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

Optimal. Leaf size=49 \[ \frac {1}{8} \text {RootSum}\left [1-4 \text {$\#$1}^4+2 \text {$\#$1}^8\& ,\frac {-\log (x)+\log \left (\sqrt [4]{x^2+x^4}-x \text {$\#$1}\right )}{\text {$\#$1}}\& \right ] \]

[Out]

Unintegrable

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Rubi [B] Leaf count is larger than twice the leaf count of optimal. \(277\) vs. \(2(49)=98\).
time = 0.12, antiderivative size = 277, normalized size of antiderivative = 5.65, number of steps used = 11, number of rules used = 7, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.368, Rules used = {2081, 1284, 1443, 385, 218, 212, 209} \begin {gather*} -\frac {\sqrt [4]{2-\sqrt {2}} \sqrt {x} \sqrt [4]{x^2+1} \text {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 {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} \tanh ^{-1}\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} \tanh ^{-1}\left (\frac {\sqrt {x}}{\sqrt [4]{2+\sqrt {2}} \sqrt [4]{x^2+1}}\right )}{4 \sqrt [4]{x^4+x^2}} \end {gather*}

Antiderivative was successfully verified.

[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*} \int \frac {1}{\left (-2+x^4\right ) \sqrt [4]{x^2+x^4}} \, dx &=\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} \tan ^{-1}\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} \tan ^{-1}\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} \tanh ^{-1}\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} \tanh ^{-1}\left (\frac {\sqrt {x}}{\sqrt [4]{2+\sqrt {2}} \sqrt [4]{1+x^2}}\right )}{4 \sqrt [4]{x^2+x^4}}\\ \end {align*}

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Mathematica [A]
time = 0.15, size = 80, normalized size = 1.63 \begin {gather*} \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}} \end {gather*}

Antiderivative was successfully verified.

[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))

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 1.
time = 44.04, size = 3152, normalized size = 64.33

method	result	size
trager	\(\text {Expression too large to display}\)	\(3152\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

