3.7.33 \(\int \frac {1}{x^5 (1-x^3)^{2/3} (1+x^3)} \, dx\) [633]

Optimal. Leaf size=124 \[ -\frac {\sqrt [3]{1-x^3}}{4 x^4}+\frac {\sqrt [3]{1-x^3}}{4 x}-\frac {\tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{2} x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{2^{2/3} \sqrt {3}}+\frac {\log \left (1+x^3\right )}{6\ 2^{2/3}}-\frac {\log \left (-\sqrt [3]{2} x-\sqrt [3]{1-x^3}\right )}{2\ 2^{2/3}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.05, antiderivative size = 124, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {491, 597, 12, 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}}{4 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}}-\frac {\sqrt [3]{1-x^3}}{4 x^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x^5*(1 - x^3)^(2/3)*(1 + x^3)),x]

[Out]

-1/4*(1 - x^3)^(1/3)/x^4 + (1 - x^3)^(1/3)/(4*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 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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]

Rule 597

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

Rubi steps

\begin {align*} \int \frac {1}{x^5 \left (1-x^3\right )^{2/3} \left (1+x^3\right )} \, dx &=\text {Subst}\left (\int \frac {\left (1+x^3\right )^2}{x^5 \left (1+2 x^3\right )} \, dx,x,\frac {x}{\sqrt [3]{1-x^3}}\right )\\ &=\text {Subst}\left (\int \left (\frac {1}{x^5}+\frac {x}{1+2 x^3}\right ) \, dx,x,\frac {x}{\sqrt [3]{1-x^3}}\right )\\ &=-\frac {\left (1-x^3\right )^{4/3}}{4 x^4}+\text {Subst}\left (\int \frac {x}{1+2 x^3} \, dx,x,\frac {x}{\sqrt [3]{1-x^3}}\right )\\ &=-\frac {\left (1-x^3\right )^{4/3}}{4 x^4}-\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 {\left (1-x^3\right )^{4/3}}{4 x^4}-\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 {\left (1-x^3\right )^{4/3}}{4 x^4}+\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 {\left (1-x^3\right )^{4/3}}{4 x^4}-\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*}

________________________________________________________________________________________

Mathematica [A]
time = 0.26, size = 141, normalized size = 1.14 \begin {gather*} \frac {1}{12} \left (-\frac {3 \left (1-x^3\right )^{4/3}}{x^4}-2 \sqrt [3]{2} \sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{x-2^{2/3} \sqrt [3]{1-x^3}}\right )-2 \sqrt [3]{2} \log \left (2 x+2^{2/3} \sqrt [3]{1-x^3}\right )+\sqrt [3]{2} \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 )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x^5*(1 - x^3)^(2/3)*(1 + x^3)),x]

[Out]

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

________________________________________________________________________________________

Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 14.06, size = 1186, normalized size = 9.56

method	result	size
trager	\(\text {Expression too large to display}\)	\(1186\)
risch	\(\text {Expression too large to display}\)	\(1478\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^5/(-x^3+1)^(2/3)/(x^3+1),x,method=_RETURNVERBOSE)

[Out]

