Optimal. Leaf size=1172 \[ \frac {4}{3} a b c x-\frac {2}{3} a b c^{3/2} \text {ArcTan}\left (\frac {x}{\sqrt {c}}\right )+\frac {4}{3} b^2 c^{3/2} \text {ArcTan}\left (\frac {x}{\sqrt {c}}\right )+\frac {1}{3} i b^2 c^{3/2} \text {ArcTan}\left (\frac {x}{\sqrt {c}}\right )^2-\frac {4}{3} b^2 c^{3/2} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right )+\frac {1}{3} b^2 c^{3/2} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right )^2-\frac {2}{3} b^2 c^{3/2} \text {ArcTan}\left (\frac {x}{\sqrt {c}}\right ) \log \left (2-\frac {2 \sqrt {c}}{\sqrt {c}-i x}\right )-\frac {2}{3} b^2 c x \log \left (1-\frac {c}{x^2}\right )+\frac {1}{3} b^2 c^{3/2} \text {ArcTan}\left (\frac {x}{\sqrt {c}}\right ) \log \left (1-\frac {c}{x^2}\right )-\frac {1}{3} b c^{3/2} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )+\frac {1}{12} x^3 \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )^2+\frac {2}{3} b^2 c x \log \left (1+\frac {c}{x^2}\right )+\frac {1}{3} a b x^3 \log \left (1+\frac {c}{x^2}\right )-\frac {1}{3} b^2 c^{3/2} \text {ArcTan}\left (\frac {x}{\sqrt {c}}\right ) \log \left (1+\frac {c}{x^2}\right )-\frac {1}{3} b^2 c^{3/2} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (1+\frac {c}{x^2}\right )-\frac {1}{6} b^2 x^3 \log \left (1-\frac {c}{x^2}\right ) \log \left (1+\frac {c}{x^2}\right )+\frac {1}{12} b^2 x^3 \log ^2\left (1+\frac {c}{x^2}\right )+\frac {2}{3} b^2 c^{3/2} \text {ArcTan}\left (\frac {x}{\sqrt {c}}\right ) \log \left (\frac {2 \sqrt {c}}{\sqrt {c}-i x}\right )-\frac {1}{3} b^2 c^{3/2} \text {ArcTan}\left (\frac {x}{\sqrt {c}}\right ) \log \left (\frac {(1+i) \left (\sqrt {c}-x\right )}{\sqrt {c}-i x}\right )-\frac {2}{3} b^2 c^{3/2} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (\frac {2 \sqrt {c}}{\sqrt {c}+x}\right )+\frac {1}{3} b^2 c^{3/2} \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}{3} b^2 c^{3/2} \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}{3} b^2 c^{3/2} \text {ArcTan}\left (\frac {x}{\sqrt {c}}\right ) \log \left (\frac {(1-i) \left (\sqrt {c}+x\right )}{\sqrt {c}-i x}\right )+\frac {2}{3} b^2 c^{3/2} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (2-\frac {2 \sqrt {c}}{\sqrt {c}+x}\right )-\frac {1}{3} i b^2 c^{3/2} \text {PolyLog}\left (2,1-\frac {2 \sqrt {c}}{\sqrt {c}-i x}\right )+\frac {1}{3} i b^2 c^{3/2} \text {PolyLog}\left (2,-1+\frac {2 \sqrt {c}}{\sqrt {c}-i x}\right )+\frac {1}{6} i b^2 c^{3/2} \text {PolyLog}\left (2,1-\frac {(1+i) \left (\sqrt {c}-x\right )}{\sqrt {c}-i x}\right )+\frac {1}{3} b^2 c^{3/2} \text {PolyLog}\left (2,-\frac {x}{\sqrt {c}}\right )+\frac {1}{3} i b^2 c^{3/2} \text {PolyLog}\left (2,-\frac {i x}{\sqrt {c}}\right )-\frac {1}{3} i b^2 c^{3/2} \text {PolyLog}\left (2,\frac {i x}{\sqrt {c}}\right )-\frac {1}{3} b^2 c^{3/2} \text {PolyLog}\left (2,\frac {x}{\sqrt {c}}\right )+\frac {1}{3} b^2 c^{3/2} \text {PolyLog}\left (2,1-\frac {2 \sqrt {c}}{\sqrt {c}+x}\right )-\frac {1}{3} b^2 c^{3/2} \text {PolyLog}\left (2,-1+\frac {2 \sqrt {c}}{\sqrt {c}+x}\right )-\frac {1}{6} b^2 c^{3/2} \text {PolyLog}\left (2,1-\frac {2 \sqrt {c} \left (\sqrt {-c}-x\right )}{\left (\sqrt {-c}-\sqrt {c}\right ) \left (\sqrt {c}+x\right )}\right )-\frac {1}{6} b^2 c^{3/2} \text {PolyLog}\left (2,1-\frac {2 \sqrt {c} \left (\sqrt {-c}+x\right )}{\left (\sqrt {-c}+\sqrt {c}\right ) \left (\sqrt {c}+x\right )}\right )+\frac {1}{6} i b^2 c^{3/2} \text {PolyLog}\left (2,1-\frac {(1-i) \left (\sqrt {c}+x\right )}{\sqrt {c}-i x}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 1.56, antiderivative size = 1172, normalized size of antiderivative = 1.00, number of steps
used = 80, number of rules used = 34, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 2.125, Rules used = {6045, 6042,
2507, 2521, 2498, 269, 213, 2520, 12, 266, 6820, 6135, 6079, 2497, 2505, 199, 327, 6874, 209, 30,
2637, 6139, 6031, 6057, 2449, 2352, 2526, 210, 5048, 4940, 2438, 4966, 5044, 4988}
\begin {gather*} \frac {1}{12} \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )^2 x^3+\frac {1}{12} b^2 \log ^2\left (\frac {c}{x^2}+1\right ) x^3+\frac {1}{3} a b \log \left (\frac {c}{x^2}+1\right ) x^3-\frac {1}{6} b^2 \log \left (1-\frac {c}{x^2}\right ) \log \left (\frac {c}{x^2}+1\right ) x^3+\frac {4}{3} a b c x-\frac {2}{3} b^2 c \log \left (1-\frac {c}{x^2}\right ) x+\frac {2}{3} b^2 c \log \left (\frac {c}{x^2}+1\right ) x+\frac {1}{3} i b^2 c^{3/2} \text {ArcTan}\left (\frac {x}{\sqrt {c}}\right )^2+\frac {1}{3} b^2 c^{3/2} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right )^2+\frac {4}{3} b^2 c^{3/2} \text {ArcTan}\left (\frac {x}{\sqrt {c}}\right )-\frac {2}{3} a b c^{3/2} \text {ArcTan}\left (\frac {x}{\sqrt {c}}\right )-\frac {4}{3} b^2 c^{3/2} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right )-\frac {2}{3} b^2 c^{3/2} \text {ArcTan}\left (\frac {x}{\sqrt {c}}\right ) \log \left (2-\frac {2 \sqrt {c}}{\sqrt {c}-i x}\right )+\frac {1}{3} b^2 c^{3/2} \text {ArcTan}\left (\frac {x}{\sqrt {c}}\right ) \log \left (1-\frac {c}{x^2}\right )-\frac {1}{3} b c^{3/2} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )-\frac {1}{3} b^2 c^{3/2} \text {ArcTan}\left (\frac {x}{\sqrt {c}}\right ) \log \left (\frac {c}{x^2}+1\right )-\frac {1}{3} b^2 c^{3/2} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (\frac {c}{x^2}+1\right )+\frac {2}{3} b^2 c^{3/2} \text {ArcTan}\left (\frac {x}{\sqrt {c}}\right ) \log \left (\frac {2 \sqrt {c}}{\sqrt {c}-i x}\right )-\frac {1}{3} b^2 c^{3/2} \text {ArcTan}\left (\frac {x}{\sqrt {c}}\right ) \log \left (\frac {(1+i) \left (\sqrt {c}-x\right )}{\sqrt {c}-i x}\right )-\frac {2}{3} b^2 c^{3/2} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (\frac {2 \sqrt {c}}{x+\sqrt {c}}\right )+\frac {1}{3} b^2 c^{3/2} \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}{3} b^2 c^{3/2} \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}{3} b^2 c^{3/2} \text {ArcTan}\left (\frac {x}{\sqrt {c}}\right ) \log \left (\frac {(1-i) \left (x+\sqrt {c}\right )}{\sqrt {c}-i x}\right )+\frac {2}{3} b^2 c^{3/2} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (2-\frac {2 \sqrt {c}}{x+\sqrt {c}}\right )-\frac {1}{3} i b^2 c^{3/2} \text {Li}_2\left (1-\frac {2 \sqrt {c}}{\sqrt {c}-i x}\right )+\frac {1}{3} i b^2 c^{3/2} \text {Li}_2\left (\frac {2 \sqrt {c}}{\sqrt {c}-i x}-1\right )+\frac {1}{6} i b^2 c^{3/2} \text {Li}_2\left (1-\frac {(1+i) \left (\sqrt {c}-x\right )}{\sqrt {c}-i x}\right )+\frac {1}{3} b^2 c^{3/2} \text {Li}_2\left (-\frac {x}{\sqrt {c}}\right )+\frac {1}{3} i b^2 c^{3/2} \text {Li}_2\left (-\frac {i x}{\sqrt {c}}\right )-\frac {1}{3} i b^2 c^{3/2} \text {Li}_2\left (\frac {i x}{\sqrt {c}}\right )-\frac {1}{3} b^2 c^{3/2} \text {Li}_2\left (\frac {x}{\sqrt {c}}\right )+\frac {1}{3} b^2 c^{3/2} \text {Li}_2\left (1-\frac {2 \sqrt {c}}{x+\sqrt {c}}\right )-\frac {1}{3} b^2 c^{3/2} \text {Li}_2\left (\frac {2 \sqrt {c}}{x+\sqrt {c}}-1\right )-\frac {1}{6} b^2 c^{3/2} \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}{6} b^2 c^{3/2} \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}{6} i b^2 c^{3/2} \text {Li}_2\left (1-\frac {(1-i) \left (x+\sqrt {c}\right )}{\sqrt {c}-i x}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 30
Rule 199
Rule 209
Rule 210
Rule 213
Rule 266
Rule 269
Rule 327
Rule 2352
Rule 2438
Rule 2449
Rule 2497
Rule 2498
Rule 2505
Rule 2507
Rule 2520
Rule 2521
Rule 2526
Rule 2637
Rule 4940
Rule 4966
Rule 4988
Rule 5044
Rule 5048
Rule 6031
Rule 6042
Rule 6045
Rule 6057
Rule 6079
Rule 6135
Rule 6139
Rule 6820
Rule 6874
Rubi steps
\begin {align*} \int x^2 \left (a+b \tanh ^{-1}\left (\frac {c}{x^2}\right )\right )^2 \, dx &=\int \left (\frac {1}{4} x^2 \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )^2-\frac {1}{2} b x^2 \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^2 \log ^2\left (1+\frac {c}{x^2}\right )\right ) \, dx\\ &=\frac {1}{4} \int x^2 \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )^2 \, dx-\frac {1}{2} b \int x^2 \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^2 \log ^2\left (1+\frac {c}{x^2}\right ) \, dx\\ &=\frac {1}{12} x^3 \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )^2+\frac {1}{12} b^2 x^3 \log ^2\left (1+\frac {c}{x^2}\right )-\frac {1}{2} b \int \left (-2 a x^2 \log \left (1+\frac {c}{x^2}\right )+b x^2 \log \left (1-\frac {c}{x^2}\right ) \log \left (1+\frac {c}{x^2}\right )\right ) \, dx+\frac {1}{3} (b c) \int \frac {2 a-b \log \left (1-\frac {c}{x^2}\right )}{1-\frac {c}{x^2}} \, dx+\frac {1}{3} \left (b^2 c\right ) \int \frac {\log \left (1+\frac {c}{x^2}\right )}{1+\frac {c}{x^2}} \, dx\\ &=\frac {1}{12} x^3 \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )^2+\frac {1}{12} b^2 x^3 \log ^2\left (1+\frac {c}{x^2}\right )+(a b) \int x^2 \log \left (1+\frac {c}{x^2}\right ) \, dx-\frac {1}{2} b^2 \int x^2 \log \left (1-\frac {c}{x^2}\right ) \log \left (1+\frac {c}{x^2}\right ) \, dx+\frac {1}{3} (b c) \int \left (2 a-b \log \left (1-\frac {c}{x^2}\right )+\frac {c \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )}{-c+x^2}\right ) \, dx+\frac {1}{3} \left (b^2 c\right ) \int \left (\log \left (1+\frac {c}{x^2}\right )-\frac {c \log \left (1+\frac {c}{x^2}\right )}{c+x^2}\right ) \, dx\\ &=\frac {2}{3} a b c x+\frac {1}{12} x^3 \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )^2+\frac {1}{3} a b x^3 \log \left (1+\frac {c}{x^2}\right )-\frac {1}{6} b^2 x^3 \log \left (1-\frac {c}{x^2}\right ) \log \left (1+\frac {c}{x^2}\right )+\frac {1}{12} b^2 x^3 \log ^2\left (1+\frac {c}{x^2}\right )+\frac {1}{2} b^2 \int \frac {2 c x^2 \log \left (1-\frac {c}{x^2}\right )}{3 \left (-c-x^2\right )} \, dx+\frac {1}{2} b^2 \int \frac {2 c \log \left (1+\frac {c}{x^2}\right )}{3 \left (1-\frac {c}{x^2}\right )} \, dx+\frac {1}{3} (2 