8388608*RootOf(RootOf(8388608*_Z^8-8192*_Z^4+1)^2+_Z^2)*RootOf(8388608*_Z^8-8192*_Z^4+1)^7*RootOf(4096*RootOf(
8388608*_Z^8-8192*_Z^4+1)^6-3*RootOf(8388608*_Z^8-8192*_Z^4+1)^2+_Z^2)*ln((2199023255552*RootOf(RootOf(8388608
*_Z^8-8192*_Z^4+1)^2+_Z^2)*RootOf(4096*RootOf(8388608*_Z^8-8192*_Z^4+1)^6-3*RootOf(8388608*_Z^8-8192*_Z^4+1)^2
+_Z^2)*RootOf(8388608*_Z^8-8192*_Z^4+1)^11*x^3-2199023255552*RootOf(RootOf(8388608*_Z^8-8192*_Z^4+1)^2+_Z^2)*R
ootOf(4096*RootOf(8388608*_Z^8-8192*_Z^4+1)^6-3*RootOf(8388608*_Z^8-8192*_Z^4+1)^2+_Z^2)*RootOf(8388608*_Z^8-8
192*_Z^4+1)^11*x+4529848320*RootOf(RootOf(8388608*_Z^8-8192*_Z^4+1)^2+_Z^2)*RootOf(4096*RootOf(8388608*_Z^8-81
92*_Z^4+1)^6-3*RootOf(8388608*_Z^8-8192*_Z^4+1)^2+_Z^2)*RootOf(8388608*_Z^8-8192*_Z^4+1)^7*x^3+187695104*(x^4+
x^2)^(1/2)*RootOf(RootOf(8388608*_Z^8-8192*_Z^4+1)^2+_Z^2)*RootOf(4096*RootOf(8388608*_Z^8-8192*_Z^4+1)^6-3*Ro
otOf(8388608*_Z^8-8192*_Z^4+1)^2+_Z^2)*RootOf(8388608*_Z^8-8192*_Z^4+1)^5*x+4596957184*RootOf(RootOf(8388608*_
Z^8-8192*_Z^4+1)^2+_Z^2)*RootOf(4096*RootOf(8388608*_Z^8-8192*_Z^4+1)^6-3*RootOf(8388608*_Z^8-8192*_Z^4+1)^2+_
Z^2)*RootOf(8388608*_Z^8-8192*_Z^4+1)^7*x+23461888*(x^4+x^2)^(1/4)*RootOf(8388608*_Z^8-8192*_Z^4+1)^6*x^2-6635
52*RootOf(RootOf(8388608*_Z^8-8192*_Z^4+1)^2+_Z^2)*RootOf(4096*RootOf(8388608*_Z^8-8192*_Z^4+1)^6-3*RootOf(838
8608*_Z^8-8192*_Z^4+1)^2+_Z^2)*RootOf(8388608*_Z^8-8192*_Z^4+1)^3*x^3+516096*(x^4+x^2)^(3/4)*RootOf(8388608*_Z
^8-8192*_Z^4+1)^4-27136*(x^4+x^2)^(1/2)*RootOf(RootOf(8388608*_Z^8-8192*_Z^4+1)^2+_Z^2)*RootOf(4096*RootOf(838
8608*_Z^8-8192*_Z^4+1)^6-3*RootOf(8388608*_Z^8-8192*_Z^4+1)^2+_Z^2)*RootOf(8388608*_Z^8-8192*_Z^4+1)*x-589824*
RootOf(RootOf(8388608*_Z^8-8192*_Z^4+1)^2+_Z^2)*RootOf(4096*RootOf(8388608*_Z^8-8192*_Z^4+1)^6-3*RootOf(838860
8*_Z^8-8192*_Z^4+1)^2+_Z^2)*RootOf(8388608*_Z^8-8192*_Z^4+1)^3*x-3392*(x^4+x^2)^(1/4)*RootOf(8388608*_Z^8-8192
*_Z^4+1)^2*x^2-73*(x^4+x^2)^(3/4))/(4096*RootOf(8388608*_Z^8-8192*_Z^4+1)^4*x^2-4096*RootOf(8388608*_Z^8-8192*
_Z^4+1)^4-x^2)/x)-8192*RootOf(RootOf(8388608*_Z^8-8192*_Z^4+1)^2+_Z^2)*RootOf(8388608*_Z^8-8192*_Z^4+1)^3*Root
Of(4096*RootOf(8388608*_Z^8-8192*_Z^4+1)^6-3*RootOf(8388608*_Z^8-8192*_Z^4+1)^2+_Z^2)*ln((2199023255552*RootOf
(RootOf(8388608*_Z^8-8192*_Z^4+1)^2+_Z^2)*RootOf(4096*RootOf(8388608*_Z^8-8192*_Z^4+1)^6-3*RootOf(8388608*_Z^8
-8192*_Z^4+1)^2+_Z^2)*RootOf(8388608*_Z^8-8192*_Z^4+1)^11*x^3-2199023255552*RootOf(RootOf(8388608*_Z^8-8192*_Z
^4+1)^2+_Z^2)*RootOf(4096*RootOf(8388608*_Z^8-8192*_Z^4+1)^6-3*RootOf(8388608*_Z^8-8192*_Z^4+1)^2+_Z^2)*RootOf
