Integrand size = 27, antiderivative size = 381 \[ \int \frac {\left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )}{x} \, dx=-\frac {1}{2} b e \log ^2\left (1+\frac {1}{c x}\right ) \log \left (-\frac {1}{c x}\right )+\frac {1}{2} b e \log ^2\left (1-\frac {1}{c x}\right ) \log \left (\frac {1}{c x}\right )+a d \log (x)-b e \log \left (\frac {c+\frac {1}{x}}{c}\right ) \operatorname {PolyLog}\left (2,\frac {c+\frac {1}{x}}{c}\right )+b e \log \left (1-\frac {1}{c x}\right ) \operatorname {PolyLog}\left (2,1-\frac {1}{c x}\right )+\frac {1}{2} b d \operatorname {PolyLog}\left (2,-\frac {1}{c x}\right )+\frac {1}{2} b e \log \left (-c^2 x^2\right ) \operatorname {PolyLog}\left (2,-\frac {1}{c x}\right )-\frac {1}{2} b e \left (\log \left (1-\frac {1}{c x}\right )+\log \left (1+\frac {1}{c x}\right )+\log \left (-c^2 x^2\right )-\log \left (1-c^2 x^2\right )\right ) \operatorname {PolyLog}\left (2,-\frac {1}{c x}\right )-\frac {1}{2} b d \operatorname {PolyLog}\left (2,\frac {1}{c x}\right )-\frac {1}{2} b e \log \left (-c^2 x^2\right ) \operatorname {PolyLog}\left (2,\frac {1}{c x}\right )+\frac {1}{2} b e \left (\log \left (1-\frac {1}{c x}\right )+\log \left (1+\frac {1}{c x}\right )+\log \left (-c^2 x^2\right )-\log \left (1-c^2 x^2\right )\right ) \operatorname {PolyLog}\left (2,\frac {1}{c x}\right )-\frac {1}{2} a e \operatorname {PolyLog}\left (2,c^2 x^2\right )+b e \operatorname {PolyLog}\left (3,\frac {c+\frac {1}{x}}{c}\right )-b e \operatorname {PolyLog}\left (3,1-\frac {1}{c x}\right )+b e \operatorname {PolyLog}\left (3,-\frac {1}{c x}\right )-b e \operatorname {PolyLog}\left (3,\frac {1}{c x}\right ) \]
-1/2*b*e*ln(1+1/c/x)^2*ln(-1/c/x)+1/2*b*e*ln(1-1/c/x)^2*ln(1/c/x)+a*d*ln(x )-b*e*ln((c+1/x)/c)*polylog(2,(c+1/x)/c)+b*e*ln(1-1/c/x)*polylog(2,1-1/c/x )+1/2*b*d*polylog(2,-1/c/x)+1/2*b*e*ln(-c^2*x^2)*polylog(2,-1/c/x)-1/2*b*e *(ln(1-1/c/x)+ln(1+1/c/x)+ln(-c^2*x^2)-ln(-c^2*x^2+1))*polylog(2,-1/c/x)-1 /2*b*d*polylog(2,1/c/x)-1/2*b*e*ln(-c^2*x^2)*polylog(2,1/c/x)+1/2*b*e*(ln( 1-1/c/x)+ln(1+1/c/x)+ln(-c^2*x^2)-ln(-c^2*x^2+1))*polylog(2,1/c/x)-1/2*a*e *polylog(2,c^2*x^2)+b*e*polylog(3,(c+1/x)/c)-b*e*polylog(3,1-1/c/x)+b*e*po lylog(3,-1/c/x)-b*e*polylog(3,1/c/x)
\[ \int \frac {\left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )}{x} \, dx=\int \frac {\left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )}{x} \, dx \]
Time = 1.92 (sec) , antiderivative size = 336, normalized size of antiderivative = 0.88, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.556, Rules used = {6642, 6447, 6640, 2838, 6638, 2904, 2843, 27, 2881, 27, 2821, 6447, 6632, 2821, 7143}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {\left (a+b \coth ^{-1}(c x)\right ) \left (e \log \left (1-c^2 x^2\right )+d\right )}{x} \, dx\) |
| \(\Big \downarrow \) 6642 |
| \(\displaystyle e \int \frac {\left (a+b \coth ^{-1}(c x)\right ) \log \left (1-c^2 x^2\right )}{x}dx+d \int \frac {a+b \coth ^{-1}(c x)}{x}dx\) |
