3.29.38 \(\int \frac {(-1+x^3)^{2/3} (4+x^6)}{x^6 (-4+x^6)} \, dx\)

Optimal. Leaf size=289 \[ \frac {\log \left (\sqrt [3]{2} \sqrt [3]{x^3-1}-x\right )}{6\ 2^{2/3}}-\frac {\log \left (\sqrt [3]{2} 3^{2/3} \sqrt [3]{x^3-1}-3 x\right )}{2\ 2^{2/3} \sqrt [3]{3}}-\frac {\tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{2} \sqrt [3]{x^3-1}+x}\right )}{2\ 2^{2/3} \sqrt {3}}+\frac {\sqrt [6]{3} \tan ^{-1}\left (\frac {3^{5/6} x}{2 \sqrt [3]{2} \sqrt [3]{x^3-1}+\sqrt [3]{3} x}\right )}{2\ 2^{2/3}}+\frac {\left (x^3-1\right )^{2/3} \left (1-x^3\right )}{5 x^5}-\frac {\log \left (\sqrt [3]{2} \sqrt [3]{x^3-1} x+2^{2/3} \left (x^3-1\right )^{2/3}+x^2\right )}{12\ 2^{2/3}}+\frac {\log \left (\sqrt [3]{2} 3^{2/3} \sqrt [3]{x^3-1} x+2^{2/3} \sqrt [3]{3} \left (x^3-1\right )^{2/3}+3 x^2\right )}{4\ 2^{2/3} \sqrt [3]{3}} \]

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Rubi [C]  time = 0.34, antiderivative size = 109, normalized size of antiderivative = 0.38, number of steps used = 7, number of rules used = 4, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {6725, 264, 430, 429} \begin {gather*} -\frac {x \left (x^3-1\right )^{2/3} F_1\left (\frac {1}{3};-\frac {2}{3},1;\frac {4}{3};x^3,-\frac {x^3}{2}\right )}{4 \left (1-x^3\right )^{2/3}}-\frac {x \left (x^3-1\right )^{2/3} F_1\left (\frac {1}{3};-\frac {2}{3},1;\frac {4}{3};x^3,\frac {x^3}{2}\right )}{4 \left (1-x^3\right )^{2/3}}-\frac {\left (x^3-1\right )^{5/3}}{5 x^5} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-1/5*(-1 + x^3)^(5/3)/x^5 - (x*(-1 + x^3)^(2/3)*AppellF1[1/3, -2/3, 1, 4/3, x^3, -1/2*x^3])/(4*(1 - x^3)^(2/3)
) - (x*(-1 + x^3)^(2/3)*AppellF1[1/3, -2/3, 1, 4/3, x^3, x^3/2])/(4*(1 - x^3)^(2/3))

Rule 264

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a
*c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[(m + 1)/n + p + 1, 0] && NeQ[m, -1]

Rule 429

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[a^p*c^q*x*AppellF1[1/n, -p,
-q, 1 + 1/n, -((b*x^n)/a), -((d*x^n)/c)], x] /; FreeQ[{a, b, c, d, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[n
, -1] && (IntegerQ[p] || GtQ[a, 0]) && (IntegerQ[q] || GtQ[c, 0])

Rule 430

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Dist[(a^IntPart[p]*(a + b*x^n)^F
racPart[p])/(1 + (b*x^n)/a)^FracPart[p], Int[(1 + (b*x^n)/a)^p*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, n,
p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[n, -1] &&  !(IntegerQ[p] || GtQ[a, 0])

Rule 6725

Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a + b*x^n), x]}, Int[v, x]
/; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ[n, 0]