(8388608*_Z^8-8192*_Z^4+1)^11*x+4529848320*RootOf(RootOf(8388608*_Z^8-8192*_Z^4+1)^2+_Z^2)*RootOf(4096*RootOf(
8388608*_Z^8-8192*_Z^4+1)^6-3*RootOf(8388608*_Z^8-8192*_Z^4+1)^2+_Z^2)*RootOf(8388608*_Z^8-8192*_Z^4+1)^7*x^3+
187695104*(x^4+x^2)^(1/2)*RootOf(RootOf(8388608*_Z^8-8192*_Z^4+1)^2+_Z^2)*RootOf(4096*RootOf(8388608*_Z^8-8192
*_Z^4+1)^6-3*RootOf(8388608*_Z^8-8192*_Z^4+1)^2+_Z^2)*RootOf(8388608*_Z^8-8192*_Z^4+1)^5*x+4596957184*RootOf(R
ootOf(8388608*_Z^8-8192*_Z^4+1)^2+_Z^2)*RootOf(4096*RootOf(8388608*_Z^8-8192*_Z^4+1)^6-3*RootOf(8388608*_Z^8-8
192*_Z^4+1)^2+_Z^2)*RootOf(8388608*_Z^8-8192*_Z^4+1)^7*x+23461888*(x^4+x^2)^(1/4)*RootOf(8388608*_Z^8-8192*_Z^
4+1)^6*x^2-663552*RootOf(RootOf(8388608*_Z^8-8192*_Z^4+1)^2+_Z^2)*RootOf(4096*RootOf(8388608*_Z^8-8192*_Z^4+1)
^6-3*RootOf(8388608*_Z^8-8192*_Z^4+1)^2+_Z^2)*RootOf(8388608*_Z^8-8192*_Z^4+1)^3*x^3+516096*(x^4+x^2)^(3/4)*Ro
otOf(8388608*_Z^8-8192*_Z^4+1)^4-27136*(x^4+x^2)^(1/2)*RootOf(RootOf(8388608*_Z^8-8192*_Z^4+1)^2+_Z^2)*RootOf(
4096*RootOf(8388608*_Z^8-8192*_Z^4+1)^6-3*RootOf(8388608*_Z^8-8192*_Z^4+1)^2+_Z^2)*RootOf(8388608*_Z^8-8192*_Z
^4+1)*x-589824*RootOf(RootOf(8388608*_Z^8-8192*_Z^4+1)^2+_Z^2)*RootOf(4096*RootOf(8388608*_Z^8-8192*_Z^4+1)^6-
3*RootOf(8388608*_Z^8-8192*_Z^4+1)^2+_Z^2)*RootOf(8388608*_Z^8-8192*_Z^4+1)^3*x-3392*(x^4+x^2)^(1/4)*RootOf(83
88608*_Z^8-8192*_Z^4+1)^2*x^2-73*(x^4+x^2)^(3/4))/(4096*RootOf(8388608*_Z^8-8192*_Z^4+1)^4*x^2-4096*RootOf(838
8608*_Z^8-8192*_Z^4+1)^4-x^2)/x)-RootOf(RootOf(8388608*_Z^8-8192*_Z^4+1)^2+_Z^2)*ln(-(-1845493760*RootOf(RootO
f(8388608*_Z^8-8192*_Z^4+1)^2+_Z^2)*RootOf(8388608*_Z^8-8192*_Z^4+1)^8*x^3-451936256*(x^4+x^2)^(1/2)*RootOf(83
88608*_Z^8-8192*_Z^4+1)^6*RootOf(RootOf(8388608*_Z^8-8192*_Z^4+1)^2+_Z^2)*x+1845493760*RootOf(RootOf(8388608*_
Z^8-8192*_Z^4+1)^2+_Z^2)*RootOf(8388608*_Z^8-8192*_Z^4+1)^8*x-56492032*(x^4+x^2)^(1/4)*RootOf(8388608*_Z^8-819
2*_Z^4+1)^6*x^2+7397376*RootOf(RootOf(8388608*_Z^8-8192*_Z^4+1)^2+_Z^2)*RootOf(8388608*_Z^8-8192*_Z^4+1)^4*x^3
+516096*(x^4+x^2)^(3/4)*RootOf(8388608*_Z^8-8192*_Z^4+1)^4+376832*(x^4+x^2)^(1/2)*RootOf(RootOf(8388608*_Z^8-8
192*_Z^4+1)^2+_Z^2)*RootOf(8388608*_Z^8-8192*_Z^4+1)^2*x+262144*RootOf(RootOf(8388608*_Z^8-8192*_Z^4+1)^2+_Z^2
)*RootOf(8388608*_Z^8-8192*_Z^4+1)^4*x+47104*(x^4+x^2)^(1/4)*RootOf(8388608*_Z^8-8192*_Z^4+1)^2*x^2-4888*RootO
f(RootOf(8388608*_Z^8-8192*_Z^4+1)^2+_Z^2)*x^3-431*(x^4+x^2)^(3/4)-1504*RootOf(RootOf(8388608*_Z^8-8192*_Z^4+1
)^2+_Z^2)*x)/(4096*RootOf(8388608*_Z^8-8192*_Z^...

