Optimal. Leaf size=1176 \[ \text{result too large to display} \]
[Out]
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Rubi [A] time = 1.98966, antiderivative size = 1176, normalized size of antiderivative = 1., number of steps used = 77, number of rules used = 24, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 1.5, Rules used = {6099, 2457, 2476, 2455, 325, 206, 207, 2470, 12, 5984, 5918, 2402, 2315, 6742, 203, 30, 2557, 5992, 5920, 2447, 4928, 4856, 4920, 4854} \[ \text{result too large to display} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 6099
Rule 2457
Rule 2476
Rule 2455
Rule 325
Rule 206
Rule 207
Rule 2470
Rule 12
Rule 5984
Rule 5918
Rule 2402
Rule 2315
Rule 6742
Rule 203
Rule 30
Rule 2557
Rule 5992
Rule 5920
Rule 2447
Rule 4928
Rule 4856
Rule 4920
Rule 4854
Rubi steps
\begin{align*} \int \frac{\left (a+b \tanh ^{-1}\left (c x^2\right )\right )^2}{x^6} \, dx &=\int \left (\frac{\left (2 a-b \log \left (1-c x^2\right )\right )^2}{4 x^6}-\frac{b \left (-2 a+b \log \left (1-c x^2\right )\right ) \log \left (1+c x^2\right )}{2 x^6}+\frac{b^2 \log ^2\left (1+c x^2\right )}{4 x^6}\right ) \, dx\\ &=\frac{1}{4} \int \frac{\left (2 a-b \log \left (1-c x^2\right )\right )^2}{x^6} \, 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^6} \, dx+\frac{1}{4} b^2 \int \frac{\log ^2\left (1+c x^2\right )}{x^6} \, dx\\ &=-\frac{\left (2 a-b \log \left (1-c x^2\right )\right )^2}{20 x^5}-\frac{b^2 \log ^2\left (1+c x^2\right )}{20 x^5}-\frac{1}{2} b \int \left (-\frac{2 a \log \left (1+c x^2\right )}{x^6}+\frac{b \log \left (1-c x^2\right ) \log \left (1+c x^2\right )}{x^6}\right ) \, dx+\frac{1}{5} (b c) \int \frac{2 a-b \log \left (1-c x^2\right )}{x^4 \left (1-c x^2\right )} \, dx+\frac{1}{5} \left (b^2 c\right ) \int \frac{\log \left (1+c x^2\right )}{x^4 \left (1+c x^2\right )} \, dx\\ &=-\frac{\left (2 a-b \log \left (1-c x^2\right )\right )^2}{20 x^5}-\frac{b^2 \log ^2\left (1+c x^2\right )}{20 x^5}+(a b) \int \frac{\log \left (1+c x^2\right )}{x^6} \, dx-\frac{1}{2} b^2 \int \frac{\log \left (1-c x^2\right ) \log \left (1+c x^2\right )}{x^6} \, dx+\frac{1}{5} (b c) \int \left (\frac{2 a-b \log \left (1-c x^2\right )}{x^4}+\frac{c \left (2 a-b \log \left (1-c x^2\right )\right )}{x^2}-\frac{c^2 \left (2 a-b \log \left (1-c x^2\right )\right )}{-1+c x^2}\right ) \, dx+\frac{1}{5} \left (b^2 c\right ) \int \left (\frac{\log \left (1+c x^2\right )}{x^4}-\frac{c \log \left (1+c x^2\right )}{x^2}+\frac{c^2 \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}{20 x^5}-\frac{a b \log \left (1+c x^2\right )}{5 x^5}+\frac{b^2 \log \left (1-c x^2\right ) \log \left (1+c x^2\right )}{10 x^5}-\frac{b^2 \log ^2\left (1+c x^2\right )}{20 x^5}+\frac{1}{2} b^2 \int -\frac{2 c \log \left (1-c x^2\right )}{5 x^4 \left (1+c x^2\right )} \, dx+\frac{1}{2} b^2 \int \frac{2 c \log \left (1+c x^2\right )}{5 x^4 \left (1-c x^2\right )} \, dx+\frac{1}{5} (b c) \int \frac{2 a-b \log \left (1-c x^2\right )}{x^4} \, dx+\frac{1}{5} (2 a b c) \int \frac{1}{x^4 \left (1+c x^2\right )} \, dx+\frac{1}{5} \left (b^2 c\right ) \int \frac{\log \left (1+c x^2\right )}{x^4} \, dx+\frac{1}{5} \left (b c^2\right ) \int \frac{2 a-b \log \left (1-c x^2\right )}{x^2} \, dx-\frac{1}{5} \left (b^2 c^2\right ) \int \frac{\log \left (1+c x^2\right )}{x^2} \, dx-\frac{1}{5} \left (b c^3\right ) \int \frac{2 a-b \log \left (1-c x^2\right )}{-1+c x^2} \, dx+\frac{1}{5} \left (b^2 c^3\right ) \int \frac{\log \left (1+c x^2\right )}{1+c x^2} \, dx\\ &=-\frac{2 a b c}{15 x^3}-\frac{b c \left (2 a-b \log \left (1-c x^2\right )\right )}{15 x^3}-\frac{b c^2 \left (2 a-b \log \left (1-c x^2\right )\right )}{5 x}+\frac{1}{5} b c^{5/2} \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}{20 x^5}-\frac{a b \log \left (1+c x^2\right )}{5 x^5}-\frac{b^2 c \log \left (1+c x^2\right )}{15 x^3}+\frac{b^2 c^2 \log \left (1+c x^2\right )}{5 x}+\frac{1}{5} b^2 c^{5/2} \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 )}{10 x^5}-\frac{b^2 \log ^2\left (1+c x^2\right )}{20 x^5}-\frac{1}{5} \left (b^2 c\right ) \int \frac{\log \left (1-c x^2\right )}{x^4 \left (1+c x^2\right )} \, dx+\frac{1}{5} \left (b^2 c\right ) \int \frac{\log \left (1+c x^2\right )}{x^4 \left (1-c x^2\right )} \, dx-\frac{1}{5} \left (2 a b c^2\right ) \int \frac{1}{x^2 \left (1+c x^2\right )} \, dx+\frac{1}{15} \left (2 b^2 c^2\right ) \int \frac{1}{x^2 \left (1-c x^2\right )} \, dx+\frac{1}{15} \left (2 b^2 c^2\right ) \int \frac{1}{x^2 \left (1+c x^2\right )} \, dx+\frac{1}{5} \left (2 b^2 c^3\right ) \int \frac{1}{1-c x^2} \, dx-\frac{1}{5} \left (2 b^2 c^3\right ) \int \frac{1}{1+c x^2} \, dx-\frac{1}{5} \left (2 b^2 c^4\right ) \int \frac{x \tan ^{-1}\left (\sqrt{c} x\right )}{\sqrt{c} \left (1+c x^2\right )} \, dx-\frac{1}{5} \left (2 b^2 c^4\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}{15 x^3}+\frac{2 a b c^2}{5 x}-\frac{4 b^2 c^2}{15 x}-\frac{2}{5} b^2 c^{5/2} \tan ^{-1}\left (\sqrt{c} x\right )+\frac{2}{5} b^2 c^{5/2} \tanh ^{-1}\left (\sqrt{c} x\right )-\frac{b c \left (2 a-b \log \left (1-c x^2\right )\right )}{15 x^3}-\frac{b c^2 \left (2 a-b \log \left (1-c x^2\right )\right )}{5 x}+\frac{1}{5} b c^{5/2} \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}{20 x^5}-\frac{a b \log \left (1+c x^2\right )}{5 x^5}-\frac{b^2 c \log \left (1+c x^2\right )}{15 x^3}+\frac{b^2 c^2 \log \left (1+c x^2\right )}{5 x}+\frac{1}{5} b^2 c^{5/2} \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 )}{10 x^5}-\frac{b^2 \log ^2\left (1+c x^2\right )}{20 x^5}-\frac{1}{5} \left (b^2 c\right ) \int \left (\frac{\log \left (1-c x^2\right )}{x^4}-\frac{c \log \left (1-c x^2\right )}{x^2}+\frac{c^2 \log \left (1-c x^2\right )}{1+c x^2}\right ) \, dx+\frac{1}{5} \left (b^2 c\right ) \int \left (\frac{\log \left (1+c