\(\int \frac {2+x^4}{\sqrt [4]{-1+x^4} (-2+x^8)} \, dx\) [1174]

Optimal result

Integrand size = 22, antiderivative size = 86 \[ \int \frac {2+x^4}{\sqrt [4]{-1+x^4} \left (-2+x^8\right )} \, dx=-\frac {1}{16} \text {RootSum}\left [1-4 \text {$\#$1}^4+2 \text {$\#$1}^8\&,\frac {-3 \log (x)+3 \log \left (\sqrt [4]{-1+x^4}-x \text {$\#$1}\right )+2 \log (x) \text {$\#$1}^4-2 \log \left (\sqrt [4]{-1+x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4}{-\text {$\#$1}+\text {$\#$1}^5}\&\right ] \]

[Out]

Unintegrable

Rubi [A] (verified)

Time = 0.28 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.99, number of steps used = 10, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {6857, 385, 218, 212, 209} \[ \int \frac {2+x^4}{\sqrt [4]{-1+x^4} \left (-2+x^8\right )} \, dx=-\frac {\sqrt [4]{58-41 \sqrt {2}} \arctan \left (\frac {x}{\sqrt [4]{2-\sqrt {2}} \sqrt [4]{x^4-1}}\right )}{4 \sqrt {2}}-\frac {1}{8} \left (2+\sqrt {2}\right )^{5/4} \arctan \left (\frac {x}{\sqrt [4]{2+\sqrt {2}} \sqrt [4]{x^4-1}}\right )-\frac {\sqrt [4]{58-41 \sqrt {2}} \text {arctanh}\left (\frac {x}{\sqrt [4]{2-\sqrt {2}} \sqrt [4]{x^4-1}}\right )}{4 \sqrt {2}}-\frac {1}{8} \left (2+\sqrt {2}\right )^{5/4} \text {arctanh}\left (\frac {x}{\sqrt [4]{2+\sqrt {2}} \sqrt [4]{x^4-1}}\right ) \]

[In]

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

[Out]

-1/4*((58 - 41*Sqrt[2])^(1/4)*ArcTan[x/((2 - Sqrt[2])^(1/4)*(-1 + x^4)^(1/4))])/Sqrt[2] - ((2 + Sqrt[2])^(5/4)
*ArcTan[x/((2 + Sqrt[2])^(1/4)*(-1 + x^4)^(1/4))])/8 - ((58 - 41*Sqrt[2])^(1/4)*ArcTanh[x/((2 - Sqrt[2])^(1/4)
*(-1 + x^4)^(1/4))])/(4*Sqrt[2]) - ((2 + Sqrt[2])^(5/4)*ArcTanh[x/((2 + Sqrt[2])^(1/4)*(-1 + x^4)^(1/4))])/8

