\(\int \frac {-1+x^6}{-1-x+x^3+x^4} \, dx\) [25]

Optimal result

Integrand size = 19, antiderivative size = 16 \[ \int \frac {-1+x^6}{-1-x+x^3+x^4} \, dx=x-\frac {x^2}{2}+\frac {x^3}{3} \]

[Out]

x-1/2*x^2+1/3*x^3

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {1600} \[ \int \frac {-1+x^6}{-1-x+x^3+x^4} \, dx=\frac {x^3}{3}-\frac {x^2}{2}+x \]

[In]

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

[Out]

x - x^2/2 + x^3/3

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 \left (1-x+x^2\right ) \, dx \\ & = x-\frac {x^2}{2}+\frac {x^3}{3} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int \frac {-1+x^6}{-1-x+x^3+x^4} \, dx=x-\frac {x^2}{2}+\frac {x^3}{3} \]

[In]

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

[Out]

x - x^2/2 + x^3/3

Maple [A] (verified)

Time = 0.02 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.81

method	result	size
default	\(x -\frac {1}{2} x^{2}+\frac {1}{3} x^{3}\)	\(13\)
norman	\(x -\frac {1}{2} x^{2}+\frac {1}{3} x^{3}\)	\(13\)
risch	\(x -\frac {1}{2} x^{2}+\frac {1}{3} x^{3}\)	\(13\)
parallelrisch	\(x -\frac {1}{2} x^{2}+\frac {1}{3} x^{3}\)	\(13\)
parts	\(x -\frac {1}{2} x^{2}+\frac {1}{3} x^{3}\)	\(13\)
gosper	\(\frac {x \left (2 x^{2}-3 x +6\right )}{6}\)	\(14\)

[In]

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

[Out]

x-1/2*x^2+1/3*x^3

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.75 \[ \int \frac {-1+x^6}{-1-x+x^3+x^4} \, dx=\frac {1}{3} \, x^{3} - \frac {1}{2} \, x^{2} + x \]

[In]

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

[Out]

1/3*x^3 - 1/2*x^2 + x

Sympy [A] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.62 \[ \int \frac {-1+x^6}{-1-x+x^3+x^4} \, dx=\frac {x^{3}}{3} - \frac {x^{2}}{2} + x \]

[In]

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

[Out]

x**3/3 - x**2/2 + x

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.75 \[ \int \frac {-1+x^6}{-1-x+x^3+x^4} \, dx=\frac {1}{3} \, x^{3} - \frac {1}{2} \, x^{2} + x \]

[In]

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

[Out]

1/3*x^3 - 1/2*x^2 + x

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.75 \[ \int \frac {-1+x^6}{-1-x+x^3+x^4} \, dx=\frac {1}{3} \, x^{3} - \frac {1}{2} \, x^{2} + x \]

[In]

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

[Out]

1/3*x^3 - 1/2*x^2 + x

Mupad [B] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.81 \[ \int \frac {-1+x^6}{-1-x+x^3+x^4} \, dx=\frac {x\,\left (2\,x^2-3\,x+6\right )}{6} \]

[In]

int(-(x^6 - 1)/(x - x^3 - x^4 + 1),x)

[Out]

(x*(2*x^2 - 3*x + 6))/6