x^2\right )}{x^4}+\frac{c \log \left (1+c x^2\right )}{x^2}-\frac{c^2 \log \left (1+c x^2\right )}{-1+c x^2}\right ) \, dx+\frac{1}{5} \left (2 a b c^3\right ) \int \frac{1}{1+c x^2} \, dx+\frac{1}{15} \left (2 b^2 c^3\right ) \int \frac{1}{1-c x^2} \, dx-\frac{1}{15} \left (2 b^2 c^3\right ) \int \frac{1}{1+c x^2} \, dx-\frac{1}{5} \left (2 b^2 c^{7/2}\right ) \int \frac{x \tan ^{-1}\left (\sqrt{c} x\right )}{1+c x^2} \, dx-\frac{1}{5} \left (2 b^2 c^{7/2}\right ) \int \frac{x \tanh ^{-1}\left (\sqrt{c} x\right )}{1-c x^2} \, dx\\ &=-\frac{2 a b c}{15 x^3}+\frac{2 a b c^2}{5 x}-\frac{4 b^2 c^2}{15 x}+\frac{2}{5} a b c^{5/2} \tan ^{-1}\left (\sqrt{c} x\right )-\frac{8}{15} b^2 c^{5/2} \tan ^{-1}\left (\sqrt{c} x\right )+\frac{1}{5} i b^2 c^{5/2} \tan ^{-1}\left (\sqrt{c} x\right )^2+\frac{8}{15} b^2 c^{5/2} \tanh ^{-1}\left (\sqrt{c} x\right )+\frac{1}{5} b^2 c^{5/2} \tanh ^{-1}\left (\sqrt{c} x\right )^2-\frac{b c \left (2 a-b \log \left (1-c x^2\right )\right )}{15 x^3}-\frac{b c^2 \left (2 a-b \log \left (1-c x^2\right )\right )}{5 x}+\frac{1}{5} b c^{5/2} \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}{20 x^5}-\frac{a b \log \left (1+c x^2\right )}{5 x^5}-\frac{b^2 c \log \left (1+c x^2\right )}{15 x^3}+\frac{b^2 c^2 \log \left (1+c x^2\right )}{5 x}+\frac{1}{5} b^2 c^{5/2} \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 )}{10 x^5}-\frac{b^2 \log ^2\left (1+c x^2\right )}{20 x^5}-\frac{1}{5} \left (b^2 c\right ) \int \frac{\log \left (1-c x^2\right )}{x^4} \, dx+\frac{1}{5} \left (b^2 c\right ) \int \frac{\log \left (1+c x^2\right )}{x^4} \, dx+\frac{1}{5} \left (b^2 c^2\right ) \int \frac{\log \left (1-c x^2\right )}{x^2} \, dx+\frac{1}{5} \left (b^2 c^2\right ) \int \frac{\log \left (1+c x^2\right )}{x^2} \, dx-\frac{1}{5} \left (b^2 c^3\right ) \int \frac{\log \left (1-c x^2\right )}{1+c x^2} \, dx-\frac{1}{5} \left (b^2 c^3\right ) \int \frac{\log \left (1+c x^2\right )}{-1+c x^2} \, dx+\frac{1}{5} \left (2 b^2 c^3\right ) \int \frac{\tan ^{-1}\left (\sqrt{c} x\right )}{i-\sqrt{c} x} \, dx-\frac{1}{5} \left (2 b^2 c^3\right ) \int \frac{\tanh ^{-1}\left (\sqrt{c} x\right )}{1-\sqrt{c} x} \, dx\\ &=-\frac{2 a b c}{15 x^3}+\frac{2 a b c^2}{5 x}-\frac{4 b^2 c^2}{15 x}+\frac{2}{5} a b c^{5/2} \tan ^{-1}\left (\sqrt{c} x\right )-\frac{8}{15} b^2 c^{5/2} \tan ^{-1}\left (\sqrt{c} x\right )+\frac{1}{5} i b^2 c^{5/2} \tan ^{-1}\left (\sqrt{c} x\right )^2+\frac{8}{15} b^2 c^{5/2} \tanh ^{-1}\left (\sqrt{c} x\right )+\frac{1}{5} b^2 c^{5/2} \tanh ^{-1}\left (\sqrt{c} x\right )^2-\frac{2}{5} b^2 c^{5/2} \tanh ^{-1}\left (\sqrt{c} x\right ) \log \left (\frac{2}{1-\sqrt{c} x}\right )+\frac{2}{5} b^2 c^{5/2} \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 )}{15 x^3}-\frac{b^2 c^2 \log \left (1-c x^2\right )}{5 x}-\frac{1}{5} b^2 c^{5/2} \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 )}{15 x^3}-\frac{b