3.7.37 \(\int \frac {1}{x^3 (1-x^3)^{2/3} (1+x^3)} \, dx\) [637]

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

[Out]

-1/2*(-x^3+1)^(1/3)/x^2-1/12*ln(2^(2/3)+(-1+x)/(-x^3+1)^(1/3))*2^(1/3)+1/12*ln(1+2^(2/3)*(1-x)^2/(-x^3+1)^(2/3
)-2^(1/3)*(1-x)/(-x^3+1)^(1/3))*2^(1/3)-1/6*ln(1+2^(1/3)*(1-x)/(-x^3+1)^(1/3))*2^(1/3)+1/24*ln(2*2^(1/3)+(1-x)
^2/(-x^3+1)^(2/3)+2^(2/3)*(1-x)/(-x^3+1)^(1/3))*2^(1/3)-1/6*arctan(1/3*(1-2*2^(1/3)*(1-x)/(-x^3+1)^(1/3))*3^(1
/2))*2^(1/3)*3^(1/2)-1/12*arctan(1/3*(1+2^(1/3)*(1-x)/(-x^3+1)^(1/3))*3^(1/2))*2^(1/3)*3^(1/2)

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Rubi [A]
time = 0.12, antiderivative size = 294, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 10, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.454, Rules used = {491, 21, 420, 493, 298, 31, 648, 631, 210, 642} \begin {gather*} -\frac {\text {ArcTan}\left (\frac {1-\frac {2 \sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{2^{2/3} \sqrt {3}}-\frac {\text {ArcTan}\left (\frac {\frac {\sqrt [3]{2} (1-x)}{\sqrt [3]{1-x^3}}+1}{\sqrt {3}}\right )}{2\ 2^{2/3} \sqrt {3}}-\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 {\sqrt [3]{1-x^3}}{2 x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 21

Int[(u_.)*((a_) + (b_.)*(v_))^(m_.)*((c_) + (d_.)*(v_))^(n_.), x_Symbol] :> Dist[(b/d)^m, Int[u*(c + d*v)^(m +
n), x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[b*c - a*d, 0] && IntegerQ[m] && ( !IntegerQ[n] || SimplerQ[c +
d*x, a + b*x])

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 210

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^(-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])
], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 298

Int[(x_)/((a_) + (b_.)*(x_)^3), x_Symbol] :> Dist[-(3*Rt[a, 3]*Rt[b, 3])^(-1), Int[1/(Rt[a, 3] + Rt[b, 3]*x),
x], x] + Dist[1/(3*Rt[a, 3]*Rt[b, 3]), Int[(Rt[a, 3] + Rt[b, 3]*x)/(Rt[a, 3]^2 - Rt[a, 3]*Rt[b, 3]*x + Rt[b, 3
]^2*x^2), x], x] /; FreeQ[{a, b}, x]

Rule 420

Int[((a_) + (b_.)*(x_)^3)^(1/3)/((c_) + (d_.)*(x_)^3), x_Symbol] :> With[{q = Rt[b/a, 3]}, Dist[9*(a/(c*q)), S
ubst[Int[x/((4 - a*x^3)*(1 + 2*a*x^3)), x], x, (1 + q*x)/(a + b*x^3)^(1/3)], x]] /; FreeQ[{a, b, c, d}, x] &&
NeQ[b*c - a*d, 0] && EqQ[b*c + a*d, 0]

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 493

Int[((e_.)*(x_))^(m_.)/(((a_) + (b_.)*(x_)^(n_))*((c_) + (d_.)*(x_)^(n_))), x_Symbol] :> Dist[b/(b*c - a*d), I
nt[(e*x)^m/(a + b*x^n), x], x] - Dist[d/(b*c - a*d), Int[(e*x)^m/(c + d*x^n), x], x] /; FreeQ[{a, b, c, d, e,
m}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0]

Rule 631

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[a*(c/b^2)]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b)], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 642

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[d*(Log[RemoveContent[a + b*x +
c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 648

