\(\int \frac {x^4}{\sqrt [4]{-1+x^4} (-1+x^8)} \, dx\) [887]

Optimal result

Integrand size = 20, antiderivative size = 67 \[ \int \frac {x^4}{\sqrt [4]{-1+x^4} \left (-1+x^8\right )} \, dx=-\frac {x}{2 \sqrt [4]{-1+x^4}}+\frac {\arctan \left (\frac {\sqrt [4]{2} x}{\sqrt [4]{-1+x^4}}\right )}{4 \sqrt [4]{2}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{-1+x^4}}\right )}{4 \sqrt [4]{2}} \]

[Out]

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

Rubi [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.

Time = 0.03 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.87, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {1493, 525, 524} \[ \int \frac {x^4}{\sqrt [4]{-1+x^4} \left (-1+x^8\right )} \, dx=-\frac {x^5 \sqrt [4]{1-x^4} \operatorname {Hypergeometric2F1}\left (\frac {5}{4},\frac {5}{4},\frac {9}{4},\frac {2 x^4}{x^4+1}\right )}{5 \sqrt [4]{x^4-1} \left (x^4+1\right )^{5/4}} \]

[In]

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

[Out]

-1/5*(x^5*(1 - x^4)^(1/4)*Hypergeometric2F1[5/4, 5/4, 9/4, (2*x^4)/(1 + x^4)])/((-1 + x^4)^(1/4)*(1 + x^4)^(5/
4))

Rule 524

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[a^p*c^q*
((e*x)^(m + 1)/(e*(m + 1)))*AppellF1[(m + 1)/n, -p, -q, 1 + (m + 1)/n, (-b)*(x^n/a), (-d)*(x^n/c)], x] /; Free
Q[{a, b, c, d, e, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1] && NeQ[m, n - 1] && (IntegerQ[p] || GtQ[a
, 0]) && (IntegerQ[q] || GtQ[c, 0])

Rule 525

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Dist[a^IntPar
t[p]*((a + b*x^n)^FracPart[p]/(1 + b*(x^n/a))^FracPart[p]), Int[(e*x)^m*(1 + b*(x^n/a))^p*(c + d*x^n)^q, x], x
] /; FreeQ[{a, b, c, d, e, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1] && NeQ[m, n - 1] &&  !(IntegerQ[
p] || GtQ[a, 0])

Rule 1493

Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(n_))^(q_.)*((a_) + (c_.)*(x_)^(n2_))^(p_.), x_Symbol] :> Int[(f*x)^
m*(d + e*x^n)^(q + p)*(a/d + (c/e)*x^n)^p, x] /; FreeQ[{a, c, d, e, f, q, m, 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 {x^4}{\left (-1+x^4\right )^{5/4} \left (1+x^4\right )} \, dx \\ & = -\frac {\sqrt [4]{1-x^4} \int \frac {x^4}{\left (1-x^4\right )^{5/4} \left (1+x^4\right )} \, dx}{\sqrt [4]{-1+x^4}} \\ & = -\frac {x^5 \sqrt [4]{1-x^4} \operatorname {Hypergeometric2F1}\left (\frac {5}{4},\frac {5}{4},\frac {9}{4},\frac {2 x^4}{1+x^4}\right )}{5 \sqrt [4]{-1+x^4} \left (1+x^4\right )^{5/4}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.28 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.94 \[ \int \frac {x^4}{\sqrt [4]{-1+x^4} \left (-1+x^8\right )} \, dx=\frac {1}{8} \left (-\frac {4 x}{\sqrt [4]{-1+x^4}}+2^{3/4} \arctan \left (\frac {\sqrt [4]{2} x}{\sqrt [4]{-1+x^4}}\right )+2^{3/4} \text {arctanh}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{-1+x^4}}\right )\right ) \]

[In]

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

[Out]

((-4*x)/(-1 + x^4)^(1/4) + 2^(3/4)*ArcTan[(2^(1/4)*x)/(-1 + x^4)^(1/4)] + 2^(3/4)*ArcTanh[(2^(1/4)*x)/(-1 + x^
4)^(1/4)])/8

Maple [A] (verified)

Time = 6.10 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.25

method	result	size
pseudoelliptic	\(\frac {\ln \left (\frac {2^{\frac {1}{4}} x +\left (x^{4}-1\right )^{\frac {1}{4}}}{-2^{\frac {1}{4}} x +\left (x^{4}-1\right )^{\frac {1}{4}}}\right ) 2^{\frac {3}{4}} \left (x^{4}-1\right )^{\frac {1}{4}}-2 \arctan \left (\frac {2^{\frac {3}{4}} \left (x^{4}-1\right )^{\frac {1}{4}}}{2 x}\right ) 2^{\frac {3}{4}} \left (x^{4}-1\right )^{\frac {1}{4}}-8 x}{16 \left (x^{4}-1\right )^{\frac {1}{4}}}\)	\(84\)
risch	\(-\frac {x}{2 \left (x^{4}-1\right )^{\frac {1}{4}}}-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right ) \ln \left (\frac {-\sqrt {x^{4}-1}\, \operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{3} x^{2}+2 \left (x^{4}-1\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2} x^{3}-3 x^{4} \operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )+4 \left (x^{4}-1\right )^{\frac {3}{4}} x +\operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )}{x^{4}+1}\right )}{16}-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2}\right ) \ln \left (\frac {\sqrt {x^{4}-1}\, \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2}\right ) \operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2} x^{2}-2 \left (x^{4}-1\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2} x^{3}-3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2}\right ) x^{4}+4 \left (x^{4}-1\right )^{\frac {3}{4}} x +\operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2}\right )}{x^{4}+1}\right )}{16}\)	\(215\)
trager	\(-\frac {x}{2 \left (x^{4}-1\right )^{\frac {1}{4}}}-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right ) \ln \left (\frac {-\sqrt {x^{4}-1}\, \operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{3} x^{2}+2 \left (x^{4}-1\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2} x^{3}-3 x^{4} \operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )+4 \left (x^{4}-1\right )^{\frac {3}{4}} x +\operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )}{x^{4}+1}\right )}{16}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2}\right ) \ln \left (\frac {-\sqrt {x^{4}-1}\, \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2}\right ) \operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2} x^{2}-2 \left (x^{4}-1\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2} x^{3}+3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2}\right ) x^{4}+4 \left (x^{4}-1\right )^{\frac {3}{4}} x -\operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-8\right )^{2}\right )}{x^{4}+1}\right )}{16}\)	\(218\)

