3.26.18 \(\int \frac {1+x^2}{(-1+x+x^2) \sqrt [3]{-1+x^6}} \, dx\) [2518]

Optimal. Leaf size=210 \[ -\frac {\text {ArcTan}\left (\frac {\sqrt {3} \sqrt [3]{-1+x^6}}{-2^{2/3}-2^{2/3} x+2^{2/3} x^2+\sqrt [3]{-1+x^6}}\right )}{2^{2/3} \sqrt {3}}-\frac {\log \left (-2^{2/3}-2^{2/3} x+2^{2/3} x^2-2 \sqrt [3]{-1+x^6}\right )}{3\ 2^{2/3}}+\frac {\log \left (\sqrt [3]{2}+2 \sqrt [3]{2} x-\sqrt [3]{2} x^2-2 \sqrt [3]{2} x^3+\sqrt [3]{2} x^4+\left (-2^{2/3}-2^{2/3} x+2^{2/3} x^2\right ) \sqrt [3]{-1+x^6}+2 \left (-1+x^6\right )^{2/3}\right )}{6\ 2^{2/3}} \]

[Out]

-1/6*arctan(3^(1/2)*(x^6-1)^(1/3)/(-2^(2/3)-2^(2/3)*x+2^(2/3)*x^2+(x^6-1)^(1/3)))*2^(1/3)*3^(1/2)-1/6*ln(-2^(2
/3)-2^(2/3)*x+2^(2/3)*x^2-2*(x^6-1)^(1/3))*2^(1/3)+1/12*ln(2^(1/3)+2*2^(1/3)*x-2^(1/3)*x^2-2*2^(1/3)*x^3+2^(1/
3)*x^4+(-2^(2/3)-2^(2/3)*x+2^(2/3)*x^2)*(x^6-1)^(1/3)+2*(x^6-1)^(2/3))*2^(1/3)

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Rubi [F]
time = 0.21, 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 {1+x^2}{\left (-1+x+x^2\right ) \sqrt [3]{-1+x^6}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(1 + x^2)/((-1 + x + x^2)*(-1 + x^6)^(1/3)),x]

[Out]

(x*(1 - x^6)^(1/3)*Hypergeometric2F1[1/6, 1/3, 7/6, x^6])/(-1 + x^6)^(1/3) - (1 - Sqrt[5])*Defer[Int][1/((1 -
Sqrt[5] + 2*x)*(-1 + x^6)^(1/3)), x] - (1 + Sqrt[5])*Defer[Int][1/((1 + Sqrt[5] + 2*x)*(-1 + x^6)^(1/3)), x]

Rubi steps

\begin {align*} \int \frac {1+x^2}{\left (-1+x+x^2\right ) \sqrt [3]{-1+x^6}} \, dx &=\int \left (\frac {1}{\sqrt [3]{-1+x^6}}+\frac {2-x}{\left (-1+x+x^2\right ) \sqrt [3]{-1+x^6}}\right ) \, dx\\ &=\int \frac {1}{\sqrt [3]{-1+x^6}} \, dx+\int \frac {2-x}{\left (-1+x+x^2\right ) \sqrt [3]{-1+x^6}} \, dx\\ &=\frac {\sqrt [3]{1-x^6} \int \frac {1}{\sqrt [3]{1-x^6}} \, dx}{\sqrt [3]{-1+x^6}}+\int \left (\frac {-1+\sqrt {5}}{\left (1-\sqrt {5}+2 x\right ) \sqrt [3]{-1+x^6}}+\frac {-1-\sqrt {5}}{\left (1+\sqrt {5}+2 x\right ) \sqrt [3]{-1+x^6}}\right ) \, dx\\ &=\frac {x \sqrt [3]{1-x^6} \, _2F_1\left (\frac {1}{6},\frac {1}{3};\frac {7}{6};x^6\right )}{\sqrt [3]{-1+x^6}}+\left (-1-\sqrt {5}\right ) \int \frac {1}{\left (1+\sqrt {5}+2 x\right ) \sqrt [3]{-1+x^6}} \, dx+\left (-1+\sqrt {5}\right ) \int \frac {1}{\left (1-\sqrt {5}+2 x\right ) \sqrt [3]{-1+x^6}} \, dx\\ \end {align*}

