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

Optimal. Leaf size=144 \[ -\frac {\log \left (x^3+1\right )}{12 \sqrt [3]{2}}+\frac {\log \left (-\sqrt [3]{1-x^3}-\sqrt [3]{2} x\right )}{4 \sqrt [3]{2}}-\frac {\tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{2} x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{2 \sqrt [3]{2} \sqrt {3}}-\frac {7 \left (1-x^3\right )^{2/3}}{10 x^5}+\frac {1}{2 x^5 \sqrt [3]{1-x^3}}-\frac {4 \left (1-x^3\right )^{2/3}}{5 x^2} \]

[Out]

1/2/x^5/(-x^3+1)^(1/3)-7/10*(-x^3+1)^(2/3)/x^5-4/5*(-x^3+1)^(2/3)/x^2-1/24*ln(x^3+1)*2^(2/3)+1/8*ln(-2^(1/3)*x
-(-x^3+1)^(1/3))*2^(2/3)-1/12*arctan(1/3*(1-2*2^(1/3)*x/(-x^3+1)^(1/3))*3^(1/2))*2^(2/3)*3^(1/2)

________________________________________________________________________________________

Rubi [C]  time = 8.32, antiderivative size = 397, normalized size of antiderivative = 2.76, number of steps used = 1, number of rules used = 1, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {510} \[ -\frac {54 \left (x^3+1\right )^2 \left (6 x^3+1\right ) x^6 \, _3F_2\left (2,2,\frac {7}{3};1,\frac {10}{3};-\frac {2 x^3}{1-x^3}\right )+54 \left (x^3+1\right )^3 x^6 \, _4F_3\left (2,2,2,\frac {7}{3};1,1,\frac {10}{3};-\frac {2 x^3}{1-x^3}\right )+567 x^{15} \, _2F_1\left (\frac {1}{3},1;\frac {4}{3};-\frac {2 x^3}{1-x^3}\right )+594 x^{15} \, _2F_1\left (2,\frac {7}{3};\frac {10}{3};-\frac {2 x^3}{1-x^3}\right )-378 x^{12} \, _2F_1\left (\frac {1}{3},1;\frac {4}{3};-\frac {2 x^3}{1-x^3}\right )+972 x^{12} \, _2F_1\left (2,\frac {7}{3};\frac {10}{3};-\frac {2 x^3}{1-x^3}\right )-819 x^9 \, _2F_1\left (\frac {1}{3},1;\frac {4}{3};-\frac {2 x^3}{1-x^3}\right )+342 x^9 \, _2F_1\left (2,\frac {7}{3};\frac {10}{3};-\frac {2 x^3}{1-x^3}\right )+476 x^6 \, _2F_1\left (\frac {1}{3},1;\frac {4}{3};-\frac {2 x^3}{1-x^3}\right )-36 x^6 \, _2F_1\left (2,\frac {7}{3};\frac {10}{3};-\frac {2 x^3}{1-x^3}\right )+182 x^3 \, _2F_1\left (\frac {1}{3},1;\frac {4}{3};-\frac {2 x^3}{1-x^3}\right )-28 \, _2F_1\left (\frac {1}{3},1;\frac {4}{3};-\frac {2 x^3}{1-x^3}\right )-567 x^{15}+378 x^{12}+819 x^9-476 x^6-182 x^3+28}{70 x^8 \left (1-x^3\right )^{7/3}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-(28 - 182*x^3 - 476*x^6 + 819*x^9 + 378*x^12 - 567*x^15 - 28*Hypergeometric2F1[1/3, 1, 4/3, (-2*x^3)/(1 - x^3
)] + 182*x^3*Hypergeometric2F1[1/3, 1, 4/3, (-2*x^3)/(1 - x^3)] + 476*x^6*Hypergeometric2F1[1/3, 1, 4/3, (-2*x
^3)/(1 - x^3)] - 819*x^9*Hypergeometric2F1[1/3, 1, 4/3, (-2*x^3)/(1 - x^3)] - 378*x^12*Hypergeometric2F1[1/3,
1, 4/3, (-2*x^3)/(1 - x^3)] + 567*x^15*Hypergeometric2F1[1/3, 1, 4/3, (-2*x^3)/(1 - x^3)] - 36*x^6*Hypergeomet
ric2F1[2, 7/3, 10/3, (-2*x^3)/(1 - x^3)] + 342*x^9*Hypergeometric2F1[2, 7/3, 10/3, (-2*x^3)/(1 - x^3)] + 972*x
^12*Hypergeometric2F1[2, 7/3, 10/3, (-2*x^3)/(1 - x^3)] + 594*x^15*Hypergeometric2F1[2, 7/3, 10/3, (-2*x^3)/(1
- x^3)] + 54*x^6*(1 + x^3)^2*(1 + 6*x^3)*HypergeometricPFQ[{2, 2, 7/3}, {1, 10/3}, (-2*x^3)/(1 - x^3)] + 54*x
^6*(1 + x^3)^3*HypergeometricPFQ[{2, 2, 2, 7/3}, {1, 1, 10/3}, (-2*x^3)/(1 - x^3)])/(70*x^8*(1 - x^3)^(7/3))