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 218

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

Rule 385

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

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 {2}}{2 \sqrt {2} \left (\sqrt {2}-x^4\right ) \sqrt [4]{-1+x^4}}+\frac {-2+\sqrt {2}}{2 \sqrt {2} \sqrt [4]{-1+x^4} \left (\sqrt {2}+x^4\right )}\right ) \, dx \\ & = \frac {1}{2} \left (1-\sqrt {2}\right ) \int \frac {1}{\sqrt [4]{-1+x^4} \left (\sqrt {2}+x^4\right )} \, dx-\frac {1}{2} \left (1+\sqrt {2}\right ) \int \frac {1}{\left (\sqrt {2}-x^4\right ) \sqrt [4]{-1+x^4}} \, dx \\ & = \frac {1}{2} \left (1-\sqrt {2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {2}-\left (1+\sqrt {2}\right ) x^4} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right )-\frac {1}{2} \left (1+\sqrt {2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {2}-\left (-1+\sqrt {2}\right ) x^4} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right ) \\ & = \frac {1}{4} \left (\left (1-\sqrt {2}\right ) \sqrt {\frac {1}{2} \left (2-\sqrt {2}\right )}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {2-\sqrt {2}}-x^2} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right )+\frac {1}{4} \left (\left (1-\sqrt {2}\right ) \sqrt {\frac {1}{2} \left (2-\sqrt {2}\right )}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {2-\sqrt {2}}+x^2} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right )-\frac {1}{4} \left (\left (1+\sqrt {2}\right ) \sqrt {\frac {1}{2} \left (2+\sqrt {2}\right )}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {2+\sqrt {2}}-x^2} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right )-\frac {1}{4} \left (\left (1+\sqrt {2}\right ) \sqrt {\frac {1}{2} \left (2+\sqrt {2}\right )}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {2+\sqrt {2}}+x^2} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right ) \\ & = -\frac {\sqrt [4]{58-41 \sqrt {2}} \arctan \left (\frac {x}{\sqrt [4]{2-\sqrt {2}} \sqrt [4]{-1+x^4}}\right )}{4 \sqrt {2}}-\frac {\sqrt [4]{58+41 \sqrt {2}} \arctan \left (\frac {x}{\sqrt [4]{2+\sqrt {2}} \sqrt [4]{-1+x^4}}\right )}{4 \sqrt {2}}-\frac {\sqrt [4]{58-41 \sqrt {2}} \text {arctanh}\left (\frac {x}{\sqrt [4]{2-\sqrt {2}} \sqrt [4]{-1+x^4}}\right )}{4 \sqrt {2}}-\frac {\sqrt [4]{58+41 \sqrt {2}} \text {arctanh}\left (\frac {x}{\sqrt [4]{2+\sqrt {2}} \sqrt [4]{-1+x^4}}\right )}{4 \sqrt {2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.26 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.00 \[ \int \frac {2+x^4}{\sqrt [4]{-1+x^4} \left (-2+x^8\right )} \, dx=-\frac {1}{16} \text {RootSum}\left [1-4 \text {$\#$1}^4+2 \text {$\#$1}^8\&,\frac {-3 \log (x)+3 \log \left (\sqrt [4]{-1+x^4}-x \text {$\#$1}\right )+2 \log (x) \text {$\#$1}^4-2 \log \left (\sqrt [4]{-1+x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4}{-\text {$\#$1}+\text {$\#$1}^5}\&\right ] \]

[In]

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

[Out]

-1/16*RootSum[1 - 4*#1^4 + 2*#1^8 & , (-3*Log[x] + 3*Log[(-1 + x^4)^(1/4) - x*#1] + 2*Log[x]*#1^4 - 2*Log[(-1
+ x^4)^(1/4) - x*#1]*#1^4)/(-#1 + #1^5) & ]

Maple [N/A] (verified)

Time = 30.12 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.62

method	result	size
pseudoelliptic	\(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (2 \textit {\_Z}^{8}-4 \textit {\_Z}^{4}+1\right )}{\sum }\frac {\left (2 \textit {\_R}^{4}-3\right ) \ln \left (\frac {-\textit {\_R} x +\left (x^{4}-1\right )^{\frac {1}{4}}}{x}\right )}{\textit {\_R}^{5}-\textit {\_R}}\right )}{16}\)	\(53\)
trager	\(\text {Expression too large to display}\)	\(3206\)

[In]

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

[Out]

1/16*sum((2*_R^4-3)*ln((-_R*x+(x^4-1)^(1/4))/x)/(_R^5-_R),_R=RootOf(2*_Z^8-4*_Z^4+1))

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 3 vs. order 1.

