\(\int \frac {-3-4 x+3 x^6}{(1+2 x+x^6) \sqrt [3]{1+2 x+2 x^3+x^6}} \, dx\) [2036]

Optimal result

Integrand size = 38, antiderivative size = 145 \[ \int \frac {-3-4 x+3 x^6}{\left (1+2 x+x^6\right ) \sqrt [3]{1+2 x+2 x^3+x^6}} \, dx=-\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x+2^{2/3} \sqrt [3]{1+2 x+2 x^3+x^6}}\right )}{\sqrt [3]{2}}+\frac {\log \left (-2 x+2^{2/3} \sqrt [3]{1+2 x+2 x^3+x^6}\right )}{\sqrt [3]{2}}-\frac {\log \left (2 x^2+2^{2/3} x \sqrt [3]{1+2 x+2 x^3+x^6}+\sqrt [3]{2} \left (1+2 x+2 x^3+x^6\right )^{2/3}\right )}{2 \sqrt [3]{2}} \]

[Out]

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

Rubi [F]

\[ \int \frac {-3-4 x+3 x^6}{\left (1+2 x+x^6\right ) \sqrt [3]{1+2 x+2 x^3+x^6}} \, dx=\int \frac {-3-4 x+3 x^6}{\left (1+2 x+x^6\right ) \sqrt [3]{1+2 x+2 x^3+x^6}} \, dx \]

[In]

Int[(-3 - 4*x + 3*x^6)/((1 + 2*x + x^6)*(1 + 2*x + 2*x^3 + x^6)^(1/3)),x]

[Out]

3*Defer[Int][(1 + 2*x + 2*x^3 + x^6)^(-1/3), x] - Defer[Int][1/((1 + x)*(1 + 2*x + 2*x^3 + x^6)^(1/3)), x] - 5
*Defer[Int][1/((1 + x - x^2 + x^3 - x^4 + x^5)*(1 + 2*x + 2*x^3 + x^6)^(1/3)), x] - 4*Defer[Int][x/((1 + x - x
^2 + x^3 - x^4 + x^5)*(1 + 2*x + 2*x^3 + x^6)^(1/3)), x] + 3*Defer[Int][x^2/((1 + x - x^2 + x^3 - x^4 + x^5)*(
1 + 2*x + 2*x^3 + x^6)^(1/3)), x] - 2*Defer[Int][x^3/((1 + x - x^2 + x^3 - x^4 + x^5)*(1 + 2*x + 2*x^3 + x^6)^
(1/3)), x] + Defer[Int][x^4/((1 + x - x^2 + x^3 - x^4 + x^5)*(1 + 2*x + 2*x^3 + x^6)^(1/3)), x]

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

Mathematica [A] (verified)

Time = 0.37 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.94 \[ \int \frac {-3-4 x+3 x^6}{\left (1+2 x+x^6\right ) \sqrt [3]{1+2 x+2 x^3+x^6}} \, dx=-\frac {2 \sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x+2^{2/3} \sqrt [3]{1+2 x+2 x^3+x^6}}\right )-2 \log \left (-2 x+2^{2/3} \sqrt [3]{1+2 x+2 x^3+x^6}\right )+\log \left (2 x^2+2^{2/3} x \sqrt [3]{1+2 x+2 x^3+x^6}+\sqrt [3]{2} \left (1+2 x+2 x^3+x^6\right )^{2/3}\right )}{2 \sqrt [3]{2}} \]

[In]

Integrate[(-3 - 4*x + 3*x^6)/((1 + 2*x + x^6)*(1 + 2*x + 2*x^3 + x^6)^(1/3)),x]

[Out]

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

Maple [A] (verified)

Time = 24.71 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.83

method	result	size
pseudoelliptic	\(\frac {2^{\frac {2}{3}} \left (2 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (x +2^{\frac {2}{3}} \left (x^{6}+2 x^{3}+2 x +1\right )^{\frac {1}{3}}\right )}{3 x}\right )+2 \ln \left (\frac {-2^{\frac {1}{3}} x +\left (x^{6}+2 x^{3}+2 x +1\right )^{\frac {1}{3}}}{x}\right )-\ln \left (\frac {2^{\frac {2}{3}} x^{2}+2^{\frac {1}{3}} \left (x^{6}+2 x^{3}+2 x +1\right )^{\frac {1}{3}} x +\left (x^{6}+2 x^{3}+2 x +1\right )^{\frac {2}{3}}}{x^{2}}\right )\right )}{4}\)	\(121\)
trager	\(\text {Expression too large to display}\)	\(606\)

[In]

int((3*x^6-4*x-3)/(x^6+2*x+1)/(x^6+2*x^3+2*x+1)^(1/3),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 478 vs. \(2 (118) = 236\).

