Integrand size = 39, antiderivative size = 119 \[ \int \frac {\left (2+x+x^2\right ) \sqrt [3]{-1+x^3}}{x \left (-1+x^2\right )^2 \left (-3-2 x+x^2+x^3\right )} \, dx=\frac {\sqrt [3]{-1+x^3}}{-1+x^2}-\frac {\arctan \left (\frac {\sqrt {3} \sqrt [3]{-1+x^3}}{-2+2 x^2+\sqrt [3]{-1+x^3}}\right )}{\sqrt {3}}+\frac {1}{3} \log \left (1-x^2+\sqrt [3]{-1+x^3}\right )-\frac {1}{6} \log \left (1-2 x^2+x^4+\left (-1+x^2\right ) \sqrt [3]{-1+x^3}+\left (-1+x^3\right )^{2/3}\right ) \] Output:
(x^3-1)^(1/3)/(x^2-1)-1/3*arctan(3^(1/2)*(x^3-1)^(1/3)/(-2+2*x^2+(x^3-1)^( 1/3)))*3^(1/2)+1/3*ln(1-x^2+(x^3-1)^(1/3))-1/6*ln(1-2*x^2+x^4+(x^2-1)*(x^3 -1)^(1/3)+(x^3-1)^(2/3))
Time = 6.00 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.00 \[ \int \frac {\left (2+x+x^2\right ) \sqrt [3]{-1+x^3}}{x \left (-1+x^2\right )^2 \left (-3-2 x+x^2+x^3\right )} \, dx=\frac {\sqrt [3]{-1+x^3}}{-1+x^2}-\frac {\arctan \left (\frac {\sqrt {3} \sqrt [3]{-1+x^3}}{-2+2 x^2+\sqrt [3]{-1+x^3}}\right )}{\sqrt {3}}+\frac {1}{3} \log \left (1-x^2+\sqrt [3]{-1+x^3}\right )-\frac {1}{6} \log \left (1-2 x^2+x^4+\left (-1+x^2\right ) \sqrt [3]{-1+x^3}+\left (-1+x^3\right )^{2/3}\right ) \] Input:
Integrate[((2 + x + x^2)*(-1 + x^3)^(1/3))/(x*(-1 + x^2)^2*(-3 - 2*x + x^2 + x^3)),x]
Output:
(-1 + x^3)^(1/3)/(-1 + x^2) - ArcTan[(Sqrt[3]*(-1 + x^3)^(1/3))/(-2 + 2*x^ 2 + (-1 + x^3)^(1/3))]/Sqrt[3] + Log[1 - x^2 + (-1 + x^3)^(1/3)]/3 - Log[1 - 2*x^2 + x^4 + (-1 + x^2)*(-1 + x^3)^(1/3) + (-1 + x^3)^(2/3)]/6
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.
| \(\displaystyle \int \frac {\left (x^2+x+2\right ) \sqrt [3]{x^3-1}}{x \left (x^2-1\right )^2 \left (x^3+x^2-2 x-3\right )} \, dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {\sqrt [3]{x^3-1}}{12 (x-1)}-\frac {2 \sqrt [3]{x^3-1}}{3 x}+\frac {\sqrt [3]{x^3-1}}{4 (x+1)}-\frac {\sqrt [3]{x^3-1}}{3 (x-1)^2}+\frac {\sqrt [3]{x^3-1}}{2 (x+1)^2}+\frac {\sqrt [3]{x^3-1} \left (x^2+x+2\right )}{3 \left (x^3+x^2-2 x-3\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {2}{3} \int \frac {\sqrt [3]{x^3-1}}{x^3+x^2-2 x-3}dx+\frac {1}{3} \int \frac {x \sqrt [3]{x^3-1}}{x^3+x^2-2 x-3}dx+\frac {1}{3} \int \frac {x^2 \sqrt [3]{x^3-1}}{x^3+x^2-2 x-3}dx-\frac {2 \arctan \left (\frac {1-2 \sqrt [3]{x^3-1}}{\sqrt {3}}\right )}{3 \sqrt {3}}-\frac {\left (1-x^3\right )^{2/3} x \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {2}{3},\frac {4}{3},x^3\right )}{6 \left (x^3-1\right )^{2/3}}+\frac {\left (1-x^3\right )^{2/3} x \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {5}{3},\frac {4}{3},x^3\right )}{3 \left (x^3-1\right )^{2/3}}+\frac {\left (1-x^3\right )^{2/3} x^4 \operatorname {Hypergeometric2F1}\left (\frac {4}{3},\frac {5}{3},\frac {7}{3},x^3\right )}{6 \left (x^3-1\right )^{2/3}}+\frac {\sqrt [3]{x^3-1} x}{2 \left (x^3+1\right )}-\frac {1}{3} \sqrt [3]{x^3-1}+\frac {1}{2 \left (x^3-1\right )^{2/3}}-\frac {\sqrt [3]{x^3-1}}{2 \left (x^3+1\right )}+\frac {1}{3} \log \left (\sqrt [3]{x^3-1}+1\right )+\frac {x^2}{2 \left (x^3-1\right )^{2/3}}-\frac {\sqrt [3]{x^3-1} x^2}{2 \left (x^3+1\right )}-\frac {\log (x)}{3}\) |
Input:
Int[((2 + x + x^2)*(-1 + x^3)^(1/3))/(x*(-1 + x^2)^2*(-3 - 2*x + x^2 + x^3 )),x]
Output:
$Aborted
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 7.25 (sec) , antiderivative size = 887, normalized size of antiderivative = 7.45
| method | result | size |
| trager | \(\text {Expression too large to display}\) | \(887\) |
| risch | \(\text {Expression too large to display}\) | \(1795\) |
Input:
int((x^2+x+2)*(x^3-1)^(1/3)/x/(x^2-1)^2/(x^3+x^2-2*x-3),x,method=_RETURNVE RBOSE)
Output:
(x^3-1)^(1/3)/(x^2-1)+1/3*ln((-1223450633004980-1223450633004980*x-5415273 29362860*x^5+23617711362609732*x^2*(x^3-1)^(1/3)-541527329362860*x^4+10830 54658725720*x^3-23617711362609732*(x^3-1)^(1/3)+401131355083600*x^2-337194 21708885792*(x^3-1)^(2/3)-45457696558242270*RootOf(81*_Z^2+18*_Z+4)*(x^3-1 )^(2/3)-151737397689986064*RootOf(81*_Z^2+18*_Z+4)*(x^3-1)^(1/3)+802670784 40654371*RootOf(81*_Z^2+18*_Z+4)^2*x^5-68168571064875648*RootOf(81*_Z^2+18 *_Z+4)*x^5-68168571064875648*RootOf(81*_Z^2+18*_Z+4)*x^4+13633714212975129 6*RootOf(81*_Z^2+18*_Z+4)*x^3+47262787656118470*RootOf(81*_Z^2+18*_Z+4)*x^ 2-157242925538508474*RootOf(81*_Z^2+18*_Z+4)*x+151737397689986064*RootOf(8 1*_Z^2+18*_Z+4)*(x^3-1)^(1/3)*x^3-45457696558242270*RootOf(81*_Z^2+18*_Z+4 )*(x^3-1)^(2/3)*x+151737397689986064*RootOf(81*_Z^2+18*_Z+4)*(x^3-1)^(1/3) *x^2-151737397689986064*RootOf(81*_Z^2+18*_Z+4)*(x^3-1)^(1/3)*x-2293345098 30441060*RootOf(81*_Z^2+18*_Z+4)^2-157242925538508474*RootOf(81*_Z^2+18*_Z +4)-23617711362609732*(x^3-1)^(1/3)*x-33719421708885792*x*(x^3-1)^(2/3)+23 617711362609732*(x^3-1)^(1/3)*x^3-470135745152404173*RootOf(81*_Z^2+18*_Z+ 4)^2*x^2-229334509830441060*RootOf(81*_Z^2+18*_Z+4)^2*x+80267078440654371* RootOf(81*_Z^2+18*_Z+4)^2*x^4-160534156881308742*RootOf(81*_Z^2+18*_Z+4)^2 *x^3)/(x^3+x^2-2*x-3)/x^2)+3/2*RootOf(81*_Z^2+18*_Z+4)*ln(-(-4111596389606 90-411159638960690*x-340961651821060*x^5-16859710854442896*x^2*(x^3-1)^(1/ 3)-340961651821060*x^4+681923303642120*x^3+16859710854442896*(x^3-1)^(1...
