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

Optimal. Leaf size=221 \[ -\frac {10}{9} \log \left (\sqrt [3]{x^3+1}+x\right )+\frac {1}{9} 2^{2/3} \log \left (2^{2/3} \sqrt [3]{x^3+1}-2 x\right )-\frac {10 \tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{x^3+1}-x}\right )}{3 \sqrt {3}}-\frac {2^{2/3} \tan ^{-1}\left (\frac {\sqrt {3} x}{2^{2/3} \sqrt [3]{x^3+1}+x}\right )}{3 \sqrt {3}}+\frac {\left (x^3+1\right )^{2/3} \left (2-13 x^3\right )}{10 x^5}+\frac {5}{9} \log \left (-\sqrt [3]{x^3+1} x+\left (x^3+1\right )^{2/3}+x^2\right )-\frac {\log \left (2^{2/3} \sqrt [3]{x^3+1} x+\sqrt [3]{2} \left (x^3+1\right )^{2/3}+2 x^2\right )}{9 \sqrt [3]{2}} \]

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Rubi [F]  time = 0.78, 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^3\right )^{2/3} \left (1-2 x^3+2 x^6\right )}{x^6 \left (-1-x^3+2 x^6\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

(-3*(1 + x^3)^(2/3))/(2*x^2) + (1 + x^3)^(5/3)/(5*x^5) - (20*x*AppellF1[1/3, -2/3, 1, 4/3, -x^3, -2*x^3])/3 +
Sqrt[3]*ArcTan[(1 + (2*x)/(1 + x^3)^(1/3))/Sqrt[3]] - (3*Log[-x + (1 + x^3)^(1/3)])/2 + Defer[Int][(1 + x^3)^(
2/3)/(-1 + x), x]/9 - ((1 - I*Sqrt[3])*Defer[Int][(1 + x^3)^(2/3)/(1 - I*Sqrt[3] + 2*x), x])/9 - ((1 + I*Sqrt[
3])*Defer[Int][(1 + x^3)^(2/3)/(1 + I*Sqrt[3] + 2*x), x])/9

Rubi steps

\begin {align*} \int \frac {\left (1+x^3\right )^{2/3} \left (1-2 x^3+2 x^6\right )}{x^6 \left (-1-x^3+2 x^6\right )} \, dx &=\int \left (\frac {\left (1+x^3\right )^{2/3}}{9 (-1+x)}-\frac {\left (1+x^3\right )^{2/3}}{x^6}+\frac {3 \left (1+x^3\right )^{2/3}}{x^3}+\frac {(-2-x) \left (1+x^3\right )^{2/3}}{9 \left (1+x+x^2\right )}-\frac {20 \left (1+x^3\right )^{2/3}}{3 \left (1+2 x^3\right )}\right ) \, dx\\ &=\frac {1}{9} \int \frac {\left (1+x^3\right )^{2/3}}{-1+x} \, dx+\frac {1}{9} \int \frac {(-2-x) \left (1+x^3\right )^{2/3}}{1+x+x^2} \, dx+3 \int \frac {\left (1+x^3\right )^{2/3}}{x^3} \, dx-\frac {20}{3} \int \frac {\left (1+x^3\right )^{2/3}}{1+2 x^3} \, dx-\int \frac {\left (1+x^3\right )^{2/3}}{x^6} \, dx\\ &=-\frac {3 \left (1+x^3\right )^{2/3}}{2 x^2}+\frac {\left (1+x^3\right )^{5/3}}{5 x^5}-\frac {20}{3} x F_1\left (\frac {1}{3};-\frac {2}{3},1;\frac {4}{3};-x^3,-2 x^3\right )+\frac {1}{9} \int \frac {\left (1+x^3\right )^{2/3}}{-1+x} \, dx+\frac {1}{9} \int \left (\frac {\left (-1+i \sqrt {3}\right ) \left (1+x^3\right )^{2/3}}{1-i \sqrt {3}+2 x}+\frac {\left (-1-i \sqrt {3}\right ) \left (1+x^3\right )^{2/3}}{1+i \sqrt {3}+2 x}\right ) \, dx+3 \int \frac {1}{\sqrt [3]{1+x^3}} \, dx\\ &=-\frac {3 \left (1+x^3\right )^{2/3}}{2 x^2}+\frac {\left (1+x^3\right )^{5/3}}{5 x^5}-\frac {20}{3} x F_1\left (\frac {1}{3};-\frac {2}{3},1;\frac {4}{3};-x^3,-2 x^3\right )+\sqrt {3} \tan ^{-1}\left (\frac {1+\frac {2 x}{\sqrt [3]{1+x^3}}}{\sqrt {3}}\right )-\frac {3}{2} \log \left (-x+\sqrt [3]{1+x^3}\right )+\frac {1}{9} \int \frac {\left (1+x^3\right )^{2/3}}{-1+x} \, dx+\frac {1}{9} \left (-1-i \sqrt {3}\right ) \int \frac {\left (1+x^3\right )^{2/3}}{1+i \sqrt {3}+2 x} \, dx+\frac {1}{9} \left (-1+i \sqrt {3}\right ) \int \frac {\left (1+x^3\right )^{2/3}}{1-i \sqrt {3}+2 x} \, dx\\ \end {align*}