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

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] Result contains higher order function than in optimal. Order 6 vs. order 3 in optimal.
time = 20.09, size = 120, normalized size = 0.41 \begin {gather*} \frac {\sqrt [3]{1-x^3} \left (-1+\frac {4 x^3 F_1\left (\frac {1}{3};-\frac {1}{3},1;\frac {4}{3};x^3,-x^3\right )}{\left (1+x^3\right ) \left (-4 F_1\left (\frac {1}{3};-\frac {1}{3},1;\frac {4}{3};x^3,-x^3\right )+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 )\right )}\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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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 35.43, size = 695, normalized size = 2.36

method	result	size
risch	\(\text {Expression too large to display}\)	\(695\)
trager	\(\text {Expression too large to display}\)	\(1729\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(x^3-1)/x^2/(-x^3+1)^(2/3)+(1/4*RootOf(RootOf(_Z^3+2)^2+3*_Z*RootOf(_Z^3+2)+9*_Z^2)*ln(-(18*RootOf(RootOf(
_Z^3+2)^2+3*_Z*RootOf(_Z^3+2)+9*_Z^2)^2*RootOf(_Z^3+2)^2*x^3+12*RootOf(RootOf(_Z^3+2)^2+3*_Z*RootOf(_Z^3+2)+9*
_Z^2)*RootOf(_Z^3+2)^3*x^3-3*RootOf(RootOf(_Z^3+2)^2+3*_Z*RootOf(_Z^3+2)+9*_Z^2)*x^6-2*RootOf(_Z^3+2)*x^6+9*Ro
otOf(_Z^3+2)^2*RootOf(RootOf(_Z^3+2)^2+3*_Z*RootOf(_Z^3+2)+9*_Z^2)*(x^6-2*x^3+1)^(2/3)*x-18*RootOf(_Z^3+2)*(x^
6-2*x^3+1)^(1/3)*RootOf(RootOf(_Z^3+2)^2+3*_Z*RootOf(_Z^3+2)+9*_Z^2)*x^2-6*RootOf(_Z^3+2)^2*(x^6-2*x^3+1)^(1/3
)*x^2+6*RootOf(RootOf(_Z^3+2)^2+3*_Z*RootOf(_Z^3+2)+9*_Z^2)*x^3+4*RootOf(_Z^3+2)*x^3-3*RootOf(RootOf(_Z^3+2)^2
+3*_Z*RootOf(_Z^3+2)+9*_Z^2)-2*RootOf(_Z^3+2))/(x+1)^2/(x^2-x+1)^2)+1/12*RootOf(_Z^3+2)*ln((36*RootOf(RootOf(_
Z^3+2)^2+3*_Z*RootOf(_Z^3+2)+9*_Z^2)^2*RootOf(_Z^3+2)^2*x^3+6*RootOf(RootOf(_Z^3+2)^2+3*_Z*RootOf(_Z^3+2)+9*_Z
^2)*RootOf(_Z^3+2)^3*x^3+6*RootOf(RootOf(_Z^3+2)^2+3*_Z*RootOf(_Z^3+2)+9*_Z^2)*x^6+RootOf(_Z^3+2)*x^6+9*RootOf
(_Z^3+2)^2*RootOf(RootOf(_Z^3+2)^2+3*_Z*RootOf(_Z^3+2)+9*_Z^2)*(x^6-2*x^3+1)^(2/3)*x-6*RootOf(_Z^3+2)^2*(x^6-2
*x^3+1)^(1/3)*x^2-36*RootOf(RootOf(_Z^3+2)^2+3*_Z*RootOf(_Z^3+2)+9*_Z^2)*x^3-6*RootOf(_Z^3+2)*x^3-6*(x^6-2*x^3
+1)^(2/3)*x+6*RootOf(RootOf(_Z^3+2)^2+3*_Z*RootOf(_Z^3+2)+9*_Z^2)+RootOf(_Z^3+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*} \text {Failed to integrate} \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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Fricas [A]
time = 5.80, 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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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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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^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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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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