Optimal. Leaf size=1102 \[ -\frac {1}{3} i c^{3/2} \tan ^{-1}\left (\sqrt {c} x\right )^2 b^2+\frac {1}{3} c^{3/2} \tanh ^{-1}\left (\sqrt {c} x\right )^2 b^2-\frac {\log ^2\left (c x^2+1\right ) b^2}{12 x^3}+\frac {4}{3} c^{3/2} \tan ^{-1}\left (\sqrt {c} x\right ) b^2+\frac {4}{3} c^{3/2} \tanh ^{-1}\left (\sqrt {c} x\right ) b^2-\frac {2}{3} c^{3/2} \tanh ^{-1}\left (\sqrt {c} x\right ) \log \left (\frac {2}{1-\sqrt {c} x}\right ) b^2+\frac {2}{3} c^{3/2} \tan ^{-1}\left (\sqrt {c} x\right ) \log \left (\frac {2}{1-i \sqrt {c} x}\right ) b^2-\frac {1}{3} c^{3/2} \tan ^{-1}\left (\sqrt {c} x\right ) \log \left (\frac {(1+i) \left (1-\sqrt {c} x\right )}{1-i \sqrt {c} x}\right ) b^2-\frac {2}{3} c^{3/2} \tan ^{-1}\left (\sqrt {c} x\right ) \log \left (\frac {2}{i \sqrt {c} x+1}\right ) b^2+\frac {2}{3} c^{3/2} \tanh ^{-1}\left (\sqrt {c} x\right ) \log \left (\frac {2}{\sqrt {c} x+1}\right ) b^2-\frac {1}{3} c^{3/2} \tanh ^{-1}\left (\sqrt {c} x\right ) \log \left (-\frac {2 \sqrt {c} \left (1-\sqrt {-c} x\right )}{\left (\sqrt {-c}-\sqrt {c}\right ) \left (\sqrt {c} x+1\right )}\right ) b^2-\frac {1}{3} c^{3/2} \tanh ^{-1}\left (\sqrt {c} x\right ) \log \left (\frac {2 \sqrt {c} \left (\sqrt {-c} x+1\right )}{\left (\sqrt {-c}+\sqrt {c}\right ) \left (\sqrt {c} x+1\right )}\right ) b^2-\frac {1}{3} c^{3/2} \tan ^{-1}\left (\sqrt {c} x\right ) \log \left (\frac {(1-i) \left (\sqrt {c} x+1\right )}{1-i \sqrt {c} x}\right ) b^2+\frac {1}{3} c^{3/2} \tan ^{-1}\left (\sqrt {c} x\right ) \log \left (1-c x^2\right ) b^2+\frac {c \log \left (1-c x^2\right ) b^2}{3 x}-\frac {1}{3} c^{3/2} \tan ^{-1}\left (\sqrt {c} x\right ) \log \left (c x^2+1\right ) b^2+\frac {1}{3} c^{3/2} \tanh ^{-1}\left (\sqrt {c} x\right ) \log \left (c x^2+1\right ) b^2+\frac {\log \left (1-c x^2\right ) \log \left (c x^2+1\right ) b^2}{6 x^3}-\frac {2 c \log \left (c x^2+1\right ) b^2}{3 x}-\frac {1}{3} c^{3/2} \text {Li}_2\left (1-\frac {2}{1-\sqrt {c} x}\right ) b^2-\frac {1}{3} i c^{3/2} \text {Li}_2\left (1-\frac {2}{1-i \sqrt {c} x}\right ) b^2+\frac {1}{6} i c^{3/2} \text {Li}_2\left (1-\frac {(1+i) \left (1-\sqrt {c} x\right )}{1-i \sqrt {c} x}\right ) b^2-\frac {1}{3} i c^{3/2} \text {Li}_2\left (1-\frac {2}{i \sqrt {c} x+1}\right ) b^2-\frac {1}{3} c^{3/2} \text {Li}_2\left (1-\frac {2}{\sqrt {c} x+1}\right ) b^2+\frac {1}{6} c^{3/2} \text {Li}_2\left (\frac {2 \sqrt {c} \left (1-\sqrt {-c} x\right )}{\left (\sqrt {-c}-\sqrt {c}\right ) \left (\sqrt {c} x+1\right )}+1\right ) b^2+\frac {1}{6} c^{3/2} \text {Li}_2\left (1-\frac {2 \sqrt {c} \left (\sqrt {-c} x+1\right )}{\left (\sqrt {-c}+\sqrt {c}\right ) \left (\sqrt {c} x+1\right )}\right ) b^2+\frac {1}{6} i c^{3/2} \text {Li}_2\left (1-\frac {(1-i) \left (\sqrt {c} x+1\right )}{1-i \sqrt {c} x}\right ) b^2-\frac {2}{3} a c^{3/2} \tan ^{-1}\left (\sqrt {c} x\right ) b+\frac {1}{3} c^{3/2} \tanh ^{-1}\left (\sqrt {c} x\right ) \left (2 a-b \log \left (1-c x^2\right )\right ) b-\frac {c \left (2 a-b \log \left (1-c x^2\right )\right ) b}{3 x}-\frac {a \log \left (c x^2+1\right ) b}{3 x^3}-\frac {2 a c b}{3 x}-\frac {\left (2 a-b \log \left (1-c x^2\right )\right )^2}{12 x^3} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 1.84, antiderivative size = 1102, normalized size of antiderivative = 1.00, number of steps used = 64, number of rules used = 24, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.500, Rules used = {6099, 2457, 2476, 2455, 206, 207, 2470, 12, 5984, 5918, 2402, 2315, 325, 6742, 203, 30, 2557, 5992, 5920, 2447, 4928, 4856, 4920, 4854} \[ -\frac {1}{3} i c^{3/2} \tan ^{-1}\left (\sqrt {c} x\right )^2 b^2+\frac {1}{3} c^{3/2} \tanh ^{-1}\left (\sqrt {c} x\right )^2 b^2-\frac {\log ^2\left (c x^2+1\right ) b^2}{12 x^3}+\frac {4}{3} c^{3/2} \tan ^{-1}\left (\sqrt {c} x\right ) b^2+\frac {4}{3} c^{3/2} \tanh ^{-1}\left (\sqrt {c} x\right ) b^2-\frac {2}{3} c^{3/2} \tanh ^{-1}\left (\sqrt {c} x\right ) \log \left (\frac {2}{1-\sqrt {c} x}\right ) b^2+\frac {2}{3} c^{3/2} \tan ^{-1}\left (\sqrt {c} x\right ) \log \left (\frac {2}{1-i \sqrt {c} x}\right ) b^2-\frac {1}{3} c^{3/2} \tan ^{-1}\left (\sqrt {c} x\right ) \log \left (\frac {(1+i) \left (1-\sqrt {c} x\right )}{1-i \sqrt {c} x}\right ) b^2-\frac {2}{3} c^{3/2} \tan ^{-1}\left (\sqrt {c} x\right ) \log \left (\frac {2}{i \sqrt {c} x+1}\right ) b^2+\frac {2}{3} c^{3/2} \tanh ^{-1}\left (\sqrt {c} x\right ) \log \left (\frac {2}{\sqrt {c} x+1}\right ) b^2-\frac {1}{3} c^{3/2} \tanh ^{-1}\left (\sqrt {c} x\right ) \log \left (-\frac {2 \sqrt {c} \left (1-\sqrt {-c} x\right )}{\left (\sqrt {-c}-\sqrt {c}\right ) \left (\sqrt {c} x+1\right )}\right ) b^2-\frac {1}{3} c^{3/2} \tanh ^{-1}\left (\sqrt {c} x\right ) \log \left (\frac {2 \sqrt {c} \left (\sqrt {-c} x+1\right )}{\left (\sqrt {-c}+\sqrt {c}\right ) \left (\sqrt {c} x+1\right )}\right ) b^2-\frac {1}{3} c^{3/2} \tan ^{-1}\left (\sqrt {c} x\right ) \log \left (\frac {(1-i) \left (\sqrt {c} x+1\right )}{1-i \sqrt {c} x}\right ) b^2+\frac {1}{3} c^{3/2} \tan ^{-1}\left (\sqrt {c} x\right ) \log \left (1-c x^2\right ) b^2+\frac {c \log \left (1-c x^2\right ) b^2}{3 x}-\frac {1}{3} c^{3/2} \tan ^{-1}\left (\sqrt {c} x\right ) \log \left (c x^2+1\right ) b^2+\frac {1}{3} c^{3/2} \tanh ^{-1}\left (\sqrt {c} x\right ) \log \left (c x^2+1\right ) b^2+\frac {\log \left (1-c x^2\right ) \log \left (c x^2+1\right ) b^2}{6 x^3}-\frac {2 c \log \left (c x^2+1\right ) b^2}{3 x}-\frac {1}{3} c^{3/2} \text {PolyLog}\left (2,1-\frac {2}{1-\sqrt {c} x}\right ) b^2-\frac {1}{3} i c^{3/2} \text {PolyLog}\left (2,1-\frac {2}{1-i \sqrt {c} x}\right ) b^2+\frac {1}{6} i c^{3/2} \text {PolyLog}\left (2,1-\frac {(1+i) \left (1-\sqrt {c} x\right )}{1-i \sqrt {c} x}\right ) b^2-\frac {1}{3} i c^{3/2} \text {PolyLog}\left (2,1-\frac {2}{i \sqrt {c} x+1}\right ) b^2-\frac {1}{3} c^{3/2} \text {PolyLog}\left (2,1-\frac {2}{\sqrt {c} x+1}\right ) b^2+\frac {1}{6} c^{3/2} \text {PolyLog}\left (2,\frac {2 \sqrt {c} \left (1-\sqrt {-c} x\right )}{\left (\sqrt {-c}-\sqrt {c}\right ) \left (\sqrt {c} x+1\right )}+1\right ) b^2+\frac {1}{6} c^{3/2} \text {PolyLog}\left (2,1-\frac {2 \sqrt {c} \left (\sqrt {-c} x+1\right )}{\left (\sqrt {-c}+\sqrt {c}\right ) \left (\sqrt {c} x+1\right )}\right ) b^2+\frac {1}{6} i c^{3/2} \text {PolyLog}\left (2,1-\frac {(1-i) \left (\sqrt {c} x+1\right )}{1-i \sqrt {c} x}\right ) b^2-\frac {2}{3} a c^{3/2} \tan ^{-1}\left (\sqrt {c} x\right ) b+\frac {1}{3} c^{3/2} \tanh ^{-1}\left (\sqrt {c} x\right ) \left (2 a-b \log \left (1-c x^2\right )\right ) b-\frac {c \left (2 a-b \log \left (1-c x^2\right )\right ) b}{3 x}-\frac {a \log \left (c x^2+1\right ) b}{3 x^3}-\frac {2 a c b}{3 x}-\frac {\left (2 a-b \log \left (1-c x^2\right )\right )^2}{12 x^3} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 30
Rule 203
Rule 206
Rule 207
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 5918
Rule 5920
Rule 5984
Rule 5992
Rule 6099
Rule 6742
Rubi steps
\begin {align*} \int \frac {\left (a+b \tanh ^{-1}\left (c x^2\right )\right )^2}{x^4} \, dx &=\int \left (\frac {\left (2 a-b \log \left (1-c x^2\right )\right )^2}{4 x^4}-\frac {b \left (-2 a+b \log \left (1-c x^2\right )\right ) \log \left (1+c x^2\right )}{2 x^4}+\frac {b^2 \log ^2\left (1+c x^2\right )}{4 x^4}\right ) \, dx\\ &=\frac {1}{4} \int \frac {\left (2 a-b \log \left (1-c x^2\right )\right )^2}{x^4} \, dx-\frac {1}{2} b \int \frac {\left (-2 a+b \log \left (1-c x^2\right )\right ) \log \left (1+c x^2\right )}{x^4} \, dx+\frac {1}{4} b^2 \int \frac {\log ^2\left (1+c x^2\right )}{x^4} \, dx\\ &=-\frac {\left (2 a-b \log \left (1-c x^2\right )\right )^2}{12 x^3}-\frac {b^2 \log ^2\left (1+c x^2\right )}{12 x^3}-\frac {1}{2} b \int \left (-\frac {2 a \log \left (1+c x^2\right )}{x^4}+\frac {b \log \left (1-c x^2\right ) \log \left (1+c x^2\right )}{x^4}\right ) \, dx+\frac {1}{3} (b c) \int \frac {2 a-b \log \left (1-c x^2\right )}{x^2 \left (1-c x^2\right )} \, dx+\frac {1}{3} \left (b^2 c\right ) \int \frac {\log \left (1+c x^2\right )}{x^2 \left (1+c x^2\right )} \, dx\\ &=-\frac {\left (2 a-b \log \left (1-c x^2\right )\right )^2}{12 x^3}-\frac {b^2 \log ^2\left (1+c x^2\right )}{12 x^3}+(a b) \int \frac {\log \left (1+c x^2\right )}{x^4} \, dx-\frac {1}{2} b^2 \int \frac {\log \left (1-c x^2\right ) \log \left (1+c x^2\right )}{x^4} \, dx+\frac {1}{3} (b c) \int \left (\frac {2 a-b \log \left (1-c x^2\right )}{x^2}-\frac {c \left (2 a-b \log \left (1-c x^2\right )\right )}{-1+c x^2}\right ) \, dx+\frac {1}{3} \left (b^2 c\right ) \int \left (\frac {\log \left (1+c x^2\right )}{x^2}-\frac {c \log \left (1+c x^2\right )}{1+c x^2}\right ) \, dx\\ &=-\frac {\left (2 a-b \log \left (1-c x^2\right )\right )^2}{12 x^3}-\frac {a b \log \left (1+c x^2\right )}{3 x^3}+\frac {b^2 \log \left (1-c x^2\right ) \log \left (1+c x^2\right )}{6 x^3}-\frac {b^2 \log ^2\left (1+c x^2\right )}{12 x^3}+\frac {1}{2} b^2 \int -\frac {2 c \log \left (1-c x^2\right )}{3 x^2 \left (1+c x^2\right )} \, dx+\frac {1}{2} b^2 \int \frac {2 c \log \left (1+c x^2\right )}{x^2 \left (3-3 c x^2\right )} \, dx+\frac {1}{3} (b c) \int \frac {2 a-b \log \left (1-c x^2\right )}{x^2} \, dx+\frac {1}{3} (2 a b c) \int \frac {1}{x^2 \left (1+c x^2\right )} \, dx+\frac {1}{3} \left (b^2 c\right ) \int \frac {\log \left (1+c x^2\right )}{x^2} \, dx-\frac {1}{3} \left (b c^2\right ) \int \frac {2 a-b \log \left (1-c x^2\right )}{-1+c x^2} \, dx-\frac {1}{3} \left (b^2 c^2\right ) \int \frac {\log \left (1+c x^2\right )}{1+c x^2} \, dx\\ &=-\frac {2 a b c}{3 x}-\frac {b c \left (2 a-b \log \left (1-c x^2\right )\right )}{3 x}+\frac {1}{3} b c^{3/2} \tanh ^{-1}\left (\sqrt {c} x\right ) \left (2 a-b \log \left (1-c x^2\right )\right )-\frac {\left (2 a-b \log \left (1-c x^2\right )\right )^2}{12 x^3}-\frac {a b \log \left (1+c x^2\right )}{3 x^3}-\frac {b^2 c \log \left (1+c x^2\right )}{3 x}-\frac {1}{3} b^2 c^{3/2} \tan ^{-1}\left (\sqrt {c} x\right ) \log \left (1+c x^2\right )+\frac {b^2 \log \left (1-c x^2\right ) \log \left (1+c x^2\right )}{6 x^3}-\frac {b^2 \log ^2\left (1+c x^2\right )}{12 x^3}-\frac {1}{3} \left (b^2 c\right ) \int \frac {\log \left (1-c x^2\right )}{x^2 \left (1+c x^2\right )} \, dx+\left (b^2 c\right ) \int \frac {\log \left (1+c x^2\right )}{x^2 \left (3-3 c x^2\right )} \, dx-\frac {1}{3} \left (2 a b c^2\right ) \int \frac {1}{1+c x^2} \, dx+\frac {1}{3} \left (2 b^2 c^2\right ) \int \frac {1}{1-c x^2} \, dx+\frac {1}{3} \left (2 b^2 c^2\right ) \int \frac {1}{1+c x^2} \, dx+\frac {1}{3} \left (2 b^2 c^3\right ) \int \frac {x \tan ^{-1}\left (\sqrt {c} x\right )}{\sqrt {c} \left (1+c x^2\right )} \, dx-\frac {1}{3} \left (2 b^2 c^3\right ) \int \frac {x \tanh ^{-1}\left (\sqrt {c} x\right )}{\sqrt {c} \left (1-c x^2\right )} \, dx\\ &=-\frac {2 a b c}{3 x}-\frac {2}{3} a b c^{3/2} \tan ^{-1}\left (\sqrt {c} x\right )+\frac {2}{3} b^2 c^{3/2} \tan ^{-1}\left (\sqrt {c} x\right )+\frac {2}{3} b^2 c^{3/2} \tanh ^{-1}\left (\sqrt {c} x\right )-\frac {b c \left (2 a-b \log \left (1-c x^2\right )\right )}{3 x}+\frac {1}{3} b c^{3/2} \tanh ^{-1}\left (\sqrt {c} x\right ) \left (2 a-b \log \left (1-c x^2\right )\right )-\frac {\left (2 a-b \log \left (1-c x^2\right )\right )^2}{12 x^3}-\frac {a b \log \left (1+c x^2\right )}{3 x^3}-\frac {b^2 c \log \left (1+c x^2\right )}{3 x}-\frac {1}{3} b^2 c^{3/2} \tan ^{-1}\left (\sqrt {c} x\right ) \log \left (1+c x^2\right )+\frac {b^2 \log \left (1-c x^2\right ) \log \left (1+c x^2\right )}{6 x^3}-\frac {b^2 \log ^2\left (1+c x^2\right )}{12 x^3}-\frac {1}{3} \left (b^2 c\right ) \int \left (\frac {\log \left (1-c x^2\right )}{x^2}-\frac {c \log \left (1-c x^2\right )}{1+c x^2}\right ) \, dx+\left (b^2 c\right ) \int \left (\frac {\log \left (1+c x^2\right )}{3 x^2}-\frac {c \log \left (1+c x^2\right )}{3 \left (-1+c x^2\right )}\right ) \, dx+\frac {1}{3} \left (2 b^2 c^{5/2}\right ) \int \frac {x \tan ^{-1}\left (\sqrt {c} x\right )}{1+c x^2} \, dx-\frac {1}{3} \left (2 b^2 c^{5/2}\right ) \int \frac {x \tanh ^{-1}\left (\sqrt {c} x\right )}{1-c x^2} \, dx\\ &=-\frac {2 a b c}{3 x}-\frac {2}{3} a b c^{3/2} \tan ^{-1}\left (\sqrt {c} x\right )+\frac {2}{3} b^2 c^{3/2} \tan ^{-1}\left (\sqrt {c} x\right )-\frac {1}{3} i b^2 c^{3/2} \tan ^{-1}\left (\sqrt {c} x\right )^2+\frac {2}{3} b^2 