c^2 \left (2 a-b \log \left (1-c x^2\right )\right )}{5 x}+\frac{1}{5} b c^{5/2} \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}{20 x^5}-\frac{a b \log \left (1+c x^2\right )}{5 x^5}-\frac{2 b^2 c \log \left (1+c x^2\right )}{15 x^3}+\frac{1}{5} b^2 c^{5/2} \tan ^{-1}\left (\sqrt{c} x\right ) \log \left (1+c x^2\right )+\frac{1}{5} b^2 c^{5/2} \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 )}{10 x^5}-\frac{b^2 \log ^2\left (1+c x^2\right )}{20 x^5}+\frac{1}{15} \left (2 b^2 c^2\right ) \int \frac{1}{x^2 \left (1-c x^2\right )} \, dx+\frac{1}{15} \left (2 b^2 c^2\right ) \int \frac{1}{x^2 \left (1+c x^2\right )} \, dx-\frac{1}{5} \left (2 b^2 c^3\right ) \int \frac{1}{1-c x^2} \, dx+\frac{1}{5} \left (2 b^2 c^3\right ) \int \frac{1}{1+c x^2} \, dx+\frac{1}{5} \left (2 b^2 c^3\right ) \int \frac{\log \left (\frac{2}{1-\sqrt{c} x}\right )}{1-c x^2} \, dx-\frac{1}{5} \left (2 b^2 c^3\right ) \int \frac{\log \left (\frac{2}{1+i \sqrt{c} x}\right )}{1+c x^2} \, dx-\frac{1}{5} \left (2 b^2 c^4\right ) \int \frac{x \tan ^{-1}\left (\sqrt{c} x\right )}{\sqrt{c} \left (1-c x^2\right )} \, dx-\frac{1}{5} \left (2 b^2 c^4\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}{15 x^3}+\frac{2 a b c^2}{5 x}-\frac{8 b^2 c^2}{15 x}+\frac{2}{5} a b c^{5/2} \tan ^{-1}\left (\sqrt{c} x\right )-\frac{2}{15} b^2 c^{5/2} \tan ^{-1}\left (\sqrt{c} x\right )+\frac{1}{5} i b^2 c^{5/2} \tan ^{-1}\left (\sqrt{c} x\right )^2+\frac{2}{15} b^2 c^{5/2} \tanh ^{-1}\left (\sqrt{c} x\right )+\frac{1}{5} b^2 c^{5/2} \tanh ^{-1}\left (\sqrt{c} x\right )^2-\frac{2}{5} b^2 c^{5/2} \tanh ^{-1}\left (\sqrt{c} x\right ) \log \left (\frac{2}{1-\sqrt{c} x}\right )+\frac{2}{5} b^2 c^{5/2} \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 )}{15 x^3}-\frac{b^2 c^2 \log \left (1-c x^2\right )}{5 x}-\frac{1}{5} b^2 c^{5/2} \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 )}{15 x^3}-\frac{b c^2 \left (2 a-b \log \left (1-c x^2\right )\right )}{5 x}+\frac{1}{5} b c^{5/2} \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}{20 x^5}-\frac{a b \log \left (1+c x^2\right )}{5 x^5}-\frac{2 b^2 c \log \left (1+c x^2\right )}{15 x^3}+\frac{1}{5} b^2 c^{5/2} \tan ^{-1}\left (\sqrt{c} x\right ) \log \left (1+c x^2\right )+\frac{1}{5} b^2 c^{5/2} \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 )}{10 x^5}-\frac{b^2 \log ^2\left (1+c x^2\right )}{20 x^5}+\frac{1}{5} \left (2 i b^2 c^{5/2}\right ) \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1+i \sqrt{c} x}\right )-\frac{1}{5} \left (2 b^2 c^{5/2}\right ) \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1-\sqrt{c} x}\right )+\frac{1}{15} \left (2 b^2 c^3\right ) \int \frac{1}{1-c x^2} \, dx-\frac{1}{15} \left (2 b^2 c^3\right ) \int \frac{1}{1+c x^2} \, dx-\frac{1}{5} \left (2 b^2 c^{7/2}\right ) \int \frac{x \tan ^{-1}\left (\sqrt{c} x\right )}{1-c x^2} \, dx-\frac{1}{5} \left (2 b^2 