Integrand size = 22, antiderivative size = 759 \[ \int x \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^3 \, dx =\text {Too large to display} \] Output:
71/40*b^3*e^5*n^3*x^(1/3)/d^5-3/10*b^3*e^4*n^3*x^(2/3)/d^4+1/20*b^3*e^3*n^ 3*x/d^3-71/40*b^3*e^6*n^3*ln(d+e/x^(1/3))/d^6-77/20*b^2*e^5*n^2*(d+e/x^(1/ 3))*x^(1/3)*(a+b*ln(c*(d+e/x^(1/3))^n))/d^6+47/40*b^2*e^4*n^2*x^(2/3)*(a+b *ln(c*(d+e/x^(1/3))^n))/d^4-9/20*b^2*e^3*n^2*x*(a+b*ln(c*(d+e/x^(1/3))^n)) /d^3+3/20*b^2*e^2*n^2*x^(4/3)*(a+b*ln(c*(d+e/x^(1/3))^n))/d^2-77/20*b^2*e^ 6*n^2*ln(1-d/(d+e/x^(1/3)))*(a+b*ln(c*(d+e/x^(1/3))^n))/d^6+3/2*b*e^5*n*(d +e/x^(1/3))*x^(1/3)*(a+b*ln(c*(d+e/x^(1/3))^n))^2/d^6-3/4*b*e^4*n*x^(2/3)* (a+b*ln(c*(d+e/x^(1/3))^n))^2/d^4+1/2*b*e^3*n*x*(a+b*ln(c*(d+e/x^(1/3))^n) )^2/d^3-3/8*b*e^2*n*x^(4/3)*(a+b*ln(c*(d+e/x^(1/3))^n))^2/d^2+3/10*b*e*n*x ^(5/3)*(a+b*ln(c*(d+e/x^(1/3))^n))^2/d+3/2*b*e^6*n*ln(1-d/(d+e/x^(1/3)))*( a+b*ln(c*(d+e/x^(1/3))^n))^2/d^6+1/2*x^2*(a+b*ln(c*(d+e/x^(1/3))^n))^3-3*b ^2*e^6*n^2*(a+b*ln(c*(d+e/x^(1/3))^n))*ln(-e/d/x^(1/3))/d^6-15/8*b^3*e^6*n ^3*ln(x)/d^6+77/20*b^3*e^6*n^3*polylog(2,d/(d+e/x^(1/3)))/d^6-3*b^2*e^6*n^ 2*(a+b*ln(c*(d+e/x^(1/3))^n))*polylog(2,d/(d+e/x^(1/3)))/d^6-3*b^3*e^6*n^3 *polylog(2,1+e/d/x^(1/3))/d^6-3*b^3*e^6*n^3*polylog(3,d/(d+e/x^(1/3)))/d^6
\[ \int x \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^3 \, dx=\int x \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^3 \, dx \] Input:
Integrate[x*(a + b*Log[c*(d + e/x^(1/3))^n])^3,x]
Output:
Integrate[x*(a + b*Log[c*(d + e/x^(1/3))^n])^3, x]
Time = 10.85 (sec) , antiderivative size = 1374, normalized size of antiderivative = 1.81, number of steps used = 28, number of rules used = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.227, Rules used = {2904, 2845, 2858, 27, 2789, 2756, 2789, 2756, 54, 2009, 2789, 2756, 54, 2009, 2789, 2756, 54, 2009, 2789, 2751, 16, 2755, 2754, 2779, 2821, 2838, 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.
| \(\displaystyle \int x \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^3 \, dx\) |
| \(\Big \downarrow \) 2904 |
| \(\displaystyle -3 \int x^{7/3} \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^3d\frac {1}{\sqrt [3]{x}}\) |
| \(\Big \downarrow \) 2845 |
| \(\displaystyle -3 \left (\frac {1}{2} b e n \int \frac {x^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^2}{d+\frac {e}{\sqrt [3]{x}}}d\frac {1}{\sqrt [3]{x}}-\frac {1}{6} x^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^3\right )\) |
| \(\Big \downarrow \) 2858 |
| \(\displaystyle -3 \left (\frac {1}{2} b n \int x^{7/3} \left (a+b \log \left (c x^{-n/3}\right )\right )^2d\left (d+\frac {e}{\sqrt [3]{x}}\right )-\frac {1}{6} x^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^3\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -3 \left (\frac {1}{2} b e^6 n \int \frac {x^{7/3} \left (a+b \log \left (c x^{-n/3}\right )\right )^2}{e^6}d\left (d+\frac {e}{\sqrt [3]{x}}\right )-\frac {1}{6} x^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^3\right )\) |
| \(\Big \downarrow \) 2789 |
| \(\displaystyle -3 \left (\frac {1}{2} b e^6 n \left (\frac {\int \frac {x^2 \left (a+b \log \left (c x^{-n/3}\right )\right )^2}{e^6}d\left (d+\frac {e}{\sqrt [3]{x}}\right )}{d}+\frac {\int -\frac {x^2 \left (a+b \log \left (c x^{-n/3}\right )\right )^2}{e^5}d\left (d+\frac {e}{\sqrt [3]{x}}\right )}{d}\right )-\frac {1}{6} x^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^3\right )\) |
| \(\Big \downarrow \) 2756 |
| \(\displaystyle -3 \left (\frac {1}{2} b e^6 n \left (\frac {-\frac {2}{5} b n \int -\frac {x^2 \left (a+b \log \left (c x^{-n/3}\right )\right )}{e^5}d\left (d+\frac {e}{\sqrt [3]{x}}\right )-\frac {x^{5/3} \left (a+b \log \left (c x^{-n/3}\right )\right )^2}{5 e^5}}{d}+\frac {\int -\frac {x^2 \left (a+b \log \left (c x^{-n/3}\right )\right )^2}{e^5}d\left (d+\frac {e}{\sqrt [3]{x}}\right )}{d}\right )-\frac {1}{6} x^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^3\right )\) |
| \(\Big \downarrow \) 2789 |
| \(\displaystyle -3 \left (\frac {1}{2} b e^6 n \left (\frac {-\frac {2}{5} b n \left (\frac {\int -\frac {x^{5/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{e^5}d\left (d+\frac {e}{\sqrt [3]{x}}\right )}{d}+\frac {\int \frac {x^{5/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{e^4}d\left (d+\frac {e}{\sqrt [3]{x}}\right )}{d}\right )-\frac {x^{5/3} \left (a+b \log \left (c x^{-n/3}\right )\right )^2}{5 e^5}}{d}+\frac {\frac {\int -\frac {x^{5/3} \left (a+b \log \left (c x^{-n/3}\right )\right )^2}{e^5}d\left (d+\frac {e}{\sqrt [3]{x}}\right )}{d}+\frac {\int \frac {x^{5/3} \left (a+b \log \left (c x^{-n/3}\right )\right )^2}{e^4}d\left (d+\frac {e}{\sqrt [3]{x}}\right )}{d}}{d}\right )-\frac {1}{6} x^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^3\right )\) |
| \(\Big \downarrow \) 2756 |
| \(\displaystyle -3 \left (\frac {1}{2} b e^6 n \left (\frac {\frac {\frac {x^{4/3} \left (a+b \log \left (c x^{-n/3}\right )\right )^2}{4 e^4}-\frac {1}{2} b n \int \frac {x^{5/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{e^4}d\left (d+\frac {e}{\sqrt [3]{x}}\right )}{d}+\frac {\int \frac {x^{5/3} \left (a+b \log \left (c x^{-n/3}\right )\right )^2}{e^4}d\left (d+\frac {e}{\sqrt [3]{x}}\right )}{d}}{d}+\frac {-\frac {2}{5} b n \left (\frac {\frac {x^{4/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{4 e^4}-\frac {1}{4} b n \int \frac {x^{5/3}}{e^4}d\left (d+\frac {e}{\sqrt [3]{x}}\right )}{d}+\frac {\int \frac {x^{5/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{e^4}d\left (d+\frac {e}{\sqrt [3]{x}}\right )}{d}\right )-\frac {x^{5/3} \left (a+b \log \left (c x^{-n/3}\right )\right )^2}{5 e^5}}{d}\right )-\frac {1}{6} x^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^3\right )\) |
| \(\Big \downarrow \) 54 |
| \(\displaystyle -3 \left (\frac {1}{2} b e^6 n \left (\frac {-\frac {2}{5} b n \left (\frac {\frac {x^{4/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{4 e^4}-\frac {1}{4} b n \int \left (\frac {x^{4/3}}{d e^4}-\frac {x}{d^2 e^3}+\frac {x^{2/3}}{d^3 e^2}-\frac {\sqrt [3]{x}}{d^4 e}+\frac {\sqrt [3]{x}}{d^4}\right )d\left (d+\frac {e}{\sqrt [3]{x}}\right )}{d}+\frac {\int \frac {x^{5/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{e^4}d\left (d+\frac {e}{\sqrt [3]{x}}\right )}{d}\right )-\frac {x^{5/3} \left (a+b \log \left (c x^{-n/3}\right )\right )^2}{5 e^5}}{d}+\frac {\frac {\frac {x^{4/3} \left (a+b \log \left (c x^{-n/3}\right )\right )^2}{4 e^4}-\frac {1}{2} b n \int \frac {x^{5/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{e^4}d\left (d+\frac {e}{\sqrt [3]{x}}\right )}{d}+\frac {\int \frac {x^{5/3} \left (a+b \log \left (c x^{-n/3}\right )\right )^2}{e^4}d\left (d+\frac {e}{\sqrt [3]{x}}\right )}{d}}{d}\right )-\frac {1}{6} x^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^3\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -3 \left (\frac {1}{2} b e^6 n \left (\frac {-\frac {2}{5} b n \left (\frac {\int \frac {x^{5/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{e^4}d\left (d+\frac {e}{\sqrt [3]{x}}\right )}{d}+\frac {\frac {x^{4/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{4 e^4}-\frac {1}{4} b n \left (\frac {\log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^4}-\frac {\log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d^4}-\frac {\sqrt [3]{x}}{d^3 e}+\frac {x^{2/3}}{2 d^2 e^2}-\frac {x}{3 d e^3}\right )}{d}\right )-\frac {x^{5/3} \left (a+b \log \left (c x^{-n/3}\right )\right )^2}{5 e^5}}{d}+\frac {\frac {\frac {x^{4/3} \left (a+b \log \left (c x^{-n/3}\right )\right )^2}{4 e^4}-\frac {1}{2} b n \int \frac {x^{5/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{e^4}d\left (d+\frac {e}{\sqrt [3]{x}}\right )}{d}+\frac {\int \frac {x^{5/3} \left (a+b \log \left (c x^{-n/3}\right )\right )^2}{e^4}d\left (d+\frac {e}{\sqrt [3]{x}}\right )}{d}}{d}\right )-\frac {1}{6} x^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^3\right )\) |
| \(\Big \downarrow \) 2789 |
| \(\displaystyle -3 \left (\frac {1}{2} b e^6 n \left (\frac {-\frac {2}{5} b n \left (\frac {\frac {\int \frac {x^{4/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{e^4}d\left (d+\frac {e}{\sqrt [3]{x}}\right )}{d}+\frac {\int -\frac {x^{4/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{e^3}d\left (d+\frac {e}{\sqrt [3]{x}}\right )}{d}}{d}+\frac {\frac {x^{4/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{4 e^4}-\frac {1}{4} b n \left (\frac {\log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^4}-\frac {\log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d^4}-\frac {\sqrt [3]{x}}{d^3 e}+\frac {x^{2/3}}{2 d^2 e^2}-\frac {x}{3 d e^3}\right )}{d}\right )-\frac {x^{5/3} \left (a+b \log \left (c x^{-n/3}\right )\right )^2}{5 e^5}}{d}+\frac {\frac {\frac {x^{4/3} \left (a+b \log \left (c x^{-n/3}\right )\right )^2}{4 e^4}-\frac {1}{2} b n \left (\frac {\int \frac {x^{4/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{e^4}d\left (d+\frac {e}{\sqrt [3]{x}}\right )}{d}+\frac {\int -\frac {x^{4/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{e^3}d\left (d+\frac {e}{\sqrt [3]{x}}\right )}{d}\right )}{d}+\frac {\frac {\int \frac {x^{4/3} \left (a+b \log \left (c x^{-n/3}\right )\right )^2}{e^4}d\left (d+\frac {e}{\sqrt [3]{x}}\right )}{d}+\frac {\int -\frac {x^{4/3} \left (a+b \log \left (c x^{-n/3}\right )\right )^2}{e^3}d\left (d+\frac {e}{\sqrt [3]{x}}\right )}{d}}{d}}{d}\right )-\frac {1}{6} x^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^3\right )\) |
| \(\Big \downarrow \) 2756 |
