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

Optimal. Leaf size=124 \[ \frac {\log \left (x^3+1\right )}{12 \sqrt [3]{2}}-\frac {\log \left (-\sqrt [3]{1-x^3}-\sqrt [3]{2} x\right )}{4 \sqrt [3]{2}}+\frac {\tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{2} x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{2 \sqrt [3]{2} \sqrt {3}}-\frac {\left (1-x^3\right )^{2/3}}{x^2}+\frac {1}{2 x^2 \sqrt [3]{1-x^3}} \]

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Rubi [C]  time = 8.12, antiderivative size = 204, normalized size of antiderivative = 1.65, 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} \begin {gather*} -\frac {-18 \left (x^3+1\right )^2 x^6 \, _3F_2\left (2,2,\frac {7}{3};1,\frac {10}{3};-\frac {2 x^3}{1-x^3}\right )-54 x^{12} \, _2F_1\left (2,\frac {7}{3};\frac {10}{3};-\frac {2 x^3}{1-x^3}\right )-84 x^9 \, _2F_1\left (2,\frac {7}{3};\frac {10}{3};-\frac {2 x^3}{1-x^3}\right )-30 x^6 \, _2F_1\left (2,\frac {7}{3};\frac {10}{3};-\frac {2 x^3}{1-x^3}\right )-7 \left (1-x^3\right )^2 \left (9 x^6+12 x^3+2\right ) \, _2F_1\left (\frac {1}{3},1;\frac {4}{3};-\frac {2 x^3}{1-x^3}\right )+63 x^{12}-42 x^9-91 x^6+56 x^3+14}{14 x^5 \left (1-x^3\right )^{7/3}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-(14 + 56*x^3 - 91*x^6 - 42*x^9 + 63*x^12 - 7*(1 - x^3)^2*(2 + 12*x^3 + 9*x^6)*Hypergeometric2F1[1/3, 1, 4/3,
(-2*x^3)/(1 - x^3)] - 30*x^6*Hypergeometric2F1[2, 7/3, 10/3, (-2*x^3)/(1 - x^3)] - 84*x^9*Hypergeometric2F1[2,
7/3, 10/3, (-2*x^3)/(1 - x^3)] - 54*x^12*Hypergeometric2F1[2, 7/3, 10/3, (-2*x^3)/(1 - x^3)] - 18*x^6*(1 + x^
3)^2*HypergeometricPFQ[{2, 2, 7/3}, {1, 10/3}, (-2*x^3)/(1 - x^3)])/(14*x^5*(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^3 \left (1-x^3\right )^{4/3} \left (1+x^3\right )} \, dx &=-\frac {14+56 x^3-91 x^6-42 x^9+63 x^{12}-7 \left (1-x^3\right )^2 \left (2+12 x^3+9 x^6\right ) \, _2F_1\left (\frac {1}{3},1;\frac {4}{3};-\frac {2 x^3}{1-x^3}\right )-30 x^6 \, _2F_1\left (2,\frac {7}{3};\frac {10}{3};-\frac {2 x^3}{1-x^3}\right )-84 x^9 \, _2F_1\left (2,\frac {7}{3};\frac {10}{3};-\frac {2 x^3}{1-x^3}\right )-54 x^{12} \, _2F_1\left (2,\frac {7}{3};\frac {10}{3};-\frac {2 x^3}{1-x^3}\right )-18 x^6 \left (1+x^3\right )^2 \, _3F_2\left (2,2,\frac {7}{3};1,\frac {10}{3};-\frac {2 x^3}{1-x^3}\right )}{14 x^5 \left (1-x^3\right )^{7/3}}\\ \end {align*}

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Mathematica [C]  time = 3.11, size = 192, normalized size = 1.55 \begin {gather*} \frac {18 \left (x^3+1\right )^2 x^6 \, _3F_2\left (2,2,\frac {7}{3};1,\frac {10}{3};\frac {2 x^3}{x^3-1}\right )+54 x^{12} \, _2F_1\left (2,\frac {7}{3};\frac {10}{3};\frac {2 x^3}{x^3-1}\right )+84 x^9 \, _2F_1\left (2,\frac {7}{3};\frac {10}{3};\frac {2 x^3}{x^3-1}\right )+30 x^6 \, _2F_1\left (2,\frac {7}{3};\frac {10}{3};\frac {2 x^3}{x^3-1}\right )+7 \left (x^3-1\right )^2 \left (9 x^6+12 x^3+2\right ) \, _2F_1\left (\frac {1}{3},1;\frac {4}{3};\frac {2 x^3}{x^3-1}\right )-63 x^{12}+42 x^9+91 x^6-56 x^3-14}{14 x^5 \left (1-x^3\right )^{7/3}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(-14 - 56*x^3 + 91*x^6 + 42*x^9 - 63*x^12 + 7*(-1 + x^3)^2*(2 + 12*x^3 + 9*x^6)*Hypergeometric2F1[1/3, 1, 4/3,
