\(\int \frac {(1-x^4)^3}{(1+x+x^2+x^3)^3} \, dx\) [179]

Optimal result

Integrand size = 21, antiderivative size = 11 \[ \int \frac {\left (1-x^4\right )^3}{\left (1+x+x^2+x^3\right )^3} \, dx=-\frac {1}{4} (1-x)^4 \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {1600, 32} \[ \int \frac {\left (1-x^4\right )^3}{\left (1+x+x^2+x^3\right )^3} \, dx=-\frac {1}{4} (1-x)^4 \]

[In]

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

[Out]

-1/4*(1 - x)^4

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

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)^3 \, dx \\ & = -\frac {1}{4} (1-x)^4 \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.82 \[ \int \frac {\left (1-x^4\right )^3}{\left (1+x+x^2+x^3\right )^3} \, dx=-\frac {1}{4} (-1+x)^4 \]

[In]

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

[Out]

-1/4*(-1 + x)^4

Maple [A] (verified)

Time = 1.46 (sec) , antiderivative size = 8, normalized size of antiderivative = 0.73

method	result	size
default	\(-\frac {\left (-1+x \right )^{4}}{4}\)	\(8\)
parallelrisch	\(-\frac {1}{4} x^{4}+x^{3}-\frac {3}{2} x^{2}+x\)	\(16\)
gosper	\(-\frac {x \left (x^{3}-4 x^{2}+6 x -4\right )}{4}\)	\(17\)
risch	\(-\frac {1}{4} x^{4}+x^{3}-\frac {3}{2} x^{2}+x -\frac {1}{4}\)	\(17\)
norman	\(\frac {-2 x^{5}-2 x^{3}-x^{4}-\frac {7}{4} x^{2}-\frac {1}{2} x -\frac {1}{4} x^{8}+\frac {1}{2} x^{9}-\frac {1}{4} x^{10}-\frac {3}{4}}{\left (x^{3}+x^{2}+x +1\right )^{2}}\)	\(53\)

[In]

int((-x^4+1)^3/(x^3+x^2+x+1)^3,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. 15 vs. \(2 (7) = 14\).

Time = 0.27 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.36 \[ \int \frac {\left (1-x^4\right )^3}{\left (1+x+x^2+x^3\right )^3} \, dx=-\frac {1}{4} \, x^{4} + x^{3} - \frac {3}{2} \, x^{2} + x \]

[In]

integrate((-x^4+1)^3/(x^3+x^2+x+1)^3,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. \(2 (7) = 14\).

Time = 0.03 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.36 \[ \int \frac {\left (1-x^4\right )^3}{\left (1+x+x^2+x^3\right )^3} \, dx=- \frac {x^{4}}{4} + x^{3} - \frac {3 x^{2}}{2} + x \]

[In]

integrate((-x**4+1)**3/(x**3+x**2+x+1)**3,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. 15 vs. \(2 (7) = 14\).

Time = 0.21 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.36 \[ \int \frac {\left (1-x^4\right )^3}{\left (1+x+x^2+x^3\right )^3} \, dx=-\frac {1}{4} \, x^{4} + x^{3} - \frac {3}{2} \, x^{2} + x \]

[In]

integrate((-x^4+1)^3/(x^3+x^2+x+1)^3,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. 15 vs. \(2 (7) = 14\).

Time = 0.28 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.36 \[ \int \frac {\left (1-x^4\right )^3}{\left (1+x+x^2+x^3\right )^3} \, dx=-\frac {1}{4} \, x^{4} + x^{3} - \frac {3}{2} \, x^{2} + x \]

[In]

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

[In]

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

[Out]

x - (3*x^2)/2 + x^3 - x^4/4