Integrand size = 24, antiderivative size = 24 \[ \int \frac {\left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right )}{x} \, dx=a d \log (x)+\frac {1}{2} a e \log \left (-\frac {g x^2}{f}\right ) \log \left (f+g x^2\right )+\frac {1}{2} b d \operatorname {PolyLog}\left (2,-\frac {1}{c x}\right )-\frac {1}{2} b d \operatorname {PolyLog}\left (2,\frac {1}{c x}\right )+\frac {1}{2} a e \operatorname {PolyLog}\left (2,1+\frac {g x^2}{f}\right )+b e \text {Int}\left (\frac {\coth ^{-1}(c x) \log \left (f+g x^2\right )}{x},x\right ) \]
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Not integrable
Time = 0.21 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 0, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {\left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right )}{x} \, dx=\int \frac {\left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right )}{x} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = d \int \frac {a+b \coth ^{-1}(c x)}{x} \, dx+e \int \frac {\left (a+b \coth ^{-1}(c x)\right ) \log \left (f+g x^2\right )}{x} \, dx \\ & = a d \log (x)+\frac {1}{2} b d \operatorname {PolyLog}\left (2,-\frac {1}{c x}\right )-\frac {1}{2} b d \operatorname {PolyLog}\left (2,\frac {1}{c x}\right )+(a e) \int \frac {\log \left (f+g x^2\right )}{x} \, dx+(b e) \int \frac {\coth ^{-1}(c x) \log \left (f+g x^2\right )}{x} \, dx \\ & = a d \log (x)+\frac {1}{2} b d \operatorname {PolyLog}\left (2,-\frac {1}{c x}\right )-\frac {1}{2} b d \operatorname {PolyLog}\left (2,\frac {1}{c x}\right )+\frac {1}{2} (a e) \text {Subst}\left (\int \frac {\log (f+g x)}{x} \, dx,x,x^2\right )+(b e) \int \frac {\coth ^{-1}(c x) \log \left (f+g x^2\right )}{x} \, dx \\ & = a d \log (x)+\frac {1}{2} a e \log \left (-\frac {g x^2}{f}\right ) \log \left (f+g x^2\right )+\frac {1}{2} b d \operatorname {PolyLog}\left (2,-\frac {1}{c x}\right )-\frac {1}{2} b d \operatorname {PolyLog}\left (2,\frac {1}{c x}\right )+(b e) \int \frac {\coth ^{-1}(c x) \log \left (f+g x^2\right )}{x} \, dx-\frac {1}{2} (a e g) \text {Subst}\left (\int \frac {\log \left (-\frac {g x}{f}\right )}{f+g x} \, dx,x,x^2\right ) \\ & = a d \log (x)+\frac {1}{2} a e \log \left (-\frac {g x^2}{f}\right ) \log \left (f+g x^2\right )+\frac {1}{2} b d \operatorname {PolyLog}\left (2,-\frac {1}{c x}\right )-\frac {1}{2} b d \operatorname {PolyLog}\left (2,\frac {1}{c x}\right )+\frac {1}{2} a e \operatorname {PolyLog}\left (2,1+\frac {g x^2}{f}\right )+(b e) \int \frac {\coth ^{-1}(c x) \log \left (f+g x^2\right )}{x} \, dx \\ \end{align*}
Not integrable
Time = 0.15 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {\left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right )}{x} \, dx=\int \frac {\left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right )}{x} \, dx \]
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Not integrable
Time = 0.57 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00
\[\int \frac {\left (a +b \,\operatorname {arccoth}\left (c x \right )\right ) \left (d +e \ln \left (g \,x^{2}+f \right )\right )}{x}d x\]
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Not integrable
Time = 0.26 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.54 \[ \int \frac {\left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right )}{x} \, dx=\int { \frac {{\left (b \operatorname {arcoth}\left (c x\right ) + a\right )} {\left (e \log \left (g x^{2} + f\right ) + d\right )}}{x} \,d x } \]
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Not integrable
Time = 113.02 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92 \[ \int \frac {\left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right )}{x} \, dx=\int \frac {\left (a + b \operatorname {acoth}{\left (c x \right )}\right ) \left (d + e \log {\left (f + g x^{2} \right )}\right )}{x}\, dx \]
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Not integrable
Time = 0.40 (sec) , antiderivative size = 93, normalized size of antiderivative = 3.88 \[ \int \frac {\left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right )}{x} \, dx=\int { \frac {{\left (b \operatorname {arcoth}\left (c x\right ) + a\right )} {\left (e \log \left (g x^{2} + f\right ) + d\right )}}{x} \,d x } \]
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Not integrable
Time = 0.30 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {\left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right )}{x} \, dx=\int { \frac {{\left (b \operatorname {arcoth}\left (c x\right ) + a\right )} {\left (e \log \left (g x^{2} + f\right ) + d\right )}}{x} \,d x } \]
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Not integrable
Time = 5.67 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {\left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (f+g x^2\right )\right )}{x} \, dx=\int \frac {\left (a+b\,\mathrm {acoth}\left (c\,x\right )\right )\,\left (d+e\,\ln \left (g\,x^2+f\right )\right )}{x} \,d x \]
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