\(\int \frac {\sqrt [4]{-1+x^4} (2-x^4+2 x^8)}{x^{10} (-1+2 x^4)} \, dx\) [1600]

Optimal result

Integrand size = 34, antiderivative size = 109 \[ \int \frac {\sqrt [4]{-1+x^4} \left (2-x^4+2 x^8\right )}{x^{10} \left (-1+2 x^4\right )} \, dx=\frac {\sqrt [4]{-1+x^4} \left (2+5 x^4+65 x^8\right )}{9 x^9}+2 \sqrt {2} \arctan \left (\frac {\sqrt {2} x \sqrt [4]{-1+x^4}}{-x^2+\sqrt {-1+x^4}}\right )-2 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} x \sqrt [4]{-1+x^4}}{x^2+\sqrt {-1+x^4}}\right ) \]

[Out]

1/9*(x^4-1)^(1/4)*(65*x^8+5*x^4+2)/x^9+2*2^(1/2)*arctan(2^(1/2)*x*(x^4-1)^(1/4)/(-x^2+(x^4-1)^(1/2)))-2*2^(1/2
)*arctanh(2^(1/2)*x*(x^4-1)^(1/4)/(x^2+(x^4-1)^(1/2)))

Rubi [C] (verified)

Result contains higher order function than in optimal. Order 6 vs. order 3 in optimal.

Time = 0.23 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.11, number of steps used = 12, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.294, Rules used = {6857, 277, 270, 283, 338, 304, 209, 212, 525, 524} \[ \int \frac {\sqrt [4]{-1+x^4} \left (2-x^4+2 x^8\right )}{x^{10} \left (-1+2 x^4\right )} \, dx=-\frac {16 \sqrt [4]{x^4-1} x^3 \operatorname {AppellF1}\left (\frac {3}{4},-\frac {1}{4},1,\frac {7}{4},x^4,2 x^4\right )}{3 \sqrt [4]{1-x^4}}+4 \arctan \left (\frac {x}{\sqrt [4]{x^4-1}}\right )-4 \text {arctanh}\left (\frac {x}{\sqrt [4]{x^4-1}}\right )+\frac {8 \sqrt [4]{x^4-1}}{x}-\frac {2 \left (x^4-1\right )^{5/4}}{9 x^9}-\frac {7 \left (x^4-1\right )^{5/4}}{9 x^5} \]

[In]

Int[((-1 + x^4)^(1/4)*(2 - x^4 + 2*x^8))/(x^10*(-1 + 2*x^4)),x]

[Out]

(8*(-1 + x^4)^(1/4))/x - (2*(-1 + x^4)^(5/4))/(9*x^9) - (7*(-1 + x^4)^(5/4))/(9*x^5) - (16*x^3*(-1 + x^4)^(1/4
)*AppellF1[3/4, -1/4, 1, 7/4, x^4, 2*x^4])/(3*(1 - x^4)^(1/4)) + 4*ArcTan[x/(-1 + x^4)^(1/4)] - 4*ArcTanh[x/(-
1 + x^4)^(1/4)]

Rule 209

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*ArcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 270

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

Rule 277

Int[(x_)^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[x^(m + 1)*((a + b*x^n)^(p + 1)/(a*(m + 1))), x]
- Dist[b*((m + n*(p + 1) + 1)/(a*(m + 1))), Int[x^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, m, n, p}, x]
&& ILtQ[Simplify[(m + 1)/n + p + 1], 0] && NeQ[m, -1]

Rule 283

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c*x)^(m + 1)*((a + b*x^n)^p/(c*(m + 1
))), x] - Dist[b*n*(p/(c^n*(m + 1))), Int[(c*x)^(m + n)*(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] &&
IGtQ[n, 0] && GtQ[p, 0] && LtQ[m, -1] &&  !ILtQ[(m + n*p + n + 1)/n, 0] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 304

Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2]], s = Denominator[Rt[-a/b, 2]]}
, Dist[s/(2*b), Int[1/(r + s*x^2), x], x] - Dist[s/(2*b), Int[1/(r - s*x^2), x], x]] /; FreeQ[{a, b}, x] &&  !
GtQ[a/b, 0]

Rule 338

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[a^(p + (m + 1)/n), Subst[Int[x^m/(1 - b*x^n)^(
p + (m + 1)/n + 1), x], x, x/(a + b*x^n)^(1/n)], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[-1, p, 0] && NeQ[
p, -2^(-1)] && IntegersQ[m, p + (m + 1)/n]

