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

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

[Out]

Unintegrable

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Rubi [A]
time = 0.19, antiderivative size = 171, normalized size of antiderivative = 1.99, number of steps used = 10, number of rules used = 5, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {6857, 385, 218, 212, 209} \begin {gather*} -\frac {\sqrt [4]{58-41 \sqrt {2}} \text {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} \text {ArcTan}\left (\frac {x}{\sqrt [4]{2+\sqrt {2}} \sqrt [4]{x^4-1}}\right )-\frac {\sqrt [4]{58-41 \sqrt {2}} \tanh ^{-1}\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} \tanh ^{-1}\left (\frac {x}{\sqrt [4]{2+\sqrt {2}} \sqrt [4]{x^4-1}}\right ) \end {gather*}

Antiderivative was successfully verified.

[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*} \int \frac {2+x^4}{\sqrt [4]{-1+x^4} \left (-2+x^8\right )} \, dx &=\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}} \tan ^{-1}\left (\frac {x}{\sqrt [4]{2-\sqrt {2}} \sqrt [4]{-1+x^4}}\right )}{4 \sqrt {2}}-\frac {\sqrt [4]{58+41 \sqrt {2}} \tan ^{-1}\left (\frac {x}{\sqrt [4]{2+\sqrt {2}} \sqrt [4]{-1+x^4}}\right )}{4 \sqrt {2}}-\frac {\sqrt [4]{58-41 \sqrt {2}} \tanh ^{-1}\left (\frac {x}{\sqrt [4]{2-\sqrt {2}} \sqrt [4]{-1+x^4}}\right )}{4 \sqrt {2}}-\frac {\sqrt [4]{58+41 \sqrt {2}} \tanh ^{-1}\left (\frac {x}{\sqrt [4]{2+\sqrt {2}} \sqrt [4]{-1+x^4}}\right )}{4 \sqrt {2}}\\ \end {align*}

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Mathematica [A]
time = 0.19, size = 86, normalized size = 1.00 \begin {gather*} -\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 ] \end {gather*}

Antiderivative was successfully verified.

[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) & ]

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 1.
time = 13.53, size = 2578, normalized size = 29.98