a b c) \int \frac {1}{1+\frac {c}{x^2}} \, dx-\frac {1}{3} \left (b^2 c\right ) \int \log \left (1-\frac {c}{x^2}\right ) \, dx+\frac {1}{3} \left (b^2 c\right ) \int \log \left (1+\frac {c}{x^2}\right ) \, dx+\frac {1}{3} \left (b c^2\right ) \int \frac {2 a-b \log \left (1-\frac {c}{x^2}\right )}{-c+x^2} \, dx-\frac {1}{3} \left (b^2 c^2\right ) \int \frac {\log \left (1+\frac {c}{x^2}\right )}{c+x^2} \, dx\\ &=\frac {2}{3} a b c x-\frac {1}{3} b^2 c x \log \left (1-\frac {c}{x^2}\right )-\frac {1}{3} b c^{3/2} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )+\frac {1}{12} x^3 \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )^2+\frac {1}{3} b^2 c x \log \left (1+\frac {c}{x^2}\right )+\frac {1}{3} a b x^3 \log \left (1+\frac {c}{x^2}\right )-\frac {1}{3} b^2 c^{3/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (1+\frac {c}{x^2}\right )-\frac {1}{6} b^2 x^3 \log \left (1-\frac {c}{x^2}\right ) \log \left (1+\frac {c}{x^2}\right )+\frac {1}{12} b^2 x^3 \log ^2\left (1+\frac {c}{x^2}\right )+\frac {1}{3} (2 a b c) \int \frac {x^2}{c+x^2} \, dx+\frac {1}{3} \left (b^2 c\right ) \int \frac {x^2 \log \left (1-\frac {c}{x^2}\right )}{-c-x^2} \, dx+\frac {1}{3} \left (b^2 c\right ) \int \frac {\log \left (1+\frac {c}{x^2}\right )}{1-\frac {c}{x^2}} \, dx+\frac {1}{3} \left (2 b^2 c^2\right ) \int \frac {1}{\left (1-\frac {c}{x^2}\right ) x^2} \, dx+\frac {1}{3} \left (2 b^2 c^2\right ) \int \frac {1}{\left (1+\frac {c}{x^2}\right ) x^2} \, dx-\frac {1}{3} \left (2 b^2 c^3\right ) \int \frac {\tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )}{\sqrt {c} \left (1+\frac {c}{x^2}\right ) x^3} \, dx-\frac {1}{3} \left (2 b^2 c^3\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}{3} a b c x-\frac {1}{3} b^2 c x \log \left (1-\frac {c}{x^2}\right )-\frac {1}{3} b c^{3/2} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )+\frac {1}{12} x^3 \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )^2+\frac {1}{3} b^2 c x \log \left (1+\frac {c}{x^2}\right )+\frac {1}{3} a b x^3 \log \left (1+\frac {c}{x^2}\right )-\frac {1}{3} b^2 c^{3/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (1+\frac {c}{x^2}\right )-\frac {1}{6} b^2 x^3 \log \left (1-\frac {c}{x^2}\right ) \log \left (1+\frac {c}{x^2}\right )+\frac {1}{12} b^2 x^3 \log ^2\left (1+\frac {c}{x^2}\right )+\frac {1}{3} \left (b^2 c\right ) \int \left (-\log \left (1-\frac {c}{x^2}\right )-\frac {c \log \left (1-\frac {c}{x^2}\right )}{-c-x^2}\right ) \, dx+\frac {1}{3} \left (b^2 c\right ) \int \left (\log \left (1+\frac {c}{x^2}\right )+\frac {c \log \left (1+\frac {c}{x^2}\right )}{-c+x^2}\right ) \, dx-\frac {1}{3} \left (2 a b c^2\right ) \int \frac {1}{c+x^2} \, dx+\frac {1}{3} \left (2 b^2 c^2\right ) \int \frac {1}{-c+x^2} \, dx+\frac {1}{3} \left (2 b^2 c^2\right ) \int \frac {1}{c+x^2} \, dx-\frac {1}{3} \left (2 b^2 c^{5/2}\right ) \int \frac {\tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )}{\left (1+\frac {c}{x^2}\right ) x^3} \, dx-\frac {1}{3} \left (2 b^2 c^{5/2}\right ) \int \frac {\tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right )}{\left (1-\frac {c}{x^2}\right ) x^3} \, dx\\ &=\frac {4}{3} a b c x-\frac {2}{3} a b c^{3/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )+\frac {2}{3} b^2 c^{3/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )-\frac {2}{3} b^2 c^{3/2} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right )-\frac {1}{3} b^2 c x \log \left (1-\frac {c}{x^2}\right )-\frac {1}{3} b c^{3/2} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )+\frac {1}{12} x^3 \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )^2+\frac {1}{3} b^2 c x \log \left (1+\frac {c}{x^2}\right )+\frac {1}{3} a b x^3 \log \left (1+\frac {c}{x^2}\right )-\frac {1}{3} b^2 c^{3/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (1+\frac {c}{x^2}\right )-\frac {1}{6} b^2 x^3 \log \left (1-\frac {c}{x^2}\right ) \log \left (1+\frac {c}{x^2}\right )+\frac {1}{12} b^2 x^3 \log ^2\left (1+\frac {c}{x^2}\right )-\frac {1}{3} \left (b^2 c\right ) \int \log \left (1-\frac {c}{x^2}\right ) \, dx+\frac {1}{3} \left (b^2 c\right ) \int \log \left (1+\frac {c}{x^2}\right ) \, dx-\frac {1}{3} \left (b^2 c^2\right ) \int \frac {\log \left (1-\frac {c}{x^2}\right )}{-c-x^2} \, dx+\frac {1}{3} \left (b^2 c^2\right ) \int \frac {\log \left (1+\frac {c}{x^2}\right )}{-c+x^2} \, dx-\frac {1}{3} \left (2 b^2 c^{5/2}\right ) \int \frac {\tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )}{x \left (c+x^2\right )} \, dx-\frac {1}{3} \left (2 b^2 c^{5/2}\right ) \int \frac {\tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right )}{x \left (-c+x^2\right )} \, dx\\ &=\frac {4}{3} a b c x-\frac {2}{3} a b c^{3/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )+\frac {2}{3} b^2 c^{3/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )+\frac {1}{3} i b^2 