Integrand size = 27, antiderivative size = 381 \[ \int \frac {\left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )}{x} \, dx=-\frac {1}{2} b e \log ^2\left (1+\frac {1}{c x}\right ) \log \left (-\frac {1}{c x}\right )+\frac {1}{2} b e \log ^2\left (1-\frac {1}{c x}\right ) \log \left (\frac {1}{c x}\right )+a d \log (x)-b e \log \left (\frac {c+\frac {1}{x}}{c}\right ) \operatorname {PolyLog}\left (2,\frac {c+\frac {1}{x}}{c}\right )+b e \log \left (1-\frac {1}{c x}\right ) \operatorname {PolyLog}\left (2,1-\frac {1}{c x}\right )+\frac {1}{2} b d \operatorname {PolyLog}\left (2,-\frac {1}{c x}\right )+\frac {1}{2} b e \log \left (-c^2 x^2\right ) \operatorname {PolyLog}\left (2,-\frac {1}{c x}\right )-\frac {1}{2} b e \left (\log \left (1-\frac {1}{c x}\right )+\log \left (1+\frac {1}{c x}\right )+\log \left (-c^2 x^2\right )-\log \left (1-c^2 x^2\right )\right ) \operatorname {PolyLog}\left (2,-\frac {1}{c x}\right )-\frac {1}{2} b d \operatorname {PolyLog}\left (2,\frac {1}{c x}\right )-\frac {1}{2} b e \log \left (-c^2 x^2\right ) \operatorname {PolyLog}\left (2,\frac {1}{c x}\right )+\frac {1}{2} b e \left (\log \left (1-\frac {1}{c x}\right )+\log \left (1+\frac {1}{c x}\right )+\log \left (-c^2 x^2\right )-\log \left (1-c^2 x^2\right )\right ) \operatorname {PolyLog}\left (2,\frac {1}{c x}\right )-\frac {1}{2} a e \operatorname {PolyLog}\left (2,c^2 x^2\right )+b e \operatorname {PolyLog}\left (3,\frac {c+\frac {1}{x}}{c}\right )-b e \operatorname {PolyLog}\left (3,1-\frac {1}{c x}\right )+b e \operatorname {PolyLog}\left (3,-\frac {1}{c x}\right )-b e \operatorname {PolyLog}\left (3,\frac {1}{c x}\right ) \]
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Time = 0.35 (sec) , antiderivative size = 381, normalized size of antiderivative = 1.00, number of steps used = 21, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.407, Rules used = {6227, 6032, 6225, 2438, 6223, 2504, 2443, 2481, 2421, 6724, 6217} \[ \int \frac {\left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )}{x} \, dx=-\frac {1}{2} a e \operatorname {PolyLog}\left (2,c^2 x^2\right )+a d \log (x)+\frac {1}{2} b e \operatorname {PolyLog}\left (2,-\frac {1}{c x}\right ) \log \left (-c^2 x^2\right )-\frac {1}{2} b e \operatorname {PolyLog}\left (2,-\frac {1}{c x}\right ) \left (\log \left (-c^2 x^2\right )-\log \left (1-c^2 x^2\right )+\log \left (1-\frac {1}{c x}\right )+\log \left (\frac {1}{c x}+1\right )\right )-\frac {1}{2} b e \operatorname {PolyLog}\left (2,\frac {1}{c x}\right ) \log \left (-c^2 x^2\right )+\frac {1}{2} b e \operatorname {PolyLog}\left (2,\frac {1}{c x}\right ) \left (\log \left (-c^2 x^2\right )-\log \left (1-c^2 x^2\right )+\log \left (1-\frac {1}{c x}\right )+\log \left (\frac {1}{c x}+1\right )\right )+\frac {1}{2} b d \operatorname {PolyLog}\left (2,-\frac {1}{c x}\right )-\frac {1}{2} b d \operatorname {PolyLog}\left (2,\frac {1}{c x}\right )+b e \operatorname {PolyLog}\left (3,\frac {c+\frac {1}{x}}{c}\right )-b e \operatorname {PolyLog}\left (3,1-\frac {1}{c x}\right )+b e \operatorname {PolyLog}\left (3,-\frac {1}{c x}\right )-b e \operatorname {PolyLog}\left (3,\frac {1}{c x}\right )+b e \operatorname {PolyLog}\left (2,1-\frac {1}{c x}\right ) \log \left (1-\frac {1}{c x}\right )-b e \operatorname {PolyLog}\left (2,\frac {c+\frac {1}{x}}{c}\right ) \log \left (\frac {c+\frac {1}{x}}{c}\right )+\frac {1}{2} b e \log \left (\frac {1}{c x}\right ) \log ^2\left (1-\frac {1}{c x}\right )-\frac {1}{2} b e \log ^2\left (\frac {1}{c x}+1\right ) \log \left (-\frac {1}{c x}\right ) \]
