Integrand size = 26, antiderivative size = 248 \[ \int \frac {(a+b \arctan (c x)) \left (d+e \log \left (1+c^2 x^2\right )\right )}{x^6} \, dx=-\frac {7 b c^3 e}{60 x^2}-\frac {2 c^2 e (a+b \arctan (c x))}{15 x^3}+\frac {2 c^4 e (a+b \arctan (c x))}{5 x}+\frac {c^5 e (a+b \arctan (c x))^2}{5 b}-\frac {5}{6} b c^5 e \log (x)+\frac {19}{60} b c^5 e \log \left (1+c^2 x^2\right )-\frac {b c \left (d+e \log \left (1+c^2 x^2\right )\right )}{20 x^4}+\frac {b c^3 \left (1+c^2 x^2\right ) \left (d+e \log \left (1+c^2 x^2\right )\right )}{10 x^2}-\frac {(a+b \arctan (c x)) \left (d+e \log \left (1+c^2 x^2\right )\right )}{5 x^5}+\frac {1}{10} b c^5 \left (d+e \log \left (1+c^2 x^2\right )\right ) \log \left (1-\frac {1}{1+c^2 x^2}\right )-\frac {1}{10} b c^5 e \operatorname {PolyLog}\left (2,\frac {1}{1+c^2 x^2}\right ) \]
-7/60*b*c^3*e/x^2-2/15*c^2*e*(a+b*arctan(c*x))/x^3+2/5*c^4*e*(a+b*arctan(c *x))/x+1/5*c^5*e*(a+b*arctan(c*x))^2/b-5/6*b*c^5*e*ln(x)+19/60*b*c^5*e*ln( c^2*x^2+1)-1/20*b*c*(d+e*ln(c^2*x^2+1))/x^4+1/10*b*c^3*(c^2*x^2+1)*(d+e*ln (c^2*x^2+1))/x^2-1/5*(a+b*arctan(c*x))*(d+e*ln(c^2*x^2+1))/x^5+1/10*b*c^5* (d+e*ln(c^2*x^2+1))*ln(1-1/(c^2*x^2+1))-1/10*b*c^5*e*polylog(2,1/(c^2*x^2+ 1))
Time = 0.19 (sec) , antiderivative size = 278, normalized size of antiderivative = 1.12 \[ \int \frac {(a+b \arctan (c x)) \left (d+e \log \left (1+c^2 x^2\right )\right )}{x^6} \, dx=\frac {1}{60} \left (-\frac {8 c^2 e (a+b \arctan (c x))}{x^3}-24 c^4 e \left (-\frac {a+b \arctan (c x)}{x}-\frac {c (a+b \arctan (c x))^2}{2 b}+b c \left (\log (x)-\frac {1}{2} \log \left (1+c^2 x^2\right )\right )\right )-6 b c^5 e \left (2 \log (x)-\log \left (1+c^2 x^2\right )\right )+7 b c^3 e \left (-\frac {1}{x^2}-2 c^2 \log (x)+c^2 \log \left (1+c^2 x^2\right )\right )-\frac {3 b c \left (d+e \log \left (1+c^2 x^2\right )\right )}{x^4}+\frac {6 b c^3 \left (d+e \log \left (1+c^2 x^2\right )\right )}{x^2}-\frac {12 (a+b \arctan (c x)) \left (d+e \log \left (1+c^2 x^2\right )\right )}{x^5}-\frac {3 b c^5 \left (d+e \log \left (1+c^2 x^2\right )\right )^2}{e}+6 b c^5 \left (\log \left (-c^2 x^2\right ) \left (d+e \log \left (1+c^2 x^2\right )\right )+e \operatorname {PolyLog}\left (2,1+c^2 x^2\right )\right )\right ) \]
((-8*c^2*e*(a + b*ArcTan[c*x]))/x^3 - 24*c^4*e*(-((a + b*ArcTan[c*x])/x) - (c*(a + b*ArcTan[c*x])^2)/(2*b) + b*c*(Log[x] - Log[1 + c^2*x^2]/2)) - 6* b*c^5*e*(2*Log[x] - Log[1 + c^2*x^2]) + 7*b*c^3*e*(-x^(-2) - 2*c^2*Log[x] + c^2*Log[1 + c^2*x^2]) - (3*b*c*(d + e*Log[1 + c^2*x^2]))/x^4 + (6*b*c^3* (d + e*Log[1 + c^2*x^2]))/x^2 - (12*(a + b*ArcTan[c*x])*(d + e*Log[1 + c^2 *x^2]))/x^5 - (3*b*c^5*(d + e*Log[1 + c^2*x^2])^2)/e + 6*b*c^5*(Log[-(c^2* x^2)]*(d + e*Log[1 + c^2*x^2]) + e*PolyLog[2, 1 + c^2*x^2]))/60
Time = 2.24 (sec) , antiderivative size = 290, normalized size of antiderivative = 1.17, number of steps used = 27, number of rules used = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {5552, 2925, 2858, 25, 27, 2789, 2756, 54, 2009, 2789, 2751, 16, 2779, 2838, 5453, 5361, 243, 54, 2009, 5453, 5361, 243, 47, 14, 16, 5419}
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.
