Integrand size = 85, antiderivative size = 282 \[ \int \frac {1-\sqrt [3]{1-3 x+3 x^3-9 x^4+3 x^6-9 x^7+x^9-3 x^{10}}}{x^2+\sqrt [3]{1-3 x+3 x^3-9 x^4+3 x^6-9 x^7+x^9-3 x^{10}}} \, dx=-x+3 \text {RootSum}\left [-3-28 \text {$\#$1}+6 \text {$\#$1}^3+3 \text {$\#$1}^4-3 \text {$\#$1}^6-3 \text {$\#$1}^7+\text {$\#$1}^{10}\&,\frac {-10 \log \left (1+x^3\right ) \text {$\#$1}^2+10 \log \left (\sqrt [3]{1-3 x+3 x^3-9 x^4+3 x^6-9 x^7+x^9-3 x^{10}}-\text {$\#$1}-x^3 \text {$\#$1}\right ) \text {$\#$1}^2+2 \log \left (1+x^3\right ) \text {$\#$1}^5-2 \log \left (\sqrt [3]{1-3 x+3 x^3-9 x^4+3 x^6-9 x^7+x^9-3 x^{10}}-\text {$\#$1}-x^3 \text {$\#$1}\right ) \text {$\#$1}^5-\log \left (1+x^3\right ) \text {$\#$1}^8+\log \left (\sqrt [3]{1-3 x+3 x^3-9 x^4+3 x^6-9 x^7+x^9-3 x^{10}}-\text {$\#$1}-x^3 \text {$\#$1}\right ) \text {$\#$1}^8}{-28+18 \text {$\#$1}^2+12 \text {$\#$1}^3-18 \text {$\#$1}^5-21 \text {$\#$1}^6+10 \text {$\#$1}^9}\&\right ] \]
Time = 0.22 (sec) , antiderivative size = 228, normalized size of antiderivative = 0.81 \[ \int \frac {1-\sqrt [3]{1-3 x+3 x^3-9 x^4+3 x^6-9 x^7+x^9-3 x^{10}}}{x^2+\sqrt [3]{1-3 x+3 x^3-9 x^4+3 x^6-9 x^7+x^9-3 x^{10}}} \, dx=\frac {1}{3}-x-3 \text {RootSum}\left [-3-28 \text {$\#$1}+6 \text {$\#$1}^3+3 \text {$\#$1}^4-3 \text {$\#$1}^6-3 \text {$\#$1}^7+\text {$\#$1}^{10}\&,\frac {10 \log \left (1+x^3\right ) \text {$\#$1}^2-10 \log \left (\sqrt [3]{-\left ((-1+3 x) \left (1+x^3\right )^3\right )}-\text {$\#$1}-x^3 \text {$\#$1}\right ) \text {$\#$1}^2-2 \log \left (1+x^3\right ) \text {$\#$1}^5+2 \log \left (\sqrt [3]{-\left ((-1+3 x) \left (1+x^3\right )^3\right )}-\text {$\#$1}-x^3 \text {$\#$1}\right ) \text {$\#$1}^5+\log \left (1+x^3\right ) \text {$\#$1}^8-\log \left (\sqrt [3]{-\left ((-1+3 x) \left (1+x^3\right )^3\right )}-\text {$\#$1}-x^3 \text {$\#$1}\right ) \text {$\#$1}^8}{-28+18 \text {$\#$1}^2+12 \text {$\#$1}^3-18 \text {$\#$1}^5-21 \text {$\#$1}^6+10 \text {$\#$1}^9}\&\right ] \]
Integrate[(1 - (1 - 3*x + 3*x^3 - 9*x^4 + 3*x^6 - 9*x^7 + x^9 - 3*x^10)^(1 /3))/(x^2 + (1 - 3*x + 3*x^3 - 9*x^4 + 3*x^6 - 9*x^7 + x^9 - 3*x^10)^(1/3) ),x]
1/3 - x - 3*RootSum[-3 - 28*#1 + 6*#1^3 + 3*#1^4 - 3*#1^6 - 3*#1^7 + #1^10 & , (10*Log[1 + x^3]*#1^2 - 10*Log[(-((-1 + 3*x)*(1 + x^3)^3))^(1/3) - #1 - x^3*#1]*#1^2 - 2*Log[1 + x^3]*#1^5 + 2*Log[(-((-1 + 3*x)*(1 + x^3)^3))^ (1/3) - #1 - x^3*#1]*#1^5 + Log[1 + x^3]*#1^8 - Log[(-((-1 + 3*x)*(1 + x^3 )^3))^(1/3) - #1 - x^3*#1]*#1^8)/(-28 + 18*#1^2 + 12*#1^3 - 18*#1^5 - 21*# 1^6 + 10*#1^9) & ]
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.