c^{3/2} \tanh ^{-1}\left (\sqrt {c} x\right )+\frac {1}{3} b^2 c^{3/2} \tanh ^{-1}\left (\sqrt {c} x\right )^2-\frac {b c \left (2 a-b \log \left (1-c x^2\right )\right )}{3 x}+\frac {1}{3} b c^{3/2} \tanh ^{-1}\left (\sqrt {c} x\right ) \left (2 a-b \log \left (1-c x^2\right )\right )-\frac {\left (2 a-b \log \left (1-c x^2\right )\right )^2}{12 x^3}-\frac {a b \log \left (1+c x^2\right )}{3 x^3}-\frac {b^2 c \log \left (1+c x^2\right )}{3 x}-\frac {1}{3} b^2 c^{3/2} \tan ^{-1}\left (\sqrt {c} x\right ) \log \left (1+c x^2\right )+\frac {b^2 \log \left (1-c x^2\right ) \log \left (1+c x^2\right )}{6 x^3}-\frac {b^2 \log ^2\left (1+c x^2\right )}{12 x^3}-\frac {1}{3} \left (b^2 c\right ) \int \frac {\log \left (1-c x^2\right )}{x^2} \, dx+\frac {1}{3} \left (b^2 c\right ) \int \frac {\log \left (1+c x^2\right )}{x^2} \, dx+\frac {1}{3} \left (b^2 c^2\right ) \int \frac {\log \left (1-c x^2\right )}{1+c x^2} \, dx-\frac {1}{3} \left (b^2 c^2\right ) \int \frac {\log \left (1+c x^2\right )}{-1+c x^2} \, dx-\frac {1}{3} \left (2 b^2 c^2\right ) \int \frac {\tan ^{-1}\left (\sqrt {c} x\right )}{i-\sqrt {c} x} \, dx-\frac {1}{3} \left (2 b^2 c^2\right ) \int \frac {\tanh ^{-1}\left (\sqrt {c} x\right )}{1-\sqrt {c} x} \, dx\\ &=-\frac {2 a b c}{3 x}-\frac {2}{3} a b c^{3/2} \tan ^{-1}\left (\sqrt {c} x\right )+\frac {2}{3} b^2 c^{3/2} \tan ^{-1}\left (\sqrt {c} x\right )-\frac {1}{3} i b^2 c^{3/2} \tan ^{-1}\left (\sqrt {c} x\right )^2+\frac {2}{3} b^2 c^{3/2} \tanh ^{-1}\left (\sqrt {c} x\right )+\frac {1}{3} b^2 c^{3/2} \tanh ^{-1}\left (\sqrt {c} x\right )^2-\frac {2}{3} b^2 c^{3/2} \tanh ^{-1}\left (\sqrt {c} x\right ) \log \left (\frac {2}{1-\sqrt {c} x}\right )-\frac {2}{3} b^2 c^{3/2} \tan ^{-1}\left (\sqrt {c} x\right ) \log \left (\frac {2}{1+i \sqrt {c} x}\right )+\frac {b^2 c \log \left (1-c x^2\right )}{3 x}+\frac {1}{3} b^2 c^{3/2} \tan ^{-1}\left (\sqrt {c} x\right ) \log \left (1-c x^2\right )-\frac {b c \left (2 a-b \log \left (1-c x^2\right )\right )}{3 x}+\frac {1}{3} b c^{3/2} \tanh ^{-1}\left (\sqrt {c} x\right ) \left (2 a-b \log \left (1-c x^2\right )\right )-\frac {\left (2 a-b \log \left (1-c x^2\right )\right )^2}{12 x^3}-\frac {a b \log \left (1+c x^2\right )}{3 x^3}-\frac {2 b^2 c \log \left (1+c x^2\right )}{3 x}-\frac {1}{3} b^2 c^{3/2} \tan ^{-1}\left (\sqrt {c} x\right ) \log \left (1+c x^2\right )+\frac {1}{3} b^2 c^{3/2} \tanh ^{-1}\left (\sqrt {c} x\right ) \log \left (1+c x^2\right )+\frac {b^2 \log \left (1-c x^2\right ) \log \left (1+c x^2\right )}{6 x^3}-\frac {b^2 \log ^2\left (1+c x^2\right )}{12 x^3}+\frac {1}{3} \left (2 b^2 c^2\right ) \int \frac {1}{1-c x^2} \, dx+\frac {1}{3} \left (2 b^2 c^2\right ) \int \frac {1}{1+c x^2} \, dx+\frac {1}{3} \left (2 b^2 c^2\right ) \int \frac {\log \left (\frac {2}{1-\sqrt {c} x}\right )}{1-c x^2} \, dx+\frac {1}{3} \left (2 b^2 c^2\right ) \int \frac {\log \left (\frac {2}{1+i \sqrt {c} x}\right )}{1+c x^2} \, dx+\frac {1}{3} \left (2 b^2 c^3\right ) \int \frac {x \tan ^{-1}\left (\sqrt {c} x\right )}{\sqrt {c} \left (1-c x^2\right )} \, dx-\frac {1}{3} \left (2 b^2 c^3\right ) \int \frac {x \tanh ^{-1}\left (\sqrt {c} x\right )}{\sqrt {c} \left (1+c x^2\right )} \, dx\\ &=-\frac {2 a b c}{3 x}-\frac {2}{3} a b c^{3/2} \tan ^{-1}\left (\sqrt {c} x\right )+\frac {4}{3} b^2 c^{3/2} \tan ^{-1}\left (\sqrt {c} x\right )-\frac {1}{3} i b^2 c^{3/2} \tan ^{-1}\left (\sqrt {c} x\right )^2+\frac {4}{3} b^2 c^{3/2} \tanh ^{-1}\left (\sqrt {c} x\right )+\frac {1}{3} b^2 c^{3/2} \tanh ^{-1}\left (\sqrt {c} x\right )^2-\frac {2}{3} b^2 c^{3/2} \tanh ^{-1}\left (\sqrt {c} x\right ) \log \left (\frac {2}{1-\sqrt {c} x}\right )-\frac {2}{3} b^2 c^{3/2} \tan ^{-1}\left (\sqrt {c} x\right ) \log \left (\frac {2}{1+i \sqrt {c} x}\right )+\frac {b^2 c \log \left (1-c x^2\right )}{3 x}+\frac {1}{3} b^2 c^{3/2} \tan ^{-1}\left (\sqrt {c} x\right ) \log \left (1-c x^2\right )-\frac {b c \left (2 a-b \log \left (1-c x^2\right )\right )}{3 x}+\frac {1}{3} b c^{3/2} \tanh ^{-1}\left (\sqrt {c} x\right ) \left (2 a-b \log \left (1-c x^2\right )\right )-\frac {\left (2 