| \(\displaystyle \int \frac {(a+b \arctan (c x)) \left (e \log \left (c^2 x^2+1\right )+d\right )}{x^6} \, dx\) |
| \(\Big \downarrow \) 5552 |
| \(\displaystyle \frac {2}{5} c^2 e \int \frac {a+b \arctan (c x)}{x^4 \left (c^2 x^2+1\right )}dx+\frac {1}{5} b c \int \frac {d+e \log \left (c^2 x^2+1\right )}{x^5 \left (c^2 x^2+1\right )}dx-\frac {(a+b \arctan (c x)) \left (e \log \left (c^2 x^2+1\right )+d\right )}{5 x^5}\) |
| \(\Big \downarrow \) 2925 |
| \(\displaystyle \frac {2}{5} c^2 e \int \frac {a+b \arctan (c x)}{x^4 \left (c^2 x^2+1\right )}dx+\frac {1}{10} b c \int \frac {d+e \log \left (c^2 x^2+1\right )}{x^6 \left (c^2 x^2+1\right )}dx^2-\frac {(a+b \arctan (c x)) \left (e \log \left (c^2 x^2+1\right )+d\right )}{5 x^5}\) |
| \(\Big \downarrow \) 2858 |
| \(\displaystyle \frac {2}{5} c^2 e \int \frac {a+b \arctan (c x)}{x^4 \left (c^2 x^2+1\right )}dx+\frac {b \int \frac {d+e \log \left (c^2 x^2+1\right )}{x^8}d\left (c^2 x^2+1\right )}{10 c}-\frac {(a+b \arctan (c x)) \left (e \log \left (c^2 x^2+1\right )+d\right )}{5 x^5}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2}{5} c^2 e \int \frac {a+b \arctan (c x)}{x^4 \left (c^2 x^2+1\right )}dx-\frac {b \int -\frac {d+e \log \left (c^2 x^2+1\right )}{x^8}d\left (c^2 x^2+1\right )}{10 c}-\frac {(a+b \arctan (c x)) \left (e \log \left (c^2 x^2+1\right )+d\right )}{5 x^5}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2}{5} c^2 e \int \frac {a+b \arctan (c x)}{x^4 \left (c^2 x^2+1\right )}dx-\frac {1}{10} b c^5 \int -\frac {d+e \log \left (c^2 x^2+1\right )}{c^6 x^8}d\left (c^2 x^2+1\right )-\frac {(a+b \arctan (c x)) \left (e \log \left (c^2 x^2+1\right )+d\right )}{5 x^5}\) |
| \(\Big \downarrow \) 2789 |
| \(\displaystyle \frac {2}{5} c^2 e \int \frac {a+b \arctan (c x)}{x^4 \left (c^2 x^2+1\right )}dx-\frac {1}{10} b c^5 \left (\int -\frac {d+e \log \left (c^2 x^2+1\right )}{c^6 x^6}d\left (c^2 x^2+1\right )+\int \frac {d+e \log \left (c^2 x^2+1\right )}{c^4 x^6}d\left (c^2 x^2+1\right )\right )-\frac {(a+b \arctan (c x)) \left (e \log \left (c^2 x^2+1\right )+d\right )}{5 x^5}\) |
| \(\Big \downarrow \) 2756 |
| \(\displaystyle \frac {2}{5} c^2 e \int \frac {a+b \arctan (c x)}{x^4 \left (c^2 x^2+1\right )}dx-\frac {1}{10} b c^5 \left (\int \frac {d+e \log \left (c^2 x^2+1\right )}{c^4 x^6}d\left (c^2 x^2+1\right )-\frac {1}{2} e \int \frac {1}{c^4 x^6}d\left (c^2 x^2+1\right )+\frac {e \log \left (c^2 x^2+1\right )+d}{2 c^4 x^4}\right )-\frac {(a+b \arctan (c x)) \left (e \log \left (c^2 x^2+1\right )+d\right )}{5 x^5}\) |
| \(\Big \downarrow \) 54 |
| \(\displaystyle \frac {2}{5} c^2 e \int \frac {a+b \arctan (c x)}{x^4 \left (c^2 x^2+1\right )}dx-\frac {1}{10} b c^5 \left (\int \frac {d+e \log \left (c^2 x^2+1\right )}{c^4 x^6}d\left (c^2 x^2+1\right )-\frac {1}{2} e \int \left (-\frac {1}{c^2 x^2}+\frac {1}{x^2}+\frac {1}{c^4 x^4}\right )d\left (c^2 x^2+1\right )+\frac {e \log \left (c^2 x^2+1\right )+d}{2 c^4 x^4}\right )-\frac {(a+b \arctan (c x)) \left (e \log \left (c^2 x^2+1\right )+d\right )}{5 x^5}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {2}{5} c^2 e \int \frac {a+b \arctan (c x)}{x^4 \left (c^2 x^2+1\right )}dx-\frac {1}{10} b c^5 \left (\int \frac {d+e \log \left (c^2 x^2+1\right )}{c^4 x^6}d\left (c^2 x^2+1\right )-\frac {1}{2} e \left (-\frac {1}{c^2 x^2}-\log \left (-c^2 x^2\right )+\log \left (c^2 x^2+1\right )\right )+\frac {e \log \left (c^2 x^2+1\right )+d}{2 c^4 x^4}\right )-\frac {(a+b \arctan (c x)) \left (e \log \left (c^2 x^2+1\right )+d\right )}{5 x^5}\) |