Rule 510

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

Rubi steps

\begin {align*} \int \frac {1}{x^6 \left (1-x^3\right )^{4/3} \left (1+x^3\right )} \, dx &=-\frac {28-182 x^3-476 x^6+819 x^9+378 x^{12}-567 x^{15}-28 \, _2F_1\left (\frac {1}{3},1;\frac {4}{3};-\frac {2 x^3}{1-x^3}\right )+182 x^3 \, _2F_1\left (\frac {1}{3},1;\frac {4}{3};-\frac {2 x^3}{1-x^3}\right )+476 x^6 \, _2F_1\left (\frac {1}{3},1;\frac {4}{3};-\frac {2 x^3}{1-x^3}\right )-819 x^9 \, _2F_1\left (\frac {1}{3},1;\frac {4}{3};-\frac {2 x^3}{1-x^3}\right )-378 x^{12} \, _2F_1\left (\frac {1}{3},1;\frac {4}{3};-\frac {2 x^3}{1-x^3}\right )+567 x^{15} \, _2F_1\left (\frac {1}{3},1;\frac {4}{3};-\frac {2 x^3}{1-x^3}\right )-36 x^6 \, _2F_1\left (2,\frac {7}{3};\frac {10}{3};-\frac {2 x^3}{1-x^3}\right )+342 x^9 \, _2F_1\left (2,\frac {7}{3};\frac {10}{3};-\frac {2 x^3}{1-x^3}\right )+972 x^{12} \, _2F_1\left (2,\frac {7}{3};\frac {10}{3};-\frac {2 x^3}{1-x^3}\right )+594 x^{15} \, _2F_1\left (2,\frac {7}{3};\frac {10}{3};-\frac {2 x^3}{1-x^3}\right )+54 x^6 \left (1+x^3\right )^2 \left (1+6 x^3\right ) \, _3F_2\left (2,2,\frac {7}{3};1,\frac {10}{3};-\frac {2 x^3}{1-x^3}\right )+54 x^6 \left (1+x^3\right )^3 \, _4F_3\left (2,2,2,\frac {7}{3};1,1,\frac {10}{3};-\frac {2 x^3}{1-x^3}\right )}{70 x^8 \left (1-x^3\right )^{7/3}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 5.16, size = 133, normalized size = 0.92 \[ \frac {2 \log \left (\frac {\sqrt [3]{2} x}{\sqrt [3]{x^3-1}}+1\right )+2 \sqrt {3} \tan ^{-1}\left (\frac {\frac {2 \sqrt [3]{2} x}{\sqrt [3]{x^3-1}}-1}{\sqrt {3}}\right )-\log \left (-\frac {\sqrt [3]{2} x}{\sqrt [3]{x^3-1}}+\frac {2^{2/3} x^2}{\left (x^3-1\right )^{2/3}}+1\right )}{12 \sqrt [3]{2}}-\frac {-8 x^6+x^3+2}{10 x^5 \sqrt [3]{1-x^3}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-1/10*(2 + x^3 - 8*x^6)/(x^5*(1 - x^3)^(1/3)) + (2*Sqrt[3]*ArcTan[(-1 + (2*2^(1/3)*x)/(-1 + x^3)^(1/3))/Sqrt[3
]] - Log[1 + (2^(2/3)*x^2)/(-1 + x^3)^(2/3) - (2^(1/3)*x)/(-1 + x^3)^(1/3)] + 2*Log[1 + (2^(1/3)*x)/(-1 + x^3)
^(1/3)])/(12*2^(1/3))