c^{7/2}\right ) \int \frac{x \tanh ^{-1}\left (\sqrt{c} x\right )}{1+c x^2} \, dx\\ &=-\frac{2 a b c}{15 x^3}+\frac{2 a b c^2}{5 x}-\frac{8 b^2 c^2}{15 x}+\frac{2}{5} a b c^{5/2} \tan ^{-1}\left (\sqrt{c} x\right )-\frac{4}{15} b^2 c^{5/2} \tan ^{-1}\left (\sqrt{c} x\right )+\frac{1}{5} i b^2 c^{5/2} \tan ^{-1}\left (\sqrt{c} x\right )^2+\frac{4}{15} b^2 c^{5/2} \tanh ^{-1}\left (\sqrt{c} x\right )+\frac{1}{5} b^2 c^{5/2} \tanh ^{-1}\left (\sqrt{c} x\right )^2-\frac{2}{5} b^2 c^{5/2} \tanh ^{-1}\left (\sqrt{c} x\right ) \log \left (\frac{2}{1-\sqrt{c} x}\right )+\frac{2}{5} b^2 c^{5/2} \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 )}{15 x^3}-\frac{b^2 c^2 \log \left (1-c x^2\right )}{5 x}-\frac{1}{5} b^2 c^{5/2} \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 )}{15 x^3}-\frac{b c^2 \left (2 a-b \log \left (1-c x^2\right )\right )}{5 x}+\frac{1}{5} b c^{5/2} \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}{20 x^5}-\frac{a b \log \left (1+c x^2\right )}{5 x^5}-\frac{2 b^2 c \log \left (1+c x^2\right )}{15 x^3}+\frac{1}{5} b^2 c^{5/2} \tan ^{-1}\left (\sqrt{c} x\right ) \log \left (1+c x^2\right )+\frac{1}{5} b^2 c^{5/2} \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 )}{10 x^5}-\frac{b^2 \log ^2\left (1+c x^2\right )}{20 x^5}-\frac{1}{5} b^2 c^{5/2} \text{Li}_2\left (1-\frac{2}{1-\sqrt{c} x}\right )+\frac{1}{5} i b^2 c^{5/2} \text{Li}_2\left (1-\frac{2}{1+i \sqrt{c} x}\right )-\frac{1}{5} \left (2 b^2 c^{7/2}\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}{5} \left (2 b^2 c^{7/2}\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}{15 x^3}+\frac{2 a b c^2}{5 x}-\frac{8 b^2 c^2}{15 x}+\frac{2}{5} a b c^{5/2} \tan ^{-1}\left (\sqrt{c} x\right )-\frac{4}{15} b^2 c^{5/2} \tan ^{-1}\left (\sqrt{c} x\right )+\frac{1}{5} i b^2 c^{5/2} \tan ^{-1}\left (\sqrt{c} x\right )^2+\frac{4}{15} b^2 c^{5/2} \tanh ^{-1}\left (\sqrt{c} x\right )+\frac{1}{5} b^2 c^{5/2} \tanh ^{-1}\left (\sqrt{c} x\right )^2-\frac{2}{5} b^2 c^{5/2} \tanh ^{-1}\left (\sqrt{c} x\right ) \log \left (\frac{2}{1-\sqrt{c} x}\right )+\frac{2}{5} b^2 c^{5/2} \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 )}{15 x^3}-\frac{b^2 c^2 \log \left (1-c x^2\right )}{5 x}-\frac{1}{5} b^2 c^{5/2} \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 )}{15 x^3}-\frac{b c^2 \left (2 a-b \log \left (1-c x^2\right )\right )}{5 x}+\frac{1}{5} b c^{5/2} \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}{20 x^5}-\frac{a b \log \left (1+c x^2\right )}{5 x^5}-\frac{2 b^2 c \log \left (1+c x^2\right )}{15 x^3}+\frac{1}{5} b^2 c^{5/2} \tan ^{-1}\left (\sqrt{c} x\right ) \log \left (1+c x^2\right )+\frac{1}{5} b^2 c^{5/2} \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 )}{10 x^5}-\frac{b^2 \log ^2\left (1+c x^2\right )}{20 x^5}-\frac{1}{5} b^2 c^{5/2} \text{Li}_2\left (1-\frac{2}{1-\sqrt{c} x}\right )+\frac{1}{5} i b^2 c^{5/2} \text{Li}_2\left (1-\frac{2}{1+i \sqrt{c} x}\right )-\frac{1}{5} \left (b^2 c^3\right ) \int \frac{\tan ^{-1}\left (\sqrt{c} x\right )}{1-\sqrt{c} x} \, dx+\frac{1}{5} \left (b^2 c^3\right ) \int \frac{\tan ^{-1}\left (\sqrt{c} x\right )}{1+\sqrt{c} x} \, dx-\frac{\left (b^2 c^{7/2}\right ) \int \frac{\tanh ^{-1}\left (\sqrt{c} x\right )}{1-\sqrt{-c} x} \, dx}{5 \sqrt{-c}}+\frac{\left (b^2 c^{7/2}\right ) \int \frac{\tanh ^{-1}\left (\sqrt{c} x\right )}{1+\sqrt{-c} x} \, dx}{5 \sqrt{-c}}\\ &=-\frac{2 a b c}{15 x^3}+\frac{2 a b c^2}{5 x}-\frac{8 b^2 c^2}{15 x}+\frac{2}{5} a b c^{5/2} \tan ^{-1}\left (\sqrt{c} x\right )-\frac{4}{15} b^2 c^{5/2} \tan ^{-1}\left (\sqrt{c} x\right )+\frac{1}{5} i b^2 c^{5/2} \tan ^{-1}\left (\sqrt{c} x\right )^2+\frac{4}{15} b^2 c^{5/2} \tanh ^{-1}\left (\sqrt{c} x\right )+\frac{1}{5} b^2 c^{5/2} \tanh ^{-1}\left (\sqrt{c} x\right )^2-\frac{2}{5} b^2 c^{5/2} \tanh ^{-1}\left (\sqrt{c} x\right ) \log \left (\frac{2}{1-\sqrt{c} x}\right )-\frac{2}{5} b^2 c^{5/2} \tan ^{-1}\left (\sqrt{c} x\right ) \log \left (\frac{2}{1-i \sqrt{c} x}\right )+\frac{1}{5} b^2 c^{5/2} \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}{5} b^2 c^{5/2} \tan ^{-1}\left (\sqrt{c} x\right ) \log \left (\frac{2}{1+i \sqrt{c} x}\right )+\frac{2}{5} b^2 c^{5/2} \tanh ^{-1}\left (\sqrt{c} x\right ) \log \left (\frac{2}{1+\sqrt{c} x}\right )-\frac{1}{5} b^2 c^{5/2} \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}{5} b^2 c^{5/2} \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}{5} b^2 c^{5/2} \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 )}{15 x^3}-\frac{b^2 c^2 \log \left (1-c x^2\right )}{5 x}-\frac{1}{5} b^2 c^{5/2} \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 )}{15 x^3}-\frac{b c^2 \left (2 a-b \log \left (1-c x^2\right )\right )}{5 x}+\frac{1}{5} b c^{5/2} \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}{20 x^5}-\frac{a b \log \left (1+c x^2\right )}{5 x^5}-\frac{2 b^2 c \log \left (1+c x^2\right )}{15 x^3}+\frac{1}{5} b^2 c^{5/2} \tan ^{-1}\left (\sqrt{c} x\right ) \log \left (1+c x^2\right )+\frac{1}{5} b^2 c^{5/2} \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 )}{10 x^5}-\frac{b^2 \log ^2\left (1+c x^2\right )}{20 x^5}-\frac{1}{5} b^2 c^{5/2} \text{Li}_2\left (1-\frac{2}{1-\sqrt{c} x}\right )+\frac{1}{5} i b^2 c^{5/2} \text{Li}_2\left (1-\frac{2}{1+i \sqrt{c} x}\right )+2 \left (\frac{1}{5} \left (b^2 c^3\right ) \int \frac{\log \left (\frac{2}{1-i \sqrt{c} x}\right )}{1+c x^2} \, dx\right )-\frac{1}{5} \left (b^2 c^3\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}{5} \left (b^2 c^3\right ) \int \frac{\log \left (\frac{2}{1+\sqrt{c} x}\right )}{1-c x^2} \, dx\right )+\frac{1}{5} \left (b^2 c^3\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}{5} \left (b^2 c^3\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}{5} \left (b^2 c^3\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}{15 