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 6857

Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a + b*x^n), x]}, Int[v, x]
/; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ[n, 0]

Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {2 \sqrt [4]{-1+x^4}}{x^{10}}-\frac {3 \sqrt [4]{-1+x^4}}{x^6}-\frac {8 \sqrt [4]{-1+x^4}}{x^2}+\frac {16 x^2 \sqrt [4]{-1+x^4}}{-1+2 x^4}\right ) \, dx \\ & = -\left (2 \int \frac {\sqrt [4]{-1+x^4}}{x^{10}} \, dx\right )-3 \int \frac {\sqrt [4]{-1+x^4}}{x^6} \, dx-8 \int \frac {\sqrt [4]{-1+x^4}}{x^2} \, dx+16 \int \frac {x^2 \sqrt [4]{-1+x^4}}{-1+2 x^4} \, dx \\ & = \frac {8 \sqrt [4]{-1+x^4}}{x}-\frac {2 \left (-1+x^4\right )^{5/4}}{9 x^9}-\frac {3 \left (-1+x^4\right )^{5/4}}{5 x^5}-\frac {8}{9} \int \frac {\sqrt [4]{-1+x^4}}{x^6} \, dx-8 \int \frac {x^2}{\left (-1+x^4\right )^{3/4}} \, dx+\frac {\left (16 \sqrt [4]{-1+x^4}\right ) \int \frac {x^2 \sqrt [4]{1-x^4}}{-1+2 x^4} \, dx}{\sqrt [4]{1-x^4}} \\ & = \frac {8 \sqrt [4]{-1+x^4}}{x}-\frac {2 \left (-1+x^4\right )^{5/4}}{9 x^9}-\frac {7 \left (-1+x^4\right )^{5/4}}{9 x^5}-\frac {16 x^3 \sqrt [4]{-1+x^4} \operatorname {AppellF1}\left (\frac {3}{4},-\frac {1}{4},1,\frac {7}{4},x^4,2 x^4\right )}{3 \sqrt [4]{1-x^4}}-8 \text {Subst}\left (\int \frac {x^2}{1-x^4} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right ) \\ & = \frac {8 \sqrt [4]{-1+x^4}}{x}-\frac {2 \left (-1+x^4\right )^{5/4}}{9 x^9}-\frac {7 \left (-1+x^4\right )^{5/4}}{9 x^5}-\frac {16 x^3 \sqrt [4]{-1+x^4} \operatorname {AppellF1}\left (\frac {3}{4},-\frac {1}{4},1,\frac {7}{4},x^4,2 x^4\right )}{3 \sqrt [4]{1-x^4}}-4 \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right )+4 \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right ) \\ & = \frac {8 \sqrt [4]{-1+x^4}}{x}-\frac {2 \left (-1+x^4\right )^{5/4}}{9 x^9}-\frac {7 \left (-1+x^4\right )^{5/4}}{9 x^5}-\frac {16 x^3 \sqrt [4]{-1+x^4} \operatorname {AppellF1}\left (\frac {3}{4},-\frac {1}{4},1,\frac {7}{4},x^4,2 x^4\right )}{3 \sqrt [4]{1-x^4}}+4 \arctan \left (\frac {x}{\sqrt [4]{-1+x^4}}\right )-4 \text {arctanh}\left (\frac {x}{\sqrt [4]{-1+x^4}}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.30 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt [4]{-1+x^4} \left (2-x^4+2 x^8\right )}{x^{10} \left (-1+2 x^4\right )} \, dx=\frac {\sqrt [4]{-1+x^4} \left (2+5 x^4+65 x^8\right )}{9 x^9}+2 \sqrt {2} \arctan \left (\frac {\sqrt {2} x \sqrt [4]{-1+x^4}}{-x^2+\sqrt {-1+x^4}}\right )-2 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} x \sqrt [4]{-1+x^4}}{x^2+\sqrt {-1+x^4}}\right ) \]

[In]

Integrate[((-1 + x^4)^(1/4)*(2 - x^4 + 2*x^8))/(x^10*(-1 + 2*x^4)),x]

[Out]

((-1 + x^4)^(1/4)*(2 + 5*x^4 + 65*x^8))/(9*x^9) + 2*Sqrt[2]*ArcTan[(Sqrt[2]*x*(-1 + x^4)^(1/4))/(-x^2 + Sqrt[-
1 + x^4])] - 2*Sqrt[2]*ArcTanh[(Sqrt[2]*x*(-1 + x^4)^(1/4))/(x^2 + Sqrt[-1 + x^4])]