________________________________________________________________________________________

fricas [B]  time = 4.99, size = 316, normalized size = 2.19 \[ -\frac {10 \, \sqrt {6} 2^{\frac {1}{6}} {\left (x^{8} - x^{5}\right )} \arctan \left (\frac {2^{\frac {1}{6}} {\left (6 \, \sqrt {6} 2^{\frac {2}{3}} {\left (5 \, x^{7} + 4 \, x^{4} - x\right )} {\left (-x^{3} + 1\right )}^{\frac {2}{3}} - \sqrt {6} 2^{\frac {1}{3}} {\left (71 \, x^{9} - 111 \, x^{6} + 33 \, x^{3} - 1\right )} + 12 \, \sqrt {6} {\left (19 \, x^{8} - 16 \, x^{5} + x^{2}\right )} {\left (-x^{3} + 1\right )}^{\frac {1}{3}}\right )}}{6 \, {\left (109 \, x^{9} - 105 \, x^{6} + 3 \, x^{3} + 1\right )}}\right ) - 10 \cdot 2^{\frac {2}{3}} {\left (x^{8} - x^{5}\right )} \log \left (\frac {6 \cdot 2^{\frac {1}{3}} {\left (-x^{3} + 1\right )}^{\frac {1}{3}} x^{2} + 2^{\frac {2}{3}} {\left (x^{3} + 1\right )} + 6 \, {\left (-x^{3} + 1\right )}^{\frac {2}{3}} x}{x^{3} + 1}\right ) + 5 \cdot 2^{\frac {2}{3}} {\left (x^{8} - x^{5}\right )} \log \left (\frac {3 \cdot 2^{\frac {2}{3}} {\left (5 \, x^{4} - x\right )} {\left (-x^{3} + 1\right )}^{\frac {2}{3}} + 2^{\frac {1}{3}} {\left (19 \, x^{6} - 16 \, x^{3} + 1\right )} - 12 \, {\left (2 \, x^{5} - x^{2}\right )} {\left (-x^{3} + 1\right )}^{\frac {1}{3}}}{x^{6} + 2 \, x^{3} + 1}\right ) + 36 \, {\left (8 \, x^{6} - x^{3} - 2\right )} {\left (-x^{3} + 1\right )}^{\frac {2}{3}}}{360 \, {\left (x^{8} - x^{5}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/360*(10*sqrt(6)*2^(1/6)*(x^8 - x^5)*arctan(1/6*2^(1/6)*(6*sqrt(6)*2^(2/3)*(5*x^7 + 4*x^4 - x)*(-x^3 + 1)^(2
/3) - sqrt(6)*2^(1/3)*(71*x^9 - 111*x^6 + 33*x^3 - 1) + 12*sqrt(6)*(19*x^8 - 16*x^5 + x^2)*(-x^3 + 1)^(1/3))/(
109*x^9 - 105*x^6 + 3*x^3 + 1)) - 10*2^(2/3)*(x^8 - x^5)*log((6*2^(1/3)*(-x^3 + 1)^(1/3)*x^2 + 2^(2/3)*(x^3 +
1) + 6*(-x^3 + 1)^(2/3)*x)/(x^3 + 1)) + 5*2^(2/3)*(x^8 - x^5)*log((3*2^(2/3)*(5*x^4 - x)*(-x^3 + 1)^(2/3) + 2^
(1/3)*(19*x^6 - 16*x^3 + 1) - 12*(2*x^5 - x^2)*(-x^3 + 1)^(1/3))/(x^6 + 2*x^3 + 1)) + 36*(8*x^6 - x^3 - 2)*(-x
