Integrand size = 83, antiderivative size = 154 \[ \int \frac {-36-6 x^2+6 x^3+x^6}{x \left (-6+x^3\right ) \sqrt [6]{\frac {6+x^3}{-6+x^3}} \left (36-90 x+122 x^2-96 x^3+51 x^4-26 x^5+15 x^6-6 x^7+x^8\right )} \, dx=-\frac {1}{3} \arctan \left (\frac {\sqrt [6]{\frac {6+x^3}{-6+x^3}}}{-1+x}\right )-\frac {1}{6} \arctan \left (\frac {-1+2 x-x^2+\sqrt [3]{\frac {6+x^3}{-6+x^3}}}{(-1+x) \sqrt [6]{\frac {6+x^3}{-6+x^3}}}\right )+\frac {\text {arctanh}\left (\frac {\left (-\sqrt {3}+\sqrt {3} x\right ) \sqrt [6]{\frac {6+x^3}{-6+x^3}}}{1-2 x+x^2+\sqrt [3]{\frac {6+x^3}{-6+x^3}}}\right )}{2 \sqrt {3}} \] Output:
-1/3*arctan(((x^3+6)/(x^3-6))^(1/6)/(-1+x))-1/6*arctan((-1+2*x-x^2+((x^3+6 )/(x^3-6))^(1/3))/(-1+x)/((x^3+6)/(x^3-6))^(1/6))+1/6*arctanh((-3^(1/2)+x* 3^(1/2))*((x^3+6)/(x^3-6))^(1/6)/(1-2*x+x^2+((x^3+6)/(x^3-6))^(1/3)))*3^(1 /2)
\[ \int \frac {-36-6 x^2+6 x^3+x^6}{x \left (-6+x^3\right ) \sqrt [6]{\frac {6+x^3}{-6+x^3}} \left (36-90 x+122 x^2-96 x^3+51 x^4-26 x^5+15 x^6-6 x^7+x^8\right )} \, dx=\int \frac {-36-6 x^2+6 x^3+x^6}{x \left (-6+x^3\right ) \sqrt [6]{\frac {6+x^3}{-6+x^3}} \left (36-90 x+122 x^2-96 x^3+51 x^4-26 x^5+15 x^6-6 x^7+x^8\right )} \, dx \] Input:
Integrate[(-36 - 6*x^2 + 6*x^3 + x^6)/(x*(-6 + x^3)*((6 + x^3)/(-6 + x^3)) ^(1/6)*(36 - 90*x + 122*x^2 - 96*x^3 + 51*x^4 - 26*x^5 + 15*x^6 - 6*x^7 + x^8)),x]
Output:
Integrate[(-36 - 6*x^2 + 6*x^3 + x^6)/(x*(-6 + x^3)*((6 + x^3)/(-6 + x^3)) ^(1/6)*(36 - 90*x + 122*x^2 - 96*x^3 + 51*x^4 - 26*x^5 + 15*x^6 - 6*x^7 + x^8)), x]
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.