c^{3/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )^2-\frac {2}{3} b^2 c^{3/2} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right )+\frac {1}{3} b^2 c^{3/2} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right )^2-\frac {2}{3} b^2 c x \log \left (1-\frac {c}{x^2}\right )+\frac {1}{3} b^2 c^{3/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (1-\frac {c}{x^2}\right )-\frac {1}{3} b c^{3/2} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )+\frac {1}{12} x^3 \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )^2+\frac {2}{3} b^2 c x \log \left (1+\frac {c}{x^2}\right )+\frac {1}{3} a b x^3 \log \left (1+\frac {c}{x^2}\right )-\frac {1}{3} b^2 c^{3/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (1+\frac {c}{x^2}\right )-\frac {1}{3} b^2 c^{3/2} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (1+\frac {c}{x^2}\right )-\frac {1}{6} b^2 x^3 \log \left (1-\frac {c}{x^2}\right ) \log \left (1+\frac {c}{x^2}\right )+\frac {1}{12} b^2 x^3 \log ^2\left (1+\frac {c}{x^2}\right )-\frac {1}{3} \left (2 i b^2 c^{3/2}\right ) \int \frac {\tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )}{x \left (i+\frac {x}{\sqrt {c}}\right )} \, dx+\frac {1}{3} \left (2 b^2 c^{3/2}\right ) \int \frac {\tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right )}{x \left (1+\frac {x}{\sqrt {c}}\right )} \, dx+\frac {1}{3} \left (2 b^2 c^2\right ) \int \frac {1}{\left (1-\frac {c}{x^2}\right ) x^2} \, dx+\frac {1}{3} \left (2 b^2 c^2\right ) \int \frac {1}{\left (1+\frac {c}{x^2}\right ) x^2} \, dx-\frac {1}{3} \left (2 b^2 c^3\right ) \int \frac {\tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )}{\sqrt {c} \left (1-\frac {c}{x^2}\right ) x^3} \, dx-\frac {1}{3} \left (2 b^2 c^3\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}{3} a b c x-\frac {2}{3} a b c^{3/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )+\frac {2}{3} b^2 c^{3/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )+\frac {1}{3} i b^2 c^{3/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )^2-\frac {2}{3} b^2 c^{3/2} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right )+\frac {1}{3} b^2 c^{3/2} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right )^2-\frac {2}{3} b^2 c^{3/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (2-\frac {2 \sqrt {c}}{\sqrt {c}-i x}\right )-\frac {2}{3} b^2 c x \log \left (1-\frac {c}{x^2}\right )+\frac {1}{3} b^2 c^{3/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (1-\frac {c}{x^2}\right )-\frac {1}{3} b c^{3/2} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )+\frac {1}{12} x^3 \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )^2+\frac {2}{3} b^2 c x \log \left (1+\frac {c}{x^2}\right )+\frac {1}{3} a b x^3 \log \left (1+\frac {c}{x^2}\right )-\frac {1}{3} b^2 c^{3/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (1+\frac {c}{x^2}\right )-\frac {1}{3} b^2 c^{3/2} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (1+\frac {c}{x^2}\right )-\frac {1}{6} b^2 x^3 \log \left (1-\frac {c}{x^2}\right ) \log \left (1+\frac {c}{x^2}\right )+\frac {1}{12} b^2 x^3 \log ^2\left (1+\frac {c}{x^2}\right )+\frac {2}{3} b^2 c^{3/2} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (2-\frac {2 \sqrt {c}}{\sqrt {c}+x}\right )+\frac {1}{3} \left (2 b^2 c\right ) \int \frac {\log \left (2-\frac {2}{1-\frac {i x}{\sqrt {c}}}\right )}{1+\frac {x^2}{c}} \, dx-\frac {1}{3} \left (2 b^2 c\right ) \int \frac {\log \left (2-\frac {2}{1+\frac {x}{\sqrt {c}}}\right )}{1-\frac {x^2}{c}} \, dx+\frac {1}{3} \left (2 b^2 c^2\right ) \int \frac {1}{-c+x^2} \, dx+\frac {1}{3} \left (2 b^2 c^2\right ) \int \frac {1}{c+x^2} \, dx-\frac {1}{3} \left (2 b^2 c^{5/2}\right ) \int \frac {\tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )}{\left (1-\frac {c}{x^2}\right ) x^3} \, dx-\frac {1}{3} \left (2 b^2 c^{5/2}\right ) \int \frac {\tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right )}{\left (1+\frac {c}{x^2}\right ) x^3} \, dx\\ &=\frac {4}{3} a b c x-\frac {2}{3} a b c^{3/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )+\frac {4}{3} b^2 c^{3/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )+\frac {1}{3} i b^2 c^{3/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )^2-\frac {4}{3} b^2 c^{3/2} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right )+\frac {1}{3} b^2 c^{3/2} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right )^2-\frac {2}{3} b^2 c^{3/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (2-\frac {2 \sqrt {c}}{\sqrt {c}-i x}\right )-\frac {2}{3} b^2 c x \log \left (1-\frac {c}{x^2}\right )+\frac {1}{3} b^2 c^{3/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (1-\frac {c}{x^2}\right )-\frac {1}{3} b c^{3/2} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )+\frac {1}{12} x^3 \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )^2+\frac {2}{3} b^2 c x \log \left (1+\frac {c}{x^2}\right )+\frac {1}{3} a