∫ 1+x2 _________________________________(−1+x+x2 ) 3√___

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

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

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Rubi [F] time = 0.32, 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 [F] time = 0.07, size = 0, normalized size = 0.00 \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]

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

[Out]

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

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IntegrateAlgebraic [A] time = 0.00, size = 210, normalized size = 1.00 \begin {gather*} -\frac {\tan ^{-1}\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}} \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

-(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^(2/3)*Sqrt[3])

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

(2/3)]/(6*2^(2/3))

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fricas [A] time = 21.07, 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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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{2} + 1}{{\left (x^{6} - 1\right )}^{\frac {1}{3}} {\left (x^{2} + x - 1\right )}}\,{d x} \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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maple [C] time = 38.99, size = 2695, normalized size = 12.83


method	result	size

trager	\(\text {Expression too large to display}\)	\(2695\)

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]

-1/6*ln(-(18*RootOf(RootOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2)*x-30*x^2*(x^6-1)^(2/3)-35*RootOf(_Z^3+2)*x^6+

45*RootOf(_Z^3+2)*x+45*RootOf(_Z^3+2)*x^5+30*x*(x^6-1)^(2/3)+30*(x^6-1)^(2/3)+35*RootOf(_Z^3+2)-75*RootOf(_Z^3

+2)*x^3+14*RootOf(RootOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2)-12*RootOf(RootOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+

4*_Z^2)^2*RootOf(_Z^3+2)^2*x-30*RootOf(RootOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2)*RootOf(_Z^3+2)^3*x+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^2-12*RootOf(RootOf(_Z^3+2)^2+

2*_Z*RootOf(_Z^3+2)+4*_Z^2)^2*RootOf(_Z^3+2)^2*x^5-30*RootOf(RootOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2)*Root

Of(_Z^3+2)^3*x^5+20*RootOf(RootOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2)^2*RootOf(_Z^3+2)^2*x^3+50*RootOf(RootO

f(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2)*RootOf(_Z^3+2)^3*x^3-14*RootOf(RootOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*

_Z^2)*x^6-30*RootOf(RootOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2)*x^3+18*RootOf(RootOf(_Z^3+2)^2+2*_Z*RootOf(_Z

^3+2)+4*_Z^2)*x^5-9*RootOf(_Z^3+2)^2*(x^6-1)^(1/3)*x^4+18*RootOf(_Z^3+2)^2*(x^6-1)^(1/3)*x^3-48*(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+9*RootOf(_Z^3+2)^2*(x^6-1)^(1/3)*x^2-18*R

ootOf(_Z^3+2)^2*(x^6-1)^(1/3)*x+30*(x^6-1)^(1/3)*RootOf(RootOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2)*RootOf(_Z

^3+2)+30*(x^6-1)^(1/3)*RootOf(RootOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2)*RootOf(_Z^3+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-60*(x^6-1)^(1/3)*RootOf(RootOf(_Z^3+2)

^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2)*RootOf(_Z^3+2)*x^3-30*(x^6-1)^(1/3)*RootOf(RootOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)

+4*_Z^2)*RootOf(_Z^3+2)*x^2+60*(x^6-1)^(1/3)*RootOf(RootOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2)*RootOf(_Z^3+2

)*x-9*RootOf(_Z^3+2)^2*(x^6-1)^(1/3))/(x^2+x-1)^3)*RootOf(_Z^3+2)-1/3*ln(-(18*RootOf(RootOf(_Z^3+2)^2+2*_Z*Roo

tOf(_Z^3+2)+4*_Z^2)*x-30*x^2*(x^6-1)^(2/3)-35*RootOf(_Z^3+2)*x^6+45*RootOf(_Z^3+2)*x+45*RootOf(_Z^3+2)*x^5+30*

x*(x^6-1)^(2/3)+30*(x^6-1)^(2/3)+35*RootOf(_Z^3+2)-75*RootOf(_Z^3+2)*x^3+14*RootOf(RootOf(_Z^3+2)^2+2*_Z*RootO

f(_Z^3+2)+4*_Z^2)-12*RootOf(RootOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2)^2*RootOf(_Z^3+2)^2*x-30*RootOf(RootOf

