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

Optimal. Leaf size=294 \[ -\frac {\log \left (2^{2/3}-\frac {1-x}{\sqrt [3]{1-x^3}}\right )}{6\ 2^{2/3}}+\frac {\log \left (\frac {2^{2/3} (1-x)^2}{\left (1-x^3\right )^{2/3}}-\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1\right )}{6\ 2^{2/3}}-\frac {\log \left (\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1\right )}{3\ 2^{2/3}}+\frac {\log \left (\frac {(1-x)^2}{\left (1-x^3\right )^{2/3}}+\frac {2^{2/3} (1-x)}{\sqrt [3]{1-x^3}}+2 \sqrt [3]{2}\right )}{12\ 2^{2/3}}-\frac {\tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{2^{2/3} \sqrt {3}}-\frac {\tan ^{-1}\left (\frac {\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1}{\sqrt {3}}\right )}{2\ 2^{2/3} \sqrt {3}}-\frac {\sqrt [3]{1-x^3}}{2 x^2} \]

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Rubi [C]  time = 0.02, antiderivative size = 26, normalized size of antiderivative = 0.09, 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 {F_1\left (-\frac {2}{3};\frac {2}{3},1;\frac {1}{3};x^3,-x^3\right )}{2 x^2} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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

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Mathematica [C]  time = 0.10, size = 120, normalized size = 0.41 \begin {gather*} \frac {\sqrt [3]{1-x^3} \left (\frac {4 x^3 F_1\left (\frac {1}{3};-\frac {1}{3},1;\frac {4}{3};x^3,-x^3\right )}{\left (x^3+1\right ) \left (x^3 \left (3 F_1\left (\frac {4}{3};-\frac {1}{3},2;\frac {7}{3};x^3,-x^3\right )+F_1\left (\frac {4}{3};\frac {2}{3},1;\frac {7}{3};x^3,-x^3\right )\right )-4 F_1\left (\frac {1}{3};-\frac {1}{3},1;\frac {4}{3};x^3,-x^3\right )\right )}-1\right )}{2 x^2} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((1 - x^3)^(1/3)*(-1 + (4*x^3*AppellF1[1/3, -1/3, 1, 4/3, x^3, -x^3])/((1 + x^3)*(-4*AppellF1[1/3, -1/3, 1, 4/
3, x^3, -x^3] + x^3*(3*AppellF1[4/3, -1/3, 2, 7/3, x^3, -x^3] + AppellF1[4/3, 2/3, 1, 7/3, x^3, -x^3])))))/(2*
x^2)

