3.26.63 \(\int \frac {(-1+x+x^3+x^6)^{2/3} (3-2 x+3 x^6)}{(-1+x+x^6) (-1+x-x^3+x^6)} \, dx\)

Optimal. Leaf size=217 \[ -\log \left (\sqrt [3]{x^6+x^3+x-1}-x\right )+2^{2/3} \log \left (2^{2/3} \sqrt [3]{x^6+x^3+x-1}-2 x\right )+\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{x^6+x^3+x-1}+x}\right )-2^{2/3} \sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{2^{2/3} \sqrt [3]{x^6+x^3+x-1}+x}\right )+\frac {1}{2} \log \left (x^2+\sqrt [3]{x^6+x^3+x-1} x+\left (x^6+x^3+x-1\right )^{2/3}\right )-\frac {\log \left (2 x^2+2^{2/3} \sqrt [3]{x^6+x^3+x-1} x+\sqrt [3]{2} \left (x^6+x^3+x-1\right )^{2/3}\right )}{\sqrt [3]{2}} \]

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Rubi [F]  time = 2.29, 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 {\left (-1+x+x^3+x^6\right )^{2/3} \left (3-2 x+3 x^6\right )}{\left (-1+x+x^6\right ) \left (-1+x-x^3+x^6\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

Defer[Int][(-1 + x + x^3 + x^6)^(2/3)/(-1 + x), x] + Defer[Int][(-1 + x + x^3 + x^6)^(2/3)/(1 + x), x] + 2*Def
er[Int][(-1 + x + x^3 + x^6)^(2/3)/(1 - x + x^2 + x^4), x] + 3*Defer[Int][(x*(-1 + x + x^3 + x^6)^(2/3))/(1 -
x + x^2 + x^4), x] + Defer[Int][(x^2*(-1 + x + x^3 + x^6)^(2/3))/(1 - x + x^2 + x^4), x] - Defer[Int][(x^3*(-1
+ x + x^3 + x^6)^(2/3))/(1 - x + x^2 + x^4), x] + Defer[Int][(-1 + x + x^3 + x^6)^(2/3)/(1 - x - x^6), x] - 6
*Defer[Int][(x^3*(-1 + x + x^3 + x^6)^(2/3))/(-1 + x + x^6), x] - Defer[Int][(x^4*(-1 + x + x^3 + x^6)^(2/3))/
(-1 + x + x^6), x] - Defer[Int][(x^5*(-1 + x + x^3 + x^6)^(2/3))/(-1 + x + x^6), x]

Rubi steps

\begin {align*} \int \frac {\left (-1+x+x^3+x^6\right )^{2/3} \left (3-2 x+3 x^6\right )}{\left (-1+x+x^6\right ) \left (-1+x-x^3+x^6\right )} \, dx &=\int \left (\frac {\left (-1+x+x^3+x^6\right )^{2/3}}{-1+x}+\frac {\left (-1+x+x^3+x^6\right )^{2/3}}{1+x}+\frac {\left (2+3 x+x^2-x^3\right ) \left (-1+x+x^3+x^6\right )^{2/3}}{1-x+x^2+x^4}+\frac {\left (-1-6 x^3-x^4-x^5\right ) \left (-1+x+x^3+x^6\right )^{2/3}}{-1+x+x^6}\right ) \, dx\\ &=\int \frac {\left (-1+x+x^3+x^6\right )^{2/3}}{-1+x} \, dx+\int \frac {\left (-1+x+x^3+x^6\right )^{2/3}}{1+x} \, dx+\int \frac {\left (2+3 x+x^2-x^3\right ) \left (-1+x+x^3+x^6\right )^{2/3}}{1-x+x^2+x^4} \, dx+\int \frac {\left (-1-6 x^3-x^4-x^5\right ) \left (-1+x+x^3+x^6\right )^{2/3}}{-1+x+x^6} \, dx\\ &=\int \frac {\left (-1+x+x^3+x^6\right )^{2/3}}{-1+x} \, dx+\int \frac {\left (-1+x+x^3+x^6\right )^{2/3}}{1+x} \, dx+\int \left (\frac {2 \left (-1+x+x^3+x^6\right )^{2/3}}{1-x+x^2+x^4}+\frac {3 x \left (-1+x+x^3+x^6\right )^{2/3}}{1-x+x^2+x^4}+\frac {x^2 \left (-1+x+x^3+x^6\right )^{2/3}}{1-x+x^2+x^4}-\frac {x^3 \left (-1+x+x^3+x^6\right )^{2/3}}{1-x+x^2+x^4}\right ) \, dx+\int \left (\frac {\left (-1+x+x^3+x^6\right )^{2/3}}{1-x-x^6}-\frac {6 x^3 \left (-1+x+x^3+x^6\right )^{2/3}}{-1+x+x^6}-\frac {x^4 \left (-1+x+x^3+x^6\right )^{2/3}}{-1+x+x^6}-\frac {x^5 \left (-1+x+x^3+x^6\right )^{2/3}}{-1+x+x^6}\right ) \, dx\\ &=2 \int \frac {\left (-1+x+x^3+x^6\right )^{2/3}}{1-x+x^2+x^4} \, dx+3 \int \frac {x \left (-1+x+x^3+x^6\right )^{2/3}}{1-x+x^2+x^4} \, dx-6 \int \frac {x^3 \left (-1+x+x^3+x^6\right )^{2/3}}{-1+x+x^6} \, dx+\int \frac {\left (-1+x+x^3+x^6\right )^{2/3}}{-1+x} \, dx+\int \frac {\left (-1+x+x^3+x^6\right )^{2/3}}{1+x} \, dx+\int \frac {x^2 \left (-1+x+x^3+x^6\right )^{2/3}}{1-x+x^2+x^4} \, dx-\int \frac {x^3 \left (-1+x+x^3+x^6\right )^{2/3}}{1-x+x^2+x^4} \, dx+\int \frac {\left (-1+x+x^3+x^6\right )^{2/3}}{1-x-x^6} \, dx-\int \frac {x^4 \left (-1+x+x^3+x^6\right )^{2/3}}{-1+x+x^6} \, dx-\int \frac {x^5 \left (-1+x+x^3+x^6\right )^{2/3}}{-1+x+x^6} \, dx\\ \end {align*}