Time = 27.36 (sec) , antiderivative size = 478, normalized size of antiderivative = 3.30 \[ \int \frac {-3-4 x+3 x^6}{\left (1+2 x+x^6\right ) \sqrt [3]{1+2 x+2 x^3+x^6}} \, dx=-\frac {1}{6} \, \sqrt {3} 2^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3} 2^{\frac {1}{6}} {\left (2^{\frac {5}{6}} {\left (x^{18} + 36 \, x^{15} + 6 \, x^{13} + 183 \, x^{12} + 144 \, x^{10} + 288 \, x^{9} + 12 \, x^{8} + 372 \, x^{7} + 183 \, x^{6} + 144 \, x^{5} + 144 \, x^{4} + 44 \, x^{3} + 12 \, x^{2} + 6 \, x + 1\right )} + 12 \, \sqrt {2} {\left (x^{14} + 18 \, x^{11} + 4 \, x^{9} + 38 \, x^{8} + 36 \, x^{6} + 18 \, x^{5} + 4 \, x^{4} + 4 \, x^{3} + x^{2}\right )} {\left (x^{6} + 2 \, x^{3} + 2 \, x + 1\right )}^{\frac {1}{3}} + 12 \cdot 2^{\frac {1}{6}} {\left (x^{13} + 6 \, x^{10} + 4 \, x^{8} + 2 \, x^{7} + 12 \, x^{5} + 6 \, x^{4} + 4 \, x^{3} + 4 \, x^{2} + x\right )} {\left (x^{6} + 2 \, x^{3} + 2 \, x + 1\right )}^{\frac {2}{3}}\right )}}{6 \, {\left (x^{18} + 6 \, x^{13} - 105 \, x^{12} - 216 \, x^{9} + 12 \, x^{8} - 204 \, x^{7} - 105 \, x^{6} + 8 \, x^{3} + 12 \, x^{2} + 6 \, x + 1\right )}}\right ) + \frac {1}{6} \cdot 2^{\frac {2}{3}} \log \left (\frac {6 \cdot 2^{\frac {1}{3}} {\left (x^{6} + 2 \, x^{3} + 2 \, x + 1\right )}^{\frac {1}{3}} x^{2} + 2^{\frac {2}{3}} {\left (x^{6} + 2 \, x + 1\right )} - 6 \, {\left (x^{6} + 2 \, x^{3} + 2 \, x + 1\right )}^{\frac {2}{3}} x}{x^{6} + 2 \, x + 1}\right ) - \frac {1}{12} \cdot 2^{\frac {2}{3}} \log \left (\frac {3 \cdot 2^{\frac {2}{3}} {\left (x^{7} + 6 \, x^{4} + 2 \, x^{2} + x\right )} {\left (x^{6} + 2 \, x^{3} + 2 \, x + 1\right )}^{\frac {2}{3}} + 2^{\frac {1}{3}} {\left (x^{12} + 18 \, x^{9} + 4 \, x^{7} + 38 \, x^{6} + 36 \, x^{4} + 18 \, x^{3} + 4 \, x^{2} + 4 \, x + 1\right )} + 12 \, {\left (x^{8} + 3 \, x^{5} + 2 \, x^{3} + x^{2}\right )} {\left (x^{6} + 2 \, x^{3} + 2 \, x + 1\right )}^{\frac {1}{3}}}{x^{12} + 4 \, x^{7} + 2 \, x^{6} + 4 \, x^{2} + 4 \, x + 1}\right ) \]

[In]

integrate((3*x^6-4*x-3)/(x^6+2*x+1)/(x^6+2*x^3+2*x+1)^(1/3),x, algorithm="fricas")

[Out]

