\(\int \frac {(-3+2 x) (1-x+x^3)^{2/3}}{1-2 x+x^2+2 x^3-2 x^4+2 x^6} \, dx\) [1495]

Optimal result

Integrand size = 43, antiderivative size = 104 \[ \int \frac {(-3+2 x) \left (1-x+x^3\right )^{2/3}}{1-2 x+x^2+2 x^3-2 x^4+2 x^6} \, dx=-\arctan \left (\frac {x}{\sqrt [3]{1-x+x^3}}\right )-\frac {1}{2} \arctan \left (\frac {x \sqrt [3]{1-x+x^3}}{-x^2+\left (1-x+x^3\right )^{2/3}}\right )-\frac {1}{2} \sqrt {3} \text {arctanh}\left (\frac {\sqrt {3} x \sqrt [3]{1-x+x^3}}{x^2+\left (1-x+x^3\right )^{2/3}}\right ) \]

[Out]

-arctan(x/(x^3-x+1)^(1/3))-1/2*arctan(x*(x^3-x+1)^(1/3)/(-x^2+(x^3-x+1)^(2/3)))-1/2*3^(1/2)*arctanh(3^(1/2)*x*
(x^3-x+1)^(1/3)/(x^2+(x^3-x+1)^(2/3)))

Rubi [F]

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

[In]

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

[Out]

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

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.36 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.02 \[ \int \frac {(-3+2 x) \left (1-x+x^3\right )^{2/3}}{1-2 x+x^2+2 x^3-2 x^4+2 x^6} \, dx=-\arctan \left (\frac {x}{\sqrt [3]{1-x+x^3}}\right )-\frac {1}{2} i \left (-i+\sqrt {3}\right ) \arctan \left (\frac {\left (1-i \sqrt {3}\right ) x}{2 \sqrt [3]{1-x+x^3}}\right )+\frac {1}{2} i \left (i+\sqrt {3}\right ) \arctan \left (\frac {\left (1+i \sqrt {3}\right ) x}{2 \sqrt [3]{1-x+x^3}}\right ) \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 24.30 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.40

method	result	size
pseudoelliptic	\(-\frac {\sqrt {3}\, \ln \left (\frac {\sqrt {3}\, \left (x^{3}-x +1\right )^{\frac {1}{3}} x +\left (x^{3}-x +1\right )^{\frac {2}{3}}+x^{2}}{x^{2}}\right )}{4}+\frac {\arctan \left (\frac {x \sqrt {3}+2 \left (x^{3}-x +1\right )^{\frac {1}{3}}}{x}\right )}{2}+\frac {\sqrt {3}\, \ln \left (\frac {-\sqrt {3}\, \left (x^{3}-x +1\right )^{\frac {1}{3}} x +\left (x^{3}-x +1\right )^{\frac {2}{3}}+x^{2}}{x^{2}}\right )}{4}-\frac {\arctan \left (\frac {x \sqrt {3}-2 \left (x^{3}-x +1\right )^{\frac {1}{3}}}{x}\right )}{2}+\arctan \left (\frac {\left (x^{3}-x +1\right )^{\frac {1}{3}}}{x}\right )\)	\(146\)
trager	\(\text {Expression too large to display}\)	\(3488\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1157 vs. \(2 (87) = 174\).

Time = 3.18 (sec) , antiderivative size = 1157, normalized size of antiderivative = 11.12 \[ \int \frac {(-3+2 x) \left (1-x+x^3\right )^{2/3}}{1-2 x+x^2+2 x^3-2 x^4+2 x^6} \, dx=\text {Too large to display} \]

[In]

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

[Out]

