\(\int \frac {2-x+x^2}{\sqrt [3]{-1+x^2} (3+4 x+x^2)} \, dx\) [1714]

Optimal result

Integrand size = 28, antiderivative size = 116 \[ \int \frac {2-x+x^2}{\sqrt [3]{-1+x^2} \left (3+4 x+x^2\right )} \, dx=\frac {3 \left (-1+x^2\right )^{2/3}}{1+x}-\frac {7}{4} \sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{-1+x^2}}{1-x+\sqrt [3]{-1+x^2}}\right )-\frac {7}{4} \log \left (-1+x+2 \sqrt [3]{-1+x^2}\right )+\frac {7}{8} \log \left (1-2 x+x^2+(2-2 x) \sqrt [3]{-1+x^2}+4 \left (-1+x^2\right )^{2/3}\right ) \]

[Out]

3*(x^2-1)^(2/3)/(1+x)-7/4*3^(1/2)*arctan(3^(1/2)*(x^2-1)^(1/3)/(1-x+(x^2-1)^(1/3)))-7/4*ln(-1+x+2*(x^2-1)^(1/3
))+7/8*ln(1-2*x+x^2+(2-2*x)*(x^2-1)^(1/3)+4*(x^2-1)^(2/3))

Rubi [F]

\[ \int \frac {2-x+x^2}{\sqrt [3]{-1+x^2} \left (3+4 x+x^2\right )} \, dx=\int \frac {2-x+x^2}{\sqrt [3]{-1+x^2} \left (3+4 x+x^2\right )} \, dx \]

[In]

Int[(2 - x + x^2)/((-1 + x^2)^(1/3)*(3 + 4*x + x^2)),x]

[Out]

(3*x)/(1 + Sqrt[3] + (-1 + x^2)^(1/3)) - (3*3^(1/4)*Sqrt[2 - Sqrt[3]]*(1 + (-1 + x^2)^(1/3))*Sqrt[(1 - (-1 + x
^2)^(1/3) + (-1 + x^2)^(2/3))/(1 + Sqrt[3] + (-1 + x^2)^(1/3))^2]*EllipticE[ArcSin[(1 - Sqrt[3] + (-1 + x^2)^(
1/3))/(1 + Sqrt[3] + (-1 + x^2)^(1/3))], -7 - 4*Sqrt[3]])/(2*x*Sqrt[(1 + (-1 + x^2)^(1/3))/(1 + Sqrt[3] + (-1
+ x^2)^(1/3))^2]) + (Sqrt[2]*3^(3/4)*(1 + (-1 + x^2)^(1/3))*Sqrt[(1 - (-1 + x^2)^(1/3) + (-1 + x^2)^(2/3))/(1
+ Sqrt[3] + (-1 + x^2)^(1/3))^2]*EllipticF[ArcSin[(1 - Sqrt[3] + (-1 + x^2)^(1/3))/(1 + Sqrt[3] + (-1 + x^2)^(
1/3))], -7 - 4*Sqrt[3]])/(x*Sqrt[(1 + (-1 + x^2)^(1/3))/(1 + Sqrt[3] + (-1 + x^2)^(1/3))^2]) - Defer[Int][(1 +
5*x)/((-1 + x^2)^(1/3)*(3 + 4*x + x^2)), x]