| \(\Big \downarrow \) 6447 |
| \(\displaystyle e \int \frac {\left (a+b \coth ^{-1}(c x)\right ) \log \left (1-c^2 x^2\right )}{x}dx+d \left (a \log (x)+\frac {1}{2} b \operatorname {PolyLog}\left (2,-\frac {1}{c x}\right )-\frac {1}{2} b \operatorname {PolyLog}\left (2,\frac {1}{c x}\right )\right )\) |
| \(\Big \downarrow \) 6640 |
| \(\displaystyle e \left (a \int \frac {\log \left (1-c^2 x^2\right )}{x}dx+b \int \frac {\coth ^{-1}(c x) \log \left (1-c^2 x^2\right )}{x}dx\right )+d \left (a \log (x)+\frac {1}{2} b \operatorname {PolyLog}\left (2,-\frac {1}{c x}\right )-\frac {1}{2} b \operatorname {PolyLog}\left (2,\frac {1}{c x}\right )\right )\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle e \left (b \int \frac {\coth ^{-1}(c x) \log \left (1-c^2 x^2\right )}{x}dx-\frac {1}{2} a \operatorname {PolyLog}\left (2,c^2 x^2\right )\right )+d \left (a \log (x)+\frac {1}{2} b \operatorname {PolyLog}\left (2,-\frac {1}{c x}\right )-\frac {1}{2} b \operatorname {PolyLog}\left (2,\frac {1}{c x}\right )\right )\) |
| \(\Big \downarrow \) 6638 |
| \(\displaystyle e \left (b \left (-\left (\left (\log \left (-c^2 x^2\right )-\log \left (1-c^2 x^2\right )+\log \left (1-\frac {1}{c x}\right )+\log \left (\frac {1}{c x}+1\right )\right ) \int \frac {\coth ^{-1}(c x)}{x}dx\right )+\int \frac {\coth ^{-1}(c x) \log \left (-c^2 x^2\right )}{x}dx-\frac {1}{2} \int \frac {\log ^2\left (1-\frac {1}{c x}\right )}{x}dx+\frac {1}{2} \int \frac {\log ^2\left (1+\frac {1}{c x}\right )}{x}dx\right )-\frac {1}{2} a \operatorname {PolyLog}\left (2,c^2 x^2\right )\right )+d \left (a \log (x)+\frac {1}{2} b \operatorname {PolyLog}\left (2,-\frac {1}{c x}\right )-\frac {1}{2} b \operatorname {PolyLog}\left (2,\frac {1}{c x}\right )\right )\) |
| \(\Big \downarrow \) 2904 |
| \(\displaystyle e \left (b \left (-\left (\left (\log \left (-c^2 x^2\right )-\log \left (1-c^2 x^2\right )+\log \left (1-\frac {1}{c x}\right )+\log \left (\frac {1}{c x}+1\right )\right ) \int \frac {\coth ^{-1}(c x)}{x}dx\right )+\int \frac {\coth ^{-1}(c x) \log \left (-c^2 x^2\right )}{x}dx+\frac {1}{2} \int x \log ^2\left (1-\frac {1}{c x}\right )d\frac {1}{x}-\frac {1}{2} \int x \log ^2\left (1+\frac {1}{c x}\right )d\frac {1}{x}\right )-\frac {1}{2} a \operatorname {PolyLog}\left (2,c^2 x^2\right )\right )+d \left (a \log (x)+\frac {1}{2} b \operatorname {PolyLog}\left (2,-\frac {1}{c x}\right )-\frac {1}{2} b \operatorname {PolyLog}\left (2,\frac {1}{c x}\right )\right )\) |
| \(\Big \downarrow \) 2843 |