method	result	size
trager	\(\text {Expression too large to display}\)	\(2578\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-134217728/41*ln(-(43980465111040*RootOf(RootOf(134217728*_Z^8-950272*_Z^4+1)^2+_Z^2)*RootOf(_Z^2+19480576*Roo
tOf(134217728*_Z^8-950272*_Z^4+1)^6-137883*RootOf(134217728*_Z^8-950272*_Z^4+1)^2)*RootOf(134217728*_Z^8-95027
2*_Z^4+1)^11*x^4-25232932864*RootOf(RootOf(134217728*_Z^8-950272*_Z^4+1)^2+_Z^2)*RootOf(_Z^2+19480576*RootOf(1
34217728*_Z^8-950272*_Z^4+1)^6-137883*RootOf(134217728*_Z^8-950272*_Z^4+1)^2)*RootOf(134217728*_Z^8-950272*_Z^
4+1)^7*x^4+1762656256*RootOf(134217728*_Z^8-950272*_Z^4+1)^6*(x^4-1)^(1/4)*x^3-220117073920*RootOf(134217728*_
Z^8-950272*_Z^4+1)^7*RootOf(RootOf(134217728*_Z^8-950272*_Z^4+1)^2+_Z^2)*RootOf(_Z^2+19480576*RootOf(134217728
*_Z^8-950272*_Z^4+1)^6-137883*RootOf(134217728*_Z^8-950272*_Z^4+1)^2)-2004156416*RootOf(RootOf(134217728*_Z^8-
950272*_Z^4+1)^2+_Z^2)*RootOf(_Z^2+19480576*RootOf(134217728*_Z^8-950272*_Z^4+1)^6-137883*RootOf(134217728*_Z^
8-950272*_Z^4+1)^2)*RootOf(134217728*_Z^8-950272*_Z^4+1)^3*x^4-27541504*RootOf(134217728*_Z^8-950272*_Z^4+1)^4
*(x^4-1)^(3/4)*x+1721344*RootOf(RootOf(134217728*_Z^8-950272*_Z^4+1)^2+_Z^2)*RootOf(134217728*_Z^8-950272*_Z^4
+1)*RootOf(_Z^2+19480576*RootOf(134217728*_Z^8-950272*_Z^4+1)^6-137883*RootOf(134217728*_Z^8-950272*_Z^4+1)^2)
*(x^4-1)^(1/2)*x^2-15061760*RootOf(134217728*_Z^8-950272*_Z^4+1)^2*(x^4-1)^(1/4)*x^3+1550385152*RootOf(1342177
28*_Z^8-950272*_Z^4+1)^3*RootOf(RootOf(134217728*_Z^8-950272*_Z^4+1)^2+_Z^2)*RootOf(_Z^2+19480576*RootOf(13421
7728*_Z^8-950272*_Z^4+1)^6-137883*RootOf(134217728*_Z^8-950272*_Z^4+1)^2)+28577*(x^4-1)^(3/4)*x)/(8192*x^4*Roo
tOf(134217728*_Z^8-950272*_Z^4+1)^4-29*x^4+41))*RootOf(134217728*_Z^8-950272*_Z^4+1)^7*RootOf(RootOf(134217728
*_Z^8-950272*_Z^4+1)^2+_Z^2)*RootOf(_Z^2+19480576*RootOf(134217728*_Z^8-950272*_Z^4+1)^6-137883*RootOf(1342177
28*_Z^8-950272*_Z^4+1)^2)+950272/41*ln(-(43980465111040*RootOf(RootOf(134217728*_Z^8-950272*_Z^4+1)^2+_Z^2)*Ro
otOf(_Z^2+19480576*RootOf(134217728*_Z^8-950272*_Z^4+1)^6-137883*RootOf(134217728*_Z^8-950272*_Z^4+1)^2)*RootO
f(134217728*_Z^8-950272*_Z^4+1)^11*x^4-25232932864*RootOf(RootOf(134217728*_Z^8-950272*_Z^4+1)^2+_Z^2)*RootOf(
_Z^2+19480576*RootOf(134217728*_Z^8-950272*_Z^4+1)^6-137883*RootOf(134217728*_Z^8-950272*_Z^4+1)^2)*RootOf(134
217728*_Z^8-950272*_Z^4+1)^7*x^4+1762656256*RootOf(134217728*_Z^8-950272*_Z^4+1)^6*(x^4-1)^(1/4)*x^3-220117073
920*RootOf(134217728*_Z^8-950272*_Z^4+1)^7*RootOf(RootOf(134217728*_Z^8-950272*_Z^4+1)^2+_Z^2)*RootOf(_Z^2+194
