\(\int \frac {-1+x^3}{1+x+x^2} \, dx\) [363]

Optimal result

Integrand size = 14, antiderivative size = 11 \[ \int \frac {-1+x^3}{1+x+x^2} \, dx=-x+\frac {x^2}{2} \]

[Out]

-x+1/2*x^2

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {1600} \[ \int \frac {-1+x^3}{1+x+x^2} \, dx=\frac {x^2}{2}-x \]

[In]

Int[(-1 + x^3)/(1 + x + x^2),x]

[Out]

-x + x^2/2

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]

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

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.00 \[ \int \frac {-1+x^3}{1+x+x^2} \, dx=-x+\frac {x^2}{2} \]

[In]

Integrate[(-1 + x^3)/(1 + x + x^2),x]

[Out]

-x + x^2/2

Maple [A] (verified)

Time = 0.91 (sec) , antiderivative size = 7, normalized size of antiderivative = 0.64

method	result	size
gosper	\(\frac {x \left (x -2\right )}{2}\)	\(7\)
default	\(-x +\frac {1}{2} x^{2}\)	\(10\)
norman	\(-x +\frac {1}{2} x^{2}\)	\(10\)
risch	\(-x +\frac {1}{2} x^{2}\)	\(10\)
parallelrisch	\(-x +\frac {1}{2} x^{2}\)	\(10\)
parts	\(-x +\frac {1}{2} x^{2}\)	\(10\)

[In]

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

[Out]

1/2*x*(x-2)

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.82 \[ \int \frac {-1+x^3}{1+x+x^2} \, dx=\frac {1}{2} \, x^{2} - x \]

[In]

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

[Out]

1/2*x^2 - x

Sympy [A] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 5, normalized size of antiderivative = 0.45 \[ \int \frac {-1+x^3}{1+x+x^2} \, dx=\frac {x^{2}}{2} - x \]

[In]

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

[Out]

x**2/2 - x

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.82 \[ \int \frac {-1+x^3}{1+x+x^2} \, dx=\frac {1}{2} \, x^{2} - x \]

[In]

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

[Out]

1/2*x^2 - x

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.82 \[ \int \frac {-1+x^3}{1+x+x^2} \, dx=\frac {1}{2} \, x^{2} - x \]

[In]

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

[Out]

1/2*x^2 - x

Mupad [B] (verification not implemented)

Time = 0.01 (sec) , antiderivative size = 6, normalized size of antiderivative = 0.55 \[ \int \frac {-1+x^3}{1+x+x^2} \, dx=\frac {x\,\left (x-2\right )}{2} \]

[In]

int((x^3 - 1)/(x + x^2 + 1),x)

[Out]

(x*(x - 2))/2