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {1}{\sqrt [3]{-1+x^2}}-\frac {1+5 x}{\sqrt [3]{-1+x^2} \left (3+4 x+x^2\right )}\right ) \, dx \\ & = \int \frac {1}{\sqrt [3]{-1+x^2}} \, dx-\int \frac {1+5 x}{\sqrt [3]{-1+x^2} \left (3+4 x+x^2\right )} \, dx \\ & = \frac {\left (3 \sqrt {x^2}\right ) \text {Subst}\left (\int \frac {x}{\sqrt {1+x^3}} \, dx,x,\sqrt [3]{-1+x^2}\right )}{2 x}-\int \frac {1+5 x}{\sqrt [3]{-1+x^2} \left (3+4 x+x^2\right )} \, dx \\ & = \frac {\left (3 \sqrt {x^2}\right ) \text {Subst}\left (\int \frac {1-\sqrt {3}+x}{\sqrt {1+x^3}} \, dx,x,\sqrt [3]{-1+x^2}\right )}{2 x}+\frac {\left (3 \left (-1+\sqrt {3}\right ) \sqrt {x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1+x^3}} \, dx,x,\sqrt [3]{-1+x^2}\right )}{2 x}-\int \frac {1+5 x}{\sqrt [3]{-1+x^2} \left (3+4 x+x^2\right )} \, dx \\ & = \frac {3 x}{1+\sqrt {3}+\sqrt [3]{-1+x^2}}-\frac {3 \sqrt [4]{3} \sqrt {2-\sqrt {3}} \left (1+\sqrt [3]{-1+x^2}\right ) \sqrt {\frac {1-\sqrt [3]{-1+x^2}+\left (-1+x^2\right )^{2/3}}{\left (1+\sqrt {3}+\sqrt [3]{-1+x^2}\right )^2}} E\left (\arcsin \left (\frac {1-\sqrt {3}+\sqrt [3]{-1+x^2}}{1+\sqrt {3}+\sqrt [3]{-1+x^2}}\right )|-7-4 \sqrt {3}\right )}{2 x \sqrt {\frac {1+\sqrt [3]{-1+x^2}}{\left (1+\sqrt {3}+\sqrt [3]{-1+x^2}\right )^2}}}+\frac {\sqrt {2} 3^{3/4} \left (1+\sqrt [3]{-1+x^2}\right ) \sqrt {\frac {1-\sqrt [3]{-1+x^2}+\left (-1+x^2\right )^{2/3}}{\left (1+\sqrt {3}+\sqrt [3]{-1+x^2}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {1-\sqrt {3}+\sqrt [3]{-1+x^2}}{1+\sqrt {3}+\sqrt [3]{-1+x^2}}\right ),-7-4 \sqrt {3}\right )}{x \sqrt {\frac {1+\sqrt [3]{-1+x^2}}{\left (1+\sqrt {3}+\sqrt [3]{-1+x^2}\right )^2}}}-\int \frac {1+5 x}{\sqrt [3]{-1+x^2} \left (3+4 x+x^2\right )} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.28 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.00 \[ \int \frac {2-x+x^2}{\sqrt [3]{-1+x^2} \left (3+4 x+x^2\right )} \, dx=\frac {3 \left (-1+x^2\right )^{2/3}}{1+x}-\frac {7}{4} \sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{-1+x^2}}{1-x+\sqrt [3]{-1+x^2}}\right )-\frac {7}{4} \log \left (-1+x+2 \sqrt [3]{-1+x^2}\right )+\frac {7}{8} \log \left (1-2 x+x^2+(2-2 x) \sqrt [3]{-1+x^2}+4 \left (-1+x^2\right )^{2/3}\right ) \]

[In]

Integrate[(2 - x + x^2)/((-1 + x^2)^(1/3)*(3 + 4*x + x^2)),x]

[Out]

(3*(-1 + x^2)^(2/3))/(1 + x) - (7*Sqrt[3]*ArcTan[(Sqrt[3]*(-1 + x^2)^(1/3))/(1 - x + (-1 + x^2)^(1/3))])/4 - (
7*Log[-1 + x + 2*(-1 + x^2)^(1/3)])/4 + (7*Log[1 - 2*x + x^2 + (2 - 2*x)*(-1 + x^2)^(1/3) + 4*(-1 + x^2)^(2/3)
])/8

Maple [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 2.06 (sec) , antiderivative size = 396, normalized size of antiderivative = 3.41

