\(\int \frac {1}{(-2+x^4) \sqrt [4]{x^2+x^4}} \, dx\) [624]

Optimal result

Integrand size = 19, antiderivative size = 49 \[ \int \frac {1}{\left (-2+x^4\right ) \sqrt [4]{x^2+x^4}} \, dx=\frac {1}{8} \text {RootSum}\left [1-4 \text {$\#$1}^4+2 \text {$\#$1}^8\&,\frac {-\log (x)+\log \left (\sqrt [4]{x^2+x^4}-x \text {$\#$1}\right )}{\text {$\#$1}}\&\right ] \]

[Out]

Unintegrable

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(277\) vs. \(2(49)=98\).

Time = 0.13 (sec) , antiderivative size = 277, normalized size of antiderivative = 5.65, number of steps used = 11, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.368, Rules used = {2081, 1284, 1443, 385, 218, 212, 209} \[ \int \frac {1}{\left (-2+x^4\right ) \sqrt [4]{x^2+x^4}} \, dx=-\frac {\sqrt [4]{2-\sqrt {2}} \sqrt {x} \sqrt [4]{x^2+1} \arctan \left (\frac {\sqrt {x}}{\sqrt [4]{2-\sqrt {2}} \sqrt [4]{x^2+1}}\right )}{4 \sqrt [4]{x^4+x^2}}-\frac {\sqrt [4]{2+\sqrt {2}} \sqrt {x} \sqrt [4]{x^2+1} \arctan \left (\frac {\sqrt {x}}{\sqrt [4]{2+\sqrt {2}} \sqrt [4]{x^2+1}}\right )}{4 \sqrt [4]{x^4+x^2}}-\frac {\sqrt [4]{2-\sqrt {2}} \sqrt {x} \sqrt [4]{x^2+1} \text {arctanh}\left (\frac {\sqrt {x}}{\sqrt [4]{2-\sqrt {2}} \sqrt [4]{x^2+1}}\right )}{4 \sqrt [4]{x^4+x^2}}-\frac {\sqrt [4]{2+\sqrt {2}} \sqrt {x} \sqrt [4]{x^2+1} \text {arctanh}\left (\frac {\sqrt {x}}{\sqrt [4]{2+\sqrt {2}} \sqrt [4]{x^2+1}}\right )}{4 \sqrt [4]{x^4+x^2}} \]

[In]

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

[Out]

-1/4*((2 - Sqrt[2])^(1/4)*Sqrt[x]*(1 + x^2)^(1/4)*ArcTan[Sqrt[x]/((2 - Sqrt[2])^(1/4)*(1 + x^2)^(1/4))])/(x^2
+ x^4)^(1/4) - ((2 + Sqrt[2])^(1/4)*Sqrt[x]*(1 + x^2)^(1/4)*ArcTan[Sqrt[x]/((2 + Sqrt[2])^(1/4)*(1 + x^2)^(1/4
))])/(4*(x^2 + x^4)^(1/4)) - ((2 - Sqrt[2])^(1/4)*Sqrt[x]*(1 + x^2)^(1/4)*ArcTanh[Sqrt[x]/((2 - Sqrt[2])^(1/4)
*(1 + x^2)^(1/4))])/(4*(x^2 + x^4)^(1/4)) - ((2 + Sqrt[2])^(1/4)*Sqrt[x]*(1 + x^2)^(1/4)*ArcTanh[Sqrt[x]/((2 +
Sqrt[2])^(1/4)*(1 + x^2)^(1/4))])/(4*(x^2 + 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 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 1284

Int[((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (c_.)*(x_)^4)^(p_), x_Symbol] :> With[{k = Denominat
or[m]}, Dist[k/f, Subst[Int[x^(k*(m + 1) - 1)*(d + e*(x^(2*k)/f))^q*(a + c*(x^(4*k)/f))^p, x], x, (f*x)^(1/k)]
, x]] /; FreeQ[{a, c, d, e, f, p, q}, x] && FractionQ[m] && IntegerQ[p]

Rule 1443

Int[((d_) + (e_.)*(x_)^(n_))^(q_)/((a_) + (c_.)*(x_)^(n2_)), x_Symbol] :> With[{r = Rt[(-a)*c, 2]}, Dist[-c/(2
*r), Int[(d + e*x^n)^q/(r - c*x^n), x], x] - Dist[c/(2*r), Int[(d + e*x^n)^q/(r + c*x^n), x], x]] /; FreeQ[{a,
c, d, e, n, q}, x] && EqQ[n2, 2*n] && NeQ[c*d^2 + a*e^2, 0] &&  !IntegerQ[q]

