\(\int \frac {-1+x^8}{\sqrt {-1+x^4} (1-2 x^4+x^8)} \, dx\) [3]

Optimal result

Integrand size = 27, antiderivative size = 12 \[ \int \frac {-1+x^8}{\sqrt {-1+x^4} \left (1-2 x^4+x^8\right )} \, dx=-\frac {x}{\sqrt {-1+x^4}} \]

[Out]

-x/(x^4-1)^(1/2)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {28, 1418, 391} \[ \int \frac {-1+x^8}{\sqrt {-1+x^4} \left (1-2 x^4+x^8\right )} \, dx=-\frac {x}{\sqrt {x^4-1}} \]

[In]

Int[(-1 + x^8)/(Sqrt[-1 + x^4]*(1 - 2*x^4 + x^8)),x]

[Out]

-(x/Sqrt[-1 + x^4])

Rule 28

Int[(u_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Dist[1/c^p, Int[u*(b/2 + c*x^n)^(2*
p), x], x] /; FreeQ[{a, b, c, n}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 391

Int[((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[c*x*((a + b*x^n)^(p + 1)/a), x]
/; FreeQ[{a, b, c, d, n, p}, x] && NeQ[b*c - a*d, 0] && EqQ[a*d - b*c*(n*(p + 1) + 1), 0]

Rule 1418

Int[((d_) + (e_.)*(x_)^(n_))^(q_.)*((a_) + (c_.)*(x_)^(n2_))^(p_.), x_Symbol] :> Int[(d + e*x^n)^(p + q)*(a/d
+ (c/e)*x^n)^p, x] /; FreeQ[{a, c, d, e, n, q}, x] && EqQ[n2, 2*n] && EqQ[c*d^2 + a*e^2, 0] && IntegerQ[p]

Rubi steps \begin{align*} \text {integral}& = \int \frac {-1+x^8}{\left (-1+x^4\right )^{5/2}} \, dx \\ & = \int \frac {1+x^4}{\left (-1+x^4\right )^{3/2}} \, dx \\ & = -\frac {x}{\sqrt {-1+x^4}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.20 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.00 \[ \int \frac {-1+x^8}{\sqrt {-1+x^4} \left (1-2 x^4+x^8\right )} \, dx=-\frac {x}{\sqrt {-1+x^4}} \]

[In]

Integrate[(-1 + x^8)/(Sqrt[-1 + x^4]*(1 - 2*x^4 + x^8)),x]

[Out]

-(x/Sqrt[-1 + x^4])

Maple [A] (verified)

Time = 0.89 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.92

method	result	size
gosper	\(-\frac {x}{\sqrt {x^{4}-1}}\)	\(11\)
default	\(-\frac {x}{\sqrt {x^{4}-1}}\)	\(11\)
trager	\(-\frac {x}{\sqrt {x^{4}-1}}\)	\(11\)
risch	\(-\frac {x}{\sqrt {x^{4}-1}}\)	\(11\)
elliptic	\(-\frac {x}{\sqrt {x^{4}-1}}\)	\(11\)
pseudoelliptic	\(-\frac {x}{\sqrt {x^{4}-1}}\)	\(11\)

[In]

int((x^8-1)/(x^4-1)^(1/2)/(x^8-2*x^4+1),x,method=_RETURNVERBOSE)

[Out]

-x/(x^4-1)^(1/2)

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.83 \[ \int \frac {-1+x^8}{\sqrt {-1+x^4} \left (1-2 x^4+x^8\right )} \, dx=-\frac {x}{\sqrt {x^{4} - 1}} \]

[In]

integrate((x^8-1)/(x^4-1)^(1/2)/(x^8-2*x^4+1),x, algorithm="fricas")

[Out]

-x/sqrt(x^4 - 1)

Sympy [F]

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

[In]

integrate((x**8-1)/(x**4-1)**(1/2)/(x**8-2*x**4+1),x)

[Out]

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 27 vs. \(2 (10) = 20\).

Time = 0.32 (sec) , antiderivative size = 27, normalized size of antiderivative = 2.25 \[ \int \frac {-1+x^8}{\sqrt {-1+x^4} \left (1-2 x^4+x^8\right )} \, dx=-\frac {\sqrt {x^{2} + 1} \sqrt {x + 1} \sqrt {x - 1} x}{x^{4} - 1} \]

[In]

integrate((x^8-1)/(x^4-1)^(1/2)/(x^8-2*x^4+1),x, algorithm="maxima")

[Out]

-sqrt(x^2 + 1)*sqrt(x + 1)*sqrt(x - 1)*x/(x^4 - 1)

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.83 \[ \int \frac {-1+x^8}{\sqrt {-1+x^4} \left (1-2 x^4+x^8\right )} \, dx=-\frac {x}{\sqrt {x^{4} - 1}} \]

[In]

integrate((x^8-1)/(x^4-1)^(1/2)/(x^8-2*x^4+1),x, algorithm="giac")

[Out]

-x/sqrt(x^4 - 1)

Mupad [B] (verification not implemented)

Time = 5.03 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.83 \[ \int \frac {-1+x^8}{\sqrt {-1+x^4} \left (1-2 x^4+x^8\right )} \, dx=-\frac {x}{\sqrt {x^4-1}} \]

[In]

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

[Out]

-x/(x^4 - 1)^(1/2)