(2*x^3)/(-1 + x^3)] + 30*x^6*Hypergeometric2F1[2, 7/3, 10/3, (2*x^3)/(-1 + x^3)] + 84*x^9*Hypergeometric2F1[2
, 7/3, 10/3, (2*x^3)/(-1 + x^3)] + 54*x^12*Hypergeometric2F1[2, 7/3, 10/3, (2*x^3)/(-1 + x^3)] + 18*x^6*(1 + x
^3)^2*HypergeometricPFQ[{2, 2, 7/3}, {1, 10/3}, (2*x^3)/(-1 + x^3)])/(14*x^5*(1 - x^3)^(7/3))

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IntegrateAlgebraic [A]  time = 0.37, size = 161, normalized size = 1.30 \begin {gather*} -\frac {\log \left (2^{2/3} \sqrt [3]{1-x^3}+2 x\right )}{6 \sqrt [3]{2}}-\frac {\tan ^{-1}\left (\frac {\sqrt {3} x}{2^{2/3} \sqrt [3]{1-x^3}-x}\right )}{2 \sqrt [3]{2} \sqrt {3}}+\frac {\left (1-x^3\right )^{2/3} \left (1-2 x^3\right )}{2 x^2 \left (x^3-1\right )}+\frac {\log \left (2^{2/3} \sqrt [3]{1-x^3} x-\sqrt [3]{2} \left (1-x^3\right )^{2/3}-2 x^2\right )}{12 \sqrt [3]{2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [B]  time = 2.60, size = 340, normalized size = 2.74 \begin {gather*} -\frac {2 \, \sqrt {6} 2^{\frac {1}{6}} \left (-1\right )^{\frac {1}{3}} {\left (x^{5} - x^{2}\right )} \arctan \left (\frac {2^{\frac {1}{6}} {\left (6 \, \sqrt {6} 2^{\frac {2}{3}} \left (-1\right )^{\frac {2}{3}} {\left (5 \, x^{7} + 4 \, x^{4} - x\right )} {\left (-x^{3} + 1\right )}^{\frac {2}{3}} - 12 \, \sqrt {6} \left (-1\right )^{\frac {1}{3}} {\left (19 \, x^{8} - 16 \, x^{5} + x^{2}\right )} {\left (-x^{3} + 1\right )}^{\frac {1}{3}} - \sqrt {6} 2^{\frac {1}{3}} {\left (71 \, x^{9} - 111 \, x^{6} + 33 \, x^{3} - 1\right )}\right )}}{6 \, {\left (109 \, x^{9} - 105 \, x^{6} + 3 \, x^{3} + 1\right )}}\right ) - 2 \cdot 2^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} {\left (x^{5} - x^{2}\right )} \log \left (\frac {6 \cdot 2^{\frac {1}{3}} \left (-1\right )^{\frac {2}{3}} {\left (-x^{3} + 1\right )}^{\frac {1}{3}} x^{2} - 2^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} {\left (x^{3} + 1\right )} + 6 \, {\left (-x^{3} + 1\right )}^{\frac {2}{3}} x}{x^{3} + 1}\right ) + 2^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} {\left (x^{5} - x^{2}\right )} \log \left (-\frac {3 \cdot 2^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} {\left (5 \, x^{4} - x\right )} {\left (-x^{3} + 1\right )}^{\frac {2}{3}} - 2^{\frac {1}{3}} \left (-1\right )^{\frac {2}{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 (2 \, x^{3} - 1\right )} {\left (-x^{3} + 1\right )}^{\frac {2}{3}}}{72 \, {\left (x^{5} - x^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/72*(2*sqrt(6)*2^(1/6)*(-1)^(1/3)*(x^5 - x^2)*arctan(1/6*2^(1/6)*(6*sqrt(6)*2^(2/3)*(-1)^(2/3)*(5*x^7 + 4*x^
4 - x)*(-x^3 + 1)^(2/3) - 12*sqrt(6)*(-1)^(1/3)*(19*x^8 - 16*x^5 + x^2)*(-x^3 + 1)^(1/3) - sqrt(6)*2^(1/3)*(71
*x^9 - 111*x^6 + 33*x^3 - 1))/(109*x^9 - 105*x^6 + 3*x^3 + 1)) - 2*2^(2/3)*(-1)^(1/3)*(x^5 - x^2)*log((6*2^(1/
3)*(-1)^(2/3)*(-x^3 + 1)^(1/3)*x^2 - 2^(2/3)*(-1)^(1/3)*(x^3 + 1) + 6*(-x^3 + 1)^(2/3)*x)/(x^3 + 1)) + 2^(2/3)
*(-1)^(1/3)*(x^5 - x^2)*log(-(3*2^(2/3)*(-1)^(1/3)*(5*x^4 - x)*(-x^3 + 1)^(2/3) - 2^(1/3)*(-1)^(2/3)*(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*(2*x^3 - 1)*(-x^3 + 1)^(2/3))/(x^5 -
