3.7.32 \(\int \frac {1}{x^2 (1-x^3)^{2/3} (1+x^3)} \, dx\) [632]

Optimal. Leaf size=103 \[ -\frac {\sqrt [3]{1-x^3}}{x}+\frac {\tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{2} x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{2^{2/3} \sqrt {3}}-\frac {\log \left (1+x^3\right )}{6\ 2^{2/3}}+\frac {\log \left (-\sqrt [3]{2} x-\sqrt [3]{1-x^3}\right )}{2\ 2^{2/3}} \]

[Out]

-(-x^3+1)^(1/3)/x-1/12*ln(x^3+1)*2^(1/3)+1/4*ln(-2^(1/3)*x-(-x^3+1)^(1/3))*2^(1/3)+1/6*arctan(1/3*(1-2*2^(1/3)
*x/(-x^3+1)^(1/3))*3^(1/2))*2^(1/3)*3^(1/2)

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Rubi [A]
time = 0.02, antiderivative size = 103, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {491, 503} \begin {gather*} \frac {\text {ArcTan}\left (\frac {1-\frac {2 \sqrt [3]{2} x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{2^{2/3} \sqrt {3}}-\frac {\sqrt [3]{1-x^3}}{x}-\frac {\log \left (x^3+1\right )}{6\ 2^{2/3}}+\frac {\log \left (-\sqrt [3]{1-x^3}-\sqrt [3]{2} x\right )}{2\ 2^{2/3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 491

Int[((e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(e*x)^(m
+ 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(a*c*e*(m + 1))), x] - Dist[1/(a*c*e^n*(m + 1)), Int[(e*x)^(m +
n)*(a + b*x^n)^p*(c + d*x^n)^q*Simp[(b*c + a*d)*(m + n + 1) + n*(b*c*p + a*d*q) + b*d*(m + n*(p + q + 2) + 1)*
x^n, x], x], x] /; FreeQ[{a, b, c, d, e, p, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && LtQ[m, -1] && IntBino
mialQ[a, b, c, d, e, m, n, p, q, x]

Rule 503

Int[(x_)/(((a_) + (b_.)*(x_)^3)^(2/3)*((c_) + (d_.)*(x_)^3)), x_Symbol] :> With[{q = Rt[(b*c - a*d)/c, 3]}, Si
mp[-ArcTan[(1 + (2*q*x)/(a + b*x^3)^(1/3))/Sqrt[3]]/(Sqrt[3]*c*q^2), x] + (-Simp[Log[q*x - (a + b*x^3)^(1/3)]/
(2*c*q^2), x] + Simp[Log[c + d*x^3]/(6*c*q^2), x])] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

