3.25.2 \(\int \frac {-1+2 x^4}{\sqrt [4]{-1+x^4} (-1-x^4+x^8)} \, dx\)
Optimal. Leaf size=193 \[ -\frac {1}{2} \sqrt {\frac {1}{2} \left (\sqrt {5}-1\right )} \tan ^{-1}\left (\frac {\sqrt {\frac {\sqrt {5}}{2}-\frac {1}{2}} x}{\sqrt [4]{x^4-1}}\right )+\frac {1}{2} \sqrt {\frac {1}{2} \left (1+\sqrt {5}\right )} \tan ^{-1}\left (\frac {\sqrt {\frac {1}{2}+\frac {\sqrt {5}}{2}} x}{\sqrt [4]{x^4-1}}\right )-\frac {1}{2} \sqrt {\frac {1}{2} \left (\sqrt {5}-1\right )} \tanh ^{-1}\left (\frac {\sqrt {\frac {\sqrt {5}}{2}-\frac {1}{2}} x}{\sqrt [4]{x^4-1}}\right )+\frac {1}{2} \sqrt {\frac {1}{2} \left (1+\sqrt {5}\right )} \tanh ^{-1}\left (\frac {\sqrt {\frac {1}{2}+\frac {\sqrt {5}}{2}} x}{\sqrt [4]{x^4-1}}\right ) \]
________________________________________________________________________________________
Rubi [A] time = 0.29, antiderivative size = 189, normalized size of antiderivative = 0.98,
number of steps used = 10, number of rules used = 5, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used
= {6728, 377, 212, 206, 203} \begin {gather*} -\frac {1}{2} \sqrt [4]{\frac {1}{2} \left (3-\sqrt {5}\right )} \tan ^{-1}\left (\frac {\sqrt [4]{\frac {2}{3+\sqrt {5}}} x}{\sqrt [4]{x^4-1}}\right )+\frac {1}{2} \sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} \tan ^{-1}\left (\frac {\sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} x}{\sqrt [4]{x^4-1}}\right )-\frac {1}{2} \sqrt [4]{\frac {1}{2} \left (3-\sqrt {5}\right )} \tanh ^{-1}\left (\frac {\sqrt [4]{\frac {2}{3+\sqrt {5}}} x}{\sqrt [4]{x^4-1}}\right )+\frac {1}{2} \sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} \tanh ^{-1}\left (\frac {\sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} x}{\sqrt [4]{x^4-1}}\right ) \end {gather*}
Warning: Unable to verify antiderivative.
[In]
Int[(-1 + 2*x^4)/((-1 + x^4)^(1/4)*(-1 - x^4 + x^8)),x]
[Out]
-1/2*(((3 - Sqrt[5])/2)^(1/4)*ArcTan[((2/(3 + Sqrt[5]))^(1/4)*x)/(-1 + x^4)^(1/4)]) + (((3 + Sqrt[5])/2)^(1/4)
*ArcTan[(((3 + Sqrt[5])/2)^(1/4)*x)/(-1 + x^4)^(1/4)])/2 - (((3 - Sqrt[5])/2)^(1/4)*ArcTanh[((2/(3 + Sqrt[5]))
^(1/4)*x)/(-1 + x^4)^(1/4)])/2 + (((3 + Sqrt[5])/2)^(1/4)*ArcTanh[(((3 + Sqrt[5])/2)^(1/4)*x)/(-1 + x^4)^(1/4)
])/2
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 6728
Int[(u_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a +
b*x^n + c*x^(2*n)), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]
Rubi steps
