3.30.8 \(\int \frac {(1+x^3)^{2/3} (-1+x^6)}{x^6 (-1-2 x^3+2 x^6)} \, dx\) [2908]

Optimal. Leaf size=325 \[ \frac {\left (1+x^3\right )^{2/3} \left (-1+4 x^3\right )}{5 x^5}-\frac {1}{2} \sqrt [3]{\frac {1}{3} \left (45+26 \sqrt {3}\right )} \text {ArcTan}\left (\frac {3^{2/3} x}{\sqrt [6]{3} x-2 \sqrt [3]{1+x^3}}\right )-\frac {\text {ArcTan}\left (\frac {3^{2/3} x}{\sqrt [6]{3} x+2 \sqrt [3]{1+x^3}}\right )}{2 \sqrt [3]{45+26 \sqrt {3}}}+\frac {\log \left (-3 x+3^{5/6} \sqrt [3]{1+x^3}\right )}{2\ 3^{2/3} \sqrt [3]{26+15 \sqrt {3}}}+\frac {\sqrt [3]{26+15 \sqrt {3}} \log \left (3 x+3^{5/6} \sqrt [3]{1+x^3}\right )}{2\ 3^{2/3}}-\frac {\sqrt [3]{26+15 \sqrt {3}} \log \left (-3 x^2+3^{5/6} x \sqrt [3]{1+x^3}-3^{2/3} \left (1+x^3\right )^{2/3}\right )}{4\ 3^{2/3}}-\frac {\log \left (3 x^2+3^{5/6} x \sqrt [3]{1+x^3}+3^{2/3} \left (1+x^3\right )^{2/3}\right )}{4\ 3^{2/3} \sqrt [3]{26+15 \sqrt {3}}} \]

[Out]

1/5*(x^3+1)^(2/3)*(4*x^3-1)/x^5-1/2*(15+26/3*3^(1/2))^(1/3)*arctan(3^(2/3)*x/(3^(1/6)*x-2*(x^3+1)^(1/3)))-1/2*
arctan(3^(2/3)*x/(3^(1/6)*x+2*(x^3+1)^(1/3)))/(45+26*3^(1/2))^(1/3)+1/6*ln(-3*x+3^(5/6)*(x^3+1)^(1/3))*3^(1/3)
/(26+15*3^(1/2))^(1/3)+1/6*(26+15*3^(1/2))^(1/3)*ln(3*x+3^(5/6)*(x^3+1)^(1/3))*3^(1/3)-1/12*(26+15*3^(1/2))^(1
/3)*ln(-3*x^2+3^(5/6)*x*(x^3+1)^(1/3)-3^(2/3)*(x^3+1)^(2/3))*3^(1/3)-1/12*ln(3*x^2+3^(5/6)*x*(x^3+1)^(1/3)+3^(
2/3)*(x^3+1)^(2/3))*3^(1/3)/(26+15*3^(1/2))^(1/3)

