3.7.24 \(\int \frac {1}{(-2+x^4) \sqrt [4]{x^2+x^4}} \, dx\)
Optimal. Leaf size=49 \[ \frac {1}{8} \text {RootSum}\left [2 \text {$\#$1}^8-4 \text {$\#$1}^4+1\& ,\frac {\log \left (\sqrt [4]{x^4+x^2}-\text {$\#$1} x\right )-\log (x)}{\text {$\#$1}}\& \right ] \]
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Rubi [B] time = 0.18, 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
= {2056, 1270, 1429, 377, 212, 206, 203} \begin {gather*} -\frac {\sqrt [4]{2-\sqrt {2}} \sqrt {x} \sqrt [4]{x^2+1} \tan ^{-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} \tan ^{-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}}-\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 203
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])
Rule 206
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])
Rule 212
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] &&
!GtQ[a/b, 0]
Rule 377
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 1270
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 1429
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 2056
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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 [B] time = 0.81, size = 162, normalized size = 3.31 \begin {gather*} -\frac {\sqrt [4]{\frac {1}{x^2}+1} x \left (-\sqrt [4]{2+\sqrt {2}} \tan ^{-1}\left (\sqrt [4]{2+\sqrt {2}} \sqrt [4]{\frac {1}{x^2}+1}\right )-\sqrt [4]{2-\sqrt {2}} \tan ^{-1}\left (\sqrt [4]{-\left (\left (\sqrt {2}-2\right ) \left (\frac {1}{x^2}+1\right )\right )}\right )+\sqrt [4]{2+\sqrt {2}} \tanh ^{-1}\left (\sqrt [4]{2+\sqrt {2}} \sqrt [4]{\frac {1}{x^2}+1}\right )+\sqrt [4]{2-\sqrt {2}} \tanh ^{-1}\left (\sqrt [4]{-\left (\left (\sqrt {2}-2\right ) \left (\frac {1}{x^2}+1\right )\right )}\right )\right )}{4 \sqrt [4]{x^4+x^2}} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[1/((-2 + x^4)*(x^2 + x^4)^(1/4)),x]
[Out]
-1/4*((1 + x^(-2))^(1/4)*x*(-((2 + Sqrt[2])^(1/4)*ArcTan[(2 + Sqrt[2])^(1/4)*(1 + x^(-2))^(1/4)]) - (2 - Sqrt[
2])^(1/4)*ArcTan[(-((-2 + Sqrt[2])*(1 + x^(-2))))^(1/4)] + (2 + Sqrt[2])^(1/4)*ArcTanh[(2 + Sqrt[2])^(1/4)*(1
+ x^(-2))^(1/4)] + (2 - Sqrt[2])^(1/4)*ArcTanh[(-((-2 + Sqrt[2])*(1 + x^(-2))))^(1/4)]))/(x^2 + x^4)^(1/4)
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IntegrateAlgebraic [A] time = 0.15, size = 49, normalized size = 1.00 \begin {gather*} \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 ] \end {gather*}
Antiderivative was successfully verified.
[In]
IntegrateAlgebraic[1/((-2 + x^4)*(x^2 + x^4)^(1/4)),x]
[Out]
RootSum[1 - 4*#1^4 + 2*#1^8 & , (-Log[x] + Log[(x^2 + x^4)^(1/4) - x*#1])/#1 & ]/8
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fricas [B] time = 4.91, size = 1118, normalized size = 22.82
result too large to display
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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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \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]
Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:Unab
le to convert to real 1/4 Error: Bad Argument ValueUnable to convert to real 1/4 Error: Bad Argument Value
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maple [B] time = 68.38, size = 3669, normalized size = 74.88
| | |
| method |
result |
size |
| | |
| trager |
\(\text {Expression too large to display}\) |
\(3669\) |
| | |
|
|
|
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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(4096*RootOf(8388608*_Z^8-8192*_Z^4+1)^6-3*RootOf(8388608*_Z^8-8192*_Z^4+1)^2+_Z^2)*ln(-(536011