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Mathematica [A]
time = 0.00, size = 186, normalized size = 0.89 \begin {gather*} \frac {2 \sqrt {3} \text {ArcTan}\left (\frac {\sqrt {3} \sqrt [3]{-1+x^6}}{2^{2/3}+2^{2/3} x-2^{2/3} x^2-\sqrt [3]{-1+x^6}}\right )-2 \log \left (-2^{2/3}-2^{2/3} x+2^{2/3} x^2-2 \sqrt [3]{-1+x^6}\right )+\log \left (\sqrt [3]{2}+2 \sqrt [3]{2} x-\sqrt [3]{2} x^2-2 \sqrt [3]{2} x^3+\sqrt [3]{2} x^4+2^{2/3} \left (-1-x+x^2\right ) \sqrt [3]{-1+x^6}+2 \left (-1+x^6\right )^{2/3}\right )}{6\ 2^{2/3}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1 + x^2)/((-1 + x + x^2)*(-1 + x^6)^(1/3)),x]

[Out]

(2*Sqrt[3]*ArcTan[(Sqrt[3]*(-1 + x^6)^(1/3))/(2^(2/3) + 2^(2/3)*x - 2^(2/3)*x^2 - (-1 + x^6)^(1/3))] - 2*Log[-
2^(2/3) - 2^(2/3)*x + 2^(2/3)*x^2 - 2*(-1 + x^6)^(1/3)] + Log[2^(1/3) + 2*2^(1/3)*x - 2^(1/3)*x^2 - 2*2^(1/3)*
x^3 + 2^(1/3)*x^4 + 2^(2/3)*(-1 - x + x^2)*(-1 + x^6)^(1/3) + 2*(-1 + x^6)^(2/3)])/(6*2^(2/3))

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 32.07, size = 1803, normalized size = 8.59

method	result	size
trager	\(\text {Expression too large to display}\)	\(1803\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^2+1)/(x^2+x-1)/(x^6-1)^(1/3),x,method=_RETURNVERBOSE)

[Out]