| \(\Big \downarrow \) 2789 |
| \(\displaystyle \frac {2}{5} c^2 e \int \frac {a+b \arctan (c x)}{x^4 \left (c^2 x^2+1\right )}dx-\frac {1}{10} b c^5 \left (\int -\frac {d+e \log \left (c^2 x^2+1\right )}{c^2 x^4}d\left (c^2 x^2+1\right )+\int \frac {d+e \log \left (c^2 x^2+1\right )}{c^4 x^4}d\left (c^2 x^2+1\right )-\frac {1}{2} e \left (-\frac {1}{c^2 x^2}-\log \left (-c^2 x^2\right )+\log \left (c^2 x^2+1\right )\right )+\frac {e \log \left (c^2 x^2+1\right )+d}{2 c^4 x^4}\right )-\frac {(a+b \arctan (c x)) \left (e \log \left (c^2 x^2+1\right )+d\right )}{5 x^5}\) |
| \(\Big \downarrow \) 2751 |
| \(\displaystyle \frac {2}{5} c^2 e \int \frac {a+b \arctan (c x)}{x^4 \left (c^2 x^2+1\right )}dx-\frac {1}{10} b c^5 \left (\int -\frac {d+e \log \left (c^2 x^2+1\right )}{c^2 x^4}d\left (c^2 x^2+1\right )-e \int -\frac {1}{c^2 x^2}d\left (c^2 x^2+1\right )-\frac {\left (c^2 x^2+1\right ) \left (e \log \left (c^2 x^2+1\right )+d\right )}{c^2 x^2}-\frac {1}{2} e \left (-\frac {1}{c^2 x^2}-\log \left (-c^2 x^2\right )+\log \left (c^2 x^2+1\right )\right )+\frac {e \log \left (c^2 x^2+1\right )+d}{2 c^4 x^4}\right )-\frac {(a+b \arctan (c x)) \left (e \log \left (c^2 x^2+1\right )+d\right )}{5 x^5}\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle \frac {2}{5} c^2 e \int \frac {a+b \arctan (c x)}{x^4 \left (c^2 x^2+1\right )}dx-\frac {1}{10} b c^5 \left (\int -\frac {d+e \log \left (c^2 x^2+1\right )}{c^2 x^4}d\left (c^2 x^2+1\right )-\frac {\left (c^2 x^2+1\right ) \left (e \log \left (c^2 x^2+1\right )+d\right )}{c^2 x^2}+e \log \left (-c^2 x^2\right )-\frac {1}{2} e \left (-\frac {1}{c^2 x^2}-\log \left (-c^2 x^2\right )+\log \left (c^2 x^2+1\right )\right )+\frac {e \log \left (c^2 x^2+1\right )+d}{2 c^4 x^4}\right )-\frac {(a+b \arctan (c x)) \left (e \log \left (c^2 x^2+1\right )+d\right )}{5 x^5}\) |
| \(\Big \downarrow \) 2779 |
| \(\displaystyle \frac {2}{5} c^2 e \int \frac {a+b \arctan (c x)}{x^4 \left (c^2 x^2+1\right )}dx-\frac {1}{10} b c^5 \left (e \int \frac {\log \left (1-\frac {1}{x^2}\right )}{x^2}d\left (c^2 x^2+1\right )-\frac {\left (c^2 x^2+1\right ) \left (e \log \left (c^2 x^2+1\right )+d\right )}{c^2 x^2}-\log \left (1-\frac {1}{x^2}\right ) \left (e \log \left (c^2 x^2+1\right )+d\right )+e \log \left (-c^2 x^2\right )-\frac {1}{2} e \left (-\frac {1}{c^2 x^2}-\log \left (-c^2 x^2\right )+\log \left (c^2 x^2+1\right )\right )+\frac {e \log \left (c^2 x^2+1\right )+d}{2 c^4 x^4}\right )-\frac {(a+b \arctan (c x)) \left (e \log \left (c^2 x^2+1\right )+d\right )}{5 x^5}\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle \frac {2}{5} c^2 e \int \frac {a+b \arctan (c x)}{x^4 \left (c^2 x^2+1\right )}dx-\frac {(a+b \arctan (c x)) \left (e \log \left (c^2 x^2+1\right )+d\right )}{5 x^5}-\frac {1}{10} b c^5 \left (-\frac {\left (c^2 x^2+1\right ) \left (e \log \left (c^2 x^2+1\right )+d\right )}{c^2 x^2}-\log \left (1-\frac {1}{x^2}\right ) \left (e \log \left (c^2 x^2+1\right )+d\right )+e \log \left (-c^2 x^2\right )-\frac {1}{2} e \left (-\frac {1}{c^2 x^2}-\log \left (-c^2 x^2\right )+\log \left (c^2 x^2+1\right )\right )+\frac {e \log \left (c^2 x^2+1\right )+d}{2 c^4 x^4}+e \operatorname {PolyLog}\left (2,\frac {1}{x^2}\right )\right )\) |
