\(\int \frac {(1+x^3)^{2/3} (-1-2 x^3+2 x^6)}{x^6 (-1+x^3+2 x^6)} \, dx\) [1894]

Optimal result

Integrand size = 37, antiderivative size = 131 \[ \int \frac {\left (1+x^3\right )^{2/3} \left (-1-2 x^3+2 x^6\right )}{x^6 \left (-1+x^3+2 x^6\right )} \, dx=\frac {\left (-2-17 x^3\right ) \left (1+x^3\right )^{2/3}}{10 x^5}+2 \sqrt [6]{3} \arctan \left (\frac {3^{5/6} x}{\sqrt [3]{3} x+2 \sqrt [3]{1+x^3}}\right )-\frac {2 \log \left (-3 x+3^{2/3} \sqrt [3]{1+x^3}\right )}{\sqrt [3]{3}}+\frac {\log \left (3 x^2+3^{2/3} x \sqrt [3]{1+x^3}+\sqrt [3]{3} \left (1+x^3\right )^{2/3}\right )}{\sqrt [3]{3}} \]

[Out]

1/10*(-17*x^3-2)*(x^3+1)^(2/3)/x^5+2*3^(1/6)*arctan(3^(5/6)*x/(3^(1/3)*x+2*(x^3+1)^(1/3)))-2/3*ln(-3*x+3^(2/3)
*(x^3+1)^(1/3))*3^(2/3)+1/3*ln(3*x^2+3^(2/3)*x*(x^3+1)^(1/3)+3^(1/3)*(x^3+1)^(2/3))*3^(2/3)

Rubi [A] (verified)

Time = 0.28 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.82, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.135, Rules used = {1600, 6857, 277, 270, 384} \[ \int \frac {\left (1+x^3\right )^{2/3} \left (-1-2 x^3+2 x^6\right )}{x^6 \left (-1+x^3+2 x^6\right )} \, dx=2 \sqrt [6]{3} \arctan \left (\frac {\frac {2 \sqrt [3]{3} x}{\sqrt [3]{x^3+1}}+1}{\sqrt {3}}\right )+\frac {\log \left (2 x^3-1\right )}{\sqrt [3]{3}}-3^{2/3} \log \left (\sqrt [3]{3} x-\sqrt [3]{x^3+1}\right )-\frac {\left (x^3+1\right )^{2/3}}{5 x^5}-\frac {17 \left (x^3+1\right )^{2/3}}{10 x^2} \]

[In]

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

[Out]

-1/5*(1 + x^3)^(2/3)/x^5 - (17*(1 + x^3)^(2/3))/(10*x^2) + 2*3^(1/6)*ArcTan[(1 + (2*3^(1/3)*x)/(1 + x^3)^(1/3)
)/Sqrt[3]] + Log[-1 + 2*x^3]/3^(1/3) - 3^(2/3)*Log[3^(1/3)*x - (1 + x^3)^(1/3)]

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 277

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

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 1600

Int[(u_.)*(Px_)^(p_.)*(Qx_)^(q_.), x_Symbol] :> Int[u*PolynomialQuotient[Px, Qx, x]^p*Qx^(p + q), x] /; FreeQ[
q, x] && PolyQ[Px, x] && PolyQ[Qx, x] && EqQ[PolynomialRemainder[Px, Qx, x], 0] && IntegerQ[p] && LtQ[p*q, 0]

Rule 6857

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

Rubi steps \begin{align*} \text {integral}& = \int \frac {-1-2 x^3+2 x^6}{x^6 \sqrt [3]{1+x^3} \left (-1+2 x^3\right )} \, dx \\ & = \int \left (\frac {1}{x^6 \sqrt [3]{1+x^3}}+\frac {4}{x^3 \sqrt [3]{1+x^3}}-\frac {6}{\sqrt [3]{1+x^3} \left (-1+2 x^3\right )}\right ) \, dx \\ & = 4 \int \frac {1}{x^3 \sqrt [3]{1+x^3}} \, dx-6 \int \frac {1}{\sqrt [3]{1+x^3} \left (-1+2 x^3\right )} \, dx+\int \frac {1}{x^6 \sqrt [3]{1+x^3}} \, dx \\ & = -\frac {\left (1+x^3\right )^{2/3}}{5 x^5}-\frac {2 \left (1+x^3\right )^{2/3}}{x^2}+2 \sqrt [6]{3} \arctan \left (\frac {1+\frac {2 \sqrt [3]{3} x}{\sqrt [3]{1+x^3}}}{\sqrt {3}}\right )+\frac {\log \left (-1+2 x^3\right )}{\sqrt [3]{3}}-3^{2/3} \log \left (\sqrt [3]{3} x-\sqrt [3]{1+x^3}\right )-\frac {3}{5} \int \frac {1}{x^3 \sqrt [3]{1+x^3}} \, dx \\ & = -\frac {\left (1+x^3\right )^{2/3}}{5 x^5}-\frac {17 \left (1+x^3\right )^{2/3}}{10 x^2}+2 \sqrt [6]{3} \arctan \left (\frac {1+\frac {2 \sqrt [3]{3} x}{\sqrt [3]{1+x^3}}}{\sqrt {3}}\right )+\frac {\log \left (-1+2 x^3\right )}{\sqrt [3]{3}}-3^{2/3} \log \left (\sqrt [3]{3} x-\sqrt [3]{1+x^3}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.33 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.00 \[ \int \frac {\left (1+x^3\right )^{2/3} \left (-1-2 x^3+2 x^6\right )}{x^6 \left (-1+x^3+2 x^6\right )} \, dx=\frac {\left (-2-17 x^3\right ) \left (1+x^3\right )^{2/3}}{10 x^5}+2 \sqrt [6]{3} \arctan \left (\frac {3^{5/6} x}{\sqrt [3]{3} x+2 \sqrt [3]{1+x^3}}\right )-\frac {2 \log \left (-3 x+3^{2/3} \sqrt [3]{1+x^3}\right )}{\sqrt [3]{3}}+\frac {\log \left (3 x^2+3^{2/3} x \sqrt [3]{1+x^3}+\sqrt [3]{3} \left (1+x^3\right )^{2/3}\right )}{\sqrt [3]{3}} \]