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Rule 2421
Rule 2438
Rule 2443
Rule 2481
Rule 2504
Rule 6032
Rule 6217
Rule 6223
Rule 6225
Rule 6227
Rule 6724
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 (1-c^2 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 (1-c^2 x^2\right )}{x} \, dx+(b e) \int \frac {\coth ^{-1}(c x) \log \left (1-c^2 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 \operatorname {PolyLog}\left (2,c^2 x^2\right )-\frac {1}{2} (b e) \int \frac {\log ^2\left (1-\frac {1}{c x}\right )}{x} \, dx+\frac {1}{2} (b e) \int \frac {\log ^2\left (1+\frac {1}{c x}\right )}{x} \, dx+(b e) \int \frac {\coth ^{-1}(c x) \log \left (-c^2 x^2\right )}{x} \, dx+\left (b e \left (-\log \left (1-\frac {1}{c x}\right )-\log \left (1+\frac {1}{c x}\right )-\log \left (-c^2 x^2\right )+\log \left (1-c^2 x^2\right )\right )\right ) \int \frac {\coth ^{-1}(c x)}{x} \, dx \\ & = a d \log (x)+\frac {1}{2} b d \operatorname {PolyLog}\left (2,-\frac {1}{c x}\right )-\frac {1}{2} b e \left (\log \left (1-\frac {1}{c x}\right )+\log \left (1+\frac {1}{c x}\right )+\log \left (-c^2 x^2\right )-\log \left (1-c^2 x^2\right )\right ) \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} b e \left (\log \left (1-\frac {1}{c x}\right )+\log \left (1+\frac {1}{c x}\right )+\log \left (-c^2 x^2\right )-\log \left (1-c^2 x^2\right )\right ) \operatorname {PolyLog}\left (2,\frac {1}{c x}\right )-\frac {1}{2} a e \operatorname {PolyLog}\left (2,c^2 x^2\right )-\frac {1}{2} (b e) \int \frac {\log \left (1-\frac {1}{c x}\right ) \log \left (-c^2 x^2\right )}{x} \, dx+\frac {1}{2} (b e) \int \frac {\log \left (1+\frac {1}{c x}\right ) \log \left (-c^2 x^2\right )}{x} \, dx+\frac {1}{2} (b e) \text {Subst}\left (\int \frac {\log ^2\left (1-\frac {x}{c}\right )}{x} \, dx,x,\frac {1}{x}\right )-\frac {1}{2} (b e) \text {Subst}\left (\int \frac {\log ^2\left (1+\frac {x}{c}\right )}{x} \, dx,x,\frac {1}{x}\right ) \\ & = -\frac {1}{2} b e \log ^2\left (1+\frac {1}{c x}\right ) \log \left (-\frac {1}{c x}\right )+\frac {1}{2} b e \log ^2\left (1-\frac {1}{c x}\right ) \log \left (\frac {1}{c x}\right )+a d \log (x)+\frac {1}{2} b d \operatorname {PolyLog}\left (2,-\frac {1}{c x}\right )+\frac {1}{2} b e \log \left (-c^2 x^2\right ) \operatorname {PolyLog}\left (2,-\frac {1}{c x}\right )-\frac {1}{2} b e \left (\log \left (1-\frac {1}{c x}\right )+\log \left (1+\frac {1}{c x}\right )+\log \left (-c^2 x^2\right )-\log \left (1-c^2 x^2\right )\right ) \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} b e \log \left (-c^2 x^2\right ) \operatorname {PolyLog}\left (2,\frac {1}{c x}\right )+\frac {1}{2} b e \left (\log \left (1-\frac {1}{c x}\right )+\log \left (1+\frac {1}{c x}\right )+\log \left (-c^2 x^2\right )-\log \left (1-c^2 x^2\right )\right ) \operatorname {PolyLog}\left (2,\frac {1}{c x}\right )-\frac {1}{2} a e \operatorname {PolyLog}\left (2,c^2 x^2\right )-(b e) \int \frac {\operatorname {PolyLog}\left (2,-\frac {1}{c x}\right )}{x} \, dx+(b e) \int \frac {\operatorname {PolyLog}\left (2,\frac {1}{c x}\right )}{x} \, dx+\frac {(b e) \text {Subst}\left (\int \frac {\log \left (\frac {x}{c}\right ) \log \left (1-\frac {x}{c}\right )}{1-\frac {x}{c}} \, dx,x,\frac {1}{x}\right )}{c}+\frac {(b e) \text {Subst}\left (\int \frac {\log \left (-\frac {x}{c}\right ) \log \left (1+\frac {x}{c}\right )}{1+\frac {x}{c}} \, dx,x,\frac {1}{x}\right )}{c} \\ & = -\frac {1}{2} b e \log ^2\left (1+\frac {1}{c x}\right ) \log \left (-\frac {1}{c x}\right )+\frac {1}{2} b e \log ^2\left (1-\frac {1}{c x}\right ) \log \left (\frac {1}{c x}\right )+a d \log (x)+\frac {1}{2} b d \operatorname {PolyLog}\left (2,-\frac {1}{c x}\right )+\frac {1}{2} b e \log \left (-c^2 x^2\right ) \operatorname {PolyLog}\left (2,-\frac {1}{c x}\right )-\frac {1}{2} b e \left (\log \left (1-\frac {1}{c x}\right )+\log \left (1+\frac {1}{c x}\right )+\log \left (-c^2 x^2\right )-\log \left (1-c^2 x^2\right )\right ) \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} b e \log \left (-c^2 x^2\right ) \operatorname {PolyLog}\left (2,\frac {1}{c x}\right )+\frac {1}{2} b e \left (\log \left (1-\frac {1}{c x}\right )+\log \left (1+\frac {1}{c x}\right )+\log \left (-c^2 x^2\right )-\log \left (1-c^2 x^2\right )\right ) \operatorname {PolyLog}\left (2,\frac {1}{c x}\right )-\frac {1}{2} a e \operatorname {PolyLog}\left (2,c^2 x^2\right )+b e \operatorname {PolyLog}\left (3,-\frac {1}{c x}\right )-b e \operatorname {PolyLog}\left (3,\frac {1}{c x}\right )-(b e) \text {Subst}\left (\int \frac {\log (x) \log \left (\frac {c-c x}{c}\right )}{x} \, dx,x,1-\frac {1}{c x}\right )+(b e) \text {Subst}\left (\int \frac {\log (x) \log \left (-\frac {-c+c x}{c}\right )}{x} \, dx,x,1+\frac {1}{c x}\right ) \\ & = -\frac {1}{2} b e \log ^2\left (1+\frac {1}{c x}\right ) \log \left (-\frac {1}{c x}\right )+\frac {1}{2} b e \log ^2\left (1-\frac {1}{c x}\right ) \log \left (\frac {1}{c x}\right )+a d \log (x)+b e \log \left (1-\frac {1}{c x}\right ) \operatorname {PolyLog}\left (2,1-\frac {1}{c x}\right )-b e \log \left (1+\frac {1}{c x}\right ) \operatorname {PolyLog}\left (2,1+\frac {1}{c x}\right )+\frac {1}{2} b d \operatorname {PolyLog}\left (2,-\frac {1}{c x}\right )+\frac {1}{2} b e \log \left (-c^2 x^2\right ) \operatorname {PolyLog}\left (2,-\frac {1}{c x}\right )-\frac {1}{2} b e \left (\log \left (1-\frac {1}{c x}\right )+\log \left (1+\frac {1}{c x}\right )+\log \left (-c^2 x^2\right )-\log \left (1-c^2 x^2\right )\right ) \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} b e \log \left (-c^2 x^2\right ) \operatorname {PolyLog}\left (2,\frac {1}{c x}\right )+\frac {1}{2} b e \left (\log \left (1-\frac {1}{c x}\right )+\log \left (1+\frac {1}{c x}\right )+\log \left (-c^2 x^2\right )-\log \left (1-c^2 x^2\right )\right ) \operatorname {PolyLog}\left (2,\frac {1}{c x}\right )-\frac {1}{2} a e \operatorname {PolyLog}\left (2,c^2 x^2\right )+b e \operatorname {PolyLog}\left (3,-\frac {1}{c x}\right )-b e \operatorname {PolyLog}\left (3,\frac {1}{c x}\right )-(b e) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,x)}{x} \, dx,x,1-\frac {1}{c x}\right )+(b e) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,x)}{x} \, dx,x,1+\frac {1}{c x}\right ) \\ & = -\frac {1}{2} b e \log ^2\left (1+\frac {1}{c x}\right ) \log \left (-\frac {1}{c x}\right )+\frac {1}{2} b e \log ^2\left (1-\frac {1}{c x}\right ) \log \left (\frac {1}{c x}\right )+a d \log (x)+b e \log \left (1-\frac {1}{c x}\right ) \operatorname {PolyLog}\left (2,1-\frac {1}{c x}\right )-b e \log \left (1+\frac {1}{c x}\right ) \operatorname {PolyLog}\left (2,1+\frac {1}{c x}\right )+\frac {1}{2} b d \operatorname {PolyLog}\left (2,-\frac {1}{c x}\right )+\frac {1}{2} b e \log \left (-c^2 x^2\right ) \operatorname {PolyLog}\left (2,-\frac {1}{c x}\right )-\frac {1}{2} b e \left (\log \left (1-\frac {1}{c x}\right )+\log \left (1+\frac {1}{c x}\right )+\log \left (-c^2 x^2\right )-\log \left (1-c^2 x^2\right )\right ) \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} b e \log \left (-c^2 x^2\right ) \operatorname {PolyLog}\left (2,\frac {1}{c x}\right )+\frac {1}{2} b e \left (\log \left (1-\frac {1}{c x}\right )+\log \left (1+\frac {1}{c x}\right )+\log \left (-c^2 x^2\right )-\log \left (1-c^2 x^2\right )\right ) \operatorname {PolyLog}\left (2,\frac {1}{c x}\right )-\frac {1}{2} a e \operatorname {PolyLog}\left (2,c^2 x^2\right )-b e \operatorname {PolyLog}\left (3,1-\frac {1}{c x}\right )+b e \operatorname {PolyLog}\left (3,1+\frac {1}{c x}\right )+b e \operatorname {PolyLog}\left (3,-\frac {1}{c x}\right )-b e \operatorname {PolyLog}\left (3,\frac {1}{c x}\right ) \\ \end{align*}
\[ \int \frac {\left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )}{x} \, dx=\int \frac {\left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )}{x} \, dx \]
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 4.72 (sec) , antiderivative size = 589, normalized size of antiderivative = 1.55
method | result | size |
risch | \(-\frac {a \left (i e \pi \,\operatorname {csgn}\left (i \left (c x -1\right )\right ) \operatorname {csgn}\left (i \left (c x +1\right )\right ) \operatorname {csgn}\left (i \left (c x -1\right ) \left (c x +1\right )\right )-i e \pi \,\operatorname {csgn}\left (i \left (c x -1\right )\right ) \operatorname {csgn}\left (i \left (c x -1\right ) \left (c x +1\right )\right )^{2}-i e \pi \,\operatorname {csgn}\left (i \left (c x +1\right )\right ) \operatorname {csgn}\left (i \left (c x -1\right ) \left (c x +1\right )\right )^{2}-i e \pi \operatorname {csgn}\left (i \left (c x -1\right ) \left (c x +1\right )\right )^{3}+2 i e \pi \operatorname {csgn}\left (i \left (c x -1\right ) \left (c x +1\right )\right )^{2}-2 i e \pi -2 d \right ) \ln \left (c x \right )}{2}-\frac {\left (-i \pi b e \,\operatorname {csgn}\left (i \left (c x -1\right )\right ) \operatorname {csgn}\left (i \left (c x +1\right )\right ) \operatorname {csgn}\left (i \left (c x -1\right ) \left (c x +1\right )\right )+i \pi b e \,\operatorname {csgn}\left (i \left (c x -1\right )\right ) \operatorname {csgn}\left (i \left (c x -1\right ) \left (c x +1\right )\right )^{2}+i \pi b e \,\operatorname {csgn}\left (i \left (c x +1\right )\right ) \operatorname {csgn}\left (i \left (c x -1\right ) \left (c x +1\right )\right )^{2}+i \pi b e \operatorname {csgn}\left (i \left (c x -1\right ) \left (c x +1\right )\right )^{3}-2 i \pi b e \operatorname {csgn}\left (i \left (c x -1\right ) \left (c x +1\right )\right )^{2}+2 i e \pi b -4 a e +2 d b \right ) \left (\operatorname {dilog}\left (c x \right )+\ln \left (c x -1\right ) \ln \left (c x \right )\right )}{4}-\frac {\ln \left (c x -1\right )^{2} \ln \left (c x \right ) b e}{2}-\ln \left (c x -1\right ) \operatorname {polylog}\left (2, -c x +1\right ) b e +\operatorname {polylog}\left (3, -c x +1\right ) b e -\left (-\frac {i \pi b e \,\operatorname {csgn}\left (i \left (c x -1\right )\right ) \operatorname {csgn}\left (i \left (c x +1\right )\right ) \operatorname {csgn}\left (i \left (c x -1\right ) \left (c x +1\right )\right )}{4}+\frac {i \pi b e \,\operatorname {csgn}\left (i \left (c x -1\right )\right ) \operatorname {csgn}\left (i \left (c x -1\right ) \left (c x +1\right )\right )^{2}}{4}+\frac {i \pi b e \,\operatorname {csgn}\left (i \left (c x +1\right )\right ) \operatorname {csgn}\left (i \left (c x -1\right ) \left (c x +1\right )\right )^{2}}{4}+\frac {i \pi b e \operatorname {csgn}\left (i \left (c x -1\right ) \left (c x +1\right )\right )^{3}}{4}-\frac {i \pi b e \operatorname {csgn}\left (i \left (c x -1\right ) \left (c x +1\right )\right )^{2}}{2}+\frac {i e \pi b}{2}+a e +\frac {d b}{2}\right ) \operatorname {dilog}\left (c x +1\right )+\frac {\ln \left (-c x \right ) \ln \left (c x +1\right )^{2} b e}{2}+\operatorname {polylog}\left (2, c x +1\right ) \ln \left (c x +1\right ) b e -\operatorname {polylog}\left (3, c x +1\right ) b e\) | \(589\) |
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\[ \int \frac {\left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )}{x} \, dx=\int { \frac {{\left (b \operatorname {arcoth}\left (c x\right ) + a\right )} {\left (e \log \left (-c^{2} x^{2} + 1\right ) + d\right )}}{x} \,d x } \]
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\[ \int \frac {\left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )}{x} \, dx=\int \frac {\left (a + b \operatorname {acoth}{\left (c x \right )}\right ) \left (d + e \log {\left (- c^{2} x^{2} + 1 \right )}\right )}{x}\, dx \]
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Result contains complex when optimal does not.
Time = 0.36 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.44 \[ \int \frac {\left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )}{x} \, dx=i \, \pi a e \log \left (x\right ) - \frac {1}{2} \, {\left (\log \left (c x - 1\right )^{2} \log \left (c x\right ) + 2 \, {\rm Li}_2\left (-c x + 1\right ) \log \left (c x - 1\right ) - 2 \, {\rm Li}_{3}(-c x + 1)\right )} b e + \frac {1}{2} \, {\left (\log \left (c x + 1\right )^{2} \log \left (-c x\right ) + 2 \, {\rm Li}_2\left (c x + 1\right ) \log \left (c x + 1\right ) - 2 \, {\rm Li}_{3}(c x + 1)\right )} b e + a d \log \left (x\right ) - \frac {1}{2} \, {\left (i \, \pi b e + b d - 2 \, a e\right )} {\left (\log \left (c x - 1\right ) \log \left (c x\right ) + {\rm Li}_2\left (-c x + 1\right )\right )} - \frac {1}{2} \, {\left (-i \, \pi b e - b d - 2 \, a e\right )} {\left (\log \left (c x + 1\right ) \log \left (-c x\right ) + {\rm Li}_2\left (c x + 1\right )\right )} \]
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\[ \int \frac {\left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )}{x} \, dx=\int { \frac {{\left (b \operatorname {arcoth}\left (c x\right ) + a\right )} {\left (e \log \left (-c^{2} x^{2} + 1\right ) + d\right )}}{x} \,d x } \]
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Timed out. \[ \int \frac {\left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )}{x} \, dx=\int \frac {\left (a+b\,\mathrm {acoth}\left (c\,x\right )\right )\,\left (d+e\,\ln \left (1-c^2\,x^2\right )\right )}{x} \,d x \]
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