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Rubi [A]
time = 0.50, antiderivative size = 421, normalized size of antiderivative = 1.30, number of steps used = 13, number of rules used = 6, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {6860, 270, 283, 245, 399, 384} \begin {gather*} \frac {1}{6} \left (3+2 \sqrt {3}\right ) \text {ArcTan}\left (\frac {\frac {2 x}{\sqrt [3]{x^3+1}}+1}{\sqrt {3}}\right )-\frac {1}{6} \left (3-2 \sqrt {3}\right ) \text {ArcTan}\left (\frac {\frac {2 x}{\sqrt [3]{x^3+1}}+1}{\sqrt {3}}\right )-\frac {2 \text {ArcTan}\left (\frac {\frac {2 x}{\sqrt [3]{x^3+1}}+1}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {\left (2+\sqrt {3}\right ) \text {ArcTan}\left (\frac {1-\frac {2 \sqrt [6]{3} x}{\sqrt [3]{x^3+1}}}{\sqrt {3}}\right )}{2 \sqrt [6]{3}}-\frac {\left (2-\sqrt {3}\right ) \text {ArcTan}\left (\frac {\frac {2 \sqrt [6]{3} x}{\sqrt [3]{x^3+1}}+1}{\sqrt {3}}\right )}{2 \sqrt [6]{3}}-\frac {\left (2+\sqrt {3}\right ) \log \left (4 x^3-2 \left (1-\sqrt {3}\right )\right )}{4\ 3^{2/3}}-\frac {\left (2-\sqrt {3}\right ) \log \left (4 x^3-2 \left (1+\sqrt {3}\right )\right )}{4\ 3^{2/3}}+\frac {1}{4} \sqrt [3]{3} \left (2+\sqrt {3}\right ) \log \left (-\sqrt [3]{x^3+1}-\sqrt [6]{3} x\right )+\frac {1}{4} \sqrt [3]{3} \left (2-\sqrt {3}\right ) \log \left (\sqrt [6]{3} x-\sqrt [3]{x^3+1}\right )-\frac {1}{4} \left (2+\sqrt {3}\right ) \log \left (\sqrt [3]{x^3+1}-x\right )-\frac {1}{4} \left (2-\sqrt {3}\right ) \log \left (\sqrt [3]{x^3+1}-x\right )+\log \left (\sqrt [3]{x^3+1}-x\right )-\frac {\left (x^3+1\right )^{5/3}}{5 x^5}+\frac {\left (x^3+1\right )^{2/3}}{x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((1 + x^3)^(2/3)*(-1 + x^6))/(x^6*(-1 - 2*x^3 + 2*x^6)),x]

[Out]

(1 + x^3)^(2/3)/x^2 - (1 + x^3)^(5/3)/(5*x^5) - (2*ArcTan[(1 + (2*x)/(1 + x^3)^(1/3))/Sqrt[3]])/Sqrt[3] - ((3
- 2*Sqrt[3])*ArcTan[(1 + (2*x)/(1 + x^3)^(1/3))/Sqrt[3]])/6 + ((3 + 2*Sqrt[3])*ArcTan[(1 + (2*x)/(1 + x^3)^(1/
3))/Sqrt[3]])/6 - ((2 + Sqrt[3])*ArcTan[(1 - (2*3^(1/6)*x)/(1 + x^3)^(1/3))/Sqrt[3]])/(2*3^(1/6)) - ((2 - Sqrt
[3])*ArcTan[(1 + (2*3^(1/6)*x)/(1 + x^3)^(1/3))/Sqrt[3]])/(2*3^(1/6)) - ((2 + Sqrt[3])*Log[-2*(1 - Sqrt[3]) +
4*x^3])/(4*3^(2/3)) - ((2 - Sqrt[3])*Log[-2*(1 + Sqrt[3]) + 4*x^3])/(4*3^(2/3)) + (3^(1/3)*(2 + Sqrt[3])*Log[-
(3^(1/6)*x) - (1 + x^3)^(1/3)])/4 + (3^(1/3)*(2 - Sqrt[3])*Log[3^(1/6)*x - (1 + x^3)^(1/3)])/4 + Log[-x + (1 +
x^3)^(1/3)] - ((2 - Sqrt[3])*Log[-x + (1 + x^3)^(1/3)])/4 - ((2 + Sqrt[3])*Log[-x + (1 + x^3)^(1/3)])/4

Rule 245

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

Rule 270

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c*x)^(m + 1)*((a + b*x^n)^(p + 1)/(a*
c*(m + 1))), x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[(m + 1)/n + p + 1, 0] && NeQ[m, -1]

Rule 283

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c*x)^(m + 1)*((a + b*x^n)^p/(c*(m + 1
))), x] - Dist[b*n*(p/(c^n*(m + 1))), Int[(c*x)^(m + n)*(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] &&
IGtQ[n, 0] && GtQ[p, 0] && LtQ[m, -1] &&  !ILtQ[(m + n*p + n + 1)/n, 0] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 384

