\(\int \frac {-1+x^4}{(1+x^4) \sqrt [4]{x^2+x^6}} \, dx\) [236]

Optimal result

Integrand size = 24, antiderivative size = 23 \[ \int \frac {-1+x^4}{\left (1+x^4\right ) \sqrt [4]{x^2+x^6}} \, dx=-\frac {2 \left (x^2+x^6\right )^{3/4}}{x \left (1+x^4\right )} \]

[Out]

-2*(x^6+x^2)^(3/4)/x/(x^4+1)

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.61, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {2081, 460} \[ \int \frac {-1+x^4}{\left (1+x^4\right ) \sqrt [4]{x^2+x^6}} \, dx=-\frac {2 x}{\sqrt [4]{x^6+x^2}} \]

[In]

Int[(-1 + x^4)/((1 + x^4)*(x^2 + x^6)^(1/4)),x]

[Out]

(-2*x)/(x^2 + x^6)^(1/4)

Rule 460

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

Rule 2081

Int[(u_.)*(P_)^(p_.), x_Symbol] :> With[{m = MinimumMonomialExponent[P, x]}, Dist[P^FracPart[p]/(x^(m*FracPart
[p])*Distrib[1/x^m, P]^FracPart[p]), Int[u*x^(m*p)*Distrib[1/x^m, P]^p, x], x]] /; FreeQ[p, x] &&  !IntegerQ[p
] && SumQ[P] && EveryQ[BinomialQ[#1, x] & , P] &&  !PolyQ[P, x, 2]

Rubi steps \begin{align*} \text {integral}& = \frac {\left (\sqrt {x} \sqrt [4]{1+x^4}\right ) \int \frac {-1+x^4}{\sqrt {x} \left (1+x^4\right )^{5/4}} \, dx}{\sqrt [4]{x^2+x^6}} \\ & = -\frac {2 x}{\sqrt [4]{x^2+x^6}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.86 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.61 \[ \int \frac {-1+x^4}{\left (1+x^4\right ) \sqrt [4]{x^2+x^6}} \, dx=-\frac {2 x}{\sqrt [4]{x^2+x^6}} \]

[In]

Integrate[(-1 + x^4)/((1 + x^4)*(x^2 + x^6)^(1/4)),x]

[Out]

(-2*x)/(x^2 + x^6)^(1/4)

Maple [A] (verified)

Time = 1.08 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.57

method	result	size
gosper	\(-\frac {2 x}{\left (x^{6}+x^{2}\right )^{\frac {1}{4}}}\)	\(13\)
risch	\(-\frac {2 x}{\left (x^{2} \left (x^{4}+1\right )\right )^{\frac {1}{4}}}\)	\(15\)
pseudoelliptic	\(-\frac {2 x}{\left (x^{2} \left (x^{4}+1\right )\right )^{\frac {1}{4}}}\)	\(15\)
trager	\(-\frac {2 \left (x^{6}+x^{2}\right )^{\frac {3}{4}}}{x \left (x^{4}+1\right )}\)	\(22\)
meijerg	\(-2 \sqrt {x}\, \operatorname {hypergeom}\left (\left [\frac {1}{8}, \frac {5}{4}\right ], \left [\frac {9}{8}\right ], -x^{4}\right )+\frac {2 x^{\frac {9}{2}} \operatorname {hypergeom}\left (\left [\frac {9}{8}, \frac {5}{4}\right ], \left [\frac {17}{8}\right ], -x^{4}\right )}{9}\)	\(34\)

[In]

int((x^4-1)/(x^4+1)/(x^6+x^2)^(1/4),x,method=_RETURNVERBOSE)

[Out]

-2*x/(x^6+x^2)^(1/4)

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.78 \[ \int \frac {-1+x^4}{\left (1+x^4\right ) \sqrt [4]{x^2+x^6}} \, dx=-\frac {2 \, {\left (x^{6} + x^{2}\right )}^{\frac {3}{4}}}{x^{5} + x} \]

[In]

integrate((x^4-1)/(x^4+1)/(x^6+x^2)^(1/4),x, algorithm="fricas")

[Out]

-2*(x^6 + x^2)^(3/4)/(x^5 + x)

Sympy [F]

\[ \int \frac {-1+x^4}{\left (1+x^4\right ) \sqrt [4]{x^2+x^6}} \, dx=\int \frac {\left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right )}{\sqrt [4]{x^{2} \left (x^{4} + 1\right )} \left (x^{4} + 1\right )}\, dx \]

[In]

integrate((x**4-1)/(x**4+1)/(x**6+x**2)**(1/4),x)

[Out]

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

Maxima [F]

\[ \int \frac {-1+x^4}{\left (1+x^4\right ) \sqrt [4]{x^2+x^6}} \, dx=\int { \frac {x^{4} - 1}{{\left (x^{6} + x^{2}\right )}^{\frac {1}{4}} {\left (x^{4} + 1\right )}} \,d x } \]

[In]

integrate((x^4-1)/(x^4+1)/(x^6+x^2)^(1/4),x, algorithm="maxima")

[Out]

integrate((x^4 - 1)/((x^6 + x^2)^(1/4)*(x^4 + 1)), x)

Giac [F]

\[ \int \frac {-1+x^4}{\left (1+x^4\right ) \sqrt [4]{x^2+x^6}} \, dx=\int { \frac {x^{4} - 1}{{\left (x^{6} + x^{2}\right )}^{\frac {1}{4}} {\left (x^{4} + 1\right )}} \,d x } \]

[In]

integrate((x^4-1)/(x^4+1)/(x^6+x^2)^(1/4),x, algorithm="giac")

[Out]

integrate((x^4 - 1)/((x^6 + x^2)^(1/4)*(x^4 + 1)), x)

Mupad [B] (verification not implemented)

Time = 5.47 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.91 \[ \int \frac {-1+x^4}{\left (1+x^4\right ) \sqrt [4]{x^2+x^6}} \, dx=-\frac {2\,{\left (x^6+x^2\right )}^{3/4}}{x\,\left (x^4+1\right )} \]

[In]

int((x^4 - 1)/((x^2 + x^6)^(1/4)*(x^4 + 1)),x)

[Out]

-(2*(x^2 + x^6)^(3/4))/(x*(x^4 + 1))