Maple [A] (verified)

Time = 5.39 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.18

method	result	size
pseudoelliptic	\(\frac {-9 x^{9} \left (\ln \left (\frac {\left (x^{4}-1\right )^{\frac {1}{4}} \sqrt {2}\, x +x^{2}+\sqrt {x^{4}-1}}{-\left (x^{4}-1\right )^{\frac {1}{4}} \sqrt {2}\, x +x^{2}+\sqrt {x^{4}-1}}\right )+2 \arctan \left (\frac {\left (x^{4}-1\right )^{\frac {1}{4}} \sqrt {2}+x}{x}\right )+2 \arctan \left (\frac {\left (x^{4}-1\right )^{\frac {1}{4}} \sqrt {2}-x}{x}\right )\right ) \sqrt {2}+\left (x^{4}-1\right )^{\frac {1}{4}} \left (65 x^{8}+5 x^{4}+2\right )}{9 x^{9}}\)	\(129\)
trager	\(\frac {\left (x^{4}-1\right )^{\frac {1}{4}} \left (65 x^{8}+5 x^{4}+2\right )}{9 x^{9}}-2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) \ln \left (\frac {2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} \left (x^{4}-1\right )^{\frac {1}{4}} x^{3}+2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) \sqrt {x^{4}-1}\, x^{2}+2 \left (x^{4}-1\right )^{\frac {3}{4}} x +\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3}}{2 x^{4}-1}\right )-2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} \ln \left (\frac {2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} \sqrt {x^{4}-1}\, x^{2}-2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} \left (x^{4}-1\right )^{\frac {1}{4}} x^{3}+2 \left (x^{4}-1\right )^{\frac {3}{4}} x +\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )}{2 x^{4}-1}\right )\)	\(180\)
risch	\(\frac {65 x^{12}-60 x^{8}-3 x^{4}-2}{9 x^{9} \left (x^{4}-1\right )^{\frac {3}{4}}}+\frac {\left (-2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} \ln \left (-\frac {-2 \left (x^{12}-3 x^{8}+3 x^{4}-1\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) x^{9}-\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} x^{8}+2 \left (x^{12}-3 x^{8}+3 x^{4}-1\right )^{\frac {3}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} x^{3}+2 \sqrt {x^{12}-3 x^{8}+3 x^{4}-1}\, x^{6}+4 \left (x^{12}-3 x^{8}+3 x^{4}-1\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) x^{5}+2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} x^{4}-2 \sqrt {x^{12}-3 x^{8}+3 x^{4}-1}\, x^{2}-2 \left (x^{12}-3 x^{8}+3 x^{4}-1\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) x -\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2}}{\left (2 x^{4}-1\right ) \left (-1+x \right )^{2} \left (1+x \right )^{2} \left (x^{2}+1\right )^{2}}\right )+2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) \ln \left (\frac {-2 \left (x^{12}-3 x^{8}+3 x^{4}-1\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} x^{9}-\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} x^{8}+4 \left (x^{12}-3 x^{8}+3 x^{4}-1\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} x^{5}-2 \sqrt {x^{12}-3 x^{8}+3 x^{4}-1}\, x^{6}+2 \left (x^{12}-3 x^{8}+3 x^{4}-1\right )^{\frac {3}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) x^{3}+2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} x^{4}-2 \left (x^{12}-3 x^{8}+3 x^{4}-1\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} x +2 \sqrt {x^{12}-3 x^{8}+3 x^{4}-1}\, x^{2}-\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2}}{\left (2 x^{4}-1\right ) \left (-1+x \right )^{2} \left (1+x \right )^{2} \left (x^{2}+1\right )^{2}}\right )\right ) {\left (\left (x^{4}-1\right )^{3}\right )}^{\frac {1}{4}}}{\left (x^{4}-1\right )^{\frac {3}{4}}}\)	\(514\)

[In]

int((x^4-1)^(1/4)*(2*x^8-x^4+2)/x^10/(2*x^4-1),x,method=_RETURNVERBOSE)

[Out]

