3.25.92 \(\int \frac {1+x^2}{(-1-x+x^2) \sqrt [3]{-1+x^6}} \, dx\) [2492]
Optimal. Leaf size=206 \[ \frac {\text {ArcTan}\left (\frac {\sqrt {3} \sqrt [3]{-1+x^6}}{-2^{2/3}+2^{2/3} x+2^{2/3} x^2+\sqrt [3]{-1+x^6}}\right )}{2^{2/3} \sqrt {3}}+\frac {\log \left (-2^{2/3}+2^{2/3} x+2^{2/3} x^2-2 \sqrt [3]{-1+x^6}\right )}{3\ 2^{2/3}}-\frac {\log \left (\sqrt [3]{2}-2 \sqrt [3]{2} x-\sqrt [3]{2} x^2+2 \sqrt [3]{2} x^3+\sqrt [3]{2} x^4+\left (-2^{2/3}+2^{2/3} x+2^{2/3} x^2\right ) \sqrt [3]{-1+x^6}+2 \left (-1+x^6\right )^{2/3}\right )}{6\ 2^{2/3}} \]
[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 = 0.32, 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 [A]
time = 4.18, size = 184, 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 = 32.65, size = 1806, normalized size = 8.77
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result |
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| trager |
\(\text {Expression too large to display}\) |
\(1806\) |
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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((-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+3*Root
Of(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*x^5-5*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*x^3+
3*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*x+9*(x^6-1)^(2/3)-6*RootOf(_Z^3-2)*x-9*x^2*(x^6-1)^(2/3)
+14*RootOf(_Z^3-2)-9*x*(x^6-1)^(2/3)-7*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)-14*RootOf(_Z^3-2)*x
^6-6*RootOf(_Z^3-2)*x^5+10*RootOf(_Z^3-2)*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*x^2+10*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)^2*RootOf(_Z^3-2)^2*x^3-6*Root
Of(RootOf(_Z^3-2)^2+2*_Z*RootOf(_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+7*x^6*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)-20*RootOf(RootOf(_Z
^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*RootOf(_Z^3-2)^3*x^3+12*RootOf(_Z^3-2)^2*(x^6-1)^(1/3)+15*(x^6-1)^(1/3)*Ro
otOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*RootOf(_Z^3-2)+12*RootOf(_Z^3-2)^2*(x^6-1)^(1/3)*x^4+24*Root
Of(_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(RootO
f(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*RootOf(_Z^3-2)*x^4-24*(x^6-1)^(2/3)*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootO
f(_Z^3-2)+4*_Z^2)*RootOf(_Z^3-2)^2*x+30*(x^6-1)^(1/3)*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*Root
Of(_Z^3-2)*x^3-15*(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^2-30*(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)/(x^2-x-1)^3)+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^5+6*RootOf(RootOf(_Z^3-2)^2
+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*RootOf(_Z^3-2)^3*x^5-72*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*x^5+1
20*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*x^3-72*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z
^2)*x-30*(x^6-1)^(2/3)+9*RootOf(_Z^3-2)*x+30*x^2*(x^6-1)^(2/3)-7*RootOf(_Z^3-2)+30*x*(x^6-1)^(2/3)+56*RootOf(R
ootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)+7*RootOf(_Z^3-2)*x^6+9*RootOf(_Z^3-2)*x^5-15*RootOf(_Z^3-2)*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*x^2+80*RootOf(RootOf(_Z^3-
2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)^2*RootOf(_Z^3-2)^2*x^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*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*RootOf(_Z^3-2)^3*x-56*x^6*RootOf(R
ootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)-10*RootOf(RootOf(_Z^3-2)^2+2*_Z*RootOf(_Z^3-2)+4*_Z^2)*RootOf(_Z^3
-2)^3*x^3-24*RootOf(_Z^3-2)^2*(x^6-1)^(1/3)-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)-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(RootOf(_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(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)/(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 = 12.85, 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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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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