80576*RootOf(134217728*_Z^8-950272*_Z^4+1)^6-137883*RootOf(134217728*_Z^8-950272*_Z^4+1)^2)-2004156416*RootOf(
RootOf(134217728*_Z^8-950272*_Z^4+1)^2+_Z^2)*RootOf(_Z^2+19480576*RootOf(134217728*_Z^8-950272*_Z^4+1)^6-13788
3*RootOf(134217728*_Z^8-950272*_Z^4+1)^2)*RootOf(134217728*_Z^8-950272*_Z^4+1)^3*x^4-27541504*RootOf(134217728
*_Z^8-950272*_Z^4+1)^4*(x^4-1)^(3/4)*x+1721344*RootOf(RootOf(134217728*_Z^8-950272*_Z^4+1)^2+_Z^2)*RootOf(1342
17728*_Z^8-950272*_Z^4+1)*RootOf(_Z^2+19480576*RootOf(134217728*_Z^8-950272*_Z^4+1)^6-137883*RootOf(134217728*
_Z^8-950272*_Z^4+1)^2)*(x^4-1)^(1/2)*x^2-15061760*RootOf(134217728*_Z^8-950272*_Z^4+1)^2*(x^4-1)^(1/4)*x^3+155
0385152*RootOf(134217728*_Z^8-950272*_Z^4+1)^3*RootOf(RootOf(134217728*_Z^8-950272*_Z^4+1)^2+_Z^2)*RootOf(_Z^2
[phone]*RootOf(134217728*_Z^8-950272*_Z^4+1)^6-137883*RootOf(134217728*_Z^8-950272*_Z^4+1)^2)+28577*(x^4-1)^
(3/4)*x)/(8192*x^4*RootOf(134217728*_Z^8-950272*_Z^4+1)^4-29*x^4+41))*RootOf(134217728*_Z^8-950272*_Z^4+1)^3*R
ootOf(RootOf(134217728*_Z^8-950272*_Z^4+1)^2+_Z^2)*RootOf(_Z^2+19480576*RootOf(134217728*_Z^8-950272*_Z^4+1)^6
-137883*RootOf(134217728*_Z^8-950272*_Z^4+1)^2)-RootOf(134217728*_Z^8-950272*_Z^4+1)*ln((2684354560*x^4*RootOf
(134217728*_Z^8-950272*_Z^4+1)^9-31641829376*(x^4-1)^(1/2)*RootOf(134217728*_Z^8-950272*_Z^4+1)^7*x^2-11607736
32*RootOf(134217728*_Z^8-950272*_Z^4+1)^6*(x^4-1)^(1/4)*x^3-53936128*x^4*RootOf(134217728*_Z^8-950272*_Z^4+1)^
5-1343488*RootOf(134217728*_Z^8-950272*_Z^4+1)^4*(x^4-1)^(3/4)*x+223900672*(x^4-1)^(1/2)*RootOf(134217728*_Z^8
-950272*_Z^4+1)^3*x^2+8197376*RootOf(134217728*_Z^8-950272*_Z^4+1)^2*(x^4-1)^(1/4)*x^3+13434880*RootOf(1342177
28*_Z^8-950272*_Z^4+1)^5+237984*x^4*RootOf(134217728*_Z^8-950272*_Z^4+1)+6437*(x^4-1)^(3/4)*x-87904*RootOf(134
217728*_Z^8-950272*_Z^4+1))/(8192*x^4*RootOf(134217728*_Z^8-950272*_Z^4+1)^4-29*x^4-41))+RootOf(RootOf(1342177
28*_Z^8-950272*_Z^4+1)^2+_Z^2)*ln(-(2684354560*x^4*RootOf(134217728*_Z^8-950272*_Z^4+1)^8*RootOf(RootOf(134217
728*_Z^8-950272*_Z^4+1)^2+_Z^2)[phone]*(x^4-1)^(1/2)*RootOf(134217728*_Z^8-950272*_Z^4+1)^6*RootOf(RootOf
(134217728*_Z^8-950272*_Z^4+1)^2+_Z^2)*x^2-1160773632*RootOf(134217728*_Z^8-950272*_Z^4+1)^6*(x^4-1)^(1/4)*x^3
-53936128*x^4*RootOf(134217728*_Z^8-950272*_Z^4+1)^4*RootOf(RootOf(134217728*_Z^8-950272*_Z^4+1)^2+_Z^2)+13434
88*RootOf(134217728*_Z^8-950272*_Z^4+1)^4*(x^4-1)^(3/4)*x-223900672*(x^4-1)^(1/2)*RootOf(RootOf(134217728*_Z^8
-950272*_Z^4+1)^2+_Z^2)*RootOf(134217728*_Z^8-950272*_Z^4+1)^2*x^2+8197376*RootOf(134217728*_Z^8-950272*_Z^4+1
)^2*(x^4-1)^(1/4)*x^3+13434880*RootOf(RootOf(13...

