\(\int \frac {-1+x^8}{x^4 \sqrt [4]{1+x^4+x^8}} \, dx\) [166]

Optimal result

Integrand size = 21, antiderivative size = 19 \[ \int \frac {-1+x^8}{x^4 \sqrt [4]{1+x^4+x^8}} \, dx=\frac {\left (1+x^4+x^8\right )^{3/4}}{3 x^3} \]

[Out]

1/3*(x^8+x^4+1)^(3/4)/x^3

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {1604} \[ \int \frac {-1+x^8}{x^4 \sqrt [4]{1+x^4+x^8}} \, dx=\frac {\left (x^8+x^4+1\right )^{3/4}}{3 x^3} \]

[In]

Int[(-1 + x^8)/(x^4*(1 + x^4 + x^8)^(1/4)),x]

[Out]

(1 + x^4 + x^8)^(3/4)/(3*x^3)

Rule 1604

Int[(Pp_)*(Qq_)^(m_.)*(Rr_)^(n_.), x_Symbol] :> With[{p = Expon[Pp, x], q = Expon[Qq, x], r = Expon[Rr, x]}, S
imp[Coeff[Pp, x, p]*x^(p - q - r + 1)*Qq^(m + 1)*(Rr^(n + 1)/((p + m*q + n*r + 1)*Coeff[Qq, x, q]*Coeff[Rr, x,
r])), x] /; NeQ[p + m*q + n*r + 1, 0] && EqQ[(p + m*q + n*r + 1)*Coeff[Qq, x, q]*Coeff[Rr, x, r]*Pp, Coeff[Pp
, x, p]*x^(p - q - r)*((p - q - r + 1)*Qq*Rr + (m + 1)*x*Rr*D[Qq, x] + (n + 1)*x*Qq*D[Rr, x])]] /; FreeQ[{m, n
}, x] && PolyQ[Pp, x] && PolyQ[Qq, x] && PolyQ[Rr, x] && NeQ[m, -1] && NeQ[n, -1]

Rubi steps \begin{align*} \text {integral}& = \frac {\left (1+x^4+x^8\right )^{3/4}}{3 x^3} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.34 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00 \[ \int \frac {-1+x^8}{x^4 \sqrt [4]{1+x^4+x^8}} \, dx=\frac {\left (1+x^4+x^8\right )^{3/4}}{3 x^3} \]

[In]

Integrate[(-1 + x^8)/(x^4*(1 + x^4 + x^8)^(1/4)),x]

[Out]

(1 + x^4 + x^8)^(3/4)/(3*x^3)

Maple [A] (verified)

Time = 0.18 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.84

method	result	size
trager	\(\frac {\left (x^{8}+x^{4}+1\right )^{\frac {3}{4}}}{3 x^{3}}\)	\(16\)
risch	\(\frac {\left (x^{8}+x^{4}+1\right )^{\frac {3}{4}}}{3 x^{3}}\)	\(16\)
pseudoelliptic	\(\frac {\left (x^{8}+x^{4}+1\right )^{\frac {3}{4}}}{3 x^{3}}\)	\(16\)
gosper	\(\frac {\left (x^{2}+x +1\right ) \left (x^{2}-x +1\right ) \left (x^{4}-x^{2}+1\right )}{3 \left (x^{8}+x^{4}+1\right )^{\frac {1}{4}} x^{3}}\)	\(40\)

[In]

int((x^8-1)/x^4/(x^8+x^4+1)^(1/4),x,method=_RETURNVERBOSE)

[Out]

1/3*(x^8+x^4+1)^(3/4)/x^3

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.79 \[ \int \frac {-1+x^8}{x^4 \sqrt [4]{1+x^4+x^8}} \, dx=\frac {{\left (x^{8} + x^{4} + 1\right )}^{\frac {3}{4}}}{3 \, x^{3}} \]

[In]

integrate((x^8-1)/x^4/(x^8+x^4+1)^(1/4),x, algorithm="fricas")

[Out]

1/3*(x^8 + x^4 + 1)^(3/4)/x^3

Sympy [F]

\[ \int \frac {-1+x^8}{x^4 \sqrt [4]{1+x^4+x^8}} \, dx=\int \frac {\left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right ) \left (x^{4} + 1\right )}{x^{4} \sqrt [4]{\left (x^{2} - x + 1\right ) \left (x^{2} + x + 1\right ) \left (x^{4} - x^{2} + 1\right )}}\, dx \]

[In]

integrate((x**8-1)/x**4/(x**8+x**4+1)**(1/4),x)

[Out]

Integral((x - 1)*(x + 1)*(x**2 + 1)*(x**4 + 1)/(x**4*((x**2 - x + 1)*(x**2 + x + 1)*(x**4 - x**2 + 1))**(1/4))
, x)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 43 vs. \(2 (15) = 30\).

Time = 0.23 (sec) , antiderivative size = 43, normalized size of antiderivative = 2.26 \[ \int \frac {-1+x^8}{x^4 \sqrt [4]{1+x^4+x^8}} \, dx=\frac {x^{8} + x^{4} + 1}{3 \, {\left (x^{4} - x^{2} + 1\right )}^{\frac {1}{4}} {\left (x^{2} + x + 1\right )}^{\frac {1}{4}} {\left (x^{2} - x + 1\right )}^{\frac {1}{4}} x^{3}} \]

[In]

integrate((x^8-1)/x^4/(x^8+x^4+1)^(1/4),x, algorithm="maxima")

[Out]

1/3*(x^8 + x^4 + 1)/((x^4 - x^2 + 1)^(1/4)*(x^2 + x + 1)^(1/4)*(x^2 - x + 1)^(1/4)*x^3)

Giac [F]

\[ \int \frac {-1+x^8}{x^4 \sqrt [4]{1+x^4+x^8}} \, dx=\int { \frac {x^{8} - 1}{{\left (x^{8} + x^{4} + 1\right )}^{\frac {1}{4}} x^{4}} \,d x } \]

[In]

integrate((x^8-1)/x^4/(x^8+x^4+1)^(1/4),x, algorithm="giac")

[Out]

integrate((x^8 - 1)/((x^8 + x^4 + 1)^(1/4)*x^4), x)

Mupad [B] (verification not implemented)

Time = 4.99 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.79 \[ \int \frac {-1+x^8}{x^4 \sqrt [4]{1+x^4+x^8}} \, dx=\frac {{\left (x^8+x^4+1\right )}^{3/4}}{3\,x^3} \]

[In]

int((x^8 - 1)/(x^4*(x^4 + x^8 + 1)^(1/4)),x)

[Out]

(x^4 + x^8 + 1)^(3/4)/(3*x^3)