a-b \log \left (1-c x^2\right )\right )^2}{12 x^3}-\frac {a b \log \left (1+c x^2\right )}{3 x^3}-\frac {2 b^2 c \log \left (1+c x^2\right )}{3 x}-\frac {1}{3} b^2 c^{3/2} \tan ^{-1}\left (\sqrt {c} x\right ) \log \left (1+c x^2\right )+\frac {1}{3} b^2 c^{3/2} \tanh ^{-1}\left (\sqrt {c} x\right ) \log \left (1+c x^2\right )+\frac {b^2 \log \left (1-c x^2\right ) \log \left (1+c x^2\right )}{6 x^3}-\frac {b^2 \log ^2\left (1+c x^2\right )}{12 x^3}-\frac {1}{3} \left (2 i b^2 c^{3/2}\right ) \operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+i \sqrt {c} x}\right )-\frac {1}{3} \left (2 b^2 c^{3/2}\right ) \operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1-\sqrt {c} x}\right )+\frac {1}{3} \left (2 b^2 c^{5/2}\right ) \int \frac {x \tan ^{-1}\left (\sqrt {c} x\right )}{1-c x^2} \, dx-\frac {1}{3} \left (2 b^2 c^{5/2}\right ) \int \frac {x \tanh ^{-1}\left (\sqrt {c} x\right )}{1+c x^2} \, dx\\ &=-\frac {2 a b c}{3 x}-\frac {2}{3} a b c^{3/2} \tan ^{-1}\left (\sqrt {c} x\right )+\frac {4}{3} b^2 c^{3/2} \tan ^{-1}\left (\sqrt {c} x\right )-\frac {1}{3} i b^2 c^{3/2} \tan ^{-1}\left (\sqrt {c} x\right )^2+\frac {4}{3} b^2 c^{3/2} \tanh ^{-1}\left (\sqrt {c} x\right )+\frac {1}{3} b^2 c^{3/2} \tanh ^{-1}\left (\sqrt {c} x\right )^2-\frac {2}{3} b^2 c^{3/2} \tanh ^{-1}\left (\sqrt {c} x\right ) \log \left (\frac {2}{1-\sqrt {c} x}\right )-\frac {2}{3} b^2 c^{3/2} \tan ^{-1}\left (\sqrt {c} x\right ) \log \left (\frac {2}{1+i \sqrt {c} x}\right )+\frac {b^2 c \log \left (1-c x^2\right )}{3 x}+\frac {1}{3} b^2 c^{3/2} \tan ^{-1}\left (\sqrt {c} x\right ) \log \left (1-c x^2\right )-\frac {b c \left (2 a-b \log \left (1-c x^2\right )\right )}{3 x}+\frac {1}{3} b c^{3/2} \tanh ^{-1}\left (\sqrt {c} x\right ) \left (2 a-b \log \left (1-c x^2\right )\right )-\frac {\left (2 a-b \log \left (1-c x^2\right )\right )^2}{12 x^3}-\frac {a b \log \left (1+c x^2\right )}{3 x^3}-\frac {2 b^2 c \log \left (1+c x^2\right )}{3 x}-\frac {1}{3} b^2 c^{3/2} \tan ^{-1}\left (\sqrt {c} x\right ) \log \left (1+c x^2\right )+\frac {1}{3} b^2 c^{3/2} \tanh ^{-1}\left (\sqrt {c} x\right ) \log \left (1+c x^2\right )+\frac {b^2 \log \left (1-c x^2\right ) \log \left (1+c x^2\right )}{6 x^3}-\frac {b^2 \log ^2\left (1+c x^2\right )}{12 x^3}-\frac {1}{3} b^2 c^{3/2} \text {Li}_2\left (1-\frac {2}{1-\sqrt {c} x}\right )-\frac {1}{3} i b^2 c^{3/2} \text {Li}_2\left (1-\frac {2}{1+i \sqrt {c} x}\right )+\frac {1}{3} \left (2 b^2 c^{5/2}\right ) \int \left (\frac {\tan ^{-1}\left (\sqrt {c} x\right )}{2 \sqrt {c} \left (1-\sqrt {c} x\right )}-\frac {\tan ^{-1}\left (\sqrt {c} x\right )}{2 \sqrt {c} \left (1+\sqrt {c} x\right )}\right ) \, dx-\frac {1}{3} \left (2 b^2 c^{5/2}\right ) \int \left (-\frac {\sqrt {-c} \tanh ^{-1}\left (\sqrt {c} x\right )}{2 c \left (1-\sqrt {-c} x\right )}+\frac {\sqrt {-c} \tanh ^{-1}\left (\sqrt {c} x\right )}{2 c \left (1+\sqrt {-c} x\right )}\right ) \, dx\\ &=-\frac {2 a b c}{3 x}-\frac {2}{3} a b c^{3/2} \tan ^{-1}\left (\sqrt {c} x\right )+\frac {4}{3} b^2 c^{3/2} \tan ^{-1}\left (\sqrt {c} x\right )-\frac {1}{3} i b^2 c^{3/2} \tan ^{-1}\left (\sqrt {c} x\right )^2+\frac {4}{3} b^2 c^{3/2} \tanh ^{-1}\left (\sqrt {c} x\right )+\frac {1}{3} b^2 c^{3/2} \tanh ^{-1}\left (\sqrt {c} x\right )^2-\frac {2}{3} b^2 c^{3/2} \tanh ^{-1}\left (\sqrt {c} x\right ) \log \left (\frac {2}{1-\sqrt {c} x}\right )-\frac {2}{3} b^2 c^{3/2} \tan ^{-1}\left (\sqrt {c} x\right ) \log \left (\frac {2}{1+i \sqrt {c} x}\right )+\frac {b^2 c \log \left (1-c x^2\right )}{3 x}+\frac {1}{3} b^2 c^{3/2} \tan ^{-1}\left (\sqrt {c} x\right ) \log \left (1-c x^2\right )-\frac {b c \left (2 a-b \log \left (1-c x^2\right )\right )}{3 x}+\frac {1}{3} b c^{3/2} \tanh ^{-1}\left (\sqrt {c} x\right ) \left (2 a-b \log \left (1-c x^2\right )\right )-\frac {\left (2 a-b \log \left (1-c x^2\right )\right )^2}{12 x^3}-\frac {a b \log \left (1+c x^2\right )}{3 x^3}-\frac {2 b^2 c \log \left (1+c x^2\right )}{3 x}-\frac {1}{3} b^2 c^{3/2} \tan ^{-1}\left (\sqrt {c} x\right ) \log \left (1+c x^2\right )+\frac {1}{3} b^2 c^{3/2} \tanh ^{-1}\left (\sqrt {c} x\right ) \log \left (1+c x^2\right )+\frac {b^2 \log \left (1-c x^2\right ) \log \left (1+c x^2\right )}{6 x^3}-\frac {b^2 \log ^2\left (1+c x^2\right )}{12 x^3}-\frac {1}{3} b^2 c^{3/2} \text {Li}_2\left (1-\frac {2}{1-\sqrt {c} x}\right )-\frac {1}{3} i b^2 c^{3/2} \text {Li}_2\left (1-\frac {2}{1+i \sqrt {c} x}\right )+\frac {1}{3} \left (b^2 c^2\right ) \int \frac {\tan ^{-1}\left (\sqrt {c} x\right )}{1-\sqrt {c} x} \, dx-\frac {1}{3} \left (b^2 c^2\right ) \int \frac {\tan ^{-1}\left (\sqrt {c} x\right )}{1+\sqrt {c} x} \, dx-\frac {\left (b^2 c^{5/2}\right ) \int \frac {\tanh ^{-1}\left (\sqrt {c} x\right )}{1-\sqrt {-c} x} \, dx}{3 \sqrt {-c}}+\frac {\left (b^2 c^{5/2}\right ) \int \frac {\tanh ^{-1}\left (\sqrt {c} x\right )}{1+\sqrt {-c} x} \, dx}{3 \sqrt {-c}}\\ &=-\frac {2 a b c}{3 x}-\frac {2}{3} a b c^{3/2} \tan ^{-1}\left (\sqrt {c} x\right )+\frac {4}{3} b^2 c^{3/2} \tan ^{-1}\left (\sqrt {c} x\right )-\frac {1}{3} i b^2 c^{3/2} \tan ^{-1}\left (\sqrt {c} x\right )^2+\frac {4}{3} b^2 c^{3/2} \tanh ^{-1}\left (\sqrt {c} x\right )+\frac {1}{3} b^2 c^{3/2} \tanh ^{-1}\left (\sqrt {c} x\right )^2-\frac {2}{3} b^2 c^{3/2} \tanh ^{-1}\left (\sqrt {c} x\right ) \log \left (\frac {2}{1-\sqrt {c} x}\right )+\frac {2}{3} b^2 c^{3/2} \tan ^{-1}\left (\sqrt {c} x\right ) \log \left (\frac {2}{1-i \sqrt {c} x}\right )-\frac {1}{3} b^2 c^{3/2} \tan ^{-1}\left (\sqrt {c} x\right ) \log \left (\frac {(1+i) \left (1-\sqrt {c} x\right )}{1-i \sqrt {c} x}\right )-\frac {2}{3} b^2 c^{3/2} \tan ^{-1}\left (\sqrt {c} x\right ) \log \left (\frac {2}{1+i \sqrt {c} x}\right )+\frac {2}{3} b^2 c^{3/2} \tanh ^{-1}\left (\sqrt {c} x\right ) \log \left (\frac {2}{1+\sqrt {c} x}\right )-\frac {1}{3} b^2 c^{3/2} \tanh ^{-1}\left (\sqrt {c} x\right ) \log \left (-\frac {2 \sqrt {c} \left (1-\sqrt {-c} x\right )}{\left (\sqrt {-c}-\sqrt {c}\right ) \left (1+\sqrt {c} x\right )}\right )-\frac {1}{3} b^2 c^{3/2} \tanh ^{-1}\left (\sqrt {c} x\right ) \log \left (\frac {2 \sqrt {c} \left (1+\sqrt {-c} x\right )}{\left (\sqrt {-c}+\sqrt {c}\right ) \left (1+\sqrt {c} x\right )}\right )-\frac {1}{3} b^2 c^{3/2} \tan ^{-1}\left (\sqrt {c} x\right ) \log \left (\frac {(1-i) \left (1+\sqrt {c} x\right )}{1-i \sqrt {c} x}\right )+\frac {b^2 c \log \left (1-c x^2\right )}{3 x}+\frac {1}{3} b^2 c^{3/2} \tan ^{-1}\left (\sqrt {c} x\right ) \log \left (1-c x^2\right )-\frac {b c \left (2 a-b \log \left (1-c x^2\right )\right )}{3 x}+\frac {1}{3} b c^{3/2} \tanh ^{-1}\left (\sqrt {c} x\right ) \left (2 a-b \log \left (1-c x^2\right )\right )-\frac {\left (2 a-b \log \left (1-c x^2\right )\right )^2}{12 x^3}-\frac {a b \log \left (1+c x^2\right )}{3 x^3}-\frac {2 b^2 c \log \left (1+c x^2\right )}{3 x}-\frac {1}{3} b^2 c^{3/2} \tan ^{-1}\left (\sqrt {c} x\right ) \log \left (1+c x^2\right )+\frac {1}{3} b^2 c^{3/2} \tanh ^{-1}\left (\sqrt {c} x\right ) \log \left (1+c x^2\right )+\frac {b^2 \log \left (1-c x^2\right ) \log \left (1+c x^2\right )}{6 x^3}-\frac {b^2 \log ^2\left (1+c x^2\right )}{12 x^3}-\frac {1}{3} b^2 c^{3/2} \text {Li}_2\left (1-\frac {2}{1-\sqrt {c} x}\right )-\frac {1}{3} i b^2 c^{3/2} \text {Li}_2\left (1-\frac {2}{1+i \sqrt {c} x}\right )-2 \left (\frac {1}{3} \left (b^2 c^2\right ) \int \frac {\log \left (\frac {2}{1-i \sqrt {c} x}\right )}{1+c x^2} \, dx\right )+\frac {1}{3} \left (b^2 c^2\right ) \int \frac {\log \left (\frac {(1+i) \left (1-\sqrt {c} x\right )}{1-i \sqrt {c} x}\right )}{1+c x^2} \, dx-2 \left (\frac {1}{3} \left (b^2 c^2\right ) \int \frac {\log \left (\frac {2}{1+\sqrt {c} x}\right )}{1-c x^2} \, dx\right )+\frac {1}{3} \left (b^2 c^2\right ) \int \frac {\log \left (\frac {2 \sqrt {c} \left (1-\sqrt {-c} x\right )}{\left (-\sqrt {-c}+\sqrt {c}\right ) \left (1+\sqrt {c} x\right )}\right )}{1-c x^2} \, dx+\frac {1}{3} \left (b^2 c^2\right ) \int \frac {\log \left (\frac {2 \sqrt {c} \left (1+\sqrt {-c} x\right )}{\left (\sqrt {-c}+\sqrt {c}\right ) \left (1+\sqrt {c} x\right )}\right )}{1-c x^2} \, dx+\frac {1}{3} \left (b^2 c^2\right ) \int \frac {\log \left (\frac {(1-i) \left (1+\sqrt {c} x\right )}{1-i \sqrt {c} x}\right )}{1+c x^2} \, dx\\ &=-\frac {2 a b c}{3 x}-\frac {2}{3} a b c^{3/2} \tan ^{-1}\left (\sqrt {c} x\right )+\frac {4}{3} b^2 c^{3/2} \tan ^{-1}\left (\sqrt {c} x\right )-\frac {1}{3} i b^2 c^{3/2} \tan ^{-1}\left (\sqrt {c} x\right )^2+\frac {4}{3} b^2 c^{3/2} \tanh ^{-1}\left (\sqrt {c} x\right )+\frac {1}{3} b^2 c^{3/2} \tanh ^{-1}\left (\sqrt {c} x\right )^2-\frac {2}{3} b^2 c^{3/2} \tanh ^{-1}\left (\sqrt {c} x\right ) \log \left (\frac {2}{1-\sqrt {c} x}\right )+\frac {2}{3} b^2 c^{3/2} \tan ^{-1}\left (\sqrt {c} x\right ) \log \left (\frac {2}{1-i \sqrt {c} x}\right )-\frac {1}{3} b^2 c^{3/2} \tan ^{-1}\left (\sqrt {c} x\right ) \log \left (\frac {(1+i) \left (1-\sqrt {c} x\right )}{1-i \sqrt {c} x}\right )-\frac {2}{3} b^2 c^{3/2} \tan ^{-1}\left (\sqrt {c} x\right ) \log \left (\frac {2}{1+i \sqrt {c} x}\right )+\frac {2}{3} b^2 c^{3/2} \tanh ^{-1}\left (\sqrt {c} x\right ) \log \left (\frac {2}{1+\sqrt {c} x}\right )-\frac {1}{3} b^2 c^{3/2} \tanh ^{-1}\left (\sqrt {c} x\right ) \log \left (-\frac {2 \sqrt {c} \left (1-\sqrt {-c} x\right )}{\left (\sqrt {-c}-\sqrt {c}\right ) \left (1+\sqrt {c} x\right )}\right )-\frac {1}{3} b^2 c^{3/2} \tanh ^{-1}\left (\sqrt {c} x\right ) \log \left (\frac {2 \sqrt {c} \left (1+\sqrt {-c} x\right )}{\left (\sqrt {-c}+\sqrt {c}\right ) \left (1+\sqrt {c} x\right )}\right )-\frac {1}{3} b^2 c^{3/2} \tan ^{-1}\left (\sqrt {c} x\right ) \log \left (\frac {(1-i) \left (1+\sqrt {c} x\right )}{1-i \sqrt {c} x}\right )+\frac {b^2 c \log \left (1-c x^2\right )}{3 x}+\frac {1}{3} b^2 c^{3/2} \tan ^{-1}\left (\sqrt {c} x\right ) \log \left (1-c x^2\right )-\frac {b c \left (2 a-b \log \left (1-c x^2\right )\right )}{3 x}+\frac {1}{3} b c^{3/2} \tanh ^{-1}\left (\sqrt {c} x\right ) \left (2 a-b \log \left (1-c x^2\right )\right )-\frac {\left (2 a-b \log \left (1-c x^2\right )\right )^2}{12 x^3}-\frac {a b \log \left (1+c x^2\right )}{3 x^3}-\frac {2 b^2 c \log \left (1+c x^2\right )}{3 x}-\frac {1}{3} b^2 c^{3/2} \tan ^{-1}\left (\sqrt {c} x\right ) \log \left (1+c x^2\right )+\frac {1}{3} b^2 c^{3/2} \tanh ^{-1}\left (\sqrt {c} x\right ) \log \left (1+c x^2\right )+\frac {b^2 \log \left (1-c x^2\right ) \log \left (1+c x^2\right )}{6 x^3}-\frac {b^2 \log ^2\left (1+c x^2\right )}{12 x^3}-\frac {1}{3} b^2 c^{3/2} \text {Li}_2\left (1-\frac {2}{1-\sqrt {c} x}\right )+\frac {1}{6} i b^2 c^{3/2} \text {Li}_2\left (1-\frac {(1+i) \left (1-\sqrt {c} x\right )}{1-i \sqrt {c} x}\right )-\frac {1}{3} i b^2 c^{3/2} \text {Li}_2\left (1-\frac {2}{1+i \sqrt {c} x}\right )+\frac {1}{6} b^2 c^{3/2} \text {Li}_2\left (1+\frac {2 \sqrt {c} \left (1-\sqrt {-c} x\right )}{\left (\sqrt {-c}-\sqrt {c}\right ) \left (1+\sqrt {c} x\right )}\right )+\frac {1}{6} b^2 c^{3/2} \text {Li}_2\left (1-\frac {2 \sqrt {c} \left (1+\sqrt {-c} x\right )}{\left (\sqrt {-c}+\sqrt {c}\right ) \left (1+\sqrt {c} x\right )}\right )+\frac {1}{6} i b^2 c^{3/2} \text {Li}_2\left (1-\frac {(1-i) \left (1+\sqrt {c} x\right )}{1-i \sqrt {c} x}\right )-2 \left (\frac {1}{3} \left (i b^2 c^{3/2}\right ) \operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1-i \sqrt {c} x}\right )\right )-2 \left (\frac {1}{3} \left (b^2 c^{3/2}\right ) \operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+\sqrt {c} x}\right )\right )\\ &=-\frac {2 a b c}{3 x}-\frac {2}{3} a b c^{3/2} \tan ^{-1}\left (\sqrt {c} x\right )+\frac {4}{3} b^2 c^{3/2} \tan ^{-1}\left (\sqrt {c} x\right )-\frac {1}{3} i b^2 c^{3/2} \tan ^{-1}\left (\sqrt {c} x\right )^2+\frac {4}{3} b^2 c^{3/2} \tanh ^{-1}\left (\sqrt {c} x\right )+\frac {1}{3} b^2 c^{3/2} \tanh ^{-1}\left (\sqrt {c} x\right )^2-\frac {2}{3} b^2 c^{3/2} \tanh ^{-1}\left (\sqrt {c} x\right ) \log \left (\frac {2}{1-\sqrt {c} x}\right )+\frac {2}{3} b^2 c^{3/2} \tan ^{-1}\left (\sqrt {c} x\right ) \log \left (\frac {2}{1-i \sqrt {c} x}\right )-\frac {1}{3} b^2 c^{3/2} \tan ^{-1}\left (\sqrt {c} x\right ) \log \left (\frac {(1+i) \left (1-\sqrt {c} x\right )}{1-i \sqrt {c} x}\right )-\frac {2}{3} b^2 c^{3/2} \tan ^{-1}\left (\sqrt {c} x\right ) \log \left (\frac {2}{1+i \sqrt {c} x}\right )+\frac {2}{3} b^2 c^{3/2} \tanh ^{-1}\left (\sqrt {c} x\right ) \log \left (\frac {2}{1+\sqrt {c} x}\right )-\frac {1}{3} b^2 c^{3/2} \tanh ^{-1}\left (\sqrt {c} x\right ) \log \left (-\frac {2 \sqrt {c} \left (1-\sqrt {-c} x\right )}{\left (\sqrt {-c}-\sqrt {c}\right ) \left (1+\sqrt {c} x\right )}\right )-\frac {1}{3} b^2 c^{3/2} \tanh ^{-1}\left (\sqrt {c} x\right ) \log \left (\frac {2 \sqrt {c} \left (1+\sqrt {-c} x\right )}{\left (\sqrt {-c}+\sqrt {c}\right ) \left (1+\sqrt {c} x\right )}\right )-\frac {1}{3} b^2 c^{3/2} \tan ^{-1}\left (\sqrt {c} x\right ) \log \left (\frac {(1-i) \left (1+\sqrt {c} x\right )}{1-i \sqrt {c} x}\right )+\frac {b^2 c \log \left (1-c x^2\right )}{3 x}+\frac {1}{3} b^2 c^{3/2} \tan ^{-1}\left (\sqrt {c} x\right ) \log \left (1-c x^2\right )-\frac {b c \left (2 a-b \log \left (1-c x^2\right )\right )}{3 x}+\frac {1}{3} b c^{3/2} \tanh ^{-1}\left (\sqrt {c} x\right ) \left (2 a-b \log \left (1-c x^2\right )\right )-\frac {\left (2 a-b \log \left (1-c x^2\right )\right )^2}{12 x^3}-\frac {a b \log \left (1+c x^2\right )}{3 x^3}-\frac {2 b^2 c \log \left (1+c x^2\right )}{3 x}-\frac {1}{3} b^2 c^{3/2} \tan ^{-1}\left (\sqrt {c} x\right ) \log \left (1+c x^2\right )+\frac {1}{3} b^2 c^{3/2} \tanh ^{-1}\left (\sqrt {c} x\right ) \log \left (1+c x^2\right )+\frac {b^2 \log \left (1-c x^2\right ) \log \left (1+c x^2\right )}{6 x^3}-\frac {b^2 \log ^2\left (1+c x^2\right )}{12 x^3}-\frac {1}{3} b^2 c^{3/2} \text {Li}_2\left (1-\frac {2}{1-\sqrt {c} x}\right )-\frac {1}{3} i b^2 c^{3/2} \text {Li}_2\left (1-\frac {2}{1-i \sqrt {c} x}\right )+\frac {1}{6} i b^2 c^{3/2} \text {Li}_2\left (1-\frac {(1+i) \left (1-\sqrt {c} x\right )}{1-i \sqrt {c} x}\right )-\frac {1}{3} i b^2 c^{3/2} \text {Li}_2\left (1-\frac {2}{1+i \sqrt {c} x}\right )-\frac {1}{3} b^2 c^{3/2} \text {Li}_2\left (1-\frac {2}{1+\sqrt {c} x}\right )+\frac {1}{6} b^2 c^{3/2} \text {Li}_2\left (1+\frac {2 \sqrt {c} \left (1-\sqrt {-c} x\right )}{\left (\sqrt {-c}-\sqrt {c}\right ) \left (1+\sqrt {c} x\right )}\right )+\frac {1}{6} b^2 c^{3/2} \text {Li}_2\left (1-\frac {2 \sqrt {c} \left (1+\sqrt {-c} x\right )}{\left (\sqrt {-c}+\sqrt {c}\right ) \left (1+\sqrt {c} x\right )}\right )+\frac {1}{6} i b^2 c^{3/2} \text {Li}_2\left (1-\frac {(1-i) \left (1+\sqrt {c} x\right )}{1-i \sqrt {c} x}\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [F] time = 2.77, size = 0, normalized size = 0.00 \[ \int \frac {\left (a+b \tanh ^{-1}\left (c x^2\right )\right )^2}{x^4} \, dx \]
Verification is Not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 2.14, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b^{2} \operatorname {artanh}\left (c x^{2}\right )^{2} + 2 \, a b \operatorname {artanh}\left (c x^{2}\right ) + a^{2}}{x^{4}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b \operatorname {artanh}\left (c x^{2}\right ) + a\right )}^{2}}{x^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.29, size = 0, normalized size = 0.00 \[ \int \frac {\left (a +b \arctanh \left (c \,x^{2}\right )\right )^{2}}{x^{4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {1}{3} \, {\left ({\left (2 \, \sqrt {c} \arctan \left (\sqrt {c} x\right ) + \sqrt {c} \log \left (\frac {c x - \sqrt {c}}{c x + \sqrt {c}}\right ) + \frac {4}{x}\right )} c + \frac {2 \, \operatorname {artanh}\left (c x^{2}\right )}{x^{3}}\right )} a b - \frac {1}{12} \, b^{2} {\left (\frac {\log \left (-c x^{2} + 1\right )^{2}}{x^{3}} + 3 \, \int -\frac {3 \, {\left (c x^{2} - 1\right )} \log \left (c x^{2} + 1\right )^{2} + 2 \, {\left (2 \, c x^{2} - 3 \, {\left (c x^{2} - 1\right )} \log \left (c x^{2} + 1\right )\right )} \log \left (-c x^{2} + 1\right )}{3 \, {\left (c x^{6} - x^{4}\right )}}\,{d x}\right )} - \frac {a^{2}}{3 \, x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (a+b\,\mathrm {atanh}\left (c\,x^2\right )\right )}^2}{x^4} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (a + b \operatorname {atanh}{\left (c x^{2} \right )}\right )^{2}}{x^{4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________