\begin {align*} \int \frac {-1+2 x^4}{\sqrt [4]{-1+x^4} \left (-1-x^4+x^8\right )} \, dx &=\int \left (\frac {2}{\sqrt [4]{-1+x^4} \left (-1-\sqrt {5}+2 x^4\right )}+\frac {2}{\sqrt [4]{-1+x^4} \left (-1+\sqrt {5}+2 x^4\right )}\right ) \, dx\\ &=2 \int \frac {1}{\sqrt [4]{-1+x^4} \left (-1-\sqrt {5}+2 x^4\right )} \, dx+2 \int \frac {1}{\sqrt [4]{-1+x^4} \left (-1+\sqrt {5}+2 x^4\right )} \, dx\\ &=2 \operatorname {Subst}\left (\int \frac {1}{-1-\sqrt {5}-\left (1-\sqrt {5}\right ) x^4} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right )+2 \operatorname {Subst}\left (\int \frac {1}{-1+\sqrt {5}-\left (1+\sqrt {5}\right ) x^4} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right )\\ &=\frac {\operatorname {Subst}\left (\int \frac {1}{\sqrt {3-\sqrt {5}}-\sqrt {2} x^2} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right )}{\sqrt {2}}-\frac {\operatorname {Subst}\left (\int \frac {1}{\sqrt {3+\sqrt {5}}-\sqrt {2} x^2} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right )}{\sqrt {2}}+\frac {\operatorname {Subst}\left (\int \frac {1}{\sqrt {3-\sqrt {5}}+\sqrt {2} x^2} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right )}{\sqrt {2}}-\frac {\operatorname {Subst}\left (\int \frac {1}{\sqrt {3+\sqrt {5}}+\sqrt {2} x^2} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right )}{\sqrt {2}}\\ &=-\frac {\tan ^{-1}\left (\frac {\sqrt [4]{\frac {2}{3+\sqrt {5}}} x}{\sqrt [4]{-1+x^4}}\right )}{2^{3/4} \sqrt [4]{3+\sqrt {5}}}+\frac {1}{2} \sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} \tan ^{-1}\left (\frac {\sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} x}{\sqrt [4]{-1+x^4}}\right )-\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{\frac {2}{3+\sqrt {5}}} x}{\sqrt [4]{-1+x^4}}\right )}{2^{3/4} \sqrt [4]{3+\sqrt {5}}}+\frac {1}{2} \sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} \tanh ^{-1}\left (\frac {\sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} x}{\sqrt [4]{-1+x^4}}\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.16, size = 172, normalized size = 0.89 \begin {gather*} \frac {-\sqrt [4]{3-\sqrt {5}} \tan ^{-1}\left (\frac {\sqrt [4]{\frac {2}{3+\sqrt {5}}} x}{\sqrt [4]{x^4-1}}\right )+\sqrt [4]{3+\sqrt {5}} \tan ^{-1}\left (\frac {\sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} x}{\sqrt [4]{x^4-1}}\right )-\sqrt [4]{3-\sqrt {5}} \tanh ^{-1}\left (\frac {\sqrt [4]{\frac {2}{3+\sqrt {5}}} x}{\sqrt [4]{x^4-1}}\right )+\sqrt [4]{3+\sqrt {5}} \tanh ^{-1}\left (\frac {\sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} x}{\sqrt [4]{x^4-1}}\right )}{2 \sqrt [4]{2}} \end {gather*}