1/9*(-9*x^9*(ln(((x^4-1)^(1/4)*2^(1/2)*x+x^2+(x^4-1)^(1/2))/(-(x^4-1)^(1/4)*2^(1/2)*x+x^2+(x^4-1)^(1/2)))+2*ar
ctan(((x^4-1)^(1/4)*2^(1/2)+x)/x)+2*arctan(((x^4-1)^(1/4)*2^(1/2)-x)/x))*2^(1/2)+(x^4-1)^(1/4)*(65*x^8+5*x^4+2
))/x^9

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 1.75 (sec) , antiderivative size = 271, normalized size of antiderivative = 2.49 \[ \int \frac {\sqrt [4]{-1+x^4} \left (2-x^4+2 x^8\right )}{x^{10} \left (-1+2 x^4\right )} \, dx=\frac {\left (9 i - 9\right ) \, \sqrt {2} x^{9} \log \left (\frac {\left (i + 1\right ) \, \sqrt {2} {\left (x^{4} - 1\right )}^{\frac {1}{4}} x^{3} + 2 i \, \sqrt {x^{4} - 1} x^{2} + \left (i - 1\right ) \, \sqrt {2} {\left (x^{4} - 1\right )}^{\frac {3}{4}} x + 1}{2 \, x^{4} - 1}\right ) - \left (9 i + 9\right ) \, \sqrt {2} x^{9} \log \left (\frac {-\left (i - 1\right ) \, \sqrt {2} {\left (x^{4} - 1\right )}^{\frac {1}{4}} x^{3} - 2 i \, \sqrt {x^{4} - 1} x^{2} - \left (i + 1\right ) \, \sqrt {2} {\left (x^{4} - 1\right )}^{\frac {3}{4}} x + 1}{2 \, x^{4} - 1}\right ) + \left (9 i + 9\right ) \, \sqrt {2} x^{9} \log \left (\frac {\left (i - 1\right ) \, \sqrt {2} {\left (x^{4} - 1\right )}^{\frac {1}{4}} x^{3} - 2 i \, \sqrt {x^{4} - 1} x^{2} + \left (i + 1\right ) \, \sqrt {2} {\left (x^{4} - 1\right )}^{\frac {3}{4}} x + 1}{2 \, x^{4} - 1}\right ) - \left (9 i - 9\right ) \, \sqrt {2} x^{9} \log \left (\frac {-\left (i + 1\right ) \, \sqrt {2} {\left (x^{4} - 1\right )}^{\frac {1}{4}} x^{3} + 2 i \, \sqrt {x^{4} - 1} x^{2} - \left (i - 1\right ) \, \sqrt {2} {\left (x^{4} - 1\right )}^{\frac {3}{4}} x + 1}{2 \, x^{4} - 1}\right ) + 2 \, {\left (65 \, x^{8} + 5 \, x^{4} + 2\right )} {\left (x^{4} - 1\right )}^{\frac {1}{4}}}{18 \, x^{9}} \]

[In]

integrate((x^4-1)^(1/4)*(2*x^8-x^4+2)/x^10/(2*x^4-1),x, algorithm="fricas")

[Out]

1/18*((9*I - 9)*sqrt(2)*x^9*log(((I + 1)*sqrt(2)*(x^4 - 1)^(1/4)*x^3 + 2*I*sqrt(x^4 - 1)*x^2 + (I - 1)*sqrt(2)
*(x^4 - 1)^(3/4)*x + 1)/(2*x^4 - 1)) - (9*I + 9)*sqrt(2)*x^9*log((-(I - 1)*sqrt(2)*(x^4 - 1)^(1/4)*x^3 - 2*I*s
qrt(x^4 - 1)*x^2 - (I + 1)*sqrt(2)*(x^4 - 1)^(3/4)*x + 1)/(2*x^4 - 1)) + (9*I + 9)*sqrt(2)*x^9*log(((I - 1)*sq
rt(2)*(x^4 - 1)^(1/4)*x^3 - 2*I*sqrt(x^4 - 1)*x^2 + (I + 1)*sqrt(2)*(x^4 - 1)^(3/4)*x + 1)/(2*x^4 - 1)) - (9*I
- 9)*sqrt(2)*x^9*log((-(I + 1)*sqrt(2)*(x^4 - 1)^(1/4)*x^3 + 2*I*sqrt(x^4 - 1)*x^2 - (I - 1)*sqrt(2)*(x^4 - 1
)^(3/4)*x + 1)/(2*x^4 - 1)) + 2*(65*x^8 + 5*x^4 + 2)*(x^4 - 1)^(1/4))/x^9