Rubi steps

\begin {align*} \int \frac {1}{x^2 \left (1-x^3\right )^{2/3} \left (1+x^3\right )} \, dx &=\text {Subst}\left (\int \frac {1+x^3}{x^2 \left (1+2 x^3\right )} \, dx,x,\frac {x}{\sqrt [3]{1-x^3}}\right )\\ &=-\frac {\sqrt [3]{1-x^3}}{x}-\text {Subst}\left (\int \frac {x}{1+2 x^3} \, dx,x,\frac {x}{\sqrt [3]{1-x^3}}\right )\\ &=-\frac {\sqrt [3]{1-x^3}}{x}+\frac {\text {Subst}\left (\int \frac {1}{1+\sqrt [3]{2} x} \, dx,x,\frac {x}{\sqrt [3]{1-x^3}}\right )}{3 \sqrt [3]{2}}-\frac {\text {Subst}\left (\int \frac {1+\sqrt [3]{2} x}{1-\sqrt [3]{2} x+2^{2/3} x^2} \, dx,x,\frac {x}{\sqrt [3]{1-x^3}}\right )}{3 \sqrt [3]{2}}\\ &=-\frac {\sqrt [3]{1-x^3}}{x}+\frac {\log \left (1+\frac {\sqrt [3]{2} x}{\sqrt [3]{1-x^3}}\right )}{3\ 2^{2/3}}-\frac {\text {Subst}\left (\int \frac {-\sqrt [3]{2}+2\ 2^{2/3} x}{1-\sqrt [3]{2} x+2^{2/3} x^2} \, dx,x,\frac {x}{\sqrt [3]{1-x^3}}\right )}{6\ 2^{2/3}}-\frac {\text {Subst}\left (\int \frac {1}{1-\sqrt [3]{2} x+2^{2/3} x^2} \, dx,x,\frac {x}{\sqrt [3]{1-x^3}}\right )}{2 \sqrt [3]{2}}\\ &=-\frac {\sqrt [3]{1-x^3}}{x}-\frac {\log \left (1+\frac {2^{2/3} x^2}{\left (1-x^3\right )^{2/3}}-\frac {\sqrt [3]{2} x}{\sqrt [3]{1-x^3}}\right )}{6\ 2^{2/3}}+\frac {\log \left (1+\frac {\sqrt [3]{2} x}{\sqrt [3]{1-x^3}}\right )}{3\ 2^{2/3}}-\frac {\text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{2} x}{\sqrt [3]{1-x^3}}\right )}{2^{2/3}}\\ &=-\frac {\sqrt [3]{1-x^3}}{x}+\frac {\tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{2} x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{2^{2/3} \sqrt {3}}-\frac {\log \left (1+\frac {2^{2/3} x^2}{\left (1-x^3\right )^{2/3}}-\frac {\sqrt [3]{2} x}{\sqrt [3]{1-x^3}}\right )}{6\ 2^{2/3}}+\frac {\log \left (1+\frac {\sqrt [3]{2} x}{\sqrt [3]{1-x^3}}\right )}{3\ 2^{2/3}}\\ \end {align*}

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Mathematica [A]
time = 0.29, size = 143, normalized size = 1.39 \begin {gather*} -\frac {\sqrt [3]{1-x^3}}{x}-\frac {\tan ^{-1}\left (\frac {\sqrt {3} x}{-x+2^{2/3} \sqrt [3]{1-x^3}}\right )}{2^{2/3} \sqrt {3}}+\frac {\log \left (2 x+2^{2/3} \sqrt [3]{1-x^3}\right )}{3\ 2^{2/3}}-\frac {\log \left (-2 x^2+2^{2/3} x \sqrt [3]{1-x^3}-\sqrt [3]{2} \left (1-x^3\right )^{2/3}\right )}{6\ 2^{2/3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 14.11, size = 1162, normalized size = 11.28