Rubi steps

\begin {align*} \int \frac {\left (-1+x^3\right )^{2/3} \left (4+x^6\right )}{x^6 \left (-4+x^6\right )} \, dx &=\int \left (-\frac {\left (-1+x^3\right )^{2/3}}{x^6}+\frac {\left (-1+x^3\right )^{2/3}}{2 \left (-2+x^3\right )}-\frac {\left (-1+x^3\right )^{2/3}}{2 \left (2+x^3\right )}\right ) \, dx\\ &=\frac {1}{2} \int \frac {\left (-1+x^3\right )^{2/3}}{-2+x^3} \, dx-\frac {1}{2} \int \frac {\left (-1+x^3\right )^{2/3}}{2+x^3} \, dx-\int \frac {\left (-1+x^3\right )^{2/3}}{x^6} \, dx\\ &=-\frac {\left (-1+x^3\right )^{5/3}}{5 x^5}+\frac {\left (-1+x^3\right )^{2/3} \int \frac {\left (1-x^3\right )^{2/3}}{-2+x^3} \, dx}{2 \left (1-x^3\right )^{2/3}}-\frac {\left (-1+x^3\right )^{2/3} \int \frac {\left (1-x^3\right )^{2/3}}{2+x^3} \, dx}{2 \left (1-x^3\right )^{2/3}}\\ &=-\frac {\left (-1+x^3\right )^{5/3}}{5 x^5}-\frac {x \left (-1+x^3\right )^{2/3} F_1\left (\frac {1}{3};-\frac {2}{3},1;\frac {4}{3};x^3,-\frac {x^3}{2}\right )}{4 \left (1-x^3\right )^{2/3}}-\frac {x \left (-1+x^3\right )^{2/3} F_1\left (\frac {1}{3};-\frac {2}{3},1;\frac {4}{3};x^3,\frac {x^3}{2}\right )}{4 \left (1-x^3\right )^{2/3}}\\ \end {align*}

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Mathematica [F]  time = 0.35, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (-1+x^3\right )^{2/3} \left (4+x^6\right )}{x^6 \left (-4+x^6\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[((-1 + x^3)^(2/3)*(4 + x^6))/(x^6*(-4 + x^6)),x]

[Out]

Integrate[((-1 + x^3)^(2/3)*(4 + x^6))/(x^6*(-4 + x^6)), x]

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IntegrateAlgebraic [A]  time = 0.67, size = 289, normalized size = 1.00 \begin {gather*} \frac {\left (1-x^3\right ) \left (-1+x^3\right )^{2/3}}{5 x^5}-\frac {\tan ^{-1}\left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{2} \sqrt [3]{-1+x^3}}\right )}{2\ 2^{2/3} \sqrt {3}}+\frac {\sqrt [6]{3} \tan ^{-1}\left (\frac {3^{5/6} x}{\sqrt [3]{3} x+2 \sqrt [3]{2} \sqrt [3]{-1+x^3}}\right )}{2\ 2^{2/3}}+\frac {\log \left (-x+\sqrt [3]{2} \sqrt [3]{-1+x^3}\right )}{6\ 2^{2/3}}-\frac {\log \left (-3 x+\sqrt [3]{2} 3^{2/3} \sqrt [3]{-1+x^3}\right )}{2\ 2^{2/3} \sqrt [3]{3}}-\frac {\log \left (x^2+\sqrt [3]{2} x \sqrt [3]{-1+x^3}+2^{2/3} \left (-1+x^3\right )^{2/3}\right )}{12\ 2^{2/3}}+\frac {\log \left (3 x^2+\sqrt [3]{2} 3^{2/3} x \sqrt [3]{-1+x^3}+2^{2/3} \sqrt [3]{3} \left (-1+x^3\right )^{2/3}\right )}{4\ 2^{2/3} \sqrt [3]{3}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((-1 + x^3)^(2/3)*(4 + x^6))/(x^6*(-4 + x^6)),x]

[Out]

((1 - x^3)*(-1 + x^3)^(2/3))/(5*x^5) - ArcTan[(Sqrt[3]*x)/(x + 2*2^(1/3)*(-1 + x^3)^(1/3))]/(2*2^(2/3)*Sqrt[3]
) + (3^(1/6)*ArcTan[(3^(5/6)*x)/(3^(1/3)*x + 2*2^(1/3)*(-1 + x^3)^(1/3))])/(2*2^(2/3)) + Log[-x + 2^(1/3)*(-1
+ x^3)^(1/3)]/(6*2^(2/3)) - Log[-3*x + 2^(1/3)*3^(2/3)*(-1 + x^3)^(1/3)]/(2*2^(2/3)*3^(1/3)) - Log[x^2 + 2^(1/
3)*x*(-1 + x^3)^(1/3) + 2^(2/3)*(-1 + x^3)^(2/3)]/(12*2^(2/3)) + Log[3*x^2 + 2^(1/3)*3^(2/3)*x*(-1 + x^3)^(1/3
) + 2^(2/3)*3^(1/3)*(-1 + x^3)^(2/3)]/(4*2^(2/3)*3^(1/3))

