Optimal. Leaf size=1444 \[ -\frac {1}{5} \sqrt [4]{-1} b^2 \tan ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right )^2 c^{5/2}+\frac {1}{5} (-1)^{3/4} b^2 \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right )^2 c^{5/2}-\frac {4}{15} (-1)^{3/4} b^2 \tan ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) c^{5/2}+\frac {4}{15} (-1)^{3/4} b^2 \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) c^{5/2}+\frac {2}{5} \sqrt [4]{-1} a b \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) c^{5/2}+\frac {2}{5} (-1)^{3/4} b^2 \tan ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (\frac {2}{1-\sqrt [4]{-1} \sqrt {c} x}\right ) c^{5/2}-\frac {2}{5} (-1)^{3/4} b^2 \tan ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (\frac {2}{\sqrt [4]{-1} \sqrt {c} x+1}\right ) c^{5/2}+\frac {1}{5} (-1)^{3/4} b^2 \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 ) c^{5/2}-\frac {2}{5} (-1)^{3/4} b^2 \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (\frac {2}{1-(-1)^{3/4} \sqrt {c} x}\right ) c^{5/2}+\frac {2}{5} (-1)^{3/4} b^2 \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (\frac {2}{(-1)^{3/4} \sqrt {c} x+1}\right ) c^{5/2}-\frac {1}{5} (-1)^{3/4} b^2 \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 ) c^{5/2}-\frac {1}{5} (-1)^{3/4} b^2 \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 ) c^{5/2}+\frac {1}{5} (-1)^{3/4} b^2 \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 ) c^{5/2}+\frac {1}{5} (-1)^{3/4} b^2 \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (1-i c x^2\right ) c^{5/2}+\frac {1}{5} \sqrt [4]{-1} b \tan ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \left (2 a+i b \log \left (1-i c x^2\right )\right ) c^{5/2}-\frac {1}{5} (-1)^{3/4} b^2 \tan ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (i c x^2+1\right ) c^{5/2}-\frac {1}{5} (-1)^{3/4} b^2 \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (i c x^2+1\right ) c^{5/2}-\frac {1}{5} \sqrt [4]{-1} b^2 \text {Li}_2\left (1-\frac {2}{1-\sqrt [4]{-1} \sqrt {c} x}\right ) c^{5/2}-\frac {1}{5} \sqrt [4]{-1} b^2 \text {Li}_2\left (1-\frac {2}{\sqrt [4]{-1} \sqrt {c} x+1}\right ) c^{5/2}+\frac {1}{10} \sqrt [4]{-1} b^2 \text {Li}_2\left (1-\frac {\sqrt {2} \left (\sqrt {c} x+\sqrt [4]{-1}\right )}{\sqrt [4]{-1} \sqrt {c} x+1}\right ) c^{5/2}-\frac {1}{5} (-1)^{3/4} b^2 \text {Li}_2\left (1-\frac {2}{1-(-1)^{3/4} \sqrt {c} x}\right ) c^{5/2}-\frac {1}{5} (-1)^{3/4} b^2 \text {Li}_2\left (1-\frac {2}{(-1)^{3/4} \sqrt {c} x+1}\right ) c^{5/2}+\frac {1}{10} (-1)^{3/4} b^2 \text {Li}_2\left (\frac {\sqrt {2} \left (\sqrt {c} x+(-1)^{3/4}\right )}{(-1)^{3/4} \sqrt {c} x+1}+1\right ) c^{5/2}+\frac {1}{10} (-1)^{3/4} b^2 \text {Li}_2\left (1-\frac {(1+i) \left (\sqrt [4]{-1} \sqrt {c} x+1\right )}{(-1)^{3/4} \sqrt {c} x+1}\right ) c^{5/2}+\frac {1}{10} \sqrt [4]{-1} b^2 \text {Li}_2\left (1-\frac {(1-i) \left ((-1)^{3/4} \sqrt {c} x+1\right )}{\sqrt [4]{-1} \sqrt {c} x+1}\right ) c^{5/2}-\frac {b^2 \log \left (1-i c x^2\right ) c^2}{5 x}-\frac {i b \left (2 a+i b \log \left (1-i c x^2\right )\right ) c^2}{5 x}-\frac {8 b^2 c^2}{15 x}+\frac {2 i a b c^2}{5 x}-\frac {i b^2 \log \left (1-i c x^2\right ) c}{15 x^3}-\frac {b \left (2 a+i b \log \left (1-i c x^2\right )\right ) c}{15 x^3}+\frac {2 i b^2 \log \left (i c x^2+1\right ) c}{15 x^3}-\frac {2 a b c}{15 x^3}-\frac {\left (2 a+i b \log \left (1-i c x^2\right )\right )^2}{20 x^5}+\frac {b^2 \log ^2\left (i c x^2+1\right )}{20 x^5}-\frac {b^2 \log \left (1-i c x^2\right ) \log \left (i c x^2+1\right )}{10 x^5}+\frac {i a b \log \left (i c x^2+1\right )}{5 x^5} \]
[Out]
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Rubi [A] time = 2.43, antiderivative size = 1444, normalized size of antiderivative = 1.00, number of steps used = 77, number of rules used = 25, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.562, Rules used = {5035, 2457, 2476, 2455, 325, 203, 205, 2470, 12, 4920, 4854, 2402, 2315, 6742, 206, 30, 2557, 4928, 4856, 2447, 208, 5992, 5920, 5984, 5918} \[ \text {result too large to display} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 30
Rule 203
Rule 205
Rule 206
Rule 208
Rule 325
Rule 2315
Rule 2402
Rule 2447
Rule 2455
Rule 2457
Rule 2470
Rule 2476
Rule 2557
Rule 4854
Rule 4856
Rule 4920
Rule 4928
Rule 5035
Rule 5918
Rule 5920
Rule 5984
Rule 5992
Rule 6742
Rubi steps