method	result	size
trager	\(\text {Expression too large to display}\)	\(1162\)
risch	\(\text {Expression too large to display}\)	\(1386\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(-x^3+1)^(1/3)/x-1/6*ln((62113615200*RootOf(RootOf(_Z^3-2)^2+84*_Z*RootOf(_Z^3-2)+7056*_Z^2)^2*RootOf(_Z^3-2)
^3*x^3+2398446876*RootOf(RootOf(_Z^3-2)^2+84*_Z*RootOf(_Z^3-2)+7056*_Z^2)*RootOf(_Z^3-2)^4*x^3+43114942080*Roo
tOf(_Z^3-2)^2*RootOf(RootOf(_Z^3-2)^2+84*_Z*RootOf(_Z^3-2)+7056*_Z^2)*(-x^3+1)^(2/3)*x-496908921600*RootOf(_Z^
3-2)^3*RootOf(RootOf(_Z^3-2)^2+84*_Z*RootOf(_Z^3-2)+7056*_Z^2)^2-19187575008*RootOf(_Z^3-2)^4*RootOf(RootOf(_Z
^3-2)^2+84*_Z*RootOf(_Z^3-2)+7056*_Z^2)[phone]*RootOf(RootOf(_Z^3-2)^2+84*_Z*RootOf(_Z^3-2)+7056*_Z^2)*Ro
otOf(_Z^3-2)*x^3+1570411645*x^3*RootOf(_Z^3-2)^2+173768795928*RootOf(RootOf(_Z^3-2)^2+84*_Z*RootOf(_Z^3-2)+705
6*_Z^2)*(-x^3+1)^(1/3)*x^2+1026546240*RootOf(_Z^3-2)*(-x^3+1)^(1/3)*x^2-1042129902*x*(-x^3+1)^(2/3)-1700729940
0*RootOf(RootOf(_Z^3-2)^2+84*_Z*RootOf(_Z^3-2)+7056*_Z^2)*RootOf(_Z^3-2)-656717597*RootOf(_Z^3-2)^2)/(x+1)/(x^
2-x+1))*RootOf(_Z^3-2)-14*ln((62113615200*RootOf(RootOf(_Z^3-2)^2+84*_Z*RootOf(_Z^3-2)+7056*_Z^2)^2*RootOf(_Z^
3-2)^3*x^3+2398446876*RootOf(RootOf(_Z^3-2)^2+84*_Z*RootOf(_Z^3-2)+7056*_Z^2)*RootOf(_Z^3-2)^4*x^3+43114942080
*RootOf(_Z^3-2)^2*RootOf(RootOf(_Z^3-2)^2+84*_Z*RootOf(_Z^3-2)+7056*_Z^2)*(-x^3+1)^(2/3)*x-496908921600*RootOf
(_Z^3-2)^3*RootOf(RootOf(_Z^3-2)^2+84*_Z*RootOf(_Z^3-2)+7056*_Z^2)^2-19187575008*RootOf(_Z^3-2)^4*RootOf(RootO
f(_Z^3-2)^2+84*_Z*RootOf(_Z^3-2)+7056*_Z^2)[phone]*RootOf(RootOf(_Z^3-2)^2+84*_Z*RootOf(_Z^3-2)+7056*_Z^2
)*RootOf(_Z^3-2)*x^3+1570411645*x^3*RootOf(_Z^3-2)^2+173768795928*RootOf(RootOf(_Z^3-2)^2+84*_Z*RootOf(_Z^3-2)
+7056*_Z^2)*(-x^3+1)^(1/3)*x^2+1026546240*RootOf(_Z^3-2)*(-x^3+1)^(1/3)*x^2-1042129902*x*(-x^3+1)^(2/3)-170072
99400*RootOf(RootOf(_Z^3-2)^2+84*_Z*RootOf(_Z^3-2)+7056*_Z^2)*RootOf(_Z^3-2)-656717597*RootOf(_Z^3-2)^2)/(x+1)
/(x^2-x+1))*RootOf(RootOf(_Z^3-2)^2+84*_Z*RootOf(_Z^3-2)+7056*_Z^2)+1/6*RootOf(_Z^3-2)*ln((949032*RootOf(RootO
f(_Z^3-2)^2+84*_Z*RootOf(_Z^3-2)+7056*_Z^2)^2*RootOf(_Z^3-2)^3*x^3+24612*RootOf(RootOf(_Z^3-2)^2+84*_Z*RootOf(
_Z^3-2)+7056*_Z^2)*RootOf(_Z^3-2)^4*x^3-620928*RootOf(_Z^3-2)^2*RootOf(RootOf(_Z^3-2)^2+84*_Z*RootOf(_Z^3-2)+7
056*_Z^2)*(-x^3+1)^(2/3)*x-7592256*RootOf(_Z^3-2)^3*RootOf(RootOf(_Z^3-2)^2+84*_Z*RootOf(_Z^3-2)+7056*_Z^2)^2-
196896*RootOf(_Z^3-2)^4*RootOf(RootOf(_Z^3-2)^2+84*_Z*RootOf(_Z^3-2)+7056*_Z^2)-598794*RootOf(RootOf(_Z^3-2)^2
+84*_Z*RootOf(_Z^3-2)+7056*_Z^2)*RootOf(_Z^3-2)*x^3-15529*x^3*RootOf(_Z^3-2)^2-1680084*RootOf(RootOf(_Z^3-2)^2
+84*_Z*RootOf(_Z^3-2)+7056*_Z^2)*(-x^3+1)^(1/3)*x^2-14784*RootOf(_Z^3-2)*(-x^3+1)^(1/3)*x^2+5217*x*(-x^3+1)^(2
/3)+79086*RootOf(RootOf(_Z^3-2)^2+84*_Z*RootOf(_Z^3-2)+7056*_Z^2)*RootOf(_Z^3-2)+2051*RootOf(_Z^3-2)^2)/(x+1)/
(x^2-x+1))