| \(\displaystyle -3 \left (\frac {1}{2} b e^6 n \left (\frac {-\frac {2}{5} b n \left (\frac {\frac {-\frac {1}{3} b n \int -\frac {x^{4/3}}{e^3}d\left (d+\frac {e}{\sqrt [3]{x}}\right )-\frac {x \left (a+b \log \left (c x^{-n/3}\right )\right )}{3 e^3}}{d}+\frac {\int -\frac {x^{4/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{e^3}d\left (d+\frac {e}{\sqrt [3]{x}}\right )}{d}}{d}+\frac {\frac {x^{4/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{4 e^4}-\frac {1}{4} b n \left (\frac {\log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^4}-\frac {\log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d^4}-\frac {\sqrt [3]{x}}{d^3 e}+\frac {x^{2/3}}{2 d^2 e^2}-\frac {x}{3 d e^3}\right )}{d}\right )-\frac {x^{5/3} \left (a+b \log \left (c x^{-n/3}\right )\right )^2}{5 e^5}}{d}+\frac {\frac {\frac {-\frac {2}{3} b n \int -\frac {x^{4/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{e^3}d\left (d+\frac {e}{\sqrt [3]{x}}\right )-\frac {x \left (a+b \log \left (c x^{-n/3}\right )\right )^2}{3 e^3}}{d}+\frac {\int -\frac {x^{4/3} \left (a+b \log \left (c x^{-n/3}\right )\right )^2}{e^3}d\left (d+\frac {e}{\sqrt [3]{x}}\right )}{d}}{d}+\frac {\frac {x^{4/3} \left (a+b \log \left (c x^{-n/3}\right )\right )^2}{4 e^4}-\frac {1}{2} b n \left (\frac {-\frac {1}{3} b n \int -\frac {x^{4/3}}{e^3}d\left (d+\frac {e}{\sqrt [3]{x}}\right )-\frac {x \left (a+b \log \left (c x^{-n/3}\right )\right )}{3 e^3}}{d}+\frac {\int -\frac {x^{4/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{e^3}d\left (d+\frac {e}{\sqrt [3]{x}}\right )}{d}\right )}{d}}{d}\right )-\frac {1}{6} x^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^3\right )\) |
| \(\Big \downarrow \) 54 |
| \(\displaystyle -3 \left (\frac {1}{2} b e^6 n \left (\frac {\frac {\frac {x^{4/3} \left (a+b \log \left (c x^{-n/3}\right )\right )^2}{4 e^4}-\frac {1}{2} b n \left (\frac {-\frac {1}{3} b n \int \left (-\frac {x}{d e^3}+\frac {x^{2/3}}{d^2 e^2}-\frac {\sqrt [3]{x}}{d^3 e}+\frac {\sqrt [3]{x}}{d^3}\right )d\left (d+\frac {e}{\sqrt [3]{x}}\right )-\frac {x \left (a+b \log \left (c x^{-n/3}\right )\right )}{3 e^3}}{d}+\frac {\int -\frac {x^{4/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{e^3}d\left (d+\frac {e}{\sqrt [3]{x}}\right )}{d}\right )}{d}+\frac {\frac {-\frac {2}{3} b n \int -\frac {x^{4/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{e^3}d\left (d+\frac {e}{\sqrt [3]{x}}\right )-\frac {x \left (a+b \log \left (c x^{-n/3}\right )\right )^2}{3 e^3}}{d}+\frac {\int -\frac {x^{4/3} \left (a+b \log \left (c x^{-n/3}\right )\right )^2}{e^3}d\left (d+\frac {e}{\sqrt [3]{x}}\right )}{d}}{d}}{d}+\frac {-\frac {2}{5} b n \left (\frac {\frac {-\frac {1}{3} b n \int \left (-\frac {x}{d e^3}+\frac {x^{2/3}}{d^2 e^2}-\frac {\sqrt [3]{x}}{d^3 e}+\frac {\sqrt [3]{x}}{d^3}\right )d\left (d+\frac {e}{\sqrt [3]{x}}\right )-\frac {x \left (a+b \log \left (c x^{-n/3}\right )\right )}{3 e^3}}{d}+\frac {\int -\frac {x^{4/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{e^3}d\left (d+\frac {e}{\sqrt [3]{x}}\right )}{d}}{d}+\frac {\frac {x^{4/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{4 e^4}-\frac {1}{4} b n \left (\frac {\log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^4}-\frac {\log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d^4}-\frac {\sqrt [3]{x}}{d^3 e}+\frac {x^{2/3}}{2 d^2 e^2}-\frac {x}{3 d e^3}\right )}{d}\right )-\frac {x^{5/3} \left (a+b \log \left (c x^{-n/3}\right )\right )^2}{5 e^5}}{d}\right )-\frac {1}{6} x^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^3\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -3 \left (\frac {1}{2} b e^6 n \left (\frac {\frac {\frac {x^{4/3} \left (a+b \log \left (c x^{-n/3}\right )\right )^2}{4 e^4}-\frac {1}{2} b n \left (\frac {\int -\frac {x^{4/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{e^3}d\left (d+\frac {e}{\sqrt [3]{x}}\right )}{d}+\frac {-\frac {x \left (a+b \log \left (c x^{-n/3}\right )\right )}{3 e^3}-\frac {1}{3} b n \left (\frac {\log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^3}-\frac {\log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d^3}-\frac {\sqrt [3]{x}}{d^2 e}+\frac {x^{2/3}}{2 d e^2}\right )}{d}\right )}{d}+\frac {\frac {-\frac {2}{3} b n \int -\frac {x^{4/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{e^3}d\left (d+\frac {e}{\sqrt [3]{x}}\right )-\frac {x \left (a+b \log \left (c x^{-n/3}\right )\right )^2}{3 e^3}}{d}+\frac {\int -\frac {x^{4/3} \left (a+b \log \left (c x^{-n/3}\right )\right )^2}{e^3}d\left (d+\frac {e}{\sqrt [3]{x}}\right )}{d}}{d}}{d}+\frac {-\frac {2}{5} b n \left (\frac {\frac {\int -\frac {x^{4/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{e^3}d\left (d+\frac {e}{\sqrt [3]{x}}\right )}{d}+\frac {-\frac {x \left (a+b \log \left (c x^{-n/3}\right )\right )}{3 e^3}-\frac {1}{3} b n \left (\frac {\log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^3}-\frac {\log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d^3}-\frac {\sqrt [3]{x}}{d^2 e}+\frac {x^{2/3}}{2 d e^2}\right )}{d}}{d}+\frac {\frac {x^{4/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{4 e^4}-\frac {1}{4} b n \left (\frac {\log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^4}-\frac {\log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d^4}-\frac {\sqrt [3]{x}}{d^3 e}+\frac {x^{2/3}}{2 d^2 e^2}-\frac {x}{3 d e^3}\right )}{d}\right )-\frac {x^{5/3} \left (a+b \log \left (c x^{-n/3}\right )\right )^2}{5 e^5}}{d}\right )-\frac {1}{6} x^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^3\right )\) |
| \(\Big \downarrow \) 2789 |
| \(\displaystyle -3 \left (\frac {1}{2} b e^6 n \left (\frac {\frac {\frac {x^{4/3} \left (a+b \log \left (c x^{-n/3}\right )\right )^2}{4 e^4}-\frac {1}{2} b n \left (\frac {\frac {\int -\frac {x \left (a+b \log \left (c x^{-n/3}\right )\right )}{e^3}d\left (d+\frac {e}{\sqrt [3]{x}}\right )}{d}+\frac {\int \frac {x \left (a+b \log \left (c x^{-n/3}\right )\right )}{e^2}d\left (d+\frac {e}{\sqrt [3]{x}}\right )}{d}}{d}+\frac {-\frac {x \left (a+b \log \left (c x^{-n/3}\right )\right )}{3 e^3}-\frac {1}{3} b n \left (\frac {\log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^3}-\frac {\log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d^3}-\frac {\sqrt [3]{x}}{d^2 e}+\frac {x^{2/3}}{2 d e^2}\right )}{d}\right )}{d}+\frac {\frac {-\frac {2}{3} b n \left (\frac {\int -\frac {x \left (a+b \log \left (c x^{-n/3}\right )\right )}{e^3}d\left (d+\frac {e}{\sqrt [3]{x}}\right )}{d}+\frac {\int \frac {x \left (a+b \log \left (c x^{-n/3}\right )\right )}{e^2}d\left (d+\frac {e}{\sqrt [3]{x}}\right )}{d}\right )-\frac {x \left (a+b \log \left (c x^{-n/3}\right )\right )^2}{3 e^3}}{d}+\frac {\frac {\int -\frac {x \left (a+b \log \left (c x^{-n/3}\right )\right )^2}{e^3}d\left (d+\frac {e}{\sqrt [3]{x}}\right )}{d}+\frac {\int \frac {x \left (a+b \log \left (c x^{-n/3}\right )\right )^2}{e^2}d\left (d+\frac {e}{\sqrt [3]{x}}\right )}{d}}{d}}{d}}{d}+\frac {-\frac {2}{5} b n \left (\frac {\frac {\frac {\int -\frac {x \left (a+b \log \left (c x^{-n/3}\right )\right )}{e^3}d\left (d+\frac {e}{\sqrt [3]{x}}\right )}{d}+\frac {\int \frac {x \left (a+b \log \left (c x^{-n/3}\right )\right )}{e^2}d\left (d+\frac {e}{\sqrt [3]{x}}\right )}{d}}{d}+\frac {-\frac {x \left (a+b \log \left (c x^{-n/3}\right )\right )}{3 e^3}-\frac {1}{3} b n \left (\frac {\log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^3}-\frac {\log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d^3}-\frac {\sqrt [3]{x}}{d^2 e}+\frac {x^{2/3}}{2 d e^2}\right )}{d}}{d}+\frac {\frac {x^{4/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{4 e^4}-\frac {1}{4} b n \left (\frac {\log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^4}-\frac {\log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d^4}-\frac {\sqrt [3]{x}}{d^3 e}+\frac {x^{2/3}}{2 d^2 e^2}-\frac {x}{3 d e^3}\right )}{d}\right )-\frac {x^{5/3} \left (a+b \log \left (c x^{-n/3}\right )\right )^2}{5 e^5}}{d}\right )-\frac {1}{6} x^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^3\right )\) |
| \(\Big \downarrow \) 2756 |
| \(\displaystyle -3 \left (\frac {1}{2} b e^6 n \left (\frac {-\frac {x^{5/3} \left (a+b \log \left (c x^{-n/3}\right )\right )^2}{5 e^5}-\frac {2}{5} b n \left (\frac {\frac {x^{4/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{4 e^4}-\frac {1}{4} b n \left (-\frac {x}{3 d e^3}+\frac {x^{2/3}}{2 d^2 e^2}-\frac {\sqrt [3]{x}}{d^3 e}+\frac {\log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^4}-\frac {\log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d^4}\right )}{d}+\frac {\frac {-\frac {1}{3} b n \left (\frac {\log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^3}-\frac {\log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d^3}+\frac {x^{2/3}}{2 d e^2}-\frac {\sqrt [3]{x}}{d^2 e}\right )-\frac {x \left (a+b \log \left (c x^{-n/3}\right )\right )}{3 e^3}}{d}+\frac {\frac {\frac {x^{2/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{2 e^2}-\frac {1}{2} b n \int \frac {x}{e^2}d\left (d+\frac {e}{\sqrt [3]{x}}\right )}{d}+\frac {\int \frac {x \left (a+b \log \left (c x^{-n/3}\right )\right )}{e^2}d\left (d+\frac {e}{\sqrt [3]{x}}\right )}{d}}{d}}{d}\right )}{d}+\frac {\frac {\frac {x^{4/3} \left (a+b \log \left (c x^{-n/3}\right )\right )^2}{4 e^4}-\frac {1}{2} b n \left (\frac {-\frac {1}{3} b n \left (\frac {\log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^3}-\frac {\log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d^3}+\frac {x^{2/3}}{2 d e^2}-\frac {\sqrt [3]{x}}{d^2 e}\right )-\frac {x \left (a+b \log \left (c x^{-n/3}\right )\right )}{3 e^3}}{d}+\frac {\frac {\frac {x^{2/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{2 e^2}-\frac {1}{2} b n \int \frac {x}{e^2}d\left (d+\frac {e}{\sqrt [3]{x}}\right )}{d}+\frac {\int \frac {x \left (a+b \log \left (c x^{-n/3}\right )\right )}{e^2}d\left (d+\frac {e}{\sqrt [3]{x}}\right )}{d}}{d}\right )}{d}+\frac {\frac {-\frac {x \left (a+b \log \left (c x^{-n/3}\right )\right )^2}{3 e^3}-\frac {2}{3} b n \left (\frac {\frac {x^{2/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{2 e^2}-\frac {1}{2} b n \int \frac {x}{e^2}d\left (d+\frac {e}{\sqrt [3]{x}}\right )}{d}+\frac {\int \frac {x \left (a+b \log \left (c x^{-n/3}\right )\right )}{e^2}d\left (d+\frac {e}{\sqrt [3]{x}}\right )}{d}\right )}{d}+\frac {\frac {\frac {x^{2/3} \left (a+b \log \left (c x^{-n/3}\right )\right )^2}{2 e^2}-b n \int \frac {x \left (a+b \log \left (c x^{-n/3}\right )\right )}{e^2}d\left (d+\frac {e}{\sqrt [3]{x}}\right )}{d}+\frac {\int \frac {x \left (a+b \log \left (c x^{-n/3}\right )\right )^2}{e^2}d\left (d+\frac {e}{\sqrt [3]{x}}\right )}{d}}{d}}{d}}{d}\right )-\frac {1}{6} x^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^3\right )\) |