| \(\Big \downarrow \) 5453 |
| \(\displaystyle \frac {2}{5} c^2 e \left (\int \frac {a+b \arctan (c x)}{x^4}dx-c^2 \int \frac {a+b \arctan (c x)}{x^2 \left (c^2 x^2+1\right )}dx\right )-\frac {(a+b \arctan (c x)) \left (e \log \left (c^2 x^2+1\right )+d\right )}{5 x^5}-\frac {1}{10} b c^5 \left (-\frac {\left (c^2 x^2+1\right ) \left (e \log \left (c^2 x^2+1\right )+d\right )}{c^2 x^2}-\log \left (1-\frac {1}{x^2}\right ) \left (e \log \left (c^2 x^2+1\right )+d\right )+e \log \left (-c^2 x^2\right )-\frac {1}{2} e \left (-\frac {1}{c^2 x^2}-\log \left (-c^2 x^2\right )+\log \left (c^2 x^2+1\right )\right )+\frac {e \log \left (c^2 x^2+1\right )+d}{2 c^4 x^4}+e \operatorname {PolyLog}\left (2,\frac {1}{x^2}\right )\right )\) |
| \(\Big \downarrow \) 5361 |
| \(\displaystyle \frac {2}{5} c^2 e \left (c^2 \left (-\int \frac {a+b \arctan (c x)}{x^2 \left (c^2 x^2+1\right )}dx\right )+\frac {1}{3} b c \int \frac {1}{x^3 \left (c^2 x^2+1\right )}dx-\frac {a+b \arctan (c x)}{3 x^3}\right )-\frac {(a+b \arctan (c x)) \left (e \log \left (c^2 x^2+1\right )+d\right )}{5 x^5}-\frac {1}{10} b c^5 \left (-\frac {\left (c^2 x^2+1\right ) \left (e \log \left (c^2 x^2+1\right )+d\right )}{c^2 x^2}-\log \left (1-\frac {1}{x^2}\right ) \left (e \log \left (c^2 x^2+1\right )+d\right )+e \log \left (-c^2 x^2\right )-\frac {1}{2} e \left (-\frac {1}{c^2 x^2}-\log \left (-c^2 x^2\right )+\log \left (c^2 x^2+1\right )\right )+\frac {e \log \left (c^2 x^2+1\right )+d}{2 c^4 x^4}+e \operatorname {PolyLog}\left (2,\frac {1}{x^2}\right )\right )\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle \frac {2}{5} c^2 e \left (c^2 \left (-\int \frac {a+b \arctan (c x)}{x^2 \left (c^2 x^2+1\right )}dx\right )+\frac {1}{6} b c \int \frac {1}{x^4 \left (c^2 x^2+1\right )}dx^2-\frac {a+b \arctan (c x)}{3 x^3}\right )-\frac {(a+b \arctan (c x)) \left (e \log \left (c^2 x^2+1\right )+d\right )}{5 x^5}-\frac {1}{10} b c^5 \left (-\frac {\left (c^2 x^2+1\right ) \left (e \log \left (c^2 x^2+1\right )+d\right )}{c^2 x^2}-\log \left (1-\frac {1}{x^2}\right ) \left (e \log \left (c^2 x^2+1\right )+d\right )+e \log \left (-c^2 x^2\right )-\frac {1}{2} e \left (-\frac {1}{c^2 x^2}-\log \left (-c^2 x^2\right )+\log \left (c^2 x^2+1\right )\right )+\frac {e \log \left (c^2 x^2+1\right )+d}{2 c^4 x^4}+e \operatorname {PolyLog}\left (2,\frac {1}{x^2}\right )\right )\) |
| \(\Big \downarrow \) 54 |
| \(\displaystyle \frac {2}{5} c^2 e \left (c^2 \left (-\int \frac {a+b \arctan (c x)}{x^2 \left (c^2 x^2+1\right )}dx\right )+\frac {1}{6} b c \int \left (\frac {c^4}{c^2 x^2+1}-\frac {c^2}{x^2}+\frac {1}{x^4}\right )dx^2-\frac {a+b \arctan (c x)}{3 x^3}\right )-\frac {(a+b \arctan (c x)) \left (e \log \left (c^2 x^2+1\right )+d\right )}{5 x^5}-\frac {1}{10} b c^5 \left (-\frac {\left (c^2 x^2+1\right ) \left (e \log \left (c^2 x^2+1\right )+d\right )}{c^2 x^2}-\log \left (1-\frac {1}{x^2}\right ) \left (e \log \left (c^2 x^2+1\right )+d\right )+e \log \left (-c^2 x^2\right )-\frac {1}{2} e \left (-\frac {1}{c^2 x^2}-\log \left (-c^2 x^2\right )+\log \left (c^2 x^2+1\right )\right )+\frac {e \log \left (c^2 x^2+1\right )+d}{2 c^4 x^4}+e \operatorname {PolyLog}\left (2,\frac {1}{x^2}\right )\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {2}{5} c^2 