^3 + 1)^(2/3))/(x^8 - x^5)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maple [C]  time = 4.02, size = 851, normalized size = 5.91 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/10*(8*x^6-x^3-2)/x^5/(-x^3+1)^(1/3)+1/12*RootOf(_Z^3-4)*ln(-(3*RootOf(RootOf(_Z^3-4)^2+6*_Z*RootOf(_Z^3-4)+3
6*_Z^2)*RootOf(_Z^3-4)^3*x^3+54*RootOf(RootOf(_Z^3-4)^2+6*_Z*RootOf(_Z^3-4)+36*_Z^2)^2*RootOf(_Z^3-4)^2*x^3+12
*(-x^3+1)^(2/3)*RootOf(RootOf(_Z^3-4)^2+6*_Z*RootOf(_Z^3-4)+36*_Z^2)*RootOf(_Z^3-4)^2*x-5*RootOf(_Z^3-4)^2*(-x
^3+1)^(1/3)*x^2-6*(-x^3+1)^(1/3)*RootOf(RootOf(_Z^3-4)^2+6*_Z*RootOf(_Z^3-4)+36*_Z^2)*RootOf(_Z^3-4)*x^2-RootO
f(_Z^3-4)*x^3-18*RootOf(RootOf(_Z^3-4)^2+6*_Z*RootOf(_Z^3-4)+36*_Z^2)*x^3-2*(-x^3+1)^(2/3)*x+RootOf(_Z^3-4)+18
*RootOf(RootOf(_Z^3-4)^2+6*_Z*RootOf(_Z^3-4)+36*_Z^2))/(x+1)/(x^2-x+1))-1/12*ln((3*RootOf(RootOf(_Z^3-4)^2+6*_
Z*RootOf(_Z^3-4)+36*_Z^2)*RootOf(_Z^3-4)^3*x^3+36*RootOf(RootOf(_Z^3-4)^2+6*_Z*RootOf(_Z^3-4)+36*_Z^2)^2*RootO
f(_Z^3-4)^2*x^3+3*RootOf(_Z^3-4)^2*(-x^3+1)^(1/3)*x^2+18*(-x^3+1)^(1/3)*RootOf(RootOf(_Z^3-4)^2+6*_Z*RootOf(_Z
^3-4)+36*_Z^2)*RootOf(_Z^3-4)*x^2+3*RootOf(_Z^3-4)*x^3+36*RootOf(RootOf(_Z^3-4)^2+6*_Z*RootOf(_Z^3-4)+36*_Z^2)
*x^3+6*(-x^3+1)^(2/3)*x-RootOf(_Z^3-4)-12*RootOf(RootOf(_Z^3-4)^2+6*_Z*RootOf(_Z^3-4)+36*_Z^2))/(x+1)/(x^2-x+1
))*RootOf(_Z^3-4)-1/2*ln((3*RootOf(RootOf(_Z^3-4)^2+6*_Z*RootOf(_Z^3-4)+36*_Z^2)*RootOf(_Z^3-4)^3*x^3+36*RootO
f(RootOf(_Z^3-4)^2+6*_Z*RootOf(_Z^3-4)+36*_Z^2)^2*RootOf(_Z^3-4)^2*x^3+3*RootOf(_Z^3-4)^2*(-x^3+1)^(1/3)*x^2+1
8*(-x^3+1)^(1/3)*RootOf(RootOf(_Z^3-4)^2+6*_Z*RootOf(_Z^3-4)+36*_Z^2)*RootOf(_Z^3-4)*x^2+3*RootOf(_Z^3-4)*x^3+
36*RootOf(RootOf(_Z^3-4)^2+6*_Z*RootOf(_Z^3-4)+36*_Z^2)*x^3+6*(-x^3+1)^(2/3)*x-RootOf(_Z^3-4)-12*RootOf(RootOf
(_Z^3-4)^2+6*_Z*RootOf(_Z^3-4)+36*_Z^2))/(x+1)/(x^2-x+1))*RootOf(RootOf(_Z^3-4)^2+6*_Z*RootOf(_Z^3-4)+36*_Z^2)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{x^6\,{\left (1-x^3\right )}^{4/3}\,\left (x^3+1\right )} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________