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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((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)

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Fricas [C] Result contains higher order function than in optimal. Order 3 vs. order 1.
time = 8.29, size = 1163, normalized size = 13.52 \begin {gather*} \frac {1}{8} \, \sqrt {2} {\left (41 \, \sqrt {2} + 58\right )}^{\frac {1}{4}} \arctan \left (-\frac {{\left (196 \, {\left (17 \, x^{5} - \sqrt {2} {\left (12 \, x^{5} - 17 \, x\right )} - 24 \, x\right )} {\left (x^{4} - 1\right )}^{\frac {3}{4}} \sqrt {41 \, \sqrt {2} + 58} + \sqrt {2} {\left (2 \, {\left (8 \, x^{6} - 18 \, x^{2} - \sqrt {2} {\left (9 \, x^{6} - 8 \, x^{2}\right )}\right )} \sqrt {x^{4} - 1} + {\left (163 \, x^{8} - 292 \, x^{4} - 2 \, \sqrt {2} {\left (58 \, x^{8} - 103 \, x^{4} + 30\right )} + 86\right )} \sqrt {41 \, \sqrt {2} + 58}\right )} \sqrt {-{\left (782 \, \sqrt {2} - 1107\right )} \sqrt {41 \, \sqrt {2} + 58}} + 196 \, {\left (3 \, x^{7} - 4 \, x^{3} - \sqrt {2} {\left (2 \, x^{7} - 3 \, x^{3}\right )}\right )} {\left (x^{4} - 1\right )}^{\frac {1}{4}}\right )} {\left (41 \, \sqrt {2} + 58\right )}^{\frac {1}{4}}}{98 \, {\left (x^{8} - 2\right )}}\right ) + \frac {1}{8} \, \sqrt {2} {\left (-41 \, \sqrt {2} + 58\right )}^{\frac {1}{4}} \arctan \left (\frac {\sqrt {2} {\left (2 \, {\left (8 \, x^{6} - 18 \, x^{2} + \sqrt {2} {\left (9 \, x^{6} - 8 \, x^{2}\right )}\right )} \sqrt {x^{4} - 1} + {\left (163 \, x^{8} - 292 \, x^{4} + 2 \, \sqrt {2} {\left (58 \, x^{8} - 103 \, x^{4} + 30\right )} + 86\right )} \sqrt {-41 \, \sqrt {2} + 58}\right )} \sqrt {{\left (782 \, \sqrt {2} + 1107\right )} \sqrt {-41 \, \sqrt {2} + 58}} {\left (-41 \, \sqrt {2} + 58\right )}^{\frac {1}{4}} + 196 \, {\left ({\left (17 \, x^{5} + \sqrt {2} {\left (12 \, x^{5} - 17 \, x\right )} - 24 \, x\right )} {\left (x^{4} - 1\right )}^{\frac {3}{4}} \sqrt {-41 \, \sqrt {2} + 58} + {\left (3 \, x^{7} - 4 \, x^{3} + \sqrt {2} {\left (2 \, x^{7} - 3 \, x^{3}\right )}\right )} {\left (x^{4} - 1\right )}^{\frac {1}{4}}\right )} {\left (-41 \, \sqrt {2} + 58\right )}^{\frac {1}{4}}}{98 \, {\left (x^{8} - 2\right )}}\right ) - \frac {1}{32} \, \sqrt {2} {\left (-41 \, \sqrt {2} + 58\right )}^{\frac {1}{4}} \log \left (\frac {4 \, {\left (11 \, x^{5} + \sqrt {2} {\left (6 \, x^{5} - 11 \, x\right )} - 12 \, x\right )} {\left (x^{4} - 1\right )}^{\frac {3}{4}} + 2 \, {\left (194 \, x^{7} - 274 \, x^{3} + \sqrt {2} {\left (137 \, x^{7} - 194 \, x^{3}\right )}\right )} {\left (x^{4} - 1\right )}^{\frac {1}{4}} \sqrt {-41 \, \sqrt {2} + 58} + {\left (126 \, x^{8} - 228 \, x^{4} + 2 \, {\left (468 \, x^{6} - 662 \, x^{2} + \sqrt {2} {\left (331 \, x^{6} - 468 \, x^{2}\right )}\right )} \sqrt {x^{4} - 1} \sqrt {-41 \, \sqrt {2} + 58} + \sqrt {2} {\left (91 \, x^{8} - 160 \, x^{4} + 46\right )} + 68\right )} {\left (-41 \, \sqrt {2} + 58\right )}^{\frac {1}{4}}}{x^{8} - 2}\right ) + \frac {1}{32} \, \sqrt {2} {\left (-41 \, \sqrt {2} + 58\right )}^{\frac {1}{4}} \log \left (\frac {4 \, {\left (11 \, x^{5} + \sqrt {2} {\left (6 \, x^{5} - 11 \, x\right )} - 12 \, x\right )} {\left (x^{4} - 1\right )}^{\frac {3}{4}} + 2 \, {\left (194 \, x^{7} - 274 \, x^{3} + \sqrt {2} {\left (137 \, x^{7} - 194 \, x^{3}\right )}\right )} {\left (x^{4} - 1\right )}^{\frac {1}{4}} \sqrt {-41 \, \sqrt {2} + 58} - {\left (126 \, x^{8} - 228 \, x^{4} + 2 \, {\left (468 \, x^{6} - 662 \, x^{2} + \sqrt {2} {\left (331 \, x^{6} - 468 \, x^{2}\right )}\right )} \sqrt {x^{4} - 1} \sqrt {-41 \, \sqrt {2} + 58} + \sqrt {2} {\left (91 \, x^{8} - 160 \, x^{4} + 46\right )} + 68\right )} {\left (-41 \, \sqrt {2} + 58\right )}^{\frac {1}{4}}}{x^{8} - 2}\right ) + \frac {1}{32} \, \sqrt {2} {\left (41 \, \sqrt {2} + 58\right )}^{\frac {1}{4}} \log \left (\frac {4 \, {\left (11 \, x^{5} - \sqrt {2} {\left (6 \, x^{5} - 11 \, x\right )} - 12 \, x\right )} {\left (x^{4} - 1\right )}^{\frac {3}{4}} + 2 \, {\left (194 \, x^{7} - 274 \, x^{3} - \sqrt {2} {\left (137 \, x^{7} - 194 \, x^{3}\right )}\right )} {\left (x^{4} - 1\right )}^{\frac {1}{4}} \sqrt {41 \, \sqrt {2} + 58} + {\left (126 \, x^{8} - 228 \, x^{4} + 2 \, {\left (468 \, x^{6} - 662 \, x^{2} - \sqrt {2} {\left (331 \, x^{6} - 468 \, x^{2}\right )}\right )} \sqrt {x^{4} - 1} \sqrt {41 \, \sqrt {2} + 58} - \sqrt {2} {\left (91 \, x^{8} - 160 \, x^{4} + 46\right )} + 68\right )} {\left (41 \, \sqrt {2} + 58\right )}^{\frac {1}{4}}}{x^{8} - 2}\right ) - \frac {1}{32} \, \sqrt {2} {\left (41 \, \sqrt {2} + 58\right )}^{\frac {1}{4}} \log \left (\frac {4 \, {\left (11 \, x^{5} - \sqrt {2} {\left (6 \, x^{5} - 11 \, x\right )} - 12 \, x\right )} {\left (x^{4} - 1\right )}^{\frac {3}{4}} + 2 \, {\left (194 \, x^{7} - 274 \, x^{3} - \sqrt {2} {\left (137 \, x^{7} - 194 \, x^{3}\right )}\right )} {\left (x^{4} - 1\right )}^{\frac {1}{4}} \sqrt {41 \, \sqrt {2} + 58} - {\left (126 \, x^{8} - 228 \, x^{4} + 2 \, {\left (468 \, x^{6} - 662 \, x^{2} - \sqrt {2} {\left (331 \, x^{6} - 468 \, x^{2}\right )}\right )} \sqrt {x^{4} - 1} \sqrt {41 \, \sqrt {2} + 58} - \sqrt {2} {\left (91 \, x^{8} - 160 \, x^{4} + 46\right )} + 68\right )} {\left (41 \, \sqrt {2} + 58\right )}^{\frac {1}{4}}}{x^{8} - 2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*sqrt(2)*(41*sqrt(2) + 58)^(1/4)*arctan(-1/98*(196*(17*x^5 - sqrt(2)*(12*x^5 - 17*x) - 24*x)*(x^4 - 1)^(3/4
)*sqrt(41*sqrt(2) + 58) + sqrt(2)*(2*(8*x^6 - 18*x^2 - sqrt(2)*(9*x^6 - 8*x^2))*sqrt(x^4 - 1) + (163*x^8 - 292
*x^4 - 2*sqrt(2)*(58*x^8 - 103*x^4 + 30) + 86)*sqrt(41*sqrt(2) + 58))*sqrt(-(782*sqrt(2) - 1107)*sqrt(41*sqrt(
2) + 58)) + 196*(3*x^7 - 4*x^3 - sqrt(2)*(2*x^7 - 3*x^3))*(x^4 - 1)^(1/4))*(41*sqrt(2) + 58)^(1/4)/(x^8 - 2))
+ 1/8*sqrt(2)*(-41*sqrt(2) + 58)^(1/4)*arctan(1/98*(sqrt(2)*(2*(8*x^6 - 18*x^2 + sqrt(2)*(9*x^6 - 8*x^2))*sqrt
(x^4 - 1) + (163*x^8 - 292*x^4 + 2*sqrt(2)*(58*x^8 - 103*x^4 + 30) + 86)*sqrt(-41*sqrt(2) + 58))*sqrt((782*sqr
t(2) + 1107)*sqrt(-41*sqrt(2) + 58))*(-41*sqrt(2) + 58)^(1/4) + 196*((17*x^5 + sqrt(2)*(12*x^5 - 17*x) - 24*x)
*(x^4 - 1)^(3/4)*sqrt(-41*sqrt(2) + 58) + (3*x^7 - 4*x^3 + sqrt(2)*(2*x^7 - 3*x^3))*(x^4 - 1)^(1/4))*(-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*sqrt(2) + 58) - (126*x^8 - 228*x^4 + 2*(468*x^6 - 662*x^2 + sqr
t(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*sq
rt(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)*(1
37*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))

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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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((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)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x^4+2}{{\left (x^4-1\right )}^{1/4}\,\left (x^8-2\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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