3 \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )-\frac {1}{5} b c^{5/2} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )+\frac {1}{20} x^5 \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )^2+\frac {2}{15} b^2 c x^3 \log \left (1+\frac {c}{x^2}\right )+\frac {1}{5} a b x^5 \log \left (1+\frac {c}{x^2}\right )+\frac {1}{5} b^2 c^{5/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (1+\frac {c}{x^2}\right )-\frac {1}{5} b^2 c^{5/2} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (1+\frac {c}{x^2}\right )-\frac {1}{10} b^2 x^5 \log \left (1-\frac {c}{x^2}\right ) \log \left (1+\frac {c}{x^2}\right )+\frac {1}{20} b^2 x^5 \log ^2\left (1+\frac {c}{x^2}\right )-\frac {2}{5} b^2 c^{5/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (\frac {2 \sqrt {c}}{\sqrt {c}-i x}\right )+\frac {1}{5} b^2 c^{5/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (\frac {(1+i) \left (\sqrt {c}-x\right )}{\sqrt {c}-i x}\right )-\frac {2}{5} b^2 c^{5/2} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (\frac {2 \sqrt {c}}{\sqrt {c}+x}\right )+\frac {1}{5} b^2 c^{5/2} \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 )+\frac {1}{5} b^2 c^{5/2} \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 )+\frac {1}{5} b^2 c^{5/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (\frac {(1-i) \left (\sqrt {c}+x\right )}{\sqrt {c}-i x}\right )+\frac {2}{5} b^2 c^{5/2} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (2-\frac {2 \sqrt {c}}{\sqrt {c}+x}\right )-\frac {1}{5} i b^2 c^{5/2} \text {Li}_2\left (-1+\frac {2 \sqrt {c}}{\sqrt {c}-i x}\right )-\frac {1}{10} i b^2 c^{5/2} \text {Li}_2\left (1-\frac {(1+i) \left (\sqrt {c}-x\right )}{\sqrt {c}-i x}\right )+\frac {1}{5} b^2 c^{5/2} \text {Li}_2\left (-\frac {x}{\sqrt {c}}\right )-\frac {1}{5} i b^2 c^{5/2} \text {Li}_2\left (-\frac {i x}{\sqrt {c}}\right )+\frac {1}{5} i b^2 c^{5/2} \text {Li}_2\left (\frac {i x}{\sqrt {c}}\right )-\frac {1}{5} b^2 c^{5/2} \text {Li}_2\left (\frac {x}{\sqrt {c}}\right )-\frac {1}{5} b^2 c^{5/2} \text {Li}_2\left (-1+\frac {2 \sqrt {c}}{\sqrt {c}+x}\right )-\frac {1}{10} b^2 c^{5/2} \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 )-\frac {1}{10} b^2 c^{5/2} \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 )-\frac {1}{10} i b^2 c^{5/2} \text {Li}_2\left (1-\frac {(1-i) \left (\sqrt {c}+x\right )}{\sqrt {c}-i x}\right )+2 \left (\frac {1}{5} \left (i b^2 c^{5/2}\right ) \operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1-\frac {i x}{\sqrt {c}}}\right )\right )+2 \left (\frac {1}{5} \left (b^2 c^{5/2}\right ) \operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+\frac {x}{\sqrt {c}}}\right )\right )\\ &=\frac {8}{15} b^2 c^2 x+\frac {2}{15} a b c x^3+\frac {2}{5} a b c^{5/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )-\frac {4}{15} b^2 c^{5/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )-\frac {1}{5} i b^2 c^{5/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )^2-\frac {4}{15} b^2 c^{5/2} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right )+\frac {1}{5} b^2 c^{5/2} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right )^2+\frac {2}{5} b^2 c^{5/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (2-\frac {2 \sqrt {c}}{\sqrt {c}-i x}\right )-\frac {1}{15} b^2 c x^3 \log \left (1-\frac {c}{x^2}\right )-\frac {1}{5} b^2 c^{5/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (1-\frac {c}{x^2}\right )+\frac {1}{15} b c x^3 \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )-\frac {1}{5} b c^{5/2} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )+\frac {1}{20} x^5 \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )^2+\frac {2}{15} b^2 c x^3 \log \left (1+\frac {c}{x^2}\right )+\frac {1}{5} a b x^5 \log \left (1+\frac {c}{x^2}\right )+\frac {1}{5} b^2 c^{5/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (1+\frac {c}{x^2}\right )-\frac {1}{5} b^2 c^{5/2} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (1+\frac {c}{x^2}\right )-\frac {1}{10} b^2 x^5 \log \left (1-\frac {c}{x^2}\right ) \log \left (1+\frac {c}{x^2}\right )+\frac {1}{20} b^2 x^5 \log ^2\left (1+\frac {c}{x^2}\right )-\frac {2}{5} b^2 c^{5/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (\frac {2 \sqrt {c}}{\sqrt {c}-i x}\right )+\frac {1}{5} b^2 c^{5/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (\frac {(1+i) \left (\sqrt {c}-x\right )}{\sqrt {c}-i x}\right )-\frac {2}{5} b^2 c^{5/2} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (\frac {2 \sqrt {c}}{\sqrt {c}+x}\right )+\frac {1}{5} b^2 c^{5/2} \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 )+\frac {1}{5} b^2 c^{5/2} \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 )+\frac {1}{5} b^2 c^{5/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (\frac {(1-i) \left (\sqrt {c}+x\right )}{\sqrt {c}-i x}\right )+\frac {2}{5} b^2 c^{5/2} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (2-\frac {2 \sqrt {c}}{\sqrt {c}+x}\right )+\frac {1}{5} i b^2 c^{5/2} \text {Li}_2\left (1-\frac {2 \sqrt {c}}{\sqrt {c}-i x}\right )-\frac {1}{5} i b^2 c^{5/2} \text {Li}_2\left (-1+\frac {2 \sqrt {c}}{\sqrt {c}-i x}\right )-\frac {1}{10} i b^2 c^{5/2} \text {Li}_2\left (1-\frac {(1+i) \left (\sqrt {c}-x\right )}{\sqrt {c}-i x}\right )+\frac {1}{5} b^2 c^{5/2} \text {Li}_2\left (-\frac {x}{\sqrt {c}}\right )-\frac {1}{5} i b^2 c^{5/2} \text {Li}_2\left (-\frac {i x}{\sqrt {c}}\right )+\frac {1}{5} i b^2 c^{5/2} \text {Li}_2\left (\frac {i x}{\sqrt {c}}\right )-\frac {1}{5} b^2 c^{5/2} \text {Li}_2\left (\frac {x}{\sqrt {c}}\right )+\frac {1}{5} b^2 c^{5/2} \text {Li}_2\left (1-\frac {2 \sqrt {c}}{\sqrt {c}+x}\right )-\frac {1}{5} b^2 c^{5/2} \text {Li}_2\left (-1+\frac {2 \sqrt {c}}{\sqrt {c}+x}\right )-\frac {1}{10} b^2 c^{5/2} \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 )-\frac {1}{10} b^2 c^{5/2} \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 )-\frac {1}{10} i b^2 c^{5/2} \text {Li}_2\left (1-\frac {(1-i) \left (\sqrt {c}+x\right )}{\sqrt {c}-i x}\right )\\ \end {align*}
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Mathematica [F] time = 5.90, size = 0, normalized size = 0.00 \[ \int x^4 \left (a+b \tanh ^{-1}\left (\frac {c}{x^2}\right )\right )^2 \, dx \]
Verification is Not applicable to the result.
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fricas [F] time = 0.57, size = 0, normalized size = 0.00 \[ {\rm integral}\left (b^{2} x^{4} \operatorname {artanh}\left (\frac {c}{x^{2}}\right )^{2} + 2 \, a b x^{4} \operatorname {artanh}\left (\frac {c}{x^{2}}\right ) + a^{2} x^{4}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b \operatorname {artanh}\left (\frac {c}{x^{2}}\right ) + a\right )}^{2} x^{4}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.01, size = 0, normalized size = 0.00 \[ \int x^{4} \left (a +b \arctanh \left (\frac {c}{x^{2}}\right )\right )^{2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {1}{5} \, a^{2} x^{5} + \frac {1}{15} \, {\left (6 \, x^{5} \operatorname {artanh}\left (\frac {c}{x^{2}}\right ) + {\left (4 \, x^{3} + 6 \, c^{\frac {3}{2}} \arctan \left (\frac {x}{\sqrt {c}}\right ) + 3 \, c^{\frac {3}{2}} \log \left (\frac {x - \sqrt {c}}{x + \sqrt {c}}\right )\right )} c\right )} a b + \frac {1}{20} \, {\left (x^{5} \log \left (x^{2} - c\right )^{2} - 5 \, \int -\frac {5 \, {\left (x^{6} - c x^{4}\right )} \log \left (x^{2} + c\right )^{2} - 2 \, {\left (2 \, x^{6} + 5 \, {\left (x^{6} - c x^{4}\right )} \log \left (x^{2} + c\right )\right )} \log \left (x^{2} - c\right )}{5 \, {\left (x^{2} - c\right )}}\,{d x}\right )} b^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int x^4\,{\left (a+b\,\mathrm {atanh}\left (\frac {c}{x^2}\right )\right )}^2 \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{4} \left (a + b \operatorname {atanh}{\left (\frac {c}{x^{2}} \right )}\right )^{2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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