| \(\displaystyle \int \frac {x^6+6 x^3-6 x^2-36}{x \left (x^3-6\right ) \sqrt [6]{\frac {x^3+6}{x^3-6}} \left (x^8-6 x^7+15 x^6-26 x^5+51 x^4-96 x^3+122 x^2-90 x+36\right )} \, dx\) |
| \(\Big \downarrow \) 2058 |
| \(\displaystyle \frac {\sqrt [6]{x^3+6} \int -\frac {-x^6-6 x^3+6 x^2+36}{x \left (x^3-6\right )^{5/6} \sqrt [6]{x^3+6} \left (x^8-6 x^7+15 x^6-26 x^5+51 x^4-96 x^3+122 x^2-90 x+36\right )}dx}{\sqrt [6]{x^3-6} \sqrt [6]{-\frac {x^3+6}{6-x^3}}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\sqrt [6]{x^3+6} \int \frac {-x^6-6 x^3+6 x^2+36}{x \left (x^3-6\right )^{5/6} \sqrt [6]{x^3+6} \left (x^8-6 x^7+15 x^6-26 x^5+51 x^4-96 x^3+122 x^2-90 x+36\right )}dx}{\sqrt [6]{x^3-6} \sqrt [6]{-\frac {x^3+6}{6-x^3}}}\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -\frac {\sqrt [6]{x^3+6} \int \left (\frac {-x^7+6 x^6-16 x^5+26 x^4-51 x^3+90 x^2-116 x+90}{\left (x^3-6\right )^{5/6} \sqrt [6]{x^3+6} \left (x^8-6 x^7+15 x^6-26 x^5+51 x^4-96 x^3+122 x^2-90 x+36\right )}+\frac {1}{x \left (x^3-6\right )^{5/6} \sqrt [6]{x^3+6}}\right )dx}{\sqrt [6]{x^3-6} \sqrt [6]{-\frac {x^3+6}{6-x^3}}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {\sqrt [6]{x^3+6} \left (90 \int \frac {1}{\left (x^3-6\right )^{5/6} \sqrt [6]{x^3+6} \left (x^8-6 x^7+15 x^6-26 x^5+51 x^4-96 x^3+122 x^2-90 x+36\right )}dx-116 \int \frac {x}{\left (x^3-6\right )^{5/6} \sqrt [6]{x^3+6} \left (x^8-6 x^7+15 x^6-26 x^5+51 x^4-96 x^3+122 x^2-90 x+36\right )}dx+90 \int \frac {x^2}{\left (x^3-6\right )^{5/6} \sqrt [6]{x^3+6} \left (x^8-6 x^7+15 x^6-26 x^5+51 x^4-96 x^3+122 x^2-90 x+36\right )}dx-51 \int \frac {x^3}{\left (x^3-6\right )^{5/6} \sqrt [6]{x^3+6} \left (x^8-6 x^7+15 x^6-26 x^5+51 x^4-96 x^3+122 x^2-90 x+36\right )}dx+26 \int \frac {x^4}{\left (x^3-6\right )^{5/6} \sqrt [6]{x^3+6} \left (x^8-6 x^7+15 x^6-26 x^5+51 x^4-96 x^3+122 x^2-90 x+36\right )}dx-16 \int \frac {x^5}{\left (x^3-6\right )^{5/6} \sqrt [6]{x^3+6} \left (x^8-6 x^7+15 x^6-26 x^5+51 x^4-96 x^3+122 x^2-90 x+36\right )}dx+6 \int \frac {x^6}{\left (x^3-6\right )^{5/6} \sqrt [6]{x^3+6} \left (x^8-6 x^7+15 x^6-26 x^5+51 x^4-96 x^3+122 x^2-90 x+36\right )}dx-\int \frac {x^7}{\left (x^3-6\right )^{5/6} \sqrt [6]{x^3+6} \left (x^8-6 x^7+15 x^6-26 x^5+51 x^4-96 x^3+122 x^2-90 x+36\right )}dx-\frac {1}{9} \arctan \left (\frac {\sqrt [6]{x^3+6}}{\sqrt [6]{x^3-6}}\right )+\frac {1}{18} \arctan \left (\sqrt {3}-\frac {2 \sqrt [6]{x^3+6}}{\sqrt [6]{x^3-6}}\right )-\frac {1}{18} \arctan \left (\frac {2 \sqrt [6]{x^3+6}}{\sqrt [6]{x^3-6}}+\sqrt {3}\right )-\frac {\log \left (\frac {\sqrt [3]{x^3+6}}{\sqrt [3]{x^3-6}}-\frac {\sqrt {3} \sqrt [6]{x^3+6}}{\sqrt [6]{x^3-6}}+1\right )}{12 \sqrt {3}}+\frac {\log \left (\frac {\sqrt [3]{x^3+6}}{\sqrt [3]{x^3-6}}+\frac {\sqrt {3} \sqrt [6]{x^3+6}}{\sqrt [6]{x^3-6}}+1\right )}{12 \sqrt {3}}\right )}{\sqrt [6]{x^3-6} \sqrt [6]{-\frac {x^3+6}{6-x^3}}}\) |
Input:
Int[(-36 - 6*x^2 + 6*x^3 + x^6)/(x*(-6 + x^3)*((6 + x^3)/(-6 + x^3))^(1/6) *(36 - 90*x + 122*x^2 - 96*x^3 + 51*x^4 - 26*x^5 + 15*x^6 - 6*x^7 + x^8)), x]
Output:
$Aborted
Timed out.
