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

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

[Out]

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

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Rubi [C]  time = 8.12141, 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} \[ -\frac{-18 \left (x^3+1\right )^2 x^6 \, _3F_2\left (2,2,\frac{7}{3};1,\frac{10}{3};-\frac{2 x^3}{1-x^3}\right )-54 x^{12} \, _2F_1\left (2,\frac{7}{3};\frac{10}{3};-\frac{2 x^3}{1-x^3}\right )-84 x^9 \, _2F_1\left (2,\frac{7}{3};\frac{10}{3};-\frac{2 x^3}{1-x^3}\right )-30 x^6 \, _2F_1\left (2,\frac{7}{3};\frac{10}{3};-\frac{2 x^3}{1-x^3}\right )-7 \left (1-x^3\right )^2 \left (9 x^6+12 x^3+2\right ) \, _2F_1\left (\frac{1}{3},1;\frac{4}{3};-\frac{2 x^3}{1-x^3}\right )+63 x^{12}-42 x^9-91 x^6+56 x^3+14}{14 x^5 \left (1-x^3\right )^{7/3}} \]

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*}

Mathematica [C]  time = 2.11795, size = 192, normalized size = 1.55 \[ \frac{18 \left (x^3+1\right )^2 x^6 \text{HypergeometricPFQ}\left (\left \{2,2,\frac{7}{3}\right \},\left \{1,\frac{10}{3}\right \},\frac{2 x^3}{x^3-1}\right )+54 x^{12} \, _2F_1\left (2,\frac{7}{3};\frac{10}{3};\frac{2 x^3}{x^3-1}\right )+84 x^9 \, _2F_1\left (2,\frac{7}{3};\frac{10}{3};\frac{2 x^3}{x^3-1}\right )+30 x^6 \, _2F_1\left (2,\frac{7}{3};\frac{10}{3};\frac{2 x^3}{x^3-1}\right )+7 \left (x^3-1\right )^2 \left (9 x^6+12 x^3+2\right ) \, _2F_1\left (\frac{1}{3},1;\frac{4}{3};\frac{2 x^3}{x^3-1}\right )-63 x^{12}+42 x^9+91 x^6-56 x^3-14}{14 x^5 \left (1-x^3\right )^{7/3}} \]

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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Maple [F]  time = 0.024, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{3} \left ({x}^{3}+1 \right ) } \left ( -{x}^{3}+1 \right ) ^{-{\frac{4}{3}}}}\, dx \end{align*}

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]

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

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

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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Fricas [B]  time = 23.2538, size = 900, normalized size = 7.26 \begin{align*} -\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{align*}

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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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \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{align*}

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

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)