| \(\displaystyle e \left (b \left (-\left (\left (\log \left (-c^2 x^2\right )-\log \left (1-c^2 x^2\right )+\log \left (1-\frac {1}{c x}\right )+\log \left (\frac {1}{c x}+1\right )\right ) \int \frac {\coth ^{-1}(c x)}{x}dx\right )+\int \frac {\coth ^{-1}(c x) \log \left (-c^2 x^2\right )}{x}dx+\frac {1}{2} \left (\frac {2 \int \frac {c \log \left (1+\frac {1}{c x}\right ) \log \left (-\frac {1}{c x}\right )}{c+\frac {1}{x}}d\frac {1}{x}}{c}-\log ^2\left (\frac {1}{c x}+1\right ) \log \left (-\frac {1}{c x}\right )\right )+\frac {1}{2} \left (\frac {2 \int \frac {c \log \left (1-\frac {1}{c x}\right ) \log \left (\frac {1}{c x}\right )}{c-\frac {1}{x}}d\frac {1}{x}}{c}+\log \left (\frac {1}{c x}\right ) \log ^2\left (1-\frac {1}{c x}\right )\right )\right )-\frac {1}{2} a \operatorname {PolyLog}\left (2,c^2 x^2\right )\right )+d \left (a \log (x)+\frac {1}{2} b \operatorname {PolyLog}\left (2,-\frac {1}{c x}\right )-\frac {1}{2} b \operatorname {PolyLog}\left (2,\frac {1}{c x}\right )\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle e \left (b \left (-\left (\left (\log \left (-c^2 x^2\right )-\log \left (1-c^2 x^2\right )+\log \left (1-\frac {1}{c x}\right )+\log \left (\frac {1}{c x}+1\right )\right ) \int \frac {\coth ^{-1}(c x)}{x}dx\right )+\int \frac {\coth ^{-1}(c x) \log \left (-c^2 x^2\right )}{x}dx+\frac {1}{2} \left (2 \int \frac {\log \left (1+\frac {1}{c x}\right ) \log \left (-\frac {1}{c x}\right )}{c+\frac {1}{x}}d\frac {1}{x}-\log ^2\left (\frac {1}{c x}+1\right ) \log \left (-\frac {1}{c x}\right )\right )+\frac {1}{2} \left (2 \int \frac {\log \left (1-\frac {1}{c x}\right ) \log \left (\frac {1}{c x}\right )}{c-\frac {1}{x}}d\frac {1}{x}+\log \left (\frac {1}{c x}\right ) \log ^2\left (1-\frac {1}{c x}\right )\right )\right )-\frac {1}{2} a \operatorname {PolyLog}\left (2,c^2 x^2\right )\right )+d \left (a \log (x)+\frac {1}{2} b \operatorname {PolyLog}\left (2,-\frac {1}{c x}\right )-\frac {1}{2} b \operatorname {PolyLog}\left (2,\frac {1}{c x}\right )\right )\) |
| \(\Big \downarrow \) 2881 |
| \(\displaystyle e \left (b \left (-\left (\left (\log \left (-c^2 x^2\right )-\log \left (1-c^2 x^2\right )+\log \left (1-\frac {1}{c x}\right )+\log \left (\frac {1}{c x}+1\right )\right ) \int \frac {\coth ^{-1}(c x)}{x}dx\right )+\int \frac {\coth ^{-1}(c x) \log \left (-c^2 x^2\right )}{x}dx+\frac {1}{2} \left (2 c \int \frac {x \log \left (1+\frac {1}{c x}\right ) \log \left (-\frac {1}{c x}\right )}{c}d\left (1+\frac {1}{c x}\right )-\log ^2\left (\frac {1}{c x}+1\right ) \log \left (-\frac {1}{c x}\right )\right )+\frac {1}{2} \left (\log ^2\left (1-\frac {1}{c x}\right ) \log \left (\frac {1}{c x}\right )-2 c \int \frac {x \log \left (1-\frac {1}{c x}\right ) \log \left (\frac {1}{c x}\right )}{c}d\left (1-\frac {1}{c x}\right )\right )\right )-\frac {1}{2} a \operatorname {PolyLog}\left (2,c^2 x^2\right )\right )+d \left (a \log (x)+\frac {1}{2} b \operatorname {PolyLog}\left (2,-\frac {1}{c x}\right )-\frac {1}{2} b \operatorname {PolyLog}\left (2,\frac {1}{c x}\right )\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle e \left (b \left (-\left (\left (\log \left (-c^2 x^2\right )-\log \left (1-c^2 x^2\right )+\log \left (1-\frac {1}{c x}\right )+\log \left (\frac {1}{c x}+1\right )\right ) \int \frac {\coth ^{-1}(c x)}{x}dx\right )+\int \frac {\coth ^{-1}(c x) \log \left (-c^2 x^2\right )}{x}dx+\frac {1}{2} \left (2 \int x \log \left (1+\frac {1}{c x}\right ) \log \left (-\frac {1}{c x}\right )d\left (1+\frac {1}{c x}\right )-\log ^2\left (\frac {1}{c x}+1\right ) \log \left (-\frac {1}{c x}\right )\right )+\frac {1}{2} \left (\log ^2\left (1-\frac {1}{c x}\right ) \log \left (\frac {1}{c x}\right )-2 \int x \log \left (1-\frac {1}{c x}\right ) \log \left (\frac {1}{c x}\right )d\left (1-\frac {1}{c x}\right )\right )\right )-\frac {1}{2} a \operatorname {PolyLog}\left (2,c^2 x^2\right )\right )+d \left (a \log (x)+\frac {1}{2} b \operatorname {PolyLog}\left (2,-\frac {1}{c x}\right )-\frac {1}{2} b \operatorname {PolyLog}\left (2,\frac {1}{c x}\right )\right )\) |
| \(\Big \downarrow \) 2821 |
| \(\displaystyle e \left (b \left (-\left (\left (\log \left (-c^2 x^2\right )-\log \left (1-c^2 x^2\right )+\log \left (1-\frac {1}{c x}\right )+\log \left (\frac {1}{c x}+1\right )\right ) \int \frac {\coth ^{-1}(c x)}{x}dx\right )+\int \frac {\coth ^{-1}(c x) \log \left (-c^2 x^2\right )}{x}dx+\frac {1}{2} \left (\log ^2\left (1-\frac {1}{c x}\right ) \log \left (\frac {1}{c x}\right )-2 \left (\int x \operatorname {PolyLog}\left (2,1-\frac {1}{c x}\right )d\left (1-\frac {1}{c x}\right )-\operatorname {PolyLog}\left (2,1-\frac {1}{c x}\right ) \log \left (1-\frac {1}{c x}\right )\right )\right )+\frac {1}{2} \left (2 \left (\int x \operatorname {PolyLog}\left (2,1+\frac {1}{c x}\right )d\left (1+\frac {1}{c x}\right )-\operatorname {PolyLog}\left (2,1+\frac {1}{c x}\right ) \log \left (\frac {1}{c x}+1\right )\right )-\log ^2\left (\frac {1}{c x}+1\right ) \log \left (-\frac {1}{c x}\right )\right )\right )-\frac {1}{2} a \operatorname {PolyLog}\left (2,c^2 x^2\right )\right )+d \left (a \log (x)+\frac {1}{2} b \operatorname {PolyLog}\left (2,-\frac {1}{c x}\right )-\frac {1}{2} b \operatorname {PolyLog}\left (2,\frac {1}{c x}\right )\right )\) |
| \(\Big \downarrow \) 6447 |
| \(\displaystyle e \left (b \left (\int \frac {\coth ^{-1}(c x) \log \left (-c^2 x^2\right )}{x}dx+\frac {1}{2} \left (\log ^2\left (1-\frac {1}{c x}\right ) \log \left (\frac {1}{c x}\right )-2 \left (\int x \operatorname {PolyLog}\left (2,1-\frac {1}{c x}\right )d\left (1-\frac {1}{c x}\right )-\operatorname {PolyLog}\left (2,1-\frac {1}{c x}\right ) \log \left (1-\frac {1}{c x}\right )\right )\right )+\frac {1}{2} \left (2 \left (\int x \operatorname {PolyLog}\left (2,1+\frac {1}{c x}\right )d\left (1+\frac {1}{c x}\right )-\operatorname {PolyLog}\left (2,1+\frac {1}{c x}\right ) \log \left (\frac {1}{c x}+1\right )\right )-\log ^2\left (\frac {1}{c x}+1\right ) \log \left (-\frac {1}{c x}\right )\right )-\left (\left (\frac {1}{2} \operatorname {PolyLog}\left (2,-\frac {1}{c x}\right )-\frac {1}{2} \operatorname {PolyLog}\left (2,\frac {1}{c x}\right )\right ) \left (\log \left (-c^2 x^2\right )-\log \left (1-c^2 x^2\right )+\log \left (1-\frac {1}{c x}\right )+\log \left (\frac {1}{c x}+1\right )\right )\right )\right )-\frac {1}{2} a \operatorname {PolyLog}\left (2,c^2 x^2\right )\right )+d \left (a \log (x)+\frac {1}{2} b \operatorname {PolyLog}\left (2,-\frac {1}{c x}\right )-\frac {1}{2} b \operatorname {PolyLog}\left (2,\frac {1}{c x}\right )\right )\) |