9185408*RootOf(-4096*RootOf(8388608*_Z^8-8192*_Z^4+1)^6+3*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^3-5360119185408*RootOf(-4096*RootOf(8388608*_Z^8-8192*_Z^4+1)^6+3*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-21474836480*RootOf(-4096*RootOf(8388608*_Z^8-8192*_Z^4+1)^6+3*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)*RootO
f(8388608*_Z^8-8192*_Z^4+1)^7*x^3+187695104*RootOf(8388608*_Z^8-8192*_Z^4+1)^5*RootOf(-4096*RootOf(8388608*_Z^
8-8192*_Z^4+1)^6+3*RootOf(8388608*_Z^8-8192*_Z^4+1)^2+_Z^2)*RootOf(4096*RootOf(8388608*_Z^8-8192*_Z^4+1)^6-3*R
ootOf(8388608*_Z^8-8192*_Z^4+1)^2+_Z^2)*(x^4+x^2)^(1/2)*x-771751936*RootOf(-4096*RootOf(8388608*_Z^8-8192*_Z^4
+1)^6+3*RootOf(8388608*_Z^8-8192*_Z^4+1)^2+_Z^2)*RootOf(4096*RootOf(8388608*_Z^8-8192*_Z^4+1)^6-3*RootOf(83886
08*_Z^8-8192*_Z^4+1)^2+_Z^2)*RootOf(8388608*_Z^8-8192*_Z^4+1)^7*x-56492032*(x^4+x^2)^(1/4)*RootOf(8388608*_Z^8
-8192*_Z^4+1)^6*x^2+14163968*RootOf(-4096*RootOf(8388608*_Z^8-8192*_Z^4+1)^6+3*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(838
8608*_Z^8-8192*_Z^4+1)^3*x^3+516096*(x^4+x^2)^(3/4)*RootOf(8388608*_Z^8-8192*_Z^4+1)^4-156160*RootOf(8388608*_
Z^8-8192*_Z^4+1)*RootOf(-4096*RootOf(8388608*_Z^8-8192*_Z^4+1)^6+3*RootOf(8388608*_Z^8-8192*_Z^4+1)^2+_Z^2)*Ro
otOf(4096*RootOf(8388608*_Z^8-8192*_Z^4+1)^6-3*RootOf(8388608*_Z^8-8192*_Z^4+1)^2+_Z^2)*(x^4+x^2)^(1/2)*x+4358
144*RootOf(-4096*RootOf(8388608*_Z^8-8192*_Z^4+1)^6+3*RootOf(8388608*_Z^8-8192*_Z^4+1)^2+_Z^2)*RootOf(4096*Roo
tOf(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+47104*(x^4+x^2)^(1/4)*RootOf(8388608*_Z^8-8192*_Z^4+1)^2*x^2-431*(x^4+x^2)^(3/4))/(4096*x^2*RootOf(8388608*_
Z^8-8192*_Z^4+1)^4-4096*RootOf(8388608*_Z^8-8192*_Z^4+1)^4-3*x^2+4)/x)*RootOf(8388608*_Z^8-8192*_Z^4+1)^7*Root
Of(-4096*RootOf(8388608*_Z^8-8192*_Z^4+1)^6+3*RootOf(8388608*_Z^8-8192*_Z^4+1)^2+_Z^2)+4096*RootOf(4096*RootOf
(8388608*_Z^8-8192*_Z^4+1)^6-3*RootOf(8388608*_Z^8-8192*_Z^4+1)^2+_Z^2)*ln(-(5360119185408*RootOf(-4096*RootOf
(8388608*_Z^8-8192*_Z^4+1)^6+3*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^3-5360119185408*Root
Of(-4096*RootOf(8388608*_Z^8-8192*_Z^4+1)^6+3*RootOf(8388608*_Z^8-8192*_Z^4+1)^2+_Z^2)*RootOf(4096*RootOf(8388
608*_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-21474
836480*RootOf(-4096*RootOf(8388608*_Z^8-8192*_Z^4+1)^6+3*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*RootOf(8388608*_Z^8-8192*_Z^4+1)^5*RootOf(-4096*RootOf(8388608*_Z^8-8192*_Z^4+1)^6+3*RootOf(8
388608*_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)*(x^4+x^2)^(1/2)*x-771751936*RootOf(-4096*RootOf(8388608*_Z^8-8192*_Z^4+1)^6+3*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-56492032*(x^4+x^2)^(1/4)*RootOf(8388608*_Z^8-8192*_Z^4+1)^6*x^2+1416396
8*RootOf(-4096*RootOf(8388608*_Z^8-8192*_Z^4+1)^6+3*RootOf(8388608*_Z^8-8192*_Z^4+1)^2+_Z^2)*RootOf(4096*RootO
f(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)*RootOf(8388608*_Z^8-8192*_Z^4+1)^4-156160*RootOf(8388608*_Z^8-8192*_Z^4+1)*RootOf(-40
96*RootOf(8388608*_Z^8-8192*_Z^4+1)^6+3*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)*(x^4+x^2)^(1/2)*x+4358144*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(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+47104*(x^4+x^2)^(1/4)*Roo
tOf(8388608*_Z^8-8192*_Z^4+1)^2*x^2-431*(x^4+x^2)^(3/4))/(4096*x^2*RootOf(8388608*_Z^8-8192*_Z^4+1)^4-4096*Roo