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fricas [B]  time = 4.91, size = 534, normalized size = 1.85 \begin {gather*} \frac {10 \cdot 12^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} x^{5} \log \left (-\frac {18 \cdot 12^{\frac {1}{3}} \left (-1\right )^{\frac {2}{3}} {\left (x^{3} - 1\right )}^{\frac {1}{3}} x^{2} + 12^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} {\left (x^{3} + 2\right )} - 36 \, {\left (x^{3} - 1\right )}^{\frac {2}{3}} x}{x^{3} + 2}\right ) - 5 \cdot 12^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} x^{5} \log \left (-\frac {6 \cdot 12^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} {\left (4 \, x^{4} - x\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}} - 12^{\frac {1}{3}} \left (-1\right )^{\frac {2}{3}} {\left (55 \, x^{6} - 50 \, x^{3} + 4\right )} - 18 \, {\left (7 \, x^{5} - 4 \, x^{2}\right )} {\left (x^{3} - 1\right )}^{\frac {1}{3}}}{x^{6} + 4 \, x^{3} + 4}\right ) + 20 \cdot 4^{\frac {1}{6}} \sqrt {3} x^{5} \arctan \left (\frac {4^{\frac {1}{6}} {\left (12 \cdot 4^{\frac {2}{3}} \sqrt {3} {\left (2 \, x^{7} - 5 \, x^{4} + 2 \, x\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}} + 4^{\frac {1}{3}} \sqrt {3} {\left (91 \, x^{9} - 168 \, x^{6} + 84 \, x^{3} - 8\right )} + 12 \, \sqrt {3} {\left (19 \, x^{8} - 22 \, x^{5} + 4 \, x^{2}\right )} {\left (x^{3} - 1\right )}^{\frac {1}{3}}\right )}}{6 \, {\left (53 \, x^{9} - 48 \, x^{6} - 12 \, x^{3} + 8\right )}}\right ) + 10 \cdot 4^{\frac {2}{3}} x^{5} \log \left (\frac {6 \cdot 4^{\frac {1}{3}} {\left (x^{3} - 1\right )}^{\frac {1}{3}} x^{2} + 4^{\frac {2}{3}} {\left (x^{3} - 2\right )} - 12 \, {\left (x^{3} - 1\right )}^{\frac {2}{3}} x}{x^{3} - 2}\right ) - 5 \cdot 4^{\frac {2}{3}} x^{5} \log \left (\frac {6 \cdot 4^{\frac {2}{3}} {\left (2 \, x^{4} - x\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}} + 4^{\frac {1}{3}} {\left (19 \, x^{6} - 22 \, x^{3} + 4\right )} + 6 \, {\left (5 \, x^{5} - 4 \, x^{2}\right )} {\left (x^{3} - 1\right )}^{\frac {1}{3}}}{x^{6} - 4 \, x^{3} + 4}\right ) - 60 \cdot 12^{\frac {1}{6}} \left (-1\right )^{\frac {1}{3}} x^{5} \arctan \left (\frac {12^{\frac {1}{6}} {\left (12 \cdot 12^{\frac {2}{3}} \left (-1\right )^{\frac {2}{3}} {\left (4 \, x^{7} + 7 \, x^{4} - 2 \, x\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}} + 36 \, \left (-1\right )^{\frac {1}{3}} {\left (55 \, x^{8} - 50 \, x^{5} + 4 \, x^{2}\right )} {\left (x^{3} - 1\right )}^{\frac {1}{3}} - 12^{\frac {1}{3}} {\left (377 \, x^{9} - 600 \, x^{6} + 204 \, x^{3} - 8\right )}\right )}}{6 \, {\left (487 \, x^{9} - 480 \, x^{6} + 12 \, x^{3} + 8\right )}}\right ) - 144 \, {\left (x^{3} - 1\right )}^{\frac {5}{3}}}{720 \, x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^3-1)^(2/3)*(x^6+4)/x^6/(x^6-4),x, algorithm="fricas")

[Out]