1/8*sqrt(2)*sqrt(-sqrt(-3) + 1)*log((2*x^6 - 4*x^4 + 4*x^3 + x^2 + sqrt(2)*(2*x^6 - x^2 + 2*x - 1)*sqrt(-sqrt(
-3) + 1) - (x^3 - x + 1)^(2/3)*(4*x^2 - sqrt(2)*(2*x^4 - x^2 - sqrt(-3)*(2*x^4 - x^2 + x) + x)*sqrt(-sqrt(-3)
+ 1) - 4*x) + sqrt(-3)*(2*x^6 - 4*x^4 + 4*x^3 + x^2 - 2*x + 1) - (4*x^5 - 2*x^3 + 2*x^2 - sqrt(2)*(x^3 - x^2 +
sqrt(-3)*(x^3 - x^2))*sqrt(-sqrt(-3) + 1) - 2*sqrt(-3)*(2*x^5 - x^3 + x^2))*(x^3 - x + 1)^(1/3) - 2*x + 1)/(2
*x^6 - 2*x^4 + 2*x^3 + x^2 - 2*x + 1)) - 1/8*sqrt(2)*sqrt(-sqrt(-3) + 1)*log((2*x^6 - 4*x^4 + 4*x^3 + x^2 - sq
rt(2)*(2*x^6 - x^2 + 2*x - 1)*sqrt(-sqrt(-3) + 1) - (x^3 - x + 1)^(2/3)*(4*x^2 + sqrt(2)*(2*x^4 - x^2 - sqrt(-
3)*(2*x^4 - x^2 + x) + x)*sqrt(-sqrt(-3) + 1) - 4*x) + sqrt(-3)*(2*x^6 - 4*x^4 + 4*x^3 + x^2 - 2*x + 1) - (4*x
^5 - 2*x^3 + 2*x^2 + sqrt(2)*(x^3 - x^2 + sqrt(-3)*(x^3 - x^2))*sqrt(-sqrt(-3) + 1) - 2*sqrt(-3)*(2*x^5 - x^3
+ x^2))*(x^3 - x + 1)^(1/3) - 2*x + 1)/(2*x^6 - 2*x^4 + 2*x^3 + x^2 - 2*x + 1)) + 1/8*sqrt(2)*sqrt(sqrt(-3) +
1)*log((2*x^6 - 4*x^4 + 4*x^3 + x^2 - 4*(x^3 - x + 1)^(2/3)*(x^2 - x) - sqrt(-3)*(2*x^6 - 4*x^4 + 4*x^3 + x^2
- 2*x + 1) + (sqrt(2)*(2*x^4 - x^2 + sqrt(-3)*(2*x^4 - x^2 + x) + x)*(x^3 - x + 1)^(2/3) + sqrt(2)*(x^3 - x^2
- sqrt(-3)*(x^3 - x^2))*(x^3 - x + 1)^(1/3) + sqrt(2)*(2*x^6 - x^2 + 2*x - 1))*sqrt(sqrt(-3) + 1) - 2*(2*x^5 -
x^3 + x^2 + sqrt(-3)*(2*x^5 - x^3 + x^2))*(x^3 - x + 1)^(1/3) - 2*x + 1)/(2*x^6 - 2*x^4 + 2*x^3 + x^2 - 2*x +
1)) - 1/8*sqrt(2)*sqrt(sqrt(-3) + 1)*log((2*x^6 - 4*x^4 + 4*x^3 + x^2 - 4*(x^3 - x + 1)^(2/3)*(x^2 - x) - sqr
t(-3)*(2*x^6 - 4*x^4 + 4*x^3 + x^2 - 2*x + 1) - (sqrt(2)*(2*x^4 - x^2 + sqrt(-3)*(2*x^4 - x^2 + x) + x)*(x^3 -
x + 1)^(2/3) + sqrt(2)*(x^3 - x^2 - sqrt(-3)*(x^3 - x^2))*(x^3 - x + 1)^(1/3) + sqrt(2)*(2*x^6 - x^2 + 2*x -
1))*sqrt(sqrt(-3) + 1) - 2*(2*x^5 - x^3 + x^2 + sqrt(-3)*(2*x^5 - x^3 + x^2))*(x^3 - x + 1)^(1/3) - 2*x + 1)/(
2*x^6 - 2*x^4 + 2*x^3 + x^2 - 2*x + 1)) - 1/2*arctan((6*x^6 - 4*x^4 + 4*x^3 - x^2 + 4*(3*x^4 - x^2 + x)*(x^3 -
x + 1)^(2/3) - 4*(x^5 - 2*x^3 + 2*x^2)*(x^3 - x + 1)^(1/3) + 2*x - 1)/(14*x^6 - 16*x^4 + 16*x^3 + x^2 - 2*x +
1))

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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