Integrand size = 27, antiderivative size = 256 \[ \int \frac {\left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )}{x^6} \, dx=\frac {7 b c^3 e}{60 x^2}+\frac {2 c^2 e \left (a+b \coth ^{-1}(c x)\right )}{15 x^3}+\frac {2 c^4 e \left (a+b \coth ^{-1}(c x)\right )}{5 x}-\frac {c^5 e \left (a+b \coth ^{-1}(c x)\right )^2}{5 b}-\frac {5}{6} b c^5 e \log (x)+\frac {19}{60} b c^5 e \log \left (1-c^2 x^2\right )-\frac {b c \left (d+e \log \left (1-c^2 x^2\right )\right )}{20 x^4}-\frac {b c^3 \left (1-c^2 x^2\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )}{10 x^2}-\frac {\left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )}{5 x^5}+\frac {1}{10} b c^5 \left (d+e \log \left (1-c^2 x^2\right )\right ) \log \left (1-\frac {1}{1-c^2 x^2}\right )-\frac {1}{10} b c^5 e \operatorname {PolyLog}\left (2,\frac {1}{1-c^2 x^2}\right ) \]
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Time = 0.47 (sec) , antiderivative size = 256, normalized size of antiderivative = 1.00, number of steps used = 24, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.593, Rules used = {6229, 2525, 2458, 2389, 2379, 2438, 2351, 31, 2356, 46, 6130, 6038, 272, 36, 29, 6096} \[ \int \frac {\left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )}{x^6} \, dx=-\frac {c^5 e \left (a+b \coth ^{-1}(c x)\right )^2}{5 b}+\frac {2 c^4 e \left (a+b \coth ^{-1}(c x)\right )}{5 x}-\frac {\left (a+b \coth ^{-1}(c x)\right ) \left (e \log \left (1-c^2 x^2\right )+d\right )}{5 x^5}+\frac {2 c^2 e \left (a+b \coth ^{-1}(c x)\right )}{15 x^3}-\frac {5}{6} b c^5 e \log (x)+\frac {7 b c^3 e}{60 x^2}-\frac {b c \left (e \log \left (1-c^2 x^2\right )+d\right )}{20 x^4}+\frac {1}{10} b c^5 \log \left (1-\frac {1}{1-c^2 x^2}\right ) \left (e \log \left (1-c^2 x^2\right )+d\right )-\frac {1}{10} b c^5 e \operatorname {PolyLog}\left (2,\frac {1}{1-c^2 x^2}\right )+\frac {19}{60} b c^5 e \log \left (1-c^2 x^2\right )-\frac {b c^3 \left (1-c^2 x^2\right ) \left (e \log \left (1-c^2 x^2\right )+d\right )}{10 x^2} \]
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Rule 29
Rule 31
Rule 36
Rule 46
Rule 272
Rule 2351
Rule 2356
Rule 2379
Rule 2389
Rule 2438
Rule 2458
Rule 2525
Rule 6038
Rule 6096
Rule 6130
Rule 6229
Rubi steps \begin{align*} \text {integral}& = -\frac {\left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )}{5 x^5}+\frac {1}{5} (b c) \int \frac {d+e \log \left (1-c^2 x^2\right )}{x^5 \left (1-c^2 x^2\right )} \, dx-\frac {1}{5} \left (2 c^2 e\right ) \int \frac {a+b \coth ^{-1}(c x)}{x^4 \left (1-c^2 x^2\right )} \, dx \\ & = -\frac {\left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )}{5 x^5}+\frac {1}{10} (b c) \text {Subst}\left (\int \frac {d+e \log \left (1-c^2 x\right )}{x^3 \left (1-c^2 x\right )} \, dx,x,x^2\right )-\frac {1}{5} \left (2 c^2 e\right ) \int \frac {a+b \coth ^{-1}(c x)}{x^4} \, dx-\frac {1}{5} \left (2 c^4 e\right ) \int \frac {a+b \coth ^{-1}(c x)}{x^2 \left (1-c^2 x^2\right )} \, dx \\ & = \frac {2 c^2 e \left (a+b \coth ^{-1}(c x)\right )}{15 x^3}-\frac {\left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )}{5 x^5}-\frac {b \text {Subst}\left (\int \frac {d+e \log (x)}{x \left (\frac {1}{c^2}-\frac {x}{c^2}\right )^3} \, dx,x,1-c^2 x^2\right )}{10 c}-\frac {1}{15} \left (2 b c^3 e\right ) \int \frac {1}{x^3 \left (1-c^2 x^2\right )} \, dx-\frac {1}{5} \left (2 c^4 e\right ) \int \frac {a+b \coth ^{-1}(c x)}{x^2} \, dx-\frac {1}{5} \left (2 c^6 e\right ) \int \frac {a+b \coth ^{-1}(c x)}{1-c^2 x^2} \, dx \\ & = \frac {2 c^2 e \left (a+b \coth ^{-1}(c x)\right )}{15 x^3}+\frac {2 c^4 e \left (a+b \coth ^{-1}(c x)\right )}{5 x}-\frac {c^5 e \left (a+b \coth ^{-1}(c x)\right )^2}{5 b}-\frac {\left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )}{5 x^5}-\frac {b \text {Subst}\left (\int \frac {d+e \log (x)}{\left (\frac {1}{c^2}-\frac {x}{c^2}\right )^3} \, dx,x,1-c^2 x^2\right )}{10 c}-\frac {1}{10} (b c) \text {Subst}\left (\int \frac {d+e \log (x)}{x \left (\frac {1}{c^2}-\frac {x}{c^2}\right )^2} \, dx,x,1-c^2 x^2\right )-\frac {1}{15} \left (b c^3 e\right ) \text {Subst}\left (\int \frac {1}{x^2 \left (1-c^2 x\right )} \, dx,x,x^2\right )-\frac {1}{5} \left (2 b c^5 e\right ) \int \frac {1}{x \left (1-c^2 x^2\right )} \, dx \\ & = \frac {2 c^2 e \left (a+b \coth ^{-1}(c x)\right )}{15 x^3}+\frac {2 c^4 e \left (a+b \coth ^{-1}(c x)\right )}{5 x}-\frac {c^5 e \left (a+b \coth ^{-1}(c x)\right )^2}{5 b}-\frac {b c \left (d+e \log \left (1-c^2 x^2\right )\right )}{20 x^4}-\frac {\left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )}{5 x^5}-\frac {1}{10} (b c) \text {Subst}\left (\int \frac {d+e \log (x)}{\left (\frac {1}{c^2}-\frac {x}{c^2}\right )^2} \, dx,x,1-c^2 x^2\right )-\frac {1}{10} \left (b c^3\right ) \text {Subst}\left (\int \frac {d+e \log (x)}{x \left (\frac {1}{c^2}-\frac {x}{c^2}\right )} \, dx,x,1-c^2 x^2\right )+\frac {1}{20} (b c e) \text {Subst}\left (\int \frac {1}{x \left (\frac {1}{c^2}-\frac {x}{c^2}\right )^2} \, dx,x,1-c^2 x^2\right )-\frac {1}{15} \left (b c^3 e\right ) \text {Subst}\left (\int \left (\frac {1}{x^2}+\frac {c^2}{x}-\frac {c^4}{-1+c^2 x}\right ) \, dx,x,x^2\right )-\frac {1}{5} \left (b c^5 e\right ) \text {Subst}\left (\int \frac {1}{x \left (1-c^2 x\right )} \, dx,x,x^2\right ) \\ & = \frac {b c^3 e}{15 x^2}+\frac {2 c^2 e \left (a+b \coth ^{-1}(c x)\right )}{15 x^3}+\frac {2 c^4 e \left (a+b \coth ^{-1}(c x)\right )}{5 x}-\frac {c^5 e \left (a+b \coth ^{-1}(c x)\right )^2}{5 b}-\frac {2}{15} b c^5 e \log (x)+\frac {1}{15} b c^5 e \log \left (1-c^2 x^2\right )-\frac {b c \left (d+e \log \left (1-c^2 x^2\right )\right )}{20 x^4}-\frac {b c^3 \left (1-c^2 x^2\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )}{10 x^2}-\frac {\left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )}{5 x^5}+\frac {1}{10} b c^5 \left (d+e \log \left (1-c^2 x^2\right )\right ) \log \left (1-\frac {1}{1-c^2 x^2}\right )+\frac {1}{20} (b c e) \text {Subst}\left (\int \left (\frac {c^4}{(-1+x)^2}-\frac {c^4}{-1+x}+\frac {c^4}{x}\right ) \, dx,x,1-c^2 x^2\right )+\frac {1}{10} \left (b c^3 e\right ) \text {Subst}\left (\int \frac {1}{\frac {1}{c^2}-\frac {x}{c^2}} \, dx,x,1-c^2 x^2\right )-\frac {1}{10} \left (b c^5 e\right ) \text {Subst}\left (\int \frac {\log \left (1-\frac {1}{x}\right )}{x} \, dx,x,1-c^2 x^2\right )-\frac {1}{5} \left (b c^5 e\right ) \text {Subst}\left (\int \frac {1}{x} \, dx,x,x^2\right )-\frac {1}{5} \left (b c^7 e\right ) \text {Subst}\left (\int \frac {1}{1-c^2 x} \, dx,x,x^2\right ) \\ & = \frac {7 b c^3 e}{60 x^2}+\frac {2 c^2 e \left (a+b \coth ^{-1}(c x)\right )}{15 x^3}+\frac {2 c^4 e \left (a+b \coth ^{-1}(c x)\right )}{5 x}-\frac {c^5 e \left (a+b \coth ^{-1}(c x)\right )^2}{5 b}-\frac {5}{6} b c^5 e \log (x)+\frac {19}{60} b c^5 e \log \left (1-c^2 x^2\right )-\frac {b c \left (d+e \log \left (1-c^2 x^2\right )\right )}{20 x^4}-\frac {b c^3 \left (1-c^2 x^2\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )}{10 x^2}-\frac {\left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )}{5 x^5}+\frac {1}{10} b c^5 \left (d+e \log \left (1-c^2 x^2\right )\right ) \log \left (1-\frac {1}{1-c^2 x^2}\right )-\frac {1}{10} b c^5 e \operatorname {PolyLog}\left (2,\frac {1}{1-c^2 x^2}\right ) \\ \end{align*}
\[ \int \frac {\left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )}{x^6} \, dx=\int \frac {\left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )}{x^6} \, dx \]
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\[\int \frac {\left (a +b \,\operatorname {arccoth}\left (c x \right )\right ) \left (d +e \ln \left (-c^{2} x^{2}+1\right )\right )}{x^{6}}d x\]
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\[ \int \frac {\left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )}{x^6} \, dx=\int { \frac {{\left (b \operatorname {arcoth}\left (c x\right ) + a\right )} {\left (e \log \left (-c^{2} x^{2} + 1\right ) + d\right )}}{x^{6}} \,d x } \]
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\[ \int \frac {\left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )}{x^6} \, dx=\int \frac {\left (a + b \operatorname {acoth}{\left (c x \right )}\right ) \left (d + e \log {\left (- c^{2} x^{2} + 1 \right )}\right )}{x^{6}}\, dx \]
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\[ \int \frac {\left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )}{x^6} \, dx=\int { \frac {{\left (b \operatorname {arcoth}\left (c x\right ) + a\right )} {\left (e \log \left (-c^{2} x^{2} + 1\right ) + d\right )}}{x^{6}} \,d x } \]
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\[ \int \frac {\left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )}{x^6} \, dx=\int { \frac {{\left (b \operatorname {arcoth}\left (c x\right ) + a\right )} {\left (e \log \left (-c^{2} x^{2} + 1\right ) + d\right )}}{x^{6}} \,d x } \]
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Timed out. \[ \int \frac {\left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )}{x^6} \, dx=\int \frac {\left (a+b\,\mathrm {acoth}\left (c\,x\right )\right )\,\left (d+e\,\ln \left (1-c^2\,x^2\right )\right )}{x^6} \,d x \]
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