1/720*(10*12^(2/3)*(-1)^(1/3)*x^5*log(-(18*12^(1/3)*(-1)^(2/3)*(x^3 - 1)^(1/3)*x^2 + 12^(2/3)*(-1)^(1/3)*(x^3
+ 2) - 36*(x^3 - 1)^(2/3)*x)/(x^3 + 2)) - 5*12^(2/3)*(-1)^(1/3)*x^5*log(-(6*12^(2/3)*(-1)^(1/3)*(4*x^4 - x)*(x
^3 - 1)^(2/3) - 12^(1/3)*(-1)^(2/3)*(55*x^6 - 50*x^3 + 4) - 18*(7*x^5 - 4*x^2)*(x^3 - 1)^(1/3))/(x^6 + 4*x^3 +
4)) + 20*4^(1/6)*sqrt(3)*x^5*arctan(1/6*4^(1/6)*(12*4^(2/3)*sqrt(3)*(2*x^7 - 5*x^4 + 2*x)*(x^3 - 1)^(2/3) + 4
^(1/3)*sqrt(3)*(91*x^9 - 168*x^6 + 84*x^3 - 8) + 12*sqrt(3)*(19*x^8 - 22*x^5 + 4*x^2)*(x^3 - 1)^(1/3))/(53*x^9
- 48*x^6 - 12*x^3 + 8)) + 10*4^(2/3)*x^5*log((6*4^(1/3)*(x^3 - 1)^(1/3)*x^2 + 4^(2/3)*(x^3 - 2) - 12*(x^3 - 1
)^(2/3)*x)/(x^3 - 2)) - 5*4^(2/3)*x^5*log((6*4^(2/3)*(2*x^4 - x)*(x^3 - 1)^(2/3) + 4^(1/3)*(19*x^6 - 22*x^3 +
4) + 6*(5*x^5 - 4*x^2)*(x^3 - 1)^(1/3))/(x^6 - 4*x^3 + 4)) - 60*12^(1/6)*(-1)^(1/3)*x^5*arctan(1/6*12^(1/6)*(1
2*12^(2/3)*(-1)^(2/3)*(4*x^7 + 7*x^4 - 2*x)*(x^3 - 1)^(2/3) + 36*(-1)^(1/3)*(55*x^8 - 50*x^5 + 4*x^2)*(x^3 - 1
)^(1/3) - 12^(1/3)*(377*x^9 - 600*x^6 + 204*x^3 - 8))/(487*x^9 - 480*x^6 + 12*x^3 + 8)) - 144*(x^3 - 1)^(5/3))
/x^5

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (x^{6} + 4\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}}}{{\left (x^{6} - 4\right )} x^{6}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^3-1)^(2/3)*(x^6+4)/x^6/(x^6-4),x, algorithm="giac")

[Out]

integrate((x^6 + 4)*(x^3 - 1)^(2/3)/((x^6 - 4)*x^6), x)

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maple [F]  time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {\left (x^{3}-1\right )^{\frac {2}{3}} \left (x^{6}+4\right )}{x^{6} \left (x^{6}-4\right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^3-1)^(2/3)*(x^6+4)/x^6/(x^6-4),x)

[Out]

int((x^3-1)^(2/3)*(x^6+4)/x^6/(x^6-4),x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (x^{6} + 4\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}}}{{\left (x^{6} - 4\right )} x^{6}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^3-1)^(2/3)*(x^6+4)/x^6/(x^6-4),x, algorithm="maxima")

[Out]

integrate((x^6 + 4)*(x^3 - 1)^(2/3)/((x^6 - 4)*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 (x^6+4\right )}{x^6\,\left (x^6-4\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((x^3 - 1)^(2/3)*(x^6 + 4))/(x^6*(x^6 - 4)),x)

[Out]

int(((x^3 - 1)^(2/3)*(x^6 + 4))/(x^6*(x^6 - 4)), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (\left (x - 1\right ) \left (x^{2} + x + 1\right )\right )^{\frac {2}{3}} \left (x^{6} + 4\right )}{x^{6} \left (x^{3} - 2\right ) \left (x^{3} + 2\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**3-1)**(2/3)*(x**6+4)/x**6/(x**6-4),x)

[Out]

Integral(((x - 1)*(x**2 + x + 1))**(2/3)*(x**6 + 4)/(x**6*(x**3 - 2)*(x**3 + 2)), x)

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