\(\int \frac {1+x^8}{\sqrt [4]{1-x^8} (-1+x^8)} \, dx\) [188]

Optimal result

Integrand size = 24, antiderivative size = 20 \[ \int \frac {1+x^8}{\sqrt [4]{1-x^8} \left (-1+x^8\right )} \, dx=\frac {x \left (1-x^8\right )^{3/4}}{-1+x^8} \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.70, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {21, 391} \[ \int \frac {1+x^8}{\sqrt [4]{1-x^8} \left (-1+x^8\right )} \, dx=-\frac {x}{\sqrt [4]{1-x^8}} \]

[In]

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

[Out]

-(x/(1 - x^8)^(1/4))

Rule 21

Int[(u_.)*((a_) + (b_.)*(v_))^(m_.)*((c_) + (d_.)*(v_))^(n_.), x_Symbol] :> Dist[(b/d)^m, Int[u*(c + d*v)^(m +
n), x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[b*c - a*d, 0] && IntegerQ[m] && ( !IntegerQ[n] || SimplerQ[c +
d*x, a + b*x])

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]

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

Mathematica [A] (verified)

Time = 0.94 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.70 \[ \int \frac {1+x^8}{\sqrt [4]{1-x^8} \left (-1+x^8\right )} \, dx=-\frac {x}{\sqrt [4]{1-x^8}} \]

[In]

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

[Out]

-(x/(1 - x^8)^(1/4))

Maple [A] (verified)

Time = 1.03 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.65

method	result	size
gosper	\(-\frac {x}{\left (-x^{8}+1\right )^{\frac {1}{4}}}\)	\(13\)
risch	\(-\frac {x}{\left (-x^{8}+1\right )^{\frac {1}{4}}}\)	\(13\)
pseudoelliptic	\(-\frac {x}{\left (-x^{8}+1\right )^{\frac {1}{4}}}\)	\(13\)
trager	\(\frac {x \left (-x^{8}+1\right )^{\frac {3}{4}}}{x^{8}-1}\)	\(19\)
meijerg	\(-x \operatorname {hypergeom}\left (\left [\frac {1}{8}, \frac {5}{4}\right ], \left [\frac {9}{8}\right ], x^{8}\right )-\frac {x^{9} \operatorname {hypergeom}\left (\left [\frac {9}{8}, \frac {5}{4}\right ], \left [\frac {17}{8}\right ], x^{8}\right )}{9}\)	\(28\)

[In]

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

[Out]

-x/(-x^8+1)^(1/4)

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.90 \[ \int \frac {1+x^8}{\sqrt [4]{1-x^8} \left (-1+x^8\right )} \, dx=\frac {{\left (-x^{8} + 1\right )}^{\frac {3}{4}} x}{x^{8} - 1} \]

[In]

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

[Out]

(-x^8 + 1)^(3/4)*x/(x^8 - 1)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.45 \[ \int \frac {1+x^8}{\sqrt [4]{1-x^8} \left (-1+x^8\right )} \, dx=-\frac {x}{{\left (x^{4} + 1\right )}^{\frac {1}{4}} {\left (x^{2} + 1\right )}^{\frac {1}{4}} {\left (x + 1\right )}^{\frac {1}{4}} {\left (-x + 1\right )}^{\frac {1}{4}}} \]

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 5.10 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.60 \[ \int \frac {1+x^8}{\sqrt [4]{1-x^8} \left (-1+x^8\right )} \, dx=-\frac {x}{{\left (1-x^8\right )}^{1/4}} \]

[In]

int(-(x^8 + 1)/(1 - x^8)^(5/4),x)

[Out]

-x/(1 - x^8)^(1/4)