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

Rule 399

Int[((a_) + (b_.)*(x_)^(n_))^(p_)/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Dist[b/d, Int[(a + b*x^n)^(p - 1), x]
, x] - Dist[(b*c - a*d)/d, Int[(a + b*x^n)^(p - 1)/(c + d*x^n), x], x] /; FreeQ[{a, b, c, d, p}, x] && NeQ[b*c
- a*d, 0] && EqQ[n*(p - 1) + 1, 0] && IntegerQ[n]

Rule 6860

Int[(u_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a +
b*x^n + c*x^(2*n)), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]

Rubi steps

\begin {align*} \int \frac {\left (1+x^3\right )^{2/3} \left (-1+x^6\right )}{x^6 \left (-1-2 x^3+2 x^6\right )} \, dx &=\int \left (\frac {\left (1+x^3\right )^{2/3}}{x^6}-\frac {2 \left (1+x^3\right )^{2/3}}{x^3}+\frac {\left (1+x^3\right )^{2/3} \left (-5+4 x^3\right )}{-1-2 x^3+2 x^6}\right ) \, dx\\ &=-\left (2 \int \frac {\left (1+x^3\right )^{2/3}}{x^3} \, dx\right )+\int \frac {\left (1+x^3\right )^{2/3}}{x^6} \, dx+\int \frac {\left (1+x^3\right )^{2/3} \left (-5+4 x^3\right )}{-1-2 x^3+2 x^6} \, dx\\ &=\frac {\left (1+x^3\right )^{2/3}}{x^2}-\frac {\left (1+x^3\right )^{5/3}}{5 x^5}-2 \int \frac {1}{\sqrt [3]{1+x^3}} \, dx+\int \left (\frac {\left (4-2 \sqrt {3}\right ) \left (1+x^3\right )^{2/3}}{-2-2 \sqrt {3}+4 x^3}+\frac {\left (4+2 \sqrt {3}\right ) \left (1+x^3\right )^{2/3}}{-2+2 \sqrt {3}+4 x^3}\right ) \, dx\\ &=\frac {\left (1+x^3\right )^{2/3}}{x^2}-\frac {\left (1+x^3\right )^{5/3}}{5 x^5}-\frac {2 \tan ^{-1}\left (\frac {1+\frac {2 x}{\sqrt [3]{1+x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}+\log \left (-x+\sqrt [3]{1+x^3}\right )+\left (2 \left (2-\sqrt {3}\right )\right ) \int \frac {\left (1+x^3\right )^{2/3}}{-2-2 \sqrt {3}+4 x^3} \, dx+\left (2 \left (2+\sqrt {3}\right )\right ) \int \frac {\left (1+x^3\right )^{2/3}}{-2+2 \sqrt {3}+4 x^3} \, dx\\ &=\frac {\left (1+x^3\right )^{2/3}}{x^2}-\frac {\left (1+x^3\right )^{5/3}}{5 x^5}-\frac {\left (2+\sqrt {3}\right ) x F_1\left (\frac {1}{3};-\frac {2}{3},1;\frac {4}{3};-x^3,\frac {2 x^3}{1-\sqrt {3}}\right )}{1-\sqrt {3}}-\frac {\left (2-\sqrt {3}\right ) x F_1\left (\frac {1}{3};-\frac {2}{3},1;\frac {4}{3};-x^3,\frac {2 x^3}{1+\sqrt {3}}\right )}{1+\sqrt {3}}-\frac {2 \tan ^{-1}\left (\frac {1+\frac {2 x}{\sqrt [3]{1+x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}+\log \left (-x+\sqrt [3]{1+x^3}\right )\\ \end {align*}

