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

Optimal. Leaf size=115 \[ \log \left (\sqrt [3]{x^2-x+1}+x-1\right )-\frac {1}{2} \log \left (x^2+\left (x^2-x+1\right )^{2/3}+(1-x) \sqrt [3]{x^2-x+1}-2 x+1\right )-\sqrt {3} \tan ^{-1}\left (\frac {\frac {\sqrt [3]{x^2-x+1}}{\sqrt {3}}-\frac {2 x}{\sqrt {3}}+\frac {2}{\sqrt {3}}}{\sqrt [3]{x^2-x+1}}\right ) \]

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Rubi [F]  time = 0.47, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {2+x^2}{x \left (2-2 x+x^2\right ) \sqrt [3]{1-x+x^2}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

(-3*(-((1 - I*Sqrt[3] - 2*x)/x))^(1/3)*(-((1 + I*Sqrt[3] - 2*x)/x))^(1/3)*AppellF1[2/3, 1/3, 1/3, 5/3, (1 - I*
Sqrt[3])/(2*x), (1 + I*Sqrt[3])/(2*x)])/(2*2^(2/3)*(1 - x + x^2)^(1/3)) + 2*Defer[Int][1/((2 - 2*x + x^2)*(1 -
x + x^2)^(1/3)), x]

Rubi steps

\begin {align*} \int \frac {2+x^2}{x \left (2-2 x+x^2\right ) \sqrt [3]{1-x+x^2}} \, dx &=\int \left (\frac {1}{x \sqrt [3]{1-x+x^2}}+\frac {2}{\left (2-2 x+x^2\right ) \sqrt [3]{1-x+x^2}}\right ) \, dx\\ &=2 \int \frac {1}{\left (2-2 x+x^2\right ) \sqrt [3]{1-x+x^2}} \, dx+\int \frac {1}{x \sqrt [3]{1-x+x^2}} \, dx\\ &=2 \int \frac {1}{\left (2-2 x+x^2\right ) \sqrt [3]{1-x+x^2}} \, dx-\frac {\left (\sqrt [3]{\frac {-1-i \sqrt {3}+2 x}{x}} \sqrt [3]{\frac {-1+i \sqrt {3}+2 x}{x}}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{x} \sqrt [3]{1-\frac {1}{2} \left (1-i \sqrt {3}\right ) x} \sqrt [3]{1-\frac {1}{2} \left (1+i \sqrt {3}\right ) x}} \, dx,x,\frac {1}{x}\right )}{2^{2/3} \left (\frac {1}{x}\right )^{2/3} \sqrt [3]{1-x+x^2}}\\ &=-\frac {3 \sqrt [3]{-\frac {1-i \sqrt {3}-2 x}{x}} \sqrt [3]{-\frac {1+i \sqrt {3}-2 x}{x}} F_1\left (\frac {2}{3};\frac {1}{3},\frac {1}{3};\frac {5}{3};\frac {1-i \sqrt {3}}{2 x},\frac {1+i \sqrt {3}}{2 x}\right )}{2\ 2^{2/3} \sqrt [3]{1-x+x^2}}+2 \int \frac {1}{\left (2-2 x+x^2\right ) \sqrt [3]{1-x+x^2}} \, dx\\ \end {align*}

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Mathematica [F]  time = 0.23, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {2+x^2}{x \left (2-2 x+x^2\right ) \sqrt [3]{1-x+x^2}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 0.12, size = 115, normalized size = 1.00 \begin {gather*} -\sqrt {3} \tan ^{-1}\left (\frac {\frac {2}{\sqrt {3}}-\frac {2 x}{\sqrt {3}}+\frac {\sqrt [3]{1-x+x^2}}{\sqrt {3}}}{\sqrt [3]{1-x+x^2}}\right )+\log \left (-1+x+\sqrt [3]{1-x+x^2}\right )-\frac {1}{2} \log \left (1-2 x+x^2+(1-x) \sqrt [3]{1-x+x^2}+\left (1-x+x^2\right )^{2/3}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 1.09, size = 147, normalized size = 1.28 \begin {gather*} -\sqrt {3} \arctan \left (\frac {4 \, \sqrt {3} {\left (x^{2} - x + 1\right )}^{\frac {2}{3}} {\left (x - 1\right )} + 2 \, \sqrt {3} {\left (x^{2} - x + 1\right )}^{\frac {1}{3}} {\left (x^{2} - 2 \, x + 1\right )} + \sqrt {3} {\left (x^{3} - 3 \, x^{2} + 3 \, x - 1\right )}}{x^{3} - 11 \, x^{2} + 11 \, x - 9}\right ) + \frac {1}{2} \, \log \left (\frac {x^{3} - 2 \, x^{2} + 3 \, {\left (x^{2} - x + 1\right )}^{\frac {2}{3}} {\left (x - 1\right )} + 3 \, {\left (x^{2} - x + 1\right )}^{\frac {1}{3}} {\left (x^{2} - 2 \, x + 1\right )} + 2 \, x}{x^{3} - 2 \, x^{2} + 2 \, x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt(3)*arctan((4*sqrt(3)*(x^2 - x + 1)^(2/3)*(x - 1) + 2*sqrt(3)*(x^2 - x + 1)^(1/3)*(x^2 - 2*x + 1) + sqrt(
3)*(x^3 - 3*x^2 + 3*x - 1))/(x^3 - 11*x^2 + 11*x - 9)) + 1/2*log((x^3 - 2*x^2 + 3*(x^2 - x + 1)^(2/3)*(x - 1)
+ 3*(x^2 - x + 1)^(1/3)*(x^2 - 2*x + 1) + 2*x)/(x^3 - 2*x^2 + 2*x))