e \left (c^2 \left (-\int \frac {a+b \arctan (c x)}{x^2 \left (c^2 x^2+1\right )}dx\right )-\frac {a+b \arctan (c x)}{3 x^3}+\frac {1}{6} b c \left (c^2 \left (-\log \left (x^2\right )\right )+c^2 \log \left (c^2 x^2+1\right )-\frac {1}{x^2}\right )\right )-\frac {(a+b \arctan (c x)) \left (e \log \left (c^2 x^2+1\right )+d\right )}{5 x^5}-\frac {1}{10} b c^5 \left (-\frac {\left (c^2 x^2+1\right ) \left (e \log \left (c^2 x^2+1\right )+d\right )}{c^2 x^2}-\log \left (1-\frac {1}{x^2}\right ) \left (e \log \left (c^2 x^2+1\right )+d\right )+e \log \left (-c^2 x^2\right )-\frac {1}{2} e \left (-\frac {1}{c^2 x^2}-\log \left (-c^2 x^2\right )+\log \left (c^2 x^2+1\right )\right )+\frac {e \log \left (c^2 x^2+1\right )+d}{2 c^4 x^4}+e \operatorname {PolyLog}\left (2,\frac {1}{x^2}\right )\right )\) |
| \(\Big \downarrow \) 5453 |
| \(\displaystyle \frac {2}{5} c^2 e \left (-\left (c^2 \left (\int \frac {a+b \arctan (c x)}{x^2}dx-c^2 \int \frac {a+b \arctan (c x)}{c^2 x^2+1}dx\right )\right )-\frac {a+b \arctan (c x)}{3 x^3}+\frac {1}{6} b c \left (c^2 \left (-\log \left (x^2\right )\right )+c^2 \log \left (c^2 x^2+1\right )-\frac {1}{x^2}\right )\right )-\frac {(a+b \arctan (c x)) \left (e \log \left (c^2 x^2+1\right )+d\right )}{5 x^5}-\frac {1}{10} b c^5 \left (-\frac {\left (c^2 x^2+1\right ) \left (e \log \left (c^2 x^2+1\right )+d\right )}{c^2 x^2}-\log \left (1-\frac {1}{x^2}\right ) \left (e \log \left (c^2 x^2+1\right )+d\right )+e \log \left (-c^2 x^2\right )-\frac {1}{2} e \left (-\frac {1}{c^2 x^2}-\log \left (-c^2 x^2\right )+\log \left (c^2 x^2+1\right )\right )+\frac {e \log \left (c^2 x^2+1\right )+d}{2 c^4 x^4}+e \operatorname {PolyLog}\left (2,\frac {1}{x^2}\right )\right )\) |
| \(\Big \downarrow \) 5361 |
| \(\displaystyle \frac {2}{5} c^2 e \left (-\left (c^2 \left (c^2 \left (-\int \frac {a+b \arctan (c x)}{c^2 x^2+1}dx\right )+b c \int \frac {1}{x \left (c^2 x^2+1\right )}dx-\frac {a+b \arctan (c x)}{x}\right )\right )-\frac {a+b \arctan (c x)}{3 x^3}+\frac {1}{6} b c \left (c^2 \left (-\log \left (x^2\right )\right )+c^2 \log \left (c^2 x^2+1\right )-\frac {1}{x^2}\right )\right )-\frac {(a+b \arctan (c x)) \left (e \log \left (c^2 x^2+1\right )+d\right )}{5 x^5}-\frac {1}{10} b c^5 \left (-\frac {\left (c^2 x^2+1\right ) \left (e \log \left (c^2 x^2+1\right )+d\right )}{c^2 x^2}-\log \left (1-\frac {1}{x^2}\right ) \left (e \log \left (c^2 x^2+1\right )+d\right )+e \log \left (-c^2 x^2\right )-\frac {1}{2} e \left (-\frac {1}{c^2 x^2}-\log \left (-c^2 x^2\right )+\log \left (c^2 x^2+1\right )\right )+\frac {e \log \left (c^2 x^2+1\right )+d}{2 c^4 x^4}+e \operatorname {PolyLog}\left (2,\frac {1}{x^2}\right )\right )\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle \frac {2}{5} c^2 e \left (-\left (c^2 \left (c^2 \left (-\int \frac {a+b \arctan (c x)}{c^2 x^2+1}dx\right )+\frac {1}{2} b c \int \frac {1}{x^2 \left (c^2 x^2+1\right )}dx^2-\frac {a+b \arctan (c x)}{x}\right )\right )-\frac {a+b \arctan (c x)}{3 x^3}+\frac {1}{6} b c \left (c^2 \left (-\log \left (x^2\right )\right )+c^2 \log \left (c^2 x^2+1\right )-\frac {1}{x^2}\right )\right )-\frac {(a+b \arctan (c x)) \left (e \log \left (c^2 x^2+1\right )+d\right )}{5 x^5}-\frac {1}{10} b c^5 \left (-\frac {\left (c^2 x^2+1\right ) \left (e \log \left (c^2 x^2+1\right )+d\right )}{c^2 x^2}-\log \left (1-\frac {1}{x^2}\right ) \left (e \log \left (c^2 x^2+1\right )+d\right )+e \log \left (-c^2 x^2\right )-\frac {1}{2} e \left (-\frac {1}{c^2 x^2}-\log \left (-c^2 x^2\right )+\log \left (c^2 x^2+1\right )\right )+\frac {e \log \left (c^2 x^2+1\right )+d}{2 c^4 x^4}+e \operatorname {PolyLog}\left (2,\frac {1}{x^2}\right )\right )\) |