b x^3 \log \left (1+\frac {c}{x^2}\right )-\frac {1}{3} b^2 c^{3/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (1+\frac {c}{x^2}\right )-\frac {1}{3} b^2 c^{3/2} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (1+\frac {c}{x^2}\right )-\frac {1}{6} b^2 x^3 \log \left (1-\frac {c}{x^2}\right ) \log \left (1+\frac {c}{x^2}\right )+\frac {1}{12} b^2 x^3 \log ^2\left (1+\frac {c}{x^2}\right )+\frac {2}{3} b^2 c^{3/2} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (2-\frac {2 \sqrt {c}}{\sqrt {c}+x}\right )+\frac {1}{3} i b^2 c^{3/2} \text {Li}_2\left (-1+\frac {2 \sqrt {c}}{\sqrt {c}-i x}\right )-\frac {1}{3} b^2 c^{3/2} \text {Li}_2\left (-1+\frac {2 \sqrt {c}}{\sqrt {c}+x}\right )-\frac {1}{3} \left (2 b^2 c^{5/2}\right ) \int \frac {\tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )}{x \left (-c+x^2\right )} \, dx-\frac {1}{3} \left (2 b^2 c^{5/2}\right ) \int \frac {\tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right )}{x \left (c+x^2\right )} \, dx\\ &=\frac {4}{3} a b c x-\frac {2}{3} a b c^{3/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )+\frac {4}{3} b^2 c^{3/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )+\frac {1}{3} i b^2 c^{3/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )^2-\frac {4}{3} b^2 c^{3/2} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right )+\frac {1}{3} b^2 c^{3/2} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right )^2-\frac {2}{3} b^2 c^{3/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (2-\frac {2 \sqrt {c}}{\sqrt {c}-i x}\right )-\frac {2}{3} b^2 c x \log \left (1-\frac {c}{x^2}\right )+\frac {1}{3} b^2 c^{3/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (1-\frac {c}{x^2}\right )-\frac {1}{3} b c^{3/2} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )+\frac {1}{12} x^3 \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )^2+\frac {2}{3} b^2 c x \log \left (1+\frac {c}{x^2}\right )+\frac {1}{3} a b x^3 \log \left (1+\frac {c}{x^2}\right )-\frac {1}{3} b^2 c^{3/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (1+\frac {c}{x^2}\right )-\frac {1}{3} b^2 c^{3/2} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (1+\frac {c}{x^2}\right )-\frac {1}{6} b^2 x^3 \log \left (1-\frac {c}{x^2}\right ) \log \left (1+\frac {c}{x^2}\right )+\frac {1}{12} b^2 x^3 \log ^2\left (1+\frac {c}{x^2}\right )+\frac {2}{3} b^2 c^{3/2} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (2-\frac {2 \sqrt {c}}{\sqrt {c}+x}\right )+\frac {1}{3} i b^2 c^{3/2} \text {Li}_2\left (-1+\frac {2 \sqrt {c}}{\sqrt {c}-i x}\right )-\frac {1}{3} b^2 c^{3/2} \text {Li}_2\left (-1+\frac {2 \sqrt {c}}{\sqrt {c}+x}\right )-\frac {1}{3} \left (2 b^2 c^{5/2}\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}{3} \left (2 b^2 c^{5/2}\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 {4}{3} a b c x-\frac {2}{3} a b c^{3/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )+\frac {4}{3} b^2 c^{3/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )+\frac {1}{3} i b^2 c^{3/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )^2-\frac {4}{3} b^2 c^{3/2} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right )+\frac {1}{3} b^2 c^{3/2} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right )^2-\frac {2}{3} b^2 c^{3/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (2-\frac {2 \sqrt {c}}{\sqrt {c}-i x}\right )-\frac {2}{3} b^2 c x \log \left (1-\frac {c}{x^2}\right )+\frac {1}{3} b^2 c^{3/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (1-\frac {c}{x^2}\right )-\frac {1}{3} b c^{3/2} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )+\frac {1}{12} x^3 \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )^2+\frac {2}{3} b^2 c x \log \left (1+\frac {c}{x^2}\right )+\frac {1}{3} a b x^3 \log \left (1+\frac {c}{x^2}\right )-\frac {1}{3} b^2 c^{3/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (1+\frac {c}{x^2}\right )-\frac {1}{3} b^2 c^{3/2} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (1+\frac {c}{x^2}\right )-\frac {1}{6} b^2 x^3 \log \left (1-\frac {c}{x^2}\right ) \log \left (1+\frac {c}{x^2}\right )+\frac {1}{12} b^2 x^3 \log ^2\left (1+\frac {c}{x^2}\right )+\frac {2}{3} b^2 c^{3/2} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (2-\frac {2 \sqrt {c}}{\sqrt {c}+x}\right )+\frac {1}{3} i b^2 c^{3/2} \text {Li}_2\left (-1+\frac {2 \sqrt {c}}{\sqrt {c}-i x}\right )-\frac {1}{3} b^2 c^{3/2} \text {Li}_2\left (-1+\frac {2 \sqrt {c}}{\sqrt {c}+x}\right )+\frac {1}{3} \left (2 b^2 c^{3/2}\right ) \int \frac {\tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )}{x} \, dx+\frac {1}{3} \left (2 b^2 c^{3/2}\right ) \int \frac {x \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )}{c-x^2} \, dx-\frac {1}{3} \left (2 b^2 c^{3/2}\right ) \int \frac {\tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right )}{x} \, dx+\frac {1}{3} \left (2 b^2 c^{3/2}\right ) \int \frac {x \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right )}{c+x^2} \, dx\\ &=\frac {4}{3} a b c x-\frac {2}{3} a b c^{3/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )+\frac {4}{3} b^2 c^{3/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )+\frac {1}{3} i b^2 c^{3/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )^2-\frac {4}{3} b^2 c^{3/2} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right )+\frac {1}{3} b^2 c^{3/2} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right )^2-\frac {2}{3} b^2 c^{3/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (2-\frac {2 \sqrt {c}}{\sqrt {c}-i x}\right )-\frac {2}{3} b^2 c x \log \left (1-\frac {c}{x^2}\right )+\frac {1}{3} b^2 c^{3/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (1-\frac {c}{x^2}\right )-\frac {1}{3} b c^{3/2} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )+\frac {1}{12} x^3 \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )^2+\frac {2}{3} b^2 c x \log \left (1+\frac {c}{x^2}\right )+\frac {1}{3} a b x^3 \log \left (1+\frac {c}{x^2}\right )-\frac {1}{3} b^2 c^{3/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (1+\frac {c}{x^2}\right )-\frac {1}{3} b^2 c^{3/2} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (1+\frac {c}{x^2}\right )-\frac {1}{6} b^2 x^3 \log \left (1-\frac {c}{x^2}\right ) \log \left (1+\frac {c}{x^2}\right )+\frac {1}{12} b^2 x^3 \log ^2\left (1+\frac {c}{x^2}\right )+\frac {2}{3} b^2 c^{3/2} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (2-\frac {2 \sqrt {c}}{\sqrt {c}+x}\right )+\frac {1}{3} i b^2 c^{3/2} \text {Li}_2\left (-1+\frac {2 \sqrt {c}}{\sqrt {c}-i x}\right )+\frac {1}{3} b^2 c^{3/2} \text {Li}_2\left (-\frac {x}{\sqrt {c}}\right )-\frac {1}{3} b^2 c^{3/2} \text {Li}_2\left (\frac {x}{\sqrt {c}}\right )-\frac {1}{3} b^2 c^{3/2} \text {Li}_2\left (-1+\frac {2 \sqrt {c}}{\sqrt {c}+x}\right )+\frac {1}{3} \left (i b^2 c^{3/2}\right ) \int \frac {\log \left (1-\frac {i x}{\sqrt {c}}\right )}{x} \, dx-\frac {1}{3} \left (i b^2 c^{3/2}\right ) \int \frac {\log \left (1+\frac {i x}{\sqrt {c}}\right )}{x} \, dx+\frac {1}{3} \left (2 b^2 c^{3/2}\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}{3} \left (2 b^2 c^{3/2}\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 {4}{3} a b c x-\frac {2}{3} a b c^{3/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )+\frac {4}{3} b^2 c^{3/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )+\frac {1}{3} i b^2 c^{3/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )^2-\frac {4}{3} b^2 c^{3/2} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right )+\frac {1}{3} b^2 c^{3/2} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right )^2-\frac {2}{3} b^2 c^{3/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (2-\frac {2 \sqrt {c}}{\sqrt {c}-i x}\right )-\frac {2}{3} b^2 c x \log \left (1-\frac {c}{x^2}\right )+\frac {1}{3} b^2 c^{3/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (1-\frac {c}{x^2}\right )-\frac {1}{3} b c^{3/2} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )+\frac {1}{12} x^3 \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )^2+\frac {2}{3} b^2 c x \log \left (1+\frac {c}{x^2}\right )+\frac {1}{3} a b x^3 \log \left (1+\frac {c}{x^2}\right )-\frac {1}{3} b^2 c^{3/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (1+\frac {c}{x^2}\right )-\frac {1}{3} b^2 c^{3/2} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (1+\frac {c}{x^2}\right )-\frac {1}{6} b^2 x^3 \log \left (1-\frac {c}{x^2}\right ) \log \left (1+\frac {c}{x^2}\right )+\frac {1}{12} b^2 x^3 \log ^2\left (1+\frac {c}{x^2}\right )+\frac {2}{3} b^2 c^{3/2} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (2-\frac {2 \sqrt {c}}{\sqrt {c}+x}\right )+\frac {1}{3} i b^2 c^{3/2} \text {Li}_2\left (-1+\frac {2 \sqrt {c}}{\sqrt {c}-i x}\right )+\frac {1}{3} b^2 c^{3/2} \text {Li}_2\left (-\frac {x}{\sqrt {c}}\right )+\frac {1}{3} i b^2 c^{3/2} \text {Li}_2\left (-\frac {i x}{\sqrt {c}}\right )-\frac {1}{3} i b^2 c^{3/2} \text {Li}_2\left (\frac {i x}{\sqrt {c}}\right )-\frac {1}{3} b^2 c^{3/2} \text {Li}_2\left (\frac {x}{\sqrt {c}}\right )-\frac {1}{3} b^2 c^{3/2} \text {Li}_2\left (-1+\frac {2 \sqrt {c}}{\sqrt {c}+x}\right )+\frac {1}{3} \left (b^2 c^{3/2}\right ) \int \frac {\tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )}{\sqrt {c}-x} \, dx-\frac {1}{3} \left (b^2 c^{3/2}\right ) \int \frac {\tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )}{\sqrt {c}+x} \, dx-\frac {1}{3} \left (b^2 c^{3/2}\right ) \int \frac {\tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right )}{\sqrt {-c}-x} \, dx+\frac {1}{3} \left (b^2 c^{3/2}\right ) \int \frac {\tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right )}{\sqrt {-c}+x} \, dx\\ &=\frac {4}{3} a b c x-\frac {2}{3} a b c^{3/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )+\frac {4}{3} b^2 