[In]

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

[Out]

((-2 - 17*x^3)*(1 + x^3)^(2/3))/(10*x^5) + 2*3^(1/6)*ArcTan[(3^(5/6)*x)/(3^(1/3)*x + 2*(1 + x^3)^(1/3))] - (2*
Log[-3*x + 3^(2/3)*(1 + x^3)^(1/3)])/3^(1/3) + Log[3*x^2 + 3^(2/3)*x*(1 + x^3)^(1/3) + 3^(1/3)*(1 + x^3)^(2/3)
]/3^(1/3)

Maple [A] (verified)

Time = 12.96 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.98

method	result	size
pseudoelliptic	\(\frac {10 \,3^{\frac {2}{3}} \ln \left (\frac {3^{\frac {2}{3}} x^{2}+3^{\frac {1}{3}} \left (x^{3}+1\right )^{\frac {1}{3}} x +\left (x^{3}+1\right )^{\frac {2}{3}}}{x^{2}}\right ) x^{5}-20 \,3^{\frac {2}{3}} \ln \left (\frac {-3^{\frac {1}{3}} x +\left (x^{3}+1\right )^{\frac {1}{3}}}{x}\right ) x^{5}-60 \,3^{\frac {1}{6}} \arctan \left (\frac {\sqrt {3}\, \left (2 \,3^{\frac {2}{3}} \left (x^{3}+1\right )^{\frac {1}{3}}+3 x \right )}{9 x}\right ) x^{5}-51 x^{3} \left (x^{3}+1\right )^{\frac {2}{3}}-6 \left (x^{3}+1\right )^{\frac {2}{3}}}{30 x^{5}}\)	\(128\)
risch	\(\text {Expression too large to display}\)	\(893\)
trager	\(\text {Expression too large to display}\)	\(1120\)

[In]

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

[Out]

1/30*(10*3^(2/3)*ln((3^(2/3)*x^2+3^(1/3)*(x^3+1)^(1/3)*x+(x^3+1)^(2/3))/x^2)*x^5-20*3^(2/3)*ln((-3^(1/3)*x+(x^
3+1)^(1/3))/x)*x^5-60*3^(1/6)*arctan(1/9*3^(1/2)*(2*3^(2/3)*(x^3+1)^(1/3)+3*x)/x)*x^5-51*x^3*(x^3+1)^(2/3)-6*(
x^3+1)^(2/3))/x^5

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 292 vs. \(2 (104) = 208\).

