Integrand size = 24, antiderivative size = 765 \[ \int \frac {\left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{x^3} \, dx =\text {Too large to display} \] Output:
-1/20*b^3*e^3*n^3/d^3/x+3/10*b^3*e^4*n^3/d^4/x^(2/3)-71/40*b^3*e^5*n^3/d^5 /x^(1/3)+71/40*b^3*e^6*n^3*ln(d+e*x^(1/3))/d^6-3/20*b^2*e^2*n^2*(a+b*ln(c* (d+e*x^(1/3))^n))/d^2/x^(4/3)+9/20*b^2*e^3*n^2*(a+b*ln(c*(d+e*x^(1/3))^n)) /d^3/x-47/40*b^2*e^4*n^2*(a+b*ln(c*(d+e*x^(1/3))^n))/d^4/x^(2/3)+77/20*b^2 *e^5*n^2*(d+e*x^(1/3))*(a+b*ln(c*(d+e*x^(1/3))^n))/d^6/x^(1/3)+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^6-3/10*b*e*n*(a +b*ln(c*(d+e*x^(1/3))^n))^2/d/x^(5/3)+3/8*b*e^2*n*(a+b*ln(c*(d+e*x^(1/3))^ n))^2/d^2/x^(4/3)-1/2*b*e^3*n*(a+b*ln(c*(d+e*x^(1/3))^n))^2/d^3/x+3/4*b*e^ 4*n*(a+b*ln(c*(d+e*x^(1/3))^n))^2/d^4/x^(2/3)-3/2*b*e^5*n*(d+e*x^(1/3))*(a +b*ln(c*(d+e*x^(1/3))^n))^2/d^6/x^(1/3)-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*(a+b*ln(c*(d+e*x^(1/3))^n))^3/x^2+3* b^2*e^6*n^2*(a+b*ln(c*(d+e*x^(1/3))^n))*ln(-e*x^(1/3)/d)/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*x^(1/3)/d)/d^6+3*b^3*e^6*n^3*polylog(3,d/(d+e*x^(1/3)))/d^ 6
Time = 1.99 (sec) , antiderivative size = 1074, normalized size of antiderivative = 1.40 \[ \int \frac {\left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{x^3} \, dx =\text {Too large to display} \] Input:
Integrate[(a + b*Log[c*(d + e*x^(1/3))^n])^3/x^3,x]
Output:
-1/40*(12*b*d^5*e*n*x^(1/3)*(a - b*n*Log[d + e*x^(1/3)] + b*Log[c*(d + e*x ^(1/3))^n])^2 - 15*b*d^4*e^2*n*x^(2/3)*(a - b*n*Log[d + e*x^(1/3)] + b*Log [c*(d + e*x^(1/3))^n])^2 + 20*b*d^3*e^3*n*x*(a - b*n*Log[d + e*x^(1/3)] + b*Log[c*(d + e*x^(1/3))^n])^2 - 30*b*d^2*e^4*n*x^(4/3)*(a - b*n*Log[d + e* x^(1/3)] + b*Log[c*(d + e*x^(1/3))^n])^2 + 60*b*d*e^5*n*x^(5/3)*(a - b*n*L og[d + e*x^(1/3)] + b*Log[c*(d + e*x^(1/3))^n])^2 + 60*b*d^6*n*Log[d + e*x ^(1/3)]*(a - b*n*Log[d + e*x^(1/3)] + b*Log[c*(d + e*x^(1/3))^n])^2 - 60*b *e^6*n*x^2*Log[d + e*x^(1/3)]*(a - b*n*Log[d + e*x^(1/3)] + b*Log[c*(d + e *x^(1/3))^n])^2 + 20*d^6*(a - b*n*Log[d + e*x^(1/3)] + b*Log[c*(d + e*x^(1 /3))^n])^3 + 20*b*e^6*n*x^2*(a - b*n*Log[d + e*x^(1/3)] + b*Log[c*(d + e*x ^(1/3))^n])^2*Log[x] + b^2*n^2*(a - b*n*Log[d + e*x^(1/3)] + b*Log[c*(d + e*x^(1/3))^n])*(6*d^4*e^2*x^(2/3) - 18*d^3*e^3*x + 47*d^2*e^4*x^(4/3) - 15 4*d*e^5*x^(5/3) + 60*(d^6 - e^6*x^2)*Log[d + e*x^(1/3)]^2 - 274*e^6*x^2*Lo g[-((e*x^(1/3))/d)] + 2*Log[d + e*x^(1/3)]*(12*d^5*e*x^(1/3) - 15*d^4*e^2* x^(2/3) + 20*d^3*e^3*x - 30*d^2*e^4*x^(4/3) + 60*d*e^5*x^(5/3) + 137*e^6*x ^2 + 60*e^6*x^2*Log[-((e*x^(1/3))/d)]) + 120*e^6*x^2*PolyLog[2, 1 + (e*x^( 1/3))/d]) + b^3*n^3*(3*d^4*e^2*x^(2/3)*(2 - 5*Log[d + e*x^(1/3)])*Log[d + e*x^(1/3)] + 12*d^5*e*x^(1/3)*Log[d + e*x^(1/3)]^2 + 20*d^6*Log[d + e*x^(1 /3)]^3 + 2*d^3*e^3*x*(1 - 9*Log[d + e*x^(1/3)] + 10*Log[d + e*x^(1/3)]^2) - d^2*e^4*x^(4/3)*(12 - 47*Log[d + e*x^(1/3)] + 30*Log[d + e*x^(1/3)]^2...
Time = 11.00 (sec) , antiderivative size = 1382, normalized size of antiderivative = 1.81, number of steps used = 28, number of rules used = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.125, 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.
| \(\displaystyle \int \frac {\left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{x^3} \, dx\) |
| \(\Big \downarrow \) 2904 |
| \(\displaystyle 3 \int \frac {\left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{x^{7/3}}d\sqrt [3]{x}\) |
| \(\Big \downarrow \) 2845 |
| \(\displaystyle 3 \left (\frac {1}{2} b e n \int \frac {\left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^2}{\left (d+e \sqrt [3]{x}\right ) x^2}d\sqrt [3]{x}-\frac {\left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{6 x^2}\right )\) |
| \(\Big \downarrow \) 2858 |
| \(\displaystyle 3 \left (\frac {1}{2} b n \int \frac {\left (a+b \log \left (c x^{n/3}\right )\right )^2}{x^{7/3}}d\left (d+e \sqrt [3]{x}\right )-\frac {\left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{6 x^2}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 3 \left (\frac {1}{2} b e^6 n \int \frac {\left (a+b \log \left (c x^{n/3}\right )\right )^2}{e^6 x^{7/3}}d\left (d+e \sqrt [3]{x}\right )-\frac {\left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{6 x^2}\right )\) |
| \(\Big \downarrow \) 2789 |
| \(\displaystyle 3 \left (\frac {1}{2} b e^6 n \left (\frac {\int \frac {\left (a+b \log \left (c x^{n/3}\right )\right )^2}{e^6 x^2}d\left (d+e \sqrt [3]{x}\right )}{d}+\frac {\int -\frac {\left (a+b \log \left (c x^{n/3}\right )\right )^2}{e^5 x^2}d\left (d+e \sqrt [3]{x}\right )}{d}\right )-\frac {\left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{6 x^2}\right )\) |
| \(\Big \downarrow \) 2756 |
| \(\displaystyle 3 \left (\frac {1}{2} b e^6 n \left (\frac {-\frac {2}{5} b n \int -\frac {a+b \log \left (c x^{n/3}\right )}{e^5 x^2}d\left (d+e \sqrt [3]{x}\right )-\frac {\left (a+b \log \left (c x^{n/3}\right )\right )^2}{5 e^5 x^{5/3}}}{d}+\frac {\int -\frac {\left (a+b \log \left (c x^{n/3}\right )\right )^2}{e^5 x^2}d\left (d+e \sqrt [3]{x}\right )}{d}\right )-\frac {\left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{6 x^2}\right )\) |
| \(\Big \downarrow \) 2789 |
| \(\displaystyle 3 \left (\frac {1}{2} b e^6 n \left (\frac {-\frac {2}{5} b n \left (\frac {\int -\frac {a+b \log \left (c x^{n/3}\right )}{e^5 x^{5/3}}d\left (d+e \sqrt [3]{x}\right )}{d}+\frac {\int \frac {a+b \log \left (c x^{n/3}\right )}{e^4 x^{5/3}}d\left (d+e \sqrt [3]{x}\right )}{d}\right )-\frac {\left (a+b \log \left (c x^{n/3}\right )\right )^2}{5 e^5 x^{5/3}}}{d}+\frac {\frac {\int -\frac {\left (a+b \log \left (c x^{n/3}\right )\right )^2}{e^5 x^{5/3}}d\left (d+e \sqrt [3]{x}\right )}{d}+\frac {\int \frac {\left (a+b \log \left (c x^{n/3}\right )\right )^2}{e^4 x^{5/3}}d\left (d+e \sqrt [3]{x}\right )}{d}}{d}\right )-\frac {\left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{6 x^2}\right )\) |
| \(\Big \downarrow \) 2756 |
| \(\displaystyle 3 \left (\frac {1}{2} b e^6 n \left (\frac {\frac {\frac {\left (a+b \log \left (c x^{n/3}\right )\right )^2}{4 e^4 x^{4/3}}-\frac {1}{2} b n \int \frac {a+b \log \left (c x^{n/3}\right )}{e^4 x^{5/3}}d\left (d+e \sqrt [3]{x}\right )}{d}+\frac {\int \frac {\left (a+b \log \left (c x^{n/3}\right )\right )^2}{e^4 x^{5/3}}d\left (d+e \sqrt [3]{x}\right )}{d}}{d}+\frac {-\frac {2}{5} b n \left (\frac {\frac {a+b \log \left (c x^{n/3}\right )}{4 e^4 x^{4/3}}-\frac {1}{4} b n \int \frac {1}{e^4 x^{5/3}}d\left (d+e \sqrt [3]{x}\right )}{d}+\frac {\int \frac {a+b \log \left (c x^{n/3}\right )}{e^4 x^{5/3}}d\left (d+e \sqrt [3]{x}\right )}{d}\right )-\frac {\left (a+b \log \left (c x^{n/3}\right )\right )^2}{5 e^5 x^{5/3}}}{d}\right )-\frac {\left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{6 x^2}\right )\) |
| \(\Big \downarrow \) 54 |
| \(\displaystyle 3 \left (\frac {1}{2} b e^6 n \left (\frac {-\frac {2}{5} b n \left (\frac {\frac {a+b \log \left (c x^{n/3}\right )}{4 e^4 x^{4/3}}-\frac {1}{4} b n \int \left (-\frac {1}{d^4 e \sqrt [3]{x}}+\frac {1}{d^4 \sqrt [3]{x}}+\frac {1}{d^3 e^2 x^{2/3}}-\frac {1}{d^2 e^3 x}+\frac {1}{d e^4 x^{4/3}}\right )d\left (d+e \sqrt [3]{x}\right )}{d}+\frac {\int \frac {a+b \log \left (c x^{n/3}\right )}{e^4 x^{5/3}}d\left (d+e \sqrt [3]{x}\right )}{d}\right )-\frac {\left (a+b \log \left (c x^{n/3}\right )\right )^2}{5 e^5 x^{5/3}}}{d}+\frac {\frac {\frac {\left (a+b \log \left (c x^{n/3}\right )\right )^2}{4 e^4 x^{4/3}}-\frac {1}{2} b n \int \frac {a+b \log \left (c x^{n/3}\right )}{e^4 x^{5/3}}d\left (d+e \sqrt [3]{x}\right )}{d}+\frac {\int \frac {\left (a+b \log \left (c x^{n/3}\right )\right )^2}{e^4 x^{5/3}}d\left (d+e \sqrt [3]{x}\right )}{d}}{d}\right )-\frac {\left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{6 x^2}\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 3 \left (\frac {1}{2} b e^6 n \left (\frac {-\frac {2}{5} b n \left (\frac {\int \frac {a+b \log \left (c x^{n/3}\right )}{e^4 x^{5/3}}d\left (d+e \sqrt [3]{x}\right )}{d}+\frac {\frac {a+b \log \left (c x^{n/3}\right )}{4 e^4 x^{4/3}}-\frac {1}{4} b n \left (\frac {\log \left (d+e \sqrt [3]{x}\right )}{d^4}-\frac {\log \left (-e \sqrt [3]{x}\right )}{d^4}-\frac {1}{d^3 e \sqrt [3]{x}}+\frac {1}{2 d^2 e^2 x^{2/3}}-\frac {1}{3 d e^3 x}\right )}{d}\right )-\frac {\left (a+b \log \left (c x^{n/3}\right )\right )^2}{5 e^5 x^{5/3}}}{d}+\frac {\frac {\frac {\left (a+b \log \left (c x^{n/3}\right )\right )^2}{4 e^4 x^{4/3}}-\frac {1}{2} b n \int \frac {a+b \log \left (c x^{n/3}\right )}{e^4 x^{5/3}}d\left (d+e \sqrt [3]{x}\right )}{d}+\frac {\int \frac {\left (a+b \log \left (c x^{n/3}\right )\right )^2}{e^4 x^{5/3}}d\left (d+e \sqrt [3]{x}\right )}{d}}{d}\right )-\frac {\left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{6 x^2}\right )\) |