Warning: Unable to verify antiderivative.
[In]
Integrate[(-1 + 2*x^4)/((-1 + x^4)^(1/4)*(-1 - x^4 + x^8)),x]
[Out]
(-((3 - Sqrt[5])^(1/4)*ArcTan[((2/(3 + Sqrt[5]))^(1/4)*x)/(-1 + x^4)^(1/4)]) + (3 + Sqrt[5])^(1/4)*ArcTan[(((3
+ Sqrt[5])/2)^(1/4)*x)/(-1 + x^4)^(1/4)] - (3 - Sqrt[5])^(1/4)*ArcTanh[((2/(3 + Sqrt[5]))^(1/4)*x)/(-1 + x^4)
^(1/4)] + (3 + Sqrt[5])^(1/4)*ArcTanh[(((3 + Sqrt[5])/2)^(1/4)*x)/(-1 + x^4)^(1/4)])/(2*2^(1/4))
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 0.75, size = 193, normalized size = 1.00 \begin {gather*} -\frac {1}{2} \sqrt {\frac {1}{2} \left (-1+\sqrt {5}\right )} \tan ^{-1}\left (\frac {\sqrt {-\frac {1}{2}+\frac {\sqrt {5}}{2}} x}{\sqrt [4]{-1+x^4}}\right )+\frac {1}{2} \sqrt {\frac {1}{2} \left (1+\sqrt {5}\right )} \tan ^{-1}\left (\frac {\sqrt {\frac {1}{2}+\frac {\sqrt {5}}{2}} x}{\sqrt [4]{-1+x^4}}\right )-\frac {1}{2} \sqrt {\frac {1}{2} \left (-1+\sqrt {5}\right )} \tanh ^{-1}\left (\frac {\sqrt {-\frac {1}{2}+\frac {\sqrt {5}}{2}} x}{\sqrt [4]{-1+x^4}}\right )+\frac {1}{2} \sqrt {\frac {1}{2} \left (1+\sqrt {5}\right )} \tanh ^{-1}\left (\frac {\sqrt {\frac {1}{2}+\frac {\sqrt {5}}{2}} x}{\sqrt [4]{-1+x^4}}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
IntegrateAlgebraic[(-1 + 2*x^4)/((-1 + x^4)^(1/4)*(-1 - x^4 + x^8)),x]
[Out]
-1/2*(Sqrt[(-1 + Sqrt[5])/2]*ArcTan[(Sqrt[-1/2 + Sqrt[5]/2]*x)/(-1 + x^4)^(1/4)]) + (Sqrt[(1 + Sqrt[5])/2]*Arc
Tan[(Sqrt[1/2 + Sqrt[5]/2]*x)/(-1 + x^4)^(1/4)])/2 - (Sqrt[(-1 + Sqrt[5])/2]*ArcTanh[(Sqrt[-1/2 + Sqrt[5]/2]*x
)/(-1 + x^4)^(1/4)])/2 + (Sqrt[(1 + Sqrt[5])/2]*ArcTanh[(Sqrt[1/2 + Sqrt[5]/2]*x)/(-1 + x^4)^(1/4)])/2
________________________________________________________________________________________
fricas [B] time = 25.05, size = 990, normalized size = 5.13
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((2*x^4-1)/(x^4-1)^(1/4)/(x^8-x^4-1),x, algorithm="fricas")
[Out]
-1/4*sqrt(2)*sqrt(sqrt(5) + 1)*arctan(-1/8*((sqrt(5)*sqrt(2)*(x^8 + x^4 - 1) - sqrt(2)*(5*x^8 - 3*x^4 - 1) - 2
*(sqrt(5)*sqrt(2)*x^6 - sqrt(2)*(x^6 + 2*x^2))*sqrt(x^4 - 1))*sqrt(2*sqrt(5) + 2)*sqrt(sqrt(5) + 1) - 4*((sqrt
(5)*sqrt(2)*x^5 - sqrt(2)*(x^5 + 2*x))*(x^4 - 1)^(3/4) - (sqrt(5)*sqrt(2)*x^3 - sqrt(2)*(2*x^7 - x^3))*(x^4 -
1)^(1/4))*sqrt(sqrt(5) + 1))/(x^8 - x^4 - 1)) + 1/4*sqrt(2)*sqrt(sqrt(5) - 1)*arctan(1/8*((sqrt(5)*sqrt(2)*(x^
8 + x^4 - 1) + sqrt(2)*(5*x^8 - 3*x^4 - 1) + 2*(sqrt(5)*sqrt(2)*x^6 + sqrt(2)*(x^6 + 2*x^2))*sqrt(x^4 - 1))*sq