1/4*(x^3-1)/x^4*(-x^3+1)^(1/3)+112*RootOf(RootOf(_Z^3+2)^2+672*_Z*RootOf(_Z^3+2)+451584*_Z^2)*ln(-(-2957791200
*RootOf(RootOf(_Z^3+2)^2+672*_Z*RootOf(_Z^3+2)+451584*_Z^2)*RootOf(_Z^3+2)^4*x^3+4459389516288*RootOf(RootOf(_
Z^3+2)^2+672*_Z*RootOf(_Z^3+2)+451584*_Z^2)^2*RootOf(_Z^3+2)^3*x^3+172459768320*(-x^3+1)^(2/3)*RootOf(_Z^3+2)^
2*RootOf(RootOf(_Z^3+2)^2+672*_Z*RootOf(_Z^3+2)+451584*_Z^2)*x+23662329600*RootOf(RootOf(_Z^3+2)^2+672*_Z*Root
Of(_Z^3+2)+451584*_Z^2)*RootOf(_Z^3+2)^4-35675116130304*RootOf(RootOf(_Z^3+2)^2+672*_Z*RootOf(_Z^3+2)+451584*_
Z^2)^2*RootOf(_Z^3+2)^3+242081125*RootOf(_Z^3+2)^2*x^3-364979796720*RootOf(RootOf(_Z^3+2)^2+672*_Z*RootOf(_Z^3
+2)+451584*_Z^2)*RootOf(_Z^3+2)*x^3+521064951*(-x^3+1)^(1/3)*RootOf(_Z^3+2)*x^2-344919536640*(-x^3+1)^(1/3)*Ro
otOf(RootOf(_Z^3+2)^2+672*_Z*RootOf(_Z^3+2)+451584*_Z^2)*x^2+521064951*x*(-x^3+1)^(2/3)-101233925*RootOf(_Z^3+
2)^2+152627914992*RootOf(RootOf(_Z^3+2)^2+672*_Z*RootOf(_Z^3+2)+451584*_Z^2)*RootOf(_Z^3+2))/(x+1)/(x^2-x+1))-
1/6*ln((-19187575008*RootOf(RootOf(_Z^3+2)^2+672*_Z*RootOf(_Z^3+2)+451584*_Z^2)*RootOf(_Z^3+2)^4*x^3-891877903
2576*RootOf(RootOf(_Z^3+2)^2+672*_Z*RootOf(_Z^3+2)+451584*_Z^2)^2*RootOf(_Z^3+2)^3*x^3+344919536640*(-x^3+1)^(
2/3)*RootOf(_Z^3+2)^2*RootOf(RootOf(_Z^3+2)^2+672*_Z*RootOf(_Z^3+2)+451584*_Z^2)*x+153500600064*RootOf(RootOf(
_Z^3+2)^2+672*_Z*RootOf(_Z^3+2)+451584*_Z^2)*RootOf(_Z^3+2)^4+71350232260608*RootOf(RootOf(_Z^3+2)^2+672*_Z*Ro
otOf(_Z^3+2)+451584*_Z^2)^2*RootOf(_Z^3+2)^3-1513305767*RootOf(_Z^3+2)^2*x^3-703415608224*RootOf(RootOf(_Z^3+2
)^2+672*_Z*RootOf(_Z^3+2)+451584*_Z^2)*RootOf(_Z^3+2)*x^3-2068676142*(-x^3+1)^(1/3)*RootOf(_Z^3+2)*x^2-6898390
73280*(-x^3+1)^(1/3)*RootOf(RootOf(_Z^3+2)^2+672*_Z*RootOf(_Z^3+2)+451584*_Z^2)*x^2-2068676142*x*(-x^3+1)^(2/3
)[phone]*RootOf(_Z^3+2)^2+92903948256*RootOf(RootOf(_Z^3+2)^2+672*_Z*RootOf(_Z^3+2)+451584*_Z^2)*RootOf(_Z^
3+2))/(x+1)/(x^2-x+1))*RootOf(_Z^3+2)-112*ln((-19187575008*RootOf(RootOf(_Z^3+2)^2+672*_Z*RootOf(_Z^3+2)+45158
4*_Z^2)*RootOf(_Z^3+2)^4*x^3-8918779032576*RootOf(RootOf(_Z^3+2)^2+672*_Z*RootOf(_Z^3+2)+451584*_Z^2)^2*RootOf
(_Z^3+2)^3*x^3+344919536640*(-x^3+1)^(2/3)*RootOf(_Z^3+2)^2*RootOf(RootOf(_Z^3+2)^2+672*_Z*RootOf(_Z^3+2)+4515
84*_Z^2)*x+153500600064*RootOf(RootOf(_Z^3+2)^2+672*_Z*RootOf(_Z^3+2)+451584*_Z^2)*RootOf(_Z^3+2)^4+7135023226
0608*RootOf(RootOf(_Z^3+2)^2+672*_Z*RootOf(_Z^3+2)+451584*_Z^2)^2*RootOf(_Z^3+2)^3-1513305767*RootOf(_Z^3+2)^2
*x^3-703415608224*RootOf(RootOf(_Z^3+2)^2+672*_Z*RootOf(_Z^3+2)+451584*_Z^2)*RootOf(_Z^3+2)*x^3-2068676142*(-x
^3+1)^(1/3)*RootOf(_Z^3+2)*x^2-689839073280*(-x^3+1)^(1/3)*RootOf(RootOf(_Z^3+2)^2+672*_Z*RootOf(_Z^3+2)+45158
4*_Z^2)*x^2-2068676142*x*(-x^3+1)^(2/3)[phone]*RootOf(_Z^3+2)^2+92903948256*RootOf(RootOf(_Z^3+2)^2+672*_Z*
RootOf(_Z^3+2)+451584*_Z^2)*RootOf(_Z^3+2))/(x+1)/(x^2-x+1))*RootOf(RootOf(_Z^3+2)^2+672*_Z*RootOf(_Z^3+2)+451
584*_Z^2)