| \(\Big \downarrow \) 54 |
| \(\displaystyle -3 \left (\frac {1}{2} b e^6 n \left (\frac {-\frac {x^{5/3} \left (a+b \log \left (c x^{-n/3}\right )\right )^2}{5 e^5}-\frac {2}{5} b n \left (\frac {\frac {x^{4/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{4 e^4}-\frac {1}{4} b n \left (-\frac {x}{3 d e^3}+\frac {x^{2/3}}{2 d^2 e^2}-\frac {\sqrt [3]{x}}{d^3 e}+\frac {\log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^4}-\frac {\log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d^4}\right )}{d}+\frac {\frac {-\frac {1}{3} b n \left (\frac {\log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^3}-\frac {\log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d^3}+\frac {x^{2/3}}{2 d e^2}-\frac {\sqrt [3]{x}}{d^2 e}\right )-\frac {x \left (a+b \log \left (c x^{-n/3}\right )\right )}{3 e^3}}{d}+\frac {\frac {\frac {x^{2/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{2 e^2}-\frac {1}{2} b n \int \left (\frac {x^{2/3}}{d e^2}-\frac {\sqrt [3]{x}}{d^2 e}+\frac {\sqrt [3]{x}}{d^2}\right )d\left (d+\frac {e}{\sqrt [3]{x}}\right )}{d}+\frac {\int \frac {x \left (a+b \log \left (c x^{-n/3}\right )\right )}{e^2}d\left (d+\frac {e}{\sqrt [3]{x}}\right )}{d}}{d}}{d}\right )}{d}+\frac {\frac {\frac {x^{4/3} \left (a+b \log \left (c x^{-n/3}\right )\right )^2}{4 e^4}-\frac {1}{2} b n \left (\frac {-\frac {1}{3} b n \left (\frac {\log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^3}-\frac {\log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d^3}+\frac {x^{2/3}}{2 d e^2}-\frac {\sqrt [3]{x}}{d^2 e}\right )-\frac {x \left (a+b \log \left (c x^{-n/3}\right )\right )}{3 e^3}}{d}+\frac {\frac {\frac {x^{2/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{2 e^2}-\frac {1}{2} b n \int \left (\frac {x^{2/3}}{d e^2}-\frac {\sqrt [3]{x}}{d^2 e}+\frac {\sqrt [3]{x}}{d^2}\right )d\left (d+\frac {e}{\sqrt [3]{x}}\right )}{d}+\frac {\int \frac {x \left (a+b \log \left (c x^{-n/3}\right )\right )}{e^2}d\left (d+\frac {e}{\sqrt [3]{x}}\right )}{d}}{d}\right )}{d}+\frac {\frac {-\frac {x \left (a+b \log \left (c x^{-n/3}\right )\right )^2}{3 e^3}-\frac {2}{3} b n \left (\frac {\frac {x^{2/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{2 e^2}-\frac {1}{2} b n \int \left (\frac {x^{2/3}}{d e^2}-\frac {\sqrt [3]{x}}{d^2 e}+\frac {\sqrt [3]{x}}{d^2}\right )d\left (d+\frac {e}{\sqrt [3]{x}}\right )}{d}+\frac {\int \frac {x \left (a+b \log \left (c x^{-n/3}\right )\right )}{e^2}d\left (d+\frac {e}{\sqrt [3]{x}}\right )}{d}\right )}{d}+\frac {\frac {\frac {x^{2/3} \left (a+b \log \left (c x^{-n/3}\right )\right )^2}{2 e^2}-b n \int \frac {x \left (a+b \log \left (c x^{-n/3}\right )\right )}{e^2}d\left (d+\frac {e}{\sqrt [3]{x}}\right )}{d}+\frac {\int \frac {x \left (a+b \log \left (c x^{-n/3}\right )\right )^2}{e^2}d\left (d+\frac {e}{\sqrt [3]{x}}\right )}{d}}{d}}{d}}{d}\right )-\frac {1}{6} x^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^3\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -3 \left (\frac {1}{2} b e^6 n \left (\frac {-\frac {x^{5/3} \left (a+b \log \left (c x^{-n/3}\right )\right )^2}{5 e^5}-\frac {2}{5} b n \left (\frac {\frac {x^{4/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{4 e^4}-\frac {1}{4} b n \left (-\frac {x}{3 d e^3}+\frac {x^{2/3}}{2 d^2 e^2}-\frac {\sqrt [3]{x}}{d^3 e}+\frac {\log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^4}-\frac {\log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d^4}\right )}{d}+\frac {\frac {-\frac {1}{3} b n \left (\frac {\log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^3}-\frac {\log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d^3}+\frac {x^{2/3}}{2 d e^2}-\frac {\sqrt [3]{x}}{d^2 e}\right )-\frac {x \left (a+b \log \left (c x^{-n/3}\right )\right )}{3 e^3}}{d}+\frac {\frac {\frac {x^{2/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{2 e^2}-\frac {1}{2} b n \left (\frac {\log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^2}-\frac {\log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d^2}-\frac {\sqrt [3]{x}}{d e}\right )}{d}+\frac {\int \frac {x \left (a+b \log \left (c x^{-n/3}\right )\right )}{e^2}d\left (d+\frac {e}{\sqrt [3]{x}}\right )}{d}}{d}}{d}\right )}{d}+\frac {\frac {\frac {x^{4/3} \left (a+b \log \left (c x^{-n/3}\right )\right )^2}{4 e^4}-\frac {1}{2} b n \left (\frac {-\frac {1}{3} b n \left (\frac {\log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^3}-\frac {\log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d^3}+\frac {x^{2/3}}{2 d e^2}-\frac {\sqrt [3]{x}}{d^2 e}\right )-\frac {x \left (a+b \log \left (c x^{-n/3}\right )\right )}{3 e^3}}{d}+\frac {\frac {\frac {x^{2/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{2 e^2}-\frac {1}{2} b n \left (\frac {\log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^2}-\frac {\log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d^2}-\frac {\sqrt [3]{x}}{d e}\right )}{d}+\frac {\int \frac {x \left (a+b \log \left (c x^{-n/3}\right )\right )}{e^2}d\left (d+\frac {e}{\sqrt [3]{x}}\right )}{d}}{d}\right )}{d}+\frac {\frac {-\frac {x \left (a+b \log \left (c x^{-n/3}\right )\right )^2}{3 e^3}-\frac {2}{3} b n \left (\frac {\frac {x^{2/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{2 e^2}-\frac {1}{2} b n \left (\frac {\log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^2}-\frac {\log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d^2}-\frac {\sqrt [3]{x}}{d e}\right )}{d}+\frac {\int \frac {x \left (a+b \log \left (c x^{-n/3}\right )\right )}{e^2}d\left (d+\frac {e}{\sqrt [3]{x}}\right )}{d}\right )}{d}+\frac {\frac {\frac {x^{2/3} \left (a+b \log \left (c x^{-n/3}\right )\right )^2}{2 e^2}-b n \int \frac {x \left (a+b \log \left (c x^{-n/3}\right )\right )}{e^2}d\left (d+\frac {e}{\sqrt [3]{x}}\right )}{d}+\frac {\int \frac {x \left (a+b \log \left (c x^{-n/3}\right )\right )^2}{e^2}d\left (d+\frac {e}{\sqrt [3]{x}}\right )}{d}}{d}}{d}}{d}\right )-\frac {1}{6} x^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^3\right )\) |
| \(\Big \downarrow \) 2789 |
| \(\displaystyle -3 \left (\frac {1}{2} b e^6 n \left (\frac {-\frac {x^{5/3} \left (a+b \log \left (c x^{-n/3}\right )\right )^2}{5 e^5}-\frac {2}{5} b n \left (\frac {\frac {x^{4/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{4 e^4}-\frac {1}{4} b n \left (-\frac {x}{3 d e^3}+\frac {x^{2/3}}{2 d^2 e^2}-\frac {\sqrt [3]{x}}{d^3 e}+\frac {\log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^4}-\frac {\log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d^4}\right )}{d}+\frac {\frac {-\frac {1}{3} b n \left (\frac {\log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^3}-\frac {\log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d^3}+\frac {x^{2/3}}{2 d e^2}-\frac {\sqrt [3]{x}}{d^2 e}\right )-\frac {x \left (a+b \log \left (c x^{-n/3}\right )\right )}{3 e^3}}{d}+\frac {\frac {\frac {x^{2/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{2 e^2}-\frac {1}{2} b n \left (\frac {\log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^2}-\frac {\log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d^2}-\frac {\sqrt [3]{x}}{d e}\right )}{d}+\frac {\frac {\int \frac {x^{2/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{e^2}d\left (d+\frac {e}{\sqrt [3]{x}}\right )}{d}+\frac {\int -\frac {x^{2/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{e}d\left (d+\frac {e}{\sqrt [3]{x}}\right )}{d}}{d}}{d}}{d}\right )}{d}+\frac {\frac {\frac {x^{4/3} \left (a+b \log \left (c x^{-n/3}\right )\right )^2}{4 e^4}-\frac {1}{2} b n \left (\frac {-\frac {1}{3} b n \left (\frac {\log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^3}-\frac {\log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d^3}+\frac {x^{2/3}}{2 d e^2}-\frac {\sqrt [3]{x}}{d^2 e}\right )-\frac {x \left (a+b \log \left (c x^{-n/3}\right )\right )}{3 e^3}}{d}+\frac {\frac {\frac {x^{2/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{2 e^2}-\frac {1}{2} b n \left (\frac {\log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^2}-\frac {\log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d^2}-\frac {\sqrt [3]{x}}{d e}\right )}{d}+\frac {\frac {\int \frac {x^{2/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{e^2}d\left (d+\frac {e}{\sqrt [3]{x}}\right )}{d}+\frac {\int -\frac {x^{2/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{e}d\left (d+\frac {e}{\sqrt [3]{x}}\right )}{d}}{d}}{d}\right )}{d}+\frac {\frac {-\frac {x \left (a+b \log \left (c x^{-n/3}\right )\right )^2}{3 e^3}-\frac {2}{3} b n \left (\frac {\frac {x^{2/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{2 e^2}-\frac {1}{2} b n \left (\frac {\log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^2}-\frac {\log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d^2}-\frac {\sqrt [3]{x}}{d e}\right )}{d}+\frac {\frac {\int \frac {x^{2/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{e^2}d\left (d+\frac {e}{\sqrt [3]{x}}\right )}{d}+\frac {\int -\frac {x^{2/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{e}d\left (d+\frac {e}{\sqrt [3]{x}}\right )}{d}}{d}\right )}{d}+\frac {\frac {\frac {x^{2/3} \left (a+b \log \left (c x^{-n/3}\right )\right )^2}{2 e^2}-b n \left (\frac {\int \frac {x^{2/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{e^2}d\left (d+\frac {e}{\sqrt [3]{x}}\right )}{d}+\frac {\int -\frac {x^{2/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{e}d\left (d+\frac {e}{\sqrt [3]{x}}\right )}{d}\right )}{d}+\frac {\frac {\int \frac {x^{2/3} \left (a+b \log \left (c x^{-n/3}\right )\right )^2}{e^2}d\left (d+\frac {e}{\sqrt [3]{x}}\right )}{d}+\frac {\int -\frac {x^{2/3} \left (a+b \log \left (c x^{-n/3}\right )\right )^2}{e}d\left (d+\frac {e}{\sqrt [3]{x}}\right )}{d}}{d}}{d}}{d}}{d}\right )-\frac {1}{6} x^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^3\right )\) |