Exception generated. \[ \int \frac {\left (2+x+x^2\right ) \sqrt [3]{-1+x^3}}{x \left (-1+x^2\right )^2 \left (-3-2 x+x^2+x^3\right )} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((x^2+x+2)*(x^3-1)^(1/3)/x/(x^2-1)^2/(x^3+x^2-2*x-3),x, algorithm ="fricas")
Output:
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (residue poly has multiple non-linear fac tors)
Timed out. \[ \int \frac {\left (2+x+x^2\right ) \sqrt [3]{-1+x^3}}{x \left (-1+x^2\right )^2 \left (-3-2 x+x^2+x^3\right )} \, dx=\text {Timed out} \] Input:
integrate((x**2+x+2)*(x**3-1)**(1/3)/x/(x**2-1)**2/(x**3+x**2-2*x-3),x)
Output:
Timed out
\[ \int \frac {\left (2+x+x^2\right ) \sqrt [3]{-1+x^3}}{x \left (-1+x^2\right )^2 \left (-3-2 x+x^2+x^3\right )} \, dx=\int { \frac {{\left (x^{3} - 1\right )}^{\frac {1}{3}} {\left (x^{2} + x + 2\right )}}{{\left (x^{3} + x^{2} - 2 \, x - 3\right )} {\left (x^{2} - 1\right )}^{2} x} \,d x } \] Input:
integrate((x^2+x+2)*(x^3-1)^(1/3)/x/(x^2-1)^2/(x^3+x^2-2*x-3),x, algorithm ="maxima")
Output:
integrate((x^3 - 1)^(1/3)*(x^2 + x + 2)/((x^3 + x^2 - 2*x - 3)*(x^2 - 1)^2 *x), x)
\[ \int \frac {\left (2+x+x^2\right ) \sqrt [3]{-1+x^3}}{x \left (-1+x^2\right )^2 \left (-3-2 x+x^2+x^3\right )} \, dx=\int { \frac {{\left (x^{3} - 1\right )}^{\frac {1}{3}} {\left (x^{2} + x + 2\right )}}{{\left (x^{3} + x^{2} - 2 \, x - 3\right )} {\left (x^{2} - 1\right )}^{2} x} \,d x } \] Input:
integrate((x^2+x+2)*(x^3-1)^(1/3)/x/(x^2-1)^2/(x^3+x^2-2*x-3),x, algorithm ="giac")
Output:
integrate((x^3 - 1)^(1/3)*(x^2 + x + 2)/((x^3 + x^2 - 2*x - 3)*(x^2 - 1)^2 *x), x)
Timed out. \[ \int \frac {\left (2+x+x^2\right ) \sqrt [3]{-1+x^3}}{x \left (-1+x^2\right )^2 \left (-3-2 x+x^2+x^3\right )} \, dx=\int -\frac {{\left (x^3-1\right )}^{1/3}\,\left (x^2+x+2\right )}{x\,{\left (x^2-1\right )}^2\,\left (-x^3-x^2+2\,x+3\right )} \,d x \] Input:
int(-((x^3 - 1)^(1/3)*(x + x^2 + 2))/(x*(x^2 - 1)^2*(2*x - x^2 - x^3 + 3)) ,x)
Output:
int(-((x^3 - 1)^(1/3)*(x + x^2 + 2))/(x*(x^2 - 1)^2*(2*x - x^2 - x^3 + 3)) , x)
\[ \int \frac {\left (2+x+x^2\right ) \sqrt [3]{-1+x^3}}{x \left (-1+x^2\right )^2 \left (-3-2 x+x^2+x^3\right )} \, dx=2 \left (\int \frac {\left (x^{3}-1\right )^{\frac {1}{3}}}{x^{8}+x^{7}-4 x^{6}-5 x^{5}+5 x^{4}+7 x^{3}-2 x^{2}-3 x}d x \right )+\int \frac {\left (x^{3}-1\right )^{\frac {1}{3}}}{x^{7}+x^{6}-4 x^{5}-5 x^{4}+5 x^{3}+7 x^{2}-2 x -3}d x +\int \frac {\left (x^{3}-1\right )^{\frac {1}{3}} x}{x^{7}+x^{6}-4 x^{5}-5 x^{4}+5 x^{3}+7 x^{2}-2 x -3}d x \] Input:
int((x^2+x+2)*(x^3-1)^(1/3)/x/(x^2-1)^2/(x^3+x^2-2*x-3),x)
Output:
2*int((x**3 - 1)**(1/3)/(x**8 + x**7 - 4*x**6 - 5*x**5 + 5*x**4 + 7*x**3 - 2*x**2 - 3*x),x) + int((x**3 - 1)**(1/3)/(x**7 + x**6 - 4*x**5 - 5*x**4 + 5*x**3 + 7*x**2 - 2*x - 3),x) + int(((x**3 - 1)**(1/3)*x)/(x**7 + x**6 - 4*x**5 - 5*x**4 + 5*x**3 + 7*x**2 - 2*x - 3),x)