| \(\Big \downarrow \) 6632 |
| \(\displaystyle e \left (b \left (-\frac {1}{2} \int \frac {\log \left (1-\frac {1}{c x}\right ) \log \left (-c^2 x^2\right )}{x}dx+\frac {1}{2} \int \frac {\log \left (1+\frac {1}{c x}\right ) \log \left (-c^2 x^2\right )}{x}dx+\frac {1}{2} \left (\log ^2\left (1-\frac {1}{c x}\right ) \log \left (\frac {1}{c x}\right )-2 \left (\int x \operatorname {PolyLog}\left (2,1-\frac {1}{c x}\right )d\left (1-\frac {1}{c x}\right )-\operatorname {PolyLog}\left (2,1-\frac {1}{c x}\right ) \log \left (1-\frac {1}{c x}\right )\right )\right )+\frac {1}{2} \left (2 \left (\int x \operatorname {PolyLog}\left (2,1+\frac {1}{c x}\right )d\left (1+\frac {1}{c x}\right )-\operatorname {PolyLog}\left (2,1+\frac {1}{c x}\right ) \log \left (\frac {1}{c x}+1\right )\right )-\log ^2\left (\frac {1}{c x}+1\right ) \log \left (-\frac {1}{c x}\right )\right )-\left (\left (\frac {1}{2} \operatorname {PolyLog}\left (2,-\frac {1}{c x}\right )-\frac {1}{2} \operatorname {PolyLog}\left (2,\frac {1}{c x}\right )\right ) \left (\log \left (-c^2 x^2\right )-\log \left (1-c^2 x^2\right )+\log \left (1-\frac {1}{c x}\right )+\log \left (\frac {1}{c x}+1\right )\right )\right )\right )-\frac {1}{2} a \operatorname {PolyLog}\left (2,c^2 x^2\right )\right )+d \left (a \log (x)+\frac {1}{2} b \operatorname {PolyLog}\left (2,-\frac {1}{c x}\right )-\frac {1}{2} b \operatorname {PolyLog}\left (2,\frac {1}{c x}\right )\right )\) |
| \(\Big \downarrow \) 2821 |
| \(\displaystyle e \left (b \left (\frac {1}{2} \left (\operatorname {PolyLog}\left (2,-\frac {1}{c x}\right ) \log \left (-c^2 x^2\right )-2 \int \frac {\operatorname {PolyLog}\left (2,-\frac {1}{c x}\right )}{x}dx\right )+\frac {1}{2} \left (2 \int \frac {\operatorname {PolyLog}\left (2,\frac {1}{c x}\right )}{x}dx-\operatorname {PolyLog}\left (2,\frac {1}{c x}\right ) \log \left (-c^2 x^2\right )\right )+\frac {1}{2} \left (\log ^2\left (1-\frac {1}{c x}\right ) \log \left (\frac {1}{c x}\right )-2 \left (\int x \operatorname {PolyLog}\left (2,1-\frac {1}{c x}\right )d\left (1-\frac {1}{c x}\right )-\operatorname {PolyLog}\left (2,1-\frac {1}{c x}\right ) \log \left (1-\frac {1}{c x}\right )\right )\right )+\frac {1}{2} \left (2 \left (\int x \operatorname {PolyLog}\left (2,1+\frac {1}{c x}\right )d\left (1+\frac {1}{c x}\right )-\operatorname {PolyLog}\left (2,1+\frac {1}{c x}\right ) \log \left (\frac {1}{c x}+1\right )\right )-\log ^2\left (\frac {1}{c x}+1\right ) \log \left (-\frac {1}{c x}\right )\right )-\left (\left (\frac {1}{2} \operatorname {PolyLog}\left (2,-\frac {1}{c x}\right )-\frac {1}{2} \operatorname {PolyLog}\left (2,\frac {1}{c x}\right )\right ) \left (\log \left (-c^2 x^2\right )-\log \left (1-c^2 x^2\right )+\log \left (1-\frac {1}{c x}\right )+\log \left (\frac {1}{c x}+1\right )\right )\right )\right )-\frac {1}{2} a \operatorname {PolyLog}\left (2,c^2 x^2\right )\right )+d \left (a \log (x)+\frac {1}{2} b \operatorname {PolyLog}\left (2,-\frac {1}{c x}\right )-\frac {1}{2} b \operatorname {PolyLog}\left (2,\frac {1}{c x}\right )\right )\) |