-1/6*sqrt(3)*2^(2/3)*arctan(1/6*sqrt(3)*2^(1/6)*(2^(5/6)*(x^18 + 36*x^15 + 6*x^13 + 183*x^12 + 144*x^10 + 288*
x^9 + 12*x^8 + 372*x^7 + 183*x^6 + 144*x^5 + 144*x^4 + 44*x^3 + 12*x^2 + 6*x + 1) + 12*sqrt(2)*(x^14 + 18*x^11
+ 4*x^9 + 38*x^8 + 36*x^6 + 18*x^5 + 4*x^4 + 4*x^3 + x^2)*(x^6 + 2*x^3 + 2*x + 1)^(1/3) + 12*2^(1/6)*(x^13 +
6*x^10 + 4*x^8 + 2*x^7 + 12*x^5 + 6*x^4 + 4*x^3 + 4*x^2 + x)*(x^6 + 2*x^3 + 2*x + 1)^(2/3))/(x^18 + 6*x^13 - 1
05*x^12 - 216*x^9 + 12*x^8 - 204*x^7 - 105*x^6 + 8*x^3 + 12*x^2 + 6*x + 1)) + 1/6*2^(2/3)*log((6*2^(1/3)*(x^6
+ 2*x^3 + 2*x + 1)^(1/3)*x^2 + 2^(2/3)*(x^6 + 2*x + 1) - 6*(x^6 + 2*x^3 + 2*x + 1)^(2/3)*x)/(x^6 + 2*x + 1)) -
1/12*2^(2/3)*log((3*2^(2/3)*(x^7 + 6*x^4 + 2*x^2 + x)*(x^6 + 2*x^3 + 2*x + 1)^(2/3) + 2^(1/3)*(x^12 + 18*x^9
+ 4*x^7 + 38*x^6 + 36*x^4 + 18*x^3 + 4*x^2 + 4*x + 1) + 12*(x^8 + 3*x^5 + 2*x^3 + x^2)*(x^6 + 2*x^3 + 2*x + 1)
^(1/3))/(x^12 + 4*x^7 + 2*x^6 + 4*x^2 + 4*x + 1))

Sympy [F]

\[ \int \frac {-3-4 x+3 x^6}{\left (1+2 x+x^6\right ) \sqrt [3]{1+2 x+2 x^3+x^6}} \, dx=\int \frac {3 x^{6} - 4 x - 3}{\sqrt [3]{\left (x^{2} + 1\right ) \left (x^{4} - x^{2} + 2 x + 1\right )} \left (x + 1\right ) \left (x^{5} - x^{4} + x^{3} - x^{2} + x + 1\right )}\, dx \]

[In]

integrate((3*x**6-4*x-3)/(x**6+2*x+1)/(x**6+2*x**3+2*x+1)**(1/3),x)

[Out]

Integral((3*x**6 - 4*x - 3)/(((x**2 + 1)*(x**4 - x**2 + 2*x + 1))**(1/3)*(x + 1)*(x**5 - x**4 + x**3 - x**2 +
x + 1)), x)

Maxima [F]

\[ \int \frac {-3-4 x+3 x^6}{\left (1+2 x+x^6\right ) \sqrt [3]{1+2 x+2 x^3+x^6}} \, dx=\int { \frac {3 \, x^{6} - 4 \, x - 3}{{\left (x^{6} + 2 \, x^{3} + 2 \, x + 1\right )}^{\frac {1}{3}} {\left (x^{6} + 2 \, x + 1\right )}} \,d x } \]

[In]

integrate((3*x^6-4*x-3)/(x^6+2*x+1)/(x^6+2*x^3+2*x+1)^(1/3),x, algorithm="maxima")

[Out]

integrate((3*x^6 - 4*x - 3)/((x^6 + 2*x^3 + 2*x + 1)^(1/3)*(x^6 + 2*x + 1)), x)

Giac [F]

\[ \int \frac {-3-4 x+3 x^6}{\left (1+2 x+x^6\right ) \sqrt [3]{1+2 x+2 x^3+x^6}} \, dx=\int { \frac {3 \, x^{6} - 4 \, x - 3}{{\left (x^{6} + 2 \, x^{3} + 2 \, x + 1\right )}^{\frac {1}{3}} {\left (x^{6} + 2 \, x + 1\right )}} \,d x } \]

[In]

integrate((3*x^6-4*x-3)/(x^6+2*x+1)/(x^6+2*x^3+2*x+1)^(1/3),x, algorithm="giac")

[Out]

integrate((3*x^6 - 4*x - 3)/((x^6 + 2*x^3 + 2*x + 1)^(1/3)*(x^6 + 2*x + 1)), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {-3-4 x+3 x^6}{\left (1+2 x+x^6\right ) \sqrt [3]{1+2 x+2 x^3+x^6}} \, dx=\int -\frac {-3\,x^6+4\,x+3}{\left (x^6+2\,x+1\right )\,{\left (x^6+2\,x^3+2\,x+1\right )}^{1/3}} \,d x \]

[In]

int(-(4*x - 3*x^6 + 3)/((2*x + x^6 + 1)*(2*x + 2*x^3 + x^6 + 1)^(1/3)),x)

[Out]

int(-(4*x - 3*x^6 + 3)/((2*x + x^6 + 1)*(2*x + 2*x^3 + x^6 + 1)^(1/3)), x)