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IntegrateAlgebraic [F]  time = 34.01, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x^3 \left (1-x^3\right )^{2/3} \left (1+x^3\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 2.42, size = 396, normalized size = 1.35 \begin {gather*} -\frac {4 \cdot 4^{\frac {1}{6}} \sqrt {3} \left (-1\right )^{\frac {1}{3}} x^{2} \arctan \left (\frac {4^{\frac {1}{6}} {\left (6 \cdot 4^{\frac {2}{3}} \sqrt {3} \left (-1\right )^{\frac {2}{3}} {\left (x^{16} - 33 \, x^{13} + 110 \, x^{10} - 110 \, x^{7} + 33 \, x^{4} - x\right )} {\left (-x^{3} + 1\right )}^{\frac {1}{3}} + 48 \, \sqrt {3} \left (-1\right )^{\frac {1}{3}} {\left (x^{14} - 2 \, x^{11} - 6 \, x^{8} - 2 \, x^{5} + x^{2}\right )} {\left (-x^{3} + 1\right )}^{\frac {2}{3}} - 4^{\frac {1}{3}} \sqrt {3} {\left (x^{18} + 42 \, x^{15} - 417 \, x^{12} + 812 \, x^{9} - 417 \, x^{6} + 42 \, x^{3} + 1\right )}\right )}}{6 \, {\left (x^{18} - 102 \, x^{15} + 447 \, x^{12} - 628 \, x^{9} + 447 \, x^{6} - 102 \, x^{3} + 1\right )}}\right ) + 4^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} x^{2} \log \left (\frac {24 \cdot 4^{\frac {1}{3}} \left (-1\right )^{\frac {2}{3}} {\left (x^{8} - 4 \, x^{5} + x^{2}\right )} {\left (-x^{3} + 1\right )}^{\frac {2}{3}} - 4^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} {\left (x^{12} - 32 \, x^{9} + 78 \, x^{6} - 32 \, x^{3} + 1\right )} + 12 \, {\left (x^{10} - 11 \, x^{7} + 11 \, x^{4} - x\right )} {\left (-x^{3} + 1\right )}^{\frac {1}{3}}}{x^{12} + 4 \, x^{9} + 6 \, x^{6} + 4 \, x^{3} + 1}\right ) - 2 \cdot 4^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} x^{2} \log \left (-\frac {12 \, {\left (-x^{3} + 1\right )}^{\frac {2}{3}} x^{2} + 3 \cdot 4^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} {\left (x^{4} - x\right )} {\left (-x^{3} + 1\right )}^{\frac {1}{3}} + 4^{\frac {1}{3}} \left (-1\right )^{\frac {2}{3}} {\left (x^{6} + 2 \, x^{3} + 1\right )}}{x^{6} + 2 \, x^{3} + 1}\right ) + 72 \, {\left (-x^{3} + 1\right )}^{\frac {1}{3}}}{144 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/144*(4*4^(1/6)*sqrt(3)*(-1)^(1/3)*x^2*arctan(1/6*4^(1/6)*(6*4^(2/3)*sqrt(3)*(-1)^(2/3)*(x^16 - 33*x^13 + 11
0*x^10 - 110*x^7 + 33*x^4 - x)*(-x^3 + 1)^(1/3) + 48*sqrt(3)*(-1)^(1/3)*(x^14 - 2*x^11 - 6*x^8 - 2*x^5 + x^2)*
(-x^3 + 1)^(2/3) - 4^(1/3)*sqrt(3)*(x^18 + 42*x^15 - 417*x^12 + 812*x^9 - 417*x^6 + 42*x^3 + 1))/(x^18 - 102*x
^15 + 447*x^12 - 628*x^9 + 447*x^6 - 102*x^3 + 1)) + 4^(2/3)*(-1)^(1/3)*x^2*log((24*4^(1/3)*(-1)^(2/3)*(x^8 -
4*x^5 + x^2)*(-x^3 + 1)^(2/3) - 4^(2/3)*(-1)^(1/3)*(x^12 - 32*x^9 + 78*x^6 - 32*x^3 + 1) + 12*(x^10 - 11*x^7 +
11*x^4 - x)*(-x^3 + 1)^(1/3))/(x^12 + 4*x^9 + 6*x^6 + 4*x^3 + 1)) - 2*4^(2/3)*(-1)^(1/3)*x^2*log(-(12*(-x^3 +
1)^(2/3)*x^2 + 3*4^(2/3)*(-1)^(1/3)*(x^4 - x)*(-x^3 + 1)^(1/3) + 4^(1/3)*(-1)^(2/3)*(x^6 + 2*x^3 + 1))/(x^6 +
2*x^3 + 1)) + 72*(-x^3 + 1)^(1/3))/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 {2}{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)^(2/3)/(x^3+1),x, algorithm="giac")

[Out]

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

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maple [C]  time = 14.43, size = 991, normalized size = 3.37