c^{3/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )+\frac {1}{3} i b^2 c^{3/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )^2-\frac {4}{3} b^2 c^{3/2} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right )+\frac {1}{3} b^2 c^{3/2} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right )^2-\frac {2}{3} b^2 c^{3/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (2-\frac {2 \sqrt {c}}{\sqrt {c}-i x}\right )-\frac {2}{3} b^2 c x \log \left (1-\frac {c}{x^2}\right )+\frac {1}{3} b^2 c^{3/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (1-\frac {c}{x^2}\right )-\frac {1}{3} b c^{3/2} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )+\frac {1}{12} x^3 \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )^2+\frac {2}{3} b^2 c x \log \left (1+\frac {c}{x^2}\right )+\frac {1}{3} a b x^3 \log \left (1+\frac {c}{x^2}\right )-\frac {1}{3} b^2 c^{3/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (1+\frac {c}{x^2}\right )-\frac {1}{3} b^2 c^{3/2} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (1+\frac {c}{x^2}\right )-\frac {1}{6} b^2 x^3 \log \left (1-\frac {c}{x^2}\right ) \log \left (1+\frac {c}{x^2}\right )+\frac {1}{12} b^2 x^3 \log ^2\left (1+\frac {c}{x^2}\right )+\frac {2}{3} b^2 c^{3/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (\frac {2 \sqrt {c}}{\sqrt {c}-i x}\right )-\frac {1}{3} b^2 c^{3/2} \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}{3} b^2 c^{3/2} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (\frac {2 \sqrt {c}}{\sqrt {c}+x}\right )+\frac {1}{3} b^2 c^{3/2} \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}{3} b^2 c^{3/2} \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}{3} b^2 c^{3/2} \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}{3} b^2 c^{3/2} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (2-\frac {2 \sqrt {c}}{\sqrt {c}+x}\right )+\frac {1}{3} i b^2 c^{3/2} \text {Li}_2\left (-1+\frac {2 \sqrt {c}}{\sqrt {c}-i x}\right )+\frac {1}{3} b^2 c^{3/2} \text {Li}_2\left (-\frac {x}{\sqrt {c}}\right )+\frac {1}{3} i b^2 c^{3/2} \text {Li}_2\left (-\frac {i x}{\sqrt {c}}\right )-\frac {1}{3} i b^2 c^{3/2} \text {Li}_2\left (\frac {i x}{\sqrt {c}}\right )-\frac {1}{3} b^2 c^{3/2} \text {Li}_2\left (\frac {x}{\sqrt {c}}\right )-\frac {1}{3} b^2 c^{3/2} \text {Li}_2\left (-1+\frac {2 \sqrt {c}}{\sqrt {c}+x}\right )-2 \left (\frac {1}{3} \left (b^2 c\right ) \int \frac {\log \left (\frac {2}{1-\frac {i x}{\sqrt {c}}}\right )}{1+\frac {x^2}{c}} \, dx\right )+\frac {1}{3} \left (b^2 c\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}{3} \left (b^2 c\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}{3} \left (b^2 c\right ) \int \frac {\log \left (\frac {2}{1+\frac {x}{\sqrt {c}}}\right )}{1-\frac {x^2}{c}} \, dx\right )-\frac {1}{3} \left (b^2 c\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}{3} \left (b^2 c\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 {4}{3} a b c x-\frac {2}{3} a b c^{3/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )+\frac {4}{3} b^2 c^{3/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )+\frac {1}{3} i b^2 c^{3/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )^2-\frac {4}{3} b^2 c^{3/2} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right )+\frac {1}{3} b^2 c^{3/2} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right )^2-\frac {2}{3} b^2 c^{3/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (2-\frac {2 \sqrt {c}}{\sqrt {c}-i x}\right )-\frac {2}{3} b^2 c x \log \left (1-\frac {c}{x^2}\right )+\frac {1}{3} b^2 c^{3/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (1-\frac {c}{x^2}\right )-\frac {1}{3} b c^{3/2} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )+\frac {1}{12} x^3 \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )^2+\frac {2}{3} b^2 c x \log \left (1+\frac {c}{x^2}\right )+\frac {1}{3} a b x^3 \log \left (1+\frac {c}{x^2}\right )-\frac {1}{3} b^2 c^{3/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (1+\frac {c}{x^2}\right )-\frac {1}{3} b^2 c^{3/2} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (1+\frac {c}{x^2}\right )-\frac {1}{6} b^2 x^3 \log \left (1-\frac {c}{x^2}\right ) \log \left (1+\frac {c}{x^2}\right )+\frac {1}{12} b^2 x^3 \log ^2\left (1+\frac {c}{x^2}\right )+\frac {2}{3} b^2 c^{3/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (\frac {2 \sqrt {c}}{\sqrt {c}-i x}\right )-\frac {1}{3} b^2 c^{3/2} \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}{3} b^2 c^{3/2} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (\frac {2 \sqrt {c}}{\sqrt {c}+x}\right )+\frac {1}{3} b^2 c^{3/2} \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}{3} b^2 c^{3/2} \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}{3} b^2 c^{3/2} \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}{3} b^2 c^{3/2} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (2-\frac {2 \sqrt {c}}{\sqrt {c}+x}\right )+\frac {1}{3} i b^2 c^{3/2} \text {Li}_2\left (-1+\frac {2 \sqrt {c}}{\sqrt {c}-i x}\right )+\frac {1}{6} i b^2 c^{3/2} \text {Li}_2\left (1-\frac {(1+i) \left (\sqrt {c}-x\right )}{\sqrt {c}-i x}\right )+\frac {1}{3} b^2 c^{3/2} \text {Li}_2\left (-\frac {x}{\sqrt {c}}\right )+\frac {1}{3} i b^2 c^{3/2} \text {Li}_2\left (-\frac {i x}{\sqrt {c}}\right )-\frac {1}{3} i b^2 c^{3/2} \text {Li}_2\left (\frac {i x}{\sqrt {c}}\right )-\frac {1}{3} b^2 c^{3/2} \text {Li}_2\left (\frac {x}{\sqrt {c}}\right )-\frac {1}{3} b^2 c^{3/2} \text {Li}_2\left (-1+\frac {2 \sqrt {c}}{\sqrt {c}+x}\right )-\frac {1}{6} b^2 c^{3/2} \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}{6} b^2 c^{3/2} \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}{6} i b^2 c^{3/2} \text {Li}_2\left (1-\frac {(1-i) \left (\sqrt {c}+x\right )}{\sqrt {c}-i x}\right )-2 \left (\frac {1}{3} \left (i b^2 c^{3/2}\right ) \text {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}{3} \left (b^2 c^{3/2}\right ) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+\frac {x}{\sqrt {c}}}\right )\right )\\ &=\frac {4}{3} a b c x-\frac {2}{3} a b c^{3/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )+\frac {4}{3} b^2 c^{3/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )+\frac {1}{3} i b^2 c^{3/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right )^2-\frac {4}{3} b^2 c^{3/2} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right )+\frac {1}{3} b^2 c^{3/2} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right )^2-\frac {2}{3} b^2 c^{3/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (2-\frac {2 \sqrt {c}}{\sqrt {c}-i x}\right )-\frac {2}{3} b^2 c x \log \left (1-\frac {c}{x^2}\right )+\frac {1}{3} b^2 c^{3/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (1-\frac {c}{x^2}\right )-\frac {1}{3} b c^{3/2} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )+\frac {1}{12} x^3 \left (2 a-b \log \left (1-\frac {c}{x^2}\right )\right )^2+\frac {2}{3} b^2 c x \log \left (1+\frac {c}{x^2}\right )+\frac {1}{3} a b x^3 \log \left (1+\frac {c}{x^2}\right )-\frac {1}{3} b^2 c^{3/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (1+\frac {c}{x^2}\right )-\frac {1}{3} b^2 c^{3/2} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (1+\frac {c}{x^2}\right )-\frac {1}{6} b^2 x^3 \log \left (1-\frac {c}{x^2}\right ) \log \left (1+\frac {c}{x^2}\right )+\frac {1}{12} b^2 x^3 \log ^2\left (1+\frac {c}{x^2}\right )+\frac {2}{3} b^2 c^{3/2} \tan ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (\frac {2 \sqrt {c}}{\sqrt {c}-i x}\right )-\frac {1}{3} b^2 c^{3/2} \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}{3} b^2 c^{3/2} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (\frac {2 \sqrt {c}}{\sqrt {c}+x}\right )+\frac {1}{3} b^2 c^{3/2} \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}{3} b^2 c^{3/2} \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}{3} b^2 c^{3/2} \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}{3} b^2 c^{3/2} \tanh ^{-1}\left (\frac {x}{\sqrt {c}}\right ) \log \left (2-\frac {2 \sqrt {c}}{\sqrt {c}+x}\right )-\frac {1}{3} i b^2 c^{3/2} \text {Li}_2\left (1-\frac {2 \sqrt {c}}{\sqrt {c}-i x}\right )+\frac {1}{3} i b^2 c^{3/2} \text {Li}_2\left (-1+\frac {2 \sqrt {c}}{\sqrt {c}-i x}\right )+\frac {1}{6} i b^2 c^{3/2} \text {Li}_2\left (1-\frac {(1+i) \left (\sqrt {c}-x\right )}{\sqrt {c}-i x}\right )+\frac {1}{3} b^2 c^{3/2} \text {Li}_2\left (-\frac {x}{\sqrt {c}}\right )+\frac {1}{3} i b^2 c^{3/2} \text {Li}_2\left (-\frac {i x}{\sqrt {c}}\right )-\frac {1}{3} i b^2 c^{3/2} \text {Li}_2\left (\frac {i x}{\sqrt {c}}\right )-\frac {1}{3} b^2 c^{3/2} \text {Li}_2\left (\frac {x}{\sqrt {c}}\right )+\frac {1}{3} b^2 c^{3/2} \text {Li}_2\left (1-\frac {2 \sqrt {c}}{\sqrt {c}+x}\right )-\frac {1}{3} b^2 c^{3/2} \text {Li}_2\left (-1+\frac {2 \sqrt {c}}{\sqrt {c}+x}\right )-\frac {1}{6} b^2 c^{3/2} \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}{6} b^2 c^{3/2} \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}{6} i b^2 c^{3/2} \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 = 7.33, size = 0, normalized size = 0.00 \begin {gather*} \int x^2 \left (a+b \tanh ^{-1}\left (\frac {c}{x^2}\right )\right )^2 \, dx \end {gather*}
Verification is not applicable to the result.
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Maple [F]
time = 0.07, size = 0, normalized size = 0.00 \[\int x^{2} \left (a +b \arctanh \left (\frac {c}{x^{2}}\right )\right )^{2}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int x^{2} \left (a + b \operatorname {atanh}{\left (\frac {c}{x^{2}} \right )}\right )^{2}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int x^2\,{\left (a+b\,\mathrm {atanh}\left (\frac {c}{x^2}\right )\right )}^2 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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