\(\displaystyle \int \frac {1-\sqrt [3]{-3 x^{10}+x^9-9 x^7+3 x^6-9 x^4+3 x^3-3 x+1}}{x^2+\sqrt [3]{-3 x^{10}+x^9-9 x^7+3 x^6-9 x^4+3 x^3-3 x+1}} \, dx\) |
\(\Big \downarrow \) 7239 |
\(\displaystyle \int \frac {1-\sqrt [3]{-\left ((3 x-1) \left (x^3+1\right )^3\right )}}{\sqrt [3]{-\left ((3 x-1) \left (x^3+1\right )^3\right )}+x^2}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {1}{\sqrt [3]{-\left ((3 x-1) \left (x^3+1\right )^3\right )}+x^2}-\frac {\sqrt [3]{-\left ((3 x-1) \left (x^3+1\right )^3\right )}}{\sqrt [3]{-\left ((3 x-1) \left (x^3+1\right )^3\right )}+x^2}\right )dx\) |
\(\Big \downarrow \) 7299 |
\(\displaystyle \int \left (\frac {1}{\sqrt [3]{-\left ((3 x-1) \left (x^3+1\right )^3\right )}+x^2}-\frac {\sqrt [3]{-\left ((3 x-1) \left (x^3+1\right )^3\right )}}{\sqrt [3]{-\left ((3 x-1) \left (x^3+1\right )^3\right )}+x^2}\right )dx\) |
Int[(1 - (1 - 3*x + 3*x^3 - 9*x^4 + 3*x^6 - 9*x^7 + x^9 - 3*x^10)^(1/3))/( x^2 + (1 - 3*x + 3*x^3 - 9*x^4 + 3*x^6 - 9*x^7 + x^9 - 3*x^10)^(1/3)),x]
3.29.23.3.1 Defintions of rubi rules used
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Timed out.
\[\int \frac {1-\left (-3 x^{10}+x^{9}-9 x^{7}+3 x^{6}-9 x^{4}+3 x^{3}-3 x +1\right )^{\frac {1}{3}}}{x^{2}+\left (-3 x^{10}+x^{9}-9 x^{7}+3 x^{6}-9 x^{4}+3 x^{3}-3 x +1\right )^{\frac {1}{3}}}d x\]
int((1-(-3*x^10+x^9-9*x^7+3*x^6-9*x^4+3*x^3-3*x+1)^(1/3))/(x^2+(-3*x^10+x^ 9-9*x^7+3*x^6-9*x^4+3*x^3-3*x+1)^(1/3)),x)
int((1-(-3*x^10+x^9-9*x^7+3*x^6-9*x^4+3*x^3-3*x+1)^(1/3))/(x^2+(-3*x^10+x^ 9-9*x^7+3*x^6-9*x^4+3*x^3-3*x+1)^(1/3)),x)
Timed out. \[ \int \frac {1-\sqrt [3]{1-3 x+3 x^3-9 x^4+3 x^6-9 x^7+x^9-3 x^{10}}}{x^2+\sqrt [3]{1-3 x+3 x^3-9 x^4+3 x^6-9 x^7+x^9-3 x^{10}}} \, dx=\text {Timed out} \]
integrate((1-(-3*x^10+x^9-9*x^7+3*x^6-9*x^4+3*x^3-3*x+1)^(1/3))/(x^2+(-3*x ^10+x^9-9*x^7+3*x^6-9*x^4+3*x^3-3*x+1)^(1/3)),x, algorithm="fricas")
Not integrable
Time = 2.31 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.45 \[ \int \frac {1-\sqrt [3]{1-3 x+3 x^3-9 x^4+3 x^6-9 x^7+x^9-3 x^{10}}}{x^2+\sqrt [3]{1-3 x+3 x^3-9 x^4+3 x^6-9 x^7+x^9-3 x^{10}}} \, dx=- \int \frac {\sqrt [3]{- 3 x^{10} + x^{9} - 9 x^{7} + 3 x^{6} - 9 x^{4} + 3 x^{3} - 3 x + 1}}{x^{2} + \sqrt [3]{- 3 x^{10} + x^{9} - 9 x^{7} + 3 x^{6} - 9 x^{4} + 3 x^{3} - 3 x + 1}}\, dx - \int \left (- \frac {1}{x^{2} + \sqrt [3]{- 3 x^{10} + x^{9} - 9 x^{7} + 3 x^{6} - 9 x^{4} + 3 x^{3} - 3 x + 1}}\right )\, dx \]