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(x^3-1)/x^2/(-x^3+1)^(2/3)+(-1/12*ln((6*RootOf(RootOf(_Z^3+2)^2+3*_Z*RootOf(_Z^3+2)+9*_Z^2)*RootOf(_Z^3+2)
^3*x^3-18*RootOf(RootOf(_Z^3+2)^2+3*_Z*RootOf(_Z^3+2)+9*_Z^2)^2*RootOf(_Z^3+2)^2*x^3+RootOf(_Z^3+2)*x^6-3*Root
Of(RootOf(_Z^3+2)^2+3*_Z*RootOf(_Z^3+2)+9*_Z^2)*x^6+9*(x^6-2*x^3+1)^(2/3)*RootOf(_Z^3+2)^2*RootOf(RootOf(_Z^3+
2)^2+3*_Z*RootOf(_Z^3+2)+9*_Z^2)*x-18*RootOf(RootOf(_Z^3+2)^2+3*_Z*RootOf(_Z^3+2)+9*_Z^2)*RootOf(_Z^3+2)*(x^6-
2*x^3+1)^(1/3)*x^2-6*RootOf(_Z^3+2)*x^3+18*RootOf(RootOf(_Z^3+2)^2+3*_Z*RootOf(_Z^3+2)+9*_Z^2)*x^3-6*(x^6-2*x^
3+1)^(2/3)*x+RootOf(_Z^3+2)-3*RootOf(RootOf(_Z^3+2)^2+3*_Z*RootOf(_Z^3+2)+9*_Z^2))/(x+1)^2/(x^2-x+1)^2)*RootOf
(_Z^3+2)-1/4*ln((6*RootOf(RootOf(_Z^3+2)^2+3*_Z*RootOf(_Z^3+2)+9*_Z^2)*RootOf(_Z^3+2)^3*x^3-18*RootOf(RootOf(_
Z^3+2)^2+3*_Z*RootOf(_Z^3+2)+9*_Z^2)^2*RootOf(_Z^3+2)^2*x^3+RootOf(_Z^3+2)*x^6-3*RootOf(RootOf(_Z^3+2)^2+3*_Z*
RootOf(_Z^3+2)+9*_Z^2)*x^6+9*(x^6-2*x^3+1)^(2/3)*RootOf(_Z^3+2)^2*RootOf(RootOf(_Z^3+2)^2+3*_Z*RootOf(_Z^3+2)+
9*_Z^2)*x-18*RootOf(RootOf(_Z^3+2)^2+3*_Z*RootOf(_Z^3+2)+9*_Z^2)*RootOf(_Z^3+2)*(x^6-2*x^3+1)^(1/3)*x^2-6*Root
Of(_Z^3+2)*x^3+18*RootOf(RootOf(_Z^3+2)^2+3*_Z*RootOf(_Z^3+2)+9*_Z^2)*x^3-6*(x^6-2*x^3+1)^(2/3)*x+RootOf(_Z^3+
2)-3*RootOf(RootOf(_Z^3+2)^2+3*_Z*RootOf(_Z^3+2)+9*_Z^2))/(x+1)^2/(x^2-x+1)^2)*RootOf(RootOf(_Z^3+2)^2+3*_Z*Ro
otOf(_Z^3+2)+9*_Z^2)+1/12*RootOf(_Z^3+2)*ln((6*RootOf(RootOf(_Z^3+2)^2+3*_Z*RootOf(_Z^3+2)+9*_Z^2)*RootOf(_Z^3
+2)^3*x^3+36*RootOf(RootOf(_Z^3+2)^2+3*_Z*RootOf(_Z^3+2)+9*_Z^2)^2*RootOf(_Z^3+2)^2*x^3-RootOf(_Z^3+2)*x^6-6*R
ootOf(RootOf(_Z^3+2)^2+3*_Z*RootOf(_Z^3+2)+9*_Z^2)*x^6-9*(x^6-2*x^3+1)^(2/3)*RootOf(_Z^3+2)^2*RootOf(RootOf(_Z
^3+2)^2+3*_Z*RootOf(_Z^3+2)+9*_Z^2)*x-6*RootOf(_Z^3+2)^2*(x^6-2*x^3+1)^(1/3)*x^2+2*RootOf(_Z^3+2)*x^3+12*RootO
f(RootOf(_Z^3+2)^2+3*_Z*RootOf(_Z^3+2)+9*_Z^2)*x^3-RootOf(_Z^3+2)-6*RootOf(RootOf(_Z^3+2)^2+3*_Z*RootOf(_Z^3+2
)+9*_Z^2))/(x+1)^2/(x^2-x+1)^2))/(-x^3+1)^(2/3)*((x^3-1)^2)^(1/3)

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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 {2}{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)^(2/3)/(x^3+1),x, algorithm="maxima")

[Out]

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

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

[Out]

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

[Out]

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

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