| \(\Big \downarrow \) 2751 |
| \(\displaystyle -3 \left (\frac {1}{2} b e^6 n \left (\frac {-\frac {x^{5/3} \left (a+b \log \left (c x^{-n/3}\right )\right )^2}{5 e^5}-\frac {2}{5} b n \left (\frac {\frac {x^{4/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{4 e^4}-\frac {1}{4} b n \left (-\frac {x}{3 d e^3}+\frac {x^{2/3}}{2 d^2 e^2}-\frac {\sqrt [3]{x}}{d^3 e}+\frac {\log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^4}-\frac {\log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d^4}\right )}{d}+\frac {\frac {-\frac {1}{3} b n \left (\frac {\log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^3}-\frac {\log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d^3}+\frac {x^{2/3}}{2 d e^2}-\frac {\sqrt [3]{x}}{d^2 e}\right )-\frac {x \left (a+b \log \left (c x^{-n/3}\right )\right )}{3 e^3}}{d}+\frac {\frac {\frac {x^{2/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{2 e^2}-\frac {1}{2} b n \left (\frac {\log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^2}-\frac {\log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d^2}-\frac {\sqrt [3]{x}}{d e}\right )}{d}+\frac {\frac {-\frac {\left (d+\frac {e}{\sqrt [3]{x}}\right ) \sqrt [3]{x} \left (a+b \log \left (c x^{-n/3}\right )\right )}{d e}-\frac {b n \int -\frac {\sqrt [3]{x}}{e}d\left (d+\frac {e}{\sqrt [3]{x}}\right )}{d}}{d}+\frac {\int -\frac {x^{2/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{e}d\left (d+\frac {e}{\sqrt [3]{x}}\right )}{d}}{d}}{d}}{d}\right )}{d}+\frac {\frac {\frac {x^{4/3} \left (a+b \log \left (c x^{-n/3}\right )\right )^2}{4 e^4}-\frac {1}{2} b n \left (\frac {-\frac {1}{3} b n \left (\frac {\log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^3}-\frac {\log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d^3}+\frac {x^{2/3}}{2 d e^2}-\frac {\sqrt [3]{x}}{d^2 e}\right )-\frac {x \left (a+b \log \left (c x^{-n/3}\right )\right )}{3 e^3}}{d}+\frac {\frac {\frac {x^{2/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{2 e^2}-\frac {1}{2} b n \left (\frac {\log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^2}-\frac {\log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d^2}-\frac {\sqrt [3]{x}}{d e}\right )}{d}+\frac {\frac {-\frac {\left (d+\frac {e}{\sqrt [3]{x}}\right ) \sqrt [3]{x} \left (a+b \log \left (c x^{-n/3}\right )\right )}{d e}-\frac {b n \int -\frac {\sqrt [3]{x}}{e}d\left (d+\frac {e}{\sqrt [3]{x}}\right )}{d}}{d}+\frac {\int -\frac {x^{2/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{e}d\left (d+\frac {e}{\sqrt [3]{x}}\right )}{d}}{d}}{d}\right )}{d}+\frac {\frac {-\frac {x \left (a+b \log \left (c x^{-n/3}\right )\right )^2}{3 e^3}-\frac {2}{3} b n \left (\frac {\frac {x^{2/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{2 e^2}-\frac {1}{2} b n \left (\frac {\log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^2}-\frac {\log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d^2}-\frac {\sqrt [3]{x}}{d e}\right )}{d}+\frac {\frac {-\frac {\left (d+\frac {e}{\sqrt [3]{x}}\right ) \sqrt [3]{x} \left (a+b \log \left (c x^{-n/3}\right )\right )}{d e}-\frac {b n \int -\frac {\sqrt [3]{x}}{e}d\left (d+\frac {e}{\sqrt [3]{x}}\right )}{d}}{d}+\frac {\int -\frac {x^{2/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{e}d\left (d+\frac {e}{\sqrt [3]{x}}\right )}{d}}{d}\right )}{d}+\frac {\frac {\frac {x^{2/3} \left (a+b \log \left (c x^{-n/3}\right )\right )^2}{2 e^2}-b n \left (\frac {-\frac {\left (d+\frac {e}{\sqrt [3]{x}}\right ) \sqrt [3]{x} \left (a+b \log \left (c x^{-n/3}\right )\right )}{d e}-\frac {b n \int -\frac {\sqrt [3]{x}}{e}d\left (d+\frac {e}{\sqrt [3]{x}}\right )}{d}}{d}+\frac {\int -\frac {x^{2/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{e}d\left (d+\frac {e}{\sqrt [3]{x}}\right )}{d}\right )}{d}+\frac {\frac {\int \frac {x^{2/3} \left (a+b \log \left (c x^{-n/3}\right )\right )^2}{e^2}d\left (d+\frac {e}{\sqrt [3]{x}}\right )}{d}+\frac {\int -\frac {x^{2/3} \left (a+b \log \left (c x^{-n/3}\right )\right )^2}{e}d\left (d+\frac {e}{\sqrt [3]{x}}\right )}{d}}{d}}{d}}{d}}{d}\right )-\frac {1}{6} x^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^3\right )\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle -3 \left (\frac {1}{2} b e^6 n \left (\frac {-\frac {x^{5/3} \left (a+b \log \left (c x^{-n/3}\right )\right )^2}{5 e^5}-\frac {2}{5} b n \left (\frac {\frac {x^{4/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{4 e^4}-\frac {1}{4} b n \left (-\frac {x}{3 d e^3}+\frac {x^{2/3}}{2 d^2 e^2}-\frac {\sqrt [3]{x}}{d^3 e}+\frac {\log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^4}-\frac {\log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d^4}\right )}{d}+\frac {\frac {-\frac {1}{3} b n \left (\frac {\log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^3}-\frac {\log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d^3}+\frac {x^{2/3}}{2 d e^2}-\frac {\sqrt [3]{x}}{d^2 e}\right )-\frac {x \left (a+b \log \left (c x^{-n/3}\right )\right )}{3 e^3}}{d}+\frac {\frac {\frac {x^{2/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{2 e^2}-\frac {1}{2} b n \left (\frac {\log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^2}-\frac {\log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d^2}-\frac {\sqrt [3]{x}}{d e}\right )}{d}+\frac {\frac {\frac {b n \log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d}-\frac {\left (d+\frac {e}{\sqrt [3]{x}}\right ) \sqrt [3]{x} \left (a+b \log \left (c x^{-n/3}\right )\right )}{d e}}{d}+\frac {\int -\frac {x^{2/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{e}d\left (d+\frac {e}{\sqrt [3]{x}}\right )}{d}}{d}}{d}}{d}\right )}{d}+\frac {\frac {\frac {x^{4/3} \left (a+b \log \left (c x^{-n/3}\right )\right )^2}{4 e^4}-\frac {1}{2} b n \left (\frac {-\frac {1}{3} b n \left (\frac {\log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^3}-\frac {\log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d^3}+\frac {x^{2/3}}{2 d e^2}-\frac {\sqrt [3]{x}}{d^2 e}\right )-\frac {x \left (a+b \log \left (c x^{-n/3}\right )\right )}{3 e^3}}{d}+\frac {\frac {\frac {x^{2/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{2 e^2}-\frac {1}{2} b n \left (\frac {\log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^2}-\frac {\log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d^2}-\frac {\sqrt [3]{x}}{d e}\right )}{d}+\frac {\frac {\frac {b n \log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d}-\frac {\left (d+\frac {e}{\sqrt [3]{x}}\right ) \sqrt [3]{x} \left (a+b \log \left (c x^{-n/3}\right )\right )}{d e}}{d}+\frac {\int -\frac {x^{2/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{e}d\left (d+\frac {e}{\sqrt [3]{x}}\right )}{d}}{d}}{d}\right )}{d}+\frac {\frac {-\frac {x \left (a+b \log \left (c x^{-n/3}\right )\right )^2}{3 e^3}-\frac {2}{3} b n \left (\frac {\frac {x^{2/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{2 e^2}-\frac {1}{2} b n \left (\frac {\log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^2}-\frac {\log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d^2}-\frac {\sqrt [3]{x}}{d e}\right )}{d}+\frac {\frac {\frac {b n \log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d}-\frac {\left (d+\frac {e}{\sqrt [3]{x}}\right ) \sqrt [3]{x} \left (a+b \log \left (c x^{-n/3}\right )\right )}{d e}}{d}+\frac {\int -\frac {x^{2/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{e}d\left (d+\frac {e}{\sqrt [3]{x}}\right )}{d}}{d}\right )}{d}+\frac {\frac {\frac {x^{2/3} \left (a+b \log \left (c x^{-n/3}\right )\right )^2}{2 e^2}-b n \left (\frac {\frac {b n \log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d}-\frac {\left (d+\frac {e}{\sqrt [3]{x}}\right ) \sqrt [3]{x} \left (a+b \log \left (c x^{-n/3}\right )\right )}{d e}}{d}+\frac {\int -\frac {x^{2/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{e}d\left (d+\frac {e}{\sqrt [3]{x}}\right )}{d}\right )}{d}+\frac {\frac {\int \frac {x^{2/3} \left (a+b \log \left (c x^{-n/3}\right )\right )^2}{e^2}d\left (d+\frac {e}{\sqrt [3]{x}}\right )}{d}+\frac {\int -\frac {x^{2/3} \left (a+b \log \left (c x^{-n/3}\right )\right )^2}{e}d\left (d+\frac {e}{\sqrt [3]{x}}\right )}{d}}{d}}{d}}{d}}{d}\right )-\frac {1}{6} x^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^3\right )\) |
| \(\Big \downarrow \) 2755 |