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Mathematica [A]
time = 1.72, size = 296, normalized size = 0.91 \begin {gather*} \frac {1}{60} \left (\frac {12 \left (1+x^3\right )^{2/3} \left (-1+4 x^3\right )}{x^5}-10\ 3^{2/3} \sqrt [3]{45+26 \sqrt {3}} \text {ArcTan}\left (\frac {3^{2/3} x}{\sqrt [6]{3} x-2 \sqrt [3]{1+x^3}}\right )-\frac {30 \text {ArcTan}\left (\frac {3^{2/3} x}{\sqrt [6]{3} x+2 \sqrt [3]{1+x^3}}\right )}{\sqrt [3]{45+26 \sqrt {3}}}+10 \sqrt [3]{78-45 \sqrt {3}} \log \left (-3 x+3^{5/6} \sqrt [3]{1+x^3}\right )+10 \sqrt [3]{78+45 \sqrt {3}} \log \left (3 x+3^{5/6} \sqrt [3]{1+x^3}\right )-5 \sqrt [3]{78+45 \sqrt {3}} \log \left (-3 x^2+3^{5/6} x \sqrt [3]{1+x^3}-3^{2/3} \left (1+x^3\right )^{2/3}\right )-5 \sqrt [3]{78-45 \sqrt {3}} \log \left (3 x^2+3^{5/6} x \sqrt [3]{1+x^3}+3^{2/3} \left (1+x^3\right )^{2/3}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((1 + x^3)^(2/3)*(-1 + x^6))/(x^6*(-1 - 2*x^3 + 2*x^6)),x]

[Out]

((12*(1 + x^3)^(2/3)*(-1 + 4*x^3))/x^5 - 10*3^(2/3)*(45 + 26*Sqrt[3])^(1/3)*ArcTan[(3^(2/3)*x)/(3^(1/6)*x - 2*
(1 + x^3)^(1/3))] - (30*ArcTan[(3^(2/3)*x)/(3^(1/6)*x + 2*(1 + x^3)^(1/3))])/(45 + 26*Sqrt[3])^(1/3) + 10*(78
- 45*Sqrt[3])^(1/3)*Log[-3*x + 3^(5/6)*(1 + x^3)^(1/3)] + 10*(78 + 45*Sqrt[3])^(1/3)*Log[3*x + 3^(5/6)*(1 + x^
3)^(1/3)] - 5*(78 + 45*Sqrt[3])^(1/3)*Log[-3*x^2 + 3^(5/6)*x*(1 + x^3)^(1/3) - 3^(2/3)*(1 + x^3)^(2/3)] - 5*(7
8 - 45*Sqrt[3])^(1/3)*Log[3*x^2 + 3^(5/6)*x*(1 + x^3)^(1/3) + 3^(2/3)*(1 + x^3)^(2/3)])/60

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 179.21, size = 6585, normalized size = 20.26

method	result	size
risch	\(\text {Expression too large to display}\)	\(6585\)
trager	\(\text {Expression too large to display}\)	\(11228\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^3+1)^(2/3)*(x^6-1)/x^6/(2*x^6-2*x^3-1),x,method=_RETURNVERBOSE)

[Out]

result too large to display

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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((x^3+1)^(2/3)*(x^6-1)/x^6/(2*x^6-2*x^3-1),x, algorithm="maxima")

[Out]

integrate((x^6 - 1)*(x^3 + 1)^(2/3)/((2*x^6 - 2*x^3 - 1)*x^6), x)

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Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^3+1)^(2/3)*(x^6-1)/x^6/(2*x^6-2*x^3-1),x, algorithm="fricas")

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (tr
ace 0)

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**3+1)**(2/3)*(x**6-1)/x**6/(2*x**6-2*x**3-1),x)

[Out]

Timed out

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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((x^3+1)^(2/3)*(x^6-1)/x^6/(2*x^6-2*x^3-1),x, algorithm="giac")

[Out]

integrate((x^6 - 1)*(x^3 + 1)^(2/3)/((2*x^6 - 2*x^3 - 1)*x^6), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} -\int \frac {{\left (x^3+1\right )}^{2/3}\,\left (x^6-1\right )}{x^6\,\left (-2\,x^6+2\,x^3+1\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-((x^3 + 1)^(2/3)*(x^6 - 1))/(x^6*(2*x^3 - 2*x^6 + 1)),x)

[Out]

-int(((x^3 + 1)^(2/3)*(x^6 - 1))/(x^6*(2*x^3 - 2*x^6 + 1)), x)

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