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [C] Result contains higher order function than in optimal. Order 3 vs. order 1.
time = 3.13, size = 1118, normalized size = 22.82 \begin {gather*} -\frac {1}{4} \, {\left (\sqrt {2} + 2\right )}^{\frac {1}{4}} \arctan \left (-\frac {{\left (196 \, {\left (x^{4} + x^{2}\right )}^{\frac {3}{4}} {\left (x^{2} - \sqrt {2} {\left (x^{2} + 1\right )} + 2\right )} \sqrt {\sqrt {2} + 2} + \sqrt {2} {\left (2 \, \sqrt {x^{4} + x^{2}} {\left (10 \, x^{3} + \sqrt {2} {\left (x^{3} - 10 \, x\right )} - 2 \, x\right )} + {\left (19 \, x^{5} + 16 \, x^{3} - \sqrt {2} {\left (3 \, x^{5} + 18 \, x^{3} + 10 \, x\right )} - 2 \, x\right )} \sqrt {\sqrt {2} + 2}\right )} \sqrt {-{\left (132 \, \sqrt {2} - 193\right )} \sqrt {\sqrt {2} + 2}} + 196 \, {\left (x^{4} + 2 \, x^{2} - \sqrt {2} {\left (x^{4} + x^{2}\right )}\right )} {\left (x^{4} + x^{2}\right )}^{\frac {1}{4}}\right )} {\left (\sqrt {2} + 2\right )}^{\frac {1}{4}}}{98 \, {\left (x^{5} - 2 \, x\right )}}\right ) - \frac {1}{4} \, {\left (-\sqrt {2} + 2\right )}^{\frac {1}{4}} \arctan \left (-\frac {\sqrt {2} {\left (2 \, \sqrt {x^{4} + x^{2}} {\left (10 \, x^{3} - \sqrt {2} {\left (x^{3} - 10 \, x\right )} - 2 \, x\right )} + {\left (19 \, x^{5} + 16 \, x^{3} + \sqrt {2} {\left (3 \, x^{5} + 18 \, x^{3} + 10 \, x\right )} - 2 \, x\right )} \sqrt {-\sqrt {2} + 2}\right )} \sqrt {{\left (132 \, \sqrt {2} + 193\right )} \sqrt {-\sqrt {2} + 2}} {\left (-\sqrt {2} + 2\right )}^{\frac {1}{4}} + 196 \, {\left ({\left (x^{4} + x^{2}\right )}^{\frac {3}{4}} {\left (x^{2} + \sqrt {2} {\left (x^{2} + 1\right )} + 2\right )} \sqrt {-\sqrt {2} + 2} + {\left (x^{4} + 2 \, x^{2} + \sqrt {2} {\left (x^{4} + x^{2}\right )}\right )} {\left (x^{4} + x^{2}\right )}^{\frac {1}{4}}\right )} {\left (-\sqrt {2} + 2\right )}^{\frac {1}{4}}}{98 \, {\left (x^{5} - 2 \, x\right )}}\right ) - \frac {1}{16} \, {\left (-\sqrt {2} + 2\right )}^{\frac {1}{4}} \log \left (\frac {4 \, {\left (x^{4} + x^{2}\right )}^{\frac {3}{4}} {\left (11 \, x^{2} + \sqrt {2} {\left (6 \, x^{2} + 11\right )} + 12\right )} + 2 \, {\left (34 \, x^{4} + 46 \, x^{2} + \sqrt {2} {\left (23 \, x^{4} + 34 \, x^{2}\right )}\right )} {\left (x^{4} + x^{2}\right )}^{\frac {1}{4}} \sqrt {-\sqrt {2} + 2} + {\left (56 \, x^{5} + 92 \, x^{3} + 2 \, \sqrt {x^{4} + x^{2}} {\left (34 \, x^{3} + \sqrt {2} {\left (23 \, x^{3} + 34 \, x\right )} + 46 \, x\right )} \sqrt {-\sqrt {2} + 2} + \sqrt {2} {\left (35 \, x^{5} + 68 \, x^{3} + 22 \, x\right )} + 24 \, x\right )} {\left (-\sqrt {2} + 2\right )}^{\frac {1}{4}}}{x^{5} - 2 \, x}\right ) + \frac {1}{16} \, {\left (-\sqrt {2} + 2\right )}^{\frac {1}{4}} \log \left (\frac {4 \, {\left (x^{4} + x^{2}\right )}^{\frac {3}{4}} {\left (11 \, x^{2} + \sqrt {2} {\left (6 \, x^{2} + 11\right )} + 12\right )} + 2 \, {\left (34 \, x^{4} + 46 \, x^{2} + \sqrt {2} {\left (23 \, x^{4} + 34 \, x^{2}\right )}\right )} {\left (x^{4} + x^{2}\right )}^{\frac {1}{4}} \sqrt {-\sqrt {2} + 2} - {\left (56 \, x^{5} + 92 \, x^{3} + 2 \, \sqrt {x^{4} + x^{2}} {\left (34 \, x^{3} + \sqrt {2} {\left (23 \, x^{3} + 34 \, x\right )} + 46 \, x\right )} \sqrt {-\sqrt {2} + 2} + \sqrt {2} {\left (35 \, x^{5} + 68 \, x^{3} + 22 \, x\right )} + 24 \, x\right )} {\left (-\sqrt {2} + 2\right )}^{\frac {1}{4}}}{x^{5} - 2 \, x}\right ) - \frac {1}{16} \, {\left (\sqrt {2} + 2\right )}^{\frac {1}{4}} \log \left (\frac {4 \, {\left (x^{4} + x^{2}\right )}^{\frac {3}{4}} {\left (11 \, x^{2} - \sqrt {2} {\left (6 \, x^{2} + 11\right )} + 12\right )} + 2 \, {\left (34 \, x^{4} + 46 \, x^{2} - \sqrt {2} {\left (23 \, x^{4} + 34 \, x^{2}\right )}\right )} {\left (x^{4} + x^{2}\right )}^{\frac {1}{4}} \sqrt {\sqrt {2} + 2} + {\left (56 \, x^{5} + 92 \, x^{3} + 2 \, \sqrt {x^{4} + x^{2}} {\left (34 \, x^{3} - \sqrt {2} {\left (23 \, x^{3} + 34 \, x\right )} + 46 \, x\right )} \sqrt {\sqrt {2} + 2} - \sqrt {2} {\left (35 \, x^{5} + 68 \, x^{3} + 22 \, x\right )} + 24 \, x\right )} {\left (\sqrt {2} + 2\right )}^{\frac {1}{4}}}{x^{5} - 2 \, x}\right ) + \frac {1}{16} \, {\left (\sqrt {2} + 2\right )}^{\frac {1}{4}} \log \left (\frac {4 \, {\left (x^{4} + x^{2}\right )}^{\frac {3}{4}} {\left (11 \, x^{2} - \sqrt {2} {\left (6 \, x^{2} + 11\right )} + 12\right )} + 2 \, {\left (34 \, x^{4} + 46 \, x^{2} - \sqrt {2} {\left (23 \, x^{4} + 34 \, x^{2}\right )}\right )} {\left (x^{4} + x^{2}\right )}^{\frac {1}{4}} \sqrt {\sqrt {2} + 2} - {\left (56 \, x^{5} + 92 \, x^{3} + 2 \, \sqrt {x^{4} + x^{2}} {\left (34 \, x^{3} - \sqrt {2} {\left (23 \, x^{3} + 34 \, x\right )} + 46 \, x\right )} \sqrt {\sqrt {2} + 2} - \sqrt {2} {\left (35 \, x^{5} + 68 \, x^{3} + 22 \, x\right )} + 24 \, x\right )} {\left (\sqrt {2} + 2\right )}^{\frac {1}{4}}}{x^{5} - 2 \, x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*(sqrt(2) + 2)^(1/4)*arctan(-1/98*(196*(x^4 + x^2)^(3/4)*(x^2 - sqrt(2)*(x^2 + 1) + 2)*sqrt(sqrt(2) + 2) +
sqrt(2)*(2*sqrt(x^4 + x^2)*(10*x^3 + sqrt(2)*(x^3 - 10*x) - 2*x) + (19*x^5 + 16*x^3 - sqrt(2)*(3*x^5 + 18*x^3
+ 10*x) - 2*x)*sqrt(sqrt(2) + 2))*sqrt(-(132*sqrt(2) - 193)*sqrt(sqrt(2) + 2)) + 196*(x^4 + 2*x^2 - sqrt(2)*(
x^4 + x^2))*(x^4 + x^2)^(1/4))*(sqrt(2) + 2)^(1/4)/(x^5 - 2*x)) - 1/4*(-sqrt(2) + 2)^(1/4)*arctan(-1/98*(sqrt(
2)*(2*sqrt(x^4 + x^2)*(10*x^3 - sqrt(2)*(x^3 - 10*x) - 2*x) + (19*x^5 + 16*x^3 + sqrt(2)*(3*x^5 + 18*x^3 + 10*
x) - 2*x)*sqrt(-sqrt(2) + 2))*sqrt((132*sqrt(2) + 193)*sqrt(-sqrt(2) + 2))*(-sqrt(2) + 2)^(1/4) + 196*((x^4 +
x^2)^(3/4)*(x^2 + sqrt(2)*(x^2 + 1) + 2)*sqrt(-sqrt(2) + 2) + (x^4 + 2*x^2 + sqrt(2)*(x^4 + x^2))*(x^4 + x^2)^
(1/4))*(-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)) + 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 - 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 - 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))

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt [4]{x^{2} \left (x^{2} + 1\right )} \left (x^{4} - 2\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {1}{{\left (x^4+x^2\right )}^{1/4}\,\left (x^4-2\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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