Time = 9.98 (sec) , antiderivative size = 1501, normalized size of antiderivative = 17.45 \[ \int \frac {2+x^4}{\sqrt [4]{-1+x^4} \left (-2+x^8\right )} \, dx=\text {Too large to display} \]

[In]

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

[Out]

-1/32*sqrt(2)*sqrt(-sqrt(-41*sqrt(2) + 58))*log((4*(11*x^5 + sqrt(2)*(6*x^5 - 11*x) - 12*x)*(x^4 - 1)^(3/4) -
2*(194*x^7 - 274*x^3 + sqrt(2)*(137*x^7 - 194*x^3))*(x^4 - 1)^(1/4)*sqrt(-41*sqrt(2) + 58) + (126*x^8 - 228*x^
4 - 2*(468*x^6 - 662*x^2 + sqrt(2)*(331*x^6 - 468*x^2))*sqrt(x^4 - 1)*sqrt(-41*sqrt(2) + 58) + sqrt(2)*(91*x^8
- 160*x^4 + 46) + 68)*sqrt(-sqrt(-41*sqrt(2) + 58)))/(x^8 - 2)) + 1/32*sqrt(2)*sqrt(-sqrt(-41*sqrt(2) + 58))*
log((4*(11*x^5 + sqrt(2)*(6*x^5 - 11*x) - 12*x)*(x^4 - 1)^(3/4) - 2*(194*x^7 - 274*x^3 + sqrt(2)*(137*x^7 - 19
4*x^3))*(x^4 - 1)^(1/4)*sqrt(-41*sqrt(2) + 58) - (126*x^8 - 228*x^4 - 2*(468*x^6 - 662*x^2 + sqrt(2)*(331*x^6
- 468*x^2))*sqrt(x^4 - 1)*sqrt(-41*sqrt(2) + 58) + sqrt(2)*(91*x^8 - 160*x^4 + 46) + 68)*sqrt(-sqrt(-41*sqrt(2
) + 58)))/(x^8 - 2)) + 1/32*sqrt(2)*sqrt(-sqrt(41*sqrt(2) + 58))*log((4*(11*x^5 - sqrt(2)*(6*x^5 - 11*x) - 12*
x)*(x^4 - 1)^(3/4) - 2*(194*x^7 - 274*x^3 - sqrt(2)*(137*x^7 - 194*x^3))*(x^4 - 1)^(1/4)*sqrt(41*sqrt(2) + 58)
+ (126*x^8 - 228*x^4 - 2*(468*x^6 - 662*x^2 - sqrt(2)*(331*x^6 - 468*x^2))*sqrt(x^4 - 1)*sqrt(41*sqrt(2) + 58
) - sqrt(2)*(91*x^8 - 160*x^4 + 46) + 68)*sqrt(-sqrt(41*sqrt(2) + 58)))/(x^8 - 2)) - 1/32*sqrt(2)*sqrt(-sqrt(4
1*sqrt(2) + 58))*log((4*(11*x^5 - sqrt(2)*(6*x^5 - 11*x) - 12*x)*(x^4 - 1)^(3/4) - 2*(194*x^7 - 274*x^3 - sqrt
(2)*(137*x^7 - 194*x^3))*(x^4 - 1)^(1/4)*sqrt(41*sqrt(2) + 58) - (126*x^8 - 228*x^4 - 2*(468*x^6 - 662*x^2 - s
qrt(2)*(331*x^6 - 468*x^2))*sqrt(x^4 - 1)*sqrt(41*sqrt(2) + 58) - sqrt(2)*(91*x^8 - 160*x^4 + 46) + 68)*sqrt(-
sqrt(41*sqrt(2) + 58)))/(x^8 - 2)) - 1/32*sqrt(2)*(-41*sqrt(2) + 58)^(1/4)*log((4*(11*x^5 + sqrt(2)*(6*x^5 - 1
1*x) - 12*x)*(x^4 - 1)^(3/4) + 2*(194*x^7 - 274*x^3 + sqrt(2)*(137*x^7 - 194*x^3))*(x^4 - 1)^(1/4)*sqrt(-41*sq
rt(2) + 58) + (126*x^8 - 228*x^4 + 2*(468*x^6 - 662*x^2 + sqrt(2)*(331*x^6 - 468*x^2))*sqrt(x^4 - 1)*sqrt(-41*
sqrt(2) + 58) + sqrt(2)*(91*x^8 - 160*x^4 + 46) + 68)*(-41*sqrt(2) + 58)^(1/4))/(x^8 - 2)) + 1/32*sqrt(2)*(-41
*sqrt(2) + 58)^(1/4)*log((4*(11*x^5 + sqrt(2)*(6*x^5 - 11*x) - 12*x)*(x^4 - 1)^(3/4) + 2*(194*x^7 - 274*x^3 +