tOf(8388608*_Z^8-8192*_Z^4+1)^4-3*x^2+4)/x)*RootOf(8388608*_Z^8-8192*_Z^4+1)^3*RootOf(-4096*RootOf(8388608*_Z^
8-8192*_Z^4+1)^6+3*RootOf(8388608*_Z^8-8192*_Z^4+1)^2+_Z^2)-RootOf(4096*RootOf(8388608*_Z^8-8192*_Z^4+1)^6-3*R
ootOf(8388608*_Z^8-8192*_Z^4+1)^2+_Z^2)*ln(-(-1845493760*RootOf(4096*RootOf(8388608*_Z^8-8192*_Z^4+1)^6-3*Root
Of(8388608*_Z^8-8192*_Z^4+1)^2+_Z^2)*RootOf(8388608*_Z^8-8192*_Z^4+1)^8*x^3+187695104*(x^4+x^2)^(1/2)*RootOf(8
388608*_Z^8-8192*_Z^4+1)^6*RootOf(4096*RootOf(8388608*_Z^8-8192*_Z^4+1)^6-3*RootOf(8388608*_Z^8-8192*_Z^4+1)^2
+_Z^2)*x+1845493760*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)^8*x+23461888*(x^4+x^2)^(1/4)*RootOf(8388608*_Z^8-8192*_Z^4+1)^6*x^2-3792896*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)^4*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(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)
^2*x-3866624*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)^4*x-3392*(x^4+x^2)^(1/4)*RootOf(8388608*_Z^8-8192*_Z^4+1)^2*x^2+576*RootOf(4096*Root
Of(8388608*_Z^8-8192*_Z^4+1)^6-3*RootOf(8388608*_Z^8-8192*_Z^4+1)^2+_Z^2)*x^3+73*(x^4+x^2)^(3/4)+512*RootOf(40
96*RootOf(8388608*_Z^8-8192*_Z^4+1)^6-3*RootOf(8388608*_Z^8-8192*_Z^4+1)^2+_Z^2)*x)/(4096*x^2*RootOf(8388608*_
Z^8-8192*_Z^4+1)^4-4096*RootOf(8388608*_Z^8-8192*_Z^4+1)^4-x^2)/x)-RootOf(-4096*RootOf(8388608*_Z^8-8192*_Z^4+
1)^6+3*RootOf(8388608*_Z^8-8192*_Z^4+1)^2+_Z^2)*ln((1845493760*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)^8*x^3+187695104*(x^4+x^2)^(1/2)*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-
8192*_Z^4+1)^6*x-1845493760*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)^8*x+23461888*(x^4+x^2)^(1/4)*RootOf(8388608*_Z^8-8192*_Z^4+1)^6*x^2+
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608*_Z^8-8192*_Z^4+1)^4*x^3+516096*(x^4+x^2)^(3/4)*RootOf(8388608*_Z^8-8192*_Z^4+1)^4-27136*(x^4+x^2)^(1/2)*Ro
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192*_Z^4+1)^2*x+3866624*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)^4*x-3392*(x^4+x^2)^(1/4)*RootOf(8388608*_Z^8-8192*_Z^4+1)^2*x^2-576*Root
Of(-4096*RootOf(8388608*_Z^8-8192*_Z^4+1)^6+3*RootOf(8388608*_Z^8-8192*_Z^4+1)^2+_Z^2)*x^3-73*(x^4+x^2)^(3/4)-
512*RootOf(-4096*RootOf(8388608*_Z^8-8192*_Z^4+1)^6+3*RootOf(8388608*_Z^8-8192*_Z^4+1)^2+_Z^2)*x)/(4096*x^2*Ro
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+1)*ln(-(1845493760*RootOf(8388608*_Z^8-8192*_Z^4+1)^9*x^3-451936256*(x^4+x^2)^(1/2)*RootOf(8388608*_Z^8-8192*
_Z^4+1)^7*x-1845493760*RootOf(8388608*_Z^8-8192*_Z^4+1)^9*x+56492032*(x^4+x^2)^(1/4)*RootOf(8388608*_Z^8-8192*
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(x^4+x^2)^(3/4)+1504*RootOf(8388608*_Z^8-8192*_Z^4+1)*x)/(4096*x^2*RootOf(8388608*_Z^8-8192*_Z^4+1)^4-4096*Roo
tOf(8388608*_Z^8-8192*_Z^4+1)^4-3*x^2+4)/x)
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {2 \, {\left (4 \, x^{5} + x^{3} - 3 \, x\right )}}{21 \, {\left (x^{\frac {9}{2}} - 2 \, \sqrt {x}\right )} {\left (x^{2} + 1\right )}^{\frac {1}{4}}} + \int \frac {16 \, {\left (4 \, x^{4} + x^{2} - 3\right )}}{21 \, {\left (x^{\frac {17}{2}} - 4 \, x^{\frac {9}{2}} + 4 \, \sqrt {x}\right )} {\left (x^{2} + 1\right )}^{\frac {1}{4}}}\,{d x} \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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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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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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