x^2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C]  time = 2.92, size = 570, normalized size = 4.60 \begin {gather*} -\frac {\RootOf \left (36 \textit {\_Z}^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}\right ) \ln \left (-\frac {-36 x^{3} \RootOf \left (36 \textit {\_Z}^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}\right )^{2} \RootOf \left (\textit {\_Z}^{3}-4\right )^{2}-9 x^{3} \RootOf \left (36 \textit {\_Z}^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-4\right )^{3}+12 x^{3} \RootOf \left (36 \textit {\_Z}^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}\right )+3 x^{3} \RootOf \left (\textit {\_Z}^{3}-4\right )+30 \left (-x^{3}+1\right )^{\frac {1}{3}} x^{2} \RootOf \left (36 \textit {\_Z}^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-4\right )+4 \left (-x^{3}+1\right )^{\frac {1}{3}} x^{2} \RootOf \left (\textit {\_Z}^{3}-4\right )^{2}+12 \left (-x^{3}+1\right )^{\frac {2}{3}} x \RootOf \left (36 \textit {\_Z}^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-4\right )^{2}-2 \left (-x^{3}+1\right )^{\frac {2}{3}} x -12 \RootOf \left (36 \textit {\_Z}^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}\right )-3 \RootOf \left (\textit {\_Z}^{3}-4\right )}{\left (x +1\right ) \left (x^{2}-x +1\right )}\right )}{2}-\frac {\RootOf \left (\textit {\_Z}^{3}-4\right ) \ln \left (\frac {-9 x^{3} \RootOf \left (36 \textit {\_Z}^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}\right )^{2} \RootOf \left (\textit {\_Z}^{3}-4\right )^{2}-3 x^{3} \RootOf \left (36 \textit {\_Z}^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-4\right )^{3}-9 x^{3} \RootOf \left (36 \textit {\_Z}^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}\right )-3 x^{3} \RootOf \left (\textit {\_Z}^{3}-4\right )-9 \left (-x^{3}+1\right )^{\frac {1}{3}} x^{2} \RootOf \left (36 \textit {\_Z}^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}\right ) \RootOf \left (\textit {\_Z}^{3}-4\right )+3 \left (-x^{3}+1\right )^{\frac {2}{3}} x +3 \RootOf \left (36 \textit {\_Z}^{2}+6 \textit {\_Z} \RootOf \left (\textit {\_Z}^{3}-4\right )+\RootOf \left (\textit {\_Z}^{3}-4\right )^{2}\right )+\RootOf \left (\textit {\_Z}^{3}-4\right )}{\left (x +1\right ) \left (x^{2}-x +1\right )}\right )}{12}+\frac {2 x^{3}-1}{2 \left (-x^{3}+1\right )^{\frac {1}{3}} x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(2*x^3-1)/x^2/(-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)+36*_Z^
2)*RootOf(_Z^3-4)^3*x^3-9*RootOf(RootOf(_Z^3-4)^2+6*_Z*RootOf(_Z^3-4)+36*_Z^2)^2*RootOf(_Z^3-4)^2*x^3-9*(-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^3-9*RootOf
(RootOf(_Z^3-4)^2+6*_Z*RootOf(_Z^3-4)+36*_Z^2)*x^3+3*(-x^3+1)^(2/3)*x+RootOf(_Z^3-4)+3*RootOf(RootOf(_Z^3-4)^2
+6*_Z*RootOf(_Z^3-4)+36*_Z^2))/(x+1)/(x^2-x+1))-1/2*RootOf(RootOf(_Z^3-4)^2+6*_Z*RootOf(_Z^3-4)+36*_Z^2)*ln(-(
-9*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*R
ootOf(_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+4*RootOf(_Z^3-4)^2*(-x^3+1)^(1/3)*x^2+30*(-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+12*RootOf(RootOf(_Z^3-4)^2+6*_Z*RootOf(_Z^3-4
)+36*_Z^2)*x^3-2*(-x^3+1)^(2/3)*x-3*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))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x^{3} \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 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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