(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2)*RootOf(_Z^3+2)^3*x+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^2-12*RootOf(RootOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2)^2*RootOf(_Z^3+2)^

2*x^5-30*RootOf(RootOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2)*RootOf(_Z^3+2)^3*x^5+20*RootOf(RootOf(_Z^3+2)^2+2

*_Z*RootOf(_Z^3+2)+4*_Z^2)^2*RootOf(_Z^3+2)^2*x^3+50*RootOf(RootOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2)*RootO

f(_Z^3+2)^3*x^3-14*RootOf(RootOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2)*x^6-30*RootOf(RootOf(_Z^3+2)^2+2*_Z*Roo

tOf(_Z^3+2)+4*_Z^2)*x^3+18*RootOf(RootOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2)*x^5-9*RootOf(_Z^3+2)^2*(x^6-1)^

(1/3)*x^4+18*RootOf(_Z^3+2)^2*(x^6-1)^(1/3)*x^3-48*(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+9*RootOf(_Z^3+2)^2*(x^6-1)^(1/3)*x^2-18*RootOf(_Z^3+2)^2*(x^6-1)^(1/3)*x+30*(x^6-1)^(1

/3)*RootOf(RootOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2)*RootOf(_Z^3+2)+30*(x^6-1)^(1/3)*RootOf(RootOf(_Z^3+2)^

2+2*_Z*RootOf(_Z^3+2)+4*_Z^2)*RootOf(_Z^3+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-60*(x^6-1)^(1/3)*RootOf(RootOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2)*RootOf(_Z^3+2)

*x^3-30*(x^6-1)^(1/3)*RootOf(RootOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2)*RootOf(_Z^3+2)*x^2+60*(x^6-1)^(1/3)*

RootOf(RootOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2)*RootOf(_Z^3+2)*x-9*RootOf(_Z^3+2)^2*(x^6-1)^(1/3))/(x^2+x-

1)^3)*RootOf(RootOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2)+1/6*RootOf(_Z^3+2)*ln((24*RootOf(RootOf(_Z^3+2)^2+2*

_Z*RootOf(_Z^3+2)+4*_Z^2)*x-18*x^2*(x^6-1)^(2/3)-35*RootOf(_Z^3+2)*x^6+15*RootOf(_Z^3+2)*x+15*RootOf(_Z^3+2)*x

^5+18*x*(x^6-1)^(2/3)+18*(x^6-1)^(2/3)+35*RootOf(_Z^3+2)-25*RootOf(_Z^3+2)*x^3+56*RootOf(RootOf(_Z^3+2)^2+2*_Z

*RootOf(_Z^3+2)+4*_Z^2)+48*RootOf(RootOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2)^2*RootOf(_Z^3+2)^2*x+30*RootOf(

RootOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2)*RootOf(_Z^3+2)^3*x+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^2+48*RootOf(RootOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2)^2*RootOf(_Z

^3+2)^2*x^5+30*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^3-50*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^6-40*RootOf(RootOf(_Z^3+2)^2+2*

_Z*RootOf(_Z^3+2)+4*_Z^2)*x^3+24*RootOf(RootOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2)*x^5-15*RootOf(_Z^3+2)^2*(

x^6-1)^(1/3)*x^4+30*RootOf(_Z^3+2)^2*(x^6-1)^(1/3)*x^3-48*(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+15*RootOf(_Z^3+2)^2*(x^6-1)^(1/3)*x^2-30*RootOf(_Z^3+2)^2*(x^6-1)^(1/3)*x+18*(x

^6-1)^(1/3)*RootOf(RootOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2)*RootOf(_Z^3+2)+18*(x^6-1)^(1/3)*RootOf(RootOf(

_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2)*RootOf(_Z^3+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)*RootOf(RootOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2)*RootOf

(_Z^3+2)*x^3-18*(x^6-1)^(1/3)*RootOf(RootOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2)*RootOf(_Z^3+2)*x^2+36*(x^6-1

)^(1/3)*RootOf(RootOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2)*RootOf(_Z^3+2)*x-15*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*} \int \frac {x^{2} + 1}{{\left (x^{6} - 1\right )}^{\frac {1}{3}} {\left (x^{2} + x - 1\right )}}\,{d x} \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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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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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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