1/3*RootOf(RootOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2)*ln((-14*RootOf(_Z^3+2)+15*(x^6-1)^(1/3)*RootOf(_Z^3+2)
*RootOf(RootOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2)*x^4-24*(x^6-1)^(2/3)*RootOf(RootOf(_Z^3+2)^2+2*_Z*RootOf(
_Z^3+2)+4*_Z^2)*RootOf(_Z^3+2)^2*x-30*(x^6-1)^(1/3)*RootOf(_Z^3+2)*RootOf(RootOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)
+4*_Z^2)*x^3-15*(x^6-1)^(1/3)*RootOf(_Z^3+2)*RootOf(RootOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2)*x^2+30*(x^6-1
)^(1/3)*RootOf(_Z^3+2)*RootOf(RootOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2)*x+3*RootOf(RootOf(_Z^3+2)^2+2*_Z*Ro
otOf(_Z^3+2)+4*_Z^2)*x^5+24*(x^6-1)^(2/3)*RootOf(RootOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2)*RootOf(_Z^3+2)^2
*x^2+9*(x^6-1)^(2/3)+7*RootOf(RootOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2)-9*x^2*(x^6-1)^(2/3)-6*RootOf(_Z^3+2
)*x+3*RootOf(RootOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2)*x+9*x*(x^6-1)^(2/3)+6*RootOf(RootOf(_Z^3+2)^2+2*_Z*R
ootOf(_Z^3+2)+4*_Z^2)^2*RootOf(_Z^3+2)^2*x-12*RootOf(RootOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2)*RootOf(_Z^3+
2)^3*x-6*RootOf(_Z^3+2)*x^5+10*RootOf(_Z^3+2)*x^3+14*RootOf(_Z^3+2)*x^6+6*RootOf(RootOf(_Z^3+2)^2+2*_Z*RootOf(
_Z^3+2)+4*_Z^2)^2*RootOf(_Z^3+2)^2*x^5-12*RootOf(RootOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2)*RootOf(_Z^3+2)^3
*x^5-10*RootOf(RootOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2)^2*RootOf(_Z^3+2)^2*x^3+20*RootOf(RootOf(_Z^3+2)^2+
2*_Z*RootOf(_Z^3+2)+4*_Z^2)*RootOf(_Z^3+2)^3*x^3-7*RootOf(RootOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2)*x^6-5*R
ootOf(RootOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2)*x^3+12*RootOf(_Z^3+2)^2*(x^6-1)^(1/3)*x^4-24*RootOf(_Z^3+2)
^2*(x^6-1)^(1/3)*x^3-24*(x^6-1)^(2/3)*RootOf(RootOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2)*RootOf(_Z^3+2)^2-12*
RootOf(_Z^3+2)^2*(x^6-1)^(1/3)*x^2+24*RootOf(_Z^3+2)^2*(x^6-1)^(1/3)*x+15*(x^6-1)^(1/3)*RootOf(_Z^3+2)*RootOf(
RootOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2)+12*RootOf(_Z^3+2)^2*(x^6-1)^(1/3))/(x^2+x-1)^3)+1/6*RootOf(_Z^3+2
)*ln(-(-7*RootOf(_Z^3+2)+18*(x^6-1)^(1/3)*RootOf(_Z^3+2)*RootOf(RootOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2)*x
^4-48*(x^6-1)^(2/3)*RootOf(RootOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2)*RootOf(_Z^3+2)^2*x-36*(x^6-1)^(1/3)*Ro
otOf(_Z^3+2)*RootOf(RootOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2)*x^3-18*(x^6-1)^(1/3)*RootOf(_Z^3+2)*RootOf(Ro
otOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2)*x^2+36*(x^6-1)^(1/3)*RootOf(_Z^3+2)*RootOf(RootOf(_Z^3+2)^2+2*_Z*Ro
otOf(_Z^3+2)+4*_Z^2)*x+72*RootOf(RootOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2)*x^5+48*(x^6-1)^(2/3)*RootOf(Root
Of(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2)*RootOf(_Z^3+2)^2*x^2+30*(x^6-1)^(2/3)+56*RootOf(RootOf(_Z^3+2)^2+2*_Z
*RootOf(_Z^3+2)+4*_Z^2)-30*x^2*(x^6-1)^(2/3)-9*RootOf(_Z^3+2)*x+72*RootOf(RootOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)
+4*_Z^2)*x+30*x*(x^6-1)^(2/3)-48*RootOf(RootOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2)^2*RootOf(_Z^3+2)^2*x+6*Ro
otOf(RootOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2)*RootOf(_Z^3+2)^3*x-9*RootOf(_Z^3+2)*x^5+15*RootOf(_Z^3+2)*x^
3+7*RootOf(_Z^3+2)*x^6-48*RootOf(RootOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2)^2*RootOf(_Z^3+2)^2*x^5+6*RootOf(
RootOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2)*RootOf(_Z^3+2)^3*x^5+80*RootOf(RootOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3+
2)+4*_Z^2)^2*RootOf(_Z^3+2)^2*x^3-10*RootOf(RootOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2)*RootOf(_Z^3+2)^3*x^3-
56*RootOf(RootOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2)*x^6-120*RootOf(RootOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_
Z^2)*x^3+24*RootOf(_Z^3+2)^2*(x^6-1)^(1/3)*x^4-48*RootOf(_Z^3+2)^2*(x^6-1)^(1/3)*x^3-48*(x^6-1)^(2/3)*RootOf(R
ootOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2)*RootOf(_Z^3+2)^2-24*RootOf(_Z^3+2)^2*(x^6-1)^(1/3)*x^2+48*RootOf(_
Z^3+2)^2*(x^6-1)^(1/3)*x+18*(x^6-1)^(1/3)*RootOf(_Z^3+2)*RootOf(RootOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2)+2
4*RootOf(_Z^3+2)^2*(x^6-1)^(1/3))/(x^2+x-1)^3)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2+1)/(x^2+x-1)/(x^6-1)^(1/3),x, algorithm="maxima")

