Integrand size = 27, antiderivative size = 105 \[ \int \frac {\left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )}{x^2} \, dx=-\frac {c e \left (a+b \coth ^{-1}(c x)\right )^2}{b}-\frac {\left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )}{x}+\frac {1}{2} b c \left (d+e \log \left (1-c^2 x^2\right )\right ) \log \left (1-\frac {1}{1-c^2 x^2}\right )-\frac {1}{2} b c e \operatorname {PolyLog}\left (2,\frac {1}{1-c^2 x^2}\right ) \]
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Time = 0.17 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {6229, 2525, 2458, 2379, 2438, 6096} \[ \int \frac {\left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )}{x^2} \, dx=-\frac {\left (a+b \coth ^{-1}(c x)\right ) \left (e \log \left (1-c^2 x^2\right )+d\right )}{x}-\frac {c e \left (a+b \coth ^{-1}(c x)\right )^2}{b}+\frac {1}{2} b c \log \left (1-\frac {1}{1-c^2 x^2}\right ) \left (e \log \left (1-c^2 x^2\right )+d\right )-\frac {1}{2} b c e \operatorname {PolyLog}\left (2,\frac {1}{1-c^2 x^2}\right ) \]
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Rule 2379
Rule 2438
Rule 2458
Rule 2525
Rule 6096
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 )}{x}+(b c) \int \frac {d+e \log \left (1-c^2 x^2\right )}{x \left (1-c^2 x^2\right )} \, dx-\left (2 c^2 e\right ) \int \frac {a+b \coth ^{-1}(c x)}{1-c^2 x^2} \, dx \\ & = -\frac {c e \left (a+b \coth ^{-1}(c x)\right )^2}{b}-\frac {\left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )}{x}+\frac {1}{2} (b c) \text {Subst}\left (\int \frac {d+e \log \left (1-c^2 x\right )}{x \left (1-c^2 x\right )} \, dx,x,x^2\right ) \\ & = -\frac {c e \left (a+b \coth ^{-1}(c x)\right )^2}{b}-\frac {\left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )}{x}-\frac {b \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 )}{2 c} \\ & = -\frac {c e \left (a+b \coth ^{-1}(c x)\right )^2}{b}-\frac {\left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )}{x}+\frac {1}{2} b c \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}{2} (b c e) \text {Subst}\left (\int \frac {\log \left (1-\frac {1}{x}\right )}{x} \, dx,x,1-c^2 x^2\right ) \\ & = -\frac {c e \left (a+b \coth ^{-1}(c x)\right )^2}{b}-\frac {\left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )}{x}+\frac {1}{2} b c \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}{2} b c e \operatorname {PolyLog}\left (2,\frac {1}{1-c^2 x^2}\right ) \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(332\) vs. \(2(105)=210\).
Time = 0.11 (sec) , antiderivative size = 332, normalized size of antiderivative = 3.16 \[ \int \frac {\left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )}{x^2} \, dx=-\frac {4 a d+4 b d \coth ^{-1}(c x)+4 b c e x \coth ^{-1}(c x)^2+8 a c e x \text {arctanh}(c x)-4 b c d x \log (x)-b c e x \log ^2\left (-\frac {1}{c}+x\right )-b c e x \log ^2\left (\frac {1}{c}+x\right )-2 b c e x \log \left (\frac {1}{c}+x\right ) \log \left (\frac {1}{2} (1-c x)\right )+4 b c e x \log (x) \log (1-c x)-2 b c e x \log \left (-\frac {1}{c}+x\right ) \log \left (\frac {1}{2} (1+c x)\right )+4 b c e x \log (x) \log (1+c x)+4 a e \log \left (1-c^2 x^2\right )+2 b c d x \log \left (1-c^2 x^2\right )+4 b e \coth ^{-1}(c x) \log \left (1-c^2 x^2\right )-4 b c e x \log (x) \log \left (1-c^2 x^2\right )+2 b c e x \log \left (-\frac {1}{c}+x\right ) \log \left (1-c^2 x^2\right )+2 b c e x \log \left (\frac {1}{c}+x\right ) \log \left (1-c^2 x^2\right )+4 b c e x \operatorname {PolyLog}(2,-c x)+4 b c e x \operatorname {PolyLog}(2,c x)-2 b c e x \operatorname {PolyLog}\left (2,\frac {1}{2}-\frac {c x}{2}\right )-2 b c e x \operatorname {PolyLog}\left (2,\frac {1}{2} (1+c x)\right )}{4 x} \]
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\[\int \frac {\left (a +b \,\operatorname {arccoth}\left (c x \right )\right ) \left (d +e \ln \left (-c^{2} x^{2}+1\right )\right )}{x^{2}}d x\]
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\[ \int \frac {\left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )}{x^2} \, dx=\int { \frac {{\left (b \operatorname {arcoth}\left (c x\right ) + a\right )} {\left (e \log \left (-c^{2} x^{2} + 1\right ) + d\right )}}{x^{2}} \,d x } \]
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\[ \int \frac {\left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )}{x^2} \, dx=\int \frac {\left (a + b \operatorname {acoth}{\left (c x \right )}\right ) \left (d + e \log {\left (- c^{2} x^{2} + 1 \right )}\right )}{x^{2}}\, dx \]
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\[ \int \frac {\left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )}{x^2} \, dx=\int { \frac {{\left (b \operatorname {arcoth}\left (c x\right ) + a\right )} {\left (e \log \left (-c^{2} x^{2} + 1\right ) + d\right )}}{x^{2}} \,d x } \]
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\[ \int \frac {\left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )}{x^2} \, dx=\int { \frac {{\left (b \operatorname {arcoth}\left (c x\right ) + a\right )} {\left (e \log \left (-c^{2} x^{2} + 1\right ) + d\right )}}{x^{2}} \,d x } \]
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Timed out. \[ \int \frac {\left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )}{x^2} \, dx=\int \frac {\left (a+b\,\mathrm {acoth}\left (c\,x\right )\right )\,\left (d+e\,\ln \left (1-c^2\,x^2\right )\right )}{x^2} \,d x \]
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