| \(\displaystyle -3 \left (\frac {1}{2} b e^6 n \left (\frac {-\frac {x^{5/3} \left (a+b \log \left (c x^{-n/3}\right )\right )^2}{5 e^5}-\frac {2}{5} b n \left (\frac {\frac {x^{4/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{4 e^4}-\frac {1}{4} b n \left (-\frac {x}{3 d e^3}+\frac {x^{2/3}}{2 d^2 e^2}-\frac {\sqrt [3]{x}}{d^3 e}+\frac {\log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^4}-\frac {\log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d^4}\right )}{d}+\frac {\frac {-\frac {1}{3} b n \left (\frac {\log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^3}-\frac {\log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d^3}+\frac {x^{2/3}}{2 d e^2}-\frac {\sqrt [3]{x}}{d^2 e}\right )-\frac {x \left (a+b \log \left (c x^{-n/3}\right )\right )}{3 e^3}}{d}+\frac {\frac {\frac {x^{2/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{2 e^2}-\frac {1}{2} b n \left (\frac {\log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^2}-\frac {\log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d^2}-\frac {\sqrt [3]{x}}{d e}\right )}{d}+\frac {\frac {\frac {b n \log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d}-\frac {\left (d+\frac {e}{\sqrt [3]{x}}\right ) \sqrt [3]{x} \left (a+b \log \left (c x^{-n/3}\right )\right )}{d e}}{d}+\frac {\int -\frac {x^{2/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{e}d\left (d+\frac {e}{\sqrt [3]{x}}\right )}{d}}{d}}{d}}{d}\right )}{d}+\frac {\frac {\frac {x^{4/3} \left (a+b \log \left (c x^{-n/3}\right )\right )^2}{4 e^4}-\frac {1}{2} b n \left (\frac {-\frac {1}{3} b n \left (\frac {\log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^3}-\frac {\log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d^3}+\frac {x^{2/3}}{2 d e^2}-\frac {\sqrt [3]{x}}{d^2 e}\right )-\frac {x \left (a+b \log \left (c x^{-n/3}\right )\right )}{3 e^3}}{d}+\frac {\frac {\frac {x^{2/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{2 e^2}-\frac {1}{2} b n \left (\frac {\log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^2}-\frac {\log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d^2}-\frac {\sqrt [3]{x}}{d e}\right )}{d}+\frac {\frac {\frac {b n \log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d}-\frac {\left (d+\frac {e}{\sqrt [3]{x}}\right ) \sqrt [3]{x} \left (a+b \log \left (c x^{-n/3}\right )\right )}{d e}}{d}+\frac {\int -\frac {x^{2/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{e}d\left (d+\frac {e}{\sqrt [3]{x}}\right )}{d}}{d}}{d}\right )}{d}+\frac {\frac {-\frac {x \left (a+b \log \left (c x^{-n/3}\right )\right )^2}{3 e^3}-\frac {2}{3} b n \left (\frac {\frac {x^{2/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{2 e^2}-\frac {1}{2} b n \left (\frac {\log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^2}-\frac {\log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d^2}-\frac {\sqrt [3]{x}}{d e}\right )}{d}+\frac {\frac {\frac {b n \log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d}-\frac {\left (d+\frac {e}{\sqrt [3]{x}}\right ) \sqrt [3]{x} \left (a+b \log \left (c x^{-n/3}\right )\right )}{d e}}{d}+\frac {\int -\frac {x^{2/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{e}d\left (d+\frac {e}{\sqrt [3]{x}}\right )}{d}}{d}\right )}{d}+\frac {\frac {\frac {x^{2/3} \left (a+b \log \left (c x^{-n/3}\right )\right )^2}{2 e^2}-b n \left (\frac {\frac {b n \log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d}-\frac {\left (d+\frac {e}{\sqrt [3]{x}}\right ) \sqrt [3]{x} \left (a+b \log \left (c x^{-n/3}\right )\right )}{d e}}{d}+\frac {\int -\frac {x^{2/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{e}d\left (d+\frac {e}{\sqrt [3]{x}}\right )}{d}\right )}{d}+\frac {\frac {-\frac {\left (d+\frac {e}{\sqrt [3]{x}}\right ) \sqrt [3]{x} \left (a+b \log \left (c x^{-n/3}\right )\right )^2}{d e}-\frac {2 b n \int -\frac {\sqrt [3]{x} \left (a+b \log \left (c x^{-n/3}\right )\right )}{e}d\left (d+\frac {e}{\sqrt [3]{x}}\right )}{d}}{d}+\frac {\int -\frac {x^{2/3} \left (a+b \log \left (c x^{-n/3}\right )\right )^2}{e}d\left (d+\frac {e}{\sqrt [3]{x}}\right )}{d}}{d}}{d}}{d}}{d}\right )-\frac {1}{6} x^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^3\right )\) |
| \(\Big \downarrow \) 2754 |
| \(\displaystyle -3 \left (\frac {1}{2} b e^6 n \left (\frac {-\frac {x^{5/3} \left (a+b \log \left (c x^{-n/3}\right )\right )^2}{5 e^5}-\frac {2}{5} b n \left (\frac {\frac {x^{4/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{4 e^4}-\frac {1}{4} b n \left (-\frac {x}{3 d e^3}+\frac {x^{2/3}}{2 d^2 e^2}-\frac {\sqrt [3]{x}}{d^3 e}+\frac {\log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^4}-\frac {\log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d^4}\right )}{d}+\frac {\frac {-\frac {1}{3} b n \left (\frac {\log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^3}-\frac {\log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d^3}+\frac {x^{2/3}}{2 d e^2}-\frac {\sqrt [3]{x}}{d^2 e}\right )-\frac {x \left (a+b \log \left (c x^{-n/3}\right )\right )}{3 e^3}}{d}+\frac {\frac {\frac {x^{2/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{2 e^2}-\frac {1}{2} b n \left (\frac {\log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^2}-\frac {\log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d^2}-\frac {\sqrt [3]{x}}{d e}\right )}{d}+\frac {\frac {\frac {b n \log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d}-\frac {\left (d+\frac {e}{\sqrt [3]{x}}\right ) \sqrt [3]{x} \left (a+b \log \left (c x^{-n/3}\right )\right )}{d e}}{d}+\frac {\int -\frac {x^{2/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{e}d\left (d+\frac {e}{\sqrt [3]{x}}\right )}{d}}{d}}{d}}{d}\right )}{d}+\frac {\frac {\frac {x^{4/3} \left (a+b \log \left (c x^{-n/3}\right )\right )^2}{4 e^4}-\frac {1}{2} b n \left (\frac {-\frac {1}{3} b n \left (\frac {\log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^3}-\frac {\log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d^3}+\frac {x^{2/3}}{2 d e^2}-\frac {\sqrt [3]{x}}{d^2 e}\right )-\frac {x \left (a+b \log \left (c x^{-n/3}\right )\right )}{3 e^3}}{d}+\frac {\frac {\frac {x^{2/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{2 e^2}-\frac {1}{2} b n \left (\frac {\log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^2}-\frac {\log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d^2}-\frac {\sqrt [3]{x}}{d e}\right )}{d}+\frac {\frac {\frac {b n \log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d}-\frac {\left (d+\frac {e}{\sqrt [3]{x}}\right ) \sqrt [3]{x} \left (a+b \log \left (c x^{-n/3}\right )\right )}{d e}}{d}+\frac {\int -\frac {x^{2/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{e}d\left (d+\frac {e}{\sqrt [3]{x}}\right )}{d}}{d}}{d}\right )}{d}+\frac {\frac {-\frac {x \left (a+b \log \left (c x^{-n/3}\right )\right )^2}{3 e^3}-\frac {2}{3} b n \left (\frac {\frac {x^{2/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{2 e^2}-\frac {1}{2} b n \left (\frac {\log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^2}-\frac {\log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d^2}-\frac {\sqrt [3]{x}}{d e}\right )}{d}+\frac {\frac {\frac {b n \log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d}-\frac {\left (d+\frac {e}{\sqrt [3]{x}}\right ) \sqrt [3]{x} \left (a+b \log \left (c x^{-n/3}\right )\right )}{d e}}{d}+\frac {\int -\frac {x^{2/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{e}d\left (d+\frac {e}{\sqrt [3]{x}}\right )}{d}}{d}\right )}{d}+\frac {\frac {\frac {x^{2/3} \left (a+b \log \left (c x^{-n/3}\right )\right )^2}{2 e^2}-b n \left (\frac {\frac {b n \log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d}-\frac {\left (d+\frac {e}{\sqrt [3]{x}}\right ) \sqrt [3]{x} \left (a+b \log \left (c x^{-n/3}\right )\right )}{d e}}{d}+\frac {\int -\frac {x^{2/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{e}d\left (d+\frac {e}{\sqrt [3]{x}}\right )}{d}\right )}{d}+\frac {\frac {-\frac {\left (d+\frac {e}{\sqrt [3]{x}}\right ) \sqrt [3]{x} \left (a+b \log \left (c x^{-n/3}\right )\right )^2}{d e}-\frac {2 b n \left (b n \int \sqrt [3]{x} \log \left (1-\frac {d+\frac {e}{\sqrt [3]{x}}}{d}\right )d\left (d+\frac {e}{\sqrt [3]{x}}\right )-\log \left (1-\frac {d+\frac {e}{\sqrt [3]{x}}}{d}\right ) \left (a+b \log \left (c x^{-n/3}\right )\right )\right )}{d}}{d}+\frac {\int -\frac {x^{2/3} \left (a+b \log \left (c x^{-n/3}\right )\right )^2}{e}d\left (d+\frac {e}{\sqrt [3]{x}}\right )}{d}}{d}}{d}}{d}}{d}\right )-\frac {1}{6} x^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^3\right )\) |
| \(\Big \downarrow \) 2779 |