| \(\Big \downarrow \) 7143 |
| \(\displaystyle e \left (b \left (-\left (\left (\frac {1}{2} \operatorname {PolyLog}\left (2,-\frac {1}{c x}\right )-\frac {1}{2} \operatorname {PolyLog}\left (2,\frac {1}{c x}\right )\right ) \left (\log \left (-c^2 x^2\right )-\log \left (1-c^2 x^2\right )+\log \left (1-\frac {1}{c x}\right )+\log \left (\frac {1}{c x}+1\right )\right )\right )+\frac {1}{2} \left (\operatorname {PolyLog}\left (2,-\frac {1}{c x}\right ) \log \left (-c^2 x^2\right )+2 \operatorname {PolyLog}\left (3,-\frac {1}{c x}\right )\right )+\frac {1}{2} \left (\operatorname {PolyLog}\left (2,\frac {1}{c x}\right ) \left (-\log \left (-c^2 x^2\right )\right )-2 \operatorname {PolyLog}\left (3,\frac {1}{c x}\right )\right )+\frac {1}{2} \left (\log ^2\left (1-\frac {1}{c x}\right ) \log \left (\frac {1}{c x}\right )-2 \left (\operatorname {PolyLog}\left (3,1-\frac {1}{c x}\right )-\operatorname {PolyLog}\left (2,1-\frac {1}{c x}\right ) \log \left (1-\frac {1}{c x}\right )\right )\right )+\frac {1}{2} \left (2 \left (\operatorname {PolyLog}\left (3,1+\frac {1}{c x}\right )-\operatorname {PolyLog}\left (2,1+\frac {1}{c x}\right ) \log \left (\frac {1}{c x}+1\right )\right )-\log ^2\left (\frac {1}{c x}+1\right ) \log \left (-\frac {1}{c x}\right )\right )\right )-\frac {1}{2} a \operatorname {PolyLog}\left (2,c^2 x^2\right )\right )+d \left (a \log (x)+\frac {1}{2} b \operatorname {PolyLog}\left (2,-\frac {1}{c x}\right )-\frac {1}{2} b \operatorname {PolyLog}\left (2,\frac {1}{c x}\right )\right )\) |
d*(a*Log[x] + (b*PolyLog[2, -(1/(c*x))])/2 - (b*PolyLog[2, 1/(c*x)])/2) + e*(-1/2*(a*PolyLog[2, c^2*x^2]) + b*(-((Log[1 - 1/(c*x)] + Log[1 + 1/(c*x) ] + Log[-(c^2*x^2)] - Log[1 - c^2*x^2])*(PolyLog[2, -(1/(c*x))]/2 - PolyLo g[2, 1/(c*x)]/2)) + (Log[1 - 1/(c*x)]^2*Log[1/(c*x)] - 2*(-(Log[1 - 1/(c*x )]*PolyLog[2, 1 - 1/(c*x)]) + PolyLog[3, 1 - 1/(c*x)]))/2 + (-(Log[1 + 1/( c*x)]^2*Log[-(1/(c*x))]) + 2*(-(Log[1 + 1/(c*x)]*PolyLog[2, 1 + 1/(c*x)]) + PolyLog[3, 1 + 1/(c*x)]))/2 + (Log[-(c^2*x^2)]*PolyLog[2, -(1/(c*x))] + 2*PolyLog[3, -(1/(c*x))])/2 + (-(Log[-(c^2*x^2)]*PolyLog[2, 1/(c*x)]) - 2* PolyLog[3, 1/(c*x)])/2))
3.3.69.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))]*((a_.) + Log[(c_.)*(x_)^(n_.)]*(b _.))^(p_.))/(x_), x_Symbol] :> Simp[(-PolyLog[2, (-d)*f*x^m])*((a + b*Log[c *x^n])^p/m), x] + Simp[b*n*(p/m) Int[PolyLog[2, (-d)*f*x^m]*((a + b*Log[c *x^n])^(p - 1)/x), x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && IGtQ[p, 0] && EqQ[d*e, 1]
Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2 , (-c)*e*x^n]/n, x] /; FreeQ[{c, d, e, n}, x] && EqQ[c*d, 1]