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^2}\right ) \int \frac {1}{\sqrt {x} \sqrt [4]{1+x^2} \left (-2+x^4\right )} \, dx}{\sqrt [4]{x^2+x^4}} \\ & = \frac {\left (2 \sqrt {x} \sqrt [4]{1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{1+x^4} \left (-2+x^8\right )} \, dx,x,\sqrt {x}\right )}{\sqrt [4]{x^2+x^4}} \\ & = -\frac {\left (\sqrt {x} \sqrt [4]{1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (\sqrt {2}-x^4\right ) \sqrt [4]{1+x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt {2} \sqrt [4]{x^2+x^4}}-\frac {\left (\sqrt {x} \sqrt [4]{1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{1+x^4} \left (\sqrt {2}+x^4\right )} \, dx,x,\sqrt {x}\right )}{\sqrt {2} \sqrt [4]{x^2+x^4}} \\ & = -\frac {\left (\sqrt {x} \sqrt [4]{1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {2}-\left (-1+\sqrt {2}\right ) x^4} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{1+x^2}}\right )}{\sqrt {2} \sqrt [4]{x^2+x^4}}-\frac {\left (\sqrt {x} \sqrt [4]{1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {2}-\left (1+\sqrt {2}\right ) x^4} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{1+x^2}}\right )}{\sqrt {2} \sqrt [4]{x^2+x^4}} \\ & = -\frac {\left (\sqrt {2-\sqrt {2}} \sqrt {x} \sqrt [4]{1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {2-\sqrt {2}}-x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{1+x^2}}\right )}{4 \sqrt [4]{x^2+x^4}}-\frac {\left (\sqrt {2-\sqrt {2}} \sqrt {x} \sqrt [4]{1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {2-\sqrt {2}}+x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{1+x^2}}\right )}{4 \sqrt [4]{x^2+x^4}}-\frac {\left (\sqrt {2+\sqrt {2}} \sqrt {x} \sqrt [4]{1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {2+\sqrt {2}}-x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{1+x^2}}\right )}{4 \sqrt [4]{x^2+x^4}}-\frac {\left (\sqrt {2+\sqrt {2}} \sqrt {x} \sqrt [4]{1+x^2}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {2+\sqrt {2}}+x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{1+x^2}}\right )}{4 \sqrt [4]{x^2+x^4}} \\ & = -\frac {\sqrt [4]{2-\sqrt {2}} \sqrt {x} \sqrt [4]{1+x^2} \arctan \left (\frac {\sqrt {x}}{\sqrt [4]{2-\sqrt {2}} \sqrt [4]{1+x^2}}\right )}{4 \sqrt [4]{x^2+x^4}}-\frac {\sqrt [4]{2+\sqrt {2}} \sqrt {x} \sqrt [4]{1+x^2} \arctan \left (\frac {\sqrt {x}}{\sqrt [4]{2+\sqrt {2}} \sqrt [4]{1+x^2}}\right )}{4 \sqrt [4]{x^2+x^4}}-\frac {\sqrt [4]{2-\sqrt {2}} \sqrt {x} \sqrt [4]{1+x^2} \text {arctanh}\left (\frac {\sqrt {x}}{\sqrt [4]{2-\sqrt {2}} \sqrt [4]{1+x^2}}\right )}{4 \sqrt [4]{x^2+x^4}}-\frac {\sqrt [4]{2+\sqrt {2}} \sqrt {x} \sqrt [4]{1+x^2} \text {arctanh}\left (\frac {\sqrt {x}}{\sqrt [4]{2+\sqrt {2}} \sqrt [4]{1+x^2}}\right )}{4 \sqrt [4]{x^2+x^4}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.22 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.63 \[ \int \frac {1}{\left (-2+x^4\right ) \sqrt [4]{x^2+x^4}} \, dx=\frac {\sqrt {x} \sqrt [4]{1+x^2} \text {RootSum}\left [1-4 \text {$\#$1}^4+2 \text {$\#$1}^8\&,\frac {-\log \left (\sqrt {x}\right )+\log \left (\sqrt [4]{1+x^2}-\sqrt {x} \text {$\#$1}\right )}{\text {$\#$1}}\&\right ]}{8 \sqrt [4]{x^2+x^4}} \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Time = 50.78 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.90

method	result	size
pseudoelliptic	\(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (2 \textit {\_Z}^{8}-4 \textit {\_Z}^{4}+1\right )}{\sum }\frac {\ln \left (\frac {-\textit {\_R} x +\left (x^{2} \left (x^{2}+1\right )\right )^{\frac {1}{4}}}{x}\right )}{\textit {\_R}}\right )}{8}\)	\(44\)
trager	\(\text {Expression too large to display}\)	\(3668\)