method	result	size
risch	\(\frac {-3+3 x}{\left (x^{2}-1\right )^{\frac {1}{3}}}-\frac {7 \ln \left (-\frac {448 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right )^{2} x^{2}+864 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right ) \left (x^{2}-1\right )^{\frac {2}{3}}-516 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right ) \left (x^{2}-1\right )^{\frac {1}{3}} x -1344 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right )^{2} x -468 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right ) x^{2}-948 \left (x^{2}-1\right )^{\frac {2}{3}}+516 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right ) \left (x^{2}-1\right )^{\frac {1}{3}}-216 x \left (x^{2}-1\right )^{\frac {1}{3}}+3000 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right ) x +119 x^{2}+216 \left (x^{2}-1\right )^{\frac {1}{3}}+1596 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right )-1326 x -969}{\left (3+x \right )^{2}}\right )}{4}+\frac {7 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right ) \ln \left (\frac {272 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right )^{2} x^{2}+432 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right ) \left (x^{2}-1\right )^{\frac {2}{3}}+474 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right ) \left (x^{2}-1\right )^{\frac {1}{3}} x -816 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right )^{2} x -129 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right ) x^{2}+258 \left (x^{2}-1\right )^{\frac {2}{3}}-474 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right ) \left (x^{2}-1\right )^{\frac {1}{3}}-108 x \left (x^{2}-1\right )^{\frac {1}{3}}-582 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right ) x +7 x^{2}+108 \left (x^{2}-1\right )^{\frac {1}{3}}-969 \operatorname {RootOf}\left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right )+378 x +399}{\left (3+x \right )^{2}}\right )}{2}\)	\(396\)
trager	\(\frac {3 \left (x^{2}-1\right )^{\frac {2}{3}}}{1+x}+\frac {7 \ln \left (\frac {-864 \operatorname {RootOf}\left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right )^{2} x^{2}+2592 \operatorname {RootOf}\left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right ) \left (x^{2}-1\right )^{\frac {2}{3}}-1296 \operatorname {RootOf}\left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right ) \left (x^{2}-1\right )^{\frac {1}{3}} x +2592 \operatorname {RootOf}\left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right )^{2} x +834 \operatorname {RootOf}\left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right ) x^{2}+516 \left (x^{2}-1\right )^{\frac {2}{3}}+1296 \operatorname {RootOf}\left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right ) \left (x^{2}-1\right )^{\frac {1}{3}}-258 x \left (x^{2}-1\right )^{\frac {1}{3}}-1476 \operatorname {RootOf}\left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right ) x -17 x^{2}+258 \left (x^{2}-1\right )^{\frac {1}{3}}+1026 \operatorname {RootOf}\left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right )-918 x -969}{\left (3+x \right )^{2}}\right )}{4}-\frac {21 \ln \left (\frac {-864 \operatorname {RootOf}\left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right )^{2} x^{2}+2592 \operatorname {RootOf}\left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right ) \left (x^{2}-1\right )^{\frac {2}{3}}-1296 \operatorname {RootOf}\left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right ) \left (x^{2}-1\right )^{\frac {1}{3}} x +2592 \operatorname {RootOf}\left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right )^{2} x +834 \operatorname {RootOf}\left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right ) x^{2}+516 \left (x^{2}-1\right )^{\frac {2}{3}}+1296 \operatorname {RootOf}\left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right ) \left (x^{2}-1\right )^{\frac {1}{3}}-258 x \left (x^{2}-1\right )^{\frac {1}{3}}-1476 \operatorname {RootOf}\left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right ) x -17 x^{2}+258 \left (x^{2}-1\right )^{\frac {1}{3}}+1026 \operatorname {RootOf}\left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right )-918 x -969}{\left (3+x \right )^{2}}\right ) \operatorname {RootOf}\left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right )}{2}+\frac {21 \operatorname {RootOf}\left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right ) \ln \left (-\frac {432 \operatorname {RootOf}\left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right )^{2} x^{2}+1296 \operatorname {RootOf}\left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right ) \left (x^{2}-1\right )^{\frac {2}{3}}-648 \operatorname {RootOf}\left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right ) \left (x^{2}-1\right )^{\frac {1}{3}} x -1296 \operatorname {RootOf}\left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right )^{2} x +273 \operatorname {RootOf}\left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right ) x^{2}-474 \left (x^{2}-1\right )^{\frac {2}{3}}+648 \operatorname {RootOf}\left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right ) \left (x^{2}-1\right )^{\frac {1}{3}}+237 x \left (x^{2}-1\right )^{\frac {1}{3}}-306 \operatorname {RootOf}\left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right ) x -49 x^{2}-237 \left (x^{2}-1\right )^{\frac {1}{3}}+513 \operatorname {RootOf}\left (36 \textit {\_Z}^{2}-6 \textit {\_Z} +1\right )+546 x +399}{\left (3+x \right )^{2}}\right )}{2}\)	\(594\)

[In]

int((x^2-x+2)/(x^2-1)^(1/3)/(x^2+4*x+3),x,method=_RETURNVERBOSE)

[Out]

3*(-1+x)/(x^2-1)^(1/3)-7/4*ln(-(448*RootOf(4*_Z^2-2*_Z+1)^2*x^2+864*RootOf(4*_Z^2-2*_Z+1)*(x^2-1)^(2/3)-516*Ro
otOf(4*_Z^2-2*_Z+1)*(x^2-1)^(1/3)*x-1344*RootOf(4*_Z^2-2*_Z+1)^2*x-468*RootOf(4*_Z^2-2*_Z+1)*x^2-948*(x^2-1)^(
2/3)+516*RootOf(4*_Z^2-2*_Z+1)*(x^2-1)^(1/3)-216*x*(x^2-1)^(1/3)+3000*RootOf(4*_Z^2-2*_Z+1)*x+119*x^2+216*(x^2
-1)^(1/3)+1596*RootOf(4*_Z^2-2*_Z+1)-1326*x-969)/(3+x)^2)+7/2*RootOf(4*_Z^2-2*_Z+1)*ln((272*RootOf(4*_Z^2-2*_Z
+1)^2*x^2+432*RootOf(4*_Z^2-2*_Z+1)*(x^2-1)^(2/3)+474*RootOf(4*_Z^2-2*_Z+1)*(x^2-1)^(1/3)*x-816*RootOf(4*_Z^2-
2*_Z+1)^2*x-129*RootOf(4*_Z^2-2*_Z+1)*x^2+258*(x^2-1)^(2/3)-474*RootOf(4*_Z^2-2*_Z+1)*(x^2-1)^(1/3)-108*x*(x^2
-1)^(1/3)-582*RootOf(4*_Z^2-2*_Z+1)*x+7*x^2+108*(x^2-1)^(1/3)-969*RootOf(4*_Z^2-2*_Z+1)+378*x+399)/(3+x)^2)