| \(\Big \downarrow \) 2789 |
| \(\displaystyle 3 \left (\frac {1}{2} b e^6 n \left (\frac {-\frac {2}{5} b n \left (\frac {\frac {\int \frac {a+b \log \left (c x^{n/3}\right )}{e^4 x^{4/3}}d\left (d+e \sqrt [3]{x}\right )}{d}+\frac {\int -\frac {a+b \log \left (c x^{n/3}\right )}{e^3 x^{4/3}}d\left (d+e \sqrt [3]{x}\right )}{d}}{d}+\frac {\frac {a+b \log \left (c x^{n/3}\right )}{4 e^4 x^{4/3}}-\frac {1}{4} b n \left (\frac {\log \left (d+e \sqrt [3]{x}\right )}{d^4}-\frac {\log \left (-e \sqrt [3]{x}\right )}{d^4}-\frac {1}{d^3 e \sqrt [3]{x}}+\frac {1}{2 d^2 e^2 x^{2/3}}-\frac {1}{3 d e^3 x}\right )}{d}\right )-\frac {\left (a+b \log \left (c x^{n/3}\right )\right )^2}{5 e^5 x^{5/3}}}{d}+\frac {\frac {\frac {\left (a+b \log \left (c x^{n/3}\right )\right )^2}{4 e^4 x^{4/3}}-\frac {1}{2} b n \left (\frac {\int \frac {a+b \log \left (c x^{n/3}\right )}{e^4 x^{4/3}}d\left (d+e \sqrt [3]{x}\right )}{d}+\frac {\int -\frac {a+b \log \left (c x^{n/3}\right )}{e^3 x^{4/3}}d\left (d+e \sqrt [3]{x}\right )}{d}\right )}{d}+\frac {\frac {\int \frac {\left (a+b \log \left (c x^{n/3}\right )\right )^2}{e^4 x^{4/3}}d\left (d+e \sqrt [3]{x}\right )}{d}+\frac {\int -\frac {\left (a+b \log \left (c x^{n/3}\right )\right )^2}{e^3 x^{4/3}}d\left (d+e \sqrt [3]{x}\right )}{d}}{d}}{d}\right )-\frac {\left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{6 x^2}\right )\) |
| \(\Big \downarrow \) 2756 |
| \(\displaystyle 3 \left (\frac {1}{2} b e^6 n \left (\frac {-\frac {2}{5} b n \left (\frac {\frac {-\frac {1}{3} b n \int -\frac {1}{e^3 x^{4/3}}d\left (d+e \sqrt [3]{x}\right )-\frac {a+b \log \left (c x^{n/3}\right )}{3 e^3 x}}{d}+\frac {\int -\frac {a+b \log \left (c x^{n/3}\right )}{e^3 x^{4/3}}d\left (d+e \sqrt [3]{x}\right )}{d}}{d}+\frac {\frac {a+b \log \left (c x^{n/3}\right )}{4 e^4 x^{4/3}}-\frac {1}{4} b n \left (\frac {\log \left (d+e \sqrt [3]{x}\right )}{d^4}-\frac {\log \left (-e \sqrt [3]{x}\right )}{d^4}-\frac {1}{d^3 e \sqrt [3]{x}}+\frac {1}{2 d^2 e^2 x^{2/3}}-\frac {1}{3 d e^3 x}\right )}{d}\right )-\frac {\left (a+b \log \left (c x^{n/3}\right )\right )^2}{5 e^5 x^{5/3}}}{d}+\frac {\frac {\frac {-\frac {2}{3} b n \int -\frac {a+b \log \left (c x^{n/3}\right )}{e^3 x^{4/3}}d\left (d+e \sqrt [3]{x}\right )-\frac {\left (a+b \log \left (c x^{n/3}\right )\right )^2}{3 e^3 x}}{d}+\frac {\int -\frac {\left (a+b \log \left (c x^{n/3}\right )\right )^2}{e^3 x^{4/3}}d\left (d+e \sqrt [3]{x}\right )}{d}}{d}+\frac {\frac {\left (a+b \log \left (c x^{n/3}\right )\right )^2}{4 e^4 x^{4/3}}-\frac {1}{2} b n \left (\frac {-\frac {1}{3} b n \int -\frac {1}{e^3 x^{4/3}}d\left (d+e \sqrt [3]{x}\right )-\frac {a+b \log \left (c x^{n/3}\right )}{3 e^3 x}}{d}+\frac {\int -\frac {a+b \log \left (c x^{n/3}\right )}{e^3 x^{4/3}}d\left (d+e \sqrt [3]{x}\right )}{d}\right )}{d}}{d}\right )-\frac {\left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{6 x^2}\right )\) |
| \(\Big \downarrow \) 54 |
| \(\displaystyle 3 \left (\frac {1}{2} b e^6 n \left (\frac {\frac {\frac {\left (a+b \log \left (c x^{n/3}\right )\right )^2}{4 e^4 x^{4/3}}-\frac {1}{2} b n \left (\frac {-\frac {1}{3} b n \int \left (-\frac {1}{d^3 e \sqrt [3]{x}}+\frac {1}{d^3 \sqrt [3]{x}}+\frac {1}{d^2 e^2 x^{2/3}}-\frac {1}{d e^3 x}\right )d\left (d+e \sqrt [3]{x}\right )-\frac {a+b \log \left (c x^{n/3}\right )}{3 e^3 x}}{d}+\frac {\int -\frac {a+b \log \left (c x^{n/3}\right )}{e^3 x^{4/3}}d\left (d+e \sqrt [3]{x}\right )}{d}\right )}{d}+\frac {\frac {-\frac {2}{3} b n \int -\frac {a+b \log \left (c x^{n/3}\right )}{e^3 x^{4/3}}d\left (d+e \sqrt [3]{x}\right )-\frac {\left (a+b \log \left (c x^{n/3}\right )\right )^2}{3 e^3 x}}{d}+\frac {\int -\frac {\left (a+b \log \left (c x^{n/3}\right )\right )^2}{e^3 x^{4/3}}d\left (d+e \sqrt [3]{x}\right )}{d}}{d}}{d}+\frac {-\frac {2}{5} b n \left (\frac {\frac {-\frac {1}{3} b n \int \left (-\frac {1}{d^3 e \sqrt [3]{x}}+\frac {1}{d^3 \sqrt [3]{x}}+\frac {1}{d^2 e^2 x^{2/3}}-\frac {1}{d e^3 x}\right )d\left (d+e \sqrt [3]{x}\right )-\frac {a+b \log \left (c x^{n/3}\right )}{3 e^3 x}}{d}+\frac {\int -\frac {a+b \log \left (c x^{n/3}\right )}{e^3 x^{4/3}}d\left (d+e \sqrt [3]{x}\right )}{d}}{d}+\frac {\frac {a+b \log \left (c x^{n/3}\right )}{4 e^4 x^{4/3}}-\frac {1}{4} b n \left (\frac {\log \left (d+e \sqrt [3]{x}\right )}{d^4}-\frac {\log \left (-e \sqrt [3]{x}\right )}{d^4}-\frac {1}{d^3 e \sqrt [3]{x}}+\frac {1}{2 d^2 e^2 x^{2/3}}-\frac {1}{3 d e^3 x}\right )}{d}\right )-\frac {\left (a+b \log \left (c x^{n/3}\right )\right )^2}{5 e^5 x^{5/3}}}{d}\right )-\frac {\left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{6 x^2}\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 3 \left (\frac {1}{2} b e^6 n \left (\frac {\frac {\frac {\left (a+b \log \left (c x^{n/3}\right )\right )^2}{4 e^4 x^{4/3}}-\frac {1}{2} b n \left (\frac {\int -\frac {a+b \log \left (c x^{n/3}\right )}{e^3 x^{4/3}}d\left (d+e \sqrt [3]{x}\right )}{d}+\frac {-\frac {a+b \log \left (c x^{n/3}\right )}{3 e^3 x}-\frac {1}{3} b n \left (\frac {\log \left (d+e \sqrt [3]{x}\right )}{d^3}-\frac {\log \left (-e \sqrt [3]{x}\right )}{d^3}-\frac {1}{d^2 e \sqrt [3]{x}}+\frac {1}{2 d e^2 x^{2/3}}\right )}{d}\right )}{d}+\frac {\frac {-\frac {2}{3} b n \int -\frac {a+b \log \left (c x^{n/3}\right )}{e^3 x^{4/3}}d\left (d+e \sqrt [3]{x}\right )-\frac {\left (a+b \log \left (c x^{n/3}\right )\right )^2}{3 e^3 x}}{d}+\frac {\int -\frac {\left (a+b \log \left (c x^{n/3}\right )\right )^2}{e^3 x^{4/3}}d\left (d+e \sqrt [3]{x}\right )}{d}}{d}}{d}+\frac {-\frac {2}{5} b n \left (\frac {\frac {\int -\frac {a+b \log \left (c x^{n/3}\right )}{e^3 x^{4/3}}d\left (d+e \sqrt [3]{x}\right )}{d}+\frac {-\frac {a+b \log \left (c x^{n/3}\right )}{3 e^3 x}-\frac {1}{3} b n \left (\frac {\log \left (d+e \sqrt [3]{x}\right )}{d^3}-\frac {\log \left (-e \sqrt [3]{x}\right )}{d^3}-\frac {1}{d^2 e \sqrt [3]{x}}+\frac {1}{2 d e^2 x^{2/3}}\right )}{d}}{d}+\frac {\frac {a+b \log \left (c x^{n/3}\right )}{4 e^4 x^{4/3}}-\frac {1}{4} b n \left (\frac {\log \left (d+e \sqrt [3]{x}\right )}{d^4}-\frac {\log \left (-e \sqrt [3]{x}\right )}{d^4}-\frac {1}{d^3 e \sqrt [3]{x}}+\frac {1}{2 d^2 e^2 x^{2/3}}-\frac {1}{3 d e^3 x}\right )}{d}\right )-\frac {\left (a+b \log \left (c x^{n/3}\right )\right )^2}{5 e^5 x^{5/3}}}{d}\right )-\frac {\left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{6 x^2}\right )\) |
| \(\Big \downarrow \) 2789 |
