\(\int (-1+3 x-3 x^2+x^3) \, dx\) [78]

Optimal result

Integrand size = 13, antiderivative size = 11 \[ \int \left (-1+3 x-3 x^2+x^3\right ) \, dx=\frac {1}{4} (1-x)^4 \]

[Out]

1/4*(1-x)^4

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(23\) vs. \(2(11)=22\).

Time = 0.00 (sec) , antiderivative size = 23, normalized size of antiderivative = 2.09, number of steps used = 1, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \left (-1+3 x-3 x^2+x^3\right ) \, dx=\frac {x^4}{4}-x^3+\frac {3 x^2}{2}-x \]

[In]

Int[-1 + 3*x - 3*x^2 + x^3,x]

[Out]

-x + (3*x^2)/2 - x^3 + x^4/4

Rubi steps \begin{align*} \text {integral}& = -x+\frac {3 x^2}{2}-x^3+\frac {x^4}{4} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(23\) vs. \(2(11)=22\).

Time = 0.00 (sec) , antiderivative size = 23, normalized size of antiderivative = 2.09 \[ \int \left (-1+3 x-3 x^2+x^3\right ) \, dx=-x+\frac {3 x^2}{2}-x^3+\frac {x^4}{4} \]

[In]

Integrate[-1 + 3*x - 3*x^2 + x^3,x]

[Out]

-x + (3*x^2)/2 - x^3 + x^4/4

Maple [A] (verified)

Time = 0.01 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.73

method	result	size
default	\(\frac {\left (-1+x \right )^{4}}{4}\)	\(8\)
gosper	\(\frac {x \left (x^{3}-4 x^{2}+6 x -4\right )}{4}\)	\(17\)
norman	\(\frac {1}{4} x^{4}-x^{3}+\frac {3}{2} x^{2}-x\)	\(20\)
risch	\(\frac {1}{4} x^{4}-x^{3}+\frac {3}{2} x^{2}-x\)	\(20\)
parallelrisch	\(\frac {1}{4} x^{4}-x^{3}+\frac {3}{2} x^{2}-x\)	\(20\)
parts	\(\frac {1}{4} x^{4}-x^{3}+\frac {3}{2} x^{2}-x\)	\(20\)

[In]

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

[Out]

1/4*(-1+x)^4

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 19 vs. \(2 (7) = 14\).

Time = 0.24 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.73 \[ \int \left (-1+3 x-3 x^2+x^3\right ) \, dx=\frac {1}{4} \, x^{4} - x^{3} + \frac {3}{2} \, x^{2} - x \]

[In]

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

[Out]

1/4*x^4 - x^3 + 3/2*x^2 - x

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 15 vs. \(2 (5) = 10\).

Time = 0.01 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.36 \[ \int \left (-1+3 x-3 x^2+x^3\right ) \, dx=\frac {x^{4}}{4} - x^{3} + \frac {3 x^{2}}{2} - x \]

[In]

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

[Out]

x**4/4 - x**3 + 3*x**2/2 - x

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 19 vs. \(2 (7) = 14\).

Time = 0.18 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.73 \[ \int \left (-1+3 x-3 x^2+x^3\right ) \, dx=\frac {1}{4} \, x^{4} - x^{3} + \frac {3}{2} \, x^{2} - x \]

[In]

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

[Out]

1/4*x^4 - x^3 + 3/2*x^2 - x

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 19 vs. \(2 (7) = 14\).

Time = 0.27 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.73 \[ \int \left (-1+3 x-3 x^2+x^3\right ) \, dx=\frac {1}{4} \, x^{4} - x^{3} + \frac {3}{2} \, x^{2} - x \]

[In]

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

[Out]

1/4*x^4 - x^3 + 3/2*x^2 - x

Mupad [B] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.73 \[ \int \left (-1+3 x-3 x^2+x^3\right ) \, dx=\frac {x^4}{4}-x^3+\frac {3\,x^2}{2}-x \]

[In]

int(3*x - 3*x^2 + x^3 - 1,x)

[Out]

(3*x^2)/2 - x - x^3 + x^4/4