| \(\displaystyle -3 \left (\frac {1}{2} b e^6 n \left (\frac {-\frac {x^{5/3} \left (a+b \log \left (c x^{-n/3}\right )\right )^2}{5 e^5}-\frac {2}{5} b n \left (\frac {\frac {x^{4/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{4 e^4}-\frac {1}{4} b n \left (-\frac {x}{3 d e^3}+\frac {x^{2/3}}{2 d^2 e^2}-\frac {\sqrt [3]{x}}{d^3 e}+\frac {\log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^4}-\frac {\log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d^4}\right )}{d}+\frac {\frac {-\frac {1}{3} b n \left (\frac {\log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^3}-\frac {\log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d^3}+\frac {x^{2/3}}{2 d e^2}-\frac {\sqrt [3]{x}}{d^2 e}\right )-\frac {x \left (a+b \log \left (c x^{-n/3}\right )\right )}{3 e^3}}{d}+\frac {\frac {\frac {x^{2/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{2 e^2}-\frac {1}{2} b n \left (\frac {\log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^2}-\frac {\log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d^2}-\frac {\sqrt [3]{x}}{d e}\right )}{d}+\frac {\frac {\frac {b n \log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d}-\frac {\left (d+\frac {e}{\sqrt [3]{x}}\right ) \sqrt [3]{x} \left (a+b \log \left (c x^{-n/3}\right )\right )}{d e}}{d}+\frac {\frac {b n \int \sqrt [3]{x} \log \left (1-d \sqrt [3]{x}\right )d\left (d+\frac {e}{\sqrt [3]{x}}\right )}{d}-\frac {\log \left (1-d \sqrt [3]{x}\right ) \left (a+b \log \left (c x^{-n/3}\right )\right )}{d}}{d}}{d}}{d}}{d}\right )}{d}+\frac {\frac {\frac {x^{4/3} \left (a+b \log \left (c x^{-n/3}\right )\right )^2}{4 e^4}-\frac {1}{2} b n \left (\frac {-\frac {1}{3} b n \left (\frac {\log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^3}-\frac {\log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d^3}+\frac {x^{2/3}}{2 d e^2}-\frac {\sqrt [3]{x}}{d^2 e}\right )-\frac {x \left (a+b \log \left (c x^{-n/3}\right )\right )}{3 e^3}}{d}+\frac {\frac {\frac {x^{2/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{2 e^2}-\frac {1}{2} b n \left (\frac {\log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^2}-\frac {\log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d^2}-\frac {\sqrt [3]{x}}{d e}\right )}{d}+\frac {\frac {\frac {b n \log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d}-\frac {\left (d+\frac {e}{\sqrt [3]{x}}\right ) \sqrt [3]{x} \left (a+b \log \left (c x^{-n/3}\right )\right )}{d e}}{d}+\frac {\frac {b n \int \sqrt [3]{x} \log \left (1-d \sqrt [3]{x}\right )d\left (d+\frac {e}{\sqrt [3]{x}}\right )}{d}-\frac {\log \left (1-d \sqrt [3]{x}\right ) \left (a+b \log \left (c x^{-n/3}\right )\right )}{d}}{d}}{d}}{d}\right )}{d}+\frac {\frac {-\frac {x \left (a+b \log \left (c x^{-n/3}\right )\right )^2}{3 e^3}-\frac {2}{3} b n \left (\frac {\frac {x^{2/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{2 e^2}-\frac {1}{2} b n \left (\frac {\log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^2}-\frac {\log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d^2}-\frac {\sqrt [3]{x}}{d e}\right )}{d}+\frac {\frac {\frac {b n \log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d}-\frac {\left (d+\frac {e}{\sqrt [3]{x}}\right ) \sqrt [3]{x} \left (a+b \log \left (c x^{-n/3}\right )\right )}{d e}}{d}+\frac {\frac {b n \int \sqrt [3]{x} \log \left (1-d \sqrt [3]{x}\right )d\left (d+\frac {e}{\sqrt [3]{x}}\right )}{d}-\frac {\log \left (1-d \sqrt [3]{x}\right ) \left (a+b \log \left (c x^{-n/3}\right )\right )}{d}}{d}}{d}\right )}{d}+\frac {\frac {\frac {x^{2/3} \left (a+b \log \left (c x^{-n/3}\right )\right )^2}{2 e^2}-b n \left (\frac {\frac {b n \log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d}-\frac {\left (d+\frac {e}{\sqrt [3]{x}}\right ) \sqrt [3]{x} \left (a+b \log \left (c x^{-n/3}\right )\right )}{d e}}{d}+\frac {\frac {b n \int \sqrt [3]{x} \log \left (1-d \sqrt [3]{x}\right )d\left (d+\frac {e}{\sqrt [3]{x}}\right )}{d}-\frac {\log \left (1-d \sqrt [3]{x}\right ) \left (a+b \log \left (c x^{-n/3}\right )\right )}{d}}{d}\right )}{d}+\frac {\frac {-\frac {\left (d+\frac {e}{\sqrt [3]{x}}\right ) \sqrt [3]{x} \left (a+b \log \left (c x^{-n/3}\right )\right )^2}{d e}-\frac {2 b n \left (b n \int \sqrt [3]{x} \log \left (1-\frac {d+\frac {e}{\sqrt [3]{x}}}{d}\right )d\left (d+\frac {e}{\sqrt [3]{x}}\right )-\log \left (1-\frac {d+\frac {e}{\sqrt [3]{x}}}{d}\right ) \left (a+b \log \left (c x^{-n/3}\right )\right )\right )}{d}}{d}+\frac {\frac {2 b n \int \sqrt [3]{x} \log \left (1-d \sqrt [3]{x}\right ) \left (a+b \log \left (c x^{-n/3}\right )\right )d\left (d+\frac {e}{\sqrt [3]{x}}\right )}{d}-\frac {\log \left (1-d \sqrt [3]{x}\right ) \left (a+b \log \left (c x^{-n/3}\right )\right )^2}{d}}{d}}{d}}{d}}{d}}{d}\right )-\frac {1}{6} x^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^3\right )\) |
| \(\Big \downarrow \) 2821 |
| \(\displaystyle -3 \left (\frac {1}{2} b e^6 n \left (\frac {-\frac {x^{5/3} \left (a+b \log \left (c x^{-n/3}\right )\right )^2}{5 e^5}-\frac {2}{5} b n \left (\frac {\frac {x^{4/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{4 e^4}-\frac {1}{4} b n \left (-\frac {x}{3 d e^3}+\frac {x^{2/3}}{2 d^2 e^2}-\frac {\sqrt [3]{x}}{d^3 e}+\frac {\log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^4}-\frac {\log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d^4}\right )}{d}+\frac {\frac {-\frac {1}{3} b n \left (\frac {\log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^3}-\frac {\log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d^3}+\frac {x^{2/3}}{2 d e^2}-\frac {\sqrt [3]{x}}{d^2 e}\right )-\frac {x \left (a+b \log \left (c x^{-n/3}\right )\right )}{3 e^3}}{d}+\frac {\frac {\frac {x^{2/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{2 e^2}-\frac {1}{2} b n \left (\frac {\log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^2}-\frac {\log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d^2}-\frac {\sqrt [3]{x}}{d e}\right )}{d}+\frac {\frac {\frac {b n \log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d}-\frac {\left (d+\frac {e}{\sqrt [3]{x}}\right ) \sqrt [3]{x} \left (a+b \log \left (c x^{-n/3}\right )\right )}{d e}}{d}+\frac {\frac {b n \int \sqrt [3]{x} \log \left (1-d \sqrt [3]{x}\right )d\left (d+\frac {e}{\sqrt [3]{x}}\right )}{d}-\frac {\log \left (1-d \sqrt [3]{x}\right ) \left (a+b \log \left (c x^{-n/3}\right )\right )}{d}}{d}}{d}}{d}}{d}\right )}{d}+\frac {\frac {\frac {x^{4/3} \left (a+b \log \left (c x^{-n/3}\right )\right )^2}{4 e^4}-\frac {1}{2} b n \left (\frac {-\frac {1}{3} b n \left (\frac {\log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^3}-\frac {\log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d^3}+\frac {x^{2/3}}{2 d e^2}-\frac {\sqrt [3]{x}}{d^2 e}\right )-\frac {x \left (a+b \log \left (c x^{-n/3}\right )\right )}{3 e^3}}{d}+\frac {\frac {\frac {x^{2/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{2 e^2}-\frac {1}{2} b n \left (\frac {\log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^2}-\frac {\log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d^2}-\frac {\sqrt [3]{x}}{d e}\right )}{d}+\frac {\frac {\frac {b n \log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d}-\frac {\left (d+\frac {e}{\sqrt [3]{x}}\right ) \sqrt [3]{x} \left (a+b \log \left (c x^{-n/3}\right )\right )}{d e}}{d}+\frac {\frac {b n \int \sqrt [3]{x} \log \left (1-d \sqrt [3]{x}\right )d\left (d+\frac {e}{\sqrt [3]{x}}\right )}{d}-\frac {\log \left (1-d \sqrt [3]{x}\right ) \left (a+b \log \left (c x^{-n/3}\right )\right )}{d}}{d}}{d}}{d}\right )}{d}+\frac {\frac {-\frac {x \left (a+b \log \left (c x^{-n/3}\right )\right )^2}{3 e^3}-\frac {2}{3} b n \left (\frac {\frac {x^{2/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{2 e^2}-\frac {1}{2} b n \left (\frac {\log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^2}-\frac {\log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d^2}-\frac {\sqrt [3]{x}}{d e}\right )}{d}+\frac {\frac {\frac {b n \log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d}-\frac {\left (d+\frac {e}{\sqrt [3]{x}}\right ) \sqrt [3]{x} \left (a+b \log \left (c x^{-n/3}\right )\right )}{d e}}{d}+\frac {\frac {b n \int \sqrt [3]{x} \log \left (1-d \sqrt [3]{x}\right )d\left (d+\frac {e}{\sqrt [3]{x}}\right )}{d}-\frac {\log \left (1-d \sqrt [3]{x}\right ) \left (a+b \log \left (c x^{-n/3}\right )\right )}{d}}{d}}{d}\right )}{d}+\frac {\frac {\frac {x^{2/3} \left (a+b \log \left (c x^{-n/3}\right )\right )^2}{2 e^2}-b n \left (\frac {\frac {b n \log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d}-\frac {\left (d+\frac {e}{\sqrt [3]{x}}\right ) \sqrt [3]{x} \left (a+b \log \left (c x^{-n/3}\right )\right )}{d e}}{d}+\frac {\frac {b n \int \sqrt [3]{x} \log \left (1-d \sqrt [3]{x}\right )d\left (d+\frac {e}{\sqrt [3]{x}}\right )}{d}-\frac {\log \left (1-d \sqrt [3]{x}\right ) \left (a+b \log \left (c x^{-n/3}\right )\right )}{d}}{d}\right )}{d}+\frac {\frac {-\frac {\left (d+\frac {e}{\sqrt [3]{x}}\right ) \sqrt [3]{x} \left (a+b \log \left (c x^{-n/3}\right )\right )^2}{d e}-\frac {2 b n \left (b n \int \sqrt [3]{x} \log \left (1-\frac {d+\frac {e}{\sqrt [3]{x}}}{d}\right )d\left (d+\frac {e}{\sqrt [3]{x}}\right )-\log \left (1-\frac {d+\frac {e}{\sqrt [3]{x}}}{d}\right ) \left (a+b \log \left (c x^{-n/3}\right )\right )\right )}{d}}{d}+\frac {\frac {2 b n \left (\left (a+b \log \left (c x^{-n/3}\right )\right ) \operatorname {PolyLog}\left (2,d \sqrt [3]{x}\right )-b n \int \sqrt [3]{x} \operatorname {PolyLog}\left (2,d \sqrt [3]{x}\right )d\left (d+\frac {e}{\sqrt [3]{x}}\right )\right )}{d}-\frac {\log \left (1-d \sqrt [3]{x}\right ) \left (a+b \log \left (c x^{-n/3}\right )\right )^2}{d}}{d}}{d}}{d}}{d}}{d}\right )-\frac {1}{6} x^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^3\right )\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle -3 \left (\frac {1}{2} b e^6 n \left (\frac {-\frac {x^{5/3} \left (a+b \log \left (c x^{-n/3}\right )\right )^2}{5 e^5}-\frac {2}{5} b n \left (\frac {\frac {x^{4/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{4 e^4}-\frac {1}{4} b n \left (-\frac {x}{3 d e^3}+\frac {x^{2/3}}{2 d^2 e^2}-\frac {\sqrt [3]{x}}{d^3 e}+\frac {\log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^4}-\frac {\log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d^4}\right )}{d}+\frac {\frac {-\frac {1}{3} b n \left (\frac {\log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^3}-\frac {\log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d^3}+\frac {x^{2/3}}{2 d e^2}-\frac {\sqrt [3]{x}}{d^2 e}\right )-\frac {x \left (a+b \log \left (c x^{-n/3}\right )\right )}{3 e^3}}{d}+\frac {\frac {\frac {x^{2/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{2 e^2}-\frac {1}{2} b n \left (\frac {\log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^2}-\frac {\log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d^2}-\frac {\sqrt [3]{x}}{d e}\right )}{d}+\frac {\frac {\frac {b n \log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d}-\frac {\left (d+\frac {e}{\sqrt [3]{x}}\right ) \sqrt [3]{x} \left (a+b \log \left (c x^{-n/3}\right )\right )}{d