\begin {align*} \int \frac {\left (a+b \tan ^{-1}\left (c x^2\right )\right )^2}{x^6} \, dx &=\int \left (\frac {\left (2 a+i b \log \left (1-i c x^2\right )\right )^2}{4 x^6}+\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^6}-\frac {b^2 \log ^2\left (1+i c x^2\right )}{4 x^6}\right ) \, dx\\ &=\frac {1}{4} \int \frac {\left (2 a+i b \log \left (1-i c x^2\right )\right )^2}{x^6} \, 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^6} \, dx-\frac {1}{4} b^2 \int \frac {\log ^2\left (1+i c x^2\right )}{x^6} \, dx\\ &=-\frac {\left (2 a+i b \log \left (1-i c x^2\right )\right )^2}{20 x^5}+\frac {b^2 \log ^2\left (1+i c x^2\right )}{20 x^5}+\frac {1}{2} b \int \left (-\frac {2 i a \log \left (1+i c x^2\right )}{x^6}+\frac {b \log \left (1-i c x^2\right ) \log \left (1+i c x^2\right )}{x^6}\right ) \, dx+\frac {1}{5} (b c) \int \frac {2 a+i b \log \left (1-i c x^2\right )}{x^4 \left (1-i c x^2\right )} \, dx-\frac {1}{5} \left (i b^2 c\right ) \int \frac {\log \left (1+i c x^2\right )}{x^4 \left (1+i c x^2\right )} \, dx\\ &=-\frac {\left (2 a+i b \log \left (1-i c x^2\right )\right )^2}{20 x^5}+\frac {b^2 \log ^2\left (1+i c x^2\right )}{20 x^5}-(i a b) \int \frac {\log \left (1+i c x^2\right )}{x^6} \, 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^6} \, dx+\frac {1}{5} (b c) \int \left (\frac {2 a+i b \log \left (1-i c x^2\right )}{x^4}+\frac {i c \left (2 a+i b \log \left (1-i c x^2\right )\right )}{x^2}-\frac {i c^2 \left (2 a+i b \log \left (1-i c x^2\right )\right )}{i+c x^2}\right ) \, dx-\frac {1}{5} \left (i b^2 c\right ) \int \left (\frac {\log \left (1+i c x^2\right )}{x^4}-\frac {i c \log \left (1+i c x^2\right )}{x^2}+\frac {i c^2 \log \left (1+i c x^2\right )}{-i+c x^2}\right ) \, dx\\ &=-\frac {\left (2 a+i b \log \left (1-i c x^2\right )\right )^2}{20 x^5}+\frac {i a b \log \left (1+i c x^2\right )}{5 x^5}-\frac {b^2 \log \left (1-i c x^2\right ) \log \left (1+i c x^2\right )}{10 x^5}+\frac {b^2 \log ^2\left (1+i c x^2\right )}{20 x^5}-\frac {1}{2} b^2 \int \frac {2 c \log \left (1-i c x^2\right )}{5 x^4 \left (i-c x^2\right )} \, dx-\frac {1}{2} b^2 \int \frac {2 c \log \left (1+i c x^2\right )}{5 x^4 \left (-i-c x^2\right )} \, dx+\frac {1}{5} (b c) \int \frac {2 a+i b \log \left (1-i c x^2\right )}{x^4} \, dx+\frac {1}{5} (2 a b c) \int \frac {1}{x^4 \left (1+i c x^2\right )} \, dx-\frac {1}{5} \left (i b^2 c\right ) \int \frac {\log \left (1+i c x^2\right )}{x^4} \, dx+\frac {1}{5} \left (i b c^2\right ) \int \frac {2 a+i b \log \left (1-i c x^2\right )}{x^2} \, dx-\frac {1}{5} \left (b^2 c^2\right ) \int \frac {\log \left (1+i c x^2\right )}{x^2} \, dx-\frac {1}{5} \left (i b c^3\right ) \int \frac {2 a+i b \log \left (1-i c x^2\right )}{i+c x^2} \, dx+\frac {1}{5} \left (b^2 c^3\right ) \int \frac {\log \left (1+i c x^2\right )}{-i+c x^2} \, dx\\ &=-\frac {2 a b c}{15 x^3}-\frac {b c \left (2 a+i b \log \left (1-i c x^2\right )\right )}{15 x^3}-\frac {i b c^2 \left (2 a+i b \log \left (1-i c x^2\right )\right )}{5 x}+\frac {1}{5} \sqrt [4]{-1} b c^{5/2} \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}{20 x^5}+\frac {i a b \log \left (1+i c x^2\right )}{5 x^5}+\frac {i b^2 c \log \left (1+i c x^2\right )}{15 x^3}+\frac {b^2 c^2 \log \left (1+i c x^2\right )}{5 x}-\frac {1}{5} (-1)^{3/4} b^2 c^{5/2} \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 )}{10 x^5}+\frac {b^2 \log ^2\left (1+i c x^2\right )}{20 x^5}-\frac {1}{5} \left (b^2 c\right ) \int \frac {\log \left (1-i c x^2\right )}{x^4 \left (i-c x^2\right )} \, dx-\frac {1}{5} \left (b^2 c\right ) \int \frac {\log \left (1+i c x^2\right )}{x^4 \left (-i-c x^2\right )} \, dx-\frac {1}{5} \left (2 i a b c^2\right ) \int \frac {1}{x^2 \left (1+i c x^2\right )} \, dx+\frac {1}{15} \left (2 b^2 c^2\right ) \int \frac {1}{x^2 \left (1-i c x^2\right )} \, dx+\frac {1}{15} \left (2 b^2 c^2\right ) \int \frac {1}{x^2 \left (1+i c x^2\right )} \, dx+\frac {1}{5} \left (2 i b^2 c^3\right ) \int \frac {1}{1-i c x^2} \, dx-\frac {1}{5} \left (2 i b^2 c^3\right ) \int \frac {1}{1+i c x^2} \, dx+\frac {1}{5} \left (2 i b^2 c^4\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+\frac {1}{5} \left (2 i b^2 c^4\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\\ &=-\frac {2 a b c}{15 x^3}+\frac {2 i a b c^2}{5 x}-\frac {4 b^2 c^2}{15 x}-\frac {2}{5} (-1)^{3/4} b^2 c^{5/2} \tan ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right )+\frac {2}{5} (-1)^{3/4} b^2 c^{5/2} \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right )-\frac {b c \left (2 a+i b \log \left (1-i c x^2\right )\right )}{15 x^3}-\frac {i b c^2 \left (2 a+i b \log \left (1-i c x^2\right )\right )}{5 x}+\frac {1}{5} \sqrt [4]{-1} b c^{5/2} \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}{20 x^5}+\frac {i a b \log \left (1+i c x^2\right )}{5 x^5}+\frac {i b^2 c \log \left (1+i c x^2\right )}{15 x^3}+\frac {b^2 c^2 \log \left (1+i c x^2\right )}{5 x}-\frac {1}{5} (-1)^{3/4} b^2 c^{5/2} \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 )}{10 x^5}+\frac {b^2 \log ^2\left (1+i c x^2\right )}{20 x^5}-\frac {1}{5} \left (b^2 c\right ) \int \left (-\frac {i \log \left (1-i c x^2\right )}{x^4}-\frac {c \log \left (1-i c x^2\right )}{x^2}+\frac {c^2 \log \left (1-i c x^2\right )}{-i+c x^2}\right ) \, dx-\frac {1}{5} \left (b^2 c\right ) \int \left (\frac {i \log \left (1+i c x^2\right )}{x^4}-\frac {c \log \left (1+i c x^2\right )}{x^2}+\frac {c^2 \log \left (1+i c x^2\right )}{i+c x^2}\right ) \, dx-\frac {1}{5} \left (2 a b c^3\right ) \int \frac {1}{1+i c x^2} \, dx+\frac {1}{15} \left (2 i b^2 c^3\right ) \int \frac {1}{1-i c x^2} \, dx-\frac {1}{15} \left (2 i b^2 c^3\right ) \int \frac {1}{1+i c x^2} \, dx-\frac {1}{5} \left (2 \sqrt [4]{-1} b^2 c^{7/2}\right ) \int \frac {x \tan ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right )}{1-i c x^2} \, dx-\frac {1}{5} \left (2 \sqrt [4]{-1} b^2 c^{7/2}\right ) \int \frac {x \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right )}{1+i c x^2} \, dx\\ &=-\frac {2 a b c}{15 x^3}+\frac {2 i a b c^2}{5 x}-\frac {4 b^2 c^2}{15 x}-\frac {8}{15} (-1)^{3/4} b^2 c^{5/2} \tan ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right )-\frac {1}{5} \sqrt [4]{-1} b^2 c^{5/2} \tan ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right )^2+\frac {2}{5} \sqrt [4]{-1} a b c^{5/2} \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right )+\frac {8}{15} (-1)^{3/4} b^2 c^{5/2} \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right )+\frac {1}{5} (-1)^{3/4} b^2 c^{5/2} \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right )^2-\frac {b c \left (2 a+i b \log \left (1-i c x^2\right )\right )}{15 x^3}-\frac {i b c^2 \left (2 a+i b \log \left (1-i c x^2\right )\right )}{5 x}+\frac {1}{5} \sqrt [4]{-1} b c^{5/2} \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}{20 x^5}+\frac {i a b \log \left (1+i c x^2\right )}{5 x^5}+\frac {i b^2 c \log \left (1+i c x^2\right )}{15 x^3}+\frac {b^2 c^2 \log \left (1+i c x^2\right )}{5 x}-\frac {1}{5} (-1)^{3/4} b^2 c^{5/2} \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 )}{10 x^5}+\frac {b^2 \log ^2\left (1+i c x^2\right )}{20 x^5}+\frac {1}{5} \left (i b^2 c\right ) \int \frac {\log \left (1-i c x^2\right )}{x^4} \, dx-\frac {1}{5} \left (i b^2 c\right ) \int \frac {\log \left (1+i c x^2\right )}{x^4} \, dx+\frac {1}{5} \left (b^2 c^2\right ) \int \frac {\log \left (1-i c x^2\right )}{x^2} \, dx+\frac {1}{5} \left (b^2 c^2\right ) \int \frac {\log \left (1+i c x^2\right )}{x^2} \, dx-\frac {1}{5} \left (2 i b^2 c^3\right ) \int \frac {\tan ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right )}{i-(-1)^{3/4} \sqrt {c} x} \, dx+\frac {1}{5} \left (2 i b^2 c^3\right ) \int \frac {\tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right )}{1-(-1)^{3/4} \sqrt {c} x} \, dx-\frac {1}{5} \left (b^2 c^3\right ) \int \frac {\log \left (1-i c x^2\right )}{-i+c x^2} \, dx-\frac {1}{5} \left (b^2 c^3\right ) \int \frac {\log \left (1+i c x^2\right )}{i+c x^2} \, dx\\ &=-\frac {2 a b c}{15 x^3}+\frac {2 i a b c^2}{5 x}-\frac {4 b^2 c^2}{15 x}-\frac {8}{15} (-1)^{3/4} b^2 c^{5/2} \tan ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right )-\frac {1}{5} \sqrt [4]{-1} b^2 c^{5/2} \tan ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right )^2+\frac {2}{5} \sqrt [4]{-1} a b c^{5/2} \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right )+\frac {8}{15} (-1)^{3/4} b^2 c^{5/2} \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right )+\frac {1}{5} (-1)^{3/4} b^2 c^{5/2} \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right )^2+\frac {2}{5} (-1)^{3/4} b^2 c^{5/2} \tan ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (\frac {2}{1-\sqrt [4]{-1} \sqrt {c} x}\right )-\frac {2}{5} (-1)^{3/4} b^2 c^{5/2} \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (\frac {2}{1-(-1)^{3/4} \sqrt {c} x}\right )-\frac {i b^2 c \log \left (1-i c x^2\right )}{15 x^3}-\frac {b^2 c^2 \log \left (1-i c x^2\right )}{5 x}+\frac {1}{5} (-1)^{3/4} b^2 c^{5/2} \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (1-i c x^2\right )-\frac {b c \left (2 a+i b \log \left (1-i c x^2\right )\right )}{15 x^3}-\frac {i b c^2 \left (2 a+i b \log \left (1-i c x^2\right )\right )}{5 x}+\frac {1}{5} \sqrt [4]{-1} b c^{5/2} \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}{20 x^5}+\frac {i a b \log \left (1+i c x^2\right )}{5 x^5}+\frac {2 i b^2 c \log \left (1+i c x^2\right )}{15 x^3}-\frac {1}{5} (-1)^{3/4} b^2 c^{5/2} \tan ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (1+i c x^2\right )-\frac {1}{5} (-1)^{3/4} b^2 c^{5/2} \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 )}{10 x^5}+\frac {b^2 \log ^2\left (1+i c x^2\right )}{20 x^5}+\frac {1}{15} \left (2 b^2 c^2\right ) \int \frac {1}{x^2 \left (1-i c x^2\right )} \, dx+\frac {1}{15} \left (2 b^2 c^2\right ) \int \frac {1}{x^2 \left (1+i c x^2\right )} \, dx-\frac {1}{5} \left (2 i b^2 c^3\right ) \int \frac {1}{1-i c x^2} \, dx+\frac {1}{5} \left (2 i b^2 c^3\right ) \int \frac {1}{1+i c x^2} \, dx+\frac {1}{5} \left (2 i b^2 c^3\right ) \int \frac {\log \left (\frac {2}{1-\sqrt [4]{-1} \sqrt {c} x}\right )}{1-i c x^2} \, dx-\frac {1}{5} \left (2 i b^2 c^3\right ) \int \frac {\log \left (\frac {2}{1-(-1)^{3/4} \sqrt {c} x}\right )}{1+i c x^2} \, dx+\frac {1}{5} \left (2 i b^2 c^4\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+\frac {1}{5} \left (2 i b^2 c^4\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\\ &=-\frac {2 a b c}{15 x^3}+\frac {2 i a b c^2}{5 x}-\frac {8 b^2 c^2}{15 x}-\frac {2}{15} (-1)^{3/4} b^2 c^{5/2} \tan ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right )-\frac {1}{5} \sqrt [4]{-1} b^2 c^{5/2} \tan ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right )^2+\frac {2}{5} \sqrt [4]{-1} a b c^{5/2} \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right )+\frac {2}{15} (-1)^{3/4} b^2 c^{5/2} \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right )+\frac {1}{5} (-1)^{3/4} b^2 c^{5/2} \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right )^2+\frac {2}{5} (-1)^{3/4} b^2 c^{5/2} \tan ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (\frac {2}{1-\sqrt [4]{-1} \sqrt {c} x}\right )-\frac {2}{5} (-1)^{3/4} b^2 c^{5/2} \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (\frac {2}{1-(-1)^{3/4} \sqrt {c} x}\right )-\frac {i b^2 c \log \left (1-i c x^2\right )}{15 x^3}-\frac {b^2 c^2 \log \left (1-i c x^2\right )}{5 x}+\frac {1}{5} (-1)^{3/4} b^2 c^{5/2} \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (1-i c x^2\right )-\frac {b c \left (2 a+i b \log \left (1-i c x^2\right )\right )}{15 x^3}-\frac {i b c^2 \left (2 a+i b \log \left (1-i c x^2\right )\right )}{5 x}+\frac {1}{5} \sqrt [4]{-1} b c^{5/2} \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}{20 x^5}+\frac {i a b \log \left (1+i c x^2\right )}{5 x^5}+\frac {2 i b^2 c \log \left (1+i c x^2\right )}{15 x^3}-\frac {1}{5} (-1)^{3/4} b^2 c^{5/2} \tan ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (1+i c x^2\right )-\frac {1}{5} (-1)^{3/4} b^2 c^{5/2} \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 )}{10 x^5}+\frac {b^2 \log ^2\left (1+i c x^2\right )}{20 x^5}-\frac {1}{5} \left (2 \sqrt [4]{-1} b^2 c^{5/2}\right ) \operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1-\sqrt [4]{-1} \sqrt {c} x}\right )-\frac {1}{5} \left (2 (-1)^{3/4} b^2 c^{5/2}\right ) \operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1-(-1)^{3/4} \sqrt {c} x}\right )+\frac {1}{15} \left (2 i b^2 c^3\right ) \int \frac {1}{1-i c x^2} \, dx-\frac {1}{15} \left (2 i b^2 c^3\right ) \int \frac {1}{1+i c x^2} \, dx-\frac {1}{5} \left (2 \sqrt [4]{-1} b^2 c^{7/2}\right ) \int \frac {x \tan ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right )}{1+i c x^2} \, dx-\frac {1}{5} \left (2 \sqrt [4]{-1} b^2 c^{7/2}\right ) \int \frac {x \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right )}{1-i c x^2} \, dx\\ &=-\frac {2 a b c}{15 x^3}+\frac {2 i a b c^2}{5 x}-\frac {8 b^2 c^2}{15 x}-\frac {4}{15} (-1)^{3/4} b^2 c^{5/2} \tan ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right )-\frac {1}{5} \sqrt [4]{-1} b^2 c^{5/2} \tan ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right )^2+\frac {2}{5} \sqrt [4]{-1} a b c^{5/2} \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right )+\frac {4}{15} (-1)^{3/4} b^2 c^{5/2} \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right )+\frac {1}{5} (-1)^{3/4} b^2 