\[\int \frac {x^{6}+6 x^{3}-6 x^{2}-36}{x \left (x^{3}-6\right ) \left (\frac {x^{3}+6}{x^{3}-6}\right )^{\frac {1}{6}} \left (x^{8}-6 x^{7}+15 x^{6}-26 x^{5}+51 x^{4}-96 x^{3}+122 x^{2}-90 x +36\right )}d x\]
Input:
int((x^6+6*x^3-6*x^2-36)/x/(x^3-6)/((x^3+6)/(x^3-6))^(1/6)/(x^8-6*x^7+15*x ^6-26*x^5+51*x^4-96*x^3+122*x^2-90*x+36),x)
Output:
int((x^6+6*x^3-6*x^2-36)/x/(x^3-6)/((x^3+6)/(x^3-6))^(1/6)/(x^8-6*x^7+15*x ^6-26*x^5+51*x^4-96*x^3+122*x^2-90*x+36),x)
Timed out. \[ \int \frac {-36-6 x^2+6 x^3+x^6}{x \left (-6+x^3\right ) \sqrt [6]{\frac {6+x^3}{-6+x^3}} \left (36-90 x+122 x^2-96 x^3+51 x^4-26 x^5+15 x^6-6 x^7+x^8\right )} \, dx=\text {Timed out} \] Input:
integrate((x^6+6*x^3-6*x^2-36)/x/(x^3-6)/((x^3+6)/(x^3-6))^(1/6)/(x^8-6*x^ 7+15*x^6-26*x^5+51*x^4-96*x^3+122*x^2-90*x+36),x, algorithm="fricas")
Output:
Timed out
Timed out. \[ \int \frac {-36-6 x^2+6 x^3+x^6}{x \left (-6+x^3\right ) \sqrt [6]{\frac {6+x^3}{-6+x^3}} \left (36-90 x+122 x^2-96 x^3+51 x^4-26 x^5+15 x^6-6 x^7+x^8\right )} \, dx=\text {Timed out} \] Input:
integrate((x**6+6*x**3-6*x**2-36)/x/(x**3-6)/((x**3+6)/(x**3-6))**(1/6)/(x **8-6*x**7+15*x**6-26*x**5+51*x**4-96*x**3+122*x**2-90*x+36),x)
Output:
Timed out
\[ \int \frac {-36-6 x^2+6 x^3+x^6}{x \left (-6+x^3\right ) \sqrt [6]{\frac {6+x^3}{-6+x^3}} \left (36-90 x+122 x^2-96 x^3+51 x^4-26 x^5+15 x^6-6 x^7+x^8\right )} \, dx=\int { \frac {x^{6} + 6 \, x^{3} - 6 \, x^{2} - 36}{{\left (x^{8} - 6 \, x^{7} + 15 \, x^{6} - 26 \, x^{5} + 51 \, x^{4} - 96 \, x^{3} + 122 \, x^{2} - 90 \, x + 36\right )} {\left (x^{3} - 6\right )} x \left (\frac {x^{3} + 6}{x^{3} - 6}\right )^{\frac {1}{6}}} \,d x } \] Input:
integrate((x^6+6*x^3-6*x^2-36)/x/(x^3-6)/((x^3+6)/(x^3-6))^(1/6)/(x^8-6*x^ 7+15*x^6-26*x^5+51*x^4-96*x^3+122*x^2-90*x+36),x, algorithm="maxima")
Output:
integrate((x^6 + 6*x^3 - 6*x^2 - 36)/((x^8 - 6*x^7 + 15*x^6 - 26*x^5 + 51* x^4 - 96*x^3 + 122*x^2 - 90*x + 36)*(x^3 - 6)*x*((x^3 + 6)/(x^3 - 6))^(1/6 )), x)