e}}{d}+\frac {\frac {b n \operatorname {PolyLog}\left (2,d \sqrt [3]{x}\right )}{d}-\frac {\log \left (1-d \sqrt [3]{x}\right ) \left (a+b \log \left (c x^{-n/3}\right )\right )}{d}}{d}}{d}}{d}}{d}\right )}{d}+\frac {\frac {\frac {x^{4/3} \left (a+b \log \left (c x^{-n/3}\right )\right )^2}{4 e^4}-\frac {1}{2} b n \left (\frac {-\frac {1}{3} b n \left (\frac {\log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^3}-\frac {\log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d^3}+\frac {x^{2/3}}{2 d e^2}-\frac {\sqrt [3]{x}}{d^2 e}\right )-\frac {x \left (a+b \log \left (c x^{-n/3}\right )\right )}{3 e^3}}{d}+\frac {\frac {\frac {x^{2/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{2 e^2}-\frac {1}{2} b n \left (\frac {\log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^2}-\frac {\log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d^2}-\frac {\sqrt [3]{x}}{d e}\right )}{d}+\frac {\frac {\frac {b n \log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d}-\frac {\left (d+\frac {e}{\sqrt [3]{x}}\right ) \sqrt [3]{x} \left (a+b \log \left (c x^{-n/3}\right )\right )}{d e}}{d}+\frac {\frac {b n \operatorname {PolyLog}\left (2,d \sqrt [3]{x}\right )}{d}-\frac {\log \left (1-d \sqrt [3]{x}\right ) \left (a+b \log \left (c x^{-n/3}\right )\right )}{d}}{d}}{d}}{d}\right )}{d}+\frac {\frac {-\frac {x \left (a+b \log \left (c x^{-n/3}\right )\right )^2}{3 e^3}-\frac {2}{3} b n \left (\frac {\frac {x^{2/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{2 e^2}-\frac {1}{2} b n \left (\frac {\log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^2}-\frac {\log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d^2}-\frac {\sqrt [3]{x}}{d e}\right )}{d}+\frac {\frac {\frac {b n \log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d}-\frac {\left (d+\frac {e}{\sqrt [3]{x}}\right ) \sqrt [3]{x} \left (a+b \log \left (c x^{-n/3}\right )\right )}{d e}}{d}+\frac {\frac {b n \operatorname {PolyLog}\left (2,d \sqrt [3]{x}\right )}{d}-\frac {\log \left (1-d \sqrt [3]{x}\right ) \left (a+b \log \left (c x^{-n/3}\right )\right )}{d}}{d}}{d}\right )}{d}+\frac {\frac {\frac {x^{2/3} \left (a+b \log \left (c x^{-n/3}\right )\right )^2}{2 e^2}-b n \left (\frac {\frac {b n \log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d}-\frac {\left (d+\frac {e}{\sqrt [3]{x}}\right ) \sqrt [3]{x} \left (a+b \log \left (c x^{-n/3}\right )\right )}{d e}}{d}+\frac {\frac {b n \operatorname {PolyLog}\left (2,d \sqrt [3]{x}\right )}{d}-\frac {\log \left (1-d \sqrt [3]{x}\right ) \left (a+b \log \left (c x^{-n/3}\right )\right )}{d}}{d}\right )}{d}+\frac {\frac {-\frac {\left (d+\frac {e}{\sqrt [3]{x}}\right ) \sqrt [3]{x} \left (a+b \log \left (c x^{-n/3}\right )\right )^2}{d e}-\frac {2 b n \left (-\log \left (1-\frac {d+\frac {e}{\sqrt [3]{x}}}{d}\right ) \left (a+b \log \left (c x^{-n/3}\right )\right )-b n \operatorname {PolyLog}\left (2,\frac {d+\frac {e}{\sqrt [3]{x}}}{d}\right )\right )}{d}}{d}+\frac {\frac {2 b n \left (\left (a+b \log \left (c x^{-n/3}\right )\right ) \operatorname {PolyLog}\left (2,d \sqrt [3]{x}\right )-b n \int \sqrt [3]{x} \operatorname {PolyLog}\left (2,d \sqrt [3]{x}\right )d\left (d+\frac {e}{\sqrt [3]{x}}\right )\right )}{d}-\frac {\log \left (1-d \sqrt [3]{x}\right ) \left (a+b \log \left (c x^{-n/3}\right )\right )^2}{d}}{d}}{d}}{d}}{d}}{d}\right )-\frac {1}{6} x^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^3\right )\) |
| \(\Big \downarrow \) 7143 |
| \(\displaystyle -3 \left (\frac {1}{2} b e^6 n \left (\frac {-\frac {x^{5/3} \left (a+b \log \left (c x^{-n/3}\right )\right )^2}{5 e^5}-\frac {2}{5} b n \left (\frac {\frac {x^{4/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{4 e^4}-\frac {1}{4} b n \left (-\frac {x}{3 d e^3}+\frac {x^{2/3}}{2 d^2 e^2}-\frac {\sqrt [3]{x}}{d^3 e}+\frac {\log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^4}-\frac {\log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d^4}\right )}{d}+\frac {\frac {-\frac {1}{3} b n \left (\frac {\log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^3}-\frac {\log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d^3}+\frac {x^{2/3}}{2 d e^2}-\frac {\sqrt [3]{x}}{d^2 e}\right )-\frac {x \left (a+b \log \left (c x^{-n/3}\right )\right )}{3 e^3}}{d}+\frac {\frac {\frac {x^{2/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{2 e^2}-\frac {1}{2} b n \left (\frac {\log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^2}-\frac {\log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d^2}-\frac {\sqrt [3]{x}}{d e}\right )}{d}+\frac {\frac {\frac {b n \log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d}-\frac {\left (d+\frac {e}{\sqrt [3]{x}}\right ) \sqrt [3]{x} \left (a+b \log \left (c x^{-n/3}\right )\right )}{d e}}{d}+\frac {\frac {b n \operatorname {PolyLog}\left (2,d \sqrt [3]{x}\right )}{d}-\frac {\log \left (1-d \sqrt [3]{x}\right ) \left (a+b \log \left (c x^{-n/3}\right )\right )}{d}}{d}}{d}}{d}}{d}\right )}{d}+\frac {\frac {\frac {x^{4/3} \left (a+b \log \left (c x^{-n/3}\right )\right )^2}{4 e^4}-\frac {1}{2} b n \left (\frac {-\frac {1}{3} b n \left (\frac {\log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^3}-\frac {\log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d^3}+\frac {x^{2/3}}{2 d e^2}-\frac {\sqrt [3]{x}}{d^2 e}\right )-\frac {x \left (a+b \log \left (c x^{-n/3}\right )\right )}{3 e^3}}{d}+\frac {\frac {\frac {x^{2/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{2 e^2}-\frac {1}{2} b n \left (\frac {\log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^2}-\frac {\log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d^2}-\frac {\sqrt [3]{x}}{d e}\right )}{d}+\frac {\frac {\frac {b n \log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d}-\frac {\left (d+\frac {e}{\sqrt [3]{x}}\right ) \sqrt [3]{x} \left (a+b \log \left (c x^{-n/3}\right )\right )}{d e}}{d}+\frac {\frac {b n \operatorname {PolyLog}\left (2,d \sqrt [3]{x}\right )}{d}-\frac {\log \left (1-d \sqrt [3]{x}\right ) \left (a+b \log \left (c x^{-n/3}\right )\right )}{d}}{d}}{d}}{d}\right )}{d}+\frac {\frac {-\frac {x \left (a+b \log \left (c x^{-n/3}\right )\right )^2}{3 e^3}-\frac {2}{3} b n \left (\frac {\frac {x^{2/3} \left (a+b \log \left (c x^{-n/3}\right )\right )}{2 e^2}-\frac {1}{2} b n \left (\frac {\log \left (d+\frac {e}{\sqrt [3]{x}}\right )}{d^2}-\frac {\log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d^2}-\frac {\sqrt [3]{x}}{d e}\right )}{d}+\frac {\frac {\frac {b n \log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d}-\frac {\left (d+\frac {e}{\sqrt [3]{x}}\right ) \sqrt [3]{x} \left (a+b \log \left (c x^{-n/3}\right )\right )}{d e}}{d}+\frac {\frac {b n \operatorname {PolyLog}\left (2,d \sqrt [3]{x}\right )}{d}-\frac {\log \left (1-d \sqrt [3]{x}\right ) \left (a+b \log \left (c x^{-n/3}\right )\right )}{d}}{d}}{d}\right )}{d}+\frac {\frac {\frac {x^{2/3} \left (a+b \log \left (c x^{-n/3}\right )\right )^2}{2 e^2}-b n \left (\frac {\frac {b n \log \left (-\frac {e}{\sqrt [3]{x}}\right )}{d}-\frac {\left (d+\frac {e}{\sqrt [3]{x}}\right ) \sqrt [3]{x} \left (a+b \log \left (c x^{-n/3}\right )\right )}{d e}}{d}+\frac {\frac {b n \operatorname {PolyLog}\left (2,d \sqrt [3]{x}\right )}{d}-\frac {\log \left (1-d \sqrt [3]{x}\right ) \left (a+b \log \left (c x^{-n/3}\right )\right )}{d}}{d}\right )}{d}+\frac {\frac {-\frac {\left (d+\frac {e}{\sqrt [3]{x}}\right ) \sqrt [3]{x} \left (a+b \log \left (c x^{-n/3}\right )\right )^2}{d e}-\frac {2 b n \left (-\log \left (1-\frac {d+\frac {e}{\sqrt [3]{x}}}{d}\right ) \left (a+b \log \left (c x^{-n/3}\right )\right )-b n \operatorname {PolyLog}\left (2,\frac {d+\frac {e}{\sqrt [3]{x}}}{d}\right )\right )}{d}}{d}+\frac {\frac {2 b n \left (\left (a+b \log \left (c x^{-n/3}\right )\right ) \operatorname {PolyLog}\left (2,d \sqrt [3]{x}\right )+b n \operatorname {PolyLog}\left (3,d \sqrt [3]{x}\right )\right )}{d}-\frac {\log \left (1-d \sqrt [3]{x}\right ) \left (a+b \log \left (c x^{-n/3}\right )\right )^2}{d}}{d}}{d}}{d}}{d}}{d}\right )-\frac {1}{6} x^2 \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^3\right )\) |
Input:
Int[x*(a + b*Log[c*(d + e/x^(1/3))^n])^3,x]
Output:
-3*(-1/6*(x^2*(a + b*Log[c*(d + e/x^(1/3))^n])^3) + (b*e^6*n*((-1/5*(x^(5/ 3)*(a + b*Log[c/x^(n/3)])^2)/e^5 - (2*b*n*((-1/4*(b*n*(-(x^(1/3)/(d^3*e)) + x^(2/3)/(2*d^2*e^2) - x/(3*d*e^3) + Log[d + e/x^(1/3)]/d^4 - Log[-(e/x^( 1/3))]/d^4)) + (x^(4/3)*(a + b*Log[c/x^(n/3)]))/(4*e^4))/d + ((-1/3*(b*n*( -(x^(1/3)/(d^2*e)) + x^(2/3)/(2*d*e^2) + Log[d + e/x^(1/3)]/d^3 - Log[-(e/ x^(1/3))]/d^3)) - (x*(a + b*Log[c/x^(n/3)]))/(3*e^3))/d + ((-1/2*(b*n*(-(x ^(1/3)/(d*e)) + Log[d + e/x^(1/3)]/d^2 - Log[-(e/x^(1/3))]/d^2)) + (x^(2/3 )*(a + b*Log[c/x^(n/3)]))/(2*e^2))/d + (((b*n*Log[-(e/x^(1/3))])/d - ((d + e/x^(1/3))*x^(1/3)*(a + b*Log[c/x^(n/3)]))/(d*e))/d + (-((Log[1 - d*x^(1/ 3)]*(a + b*Log[c/x^(n/3)]))/d) + (b*n*PolyLog[2, d*x^(1/3)])/d)/d)/d)/d)/d ))/5)/d + (((x^(4/3)*(a + b*Log[c/x^(n/3)])^2)/(4*e^4) - (b*n*((-1/3*(b*n* (-(x^(1/3)/(d^2*e)) + x^(2/3)/(2*d*e^2) + Log[d + e/x^(1/3)]/d^3 - Log[-(e /x^(1/3))]/d^3)) - (x*(a + b*Log[c/x^(n/3)]))/(3*e^3))/d + ((-1/2*(b*n*(-( x^(1/3)/(d*e)) + Log[d + e/x^(1/3)]/d^2 - Log[-(e/x^(1/3))]/d^2)) + (x^(2/ 3)*(a + b*Log[c/x^(n/3)]))/(2*e^2))/d + (((b*n*Log[-(e/x^(1/3))])/d - ((d + e/x^(1/3))*x^(1/3)*(a + b*Log[c/x^(n/3)]))/(d*e))/d + (-((Log[1 - d*x^(1 /3)]*(a + b*Log[c/x^(n/3)]))/d) + (b*n*PolyLog[2, d*x^(1/3)])/d)/d)/d)/d)) /2)/d + ((-1/3*(x*(a + b*Log[c/x^(n/3)])^2)/e^3 - (2*b*n*((-1/2*(b*n*(-(x^ (1/3)/(d*e)) + Log[d + e/x^(1/3)]/d^2 - Log[-(e/x^(1/3))]/d^2)) + (x^(2/3) *(a + b*Log[c/x^(n/3)]))/(2*e^2))/d + (((b*n*Log[-(e/x^(1/3))])/d - ((d...