| \(\Big \downarrow \) 47 |
| \(\displaystyle \frac {2}{5} c^2 e \left (-\left (c^2 \left (c^2 \left (-\int \frac {a+b \arctan (c x)}{c^2 x^2+1}dx\right )+\frac {1}{2} b c \left (\int \frac {1}{x^2}dx^2-c^2 \int \frac {1}{c^2 x^2+1}dx^2\right )-\frac {a+b \arctan (c x)}{x}\right )\right )-\frac {a+b \arctan (c x)}{3 x^3}+\frac {1}{6} b c \left (c^2 \left (-\log \left (x^2\right )\right )+c^2 \log \left (c^2 x^2+1\right )-\frac {1}{x^2}\right )\right )-\frac {(a+b \arctan (c x)) \left (e \log \left (c^2 x^2+1\right )+d\right )}{5 x^5}-\frac {1}{10} b c^5 \left (-\frac {\left (c^2 x^2+1\right ) \left (e \log \left (c^2 x^2+1\right )+d\right )}{c^2 x^2}-\log \left (1-\frac {1}{x^2}\right ) \left (e \log \left (c^2 x^2+1\right )+d\right )+e \log \left (-c^2 x^2\right )-\frac {1}{2} e \left (-\frac {1}{c^2 x^2}-\log \left (-c^2 x^2\right )+\log \left (c^2 x^2+1\right )\right )+\frac {e \log \left (c^2 x^2+1\right )+d}{2 c^4 x^4}+e \operatorname {PolyLog}\left (2,\frac {1}{x^2}\right )\right )\) |
| \(\Big \downarrow \) 14 |
| \(\displaystyle \frac {2}{5} c^2 e \left (-\left (c^2 \left (c^2 \left (-\int \frac {a+b \arctan (c x)}{c^2 x^2+1}dx\right )+\frac {1}{2} b c \left (\log \left (x^2\right )-c^2 \int \frac {1}{c^2 x^2+1}dx^2\right )-\frac {a+b \arctan (c x)}{x}\right )\right )-\frac {a+b \arctan (c x)}{3 x^3}+\frac {1}{6} b c \left (c^2 \left (-\log \left (x^2\right )\right )+c^2 \log \left (c^2 x^2+1\right )-\frac {1}{x^2}\right )\right )-\frac {(a+b \arctan (c x)) \left (e \log \left (c^2 x^2+1\right )+d\right )}{5 x^5}-\frac {1}{10} b c^5 \left (-\frac {\left (c^2 x^2+1\right ) \left (e \log \left (c^2 x^2+1\right )+d\right )}{c^2 x^2}-\log \left (1-\frac {1}{x^2}\right ) \left (e \log \left (c^2 x^2+1\right )+d\right )+e \log \left (-c^2 x^2\right )-\frac {1}{2} e \left (-\frac {1}{c^2 x^2}-\log \left (-c^2 x^2\right )+\log \left (c^2 x^2+1\right )\right )+\frac {e \log \left (c^2 x^2+1\right )+d}{2 c^4 x^4}+e \operatorname {PolyLog}\left (2,\frac {1}{x^2}\right )\right )\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle \frac {2}{5} c^2 e \left (-\left (c^2 \left (c^2 \left (-\int \frac {a+b \arctan (c x)}{c^2 x^2+1}dx\right )-\frac {a+b \arctan (c x)}{x}+\frac {1}{2} b c \left (\log \left (x^2\right )-\log \left (c^2 x^2+1\right )\right )\right )\right )-\frac {a+b \arctan (c x)}{3 x^3}+\frac {1}{6} b c \left (c^2 \left (-\log \left (x^2\right )\right )+c^2 \log \left (c^2 x^2+1\right )-\frac {1}{x^2}\right )\right )-\frac {(a+b \arctan (c x)) \left (e \log \left (c^2 x^2+1\right )+d\right )}{5 x^5}-\frac {1}{10} b c^5 \left (-\frac {\left (c^2 x^2+1\right ) \left (e \log \left (c^2 x^2+1\right )+d\right )}{c^2 x^2}-\log \left (1-\frac {1}{x^2}\right ) \left (e \log \left (c^2 x^2+1\right )+d\right )+e \log \left (-c^2 x^2\right )-\frac {1}{2} e \left (-\frac {1}{c^2 x^2}-\log \left (-c^2 x^2\right )+\log \left (c^2 x^2+1\right )\right )+\frac {e \log \left (c^2 x^2+1\right )+d}{2 c^4 x^4}+e \operatorname {PolyLog}\left (2,\frac {1}{x^2}\right )\right )\) |