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Mathematica [F]  time = 0.61, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (-1+x+x^3+x^6\right )^{2/3} \left (3-2 x+3 x^6\right )}{\left (-1+x+x^6\right ) \left (-1+x-x^3+x^6\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 0.75, size = 217, normalized size = 1.00 \begin {gather*} \sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{-1+x+x^3+x^6}}\right )-2^{2/3} \sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{x+2^{2/3} \sqrt [3]{-1+x+x^3+x^6}}\right )-\log \left (-x+\sqrt [3]{-1+x+x^3+x^6}\right )+2^{2/3} \log \left (-2 x+2^{2/3} \sqrt [3]{-1+x+x^3+x^6}\right )+\frac {1}{2} \log \left (x^2+x \sqrt [3]{-1+x+x^3+x^6}+\left (-1+x+x^3+x^6\right )^{2/3}\right )-\frac {\log \left (2 x^2+2^{2/3} x \sqrt [3]{-1+x+x^3+x^6}+\sqrt [3]{2} \left (-1+x+x^3+x^6\right )^{2/3}\right )}{\sqrt [3]{2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [B]  time = 53.53, size = 615, normalized size = 2.83 \begin {gather*} -\frac {1}{3} \cdot 4^{\frac {1}{3}} \sqrt {3} \arctan \left (\frac {3 \cdot 4^{\frac {2}{3}} \sqrt {3} {\left (x^{13} + 4 \, x^{10} + 2 \, x^{8} - 7 \, x^{7} + 4 \, x^{5} - 4 \, x^{4} + x^{3} - 2 \, x^{2} + x\right )} {\left (x^{6} + x^{3} + x - 1\right )}^{\frac {2}{3}} + 6 \cdot 4^{\frac {1}{3}} \sqrt {3} {\left (x^{14} + 16 \, x^{11} + 2 \, x^{9} + 17 \, x^{8} + 16 \, x^{6} - 16 \, x^{5} + x^{4} - 2 \, x^{3} + x^{2}\right )} {\left (x^{6} + x^{3} + x - 1\right )}^{\frac {1}{3}} + \sqrt {3} {\left (x^{18} + 33 \, x^{15} + 3 \, x^{13} + 108 \, x^{12} + 66 \, x^{10} + 5 \, x^{9} + 3 \, x^{8} + 105 \, x^{7} - 108 \, x^{6} + 33 \, x^{5} - 66 \, x^{4} + 34 \, x^{3} - 3 \, x^{2} + 3 \, x - 1\right )}}{3 \, {\left (x^{18} - 3 \, x^{15} + 3 \, x^{13} - 108 \, x^{12} - 6 \, x^{10} - 103 \, x^{9} + 3 \, x^{8} - 111 \, x^{7} + 108 \, x^{6} - 3 \, x^{5} + 6 \, x^{4} - 2 \, x^{3} - 3 \, x^{2} + 3 \, x - 1\right )}}\right ) + \sqrt {3} \arctan \left (-\frac {682 \, \sqrt {3} {\left (x^{6} + x^{3} + x - 1\right )}^{\frac {1}{3}} x^{2} - 248 \, \sqrt {3} {\left (x^{6} + x^{3} + x - 1\right )}^{\frac {2}{3}} x + \sqrt {3} {\left (96 \, x^{6} + 217 \, x^{3} + 96 \, x - 96\right )}}{64 \, x^{6} + 1395 \, x^{3} + 64 \, x - 64}\right ) + \frac {1}{3} \cdot 4^{\frac {1}{3}} \log \left (\frac {3 \cdot 4^{\frac {2}{3}} {\left (x^{6} + x^{3} + x - 1\right )}^{\frac {1}{3}} x^{2} - 6 \, {\left (x^{6} + x^{3} + x - 1\right )}^{\frac {2}{3}} x + 4^{\frac {1}{3}} {\left (x^{6} - x^{3} + x - 1\right )}}{x^{6} - x^{3} + x - 1}\right ) - \frac {1}{6} \cdot 4^{\frac {1}{3}} \log \left (\frac {6 \cdot 4^{\frac {1}{3}} {\left (x^{7} + 5 \, x^{4} + x^{2} - x\right )} {\left (x^{6} + x^{3} + x - 1\right )}^{\frac {2}{3}} + 4^{\frac {2}{3}} {\left (x^{12} + 16 \, x^{9} + 2 \, x^{7} + 17 \, x^{6} + 16 \, x^{4} - 16 \, x^{3} + x^{2} - 2 \, x + 1\right )} + 24 \, {\left (x^{8} + 2 \, x^{5} + x^{3} - x^{2}\right )} {\left (x^{6} + x^{3} + x - 1\right )}^{\frac {1}{3}}}{x^{12} - 2 \, x^{9} + 2 \, x^{7} - x^{6} - 2 \, x^{4} + 2 \, x^{3} + x^{2} - 2 \, x + 1}\right ) - \frac {1}{2} \, \log \left (\frac {x^{6} + 3 \, {\left (x^{6} + x^{3} + x - 1\right )}^{\frac {1}{3}} x^{2} - 3 \, {\left (x^{6} + x^{3} + x - 1\right )}^{\frac {2}{3}} x + x - 1}{x^{6} + x - 1}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*4^(1/3)*sqrt(3)*arctan(1/3*(3*4^(2/3)*sqrt(3)*(x^13 + 4*x^10 + 2*x^8 - 7*x^7 + 4*x^5 - 4*x^4 + x^3 - 2*x^
2 + x)*(x^6 + x^3 + x - 1)^(2/3) + 6*4^(1/3)*sqrt(3)*(x^14 + 16*x^11 + 2*x^9 + 17*x^8 + 16*x^6 - 16*x^5 + x^4
- 2*x^3 + x^2)*(x^6 + x^3 + x - 1)^(1/3) + sqrt(3)*(x^18 + 33*x^15 + 3*x^13 + 108*x^12 + 66*x^10 + 5*x^9 + 3*x
^8 + 105*x^7 - 108*x^6 + 33*x^5 - 66*x^4 + 34*x^3 - 3*x^2 + 3*x - 1))/(x^18 - 3*x^15 + 3*x^13 - 108*x^12 - 6*x
^10 - 103*x^9 + 3*x^8 - 111*x^7 + 108*x^6 - 3*x^5 + 6*x^4 - 2*x^3 - 3*x^2 + 3*x - 1)) + sqrt(3)*arctan(-(682*s
qrt(3)*(x^6 + x^3 + x - 1)^(1/3)*x^2 - 248*sqrt(3)*(x^6 + x^3 + x - 1)^(2/3)*x + sqrt(3)*(96*x^6 + 217*x^3 + 9
6*x - 96))/(64*x^6 + 1395*x^3 + 64*x - 64)) + 1/3*4^(1/3)*log((3*4^(2/3)*(x^6 + x^3 + x - 1)^(1/3)*x^2 - 6*(x^
6 + x^3 + x - 1)^(2/3)*x + 4^(1/3)*(x^6 - x^3 + x - 1))/(x^6 - x^3 + x - 1)) - 1/6*4^(1/3)*log((6*4^(1/3)*(x^7
+ 5*x^4 + x^2 - x)*(x^6 + x^3 + x - 1)^(2/3) + 4^(2/3)*(x^12 + 16*x^9 + 2*x^7 + 17*x^6 + 16*x^4 - 16*x^3 + x^
2 - 2*x + 1) + 24*(x^8 + 2*x^5 + x^3 - x^2)*(x^6 + x^3 + x - 1)^(1/3))/(x^12 - 2*x^9 + 2*x^7 - x^6 - 2*x^4 + 2
*x^3 + x^2 - 2*x + 1)) - 1/2*log((x^6 + 3*(x^6 + x^3 + x - 1)^(1/3)*x^2 - 3*(x^6 + x^3 + x - 1)^(2/3)*x + x -
1)/(x^6 + x - 1))

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (3 \, x^{6} - 2 \, x + 3\right )} {\left (x^{6} + x^{3} + x - 1\right )}^{\frac {2}{3}}}{{\left (x^{6} - x^{3} + x - 1\right )} {\left (x^{6} + x - 1\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F]  time = 0.03, size = 0, normalized size = 0.00 \[\int \frac {\left (x^{6}+x^{3}+x -1\right )^{\frac {2}{3}} \left (3 x^{6}-2 x +3\right )}{\left (x^{6}+x -1\right ) \left (x^{6}-x^{3}+x -1\right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (3 \, x^{6} - 2 \, x + 3\right )} {\left (x^{6} + x^{3} + x - 1\right )}^{\frac {2}{3}}}{{\left (x^{6} - x^{3} + x - 1\right )} {\left (x^{6} + x - 1\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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