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 272 vs. \(2 (81) = 162\).
time = 8.34, size = 272, normalized size = 2.64 \begin {gather*} \frac {4 \cdot 4^{\frac {1}{6}} \sqrt {3} x \arctan \left (\frac {4^{\frac {1}{6}} {\left (6 \cdot 4^{\frac {2}{3}} \sqrt {3} {\left (19 \, x^{8} - 16 \, x^{5} + x^{2}\right )} {\left (-x^{3} + 1\right )}^{\frac {1}{3}} + 12 \, \sqrt {3} {\left (5 \, x^{7} + 4 \, x^{4} - x\right )} {\left (-x^{3} + 1\right )}^{\frac {2}{3}} - 4^{\frac {1}{3}} \sqrt {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 4^{\frac {2}{3}} x \log \left (\frac {3 \cdot 4^{\frac {2}{3}} {\left (-x^{3} + 1\right )}^{\frac {1}{3}} x^{2} + 6 \, {\left (-x^{3} + 1\right )}^{\frac {2}{3}} x + 4^{\frac {1}{3}} {\left (x^{3} + 1\right )}}{x^{3} + 1}\right ) - 4^{\frac {2}{3}} x \log \left (\frac {6 \cdot 4^{\frac {1}{3}} {\left (5 \, x^{4} - x\right )} {\left (-x^{3} + 1\right )}^{\frac {2}{3}} + 4^{\frac {2}{3}} {\left (19 \, x^{6} - 16 \, x^{3} + 1\right )} - 24 \, {\left (2 \, x^{5} - x^{2}\right )} {\left (-x^{3} + 1\right )}^{\frac {1}{3}}}{x^{6} + 2 \, x^{3} + 1}\right ) - 72 \, {\left (-x^{3} + 1\right )}^{\frac {1}{3}}}{72 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/72*(4*4^(1/6)*sqrt(3)*x*arctan(1/6*4^(1/6)*(6*4^(2/3)*sqrt(3)*(19*x^8 - 16*x^5 + x^2)*(-x^3 + 1)^(1/3) + 12*
sqrt(3)*(5*x^7 + 4*x^4 - x)*(-x^3 + 1)^(2/3) - 4^(1/3)*sqrt(3)*(71*x^9 - 111*x^6 + 33*x^3 - 1))/(109*x^9 - 105
*x^6 + 3*x^3 + 1)) + 2*4^(2/3)*x*log((3*4^(2/3)*(-x^3 + 1)^(1/3)*x^2 + 6*(-x^3 + 1)^(2/3)*x + 4^(1/3)*(x^3 + 1
))/(x^3 + 1)) - 4^(2/3)*x*log((6*4^(1/3)*(5*x^4 - x)*(-x^3 + 1)^(2/3) + 4^(2/3)*(19*x^6 - 16*x^3 + 1) - 24*(2*
x^5 - x^2)*(-x^3 + 1)^(1/3))/(x^6 + 2*x^3 + 1)) - 72*(-x^3 + 1)^(1/3))/x

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

[Out]

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{x^2\,{\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^2*(1 - x^3)^(2/3)*(x^3 + 1)),x)

[Out]

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

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