| \(\Big \downarrow \) 5419 |
| \(\displaystyle -\frac {(a+b \arctan (c x)) \left (e \log \left (c^2 x^2+1\right )+d\right )}{5 x^5}+\frac {2}{5} c^2 e \left (-\left (c^2 \left (-\frac {c (a+b \arctan (c x))^2}{2 b}-\frac {a+b \arctan (c x)}{x}+\frac {1}{2} b c \left (\log \left (x^2\right )-\log \left (c^2 x^2+1\right )\right )\right )\right )-\frac {a+b \arctan (c x)}{3 x^3}+\frac {1}{6} b c \left (c^2 \left (-\log \left (x^2\right )\right )+c^2 \log \left (c^2 x^2+1\right )-\frac {1}{x^2}\right )\right )-\frac {1}{10} b c^5 \left (-\frac {\left (c^2 x^2+1\right ) \left (e \log \left (c^2 x^2+1\right )+d\right )}{c^2 x^2}-\log \left (1-\frac {1}{x^2}\right ) \left (e \log \left (c^2 x^2+1\right )+d\right )+e \log \left (-c^2 x^2\right )-\frac {1}{2} e \left (-\frac {1}{c^2 x^2}-\log \left (-c^2 x^2\right )+\log \left (c^2 x^2+1\right )\right )+\frac {e \log \left (c^2 x^2+1\right )+d}{2 c^4 x^4}+e \operatorname {PolyLog}\left (2,\frac {1}{x^2}\right )\right )\) |
-1/5*((a + b*ArcTan[c*x])*(d + e*Log[1 + c^2*x^2]))/x^5 + (2*c^2*e*(-1/3*( a + b*ArcTan[c*x])/x^3 - c^2*(-((a + b*ArcTan[c*x])/x) - (c*(a + b*ArcTan[ c*x])^2)/(2*b) + (b*c*(Log[x^2] - Log[1 + c^2*x^2]))/2) + (b*c*(-x^(-2) - c^2*Log[x^2] + c^2*Log[1 + c^2*x^2]))/6))/5 - (b*c^5*(e*Log[-(c^2*x^2)] - (e*(-(1/(c^2*x^2)) - Log[-(c^2*x^2)] + Log[1 + c^2*x^2]))/2 + (d + e*Log[1 + c^2*x^2])/(2*c^4*x^4) - ((1 + c^2*x^2)*(d + e*Log[1 + c^2*x^2]))/(c^2*x ^2) - Log[1 - x^(-2)]*(d + e*Log[1 + c^2*x^2]) + e*PolyLog[2, x^(-2)]))/10
3.13.96.3.1 Defintions of rubi rules used
Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Simp[b/(b*c - a*d) Int[1/(a + b*x), x], x] - Simp[d/(b*c - a*d) Int[1/(c + d*x), x ], x] /; FreeQ[{a, b, c, d}, x]
Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[E xpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && ILtQ[m, 0] && IntegerQ[n] && !(IGtQ[n, 0] && LtQ[m + n + 2, 0])
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2 Subst[In t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I ntegerQ[(m - 1)/2]
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_) + (e_.)*(x_)^(r_.))^(q_), x _Symbol] :> Simp[x*(d + e*x^r)^(q + 1)*((a + b*Log[c*x^n])/d), x] - Simp[b* (n/d) Int[(d + e*x^r)^(q + 1), x], x] /; FreeQ[{a, b, c, d, e, n, q, r}, x] && EqQ[r*(q + 1) + 1, 0]
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_))^(q_.), x_Symbol] :> Simp[(d + e*x)^(q + 1)*((a + b*Log[c*x^n])^p/(e*(q + 1))), x] - Simp[b*n*(p/(e*(q + 1))) Int[((d + e*x)^(q + 1)*(a + b*Log[c*x^n])^(p - 1))/x, x], x] /; FreeQ[{a, b, c, d, e, n, p, q}, x] && GtQ[p, 0] && NeQ[q, -1] && (EqQ[p, 1] || (IntegersQ[2*p, 2*q] && !IGtQ[q, 0]) || (EqQ[p, 2] & & NeQ[q, 1]))
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/((x_)*((d_) + (e_.)*(x_)^(r _.))), x_Symbol] :> Simp[(-Log[1 + d/(e*x^r)])*((a + b*Log[c*x^n])^p/(d*r)) , x] + Simp[b*n*(p/(d*r)) Int[Log[1 + d/(e*x^r)]*((a + b*Log[c*x^n])^(p - 1)/x), x], x] /; FreeQ[{a, b, c, d, e, n, r}, x] && IGtQ[p, 0]
Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_))^(q_))/ (x_), x_Symbol] :> Simp[1/d Int[(d + e*x)^(q + 1)*((a + b*Log[c*x^n])^p/x ), x], x] - Simp[e/d Int[(d + e*x)^q*(a + b*Log[c*x^n])^p, x], x] /; Free Q[{a, b, c, d, e, n}, x] && IGtQ[p, 0] && LtQ[q, -1] && IntegerQ[2*q]