| \(\displaystyle 3 \left (\frac {1}{2} b e^6 n \left (\frac {\frac {\frac {\left (a+b \log \left (c x^{n/3}\right )\right )^2}{4 e^4 x^{4/3}}-\frac {1}{2} b n \left (\frac {\frac {\int -\frac {a+b \log \left (c x^{n/3}\right )}{e^3 x}d\left (d+e \sqrt [3]{x}\right )}{d}+\frac {\int \frac {a+b \log \left (c x^{n/3}\right )}{e^2 x}d\left (d+e \sqrt [3]{x}\right )}{d}}{d}+\frac {-\frac {a+b \log \left (c x^{n/3}\right )}{3 e^3 x}-\frac {1}{3} b n \left (\frac {\log \left (d+e \sqrt [3]{x}\right )}{d^3}-\frac {\log \left (-e \sqrt [3]{x}\right )}{d^3}-\frac {1}{d^2 e \sqrt [3]{x}}+\frac {1}{2 d e^2 x^{2/3}}\right )}{d}\right )}{d}+\frac {\frac {-\frac {2}{3} b n \left (\frac {\int -\frac {a+b \log \left (c x^{n/3}\right )}{e^3 x}d\left (d+e \sqrt [3]{x}\right )}{d}+\frac {\int \frac {a+b \log \left (c x^{n/3}\right )}{e^2 x}d\left (d+e \sqrt [3]{x}\right )}{d}\right )-\frac {\left (a+b \log \left (c x^{n/3}\right )\right )^2}{3 e^3 x}}{d}+\frac {\frac {\int -\frac {\left (a+b \log \left (c x^{n/3}\right )\right )^2}{e^3 x}d\left (d+e \sqrt [3]{x}\right )}{d}+\frac {\int \frac {\left (a+b \log \left (c x^{n/3}\right )\right )^2}{e^2 x}d\left (d+e \sqrt [3]{x}\right )}{d}}{d}}{d}}{d}+\frac {-\frac {2}{5} b n \left (\frac {\frac {\frac {\int -\frac {a+b \log \left (c x^{n/3}\right )}{e^3 x}d\left (d+e \sqrt [3]{x}\right )}{d}+\frac {\int \frac {a+b \log \left (c x^{n/3}\right )}{e^2 x}d\left (d+e \sqrt [3]{x}\right )}{d}}{d}+\frac {-\frac {a+b \log \left (c x^{n/3}\right )}{3 e^3 x}-\frac {1}{3} b n \left (\frac {\log \left (d+e \sqrt [3]{x}\right )}{d^3}-\frac {\log \left (-e \sqrt [3]{x}\right )}{d^3}-\frac {1}{d^2 e \sqrt [3]{x}}+\frac {1}{2 d e^2 x^{2/3}}\right )}{d}}{d}+\frac {\frac {a+b \log \left (c x^{n/3}\right )}{4 e^4 x^{4/3}}-\frac {1}{4} b n \left (\frac {\log \left (d+e \sqrt [3]{x}\right )}{d^4}-\frac {\log \left (-e \sqrt [3]{x}\right )}{d^4}-\frac {1}{d^3 e \sqrt [3]{x}}+\frac {1}{2 d^2 e^2 x^{2/3}}-\frac {1}{3 d e^3 x}\right )}{d}\right )-\frac {\left (a+b \log \left (c x^{n/3}\right )\right )^2}{5 e^5 x^{5/3}}}{d}\right )-\frac {\left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{6 x^2}\right )\) |
| \(\Big \downarrow \) 2756 |
| \(\displaystyle 3 \left (\frac {1}{2} b e^6 n \left (\frac {-\frac {\left (a+b \log \left (c x^{n/3}\right )\right )^2}{5 e^5 x^{5/3}}-\frac {2}{5} b n \left (\frac {\frac {a+b \log \left (c x^{n/3}\right )}{4 e^4 x^{4/3}}-\frac {1}{4} b n \left (\frac {\log \left (d+e \sqrt [3]{x}\right )}{d^4}-\frac {\log \left (-e \sqrt [3]{x}\right )}{d^4}-\frac {1}{d^3 e \sqrt [3]{x}}+\frac {1}{2 d^2 e^2 x^{2/3}}-\frac {1}{3 d e^3 x}\right )}{d}+\frac {\frac {-\frac {1}{3} b n \left (\frac {\log \left (d+e \sqrt [3]{x}\right )}{d^3}-\frac {\log \left (-e \sqrt [3]{x}\right )}{d^3}-\frac {1}{d^2 e \sqrt [3]{x}}+\frac {1}{2 d e^2 x^{2/3}}\right )-\frac {a+b \log \left (c x^{n/3}\right )}{3 e^3 x}}{d}+\frac {\frac {\frac {a+b \log \left (c x^{n/3}\right )}{2 e^2 x^{2/3}}-\frac {1}{2} b n \int \frac {1}{e^2 x}d\left (d+e \sqrt [3]{x}\right )}{d}+\frac {\int \frac {a+b \log \left (c x^{n/3}\right )}{e^2 x}d\left (d+e \sqrt [3]{x}\right )}{d}}{d}}{d}\right )}{d}+\frac {\frac {\frac {\left (a+b \log \left (c x^{n/3}\right )\right )^2}{4 e^4 x^{4/3}}-\frac {1}{2} b n \left (\frac {-\frac {1}{3} b n \left (\frac {\log \left (d+e \sqrt [3]{x}\right )}{d^3}-\frac {\log \left (-e \sqrt [3]{x}\right )}{d^3}-\frac {1}{d^2 e \sqrt [3]{x}}+\frac {1}{2 d e^2 x^{2/3}}\right )-\frac {a+b \log \left (c x^{n/3}\right )}{3 e^3 x}}{d}+\frac {\frac {\frac {a+b \log \left (c x^{n/3}\right )}{2 e^2 x^{2/3}}-\frac {1}{2} b n \int \frac {1}{e^2 x}d\left (d+e \sqrt [3]{x}\right )}{d}+\frac {\int \frac {a+b \log \left (c x^{n/3}\right )}{e^2 x}d\left (d+e \sqrt [3]{x}\right )}{d}}{d}\right )}{d}+\frac {\frac {-\frac {\left (a+b \log \left (c x^{n/3}\right )\right )^2}{3 e^3 x}-\frac {2}{3} b n \left (\frac {\frac {a+b \log \left (c x^{n/3}\right )}{2 e^2 x^{2/3}}-\frac {1}{2} b n \int \frac {1}{e^2 x}d\left (d+e \sqrt [3]{x}\right )}{d}+\frac {\int \frac {a+b \log \left (c x^{n/3}\right )}{e^2 x}d\left (d+e \sqrt [3]{x}\right )}{d}\right )}{d}+\frac {\frac {\frac {\left (a+b \log \left (c x^{n/3}\right )\right )^2}{2 e^2 x^{2/3}}-b n \int \frac {a+b \log \left (c x^{n/3}\right )}{e^2 x}d\left (d+e \sqrt [3]{x}\right )}{d}+\frac {\int \frac {\left (a+b \log \left (c x^{n/3}\right )\right )^2}{e^2 x}d\left (d+e \sqrt [3]{x}\right )}{d}}{d}}{d}}{d}\right )-\frac {\left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{6 x^2}\right )\) |
| \(\Big \downarrow \) 54 |
| \(\displaystyle 3 \left (\frac {1}{2} b e^6 n \left (\frac {-\frac {\left (a+b \log \left (c x^{n/3}\right )\right )^2}{5 e^5 x^{5/3}}-\frac {2}{5} b n \left (\frac {\frac {a+b \log \left (c x^{n/3}\right )}{4 e^4 x^{4/3}}-\frac {1}{4} b n \left (\frac {\log \left (d+e \sqrt [3]{x}\right )}{d^4}-\frac {\log \left (-e \sqrt [3]{x}\right )}{d^4}-\frac {1}{d^3 e \sqrt [3]{x}}+\frac {1}{2 d^2 e^2 x^{2/3}}-\frac {1}{3 d e^3 x}\right )}{d}+\frac {\frac {-\frac {1}{3} b n \left (\frac {\log \left (d+e \sqrt [3]{x}\right )}{d^3}-\frac {\log \left (-e \sqrt [3]{x}\right )}{d^3}-\frac {1}{d^2 e \sqrt [3]{x}}+\frac {1}{2 d e^2 x^{2/3}}\right )-\frac {a+b \log \left (c x^{n/3}\right )}{3 e^3 x}}{d}+\frac {\frac {\frac {a+b \log \left (c x^{n/3}\right )}{2 e^2 x^{2/3}}-\frac {1}{2} b n \int \left (-\frac {1}{d^2 e \sqrt [3]{x}}+\frac {1}{d^2 \sqrt [3]{x}}+\frac {1}{d e^2 x^{2/3}}\right )d\left (d+e \sqrt [3]{x}\right )}{d}+\frac {\int \frac {a+b \log \left (c x^{n/3}\right )}{e^2 x}d\left (d+e \sqrt [3]{x}\right )}{d}}{d}}{d}\right )}{d}+\frac {\frac {\frac {\left (a+b \log \left (c x^{n/3}\right )\right )^2}{4 e^4 x^{4/3}}-\frac {1}{2} b n \left (\frac {-\frac {1}{3} b n \left (\frac {\log \left (d+e \sqrt [3]{x}\right )}{d^3}-\frac {\log \left (-e \sqrt [3]{x}\right )}{d^3}-\frac {1}{d^2 e \sqrt [3]{x}}+\frac {1}{2 d e^2 x^{2/3}}\right )-\frac {a+b \log \left (c x^{n/3}\right )}{3 e^3 x}}{d}+\frac {\frac {\frac {a+b \log \left (c x^{n/3}\right )}{2 e^2 x^{2/3}}-\frac {1}{2} b n \int \left (-\frac {1}{d^2 e \sqrt [3]{x}}+\frac {1}{d^2 \sqrt [3]{x}}+\frac {1}{d e^2 x^{2/3}}\right )d\left (d+e \sqrt [3]{x}\right )}{d}+\frac {\int \frac {a+b \log \left (c x^{n/3}\right )}{e^2 x}d\left (d+e \sqrt [3]{x}\right )}{d}}{d}\right )}{d}+\frac {\frac {-\frac {\left (a+b \log \left (c x^{n/3}\right )\right )^2}{3 e^3 x}-\frac {2}{3} b n \left (\frac {\frac {a+b \log \left (c x^{n/3}\right )}{2 e^2 x^{2/3}}-\frac {1}{2} b n \int \left (-\frac {1}{d^2 e \sqrt [3]{x}}+\frac {1}{d^2 \sqrt [3]{x}}+\frac {1}{d e^2 x^{2/3}}\right )d\left (d+e \sqrt [3]{x}\right )}{d}+\frac {\int \frac {a+b \log \left (c x^{n/3}\right )}{e^2 x}d\left (d+e \sqrt [3]{x}\right )}{d}\right )}{d}+\frac {\frac {\frac {\left (a+b \log \left (c x^{n/3}\right )\right )^2}{2 e^2 x^{2/3}}-b n \int \frac {a+b \log \left (c x^{n/3}\right )}{e^2 x}d\left (d+e \sqrt [3]{x}\right )}{d}+\frac {\int \frac {\left (a+b \log \left (c x^{n/3}\right )\right )^2}{e^2 x}d\left (d+e \sqrt [3]{x}\right )}{d}}{d}}{d}}{d}\right )-\frac {\left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{6 x^2}\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 3 \left (\frac {1}{2} b e^6 n \left (\frac {-\frac {\left (a+b \log \left (c x^{n/3}\right )\right )^2}{5 e^5 x^{5/3}}-\frac {2}{5} b n \left (\frac {\frac {a+b \log \left (c x^{n/3}\right )}{4 e^4 x^{4/3}}-\frac {1}{4} b n \left (\frac {\log \left (d+e \sqrt [3]{x}\right )}{d^4}-\frac {\log \left (-e \sqrt [3]{x}\right )}{d^4}-\frac {1}{d^3 e \sqrt [3]{x}}+\frac {1}{2 d^2 e^2 x^{2/3}}-\frac {1}{3 d e^3 x}\right )}{d}+\frac {\frac {-\frac {1}{3} b n \left (\frac {\log \left (d+e \sqrt [3]{x}\right )}{d^3}-\frac {\log \left (-e \sqrt [3]{x}\right )}{d^3}-\frac {1}{d^2 e \sqrt [3]{x}}+\frac {1}{2 d e^2 x^{2/3}}\right )-\frac {a+b \log \left (c x^{n/3}\right )}{3 e^3 x}}{d}+\frac {\frac {\frac {a+b \log \left (c x^{n/3}\right )}{2 e^2 x^{2/3}}-\frac {1}{2} b n \left (\frac {\log \left (d+e \sqrt [3]{x}\right )}{d^2}-\frac {\log \left (-e \sqrt [3]{x}\right )}{d^2}-\frac {1}{d e \sqrt [3]{x}}\right )}{d}+\frac {\int \frac {a+b \log \left (c x^{n/3}\right )}{e^2 x}d\left (d+e \sqrt [3]{x}\right )}{d}}{d}}{d}\right )}{d}+\frac {\frac {\frac {\left (a+b \log \left (c x^{n/3}\right )\right )^2}{4 e^4 x^{4/3}}-\frac {1}{2} b n \left (\frac {-\frac {1}{3} b n \left (\frac {\log \left (d+e \sqrt [3]{x}\right )}{d^3}-\frac {\log \left (-e \sqrt [3]{x}\right )}{d^3}-\frac {1}{d^2 e \sqrt [3]{x}}+\frac {1}{2 d e^2 x^{2/3}}\right )-\frac {a+b \log \left (c x^{n/3}\right )}{3 e^3 x}}{d}+\frac {\frac {\frac {a+b \log \left (c x^{n/3}\right )}{2 e^2 x^{2/3}}-\frac {1}{2} b n \left (\frac {\log \left (d+e \sqrt [3]{x}\right )}{d^2}-\frac {\log \left (-e \sqrt [3]{x}\right )}{d^2}-\frac {1}{d e \sqrt [3]{x}}\right )}{d}+\frac {\int \frac {a+b \log \left (c x^{n/3}\right )}{e^2 x}d\left (d+e \sqrt [3]{x}\right )}{d}}{d}\right )}{d}+\frac {\frac {-\frac {\left (a+b \log \left (c x^{n/3}\right )\right )^2}{3 e^3 x}-\frac {2}{3} b n \left (\frac {\frac {a+b \log \left (c x^{n/3}\right )}{2 e^2 x^{2/3}}-\frac {1}{2} b n \left (\frac {\log \left (d+e \sqrt [3]{x}\right )}{d^2}-\frac {\log \left (-e \sqrt [3]{x}\right )}{d^2}-\frac {1}{d e \sqrt [3]{x}}\right )}{d}+\frac {\int \frac {a+b \log \left (c x^{n/3}\right )}{e^2 x}d\left (d+e \sqrt [3]{x}\right )}{d}\right )}{d}+\frac {\frac {\frac {\left (a+b \log \left (c x^{n/3}\right )\right )^2}{2 e^2 x^{2/3}}-b n \int \frac {a+b \log \left (c x^{n/3}\right )}{e^2 x}d\left (d+e \sqrt [3]{x}\right )}{d}+\frac {\int \frac {\left (a+b \log \left (c x^{n/3}\right )\right )^2}{e^2 x}d\left (d+e \sqrt [3]{x}\right )}{d}}{d}}{d}}{d}\right )-\frac {\left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{6 x^2}\right )\) |