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Mathematica [A]  time = 0.52, size = 214, normalized size = 0.97 \begin {gather*} \frac {1}{18} \left (-20 \log \left (\frac {x}{\sqrt [3]{x^3+1}}+1\right )+2\ 2^{2/3} \log \left (1-\frac {\sqrt [3]{2} x}{\sqrt [3]{x^3+1}}\right )+20 \sqrt {3} \tan ^{-1}\left (\frac {1-\frac {2 x}{\sqrt [3]{x^3+1}}}{\sqrt {3}}\right )-2\ 2^{2/3} \sqrt {3} \tan ^{-1}\left (\frac {\frac {2 \sqrt [3]{2} x}{\sqrt [3]{x^3+1}}+1}{\sqrt {3}}\right )+10 \log \left (-\frac {x}{\sqrt [3]{x^3+1}}+\frac {x^2}{\left (x^3+1\right )^{2/3}}+1\right )-2^{2/3} \log \left (\frac {\sqrt [3]{2} x}{\sqrt [3]{x^3+1}}+\frac {2^{2/3} x^2}{\left (x^3+1\right )^{2/3}}+1\right )\right )+\left (x^3+1\right )^{2/3} \left (\frac {1}{5 x^5}-\frac {13}{10 x^2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(1/(5*x^5) - 13/(10*x^2))*(1 + x^3)^(2/3) + (20*Sqrt[3]*ArcTan[(1 - (2*x)/(1 + x^3)^(1/3))/Sqrt[3]] - 2*2^(2/3
)*Sqrt[3]*ArcTan[(1 + (2*2^(1/3)*x)/(1 + x^3)^(1/3))/Sqrt[3]] + 10*Log[1 + x^2/(1 + x^3)^(2/3) - x/(1 + x^3)^(
1/3)] - 20*Log[1 + x/(1 + x^3)^(1/3)] + 2*2^(2/3)*Log[1 - (2^(1/3)*x)/(1 + x^3)^(1/3)] - 2^(2/3)*Log[1 + (2^(2
/3)*x^2)/(1 + x^3)^(2/3) + (2^(1/3)*x)/(1 + x^3)^(1/3)])/18

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IntegrateAlgebraic [A]  time = 0.51, size = 221, normalized size = 1.00 \begin {gather*} \frac {\left (2-13 x^3\right ) \left (1+x^3\right )^{2/3}}{10 x^5}-\frac {10 \tan ^{-1}\left (\frac {\sqrt {3} x}{-x+2 \sqrt [3]{1+x^3}}\right )}{3 \sqrt {3}}-\frac {2^{2/3} \tan ^{-1}\left (\frac {\sqrt {3} x}{x+2^{2/3} \sqrt [3]{1+x^3}}\right )}{3 \sqrt {3}}-\frac {10}{9} \log \left (x+\sqrt [3]{1+x^3}\right )+\frac {1}{9} 2^{2/3} \log \left (-2 x+2^{2/3} \sqrt [3]{1+x^3}\right )+\frac {5}{9} \log \left (x^2-x \sqrt [3]{1+x^3}+\left (1+x^3\right )^{2/3}\right )-\frac {\log \left (2 x^2+2^{2/3} x \sqrt [3]{1+x^3}+\sqrt [3]{2} \left (1+x^3\right )^{2/3}\right )}{9 \sqrt [3]{2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((2 - 13*x^3)*(1 + x^3)^(2/3))/(10*x^5) - (10*ArcTan[(Sqrt[3]*x)/(-x + 2*(1 + x^3)^(1/3))])/(3*Sqrt[3]) - (2^(
2/3)*ArcTan[(Sqrt[3]*x)/(x + 2^(2/3)*(1 + x^3)^(1/3))])/(3*Sqrt[3]) - (10*Log[x + (1 + x^3)^(1/3)])/9 + (2^(2/
3)*Log[-2*x + 2^(2/3)*(1 + x^3)^(1/3)])/9 + (5*Log[x^2 - x*(1 + x^3)^(1/3) + (1 + x^3)^(2/3)])/9 - Log[2*x^2 +
2^(2/3)*x*(1 + x^3)^(1/3) + 2^(1/3)*(1 + x^3)^(2/3)]/(9*2^(1/3))