x^3}+\frac{2 a b c^2}{5 x}-\frac{8 b^2 c^2}{15 x}+\frac{2}{5} a b c^{5/2} \tan ^{-1}\left (\sqrt{c} x\right )-\frac{4}{15} b^2 c^{5/2} \tan ^{-1}\left (\sqrt{c} x\right )+\frac{1}{5} i b^2 c^{5/2} \tan ^{-1}\left (\sqrt{c} x\right )^2+\frac{4}{15} b^2 c^{5/2} \tanh ^{-1}\left (\sqrt{c} x\right )+\frac{1}{5} b^2 c^{5/2} \tanh ^{-1}\left (\sqrt{c} x\right )^2-\frac{2}{5} b^2 c^{5/2} \tanh ^{-1}\left (\sqrt{c} x\right ) \log \left (\frac{2}{1-\sqrt{c} x}\right )-\frac{2}{5} b^2 c^{5/2} \tan ^{-1}\left (\sqrt{c} x\right ) \log \left (\frac{2}{1-i \sqrt{c} x}\right )+\frac{1}{5} b^2 c^{5/2} \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}{5} b^2 c^{5/2} \tan ^{-1}\left (\sqrt{c} x\right ) \log \left (\frac{2}{1+i \sqrt{c} x}\right )+\frac{2}{5} b^2 c^{5/2} \tanh ^{-1}\left (\sqrt{c} x\right ) \log \left (\frac{2}{1+\sqrt{c} x}\right )-\frac{1}{5} b^2 c^{5/2} \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}{5} b^2 c^{5/2} \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}{5} b^2 c^{5/2} \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 )}{15 x^3}-\frac{b^2 c^2 \log \left (1-c x^2\right )}{5 x}-\frac{1}{5} b^2 c^{5/2} \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 )}{15 x^3}-\frac{b c^2 \left (2 a-b \log \left (1-c x^2\right )\right )}{5 x}+\frac{1}{5} b c^{5/2} \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}{20 x^5}-\frac{a b \log \left (1+c x^2\right )}{5 x^5}-\frac{2 b^2 c \log \left (1+c x^2\right )}{15 x^3}+\frac{1}{5} b^2 c^{5/2} \tan ^{-1}\left (\sqrt{c} x\right ) \log \left (1+c x^2\right )+\frac{1}{5} b^2 c^{5/2} \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 )}{10 x^5}-\frac{b^2 \log ^2\left (1+c x^2\right )}{20 x^5}-\frac{1}{5} b^2 c^{5/2} \text{Li}_2\left (1-\frac{2}{1-\sqrt{c} x}\right )-\frac{1}{10} i b^2 c^{5/2} \text{Li}_2\left (1-\frac{(1+i) \left (1-\sqrt{c} x\right )}{1-i \sqrt{c} x}\right )+\frac{1}{5} i b^2 c^{5/2} \text{Li}_2\left (1-\frac{2}{1+i \sqrt{c} x}\right )+\frac{1}{10} b^2 c^{5/2} \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}{10} b^2 c^{5/2} \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}{10} i b^2 c^{5/2} \text{Li}_2\left (1-\frac{(1-i) \left (1+\sqrt{c} x\right )}{1-i \sqrt{c} x}\right )+2 \left (\frac{1}{5} \left (i b^2 c^{5/2}\right ) \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1-i \sqrt{c} x}\right )\right )-2 \left (\frac{1}{5} \left (b^2 c^{5/2}\right ) \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1+\sqrt{c} x}\right )\right )\\ &=-\frac{2 a b c}{15 x^3}+\frac{2 a b c^2}{5 x}-\frac{8 b^2 c^2}{15 x}+\frac{2}{5} a b c^{5/2} \tan ^{-1}\left (\sqrt{c} x\right )-\frac{4}{15} b^2 c^{5/2} \tan ^{-1}\left (\sqrt{c} x\right )+\frac{1}{5} i b^2 c^{5/2} \tan ^{-1}\left (\sqrt{c} x\right )^2+\frac{4}{15} b^2 c^{5/2} \tanh ^{-1}\left (\sqrt{c} x\right )+\frac{1}{5} b^2 c^{5/2} \tanh ^{-1}\left (\sqrt{c} x\right )^2-\frac{2}{5} b^2 c^{5/2} \tanh ^{-1}\left (\sqrt{c} x\right ) \log \left (\frac{2}{1-\sqrt{c} x}\right )-\frac{2}{5} b^2 c^{5/2} \tan ^{-1}\left (\sqrt{c} x\right ) \log \left (\frac{2}{1-i \sqrt{c} x}\right )+\frac{1}{5} b^2 c^{5/2} \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}{5} b^2 c^{5/2} \tan ^{-1}\left (\sqrt{c} x\right ) \log \left (\frac{2}{1+i \sqrt{c} x}\right )+\frac{2}{5} b^2 c^{5/2} \tanh ^{-1}\left (\sqrt{c} x\right ) \log \left (\frac{2}{1+\sqrt{c} x}\right )-\frac{1}{5} b^2 c^{5/2} \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}{5} b^2 c^{5/2} \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}{5} b^2 c^{5/2} \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 )}{15 x^3}-\frac{b^2 c^2 \log \left (1-c x^2\right )}{5 x}-\frac{1}{5} b^2 c^{5/2} \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 )}{15 x^3}-\frac{b c^2 \left (2 a-b \log \left (1-c x^2\right )\right )}{5 x}+\frac{1}{5} b c^{5/2} \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}{20 x^5}-\frac{a b \log \left (1+c x^2\right )}{5 x^5}-\frac{2 b^2 c \log \left (1+c x^2\right )}{15 x^3}+\frac{1}{5} b^2 c^{5/2} \tan ^{-1}\left (\sqrt{c} x\right ) \log \left (1+c x^2\right )+\frac{1}{5} b^2 c^{5/2} \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 )}{10 x^5}-\frac{b^2 \log ^2\left (1+c x^2\right )}{20 x^5}-\frac{1}{5} b^2 c^{5/2} \text{Li}_2\left (1-\frac{2}{1-\sqrt{c} x}\right )+\frac{1}{5} i b^2 c^{5/2} \text{Li}_2\left (1-\frac{2}{1-i \sqrt{c} x}\right )-\frac{1}{10} i b^2 c^{5/2} \text{Li}_2\left (1-\frac{(1+i) \left (1-\sqrt{c} x\right )}{1-i \sqrt{c} x}\right )+\frac{1}{5} i b^2 c^{5/2} \text{Li}_2\left (1-\frac{2}{1+i \sqrt{c} x}\right )-\frac{1}{5} b^2 c^{5/2} \text{Li}_2\left (1-\frac{2}{1+\sqrt{c} x}\right )+\frac{1}{10} b^2 c^{5/2} \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}{10} b^2 c^{5/2} \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}{10} i b^2 c^{5/2} \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 = 3.01188, size = 0, normalized size = 0. \[ \int \frac{\left (a+b \tanh ^{-1}\left (c x^2\right )\right )^2}{x^6} \, dx \]
Verification is Not applicable to the result.
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Maple [F] time = 0.167, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( a+b{\it Artanh} \left ( c{x}^{2} \right ) \right ) ^{2}}{{x}^{6}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{b^{2} \operatorname{artanh}\left (c x^{2}\right )^{2} + 2 \, a b \operatorname{artanh}\left (c x^{2}\right ) + a^{2}}{x^{6}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (a + b \operatorname{atanh}{\left (c x^{2} \right )}\right )^{2}}{x^{6}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \operatorname{artanh}\left (c x^{2}\right ) + a\right )}^{2}}{x^{6}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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