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{2} + 2}{{\left (x^{2} - x + 1\right )}^{\frac {1}{3}} {\left (x^{2} - 2 \, x + 2\right )} x}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C]  time = 1.85, size = 492, normalized size = 4.28

method	result	size
trager	\(\ln \left (\frac {-\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{3}+\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{2}-x +1\right )^{\frac {2}{3}} x -\left (x^{2}-x +1\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{2}+3 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{2}-2 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{3}-\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \left (x^{2}-x +1\right )^{\frac {2}{3}}+2 x \left (x^{2}-x +1\right )^{\frac {2}{3}}+2 \left (x^{2}-x +1\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x +\left (x^{2}-x +1\right )^{\frac {1}{3}} x^{2}-3 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x +7 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{2}-x^{3}-2 \left (x^{2}-x +1\right )^{\frac {2}{3}}-\left (x^{2}-x +1\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-2 x \left (x^{2}-x +1\right )^{\frac {1}{3}}+\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2}-7 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x +4 x^{2}+\left (x^{2}-x +1\right )^{\frac {1}{3}}+3 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-4 x +2}{x \left (x^{2}-2 x +2\right )}\right )+\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \ln \left (\frac {2 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{3}-3 \left (x^{2}-x +1\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{2}-6 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{2}+\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{3}+3 x \left (x^{2}-x +1\right )^{\frac {2}{3}}+6 \left (x^{2}-x +1\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x +6 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x -\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{2}-3 \left (x^{2}-x +1\right )^{\frac {2}{3}}-3 \left (x^{2}-x +1\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-2 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2}+\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x +x^{2}+\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-x +1}{x \left (x^{2}-2 x +2\right )}\right )\)	\(492\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

ln((-RootOf(_Z^2+_Z+1)^2*x^3+RootOf(_Z^2+_Z+1)*(x^2-x+1)^(2/3)*x-(x^2-x+1)^(1/3)*RootOf(_Z^2+_Z+1)*x^2+3*RootO
f(_Z^2+_Z+1)^2*x^2-2*RootOf(_Z^2+_Z+1)*x^3-RootOf(_Z^2+_Z+1)*(x^2-x+1)^(2/3)+2*x*(x^2-x+1)^(2/3)+2*(x^2-x+1)^(
1/3)*RootOf(_Z^2+_Z+1)*x+(x^2-x+1)^(1/3)*x^2-3*RootOf(_Z^2+_Z+1)^2*x+7*RootOf(_Z^2+_Z+1)*x^2-x^3-2*(x^2-x+1)^(
2/3)-(x^2-x+1)^(1/3)*RootOf(_Z^2+_Z+1)-2*x*(x^2-x+1)^(1/3)+RootOf(_Z^2+_Z+1)^2-7*RootOf(_Z^2+_Z+1)*x+4*x^2+(x^
2-x+1)^(1/3)+3*RootOf(_Z^2+_Z+1)-4*x+2)/x/(x^2-2*x+2))+RootOf(_Z^2+_Z+1)*ln((2*RootOf(_Z^2+_Z+1)^2*x^3-3*(x^2-
x+1)^(1/3)*RootOf(_Z^2+_Z+1)*x^2-6*RootOf(_Z^2+_Z+1)^2*x^2+RootOf(_Z^2+_Z+1)*x^3+3*x*(x^2-x+1)^(2/3)+6*(x^2-x+
1)^(1/3)*RootOf(_Z^2+_Z+1)*x+6*RootOf(_Z^2+_Z+1)^2*x-RootOf(_Z^2+_Z+1)*x^2-3*(x^2-x+1)^(2/3)-3*(x^2-x+1)^(1/3)
*RootOf(_Z^2+_Z+1)-2*RootOf(_Z^2+_Z+1)^2+RootOf(_Z^2+_Z+1)*x+x^2+RootOf(_Z^2+_Z+1)-x+1)/x/(x^2-2*x+2))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{2} + 2}{{\left (x^{2} - x + 1\right )}^{\frac {1}{3}} {\left (x^{2} - 2 \, x + 2\right )} x}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x^2+2}{x\,{\left (x^2-x+1\right )}^{1/3}\,\left (x^2-2\,x+2\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{2} + 2}{x \left (x^{2} - 2 x + 2\right ) \sqrt [3]{x^{2} - x + 1}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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