integrate((1-(-3*x**10+x**9-9*x**7+3*x**6-9*x**4+3*x**3-3*x+1)**(1/3))/(x* *2+(-3*x**10+x**9-9*x**7+3*x**6-9*x**4+3*x**3-3*x+1)**(1/3)),x)
-Integral((-3*x**10 + x**9 - 9*x**7 + 3*x**6 - 9*x**4 + 3*x**3 - 3*x + 1)* *(1/3)/(x**2 + (-3*x**10 + x**9 - 9*x**7 + 3*x**6 - 9*x**4 + 3*x**3 - 3*x + 1)**(1/3)), x) - Integral(-1/(x**2 + (-3*x**10 + x**9 - 9*x**7 + 3*x**6 - 9*x**4 + 3*x**3 - 3*x + 1)**(1/3)), x)
Not integrable
Time = 0.30 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.12 \[ \int \frac {1-\sqrt [3]{1-3 x+3 x^3-9 x^4+3 x^6-9 x^7+x^9-3 x^{10}}}{x^2+\sqrt [3]{1-3 x+3 x^3-9 x^4+3 x^6-9 x^7+x^9-3 x^{10}}} \, dx=\int { -\frac {{\left (-3 \, x^{10} + x^{9} - 9 \, x^{7} + 3 \, x^{6} - 9 \, x^{4} + 3 \, x^{3} - 3 \, x + 1\right )}^{\frac {1}{3}} - 1}{x^{2} + {\left (-3 \, x^{10} + x^{9} - 9 \, x^{7} + 3 \, x^{6} - 9 \, x^{4} + 3 \, x^{3} - 3 \, x + 1\right )}^{\frac {1}{3}}} \,d x } \]
integrate((1-(-3*x^10+x^9-9*x^7+3*x^6-9*x^4+3*x^3-3*x+1)^(1/3))/(x^2+(-3*x ^10+x^9-9*x^7+3*x^6-9*x^4+3*x^3-3*x+1)^(1/3)),x, algorithm="maxima")
Not integrable
Time = 0.41 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.29 \[ \int \frac {1-\sqrt [3]{1-3 x+3 x^3-9 x^4+3 x^6-9 x^7+x^9-3 x^{10}}}{x^2+\sqrt [3]{1-3 x+3 x^3-9 x^4+3 x^6-9 x^7+x^9-3 x^{10}}} \, dx=\int { -\frac {{\left (-3 \, x^{10} + x^{9} - 9 \, x^{7} + 3 \, x^{6} - 9 \, x^{4} + 3 \, x^{3} - 3 \, x + 1\right )}^{\frac {1}{3}} - 1}{x^{2} + {\left (-3 \, x^{10} + x^{9} - 9 \, x^{7} + 3 \, x^{6} - 9 \, x^{4} + 3 \, x^{3} - 3 \, x + 1\right )}^{\frac {1}{3}}} \,d x } \]
integrate((1-(-3*x^10+x^9-9*x^7+3*x^6-9*x^4+3*x^3-3*x+1)^(1/3))/(x^2+(-3*x ^10+x^9-9*x^7+3*x^6-9*x^4+3*x^3-3*x+1)^(1/3)),x, algorithm="giac")
integrate(-((-3*x^10 + x^9 - 9*x^7 + 3*x^6 - 9*x^4 + 3*x^3 - 3*x + 1)^(1/3 ) - 1)/(x^2 + (-3*x^10 + x^9 - 9*x^7 + 3*x^6 - 9*x^4 + 3*x^3 - 3*x + 1)^(1 /3)), x)
Not integrable
Time = 7.26 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.29 \[ \int \frac {1-\sqrt [3]{1-3 x+3 x^3-9 x^4+3 x^6-9 x^7+x^9-3 x^{10}}}{x^2+\sqrt [3]{1-3 x+3 x^3-9 x^4+3 x^6-9 x^7+x^9-3 x^{10}}} \, dx=-\int \frac {{\left (-3\,x^{10}+x^9-9\,x^7+3\,x^6-9\,x^4+3\,x^3-3\,x+1\right )}^{1/3}-1}{{\left (-3\,x^{10}+x^9-9\,x^7+3\,x^6-9\,x^4+3\,x^3-3\,x+1\right )}^{1/3}+x^2} \,d x \]
int(-((3*x^3 - 3*x - 9*x^4 + 3*x^6 - 9*x^7 + x^9 - 3*x^10 + 1)^(1/3) - 1)/ ((3*x^3 - 3*x - 9*x^4 + 3*x^6 - 9*x^7 + x^9 - 3*x^10 + 1)^(1/3) + x^2),x)