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_))^(q_.)*((h_.) + (i_.)*(x_))^(r_.), x_Symbol] :> Simp[1/e Subst[In t[(g*(x/e))^q*((e*h - d*i)/e + i*(x/e))^r*(a + b*Log[c*x^n])^p, x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, h, i, n, p, q, r}, x] && EqQ[e*f - d*g, 0] && (IGtQ[p, 0] || IGtQ[r, 0]) && IntegerQ[2*r]
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*(x_)^(m _.)*((f_) + (g_.)*(x_)^(s_))^(r_.), x_Symbol] :> Simp[1/n Subst[Int[x^(Si mplify[(m + 1)/n] - 1)*(f + g*x^(s/n))^r*(a + b*Log[c*(d + e*x)^p])^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, e, f, g, m, n, p, q, r, s}, x] && Integer Q[r] && IntegerQ[s/n] && IntegerQ[Simplify[(m + 1)/n]] && (GtQ[(m + 1)/n, 0 ] || IGtQ[q, 0])
Int[((a_.) + ArcTan[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)*((a + b*ArcTan[c*x^n])^p/(m + 1)), x] - Simp[b*c*n*(p/(m + 1)) Int[x^(m + n)*((a + b*ArcTan[c*x^n])^(p - 1)/(1 + c^2*x^(2*n))), x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0] && (EqQ[p, 1] || (EqQ[n, 1] & & IntegerQ[m])) && NeQ[m, -1]
Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)^2), x_Symbo l] :> Simp[(a + b*ArcTan[c*x])^(p + 1)/(b*c*d*(p + 1)), x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[e, c^2*d] && NeQ[p, -1]
Int[(((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_))/((d_) + (e _.)*(x_)^2), x_Symbol] :> Simp[1/d Int[(f*x)^m*(a + b*ArcTan[c*x])^p, x], x] - Simp[e/(d*f^2) Int[(f*x)^(m + 2)*((a + b*ArcTan[c*x])^p/(d + e*x^2) ), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[p, 0] && LtQ[m, -1]
Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))*((d_.) + Log[(f_.) + (g_.)*(x_)^2]*( e_.))*(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)*(d + e*Log[f + g*x^2])*((a + b*ArcTan[c*x])/(m + 1)), x] + (-Simp[b*(c/(m + 1)) Int[x^(m + 1)*((d + e* Log[f + g*x^2])/(1 + c^2*x^2)), x], x] - Simp[2*e*(g/(m + 1)) Int[x^(m + 2)*((a + b*ArcTan[c*x])/(f + g*x^2)), x], x]) /; FreeQ[{a, b, c, d, e, f, g }, x] && ILtQ[m/2, 0]
\[\int \frac {\left (a +b \arctan \left (c x \right )\right ) \left (d +e \ln \left (c^{2} x^{2}+1\right )\right )}{x^{6}}d x\]
\[ \int \frac {(a+b \arctan (c x)) \left (d+e \log \left (1+c^2 x^2\right )\right )}{x^6} \, dx=\int { \frac {{\left (b \arctan \left (c x\right ) + a\right )} {\left (e \log \left (c^{2} x^{2} + 1\right ) + d\right )}}{x^{6}} \,d x } \]
Exception generated. \[ \int \frac {(a+b \arctan (c x)) \left (d+e \log \left (1+c^2 x^2\right )\right )}{x^6} \, dx=\text {Exception raised: TypeError} \]
\[ \int \frac {(a+b \arctan (c x)) \left (d+e \log \left (1+c^2 x^2\right )\right )}{x^6} \, dx=\int { \frac {{\left (b \arctan \left (c x\right ) + a\right )} {\left (e \log \left (c^{2} x^{2} + 1\right ) + d\right )}}{x^{6}} \,d x } \]
-1/20*((2*c^4*log(c^2*x^2 + 1) - 2*c^4*log(x^2) - (2*c^2*x^2 - 1)/x^4)*c + 4*arctan(c*x)/x^5)*b*d + 1/15*(2*(3*c^3*arctan(c*x) + (3*c^2*x^2 - 1)/x^3 )*c^2 - 3*log(c^2*x^2 + 1)/x^5)*a*e + b*e*integrate(arctan(c*x)*log(c^2*x^ 2 + 1)/x^6, x) - 1/5*a*d/x^5
Timed out. \[ \int \frac {(a+b \arctan (c x)) \left (d+e \log \left (1+c^2 x^2\right )\right )}{x^6} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {(a+b \arctan (c x)) \left (d+e \log \left (1+c^2 x^2\right )\right )}{x^6} \, dx=\int \frac {\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )\,\left (d+e\,\ln \left (c^2\,x^2+1\right )\right )}{x^6} \,d x \]