[Out]

integrate((x^2 + 1)/((x^6 - 1)^(1/3)*(x^2 + x - 1)), x)

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Fricas [A]
time = 12.64, size = 251, normalized size = 1.20 \begin {gather*} -\frac {1}{6} \cdot 4^{\frac {1}{6}} \sqrt {3} \left (-1\right )^{\frac {1}{3}} \arctan \left (\frac {4^{\frac {1}{6}} \sqrt {3} {\left (2 \cdot 4^{\frac {2}{3}} \left (-1\right )^{\frac {2}{3}} {\left (x^{6} - 1\right )}^{\frac {2}{3}} {\left (x^{2} - x - 1\right )} + 4 \, \left (-1\right )^{\frac {1}{3}} {\left (x^{6} - 1\right )}^{\frac {1}{3}} {\left (x^{4} - 2 \, x^{3} - x^{2} + 2 \, x + 1\right )} - 4^{\frac {1}{3}} {\left (x^{6} + 3 \, x^{5} - 5 \, x^{3} + 3 \, x - 1\right )}\right )}}{6 \, {\left (3 \, x^{6} - 3 \, x^{5} + 5 \, x^{3} - 3 \, x - 3\right )}}\right ) - \frac {1}{24} \cdot 4^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} \log \left (\frac {4^{\frac {1}{3}} \left (-1\right )^{\frac {2}{3}} {\left (x^{4} - 2 \, x^{3} - x^{2} + 2 \, x + 1\right )} - 4^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} {\left (x^{6} - 1\right )}^{\frac {2}{3}} + 2 \, {\left (x^{6} - 1\right )}^{\frac {1}{3}} {\left (x^{2} - x - 1\right )}}{x^{4} + 2 \, x^{3} - x^{2} - 2 \, x + 1}\right ) + \frac {1}{12} \cdot 4^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} \log \left (\frac {4^{\frac {1}{3}} {\left (x^{2} - x - 1\right )} + 2 \, \left (-1\right )^{\frac {1}{3}} {\left (x^{6} - 1\right )}^{\frac {1}{3}}}{x^{2} + x - 1}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2+1)/(x^2+x-1)/(x^6-1)^(1/3),x, algorithm="fricas")

[Out]

-1/6*4^(1/6)*sqrt(3)*(-1)^(1/3)*arctan(1/6*4^(1/6)*sqrt(3)*(2*4^(2/3)*(-1)^(2/3)*(x^6 - 1)^(2/3)*(x^2 - x - 1)
+ 4*(-1)^(1/3)*(x^6 - 1)^(1/3)*(x^4 - 2*x^3 - x^2 + 2*x + 1) - 4^(1/3)*(x^6 + 3*x^5 - 5*x^3 + 3*x - 1))/(3*x^
6 - 3*x^5 + 5*x^3 - 3*x - 3)) - 1/24*4^(2/3)*(-1)^(1/3)*log((4^(1/3)*(-1)^(2/3)*(x^4 - 2*x^3 - x^2 + 2*x + 1)
- 4^(2/3)*(-1)^(1/3)*(x^6 - 1)^(2/3) + 2*(x^6 - 1)^(1/3)*(x^2 - x - 1))/(x^4 + 2*x^3 - x^2 - 2*x + 1)) + 1/12*
4^(2/3)*(-1)^(1/3)*log((4^(1/3)*(x^2 - x - 1) + 2*(-1)^(1/3)*(x^6 - 1)^(1/3))/(x^2 + x - 1))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**2+1)/(x**2+x-1)/(x**6-1)**(1/3),x)

[Out]

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2+1)/(x^2+x-1)/(x^6-1)^(1/3),x, algorithm="giac")

[Out]

integrate((x^2 + 1)/((x^6 - 1)^(1/3)*(x^2 + x - 1)), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {x^2+1}{{\left (x^6-1\right )}^{1/3}\,\left (x^2+x-1\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^2 + 1)/((x^6 - 1)^(1/3)*(x + x^2 - 1)),x)

[Out]

int((x^2 + 1)/((x^6 - 1)^(1/3)*(x + x^2 - 1)), x)

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