[In]

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

[Out]

1/8*sum(ln((-_R*x+(x^2*(x^2+1))^(1/4))/x)/_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 = 3.01 (sec) , antiderivative size = 1477, normalized size of antiderivative = 30.14 \[ \int \frac {1}{\left (-2+x^4\right ) \sqrt [4]{x^2+x^4}} \, dx=\text {Too large to display} \]

[In]

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

[Out]

-1/16*sqrt(-sqrt(-sqrt(2) + 2))*log((4*(x^4 + x^2)^(3/4)*(11*x^2 + sqrt(2)*(6*x^2 + 11) + 12) - 2*(34*x^4 + 46
*x^2 + sqrt(2)*(23*x^4 + 34*x^2))*(x^4 + x^2)^(1/4)*sqrt(-sqrt(2) + 2) + (56*x^5 + 92*x^3 - 2*sqrt(x^4 + x^2)*
(34*x^3 + sqrt(2)*(23*x^3 + 34*x) + 46*x)*sqrt(-sqrt(2) + 2) + sqrt(2)*(35*x^5 + 68*x^3 + 22*x) + 24*x)*sqrt(-
sqrt(-sqrt(2) + 2)))/(x^5 - 2*x)) + 1/16*sqrt(-sqrt(-sqrt(2) + 2))*log((4*(x^4 + x^2)^(3/4)*(11*x^2 + sqrt(2)*
(6*x^2 + 11) + 12) - 2*(34*x^4 + 46*x^2 + sqrt(2)*(23*x^4 + 34*x^2))*(x^4 + x^2)^(1/4)*sqrt(-sqrt(2) + 2) - (5
6*x^5 + 92*x^3 - 2*sqrt(x^4 + x^2)*(34*x^3 + sqrt(2)*(23*x^3 + 34*x) + 46*x)*sqrt(-sqrt(2) + 2) + sqrt(2)*(35*
x^5 + 68*x^3 + 22*x) + 24*x)*sqrt(-sqrt(-sqrt(2) + 2)))/(x^5 - 2*x)) - 1/16*sqrt(-sqrt(sqrt(2) + 2))*log((4*(x
^4 + x^2)^(3/4)*(11*x^2 - sqrt(2)*(6*x^2 + 11) + 12) - 2*(34*x^4 + 46*x^2 - sqrt(2)*(23*x^4 + 34*x^2))*(x^4 +
x^2)^(1/4)*sqrt(sqrt(2) + 2) + (56*x^5 + 92*x^3 - 2*sqrt(x^4 + x^2)*(34*x^3 - sqrt(2)*(23*x^3 + 34*x) + 46*x)*
sqrt(sqrt(2) + 2) - sqrt(2)*(35*x^5 + 68*x^3 + 22*x) + 24*x)*sqrt(-sqrt(sqrt(2) + 2)))/(x^5 - 2*x)) + 1/16*sqr
t(-sqrt(sqrt(2) + 2))*log((4*(x^4 + x^2)^(3/4)*(11*x^2 - sqrt(2)*(6*x^2 + 11) + 12) - 2*(34*x^4 + 46*x^2 - sqr
t(2)*(23*x^4 + 34*x^2))*(x^4 + x^2)^(1/4)*sqrt(sqrt(2) + 2) - (56*x^5 + 92*x^3 - 2*sqrt(x^4 + x^2)*(34*x^3 - s
qrt(2)*(23*x^3 + 34*x) + 46*x)*sqrt(sqrt(2) + 2) - sqrt(2)*(35*x^5 + 68*x^3 + 22*x) + 24*x)*sqrt(-sqrt(sqrt(2)
+ 2)))/(x^5 - 2*x)) - 1/16*(-sqrt(2) + 2)^(1/4)*log((4*(x^4 + x^2)^(3/4)*(11*x^2 + sqrt(2)*(6*x^2 + 11) + 12)
+ 2*(34*x^4 + 46*x^2 + sqrt(2)*(23*x^4 + 34*x^2))*(x^4 + x^2)^(1/4)*sqrt(-sqrt(2) + 2) + (56*x^5 + 92*x^3 + 2
*sqrt(x^4 + x^2)*(34*x^3 + sqrt(2)*(23*x^3 + 34*x) + 46*x)*sqrt(-sqrt(2) + 2) + sqrt(2)*(35*x^5 + 68*x^3 + 22*
x) + 24*x)*(-sqrt(2) + 2)^(1/4))/(x^5 - 2*x)) + 1/16*(-sqrt(2) + 2)^(1/4)*log((4*(x^4 + x^2)^(3/4)*(11*x^2 + s
qrt(2)*(6*x^2 + 11) + 12) + 2*(34*x^4 + 46*x^2 + sqrt(2)*(23*x^4 + 34*x^2))*(x^4 + x^2)^(1/4)*sqrt(-sqrt(2) +
2) - (56*x^5 + 92*x^3 + 2*sqrt(x^4 + x^2)*(34*x^3 + sqrt(2)*(23*x^3 + 34*x) + 46*x)*sqrt(-sqrt(2) + 2) + sqrt(
2)*(35*x^5 + 68*x^3 + 22*x) + 24*x)*(-sqrt(2) + 2)^(1/4))/(x^5 - 2*x)) - 1/16*(sqrt(2) + 2)^(1/4)*log((4*(x^4
+ x^2)^(3/4)*(11*x^2 - sqrt(2)*(6*x^2 + 11) + 12) + 2*(34*x^4 + 46*x^2 - sqrt(2)*(23*x^4 + 34*x^2))*(x^4 + x^2
)^(1/4)*sqrt(sqrt(2) + 2) + (56*x^5 + 92*x^3 + 2*sqrt(x^4 + x^2)*(34*x^3 - sqrt(2)*(23*x^3 + 34*x) + 46*x)*sqr
t(sqrt(2) + 2) - sqrt(2)*(35*x^5 + 68*x^3 + 22*x) + 24*x)*(sqrt(2) + 2)^(1/4))/(x^5 - 2*x)) + 1/16*(sqrt(2) +
2)^(1/4)*log((4*(x^4 + x^2)^(3/4)*(11*x^2 - sqrt(2)*(6*x^2 + 11) + 12) + 2*(34*x^4 + 46*x^2 - sqrt(2)*(23*x^4
+ 34*x^2))*(x^4 + x^2)^(1/4)*sqrt(sqrt(2) + 2) - (56*x^5 + 92*x^3 + 2*sqrt(x^4 + x^2)*(34*x^3 - sqrt(2)*(23*x^
3 + 34*x) + 46*x)*sqrt(sqrt(2) + 2) - sqrt(2)*(35*x^5 + 68*x^3 + 22*x) + 24*x)*(sqrt(2) + 2)^(1/4))/(x^5 - 2*x
))