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_)/((f_.) + (g_. )*(x_)), x_Symbol] :> Simp[Log[e*((f + g*x)/(e*f - d*g))]*((a + b*Log[c*(d + e*x)^n])^p/g), x] - Simp[b*e*n*(p/g) Int[Log[(e*(f + g*x))/(e*f - d*g)] *((a + b*Log[c*(d + e*x)^n])^(p - 1)/(d + e*x)), x], x] /; FreeQ[{a, b, c, d, e, f, g, n, p}, x] && NeQ[e*f - d*g, 0] && IGtQ[p, 1]
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_.) + Log [(h_.)*((i_.) + (j_.)*(x_))^(m_.)]*(g_.))*((k_.) + (l_.)*(x_))^(r_.), x_Sym bol] :> Simp[1/e Subst[Int[(k*(x/d))^r*(a + b*Log[c*x^n])^p*(f + g*Log[h* ((e*i - d*j)/e + j*(x/e))^m]), x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, h, i, j, k, l, n, p, r}, x] && EqQ[e*k - d*l, 0]
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*(x_)^(m _.), x_Symbol] :> Simp[1/n Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*L og[c*(d + e*x)^p])^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, e, m, n, p, q}, x] && IntegerQ[Simplify[(m + 1)/n]] && (GtQ[(m + 1)/n, 0] || IGtQ[q, 0]) & & !(EqQ[q, 1] && ILtQ[n, 0] && IGtQ[m, 0])
Int[((a_.) + ArcCoth[(c_.)*(x_)]*(b_.))/(x_), x_Symbol] :> Simp[a*Log[x], x ] + (Simp[(b/2)*PolyLog[2, -(c*x)^(-1)], x] - Simp[(b/2)*PolyLog[2, 1/(c*x) ], x]) /; FreeQ[{a, b, c}, x]
Int[(ArcCoth[(c_.)*(x_)^(n_.)]*Log[(d_.)*(x_)^(m_.)])/(x_), x_Symbol] :> Si mp[1/2 Int[Log[d*x^m]*(Log[1 + 1/(c*x^n)]/x), x], x] - Simp[1/2 Int[Log [d*x^m]*(Log[1 - 1/(c*x^n)]/x), x], x] /; FreeQ[{c, d, m, n}, x]
Int[(ArcCoth[(c_.)*(x_)]*Log[(f_.) + (g_.)*(x_)^2])/(x_), x_Symbol] :> Simp [(Log[f + g*x^2] - Log[(-c^2)*x^2] - Log[1 - 1/(c*x)] - Log[1 + 1/(c*x)]) Int[ArcCoth[c*x]/x, x], x] + (Int[Log[(-c^2)*x^2]*(ArcCoth[c*x]/x), x] + S imp[1/2 Int[Log[1 + 1/(c*x)]^2/x, x], x] - Simp[1/2 Int[Log[1 - 1/(c*x) ]^2/x, x], x]) /; FreeQ[{c, f, g}, x] && EqQ[c^2*f + g, 0]
Int[(Log[(f_.) + (g_.)*(x_)^2]*(ArcCoth[(c_.)*(x_)]*(b_.) + (a_)))/(x_), x_ Symbol] :> Simp[a Int[Log[f + g*x^2]/x, x], x] + Simp[b Int[Log[f + g*x ^2]*(ArcCoth[c*x]/x), x], x] /; FreeQ[{a, b, c, f, g}, x]
Int[(((a_.) + ArcCoth[(c_.)*(x_)]*(b_.))*(Log[(f_.) + (g_.)*(x_)^2]*(e_.) + (d_)))/(x_), x_Symbol] :> Simp[d Int[(a + b*ArcCoth[c*x])/x, x], x] + Si mp[e Int[Log[f + g*x^2]*((a + b*ArcCoth[c*x])/x), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x]
Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_S ymbol] :> Simp[PolyLog[n + 1, c*(a + b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d , e, n, p}, x] && EqQ[b*d, a*e]
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\) | \(589\) |