c^{5/2} \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right )^2+\frac {2}{5} (-1)^{3/4} b^2 c^{5/2} \tan ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (\frac {2}{1-\sqrt [4]{-1} \sqrt {c} x}\right )-\frac {2}{5} (-1)^{3/4} b^2 c^{5/2} \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (\frac {2}{1-(-1)^{3/4} \sqrt {c} x}\right )-\frac {i b^2 c \log \left (1-i c x^2\right )}{15 x^3}-\frac {b^2 c^2 \log \left (1-i c x^2\right )}{5 x}+\frac {1}{5} (-1)^{3/4} b^2 c^{5/2} \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (1-i c x^2\right )-\frac {b c \left (2 a+i b \log \left (1-i c x^2\right )\right )}{15 x^3}-\frac {i b c^2 \left (2 a+i b \log \left (1-i c x^2\right )\right )}{5 x}+\frac {1}{5} \sqrt [4]{-1} b c^{5/2} \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}{20 x^5}+\frac {i a b \log \left (1+i c x^2\right )}{5 x^5}+\frac {2 i b^2 c \log \left (1+i c x^2\right )}{15 x^3}-\frac {1}{5} (-1)^{3/4} b^2 c^{5/2} \tan ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (1+i c x^2\right )-\frac {1}{5} (-1)^{3/4} b^2 c^{5/2} \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 )}{10 x^5}+\frac {b^2 \log ^2\left (1+i c x^2\right )}{20 x^5}-\frac {1}{5} \sqrt [4]{-1} b^2 c^{5/2} \text {Li}_2\left (1-\frac {2}{1-\sqrt [4]{-1} \sqrt {c} x}\right )-\frac {1}{5} (-1)^{3/4} b^2 c^{5/2} \text {Li}_2\left (1-\frac {2}{1-(-1)^{3/4} \sqrt {c} x}\right )-\frac {1}{5} \left (2 \sqrt [4]{-1} b^2 c^{7/2}\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-\frac {1}{5} \left (2 \sqrt [4]{-1} b^2 c^{7/2}\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\\ &=-\frac {2 a b c}{15 x^3}+\frac {2 i a b c^2}{5 x}-\frac {8 b^2 c^2}{15 x}-\frac {4}{15} (-1)^{3/4} b^2 c^{5/2} \tan ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right )-\frac {1}{5} \sqrt [4]{-1} b^2 c^{5/2} \tan ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right )^2+\frac {2}{5} \sqrt [4]{-1} a b c^{5/2} \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right )+\frac {4}{15} (-1)^{3/4} b^2 c^{5/2} \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right )+\frac {1}{5} (-1)^{3/4} b^2 c^{5/2} \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right )^2+\frac {2}{5} (-1)^{3/4} b^2 c^{5/2} \tan ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (\frac {2}{1-\sqrt [4]{-1} \sqrt {c} x}\right )-\frac {2}{5} (-1)^{3/4} b^2 c^{5/2} \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (\frac {2}{1-(-1)^{3/4} \sqrt {c} x}\right )-\frac {i b^2 c \log \left (1-i c x^2\right )}{15 x^3}-\frac {b^2 c^2 \log \left (1-i c x^2\right )}{5 x}+\frac {1}{5} (-1)^{3/4} b^2 c^{5/2} \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (1-i c x^2\right )-\frac {b c \left (2 a+i b \log \left (1-i c x^2\right )\right )}{15 x^3}-\frac {i b c^2 \left (2 a+i b \log \left (1-i c x^2\right )\right )}{5 x}+\frac {1}{5} \sqrt [4]{-1} b c^{5/2} \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}{20 x^5}+\frac {i a b \log \left (1+i c x^2\right )}{5 x^5}+\frac {2 i b^2 c \log \left (1+i c x^2\right )}{15 x^3}-\frac {1}{5} (-1)^{3/4} b^2 c^{5/2} \tan ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (1+i c x^2\right )-\frac {1}{5} (-1)^{3/4} b^2 c^{5/2} \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 )}{10 x^5}+\frac {b^2 \log ^2\left (1+i c x^2\right )}{20 x^5}-\frac {1}{5} \sqrt [4]{-1} b^2 c^{5/2} \text {Li}_2\left (1-\frac {2}{1-\sqrt [4]{-1} \sqrt {c} x}\right )-\frac {1}{5} (-1)^{3/4} b^2 c^{5/2} \text {Li}_2\left (1-\frac {2}{1-(-1)^{3/4} \sqrt {c} x}\right )-\frac {1}{5} \left ((-1)^{3/4} b^2 c^3\right ) \int \frac {\tan ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right )}{\sqrt [4]{-1}-\sqrt {c} x} \, dx+\frac {1}{5} \left ((-1)^{3/4} b^2 c^3\right ) \int \frac {\tan ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right )}{\sqrt [4]{-1}+\sqrt {c} x} \, dx+\frac {1}{5} \left ((-1)^{3/4} b^2 c^3\right ) \int \frac {\tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right )}{-(-1)^{3/4}-\sqrt {c} x} \, dx-\frac {1}{5} \left ((-1)^{3/4} b^2 c^3\right ) \int \frac {\tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right )}{-(-1)^{3/4}+\sqrt {c} x} \, dx\\ &=-\frac {2 a b c}{15 x^3}+\frac {2 i a b c^2}{5 x}-\frac {8 b^2 c^2}{15 x}-\frac {4}{15} (-1)^{3/4} b^2 c^{5/2} \tan ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right )-\frac {1}{5} \sqrt [4]{-1} b^2 c^{5/2} \tan ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right )^2+\frac {2}{5} \sqrt [4]{-1} a b c^{5/2} \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right )+\frac {4}{15} (-1)^{3/4} b^2 c^{5/2} \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right )+\frac {1}{5} (-1)^{3/4} b^2 c^{5/2} \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right )^2+\frac {2}{5} (-1)^{3/4} b^2 c^{5/2} \tan ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (\frac {2}{1-\sqrt [4]{-1} \sqrt {c} x}\right )-\frac {2}{5} (-1)^{3/4} b^2 c^{5/2} \tan ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (\frac {2}{1+\sqrt [4]{-1} \sqrt {c} x}\right )+\frac {1}{5} (-1)^{3/4} b^2 c^{5/2} \tan ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (\frac {\sqrt {2} \left (\sqrt [4]{-1}+\sqrt {c} x\right )}{1+\sqrt [4]{-1} \sqrt {c} x}\right )-\frac {2}{5} (-1)^{3/4} b^2 c^{5/2} \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (\frac {2}{1-(-1)^{3/4} \sqrt {c} x}\right )+\frac {2}{5} (-1)^{3/4} b^2 c^{5/2} \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (\frac {2}{1+(-1)^{3/4} \sqrt {c} x}\right )-\frac {1}{5} (-1)^{3/4} b^2 c^{5/2} \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (-\frac {\sqrt {2} \left ((-1)^{3/4}+\sqrt {c} x\right )}{1+(-1)^{3/4} \sqrt {c} x}\right )-\frac {1}{5} (-1)^{3/4} b^2 c^{5/2} \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (\frac {\sqrt [4]{-1} \sqrt {2} \left (1+\sqrt [4]{-1} \sqrt {c} x\right )}{1+(-1)^{3/4} \sqrt {c} x}\right )+\frac {1}{5} (-1)^{3/4} b^2 c^{5/2} \tan ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (\frac {(1-i) \left (1+(-1)^{3/4} \sqrt {c} x\right )}{1+\sqrt [4]{-1} \sqrt {c} x}\right )-\frac {i b^2 c \log \left (1-i c x^2\right )}{15 x^3}-\frac {b^2 c^2 \log \left (1-i c x^2\right )}{5 x}+\frac {1}{5} (-1)^{3/4} b^2 c^{5/2} \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (1-i c x^2\right )-\frac {b c \left (2 a+i b \log \left (1-i c x^2\right )\right )}{15 x^3}-\frac {i b c^2 \left (2 a+i b \log \left (1-i c x^2\right )\right )}{5 x}+\frac {1}{5} \sqrt [4]{-1} b c^{5/2} \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}{20 x^5}+\frac {i a b \log \left (1+i c x^2\right )}{5 x^5}+\frac {2 i b^2 c \log \left (1+i c x^2\right )}{15 x^3}-\frac {1}{5} (-1)^{3/4} b^2 c^{5/2} \tan ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (1+i c x^2\right )-\frac {1}{5} (-1)^{3/4} b^2 c^{5/2} \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 )}{10 x^5}+\frac {b^2 \log ^2\left (1+i c x^2\right )}{20 x^5}-\frac {1}{5} \sqrt [4]{-1} b^2 c^{5/2} \text {Li}_2\left (1-\frac {2}{1-\sqrt [4]{-1} \sqrt {c} x}\right )-\frac {1}{5} (-1)^{3/4} b^2 c^{5/2} \text {Li}_2\left (1-\frac {2}{1-(-1)^{3/4} \sqrt {c} x}\right )-2 \left (\frac {1}{5} \left (i b^2 c^3\right ) \int \frac {\log \left (\frac {2}{1+\sqrt [4]{-1} \sqrt {c} x}\right )}{1-i c x^2} \, dx\right )+\frac {1}{5} \left (i b^2 c^3\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+\frac {1}{5} \left (i b^2 c^3\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 (\frac {1}{5} \left (i b^2 c^3\right ) \int \frac {\log \left (\frac {2}{1+(-1)^{3/4} \sqrt {c} x}\right )}{1+i c x^2} \, dx\right )-\frac {1}{5} \left (i b^2 c^3\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-\frac {1}{5} \left (i b^2 c^3\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\\ &=-\frac {2 a b c}{15 x^3}+\frac {2 i a b c^2}{5 x}-\frac {8 b^2 c^2}{15 x}-\frac {4}{15} (-1)^{3/4} b^2 c^{5/2} \tan ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right )-\frac {1}{5} \sqrt [4]{-1} b^2 c^{5/2} \tan ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right )^2+\frac {2}{5} \sqrt [4]{-1} a b c^{5/2} \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right )+\frac {4}{15} (-1)^{3/4} b^2 c^{5/2} \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right )+\frac {1}{5} (-1)^{3/4} b^2 c^{5/2} \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right )^2+\frac {2}{5} (-1)^{3/4} b^2 c^{5/2} \tan ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (\frac {2}{1-\sqrt [4]{-1} \sqrt {c} x}\right )-\frac {2}{5} (-1)^{3/4} b^2 c^{5/2} \tan ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (\frac {2}{1+\sqrt [4]{-1} \sqrt {c} x}\right )+\frac {1}{5} (-1)^{3/4} b^2 c^{5/2} \tan ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (\frac {\sqrt {2} \left (\sqrt [4]{-1}+\sqrt {c} x\right )}{1+\sqrt [4]{-1} \sqrt {c} x}\right )-\frac {2}{5} (-1)^{3/4} b^2 c^{5/2} \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (\frac {2}{1-(-1)^{3/4} \sqrt {c} x}\right )+\frac {2}{5} (-1)^{3/4} b^2 c^{5/2} \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (\frac {2}{1+(-1)^{3/4} \sqrt {c} x}\right )-\frac {1}{5} (-1)^{3/4} b^2 c^{5/2} \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (-\frac {\sqrt {2} \left ((-1)^{3/4}+\sqrt {c} x\right )}{1+(-1)^{3/4} \sqrt {c} x}\right )-\frac {1}{5} (-1)^{3/4} b^2 c^{5/2} \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (\frac {\sqrt [4]{-1} \sqrt {2} \left (1+\sqrt [4]{-1} \sqrt {c} x\right )}{1+(-1)^{3/4} \sqrt {c} x}\right )+\frac {1}{5} (-1)^{3/4} b^2 c^{5/2} \tan ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (\frac {(1-i) \left (1+(-1)^{3/4} \sqrt {c} x\right )}{1+\sqrt [4]{-1} \sqrt {c} x}\right )-\frac {i b^2 c \log \left (1-i c x^2\right )}{15 x^3}-\frac {b^2 c^2 \log \left (1-i c x^2\right )}{5 x}+\frac {1}{5} (-1)^{3/4} b^2 c^{5/2} \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (1-i c x^2\right )-\frac {b c \left (2 a+i b \log \left (1-i c x^2\right )\right )}{15 x^3}-\frac {i b c^2 \left (2 a+i b \log \left (1-i c x^2\right )\right )}{5 x}+\frac {1}{5} \sqrt [4]{-1} b c^{5/2} \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}{20 x^5}+\frac {i a b \log \left (1+i c x^2\right )}{5 x^5}+\frac {2 i b^2 c \log \left (1+i c x^2\right )}{15 x^3}-\frac {1}{5} (-1)^{3/4} b^2 c^{5/2} \tan ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (1+i c x^2\right )-\frac {1}{5} (-1)^{3/4} b^2 c^{5/2} \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 )}{10 x^5}+\frac {b^2 \log ^2\left (1+i c x^2\right )}{20 x^5}-\frac {1}{5} \sqrt [4]{-1} b^2 c^{5/2} \text {Li}_2\left (1-\frac {2}{1-\sqrt [4]{-1} \sqrt {c} x}\right )+\frac {1}{10} \sqrt [4]{-1} b^2 c^{5/2} \text {Li}_2\left (1-\frac {\sqrt {2} \left (\sqrt [4]{-1}+\sqrt {c} x\right )}{1+\sqrt [4]{-1} \sqrt {c} x}\right )-\frac {1}{5} (-1)^{3/4} b^2 c^{5/2} \text {Li}_2\left (1-\frac {2}{1-(-1)^{3/4} \sqrt {c} x}\right )+\frac {1}{10} (-1)^{3/4} b^2 c^{5/2} \text {Li}_2\left (1+\frac {\sqrt {2} \left ((-1)^{3/4}+\sqrt {c} x\right )}{1+(-1)^{3/4} \sqrt {c} x}\right )+\frac {1}{10} (-1)^{3/4} b^2 c^{5/2} \text {Li}_2\left (1-\frac {\sqrt [4]{-1} \sqrt {2} \left (1+\sqrt [4]{-1} \sqrt {c} x\right )}{1+(-1)^{3/4} \sqrt {c} x}\right )+\frac {1}{10} \sqrt [4]{-1} b^2 c^{5/2} \text {Li}_2\left (1-\frac {(1-i) \left (1+(-1)^{3/4} \sqrt {c} x\right )}{1+\sqrt [4]{-1} \sqrt {c} x}\right )-2 \left (\frac {1}{5} \left (\sqrt [4]{-1} b^2 c^{5/2}\right ) \operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+\sqrt [4]{-1} \sqrt {c} x}\right )\right )-2 \left (\frac {1}{5} \left ((-1)^{3/4} b^2 c^{5/2}\right ) \operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+(-1)^{3/4} \sqrt {c} x}\right )\right )\\ &=-\frac {2 a b c}{15 x^3}+\frac {2 i a b c^2}{5 x}-\frac {8 b^2 c^2}{15 x}-\frac {4}{15} (-1)^{3/4} b^2 c^{5/2} \tan ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right )-\frac {1}{5} \sqrt [4]{-1} b^2 c^{5/2} \tan ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right )^2+\frac {2}{5} \sqrt [4]{-1} a b c^{5/2} \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right )+\frac {4}{15} (-1)^{3/4} b^2 c^{5/2} \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right )+\frac {1}{5} (-1)^{3/4} b^2 c^{5/2} \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right )^2+\frac {2}{5} (-1)^{3/4} b^2 c^{5/2} \tan ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (\frac {2}{1-\sqrt [4]{-1} \sqrt {c} x}\right )-\frac {2}{5} (-1)^{3/4} b^2 c^{5/2} \tan ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (\frac {2}{1+\sqrt [4]{-1} \sqrt {c} x}\right )+\frac {1}{5} (-1)^{3/4} b^2 c^{5/2} \tan ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (\frac {\sqrt {2} \left (\sqrt [4]{-1}+\sqrt {c} x\right )}{1+\sqrt [4]{-1} \sqrt {c} x}\right )-\frac {2}{5} (-1)^{3/4} b^2 c^{5/2} \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (\frac {2}{1-(-1)^{3/4} \sqrt {c} x}\right )+\frac {2}{5} (-1)^{3/4} b^2 c^{5/2} \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (\frac {2}{1+(-1)^{3/4} \sqrt {c} x}\right )-\frac {1}{5} (-1)^{3/4} b^2 c^{5/2} \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (-\frac {\sqrt {2} \left ((-1)^{3/4}+\sqrt {c} x\right )}{1+(-1)^{3/4} \sqrt {c} x}\right )-\frac {1}{5} (-1)^{3/4} b^2 c^{5/2} \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (\frac {\sqrt [4]{-1} \sqrt {2} \left (1+\sqrt [4]{-1} \sqrt {c} x\right )}{1+(-1)^{3/4} \sqrt {c} x}\right )+\frac {1}{5} (-1)^{3/4} b^2 c^{5/2} \tan ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (\frac {(1-i) \left (1+(-1)^{3/4} \sqrt {c} x\right )}{1+\sqrt [4]{-1} \sqrt {c} x}\right )-\frac {i b^2 c \log \left (1-i c x^2\right )}{15 x^3}-\frac {b^2 c^2 \log \left (1-i c x^2\right )}{5 x}+\frac {1}{5} (-1)^{3/4} b^2 c^{5/2} \tanh ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (1-i c x^2\right )-\frac {b c \left (2 a+i b \log \left (1-i c x^2\right )\right )}{15 x^3}-\frac {i b c^2 \left (2 a+i b \log \left (1-i c x^2\right )\right )}{5 x}+\frac {1}{5} \sqrt [4]{-1} b c^{5/2} \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}{20 x^5}+\frac {i a b \log \left (1+i c x^2\right )}{5 x^5}+\frac {2 i b^2 c \log \left (1+i c x^2\right )}{15 x^3}-\frac {1}{5} (-1)^{3/4} b^2 c^{5/2} \tan ^{-1}\left ((-1)^{3/4} \sqrt {c} x\right ) \log \left (1+i c x^2\right )-\frac {1}{5} (-1)^{3/4} b^2 c^{5/2} \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 )}{10 x^5}+\frac {b^2 \log ^2\left (1+i c x^2\right )}{20 x^5}-\frac {1}{5} \sqrt [4]{-1} b^2 c^{5/2} \text {Li}_2\left (1-\frac {2}{1-\sqrt [4]{-1} \sqrt {c} x}\right )-\frac {1}{5} \sqrt [4]{-1} b^2 c^{5/2} \text {Li}_2\left (1-\frac {2}{1+\sqrt [4]{-1} \sqrt {c} x}\right )+\frac {1}{10} \sqrt [4]{-1} b^2 c^{5/2} \text {Li}_2\left (1-\frac {\sqrt {2} \left (\sqrt [4]{-1}+\sqrt {c} x\right )}{1+\sqrt [4]{-1} \sqrt {c} x}\right )-\frac {1}{5} (-1)^{3/4} b^2 c^{5/2} \text {Li}_2\left (1-\frac {2}{1-(-1)^{3/4} \sqrt {c} x}\right )-\frac {1}{5} (-1)^{3/4} b^2 c^{5/2} \text {Li}_2\left (1-\frac {2}{1+(-1)^{3/4} \sqrt {c} x}\right )+\frac {1}{10} (-1)^{3/4} b^2 c^{5/2} \text {Li}_2\left (1+\frac {\sqrt {2} \left ((-1)^{3/4}+\sqrt {c} x\right )}{1+(-1)^{3/4} \sqrt {c} x}\right )+\frac {1}{10} (-1)^{3/4} b^2 c^{5/2} \text {Li}_2\left (1-\frac {\sqrt [4]{-1} \sqrt {2} \left (1+\sqrt [4]{-1} \sqrt {c} x\right )}{1+(-1)^{3/4} \sqrt {c} x}\right )+\frac {1}{10} \sqrt [4]{-1} b^2 c^{5/2} \text {Li}_2\left (1-\frac {(1-i) \left (1+(-1)^{3/4} \sqrt {c} x\right )}{1+\sqrt [4]{-1} \sqrt {c} x}\right )\\ \end {align*}
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Mathematica [F] time = 2.89, size = 0, normalized size = 0.00 \[ \int \frac {\left (a+b \tan ^{-1}\left (c x^2\right )\right )^2}{x^6} \, dx \]
Verification is Not applicable to the result.
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fricas [F] time = 0.43, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b^{2} \arctan \left (c x^{2}\right )^{2} + 2 \, a b \arctan \left (c x^{2}\right ) + a^{2}}{x^{6}}, x\right ) \]
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 \[ \int \frac {{\left (b \arctan \left (c x^{2}\right ) + a\right )}^{2}}{x^{6}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.35, size = 0, normalized size = 0.00 \[ \int \frac {\left (a +b \arctan \left (c \,x^{2}\right )\right )^{2}}{x^{6}}\, dx \]
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 \[ -\frac {1}{30} \, {\left ({\left (6 \, \sqrt {2} c^{\frac {3}{2}} \arctan \left (\frac {\sqrt {2} {\left (2 \, c x + \sqrt {2} \sqrt {c}\right )}}{2 \, \sqrt {c}}\right ) + 6 \, \sqrt {2} c^{\frac {3}{2}} \arctan \left (\frac {\sqrt {2} {\left (2 \, c x - \sqrt {2} \sqrt {c}\right )}}{2 \, \sqrt {c}}\right ) + 3 \, \sqrt {2} c^{\frac {3}{2}} \log \left (c x^{2} + \sqrt {2} \sqrt {c} x + 1\right ) - 3 \, \sqrt {2} c^{\frac {3}{2}} \log \left (c x^{2} - \sqrt {2} \sqrt {c} x + 1\right ) + \frac {8}{x^{3}}\right )} c + \frac {12 \, \arctan \left (c x^{2}\right )}{x^{5}}\right )} a b + \frac {\frac {1}{4} \, {\left (4 \, x^{5} \int -\frac {24 \, c^{2} x^{4} \log \left (c^{2} x^{4} + 1\right ) - 112 \, c x^{2} \arctan \left (c x^{2}\right ) - 180 \, {\left (c^{2} x^{4} + 1\right )} \arctan \left (c x^{2}\right )^{2} - 15 \, {\left (c^{2} x^{4} + 1\right )} \log \left (c^{2} x^{4} + 1\right )^{2}}{4 \, {\left (c^{2} x^{10} + x^{6}\right )}}\,{d x} - 28 \, \arctan \left (c x^{2}\right )^{2} + 3 \, \log \left (c^{2} x^{4} + 1\right )^{2}\right )} b^{2}}{80 \, x^{5}} - \frac {a^{2}}{5 \, x^{5}} \]
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 \[ \int \frac {{\left (a+b\,\mathrm {atan}\left (c\,x^2\right )\right )}^2}{x^6} \,d x \]
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 \[ \int \frac {\left (a + b \operatorname {atan}{\left (c x^{2} \right )}\right )^{2}}{x^{6}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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