Sympy [F]

\[ \int \frac {\sqrt [4]{-1+x^4} \left (2-x^4+2 x^8\right )}{x^{10} \left (-1+2 x^4\right )} \, dx=\int \frac {\sqrt [4]{\left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right )} \left (2 x^{8} - x^{4} + 2\right )}{x^{10} \cdot \left (2 x^{4} - 1\right )}\, dx \]

[In]

integrate((x**4-1)**(1/4)*(2*x**8-x**4+2)/x**10/(2*x**4-1),x)

[Out]

Integral(((x - 1)*(x + 1)*(x**2 + 1))**(1/4)*(2*x**8 - x**4 + 2)/(x**10*(2*x**4 - 1)), x)

Maxima [F]

\[ \int \frac {\sqrt [4]{-1+x^4} \left (2-x^4+2 x^8\right )}{x^{10} \left (-1+2 x^4\right )} \, dx=\int { \frac {{\left (2 \, x^{8} - x^{4} + 2\right )} {\left (x^{4} - 1\right )}^{\frac {1}{4}}}{{\left (2 \, x^{4} - 1\right )} x^{10}} \,d x } \]

[In]

integrate((x^4-1)^(1/4)*(2*x^8-x^4+2)/x^10/(2*x^4-1),x, algorithm="maxima")

[Out]

integrate((2*x^8 - x^4 + 2)*(x^4 - 1)^(1/4)/((2*x^4 - 1)*x^10), x)

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.57 \[ \int \frac {\sqrt [4]{-1+x^4} \left (2-x^4+2 x^8\right )}{x^{10} \left (-1+2 x^4\right )} \, dx=-2 \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + \frac {2 \, {\left (x^{4} - 1\right )}^{\frac {1}{4}}}{x}\right )}\right ) - 2 \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - \frac {2 \, {\left (x^{4} - 1\right )}^{\frac {1}{4}}}{x}\right )}\right ) - \sqrt {2} \log \left (\frac {\sqrt {2} {\left (x^{4} - 1\right )}^{\frac {1}{4}}}{x} + \frac {\sqrt {x^{4} - 1}}{x^{2}} + 1\right ) + \sqrt {2} \log \left (-\frac {\sqrt {2} {\left (x^{4} - 1\right )}^{\frac {1}{4}}}{x} + \frac {\sqrt {x^{4} - 1}}{x^{2}} + 1\right ) + \frac {{\left (x^{4} - 1\right )}^{\frac {1}{4}} {\left (\frac {1}{x^{4}} - 1\right )}}{x} + \frac {8 \, {\left (x^{4} - 1\right )}^{\frac {1}{4}}}{x} + \frac {2 \, {\left (x^{8} - 2 \, x^{4} + 1\right )} {\left (x^{4} - 1\right )}^{\frac {1}{4}}}{9 \, x^{9}} \]

[In]

integrate((x^4-1)^(1/4)*(2*x^8-x^4+2)/x^10/(2*x^4-1),x, algorithm="giac")

[Out]

-2*sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2) + 2*(x^4 - 1)^(1/4)/x)) - 2*sqrt(2)*arctan(-1/2*sqrt(2)*(sqrt(2) - 2*(x
^4 - 1)^(1/4)/x)) - sqrt(2)*log(sqrt(2)*(x^4 - 1)^(1/4)/x + sqrt(x^4 - 1)/x^2 + 1) + sqrt(2)*log(-sqrt(2)*(x^4
- 1)^(1/4)/x + sqrt(x^4 - 1)/x^2 + 1) + (x^4 - 1)^(1/4)*(1/x^4 - 1)/x + 8*(x^4 - 1)^(1/4)/x + 2/9*(x^8 - 2*x^
4 + 1)*(x^4 - 1)^(1/4)/x^9

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt [4]{-1+x^4} \left (2-x^4+2 x^8\right )}{x^{10} \left (-1+2 x^4\right )} \, dx=\int \frac {{\left (x^4-1\right )}^{1/4}\,\left (2\,x^8-x^4+2\right )}{x^{10}\,\left (2\,x^4-1\right )} \,d x \]

[In]

int(((x^4 - 1)^(1/4)*(2*x^8 - x^4 + 2))/(x^10*(2*x^4 - 1)),x)

[Out]

int(((x^4 - 1)^(1/4)*(2*x^8 - x^4 + 2))/(x^10*(2*x^4 - 1)), x)