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fricas [B]  time = 3.17, size = 367, normalized size = 1.66 \begin {gather*} -\frac {10 \cdot 4^{\frac {1}{3}} \sqrt {3} x^{5} \arctan \left (\frac {3 \cdot 4^{\frac {2}{3}} \sqrt {3} {\left (5 \, x^{7} - 4 \, x^{4} - x\right )} {\left (x^{3} + 1\right )}^{\frac {2}{3}} - 6 \cdot 4^{\frac {1}{3}} \sqrt {3} {\left (19 \, x^{8} + 16 \, x^{5} + x^{2}\right )} {\left (x^{3} + 1\right )}^{\frac {1}{3}} - \sqrt {3} {\left (71 \, x^{9} + 111 \, x^{6} + 33 \, x^{3} + 1\right )}}{3 \, {\left (109 \, x^{9} + 105 \, x^{6} + 3 \, x^{3} - 1\right )}}\right ) - 300 \, \sqrt {3} x^{5} \arctan \left (\frac {4 \, \sqrt {3} {\left (x^{3} + 1\right )}^{\frac {1}{3}} x^{2} + 2 \, \sqrt {3} {\left (x^{3} + 1\right )}^{\frac {2}{3}} x + \sqrt {3} {\left (x^{3} + 1\right )}}{7 \, x^{3} - 1}\right ) - 10 \cdot 4^{\frac {1}{3}} x^{5} \log \left (\frac {3 \cdot 4^{\frac {2}{3}} {\left (x^{3} + 1\right )}^{\frac {1}{3}} x^{2} - 6 \, {\left (x^{3} + 1\right )}^{\frac {2}{3}} x - 4^{\frac {1}{3}} {\left (x^{3} - 1\right )}}{x^{3} - 1}\right ) + 5 \cdot 4^{\frac {1}{3}} x^{5} \log \left (\frac {6 \cdot 4^{\frac {1}{3}} {\left (5 \, x^{4} + x\right )} {\left (x^{3} + 1\right )}^{\frac {2}{3}} + 4^{\frac {2}{3}} {\left (19 \, x^{6} + 16 \, x^{3} + 1\right )} + 24 \, {\left (2 \, x^{5} + x^{2}\right )} {\left (x^{3} + 1\right )}^{\frac {1}{3}}}{x^{6} - 2 \, x^{3} + 1}\right ) + 150 \, x^{5} \log \left (\frac {2 \, x^{3} + 3 \, {\left (x^{3} + 1\right )}^{\frac {1}{3}} x^{2} + 3 \, {\left (x^{3} + 1\right )}^{\frac {2}{3}} x + 1}{2 \, x^{3} + 1}\right ) + 27 \, {\left (13 \, x^{3} - 2\right )} {\left (x^{3} + 1\right )}^{\frac {2}{3}}}{270 \, x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/270*(10*4^(1/3)*sqrt(3)*x^5*arctan(1/3*(3*4^(2/3)*sqrt(3)*(5*x^7 - 4*x^4 - x)*(x^3 + 1)^(2/3) - 6*4^(1/3)*s
qrt(3)*(19*x^8 + 16*x^5 + x^2)*(x^3 + 1)^(1/3) - sqrt(3)*(71*x^9 + 111*x^6 + 33*x^3 + 1))/(109*x^9 + 105*x^6 +
3*x^3 - 1)) - 300*sqrt(3)*x^5*arctan((4*sqrt(3)*(x^3 + 1)^(1/3)*x^2 + 2*sqrt(3)*(x^3 + 1)^(2/3)*x + sqrt(3)*(
x^3 + 1))/(7*x^3 - 1)) - 10*4^(1/3)*x^5*log((3*4^(2/3)*(x^3 + 1)^(1/3)*x^2 - 6*(x^3 + 1)^(2/3)*x - 4^(1/3)*(x^
3 - 1))/(x^3 - 1)) + 5*4^(1/3)*x^5*log((6*4^(1/3)*(5*x^4 + x)*(x^3 + 1)^(2/3) + 4^(2/3)*(19*x^6 + 16*x^3 + 1)
+ 24*(2*x^5 + x^2)*(x^3 + 1)^(1/3))/(x^6 - 2*x^3 + 1)) + 150*x^5*log((2*x^3 + 3*(x^3 + 1)^(1/3)*x^2 + 3*(x^3 +
1)^(2/3)*x + 1)/(2*x^3 + 1)) + 27*(13*x^3 - 2)*(x^3 + 1)^(2/3))/x^5

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F]  time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {\left (x^{3}+1\right )^{\frac {2}{3}} \left (2 x^{6}-2 x^{3}+1\right )}{x^{6} \left (2 x^{6}-x^{3}-1\right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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