[In]

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

[Out]

1/16*(ln((2^(1/4)*x+(x^4-1)^(1/4))/(-2^(1/4)*x+(x^4-1)^(1/4)))*2^(3/4)*(x^4-1)^(1/4)-2*arctan(1/2*2^(3/4)/x*(x
^4-1)^(1/4))*2^(3/4)*(x^4-1)^(1/4)-8*x)/(x^4-1)^(1/4)

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 2.93 (sec) , antiderivative size = 310, normalized size of antiderivative = 4.63 \[ \int \frac {x^4}{\sqrt [4]{-1+x^4} \left (-1+x^8\right )} \, dx=\frac {2^{\frac {3}{4}} {\left (x^{4} - 1\right )} \log \left (\frac {4 \, \sqrt {2} {\left (x^{4} - 1\right )}^{\frac {1}{4}} x^{3} + 4 \cdot 2^{\frac {1}{4}} \sqrt {x^{4} - 1} x^{2} + 2^{\frac {3}{4}} {\left (3 \, x^{4} - 1\right )} + 4 \, {\left (x^{4} - 1\right )}^{\frac {3}{4}} x}{x^{4} + 1}\right ) + 2^{\frac {3}{4}} {\left (i \, x^{4} - i\right )} \log \left (-\frac {4 \, \sqrt {2} {\left (x^{4} - 1\right )}^{\frac {1}{4}} x^{3} + 4 i \cdot 2^{\frac {1}{4}} \sqrt {x^{4} - 1} x^{2} - 2^{\frac {3}{4}} {\left (3 i \, x^{4} - i\right )} - 4 \, {\left (x^{4} - 1\right )}^{\frac {3}{4}} x}{x^{4} + 1}\right ) + 2^{\frac {3}{4}} {\left (-i \, x^{4} + i\right )} \log \left (-\frac {4 \, \sqrt {2} {\left (x^{4} - 1\right )}^{\frac {1}{4}} x^{3} - 4 i \cdot 2^{\frac {1}{4}} \sqrt {x^{4} - 1} x^{2} - 2^{\frac {3}{4}} {\left (-3 i \, x^{4} + i\right )} - 4 \, {\left (x^{4} - 1\right )}^{\frac {3}{4}} x}{x^{4} + 1}\right ) - 2^{\frac {3}{4}} {\left (x^{4} - 1\right )} \log \left (\frac {4 \, \sqrt {2} {\left (x^{4} - 1\right )}^{\frac {1}{4}} x^{3} - 4 \cdot 2^{\frac {1}{4}} \sqrt {x^{4} - 1} x^{2} - 2^{\frac {3}{4}} {\left (3 \, x^{4} - 1\right )} + 4 \, {\left (x^{4} - 1\right )}^{\frac {3}{4}} x}{x^{4} + 1}\right ) - 16 \, {\left (x^{4} - 1\right )}^{\frac {3}{4}} x}{32 \, {\left (x^{4} - 1\right )}} \]

[In]

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

[Out]

1/32*(2^(3/4)*(x^4 - 1)*log((4*sqrt(2)*(x^4 - 1)^(1/4)*x^3 + 4*2^(1/4)*sqrt(x^4 - 1)*x^2 + 2^(3/4)*(3*x^4 - 1)
+ 4*(x^4 - 1)^(3/4)*x)/(x^4 + 1)) + 2^(3/4)*(I*x^4 - I)*log(-(4*sqrt(2)*(x^4 - 1)^(1/4)*x^3 + 4*I*2^(1/4)*sqr
t(x^4 - 1)*x^2 - 2^(3/4)*(3*I*x^4 - I) - 4*(x^4 - 1)^(3/4)*x)/(x^4 + 1)) + 2^(3/4)*(-I*x^4 + I)*log(-(4*sqrt(2
)*(x^4 - 1)^(1/4)*x^3 - 4*I*2^(1/4)*sqrt(x^4 - 1)*x^2 - 2^(3/4)*(-3*I*x^4 + I) - 4*(x^4 - 1)^(3/4)*x)/(x^4 + 1
)) - 2^(3/4)*(x^4 - 1)*log((4*sqrt(2)*(x^4 - 1)^(1/4)*x^3 - 4*2^(1/4)*sqrt(x^4 - 1)*x^2 - 2^(3/4)*(3*x^4 - 1)
+ 4*(x^4 - 1)^(3/4)*x)/(x^4 + 1)) - 16*(x^4 - 1)^(3/4)*x)/(x^4 - 1)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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