Fricas [A] (verification not implemented)

none

Time = 0.53 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.09 \[ \int \frac {2-x+x^2}{\sqrt [3]{-1+x^2} \left (3+4 x+x^2\right )} \, dx=\frac {14 \, \sqrt {3} {\left (x + 1\right )} \arctan \left (\frac {286273 \, \sqrt {3} {\left (x^{2} - 1\right )}^{\frac {1}{3}} {\left (x - 1\right )} + \sqrt {3} {\left (66978 \, x^{2} + 434719 \, x + 635653\right )} + 539695 \, \sqrt {3} {\left (x^{2} - 1\right )}^{\frac {2}{3}}}{226981 \, x^{2} - 1974837 \, x - 1293894}\right ) - 7 \, {\left (x + 1\right )} \log \left (\frac {x^{2} + 6 \, {\left (x^{2} - 1\right )}^{\frac {1}{3}} {\left (x - 1\right )} + 6 \, x + 12 \, {\left (x^{2} - 1\right )}^{\frac {2}{3}} + 9}{x^{2} + 6 \, x + 9}\right ) + 24 \, {\left (x^{2} - 1\right )}^{\frac {2}{3}}}{8 \, {\left (x + 1\right )}} \]

[In]

integrate((x^2-x+2)/(x^2-1)^(1/3)/(x^2+4*x+3),x, algorithm="fricas")

[Out]

1/8*(14*sqrt(3)*(x + 1)*arctan((286273*sqrt(3)*(x^2 - 1)^(1/3)*(x - 1) + sqrt(3)*(66978*x^2 + 434719*x + 63565
3) + 539695*sqrt(3)*(x^2 - 1)^(2/3))/(226981*x^2 - 1974837*x - 1293894)) - 7*(x + 1)*log((x^2 + 6*(x^2 - 1)^(1
/3)*(x - 1) + 6*x + 12*(x^2 - 1)^(2/3) + 9)/(x^2 + 6*x + 9)) + 24*(x^2 - 1)^(2/3))/(x + 1)

Sympy [F]

\[ \int \frac {2-x+x^2}{\sqrt [3]{-1+x^2} \left (3+4 x+x^2\right )} \, dx=\int \frac {x^{2} - x + 2}{\sqrt [3]{\left (x - 1\right ) \left (x + 1\right )} \left (x + 1\right ) \left (x + 3\right )}\, dx \]

[In]

integrate((x**2-x+2)/(x**2-1)**(1/3)/(x**2+4*x+3),x)

[Out]

Integral((x**2 - x + 2)/(((x - 1)*(x + 1))**(1/3)*(x + 1)*(x + 3)), x)

Maxima [F]

\[ \int \frac {2-x+x^2}{\sqrt [3]{-1+x^2} \left (3+4 x+x^2\right )} \, dx=\int { \frac {x^{2} - x + 2}{{\left (x^{2} + 4 \, x + 3\right )} {\left (x^{2} - 1\right )}^{\frac {1}{3}}} \,d x } \]

[In]

integrate((x^2-x+2)/(x^2-1)^(1/3)/(x^2+4*x+3),x, algorithm="maxima")

[Out]

integrate((x^2 - x + 2)/((x^2 + 4*x + 3)*(x^2 - 1)^(1/3)), x)

Giac [F]

\[ \int \frac {2-x+x^2}{\sqrt [3]{-1+x^2} \left (3+4 x+x^2\right )} \, dx=\int { \frac {x^{2} - x + 2}{{\left (x^{2} + 4 \, x + 3\right )} {\left (x^{2} - 1\right )}^{\frac {1}{3}}} \,d x } \]

[In]

integrate((x^2-x+2)/(x^2-1)^(1/3)/(x^2+4*x+3),x, algorithm="giac")

[Out]

integrate((x^2 - x + 2)/((x^2 + 4*x + 3)*(x^2 - 1)^(1/3)), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {2-x+x^2}{\sqrt [3]{-1+x^2} \left (3+4 x+x^2\right )} \, dx=\int \frac {x^2-x+2}{{\left (x^2-1\right )}^{1/3}\,\left (x^2+4\,x+3\right )} \,d x \]

[In]

int((x^2 - x + 2)/((x^2 - 1)^(1/3)*(4*x + x^2 + 3)),x)

[Out]

int((x^2 - x + 2)/((x^2 - 1)^(1/3)*(4*x + x^2 + 3)), x)