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

Optimal. Leaf size=16 \[ x-\frac {x^2}{2}+\frac {x^3}{3} \]

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {1600} \begin {gather*} \frac {x^3}{3}-\frac {x^2}{2}+x \end {gather*}

Antiderivative was successfully verified.

[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 {gather*} \begin {aligned} \text {Integral} &=\int \left (1-x+x^2\right ) \, dx\\ &=x-\frac {x^2}{2}+\frac {x^3}{3}\\ \end {aligned} \end {gather*}

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Mathematica [A]
time = 0.00, size = 16, normalized size = 1.00 \begin {gather*} x-\frac {x^2}{2}+\frac {x^3}{3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

x - x^2/2 + x^3/3

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Maple [A]
time = 0.01, size = 13, normalized size = 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	\(\frac {x \left (2 x^{2}-3 x +6\right )}{6}\)	\(14\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.33, size = 12, normalized size = 0.75 \begin {gather*} \frac {1}{3} \, x^{3} - \frac {1}{2} \, x^{2} + x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*x^3 - 1/2*x^2 + x

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Fricas [A]
time = 0.56, size = 12, normalized size = 0.75 \begin {gather*} \frac {1}{3} \, x^{3} - \frac {1}{2} \, x^{2} + x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*x^3 - 1/2*x^2 + x

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Sympy [A]
time = 0.02, size = 10, normalized size = 0.62 \begin {gather*} \frac {x^{3}}{3} - \frac {x^{2}}{2} + x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x**3/3 - x**2/2 + x

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Giac [A]
time = 0.46, size = 12, normalized size = 0.75 \begin {gather*} \frac {1}{3} \, x^{3} - \frac {1}{2} \, x^{2} + x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*x^3 - 1/2*x^2 + x

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Mupad [B]
time = 0.02, size = 13, normalized size = 0.81 \begin {gather*} \frac {x\,\left (2\,x^2-3\,x+6\right )}{6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(x^6 - 1)/(x - x^3 - x^4 + 1),x)

[Out]

(x*(2*x^2 - 3*x + 6))/6

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Chatgpt [F] Failed to verify
time = 1.00, size = 11, normalized size = 0.69 \begin {gather*} \frac {1}{3} x^{3}+\frac {1}{2} x^{2} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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