Sympy [N/A]

Not integrable

Time = 0.97 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.39 \[ \int \frac {1}{\left (-2+x^4\right ) \sqrt [4]{x^2+x^4}} \, dx=\int \frac {1}{\sqrt [4]{x^{2} \left (x^{2} + 1\right )} \left (x^{4} - 2\right )}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.29 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.43 \[ \int \frac {1}{\left (-2+x^4\right ) \sqrt [4]{x^2+x^4}} \, dx=\int { \frac {1}{{\left (x^{4} + x^{2}\right )}^{\frac {1}{4}} {\left (x^{4} - 2\right )}} \,d x } \]

[In]

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

[Out]

2/21*(4*x^5 + x^3 - 3*x)/((x^(9/2) - 2*sqrt(x))*(x^2 + 1)^(1/4)) + integrate(16/21*(4*x^4 + x^2 - 3)/((x^(17/2
) - 4*x^(9/2) + 4*sqrt(x))*(x^2 + 1)^(1/4)), x)

Giac [N/A]

Not integrable

Time = 0.32 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.39 \[ \int \frac {1}{\left (-2+x^4\right ) \sqrt [4]{x^2+x^4}} \, dx=\int { \frac {1}{{\left (x^{4} + x^{2}\right )}^{\frac {1}{4}} {\left (x^{4} - 2\right )}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 5.57 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.39 \[ \int \frac {1}{\left (-2+x^4\right ) \sqrt [4]{x^2+x^4}} \, dx=\int \frac {1}{{\left (x^4+x^2\right )}^{1/4}\,\left (x^4-2\right )} \,d x \]

[In]

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

[Out]

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