-1/2*a*(I*e*Pi*csgn(I*(c*x-1))*csgn(I*(c*x+1))*csgn(I*(c*x-1)*(c*x+1))-I*e *Pi*csgn(I*(c*x-1))*csgn(I*(c*x-1)*(c*x+1))^2-I*e*Pi*csgn(I*(c*x+1))*csgn( I*(c*x-1)*(c*x+1))^2-I*e*Pi*csgn(I*(c*x-1)*(c*x+1))^3+2*I*e*Pi*csgn(I*(c*x -1)*(c*x+1))^2-2*I*e*Pi-2*d)*ln(c*x)-1/4*(-I*Pi*b*e*csgn(I*(c*x-1))*csgn(I *(c*x+1))*csgn(I*(c*x-1)*(c*x+1))+I*Pi*b*e*csgn(I*(c*x-1))*csgn(I*(c*x-1)* (c*x+1))^2+I*Pi*b*e*csgn(I*(c*x+1))*csgn(I*(c*x-1)*(c*x+1))^2+I*Pi*b*e*csg n(I*(c*x-1)*(c*x+1))^3-2*I*Pi*b*e*csgn(I*(c*x-1)*(c*x+1))^2+2*I*e*Pi*b-4*a *e+2*d*b)*(dilog(c*x)+ln(c*x-1)*ln(c*x))-1/2*ln(c*x-1)^2*ln(c*x)*b*e-ln(c* x-1)*polylog(2,-c*x+1)*b*e+polylog(3,-c*x+1)*b*e-(-1/4*I*Pi*b*e*csgn(I*(c* x-1))*csgn(I*(c*x+1))*csgn(I*(c*x-1)*(c*x+1))+1/4*I*Pi*b*e*csgn(I*(c*x-1)) *csgn(I*(c*x-1)*(c*x+1))^2+1/4*I*Pi*b*e*csgn(I*(c*x+1))*csgn(I*(c*x-1)*(c* x+1))^2+1/4*I*Pi*b*e*csgn(I*(c*x-1)*(c*x+1))^3-1/2*I*Pi*b*e*csgn(I*(c*x-1) *(c*x+1))^2+1/2*I*e*Pi*b+a*e+1/2*d*b)*dilog(c*x+1)+1/2*ln(-c*x)*ln(c*x+1)^ 2*b*e+polylog(2,c*x+1)*ln(c*x+1)*b*e-polylog(3,c*x+1)*b*e
\[ \int \frac {\left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )}{x} \, dx=\int { \frac {{\left (b \operatorname {arcoth}\left (c x\right ) + a\right )} {\left (e \log \left (-c^{2} x^{2} + 1\right ) + d\right )}}{x} \,d x } \]
\[ \int \frac {\left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )}{x} \, dx=\int \frac {\left (a + b \operatorname {acoth}{\left (c x \right )}\right ) \left (d + e \log {\left (- c^{2} x^{2} + 1 \right )}\right )}{x}\, dx \]
Result contains complex when optimal does not.
Time = 0.36 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.44 \[ \int \frac {\left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )}{x} \, dx=i \, \pi a e \log \left (x\right ) - \frac {1}{2} \, {\left (\log \left (c x - 1\right )^{2} \log \left (c x\right ) + 2 \, {\rm Li}_2\left (-c x + 1\right ) \log \left (c x - 1\right ) - 2 \, {\rm Li}_{3}(-c x + 1)\right )} b e + \frac {1}{2} \, {\left (\log \left (c x + 1\right )^{2} \log \left (-c x\right ) + 2 \, {\rm Li}_2\left (c x + 1\right ) \log \left (c x + 1\right ) - 2 \, {\rm Li}_{3}(c x + 1)\right )} b e + a d \log \left (x\right ) - \frac {1}{2} \, {\left (i \, \pi b e + b d - 2 \, a e\right )} {\left (\log \left (c x - 1\right ) \log \left (c x\right ) + {\rm Li}_2\left (-c x + 1\right )\right )} - \frac {1}{2} \, {\left (-i \, \pi b e - b d - 2 \, a e\right )} {\left (\log \left (c x + 1\right ) \log \left (-c x\right ) + {\rm Li}_2\left (c x + 1\right )\right )} \]
I*pi*a*e*log(x) - 1/2*(log(c*x - 1)^2*log(c*x) + 2*dilog(-c*x + 1)*log(c*x - 1) - 2*polylog(3, -c*x + 1))*b*e + 1/2*(log(c*x + 1)^2*log(-c*x) + 2*di log(c*x + 1)*log(c*x + 1) - 2*polylog(3, c*x + 1))*b*e + a*d*log(x) - 1/2* (I*pi*b*e + b*d - 2*a*e)*(log(c*x - 1)*log(c*x) + dilog(-c*x + 1)) - 1/2*( -I*pi*b*e - b*d - 2*a*e)*(log(c*x + 1)*log(-c*x) + dilog(c*x + 1))
\[ \int \frac {\left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )}{x} \, dx=\int { \frac {{\left (b \operatorname {arcoth}\left (c x\right ) + a\right )} {\left (e \log \left (-c^{2} x^{2} + 1\right ) + d\right )}}{x} \,d x } \]
Timed out. \[ \int \frac {\left (a+b \coth ^{-1}(c x)\right ) \left (d+e \log \left (1-c^2 x^2\right )\right )}{x} \, dx=\int \frac {\left (a+b\,\mathrm {acoth}\left (c\,x\right )\right )\,\left (d+e\,\ln \left (1-c^2\,x^2\right )\right )}{x} \,d x \]