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[((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[((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_)), x_Symb ol] :> Simp[Log[1 + e*(x/d)]*((a + b*Log[c*x^n])^p/e), x] - Simp[b*n*(p/e) Int[Log[1 + e*(x/d)]*((a + b*Log[c*x^n])^(p - 1)/x), x], x] /; FreeQ[{a, b, c, d, e, n}, x] && IGtQ[p, 0]
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/((d_) + (e_.)*(x_))^2, x_Sy mbol] :> Simp[x*((a + b*Log[c*x^n])^p/(d*(d + e*x))), x] - Simp[b*n*(p/d) Int[(a + b*Log[c*x^n])^(p - 1)/(d + e*x), x], x] /; FreeQ[{a, b, c, d, e, n, p}, x] && GtQ[p, 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[(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_))^(q_.), x_Symbol] :> Simp[(f + g*x)^(q + 1)*((a + b*Log[c*(d + e*x)^ n])^p/(g*(q + 1))), x] - Simp[b*e*n*(p/(g*(q + 1))) Int[(f + g*x)^(q + 1) *((a + b*Log[c*(d + e*x)^n])^(p - 1)/(d + e*x)), x], x] /; FreeQ[{a, b, c, d, e, f, g, n, q}, x] && NeQ[e*f - d*g, 0] && GtQ[p, 0] && NeQ[q, -1] && In tegersQ[2*p, 2*q] && ( !IGtQ[q, 0] || (EqQ[p, 2] && NeQ[q, 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 _.), 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[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]
\[\int x {\left (a +b \ln \left (c \left (d +\frac {e}{x^{\frac {1}{3}}}\right )^{n}\right )\right )}^{3}d x\]
Input:
int(x*(a+b*ln(c*(d+e/x^(1/3))^n))^3,x)
Output:
int(x*(a+b*ln(c*(d+e/x^(1/3))^n))^3,x)
\[ \int x \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^3 \, dx=\int { {\left (b \log \left (c {\left (d + \frac {e}{x^{\frac {1}{3}}}\right )}^{n}\right ) + a\right )}^{3} x \,d x } \] Input:
integrate(x*(a+b*log(c*(d+e/x^(1/3))^n))^3,x, algorithm="fricas")
Output:
integral(b^3*x*log(c*((d*x + e*x^(2/3))/x)^n)^3 + 3*a*b^2*x*log(c*((d*x + e*x^(2/3))/x)^n)^2 + 3*a^2*b*x*log(c*((d*x + e*x^(2/3))/x)^n) + a^3*x, x)
\[ \int x \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^3 \, dx=\int x \left (a + b \log {\left (c \left (d + \frac {e}{\sqrt [3]{x}}\right )^{n} \right )}\right )^{3}\, dx \] Input:
integrate(x*(a+b*ln(c*(d+e/x**(1/3))**n))**3,x)
Output:
Integral(x*(a + b*log(c*(d + e/x**(1/3))**n))**3, x)
\[ \int x \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^3 \, dx=\int { {\left (b \log \left (c {\left (d + \frac {e}{x^{\frac {1}{3}}}\right )}^{n}\right ) + a\right )}^{3} x \,d x } \] Input:
integrate(x*(a+b*log(c*(d+e/x^(1/3))^n))^3,x, algorithm="maxima")
Output:
1/2*b^3*x^2*log((d*x^(1/3) + e)^n)^3 - integrate(1/2*(2*(b^3*d*x^2 + b^3*e *x^(5/3))*log(x^(1/3*n))^3 - 2*(b^3*d*log(c)^3 + 3*a*b^2*d*log(c)^2 + 3*a^ 2*b*d*log(c) + a^3*d)*x^2 + (b^3*d*n*x^2 - 6*(b^3*d*log(c) + a*b^2*d)*x^2 - 6*(b^3*e*log(c) + a*b^2*e)*x^(5/3) + 6*(b^3*d*x^2 + b^3*e*x^(5/3))*log(x ^(1/3*n)))*log((d*x^(1/3) + e)^n)^2 - 6*((b^3*d*log(c) + a*b^2*d)*x^2 + (b ^3*e*log(c) + a*b^2*e)*x^(5/3))*log(x^(1/3*n))^2 - 2*(b^3*e*log(c)^3 + 3*a *b^2*e*log(c)^2 + 3*a^2*b*e*log(c) + a^3*e)*x^(5/3) - 6*((b^3*d*log(c)^2 + 2*a*b^2*d*log(c) + a^2*b*d)*x^2 + (b^3*d*x^2 + b^3*e*x^(5/3))*log(x^(1/3* n))^2 + (b^3*e*log(c)^2 + 2*a*b^2*e*log(c) + a^2*b*e)*x^(5/3) - 2*((b^3*d* log(c) + a*b^2*d)*x^2 + (b^3*e*log(c) + a*b^2*e)*x^(5/3))*log(x^(1/3*n)))* log((d*x^(1/3) + e)^n) + 6*((b^3*d*log(c)^2 + 2*a*b^2*d*log(c) + a^2*b*d)* x^2 + (b^3*e*log(c)^2 + 2*a*b^2*e*log(c) + a^2*b*e)*x^(5/3))*log(x^(1/3*n) ))/(d*x + e*x^(2/3)), x)
\[ \int x \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^3 \, dx=\int { {\left (b \log \left (c {\left (d + \frac {e}{x^{\frac {1}{3}}}\right )}^{n}\right ) + a\right )}^{3} x \,d x } \] Input:
integrate(x*(a+b*log(c*(d+e/x^(1/3))^n))^3,x, algorithm="giac")
Output:
integrate((b*log(c*(d + e/x^(1/3))^n) + a)^3*x, x)
Timed out. \[ \int x \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^3 \, dx=\int x\,{\left (a+b\,\ln \left (c\,{\left (d+\frac {e}{x^{1/3}}\right )}^n\right )\right )}^3 \,d x \] Input:
int(x*(a + b*log(c*(d + e/x^(1/3))^n))^3,x)
Output:
int(x*(a + b*log(c*(d + e/x^(1/3))^n))^3, x)
\[ \int x \left (a+b \log \left (c \left (d+\frac {e}{\sqrt [3]{x}}\right )^n\right )\right )^3 \, dx =\text {Too large to display} \] Input:
int(x*(a+b*log(c*(d+e/x^(1/3))^n))^3,x)
Output:
(36*x**(2/3)*log(((x**(1/3)*d + e)**n*c)/x**(n/3))**2*b**3*d**5*e*n*x - 90 *x**(2/3)*log(((x**(1/3)*d + e)**n*c)/x**(n/3))**2*b**3*d**2*e**4*n + 72*x **(2/3)*log(((x**(1/3)*d + e)**n*c)/x**(n/3))*a*b**2*d**5*e*n*x - 180*x**( 2/3)*log(((x**(1/3)*d + e)**n*c)/x**(n/3))*a*b**2*d**2*e**4*n + 141*x**(2/ 3)*log(((x**(1/3)*d + e)**n*c)/x**(n/3))*b**3*d**2*e**4*n**2 + 36*x**(2/3) *a**2*b*d**5*e*n*x - 90*x**(2/3)*a**2*b*d**2*e**4*n + 141*x**(2/3)*a*b**2* d**2*e**4*n**2 - 36*x**(2/3)*b**3*d**2*e**4*n**3 + 60*x**(1/3)*log(((x**(1 /3)*d + e)**n*c)/x**(n/3))**3*b**3*d*e**5 - 45*x**(1/3)*log(((x**(1/3)*d + e)**n*c)/x**(n/3))**2*b**3*d**4*e**2*n*x + 180*x**(1/3)*log(((x**(1/3)*d + e)**n*c)/x**(n/3))**2*b**3*d*e**5*n - 90*x**(1/3)*log(((x**(1/3)*d + e)* *n*c)/x**(n/3))*a*b**2*d**4*e**2*n*x + 360*x**(1/3)*log(((x**(1/3)*d + e)* *n*c)/x**(n/3))*a*b**2*d*e**5*n + 18*x**(1/3)*log(((x**(1/3)*d + e)**n*c)/ x**(n/3))*b**3*d**4*e**2*n**2*x - 462*x**(1/3)*log(((x**(1/3)*d + e)**n*c) /x**(n/3))*b**3*d*e**5*n**2 - 45*x**(1/3)*a**2*b*d**4*e**2*n*x + 180*x**(1 /3)*a**2*b*d*e**5*n + 18*x**(1/3)*a*b**2*d**4*e**2*n**2*x - 462*x**(1/3)*a *b**2*d*e**5*n**2 + 213*x**(1/3)*b**3*d*e**5*n**3 - 20*int(log(((x**(1/3)* d + e)**n*c)/x**(n/3))**3/(x**(2/3)*e + d*x),x)*b**3*d*e**6 - 120*int(log( ((x**(1/3)*d + e)**n*c)/x**(n/3))/(x**(2/3)*e + d*x),x)*a*b**2*d*e**6*n + 274*int(log(((x**(1/3)*d + e)**n*c)/x**(n/3))/(x**(2/3)*e + d*x),x)*b**3*d *e**6*n**2 - 20*int((x**(1/3)*log(((x**(1/3)*d + e)**n*c)/x**(n/3))**3)...