3.21.10 \(\int \frac {1+x^2}{(-1-x+x^2) \sqrt [3]{-1+x^6}} \, dx\)
Optimal. Leaf size=206 \[ \frac {\log \left (-2 \sqrt [3]{x^6-1}+2^{2/3} x^2+2^{2/3} x-2^{2/3}\right )}{3\ 2^{2/3}}+\frac {\tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{x^6-1}}{\sqrt [3]{x^6-1}+2^{2/3} x^2+2^{2/3} x-2^{2/3}}\right )}{2^{2/3} \sqrt {3}}-\frac {\log \left (2 \left (x^6-1\right )^{2/3}+\sqrt [3]{2} x^4+2 \sqrt [3]{2} x^3-\sqrt [3]{2} x^2+\left (2^{2/3} x^2+2^{2/3} x-2^{2/3}\right ) \sqrt [3]{x^6-1}-2 \sqrt [3]{2} x+\sqrt [3]{2}\right )}{6\ 2^{2/3}} \]
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Rubi [F] time = 0.33, 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 = 0.08, 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 = 206, normalized size = 1.00 \begin {gather*} \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}} \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 = 37.05, size = 223, normalized size = 1.08 \begin {gather*} -\frac {1}{6} \cdot 4^{\frac {1}{6}} \sqrt {3} \arctan \left (\frac {4^{\frac {1}{6}} \sqrt {3} {\left (2 \cdot 4^{\frac {2}{3}} {\left (x^{6} - 1\right )}^{\frac {2}{3}} {\left (x^{2} + x - 1\right )} - 4^{\frac {1}{3}} {\left (x^{6} - 3 \, x^{5} + 5 \, x^{3} - 3 \, x - 1\right )} - 4 \, {\left (x^{6} - 1\right )}^{\frac {1}{3}} {\left (x^{4} + 2 \, x^{3} - x^{2} - 2 \, 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}} \log \left (\frac {4^{\frac {2}{3}} {\left (x^{6} - 1\right )}^{\frac {2}{3}} + 4^{\frac {1}{3}} {\left (x^{4} + 2 \, x^{3} - x^{2} - 2 \, x + 1\right )} + 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}} \log \left (-\frac {4^{\frac {1}{3}} {\left (x^{2} + x - 1\right )} - 2 \, {\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)*arctan(1/6*4^(1/6)*sqrt(3)*(2*4^(2/3)*(x^6 - 1)^(2/3)*(x^2 + x - 1) - 4^(1/3)*(x^6 - 3*x^
5 + 5*x^3 - 3*x - 1) - 4*(x^6 - 1)^(1/3)*(x^4 + 2*x^3 - x^2 - 2*x + 1))/(3*x^6 + 3*x^5 - 5*x^3 + 3*x - 3)) - 1
/24*4^(2/3)*log((4^(2/3)*(x^6 - 1)^(2/3) + 4^(1/3)*(x^4 + 2*x^3 - x^2 - 2*x + 1) + 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)*log(-(4^(1/3)*(x^2 + x - 1) - 2*(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 = 30.29, size = 2702, normalized size = 13.12 \begin {gather*} \text {Expression too large to display} \end {gather*}
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)
[Out]
1/6*RootOf(_Z^3-2)*ln(-(48*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)^2*RootOf(_Z^3-2)^2*x+18*x^2*(x^
6-1)^(2/3)+56*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)+18*x*(x^6-1)^(2/3)-18*(x^6-1)^(2/3)+25*RootO
f(_Z^3-2)*x^3-15*RootOf(_Z^3-2)*x+40*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*x^3-56*x^6*RootOf(Roo
tOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)-24*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*x^5-24*RootOf
(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*x+35*RootOf(_Z^3-2)-35*RootOf(_Z^3-2)*x^6-15*RootOf(_Z^3-2)*x^5+
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*RootO
f(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*RootOf(_Z^3-2)^3*x^3+30*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^
3-2)+4*_Z^2)*RootOf(_Z^3-2)^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)*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)+15*RootOf(_Z^3-2)^2*(x^6-1)^(1/3)*x^4+30*RootO
f(_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))/(x^2-x-1)^3)-1/6*ln(-(12*RootOf(RootOf(_Z^3-2)^2+2*_Z*R
ootOf(_Z^3-2)+4*_Z^2)^2*RootOf(_Z^3-2)^2*x-30*x^2*(x^6-1)^(2/3)-14*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)
+4*_Z^2)-30*x*(x^6-1)^(2/3)+30*(x^6-1)^(2/3)-75*RootOf(_Z^3-2)*x^3+45*RootOf(_Z^3-2)*x-30*RootOf(RootOf(_Z^3-2
)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*x^3+14*x^6*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)+18*RootOf(RootO
f(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*x^5+18*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*x-35*RootOf
(_Z^3-2)+35*RootOf(_Z^3-2)*x^6+45*RootOf(_Z^3-2)*x^5-48*(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*R
ootOf(_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+30*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*RootOf(_Z^3-2)^3*x+30*(x^6-1)^(1/3)*RootOf(R
ootOf(_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*R
ootOf(_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)-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(RootO
f(_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))/(x^2-
x-1)^3)*RootOf(_Z^3-2)-1/3*ln(-(12*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)^2*RootOf(_Z^3-2)^2*x-30
*x^2*(x^6-1)^(2/3)-14*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)-30*x*(x^6-1)^(2/3)+30*(x^6-1)^(2/3)-
75*RootOf(_Z^3-2)*x^3+45*RootOf(_Z^3-2)*x-30*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*x^3+14*x^6*Ro
otOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)+18*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*x^5+1
8*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*x-35*RootOf(_Z^3-2)+35*RootOf(_Z^3-2)*x^6+45*RootOf(_Z^3
-2)*x^5-48*(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(Ro
otOf(_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)*RootOf(_Z^3-2)^3*x^3+30*RootOf(RootOf(_Z^3-2)^2+2*_Z*Ro
otOf(_Z^3-2)+4*_Z^2)*RootOf(_Z^3-2)^3*x+30*(x^6-1)^(1/3)*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*R
ootOf(_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(Ro
otOf(_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*Ro
otOf(_Z^3-2)+4*_Z^2)*RootOf(_Z^3-2)*x-9*RootOf(_Z^3-2)^2*(x^6-1)^(1/3)-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)*RootO
f(_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(R
ootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*RootOf(_Z^3-2))/(x^2-x-1)^3)*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_
Z^3-2)+4*_Z^2)
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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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