Time = 1.80 (sec) , antiderivative size = 292, normalized size of antiderivative = 2.23 \[ \int \frac {\left (1+x^3\right )^{2/3} \left (-1-2 x^3+2 x^6\right )}{x^6 \left (-1+x^3+2 x^6\right )} \, dx=\frac {20 \cdot 3^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} x^{5} \log \left (\frac {9 \cdot 3^{\frac {1}{3}} \left (-1\right )^{\frac {2}{3}} {\left (x^{3} + 1\right )}^{\frac {1}{3}} x^{2} + 3^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} {\left (2 \, x^{3} - 1\right )} - 9 \, {\left (x^{3} + 1\right )}^{\frac {2}{3}} x}{2 \, x^{3} - 1}\right ) - 10 \cdot 3^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} x^{5} \log \left (-\frac {3 \cdot 3^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} {\left (7 \, x^{4} + x\right )} {\left (x^{3} + 1\right )}^{\frac {2}{3}} - 3^{\frac {1}{3}} \left (-1\right )^{\frac {2}{3}} {\left (31 \, x^{6} + 23 \, x^{3} + 1\right )} - 9 \, {\left (5 \, x^{5} + 2 \, x^{2}\right )} {\left (x^{3} + 1\right )}^{\frac {1}{3}}}{4 \, x^{6} - 4 \, x^{3} + 1}\right ) - 60 \cdot 3^{\frac {1}{6}} \left (-1\right )^{\frac {1}{3}} x^{5} \arctan \left (\frac {3^{\frac {1}{6}} {\left (6 \cdot 3^{\frac {2}{3}} \left (-1\right )^{\frac {2}{3}} {\left (14 \, x^{7} - 5 \, x^{4} - x\right )} {\left (x^{3} + 1\right )}^{\frac {2}{3}} + 18 \, \left (-1\right )^{\frac {1}{3}} {\left (31 \, x^{8} + 23 \, x^{5} + x^{2}\right )} {\left (x^{3} + 1\right )}^{\frac {1}{3}} - 3^{\frac {1}{3}} {\left (127 \, x^{9} + 201 \, x^{6} + 48 \, x^{3} + 1\right )}\right )}}{3 \, {\left (251 \, x^{9} + 231 \, x^{6} + 6 \, x^{3} - 1\right )}}\right ) - 9 \, {\left (17 \, x^{3} + 2\right )} {\left (x^{3} + 1\right )}^{\frac {2}{3}}}{90 \, x^{5}} \]

[In]

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

[Out]

1/90*(20*3^(2/3)*(-1)^(1/3)*x^5*log((9*3^(1/3)*(-1)^(2/3)*(x^3 + 1)^(1/3)*x^2 + 3^(2/3)*(-1)^(1/3)*(2*x^3 - 1)
- 9*(x^3 + 1)^(2/3)*x)/(2*x^3 - 1)) - 10*3^(2/3)*(-1)^(1/3)*x^5*log(-(3*3^(2/3)*(-1)^(1/3)*(7*x^4 + x)*(x^3 +
1)^(2/3) - 3^(1/3)*(-1)^(2/3)*(31*x^6 + 23*x^3 + 1) - 9*(5*x^5 + 2*x^2)*(x^3 + 1)^(1/3))/(4*x^6 - 4*x^3 + 1))
- 60*3^(1/6)*(-1)^(1/3)*x^5*arctan(1/3*3^(1/6)*(6*3^(2/3)*(-1)^(2/3)*(14*x^7 - 5*x^4 - x)*(x^3 + 1)^(2/3) + 1
8*(-1)^(1/3)*(31*x^8 + 23*x^5 + x^2)*(x^3 + 1)^(1/3) - 3^(1/3)*(127*x^9 + 201*x^6 + 48*x^3 + 1))/(251*x^9 + 23
1*x^6 + 6*x^3 - 1)) - 9*(17*x^3 + 2)*(x^3 + 1)^(2/3))/x^5

Sympy [F]

\[ \int \frac {\left (1+x^3\right )^{2/3} \left (-1-2 x^3+2 x^6\right )}{x^6 \left (-1+x^3+2 x^6\right )} \, dx=\int \frac {\left (\left (x + 1\right ) \left (x^{2} - x + 1\right )\right )^{\frac {2}{3}} \cdot \left (2 x^{6} - 2 x^{3} - 1\right )}{x^{6} \left (x + 1\right ) \left (2 x^{3} - 1\right ) \left (x^{2} - x + 1\right )}\, dx \]

[In]

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

[Out]

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

Maxima [F]

\[ \int \frac {\left (1+x^3\right )^{2/3} \left (-1-2 x^3+2 x^6\right )}{x^6 \left (-1+x^3+2 x^6\right )} \, dx=\int { \frac {{\left (2 \, x^{6} - 2 \, x^{3} - 1\right )} {\left (x^{3} + 1\right )}^{\frac {2}{3}}}{{\left (2 \, x^{6} + x^{3} - 1\right )} x^{6}} \,d x } \]

[In]

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

[Out]

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

Giac [F]

\[ \int \frac {\left (1+x^3\right )^{2/3} \left (-1-2 x^3+2 x^6\right )}{x^6 \left (-1+x^3+2 x^6\right )} \, dx=\int { \frac {{\left (2 \, x^{6} - 2 \, x^{3} - 1\right )} {\left (x^{3} + 1\right )}^{\frac {2}{3}}}{{\left (2 \, x^{6} + x^{3} - 1\right )} x^{6}} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (1+x^3\right )^{2/3} \left (-1-2 x^3+2 x^6\right )}{x^6 \left (-1+x^3+2 x^6\right )} \, dx=\int -\frac {{\left (x^3+1\right )}^{2/3}\,\left (-2\,x^6+2\,x^3+1\right )}{x^6\,\left (2\,x^6+x^3-1\right )} \,d x \]

[In]

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

[Out]

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