\[ \int \frac {-36-6 x^2+6 x^3+x^6}{x \left (-6+x^3\right ) \sqrt [6]{\frac {6+x^3}{-6+x^3}} \left (36-90 x+122 x^2-96 x^3+51 x^4-26 x^5+15 x^6-6 x^7+x^8\right )} \, dx=\int { \frac {x^{6} + 6 \, x^{3} - 6 \, x^{2} - 36}{{\left (x^{8} - 6 \, x^{7} + 15 \, x^{6} - 26 \, x^{5} + 51 \, x^{4} - 96 \, x^{3} + 122 \, x^{2} - 90 \, x + 36\right )} {\left (x^{3} - 6\right )} x \left (\frac {x^{3} + 6}{x^{3} - 6}\right )^{\frac {1}{6}}} \,d x } \] Input:
integrate((x^6+6*x^3-6*x^2-36)/x/(x^3-6)/((x^3+6)/(x^3-6))^(1/6)/(x^8-6*x^ 7+15*x^6-26*x^5+51*x^4-96*x^3+122*x^2-90*x+36),x, algorithm="giac")
Output:
integrate((x^6 + 6*x^3 - 6*x^2 - 36)/((x^8 - 6*x^7 + 15*x^6 - 26*x^5 + 51* x^4 - 96*x^3 + 122*x^2 - 90*x + 36)*(x^3 - 6)*x*((x^3 + 6)/(x^3 - 6))^(1/6 )), x)
Timed out. \[ \int \frac {-36-6 x^2+6 x^3+x^6}{x \left (-6+x^3\right ) \sqrt [6]{\frac {6+x^3}{-6+x^3}} \left (36-90 x+122 x^2-96 x^3+51 x^4-26 x^5+15 x^6-6 x^7+x^8\right )} \, dx=-\int \frac {-x^6-6\,x^3+6\,x^2+36}{x\,{\left (\frac {x^3+6}{x^3-6}\right )}^{1/6}\,\left (x^3-6\right )\,\left (x^8-6\,x^7+15\,x^6-26\,x^5+51\,x^4-96\,x^3+122\,x^2-90\,x+36\right )} \,d x \] Input:
int(-(6*x^2 - 6*x^3 - x^6 + 36)/(x*((x^3 + 6)/(x^3 - 6))^(1/6)*(x^3 - 6)*( 122*x^2 - 90*x - 96*x^3 + 51*x^4 - 26*x^5 + 15*x^6 - 6*x^7 + x^8 + 36)),x)
Output:
-int((6*x^2 - 6*x^3 - x^6 + 36)/(x*((x^3 + 6)/(x^3 - 6))^(1/6)*(x^3 - 6)*( 122*x^2 - 90*x - 96*x^3 + 51*x^4 - 26*x^5 + 15*x^6 - 6*x^7 + x^8 + 36)), x )
\[ \int \frac {-36-6 x^2+6 x^3+x^6}{x \left (-6+x^3\right ) \sqrt [6]{\frac {6+x^3}{-6+x^3}} \left (36-90 x+122 x^2-96 x^3+51 x^4-26 x^5+15 x^6-6 x^7+x^8\right )} \, dx=\frac {\left (22680 \,3^{\frac {5}{6}} \left (\int \frac {\left (x^{3}+6\right )^{\frac {5}{6}} \left (x^{3}-6\right )^{\frac {1}{6}} x^{5}}{x^{14}-6 x^{13}+15 x^{12}-26 x^{11}+51 x^{10}-96 x^{9}+86 x^{8}+126 x^{7}-504 x^{6}+936 x^{5}-1836 x^{4}+3456 x^{3}-4392 x^{2}+3240 x -1296}d x \right )+136080 \,3^{\frac {5}{6}} \left (\int \frac {\left (x^{3}+6\right )^{\frac {5}{6}} \left (x^{3}-6\right )^{\frac {1}{6}} x^{2}}{x^{14}-6 x^{13}+15 x^{12}-26 x^{11}+51 x^{10}-96 x^{9}+86 x^{8}+126 x^{7}-504 x^{6}+936 x^{5}-1836 x^{4}+3456 x^{3}-4392 x^{2}+3240 x -1296}d x \right )-136080 \,3^{\frac {5}{6}} \left (\int \frac {\left (x^{3}+6\right )^{\frac {5}{6}} \left (x^{3}-6\right )^{\frac {1}{6}} x}{x^{14}-6 x^{13}+15 x^{12}-26 x^{11}+51 x^{10}-96 x^{9}+86 x^{8}+126 x^{7}-504 x^{6}+936 x^{5}-1836 x^{4}+3456 x^{3}-4392 x^{2}+3240 x -1296}d x \right )-816480 \,3^{\frac {5}{6}} \left (\int \frac {\left (x^{3}+6\right )^{\frac {5}{6}} \left (x^{3}-6\right )^{\frac {1}{6}}}{x^{15}-6 x^{14}+15 x^{13}-26 x^{12}+51 x^{11}-96 x^{10}+86 x^{9}+126 x^{8}-504 x^{7}+936 x^{6}-1836 x^{5}+3456 x^{4}-4392 x^{3}+3240 x^{2}-1296 x}d x \right )\right ) 3^{\frac {1}{6}}}{68040} \] Input:
int((x^6+6*x^3-6*x^2-36)/x/(x^3-6)/((x^3+6)/(x^3-6))^(1/6)/(x^8-6*x^7+15*x ^6-26*x^5+51*x^4-96*x^3+122*x^2-90*x+36),x)
Output:
(14*(x**3 + 6)**(5/6)*(x**3 - 6)**(1/6)*3**(2/3)*3**(1/6)*3**(7/9)*3**(1/1 8)*3**(4/27)*3**(1/54)*x + 211*(x**3 + 6)**(5/6)*(x**3 - 6)**(1/6)*3**(2/3 )*3**(1/6)*3**(7/9)*3**(1/18)*3**(4/27)*3**(1/54) - 42*(x**3 + 6)**(5/6)*( x**3 - 6)**(1/6)*3**(2/3)*3**(1/6)*x - 633*(x**3 + 6)**(5/6)*(x**3 - 6)**( 1/6)*3**(2/3)*3**(1/6) + 7560*3**(2/3)*3**(1/6)*3**(7/9)*3**(1/18)*3**(4/2 7)*3**(1/54)*int(((x**3 + 6)**(5/6)*(x**3 - 6)**(1/6)*x**5)/(x**14 - 6*x** 13 + 15*x**12 - 26*x**11 + 51*x**10 - 96*x**9 + 86*x**8 + 126*x**7 - 504*x **6 + 936*x**5 - 1836*x**4 + 3456*x**3 - 4392*x**2 + 3240*x - 1296),x) + 4 5360*3**(2/3)*3**(1/6)*3**(7/9)*3**(1/18)*3**(4/27)*3**(1/54)*int(((x**3 + 6)**(5/6)*(x**3 - 6)**(1/6)*x**2)/(x**14 - 6*x**13 + 15*x**12 - 26*x**11 + 51*x**10 - 96*x**9 + 86*x**8 + 126*x**7 - 504*x**6 + 936*x**5 - 1836*x** 4 + 3456*x**3 - 4392*x**2 + 3240*x - 1296),x) - 45360*3**(2/3)*3**(1/6)*3* *(7/9)*3**(1/18)*3**(4/27)*3**(1/54)*int(((x**3 + 6)**(5/6)*(x**3 - 6)**(1 /6)*x)/(x**14 - 6*x**13 + 15*x**12 - 26*x**11 + 51*x**10 - 96*x**9 + 86*x* *8 + 126*x**7 - 504*x**6 + 936*x**5 - 1836*x**4 + 3456*x**3 - 4392*x**2 + 3240*x - 1296),x) - 816480*3**(7/9)*3**(1/18)*int(((x**3 + 6)**(5/6)*(x**3 - 6)**(1/6))/(x**15 - 6*x**14 + 15*x**13 - 26*x**12 + 51*x**11 - 96*x**10 + 86*x**9 + 126*x**8 - 504*x**7 + 936*x**6 - 1836*x**5 + 3456*x**4 - 4392 *x**3 + 3240*x**2 - 1296*x),x))/(22680*3**(2/3)*3**(1/6))