sqrt(2)*(137*x^7 - 194*x^3))*(x^4 - 1)^(1/4)*sqrt(-41*sqrt(2) + 58) - (126*x^8 - 228*x^4 + 2*(468*x^6 - 662*x^
2 + sqrt(2)*(331*x^6 - 468*x^2))*sqrt(x^4 - 1)*sqrt(-41*sqrt(2) + 58) + sqrt(2)*(91*x^8 - 160*x^4 + 46) + 68)*
(-41*sqrt(2) + 58)^(1/4))/(x^8 - 2)) + 1/32*sqrt(2)*(41*sqrt(2) + 58)^(1/4)*log((4*(11*x^5 - sqrt(2)*(6*x^5 -
11*x) - 12*x)*(x^4 - 1)^(3/4) + 2*(194*x^7 - 274*x^3 - sqrt(2)*(137*x^7 - 194*x^3))*(x^4 - 1)^(1/4)*sqrt(41*sq
rt(2) + 58) + (126*x^8 - 228*x^4 + 2*(468*x^6 - 662*x^2 - sqrt(2)*(331*x^6 - 468*x^2))*sqrt(x^4 - 1)*sqrt(41*s
qrt(2) + 58) - sqrt(2)*(91*x^8 - 160*x^4 + 46) + 68)*(41*sqrt(2) + 58)^(1/4))/(x^8 - 2)) - 1/32*sqrt(2)*(41*sq
rt(2) + 58)^(1/4)*log((4*(11*x^5 - sqrt(2)*(6*x^5 - 11*x) - 12*x)*(x^4 - 1)^(3/4) + 2*(194*x^7 - 274*x^3 - sqr
t(2)*(137*x^7 - 194*x^3))*(x^4 - 1)^(1/4)*sqrt(41*sqrt(2) + 58) - (126*x^8 - 228*x^4 + 2*(468*x^6 - 662*x^2 -
sqrt(2)*(331*x^6 - 468*x^2))*sqrt(x^4 - 1)*sqrt(41*sqrt(2) + 58) - sqrt(2)*(91*x^8 - 160*x^4 + 46) + 68)*(41*s
qrt(2) + 58)^(1/4))/(x^8 - 2))

Sympy [N/A]

Not integrable

Time = 7.70 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.30 \[ \int \frac {2+x^4}{\sqrt [4]{-1+x^4} \left (-2+x^8\right )} \, dx=\int \frac {x^{4} + 2}{\sqrt [4]{\left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right )} \left (x^{8} - 2\right )}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.30 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.26 \[ \int \frac {2+x^4}{\sqrt [4]{-1+x^4} \left (-2+x^8\right )} \, dx=\int { \frac {x^{4} + 2}{{\left (x^{8} - 2\right )} {\left (x^{4} - 1\right )}^{\frac {1}{4}}} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 0.28 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.26 \[ \int \frac {2+x^4}{\sqrt [4]{-1+x^4} \left (-2+x^8\right )} \, dx=\int { \frac {x^{4} + 2}{{\left (x^{8} - 2\right )} {\left (x^{4} - 1\right )}^{\frac {1}{4}}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 5.55 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.26 \[ \int \frac {2+x^4}{\sqrt [4]{-1+x^4} \left (-2+x^8\right )} \, dx=\int \frac {x^4+2}{{\left (x^4-1\right )}^{1/4}\,\left (x^8-2\right )} \,d x \]

[In]

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

[Out]

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