| \(\Big \downarrow \) 2789 |
| \(\displaystyle 3 \left (\frac {1}{2} b e^6 n \left (\frac {-\frac {\left (a+b \log \left (c x^{n/3}\right )\right )^2}{5 e^5 x^{5/3}}-\frac {2}{5} b n \left (\frac {\frac {a+b \log \left (c x^{n/3}\right )}{4 e^4 x^{4/3}}-\frac {1}{4} b n \left (\frac {\log \left (d+e \sqrt [3]{x}\right )}{d^4}-\frac {\log \left (-e \sqrt [3]{x}\right )}{d^4}-\frac {1}{d^3 e \sqrt [3]{x}}+\frac {1}{2 d^2 e^2 x^{2/3}}-\frac {1}{3 d e^3 x}\right )}{d}+\frac {\frac {-\frac {1}{3} b n \left (\frac {\log \left (d+e \sqrt [3]{x}\right )}{d^3}-\frac {\log \left (-e \sqrt [3]{x}\right )}{d^3}-\frac {1}{d^2 e \sqrt [3]{x}}+\frac {1}{2 d e^2 x^{2/3}}\right )-\frac {a+b \log \left (c x^{n/3}\right )}{3 e^3 x}}{d}+\frac {\frac {\frac {a+b \log \left (c x^{n/3}\right )}{2 e^2 x^{2/3}}-\frac {1}{2} b n \left (\frac {\log \left (d+e \sqrt [3]{x}\right )}{d^2}-\frac {\log \left (-e \sqrt [3]{x}\right )}{d^2}-\frac {1}{d e \sqrt [3]{x}}\right )}{d}+\frac {\frac {\int \frac {a+b \log \left (c x^{n/3}\right )}{e^2 x^{2/3}}d\left (d+e \sqrt [3]{x}\right )}{d}+\frac {\int -\frac {a+b \log \left (c x^{n/3}\right )}{e x^{2/3}}d\left (d+e \sqrt [3]{x}\right )}{d}}{d}}{d}}{d}\right )}{d}+\frac {\frac {\frac {\left (a+b \log \left (c x^{n/3}\right )\right )^2}{4 e^4 x^{4/3}}-\frac {1}{2} b n \left (\frac {-\frac {1}{3} b n \left (\frac {\log \left (d+e \sqrt [3]{x}\right )}{d^3}-\frac {\log \left (-e \sqrt [3]{x}\right )}{d^3}-\frac {1}{d^2 e \sqrt [3]{x}}+\frac {1}{2 d e^2 x^{2/3}}\right )-\frac {a+b \log \left (c x^{n/3}\right )}{3 e^3 x}}{d}+\frac {\frac {\frac {a+b \log \left (c x^{n/3}\right )}{2 e^2 x^{2/3}}-\frac {1}{2} b n \left (\frac {\log \left (d+e \sqrt [3]{x}\right )}{d^2}-\frac {\log \left (-e \sqrt [3]{x}\right )}{d^2}-\frac {1}{d e \sqrt [3]{x}}\right )}{d}+\frac {\frac {\int \frac {a+b \log \left (c x^{n/3}\right )}{e^2 x^{2/3}}d\left (d+e \sqrt [3]{x}\right )}{d}+\frac {\int -\frac {a+b \log \left (c x^{n/3}\right )}{e x^{2/3}}d\left (d+e \sqrt [3]{x}\right )}{d}}{d}}{d}\right )}{d}+\frac {\frac {-\frac {\left (a+b \log \left (c x^{n/3}\right )\right )^2}{3 e^3 x}-\frac {2}{3} b n \left (\frac {\frac {a+b \log \left (c x^{n/3}\right )}{2 e^2 x^{2/3}}-\frac {1}{2} b n \left (\frac {\log \left (d+e \sqrt [3]{x}\right )}{d^2}-\frac {\log \left (-e \sqrt [3]{x}\right )}{d^2}-\frac {1}{d e \sqrt [3]{x}}\right )}{d}+\frac {\frac {\int \frac {a+b \log \left (c x^{n/3}\right )}{e^2 x^{2/3}}d\left (d+e \sqrt [3]{x}\right )}{d}+\frac {\int -\frac {a+b \log \left (c x^{n/3}\right )}{e x^{2/3}}d\left (d+e \sqrt [3]{x}\right )}{d}}{d}\right )}{d}+\frac {\frac {\frac {\left (a+b \log \left (c x^{n/3}\right )\right )^2}{2 e^2 x^{2/3}}-b n \left (\frac {\int \frac {a+b \log \left (c x^{n/3}\right )}{e^2 x^{2/3}}d\left (d+e \sqrt [3]{x}\right )}{d}+\frac {\int -\frac {a+b \log \left (c x^{n/3}\right )}{e x^{2/3}}d\left (d+e \sqrt [3]{x}\right )}{d}\right )}{d}+\frac {\frac {\int \frac {\left (a+b \log \left (c x^{n/3}\right )\right )^2}{e^2 x^{2/3}}d\left (d+e \sqrt [3]{x}\right )}{d}+\frac {\int -\frac {\left (a+b \log \left (c x^{n/3}\right )\right )^2}{e x^{2/3}}d\left (d+e \sqrt [3]{x}\right )}{d}}{d}}{d}}{d}}{d}\right )-\frac {\left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{6 x^2}\right )\) |
| \(\Big \downarrow \) 2751 |
| \(\displaystyle 3 \left (\frac {1}{2} b e^6 n \left (\frac {-\frac {\left (a+b \log \left (c x^{n/3}\right )\right )^2}{5 e^5 x^{5/3}}-\frac {2}{5} b n \left (\frac {\frac {a+b \log \left (c x^{n/3}\right )}{4 e^4 x^{4/3}}-\frac {1}{4} b n \left (\frac {\log \left (d+e \sqrt [3]{x}\right )}{d^4}-\frac {\log \left (-e \sqrt [3]{x}\right )}{d^4}-\frac {1}{d^3 e \sqrt [3]{x}}+\frac {1}{2 d^2 e^2 x^{2/3}}-\frac {1}{3 d e^3 x}\right )}{d}+\frac {\frac {-\frac {1}{3} b n \left (\frac {\log \left (d+e \sqrt [3]{x}\right )}{d^3}-\frac {\log \left (-e \sqrt [3]{x}\right )}{d^3}-\frac {1}{d^2 e \sqrt [3]{x}}+\frac {1}{2 d e^2 x^{2/3}}\right )-\frac {a+b \log \left (c x^{n/3}\right )}{3 e^3 x}}{d}+\frac {\frac {\frac {a+b \log \left (c x^{n/3}\right )}{2 e^2 x^{2/3}}-\frac {1}{2} b n \left (\frac {\log \left (d+e \sqrt [3]{x}\right )}{d^2}-\frac {\log \left (-e \sqrt [3]{x}\right )}{d^2}-\frac {1}{d e \sqrt [3]{x}}\right )}{d}+\frac {\frac {-\frac {\left (d+e \sqrt [3]{x}\right ) \left (a+b \log \left (c x^{n/3}\right )\right )}{d e \sqrt [3]{x}}-\frac {b n \int -\frac {1}{e \sqrt [3]{x}}d\left (d+e \sqrt [3]{x}\right )}{d}}{d}+\frac {\int -\frac {a+b \log \left (c x^{n/3}\right )}{e x^{2/3}}d\left (d+e \sqrt [3]{x}\right )}{d}}{d}}{d}}{d}\right )}{d}+\frac {\frac {\frac {\left (a+b \log \left (c x^{n/3}\right )\right )^2}{4 e^4 x^{4/3}}-\frac {1}{2} b n \left (\frac {-\frac {1}{3} b n \left (\frac {\log \left (d+e \sqrt [3]{x}\right )}{d^3}-\frac {\log \left (-e \sqrt [3]{x}\right )}{d^3}-\frac {1}{d^2 e \sqrt [3]{x}}+\frac {1}{2 d e^2 x^{2/3}}\right )-\frac {a+b \log \left (c x^{n/3}\right )}{3 e^3 x}}{d}+\frac {\frac {\frac {a+b \log \left (c x^{n/3}\right )}{2 e^2 x^{2/3}}-\frac {1}{2} b n \left (\frac {\log \left (d+e \sqrt [3]{x}\right )}{d^2}-\frac {\log \left (-e \sqrt [3]{x}\right )}{d^2}-\frac {1}{d e \sqrt [3]{x}}\right )}{d}+\frac {\frac {-\frac {\left (d+e \sqrt [3]{x}\right ) \left (a+b \log \left (c x^{n/3}\right )\right )}{d e \sqrt [3]{x}}-\frac {b n \int -\frac {1}{e \sqrt [3]{x}}d\left (d+e \sqrt [3]{x}\right )}{d}}{d}+\frac {\int -\frac {a+b \log \left (c x^{n/3}\right )}{e x^{2/3}}d\left (d+e \sqrt [3]{x}\right )}{d}}{d}}{d}\right )}{d}+\frac {\frac {-\frac {\left (a+b \log \left (c x^{n/3}\right )\right )^2}{3 e^3 x}-\frac {2}{3} b n \left (\frac {\frac {a+b \log \left (c x^{n/3}\right )}{2 e^2 x^{2/3}}-\frac {1}{2} b n \left (\frac {\log \left (d+e \sqrt [3]{x}\right )}{d^2}-\frac {\log \left (-e \sqrt [3]{x}\right )}{d^2}-\frac {1}{d e \sqrt [3]{x}}\right )}{d}+\frac {\frac {-\frac {\left (d+e \sqrt [3]{x}\right ) \left (a+b \log \left (c x^{n/3}\right )\right )}{d e \sqrt [3]{x}}-\frac {b n \int -\frac {1}{e \sqrt [3]{x}}d\left (d+e \sqrt [3]{x}\right )}{d}}{d}+\frac {\int -\frac {a+b \log \left (c x^{n/3}\right )}{e x^{2/3}}d\left (d+e \sqrt [3]{x}\right )}{d}}{d}\right )}{d}+\frac {\frac {\frac {\left (a+b \log \left (c x^{n/3}\right )\right )^2}{2 e^2 x^{2/3}}-b n \left (\frac {-\frac {\left (d+e \sqrt [3]{x}\right ) \left (a+b \log \left (c x^{n/3}\right )\right )}{d e \sqrt [3]{x}}-\frac {b n \int -\frac {1}{e \sqrt [3]{x}}d\left (d+e \sqrt [3]{x}\right )}{d}}{d}+\frac {\int -\frac {a+b \log \left (c x^{n/3}\right )}{e x^{2/3}}d\left (d+e \sqrt [3]{x}\right )}{d}\right )}{d}+\frac {\frac {\int \frac {\left (a+b \log \left (c x^{n/3}\right )\right )^2}{e^2 x^{2/3}}d\left (d+e \sqrt [3]{x}\right )}{d}+\frac {\int -\frac {\left (a+b \log \left (c x^{n/3}\right )\right )^2}{e x^{2/3}}d\left (d+e \sqrt [3]{x}\right )}{d}}{d}}{d}}{d}}{d}\right )-\frac {\left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{6 x^2}\right )\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle 3 \left (\frac {1}{2} b e^6 n \left (\frac {-\frac {\left (a+b \log \left (c x^{n/3}\right )\right )^2}{5 e^5 x^{5/3}}-\frac {2}{5} b n \left (\frac {\frac {a+b \log \left (c x^{n/3}\right )}{4 e^4 x^{4/3}}-\frac {1}{4} b n \left (\frac {\log \left (d+e \sqrt [3]{x}\right )}{d^4}-\frac {\log \left (-e \sqrt [3]{x}\right )}{d^4}-\frac {1}{d^3 e \sqrt [3]{x}}+\frac {1}{2 d^2 e^2 x^{2/3}}-\frac {1}{3 d e^3 x}\right )}{d}+\frac {\frac {-\frac {1}{3} b n \left (\frac {\log \left (d+e \sqrt [3]{x}\right )}{d^3}-\frac {\log \left (-e \sqrt [3]{x}\right )}{d^3}-\frac {1}{d^2 e \sqrt [3]{x}}+\frac {1}{2 d e^2 x^{2/3}}\right )-\frac {a+b \log \left (c x^{n/3}\right )}{3 e^3 x}}{d}+\frac {\frac {\frac {a+b \log \left (c x^{n/3}\right )}{2 e^2 x^{2/3}}-\frac {1}{2} b n \left (\frac {\log \left (d+e \sqrt [3]{x}\right )}{d^2}-\frac {\log \left (-e \sqrt [3]{x}\right )}{d^2}-\frac {1}{d e \sqrt [3]{x}}\right )}{d}+\frac {\frac {\frac {b n \log \left (-e \sqrt [3]{x}\right )}{d}-\frac {\left (d+e \sqrt [3]{x}\right ) \left (a+b \log \left (c x^{n/3}\right )\right )}{d e \sqrt [3]{x}}}{d}+\frac {\int -\frac {a+b \log \left (c x^{n/3}\right )}{e x^{2/3}}d\left (d+e \sqrt [3]{x}\right )}{d}}{d}}{d}}{d}\right )}{d}+\frac {\frac {\frac {\left (a+b \log \left (c x^{n/3}\right )\right )^2}{4 e^4 x^{4/3}}-\frac {1}{2} b n \left (\frac {-\frac {1}{3} b n \left (\frac {\log \left (d+e \sqrt [3]{x}\right )}{d^3}-\frac {\log \left (-e \sqrt [3]{x}\right )}{d^3}-\frac {1}{d^2 e \sqrt [3]{x}}+\frac {1}{2 d e^2 x^{2/3}}\right )-\frac {a+b \log \left (c x^{n/3}\right )}{3 e^3 x}}{d}+\frac {\frac {\frac {a+b \log \left (c x^{n/3}\right )}{2 e^2 x^{2/3}}-\frac {1}{2} b n \left (\frac {\log \left (d+e \sqrt [3]{x}\right )}{d^2}-\frac {\log \left (-e \sqrt [3]{x}\right )}{d^2}-\frac {1}{d e \sqrt [3]{x}}\right )}{d}+\frac {\frac {\frac {b n \log \left (-e \sqrt [3]{x}\right )}{d}-\frac {\left (d+e \sqrt [3]{x}\right ) \left (a+b \log \left (c x^{n/3}\right )\right )}{d e \sqrt [3]{x}}}{d}+\frac {\int -\frac {a+b \log \left (c x^{n/3}\right )}{e x^{2/3}}d\left (d+e \sqrt [3]{x}\right )}{d}}{d}}{d}\right )}{d}+\frac {\frac {-\frac {\left (a+b \log \left (c x^{n/3}\right )\right )^2}{3 e^3 x}-\frac {2}{3} b n \left (\frac {\frac {a+b \log \left (c x^{n/3}\right )}{2 e^2 x^{2/3}}-\frac {1}{2} b n \left (\frac {\log \left (d+e \sqrt [3]{x}\right )}{d^2}-\frac {\log \left (-e \sqrt [3]{x}\right )}{d^2}-\frac {1}{d e \sqrt [3]{x}}\right )}{d}+\frac {\frac {\frac {b n \log \left (-e \sqrt [3]{x}\right )}{d}-\frac {\left (d+e \sqrt [3]{x}\right ) \left (a+b \log \left (c x^{n/3}\right )\right )}{d e \sqrt [3]{x}}}{d}+\frac {\int -\frac {a+b \log \left (c x^{n/3}\right )}{e x^{2/3}}d\left (d+e \sqrt [3]{x}\right )}{d}}{d}\right )}{d}+\frac {\frac {\frac {\left (a+b \log \left (c x^{n/3}\right )\right )^2}{2 e^2 x^{2/3}}-b n \left (\frac {\frac {b n \log \left (-e \sqrt [3]{x}\right )}{d}-\frac {\left (d+e \sqrt [3]{x}\right ) \left (a+b \log \left (c x^{n/3}\right )\right )}{d e \sqrt [3]{x}}}{d}+\frac {\int -\frac {a+b \log \left (c x^{n/3}\right )}{e x^{2/3}}d\left (d+e \sqrt [3]{x}\right )}{d}\right )}{d}+\frac {\frac {\int \frac {\left (a+b \log \left (c x^{n/3}\right )\right )^2}{e^2 x^{2/3}}d\left (d+e \sqrt [3]{x}\right )}{d}+\frac {\int -\frac {\left (a+b \log \left (c x^{n/3}\right )\right )^2}{e x^{2/3}}d\left (d+e \sqrt [3]{x}\right )}{d}}{d}}{d}}{d}}{d}\right )-\frac {\left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{6 x^2}\right )\) |
| \(\Big \downarrow \) 2755 |
| \(\displaystyle 3 \left (\frac {1}{2} b e^6 n \left (\frac {-\frac {\left (a+b \log \left (c x^{n/3}\right )\right )^2}{5 e^5 x^{5/3}}-\frac {2}{5} b n \left (\frac {\frac {a+b \log \left (c x^{n/3}\right )}{4 e^4 x^{4/3}}-\frac {1}{4} b n \left (\frac {\log \left (d+e \sqrt [3]{x}\right )}{d^4}-\frac {\log \left (-e \sqrt [3]{x}\right )}{d^4}-\frac {1}{d^3 e \sqrt [3]{x}}+\frac {1}{2 d^2 e^2 x^{2/3}}-\frac {1}{3 d e^3 x}\right )}{d}+\frac {\frac {-\frac {1}{3} b n \left (\frac {\log \left (d+e \sqrt [3]{x}\right )}{d^3}-\frac {\log \left (-e \sqrt [3]{x}\right )}{d^3}-\frac {1}{d^2 e \sqrt [3]{x}}+\frac {1}{2 d e^2 x^{2/3}}\right )-\frac {a+b \log \left (c x^{n/3}\right )}{3 e^3 x}}{d}+\frac {\frac {\frac {a+b \log \left (c x^{n/3}\right )}{2 e^2 x^{2/3}}-\frac {1}{2} b n \left (\frac {\log \left (d+e \sqrt [3]{x}\right )}{d^2}-\frac {\log \left (-e \sqrt [3]{x}\right )}{d^2}-\frac {1}{d e \sqrt [3]{x}}\right )}{d}+\frac {\frac {\frac {b n \log \left (-e \sqrt [3]{x}\right )}{d}-\frac {\left (d+e \sqrt [3]{x}\right ) \left (a+b \log \left (c x^{n/3}\right )\right )}{d e \sqrt [3]{x}}}{d}+\frac {\int -\frac {a+b \log \left (c x^{n/3}\right )}{e x^{2/3}}d\left (d+e \sqrt [3]{x}\right )}{d}}{d}}{d}}{d}\right )}{d}+\frac {\frac {\frac {\left (a+b \log \left (c x^{n/3}\right )\right )^2}{4 e^4 x^{4/3}}-\frac {1}{2} b n \left (\frac {-\frac {1}{3} b n \left (\frac {\log \left (d+e \sqrt [3]{x}\right )}{d^3}-\frac {\log \left (-e \sqrt [3]{x}\right )}{d^3}-\frac {1}{d^2 e \sqrt [3]{x}}+\frac {1}{2 d e^2 x^{2/3}}\right )-\frac {a+b \log \left (c x^{n/3}\right )}{3 e^3 x}}{d}+\frac {\frac {\frac {a+b \log \left (c x^{n/3}\right )}{2 e^2 x^{2/3}}-\frac {1}{2} b n \left (\frac {\log \left (d+e \sqrt [3]{x}\right )}{d^2}-\frac {\log \left (-e \sqrt [3]{x}\right )}{d^2}-\frac {1}{d e \sqrt [3]{x}}\right )}{d}+\frac {\frac {\frac {b n \log \left (-e \sqrt [3]{x}\right )}{d}-\frac {\left (d+e \sqrt [3]{x}\right ) \left (a+b \log \left (c x^{n/3}\right )\right )}{d e \sqrt [3]{x}}}{d}+\frac {\int -\frac {a+b \log \left (c x^{n/3}\right )}{e x^{2/3}}d\left (d+e \sqrt [3]{x}\right )}{d}}{d}}{d}\right )}{d}+\frac {\frac {-\frac {\left (a+b \log \left (c x^{n/3}\right )\right )^2}{3 e^3 x}-\frac {2}{3} b n \left (\frac {\frac {a+b \log \left (c x^{n/3}\right )}{2 e^2 x^{2/3}}-\frac {1}{2} b n \left (\frac {\log \left (d+e \sqrt [3]{x}\right )}{d^2}-\frac {\log \left (-e \sqrt [3]{x}\right )}{d^2}-\frac {1}{d e \sqrt [3]{x}}\right )}{d}+\frac {\frac {\frac {b n \log \left (-e \sqrt [3]{x}\right )}{d}-\frac {\left (d+e \sqrt [3]{x}\right ) \left (a+b \log \left (c x^{n/3}\right )\right )}{d e \sqrt [3]{x}}}{d}+\frac {\int -\frac {a+b \log \left (c x^{n/3}\right )}{e x^{2/3}}d\left (d+e \sqrt [3]{x}\right )}{d}}{d}\right )}{d}+\frac {\frac {\frac {\left (a+b \log \left (c x^{n/3}\right )\right )^2}{2 e^2 x^{2/3}}-b n \left (\frac {\frac {b n \log \left (-e \sqrt [3]{x}\right )}{d}-\frac {\left (d+e \sqrt [3]{x}\right ) \left (a+b \log \left (c x^{n/3}\right )\right )}{d e \sqrt [3]{x}}}{d}+\frac {\int -\frac {a+b \log \left (c x^{n/3}\right )}{e x^{2/3}}d\left (d+e \sqrt [3]{x}\right )}{d}\right )}{d}+\frac {\frac {-\frac {\left (d+e \sqrt [3]{x}\right ) \left (a+b \log \left (c x^{n/3}\right )\right )^2}{d e \sqrt [3]{x}}-\frac {2 b n \int -\frac {a+b \log \left (c x^{n/3}\right )}{e \sqrt [3]{x}}d\left (d+e \sqrt [3]{x}\right )}{d}}{d}+\frac {\int -\frac {\left (a+b \log \left (c x^{n/3}\right )\right )^2}{e x^{2/3}}d\left (d+e \sqrt [3]{x}\right )}{d}}{d}}{d}}{d}}{d}\right )-\frac {\left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{6 x^2}\right )\) |
| \(\Big \downarrow \) 2754 |
| \(\displaystyle 3 \left (\frac {1}{2} b e^6 n \left (\frac {-\frac {\left (a+b \log \left (c x^{n/3}\right )\right )^2}{5 e^5 x^{5/3}}-\frac {2}{5} b n \left (\frac {\frac {a+b \log \left (c x^{n/3}\right )}{4 e^4 x^{4/3}}-\frac {1}{4} b n \left (\frac {\log \left (d+e \sqrt [3]{x}\right )}{d^4}-\frac {\log \left (-e \sqrt [3]{x}\right )}{d^4}-\frac {1}{d^3 e \sqrt [3]{x}}+\frac {1}{2 d^2 e^2 x^{2/3}}-\frac {1}{3 d e^3 x}\right )}{d}+\frac {\frac {-\frac {1}{3} b n \left (\frac {\log \left (d+e \sqrt [3]{x}\right )}{d^3}-\frac {\log \left (-e \sqrt [3]{x}\right )}{d^3}-\frac {1}{d^2 e \sqrt [3]{x}}+\frac {1}{2 d e^2 x^{2/3}}\right )-\frac {a+b \log \left (c x^{n/3}\right )}{3 e^3 x}}{d}+\frac {\frac {\frac {a+b \log \left (c x^{n/3}\right )}{2 e^2 x^{2/3}}-\frac {1}{2} b n \left (\frac {\log \left (d+e \sqrt [3]{x}\right )}{d^2}-\frac {\log \left (-e \sqrt [3]{x}\right )}{d^2}-\frac {1}{d e \sqrt [3]{x}}\right )}{d}+\frac {\frac {\frac {b n \log \left (-e \sqrt [3]{x}\right )}{d}-\frac {\left (d+e \sqrt [3]{x}\right ) \left (a+b \log \left (c x^{n/3}\right )\right )}{d e \sqrt [3]{x}}}{d}+\frac {\int -\frac {a+b \log \left (c x^{n/3}\right )}{e x^{2/3}}d\left (d+e \sqrt [3]{x}\right )}{d}}{d}}{d}}{d}\right )}{d}+\frac {\frac {\frac {\left (a+b \log \left (c x^{n/3}\right )\right )^2}{4 e^4 x^{4/3}}-\frac {1}{2} b n \left (\frac {-\frac {1}{3} b n \left (\frac {\log \left (d+e \sqrt [3]{x}\right )}{d^3}-\frac {\log \left (-e \sqrt [3]{x}\right )}{d^3}-\frac {1}{d^2 e \sqrt [3]{x}}+\frac {1}{2 d e^2 x^{2/3}}\right )-\frac {a+b \log \left (c x^{n/3}\right )}{3 e^3 x}}{d}+\frac {\frac {\frac {a+b \log \left (c x^{n/3}\right )}{2 e^2 x^{2/3}}-\frac {1}{2} b n \left (\frac {\log \left (d+e \sqrt [3]{x}\right )}{d^2}-\frac {\log \left (-e \sqrt [3]{x}\right )}{d^2}-\frac {1}{d e \sqrt [3]{x}}\right )}{d}+\frac {\frac {\frac {b n \log \left (-e \sqrt [3]{x}\right )}{d}-\frac {\left (d+e \sqrt [3]{x}\right ) \left (a+b \log \left (c x^{n/3}\right )\right )}{d e \sqrt [3]{x}}}{d}+\frac {\int -\frac {a+b \log \left (c x^{n/3}\right )}{e x^{2/3}}d\left (d+e \sqrt [3]{x}\right )}{d}}{d}}{d}\right )}{d}+\frac {\frac {-\frac {\left (a+b \log \left (c x^{n/3}\right )\right )^2}{3 e^3 x}-\frac {2}{3} b n \left (\frac {\frac {a+b \log \left (c x^{n/3}\right )}{2 e^2 x^{2/3}}-\frac {1}{2} b n \left (\frac {\log \left (d+e \sqrt [3]{x}\right )}{d^2}-\frac {\log \left (-e \sqrt [3]{x}\right )}{d^2}-\frac {1}{d e \sqrt [3]{x}}\right )}{d}+\frac {\frac {\frac {b n \log \left (-e \sqrt [3]{x}\right )}{d}-\frac {\left (d+e \sqrt [3]{x}\right ) \left (a+b \log \left (c x^{n/3}\right )\right )}{d e \sqrt [3]{x}}}{d}+\frac {\int -\frac {a+b \log \left (c x^{n/3}\right )}{e x^{2/3}}d\left (d+e \sqrt [3]{x}\right )}{d}}{d}\right )}{d}+\frac {\frac {\frac {\left (a+b \log \left (c x^{n/3}\right )\right )^2}{2 e^2 x^{2/3}}-b n \left (\frac {\frac {b n \log \left (-e \sqrt [3]{x}\right )}{d}-\frac {\left (d+e \sqrt [3]{x}\right ) \left (a+b \log \left (c x^{n/3}\right )\right )}{d e \sqrt [3]{x}}}{d}+\frac {\int -\frac {a+b \log \left (c x^{n/3}\right )}{e x^{2/3}}d\left (d+e \sqrt [3]{x}\right )}{d}\right )}{d}+\frac {\frac {-\frac {\left (d+e \sqrt [3]{x}\right ) \left (a+b \log \left (c x^{n/3}\right )\right )^2}{d e \sqrt [3]{x}}-\frac {2 b n \left (b n \int \frac {\log \left (1-\frac {d+e \sqrt [3]{x}}{d}\right )}{\sqrt [3]{x}}d\left (d+e \sqrt [3]{x}\right )-\log \left (1-\frac {d+e \sqrt [3]{x}}{d}\right ) \left (a+b \log \left (c x^{n/3}\right )\right )\right )}{d}}{d}+\frac {\int -\frac {\left (a+b \log \left (c x^{n/3}\right )\right )^2}{e x^{2/3}}d\left (d+e \sqrt [3]{x}\right )}{d}}{d}}{d}}{d}}{d}\right )-\frac {\left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{6 x^2}\right )\) |
| \(\Big \downarrow \) 2779 |
| \(\displaystyle 3 \left (\frac {1}{2} b e^6 n \left (\frac {-\frac {\left (a+b \log \left (c x^{n/3}\right )\right )^2}{5 e^5 x^{5/3}}-\frac {2}{5} b n \left (\frac {\frac {a+b \log \left (c x^{n/3}\right )}{4 e^4 x^{4/3}}-\frac {1}{4} b n \left (\frac {\log \left (d+e \sqrt [3]{x}\right )}{d^4}-\frac {\log \left (-e \sqrt [3]{x}\right )}{d^4}-\frac {1}{d^3 e \sqrt [3]{x}}+\frac {1}{2 d^2 e^2 x^{2/3}}-\frac {1}{3 d e^3 x}\right )}{d}+\frac {\frac {-\frac {1}{3} b n \left (\frac {\log \left (d+e \sqrt [3]{x}\right )}{d^3}-\frac {\log \left (-e \sqrt [3]{x}\right )}{d^3}-\frac {1}{d^2 e \sqrt [3]{x}}+\frac {1}{2 d e^2 x^{2/3}}\right )-\frac {a+b \log \left (c x^{n/3}\right )}{3 e^3 x}}{d}+\frac {\frac {\frac {a+b \log \left (c x^{n/3}\right )}{2 e^2 x^{2/3}}-\frac {1}{2} b n \left (\frac {\log \left (d+e \sqrt [3]{x}\right )}{d^2}-\frac {\log \left (-e \sqrt [3]{x}\right )}{d^2}-\frac {1}{d e \sqrt [3]{x}}\right )}{d}+\frac {\frac {\frac {b n \log \left (-e \sqrt [3]{x}\right )}{d}-\frac {\left (d+e \sqrt [3]{x}\right ) \left (a+b \log \left (c x^{n/3}\right )\right )}{d e \sqrt [3]{x}}}{d}+\frac {\frac {b n \int \frac {\log \left (1-\frac {d}{\sqrt [3]{x}}\right )}{\sqrt [3]{x}}d\left (d+e \sqrt [3]{x}\right )}{d}-\frac {\log \left (1-\frac {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 {\left (a+b \log \left (c x^{n/3}\right )\right )^2}{4 e^4 x^{4/3}}-\frac {1}{2} b n \left (\frac {-\frac {1}{3} b n \left (\frac {\log \left (d+e \sqrt [3]{x}\right )}{d^3}-\frac {\log \left (-e \sqrt [3]{x}\right )}{d^3}-\frac {1}{d^2 e \sqrt [3]{x}}+\frac {1}{2 d e^2 x^{2/3}}\right )-\frac {a+b \log \left (c x^{n/3}\right )}{3 e^3 x}}{d}+\frac {\frac {\frac {a+b \log \left (c x^{n/3}\right )}{2 e^2 x^{2/3}}-\frac {1}{2} b n \left (\frac {\log \left (d+e \sqrt [3]{x}\right )}{d^2}-\frac {\log \left (-e \sqrt [3]{x}\right )}{d^2}-\frac {1}{d e \sqrt [3]{x}}\right )}{d}+\frac {\frac {\frac {b n \log \left (-e \sqrt [3]{x}\right )}{d}-\frac {\left (d+e \sqrt [3]{x}\right ) \left (a+b \log \left (c x^{n/3}\right )\right )}{d e \sqrt [3]{x}}}{d}+\frac {\frac {b n \int \frac {\log \left (1-\frac {d}{\sqrt [3]{x}}\right )}{\sqrt [3]{x}}d\left (d+e \sqrt [3]{x}\right )}{d}-\frac {\log \left (1-\frac {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 {\left (a+b \log \left (c x^{n/3}\right )\right )^2}{3 e^3 x}-\frac {2}{3} b n \left (\frac {\frac {a+b \log \left (c x^{n/3}\right )}{2 e^2 x^{2/3}}-\frac {1}{2} b n \left (\frac {\log \left (d+e \sqrt [3]{x}\right )}{d^2}-\frac {\log \left (-e \sqrt [3]{x}\right )}{d^2}-\frac {1}{d e \sqrt [3]{x}}\right )}{d}+\frac {\frac {\frac {b n \log \left (-e \sqrt [3]{x}\right )}{d}-\frac {\left (d+e \sqrt [3]{x}\right ) \left (a+b \log \left (c x^{n/3}\right )\right )}{d e \sqrt [3]{x}}}{d}+\frac {\frac {b n \int \frac {\log \left (1-\frac {d}{\sqrt [3]{x}}\right )}{\sqrt [3]{x}}d\left (d+e \sqrt [3]{x}\right )}{d}-\frac {\log \left (1-\frac {d}{\sqrt [3]{x}}\right ) \left (a+b \log \left (c x^{n/3}\right )\right )}{d}}{d}}{d}\right )}{d}+\frac {\frac {\frac {\left (a+b \log \left (c x^{n/3}\right )\right )^2}{2 e^2 x^{2/3}}-b n \left (\frac {\frac {b n \log \left (-e \sqrt [3]{x}\right )}{d}-\frac {\left (d+e \sqrt [3]{x}\right ) \left (a+b \log \left (c x^{n/3}\right )\right )}{d e \sqrt [3]{x}}}{d}+\frac {\frac {b n \int \frac {\log \left (1-\frac {d}{\sqrt [3]{x}}\right )}{\sqrt [3]{x}}d\left (d+e \sqrt [3]{x}\right )}{d}-\frac {\log \left (1-\frac {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+e \sqrt [3]{x}\right ) \left (a+b \log \left (c x^{n/3}\right )\right )^2}{d e \sqrt [3]{x}}-\frac {2 b n \left (b n \int \frac {\log \left (1-\frac {d+e \sqrt [3]{x}}{d}\right )}{\sqrt [3]{x}}d\left (d+e \sqrt [3]{x}\right )-\log \left (1-\frac {d+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 \frac {\log \left (1-\frac {d}{\sqrt [3]{x}}\right ) \left (a+b \log \left (c x^{n/3}\right )\right )}{\sqrt [3]{x}}d\left (d+e \sqrt [3]{x}\right )}{d}-\frac {\log \left (1-\frac {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 {\left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{6 x^2}\right )\) |
| \(\Big \downarrow \) 2821 |
| \(\displaystyle 3 \left (\frac {1}{2} b e^6 n \left (\frac {-\frac {\left (a+b \log \left (c x^{n/3}\right )\right )^2}{5 e^5 x^{5/3}}-\frac {2}{5} b n \left (\frac {\frac {a+b \log \left (c x^{n/3}\right )}{4 e^4 x^{4/3}}-\frac {1}{4} b n \left (\frac {\log \left (d+e \sqrt [3]{x}\right )}{d^4}-\frac {\log \left (-e \sqrt [3]{x}\right )}{d^4}-\frac {1}{d^3 e \sqrt [3]{x}}+\frac {1}{2 d^2 e^2 x^{2/3}}-\frac {1}{3 d e^3 x}\right )}{d}+\frac {\frac {-\frac {1}{3} b n \left (\frac {\log \left (d+e \sqrt [3]{x}\right )}{d^3}-\frac {\log \left (-e \sqrt [3]{x}\right )}{d^3}-\frac {1}{d^2 e \sqrt [3]{x}}+\frac {1}{2 d e^2 x^{2/3}}\right )-\frac {a+b \log \left (c x^{n/3}\right )}{3 e^3 x}}{d}+\frac {\frac {\frac {a+b \log \left (c x^{n/3}\right )}{2 e^2 x^{2/3}}-\frac {1}{2} b n \left (\frac {\log \left (d+e \sqrt [3]{x}\right )}{d^2}-\frac {\log \left (-e \sqrt [3]{x}\right )}{d^2}-\frac {1}{d e \sqrt [3]{x}}\right )}{d}+\frac {\frac {\frac {b n \log \left (-e \sqrt [3]{x}\right )}{d}-\frac {\left (d+e \sqrt [3]{x}\right ) \left (a+b \log \left (c x^{n/3}\right )\right )}{d e \sqrt [3]{x}}}{d}+\frac {\frac {b n \int \frac {\log \left (1-\frac {d}{\sqrt [3]{x}}\right )}{\sqrt [3]{x}}d\left (d+e \sqrt [3]{x}\right )}{d}-\frac {\log \left (1-\frac {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 {\left (a+b \log \left (c x^{n/3}\right )\right )^2}{4 e^4 x^{4/3}}-\frac {1}{2} b n \left (\frac {-\frac {1}{3} b n \left (\frac {\log \left (d+e \sqrt [3]{x}\right )}{d^3}-\frac {\log \left (-e \sqrt [3]{x}\right )}{d^3}-\frac {1}{d^2 e \sqrt [3]{x}}+\frac {1}{2 d e^2 x^{2/3}}\right )-\frac {a+b \log \left (c x^{n/3}\right )}{3 e^3 x}}{d}+\frac {\frac {\frac {a+b \log \left (c x^{n/3}\right )}{2 e^2 x^{2/3}}-\frac {1}{2} b n \left (\frac {\log \left (d+e \sqrt [3]{x}\right )}{d^2}-\frac {\log \left (-e \sqrt [3]{x}\right )}{d^2}-\frac {1}{d e \sqrt [3]{x}}\right )}{d}+\frac {\frac {\frac {b n \log \left (-e \sqrt [3]{x}\right )}{d}-\frac {\left (d+e \sqrt [3]{x}\right ) \left (a+b \log \left (c x^{n/3}\right )\right )}{d e \sqrt [3]{x}}}{d}+\frac {\frac {b n \int \frac {\log \left (1-\frac {d}{\sqrt [3]{x}}\right )}{\sqrt [3]{x}}d\left (d+e \sqrt [3]{x}\right )}{d}-\frac {\log \left (1-\frac {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 {\left (a+b \log \left (c x^{n/3}\right )\right )^2}{3 e^3 x}-\frac {2}{3} b n \left (\frac {\frac {a+b \log \left (c x^{n/3}\right )}{2 e^2 x^{2/3}}-\frac {1}{2} b n \left (\frac {\log \left (d+e \sqrt [3]{x}\right )}{d^2}-\frac {\log \left (-e \sqrt [3]{x}\right )}{d^2}-\frac {1}{d e \sqrt [3]{x}}\right )}{d}+\frac {\frac {\frac {b n \log \left (-e \sqrt [3]{x}\right )}{d}-\frac {\left (d+e \sqrt [3]{x}\right ) \left (a+b \log \left (c x^{n/3}\right )\right )}{d e \sqrt [3]{x}}}{d}+\frac {\frac {b n \int \frac {\log \left (1-\frac {d}{\sqrt [3]{x}}\right )}{\sqrt [3]{x}}d\left (d+e \sqrt [3]{x}\right )}{d}-\frac {\log \left (1-\frac {d}{\sqrt [3]{x}}\right ) \left (a+b \log \left (c x^{n/3}\right )\right )}{d}}{d}}{d}\right )}{d}+\frac {\frac {\frac {\left (a+b \log \left (c x^{n/3}\right )\right )^2}{2 e^2 x^{2/3}}-b n \left (\frac {\frac {b n \log \left (-e \sqrt [3]{x}\right )}{d}-\frac {\left (d+e \sqrt [3]{x}\right ) \left (a+b \log \left (c x^{n/3}\right )\right )}{d e \sqrt [3]{x}}}{d}+\frac {\frac {b n \int \frac {\log \left (1-\frac {d}{\sqrt [3]{x}}\right )}{\sqrt [3]{x}}d\left (d+e \sqrt [3]{x}\right )}{d}-\frac {\log \left (1-\frac {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+e \sqrt [3]{x}\right ) \left (a+b \log \left (c x^{n/3}\right )\right )^2}{d e \sqrt [3]{x}}-\frac {2 b n \left (b n \int \frac {\log \left (1-\frac {d+e \sqrt [3]{x}}{d}\right )}{\sqrt [3]{x}}d\left (d+e \sqrt [3]{x}\right )-\log \left (1-\frac {d+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,\frac {d}{\sqrt [3]{x}}\right )-b n \int \frac {\operatorname {PolyLog}\left (2,\frac {d}{\sqrt [3]{x}}\right )}{\sqrt [3]{x}}d\left (d+e \sqrt [3]{x}\right )\right )}{d}-\frac {\log \left (1-\frac {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 {\left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{6 x^2}\right )\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle 3 \left (\frac {1}{2} b e^6 n \left (\frac {-\frac {\left (a+b \log \left (c x^{n/3}\right )\right )^2}{5 e^5 x^{5/3}}-\frac {2}{5} b n \left (\frac {\frac {a+b \log \left (c x^{n/3}\right )}{4 e^4 x^{4/3}}-\frac {1}{4} b n \left (\frac {\log \left (d+e \sqrt [3]{x}\right )}{d^4}-\frac {\log \left (-e \sqrt [3]{x}\right )}{d^4}-\frac {1}{d^3 e \sqrt [3]{x}}+\frac {1}{2 d^2 e^2 x^{2/3}}-\frac {1}{3 d e^3 x}\right )}{d}+\frac {\frac {-\frac {1}{3} b n \left (\frac {\log \left (d+e \sqrt [3]{x}\right )}{d^3}-\frac {\log \left (-e \sqrt [3]{x}\right )}{d^3}-\frac {1}{d^2 e \sqrt [3]{x}}+\frac {1}{2 d e^2 x^{2/3}}\right )-\frac {a+b \log \left (c x^{n/3}\right )}{3 e^3 x}}{d}+\frac {\frac {\frac {a+b \log \left (c x^{n/3}\right )}{2 e^2 x^{2/3}}-\frac {1}{2} b n \left (\frac {\log \left (d+e \sqrt [3]{x}\right )}{d^2}-\frac {\log \left (-e \sqrt [3]{x}\right )}{d^2}-\frac {1}{d e \sqrt [3]{x}}\right )}{d}+\frac {\frac {\frac {b n \log \left (-e \sqrt [3]{x}\right )}{d}-\frac {\left (d+e \sqrt [3]{x}\right ) \left (a+b \log \left (c x^{n/3}\right )\right )}{d e \sqrt [3]{x}}}{d}+\frac {\frac {b n \operatorname {PolyLog}\left (2,\frac {d}{\sqrt [3]{x}}\right )}{d}-\frac {\log \left (1-\frac {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 {\left (a+b \log \left (c x^{n/3}\right )\right )^2}{4 e^4 x^{4/3}}-\frac {1}{2} b n \left (\frac {-\frac {1}{3} b n \left (\frac {\log \left (d+e \sqrt [3]{x}\right )}{d^3}-\frac {\log \left (-e \sqrt [3]{x}\right )}{d^3}-\frac {1}{d^2 e \sqrt [3]{x}}+\frac {1}{2 d e^2 x^{2/3}}\right )-\frac {a+b \log \left (c x^{n/3}\right )}{3 e^3 x}}{d}+\frac {\frac {\frac {a+b \log \left (c x^{n/3}\right )}{2 e^2 x^{2/3}}-\frac {1}{2} b n \left (\frac {\log \left (d+e \sqrt [3]{x}\right )}{d^2}-\frac {\log \left (-e \sqrt [3]{x}\right )}{d^2}-\frac {1}{d e \sqrt [3]{x}}\right )}{d}+\frac {\frac {\frac {b n \log \left (-e \sqrt [3]{x}\right )}{d}-\frac {\left (d+e \sqrt [3]{x}\right ) \left (a+b \log \left (c x^{n/3}\right )\right )}{d e \sqrt [3]{x}}}{d}+\frac {\frac {b n \operatorname {PolyLog}\left (2,\frac {d}{\sqrt [3]{x}}\right )}{d}-\frac {\log \left (1-\frac {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 {\left (a+b \log \left (c x^{n/3}\right )\right )^2}{3 e^3 x}-\frac {2}{3} b n \left (\frac {\frac {a+b \log \left (c x^{n/3}\right )}{2 e^2 x^{2/3}}-\frac {1}{2} b n \left (\frac {\log \left (d+e \sqrt [3]{x}\right )}{d^2}-\frac {\log \left (-e \sqrt [3]{x}\right )}{d^2}-\frac {1}{d e \sqrt [3]{x}}\right )}{d}+\frac {\frac {\frac {b n \log \left (-e \sqrt [3]{x}\right )}{d}-\frac {\left (d+e \sqrt [3]{x}\right ) \left (a+b \log \left (c x^{n/3}\right )\right )}{d e \sqrt [3]{x}}}{d}+\frac {\frac {b n \operatorname {PolyLog}\left (2,\frac {d}{\sqrt [3]{x}}\right )}{d}-\frac {\log \left (1-\frac {d}{\sqrt [3]{x}}\right ) \left (a+b \log \left (c x^{n/3}\right )\right )}{d}}{d}}{d}\right )}{d}+\frac {\frac {\frac {\left (a+b \log \left (c x^{n/3}\right )\right )^2}{2 e^2 x^{2/3}}-b n \left (\frac {\frac {b n \log \left (-e \sqrt [3]{x}\right )}{d}-\frac {\left (d+e \sqrt [3]{x}\right ) \left (a+b \log \left (c x^{n/3}\right )\right )}{d e \sqrt [3]{x}}}{d}+\frac {\frac {b n \operatorname {PolyLog}\left (2,\frac {d}{\sqrt [3]{x}}\right )}{d}-\frac {\log \left (1-\frac {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+e \sqrt [3]{x}\right ) \left (a+b \log \left (c x^{n/3}\right )\right )^2}{d e \sqrt [3]{x}}-\frac {2 b n \left (-\log \left (1-\frac {d+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+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,\frac {d}{\sqrt [3]{x}}\right )-b n \int \frac {\operatorname {PolyLog}\left (2,\frac {d}{\sqrt [3]{x}}\right )}{\sqrt [3]{x}}d\left (d+e \sqrt [3]{x}\right )\right )}{d}-\frac {\log \left (1-\frac {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 {\left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{6 x^2}\right )\) |
| \(\Big \downarrow \) 7143 |
| \(\displaystyle 3 \left (\frac {1}{2} b e^6 n \left (\frac {-\frac {\left (a+b \log \left (c x^{n/3}\right )\right )^2}{5 e^5 x^{5/3}}-\frac {2}{5} b n \left (\frac {\frac {a+b \log \left (c x^{n/3}\right )}{4 e^4 x^{4/3}}-\frac {1}{4} b n \left (\frac {\log \left (d+e \sqrt [3]{x}\right )}{d^4}-\frac {\log \left (-e \sqrt [3]{x}\right )}{d^4}-\frac {1}{d^3 e \sqrt [3]{x}}+\frac {1}{2 d^2 e^2 x^{2/3}}-\frac {1}{3 d e^3 x}\right )}{d}+\frac {\frac {-\frac {1}{3} b n \left (\frac {\log \left (d+e \sqrt [3]{x}\right )}{d^3}-\frac {\log \left (-e \sqrt [3]{x}\right )}{d^3}-\frac {1}{d^2 e \sqrt [3]{x}}+\frac {1}{2 d e^2 x^{2/3}}\right )-\frac {a+b \log \left (c x^{n/3}\right )}{3 e^3 x}}{d}+\frac {\frac {\frac {a+b \log \left (c x^{n/3}\right )}{2 e^2 x^{2/3}}-\frac {1}{2} b n \left (\frac {\log \left (d+e \sqrt [3]{x}\right )}{d^2}-\frac {\log \left (-e \sqrt [3]{x}\right )}{d^2}-\frac {1}{d e \sqrt [3]{x}}\right )}{d}+\frac {\frac {\frac {b n \log \left (-e \sqrt [3]{x}\right )}{d}-\frac {\left (d+e \sqrt [3]{x}\right ) \left (a+b \log \left (c x^{n/3}\right )\right )}{d e \sqrt [3]{x}}}{d}+\frac {\frac {b n \operatorname {PolyLog}\left (2,\frac {d}{\sqrt [3]{x}}\right )}{d}-\frac {\log \left (1-\frac {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 {\left (a+b \log \left (c x^{n/3}\right )\right )^2}{4 e^4 x^{4/3}}-\frac {1}{2} b n \left (\frac {-\frac {1}{3} b n \left (\frac {\log \left (d+e \sqrt [3]{x}\right )}{d^3}-\frac {\log \left (-e \sqrt [3]{x}\right )}{d^3}-\frac {1}{d^2 e \sqrt [3]{x}}+\frac {1}{2 d e^2 x^{2/3}}\right )-\frac {a+b \log \left (c x^{n/3}\right )}{3 e^3 x}}{d}+\frac {\frac {\frac {a+b \log \left (c x^{n/3}\right )}{2 e^2 x^{2/3}}-\frac {1}{2} b n \left (\frac {\log \left (d+e \sqrt [3]{x}\right )}{d^2}-\frac {\log \left (-e \sqrt [3]{x}\right )}{d^2}-\frac {1}{d e \sqrt [3]{x}}\right )}{d}+\frac {\frac {\frac {b n \log \left (-e \sqrt [3]{x}\right )}{d}-\frac {\left (d+e \sqrt [3]{x}\right ) \left (a+b \log \left (c x^{n/3}\right )\right )}{d e \sqrt [3]{x}}}{d}+\frac {\frac {b n \operatorname {PolyLog}\left (2,\frac {d}{\sqrt [3]{x}}\right )}{d}-\frac {\log \left (1-\frac {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 {\left (a+b \log \left (c x^{n/3}\right )\right )^2}{3 e^3 x}-\frac {2}{3} b n \left (\frac {\frac {a+b \log \left (c x^{n/3}\right )}{2 e^2 x^{2/3}}-\frac {1}{2} b n \left (\frac {\log \left (d+e \sqrt [3]{x}\right )}{d^2}-\frac {\log \left (-e \sqrt [3]{x}\right )}{d^2}-\frac {1}{d e \sqrt [3]{x}}\right )}{d}+\frac {\frac {\frac {b n \log \left (-e \sqrt [3]{x}\right )}{d}-\frac {\left (d+e \sqrt [3]{x}\right ) \left (a+b \log \left (c x^{n/3}\right )\right )}{d e \sqrt [3]{x}}}{d}+\frac {\frac {b n \operatorname {PolyLog}\left (2,\frac {d}{\sqrt [3]{x}}\right )}{d}-\frac {\log \left (1-\frac {d}{\sqrt [3]{x}}\right ) \left (a+b \log \left (c x^{n/3}\right )\right )}{d}}{d}}{d}\right )}{d}+\frac {\frac {\frac {\left (a+b \log \left (c x^{n/3}\right )\right )^2}{2 e^2 x^{2/3}}-b n \left (\frac {\frac {b n \log \left (-e \sqrt [3]{x}\right )}{d}-\frac {\left (d+e \sqrt [3]{x}\right ) \left (a+b \log \left (c x^{n/3}\right )\right )}{d e \sqrt [3]{x}}}{d}+\frac {\frac {b n \operatorname {PolyLog}\left (2,\frac {d}{\sqrt [3]{x}}\right )}{d}-\frac {\log \left (1-\frac {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+e \sqrt [3]{x}\right ) \left (a+b \log \left (c x^{n/3}\right )\right )^2}{d e \sqrt [3]{x}}-\frac {2 b n \left (-\log \left (1-\frac {d+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+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,\frac {d}{\sqrt [3]{x}}\right )+b n \operatorname {PolyLog}\left (3,\frac {d}{\sqrt [3]{x}}\right )\right )}{d}-\frac {\log \left (1-\frac {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 {\left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{6 x^2}\right )\) |
Input:
Int[(a + b*Log[c*(d + e*x^(1/3))^n])^3/x^3,x]
Output:
3*(-1/6*(a + b*Log[c*(d + e*x^(1/3))^n])^3/x^2 + (b*e^6*n*((-1/5*(a + b*Lo g[c*x^(n/3)])^2/(e^5*x^(5/3)) - (2*b*n*((-1/4*(b*n*(-1/3*1/(d*e^3*x) + 1/( 2*d^2*e^2*x^(2/3)) - 1/(d^3*e*x^(1/3)) + Log[d + e*x^(1/3)]/d^4 - Log[-(e* x^(1/3))]/d^4)) + (a + b*Log[c*x^(n/3)])/(4*e^4*x^(4/3)))/d + ((-1/3*(b*n* (1/(2*d*e^2*x^(2/3)) - 1/(d^2*e*x^(1/3)) + Log[d + e*x^(1/3)]/d^3 - Log[-( e*x^(1/3))]/d^3)) - (a + b*Log[c*x^(n/3)])/(3*e^3*x))/d + ((-1/2*(b*n*(-(1 /(d*e*x^(1/3))) + Log[d + e*x^(1/3)]/d^2 - Log[-(e*x^(1/3))]/d^2)) + (a + b*Log[c*x^(n/3)])/(2*e^2*x^(2/3)))/d + (((b*n*Log[-(e*x^(1/3))])/d - ((d + e*x^(1/3))*(a + b*Log[c*x^(n/3)]))/(d*e*x^(1/3)))/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 + (((a + b*Log[c*x^(n/3)])^2/(4*e^4*x^(4/3)) - (b*n*((-1/3*(b*n*(1 /(2*d*e^2*x^(2/3)) - 1/(d^2*e*x^(1/3)) + Log[d + e*x^(1/3)]/d^3 - Log[-(e* x^(1/3))]/d^3)) - (a + b*Log[c*x^(n/3)])/(3*e^3*x))/d + ((-1/2*(b*n*(-(1/( d*e*x^(1/3))) + Log[d + e*x^(1/3)]/d^2 - Log[-(e*x^(1/3))]/d^2)) + (a + b* Log[c*x^(n/3)])/(2*e^2*x^(2/3)))/d + (((b*n*Log[-(e*x^(1/3))])/d - ((d + e *x^(1/3))*(a + b*Log[c*x^(n/3)]))/(d*e*x^(1/3)))/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*(a + b*Log[c*x^(n/3)])^2/(e^3*x) - (2*b*n*((-1/2*(b*n*(-(1/(d* e*x^(1/3))) + Log[d + e*x^(1/3)]/d^2 - Log[-(e*x^(1/3))]/d^2)) + (a + b*Lo g[c*x^(n/3)])/(2*e^2*x^(2/3)))/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]
\[\int \frac {{\left (a +b \ln \left (c \left (d +e \,x^{\frac {1}{3}}\right )^{n}\right )\right )}^{3}}{x^{3}}d x\]
Input:
int((a+b*ln(c*(d+e*x^(1/3))^n))^3/x^3,x)
Output:
int((a+b*ln(c*(d+e*x^(1/3))^n))^3/x^3,x)
\[ \int \frac {\left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{x^3} \, dx=\int { \frac {{\left (b \log \left ({\left (e x^{\frac {1}{3}} + d\right )}^{n} c\right ) + a\right )}^{3}}{x^{3}} \,d x } \] Input:
integrate((a+b*log(c*(d+e*x^(1/3))^n))^3/x^3,x, algorithm="fricas")
Output:
integral((b^3*log((e*x^(1/3) + d)^n*c)^3 + 3*a*b^2*log((e*x^(1/3) + d)^n*c )^2 + 3*a^2*b*log((e*x^(1/3) + d)^n*c) + a^3)/x^3, x)
\[ \int \frac {\left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{x^3} \, dx=\int \frac {\left (a + b \log {\left (c \left (d + e \sqrt [3]{x}\right )^{n} \right )}\right )^{3}}{x^{3}}\, dx \] Input:
integrate((a+b*ln(c*(d+e*x**(1/3))**n))**3/x**3,x)
Output:
Integral((a + b*log(c*(d + e*x**(1/3))**n))**3/x**3, x)
\[ \int \frac {\left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{x^3} \, dx=\int { \frac {{\left (b \log \left ({\left (e x^{\frac {1}{3}} + d\right )}^{n} c\right ) + a\right )}^{3}}{x^{3}} \,d x } \] Input:
integrate((a+b*log(c*(d+e*x^(1/3))^n))^3/x^3,x, algorithm="maxima")
Output:
-1/2*b^3*log((e*x^(1/3) + d)^n)^3/x^2 + integrate(1/2*((b^3*e*n*x + 6*(b^3 *e*log(c) + a*b^2*e)*x + 6*(b^3*d*log(c) + a*b^2*d)*x^(2/3))*log((e*x^(1/3 ) + d)^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 + 6*((b^3*e*log(c)^2 + 2*a*b^2*e*log(c) + a^2*b*e)*x + (b^3*d*log (c)^2 + 2*a*b^2*d*log(c) + a^2*b*d)*x^(2/3))*log((e*x^(1/3) + d)^n) + 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^(2/3))/(e *x^4 + d*x^(11/3)), x)
\[ \int \frac {\left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{x^3} \, dx=\int { \frac {{\left (b \log \left ({\left (e x^{\frac {1}{3}} + d\right )}^{n} c\right ) + a\right )}^{3}}{x^{3}} \,d x } \] Input:
integrate((a+b*log(c*(d+e*x^(1/3))^n))^3/x^3,x, algorithm="giac")
Output:
integrate((b*log((e*x^(1/3) + d)^n*c) + a)^3/x^3, x)
Timed out. \[ \int \frac {\left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{x^3} \, dx=\int \frac {{\left (a+b\,\ln \left (c\,{\left (d+e\,x^{1/3}\right )}^n\right )\right )}^3}{x^3} \,d x \] Input:
int((a + b*log(c*(d + e*x^(1/3))^n))^3/x^3,x)
Output:
int((a + b*log(c*(d + e*x^(1/3))^n))^3/x^3, x)
\[ \int \frac {\left (a+b \log \left (c \left (d+e \sqrt [3]{x}\right )^n\right )\right )^3}{x^3} \, dx =\text {Too large to display} \] Input:
int((a+b*log(c*(d+e*x^(1/3))^n))^3/x^3,x)
Output:
(45*x**(2/3)*log((x**(1/3)*e + d)**n*c)**2*b**3*d**4*e**2*n - 180*x**(2/3) *log((x**(1/3)*e + d)**n*c)**2*b**3*d*e**5*n*x + 90*x**(2/3)*log((x**(1/3) *e + d)**n*c)*a*b**2*d**4*e**2*n - 360*x**(2/3)*log((x**(1/3)*e + d)**n*c) *a*b**2*d*e**5*n*x - 18*x**(2/3)*log((x**(1/3)*e + d)**n*c)*b**3*d**4*e**2 *n**2 + 462*x**(2/3)*log((x**(1/3)*e + d)**n*c)*b**3*d*e**5*n**2*x + 45*x* *(2/3)*a**2*b*d**4*e**2*n - 180*x**(2/3)*a**2*b*d*e**5*n*x - 18*x**(2/3)*a *b**2*d**4*e**2*n**2 + 462*x**(2/3)*a*b**2*d*e**5*n**2*x - 213*x**(2/3)*b* *3*d*e**5*n**3*x - 36*x**(1/3)*log((x**(1/3)*e + d)**n*c)**2*b**3*d**5*e*n + 90*x**(1/3)*log((x**(1/3)*e + d)**n*c)**2*b**3*d**2*e**4*n*x - 72*x**(1 /3)*log((x**(1/3)*e + d)**n*c)*a*b**2*d**5*e*n + 180*x**(1/3)*log((x**(1/3 )*e + d)**n*c)*a*b**2*d**2*e**4*n*x - 141*x**(1/3)*log((x**(1/3)*e + d)**n *c)*b**3*d**2*e**4*n**2*x - 36*x**(1/3)*a**2*b*d**5*e*n + 90*x**(1/3)*a**2 *b*d**2*e**4*n*x - 141*x**(1/3)*a*b**2*d**2*e**4*n**2*x + 36*x**(1/3)*b**3 *d**2*e**4*n**3*x - 60*int(log((x**(1/3)*e + d)**n*c)**2/(x**(1/3)*e*x + d *x),x)*b**3*d*e**6*n*x**2 - 120*int(log((x**(1/3)*e + d)**n*c)/(x**(1/3)*e *x + d*x),x)*a*b**2*d*e**6*n*x**2 + 274*int(log((x**(1/3)*e + d)**n*c)/(x* *(1/3)*e*x + d*x),x)*b**3*d*e**6*n**2*x**2 - 180*log(x**(1/3))*a**2*b*e**6 *n*x**2 + 822*log(x**(1/3))*a*b**2*e**6*n**2*x**2 - 675*log(x**(1/3))*b**3 *e**6*n**3*x**2 - 60*log((x**(1/3)*e + d)**n*c)**3*b**3*d**6 - 180*log((x* *(1/3)*e + d)**n*c)**2*a*b**2*d**6 - 60*log((x**(1/3)*e + d)**n*c)**2*b...