________________________________________________________________________________________

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^5/(-x^3+1)^(2/3)/(x^3+1),x, algorithm="maxima")

[Out]

integrate(1/((x^3 + 1)*(-x^3 + 1)^(2/3)*x^5), x)

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 312 vs. \(2 (95) = 190\).
time = 5.68, size = 312, normalized size = 2.52 \begin {gather*} -\frac {4 \cdot 4^{\frac {1}{6}} \sqrt {3} \left (-1\right )^{\frac {1}{3}} x^{4} \arctan \left (-\frac {4^{\frac {1}{6}} {\left (6 \cdot 4^{\frac {2}{3}} \sqrt {3} \left (-1\right )^{\frac {2}{3}} {\left (19 \, x^{8} - 16 \, x^{5} + x^{2}\right )} {\left (-x^{3} + 1\right )}^{\frac {1}{3}} - 12 \, \sqrt {3} \left (-1\right )^{\frac {1}{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}} \left (-1\right )^{\frac {1}{3}} x^{4} \log \left (-\frac {3 \cdot 4^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} {\left (-x^{3} + 1\right )}^{\frac {1}{3}} x^{2} - 4^{\frac {1}{3}} \left (-1\right )^{\frac {2}{3}} {\left (x^{3} + 1\right )} - 6 \, {\left (-x^{3} + 1\right )}^{\frac {2}{3}} x}{x^{3} + 1}\right ) + 4^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} x^{4} \log \left (\frac {6 \cdot 4^{\frac {1}{3}} \left (-1\right )^{\frac {2}{3}} {\left (5 \, x^{4} - x\right )} {\left (-x^{3} + 1\right )}^{\frac {2}{3}} - 4^{\frac {2}{3}} \left (-1\right )^{\frac {1}{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 ) - 18 \, {\left (x^{3} - 1\right )} {\left (-x^{3} + 1\right )}^{\frac {1}{3}}}{72 \, x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^5/(-x^3+1)^(2/3)/(x^3+1),x, algorithm="fricas")

[Out]

-1/72*(4*4^(1/6)*sqrt(3)*(-1)^(1/3)*x^4*arctan(-1/6*4^(1/6)*(6*4^(2/3)*sqrt(3)*(-1)^(2/3)*(19*x^8 - 16*x^5 + x
^2)*(-x^3 + 1)^(1/3) - 12*sqrt(3)*(-1)^(1/3)*(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)*(-1)^(1/3)*x^4*log(-(3*4^(2/3)*(-1)^(1/3)*
(-x^3 + 1)^(1/3)*x^2 - 4^(1/3)*(-1)^(2/3)*(x^3 + 1) - 6*(-x^3 + 1)^(2/3)*x)/(x^3 + 1)) + 4^(2/3)*(-1)^(1/3)*x^
4*log((6*4^(1/3)*(-1)^(2/3)*(5*x^4 - x)*(-x^3 + 1)^(2/3) - 4^(2/3)*(-1)^(1/3)*(19*x^6 - 16*x^3 + 1) - 24*(2*x^
5 - x^2)*(-x^3 + 1)^(1/3))/(x^6 + 2*x^3 + 1)) - 18*(x^3 - 1)*(-x^3 + 1)^(1/3))/x^4

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x^{5} \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**5/(-x**3+1)**(2/3)/(x**3+1),x)

[Out]

Integral(1/(x**5*(-(x - 1)*(x**2 + x + 1))**(2/3)*(x + 1)*(x**2 - x + 1)), x)

________________________________________________________________________________________

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^5/(-x^3+1)^(2/3)/(x^3+1),x, algorithm="giac")

[Out]

integrate(1/((x^3 + 1)*(-x^3 + 1)^(2/3)*x^5), x)

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{x^5\,{\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^5*(1 - x^3)^(2/3)*(x^3 + 1)),x)

[Out]

int(1/(x^5*(1 - x^3)^(2/3)*(x^3 + 1)), x)

________________________________________________________________________________________