rt(2*sqrt(5) - 2)*sqrt(sqrt(5) - 1) - 4*((sqrt(5)*sqrt(2)*x^5 + sqrt(2)*(x^5 + 2*x))*(x^4 - 1)^(3/4) + (sqrt(5
)*sqrt(2)*x^3 + sqrt(2)*(2*x^7 - x^3))*(x^4 - 1)^(1/4))*sqrt(sqrt(5) - 1))/(x^8 - x^4 - 1)) - 1/16*sqrt(2)*sqr
t(sqrt(5) - 1)*log((2*(2*x^5 + sqrt(5)*x - x)*(x^4 - 1)^(3/4) + (sqrt(5)*sqrt(2)*(x^8 - x^4) + sqrt(2)*(2*x^4
- 1) + sqrt(x^4 - 1)*(sqrt(5)*sqrt(2)*x^2 + sqrt(2)*(2*x^6 - x^2)))*sqrt(sqrt(5) - 1) - 2*(x^7 - 3*x^3 - sqrt(
5)*(x^7 - x^3))*(x^4 - 1)^(1/4))/(x^8 - x^4 - 1)) + 1/16*sqrt(2)*sqrt(sqrt(5) - 1)*log((2*(2*x^5 + sqrt(5)*x -
x)*(x^4 - 1)^(3/4) - (sqrt(5)*sqrt(2)*(x^8 - x^4) + sqrt(2)*(2*x^4 - 1) + sqrt(x^4 - 1)*(sqrt(5)*sqrt(2)*x^2
+ sqrt(2)*(2*x^6 - x^2)))*sqrt(sqrt(5) - 1) - 2*(x^7 - 3*x^3 - sqrt(5)*(x^7 - x^3))*(x^4 - 1)^(1/4))/(x^8 - x^
4 - 1)) + 1/16*sqrt(2)*sqrt(sqrt(5) + 1)*log((2*(2*x^5 - sqrt(5)*x - x)*(x^4 - 1)^(3/4) + (sqrt(5)*sqrt(2)*(x^
8 - x^4) - sqrt(2)*(2*x^4 - 1) - sqrt(x^4 - 1)*(sqrt(5)*sqrt(2)*x^2 - sqrt(2)*(2*x^6 - x^2)))*sqrt(sqrt(5) + 1
) + 2*(x^7 - 3*x^3 + sqrt(5)*(x^7 - x^3))*(x^4 - 1)^(1/4))/(x^8 - x^4 - 1)) - 1/16*sqrt(2)*sqrt(sqrt(5) + 1)*l
og((2*(2*x^5 - sqrt(5)*x - x)*(x^4 - 1)^(3/4) - (sqrt(5)*sqrt(2)*(x^8 - x^4) - sqrt(2)*(2*x^4 - 1) - sqrt(x^4
- 1)*(sqrt(5)*sqrt(2)*x^2 - sqrt(2)*(2*x^6 - x^2)))*sqrt(sqrt(5) + 1) + 2*(x^7 - 3*x^3 + sqrt(5)*(x^7 - x^3))*
(x^4 - 1)^(1/4))/(x^8 - x^4 - 1))
________________________________________________________________________________________
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((2*x^4-1)/(x^4-1)^(1/4)/(x^8-x^4-1),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
________________________________________________________________________________________
maple [C] time = 11.14, size = 1750, normalized size = 9.07
| | |
| method |
result |
size |
| | |
| trager |
\(\text {Expression too large to display}\) |
\(1750\) |
| | |
|
|
|
|
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((2*x^4-1)/(x^4-1)^(1/4)/(x^8-x^4-1),x,method=_RETURNVERBOSE)
[Out]
RootOf(256*_Z^4-16*_Z^2-1)*ln((512*RootOf(256*_Z^4-16*_Z^2-1)^5*x^4+192*RootOf(256*_Z^4-16*_Z^2-1)^3*(x^4-1)^(
1/2)*x^2+128*x^4*RootOf(256*_Z^4-16*_Z^2-1)^3+48*RootOf(256*_Z^4-16*_Z^2-1)^2*(x^4-1)^(3/4)*x+64*(x^4-1)^(1/4)
*RootOf(256*_Z^4-16*_Z^2-1)^2*x^3+4*RootOf(256*_Z^4-16*_Z^2-1)*(x^4-1)^(1/2)*x^2+8*x^4*RootOf(256*_Z^4-16*_Z^2
-1)+(x^4-1)^(3/4)*x+3*x^3*(x^4-1)^(1/4)-32*RootOf(256*_Z^4-16*_Z^2-1)^3-4*RootOf(256*_Z^4-16*_Z^2-1))/(16*Root
Of(256*_Z^4-16*_Z^2-1)^2*x^4+1))+1/4*RootOf(_Z^2+16*RootOf(256*_Z^4-16*_Z^2-1)^2-1)*ln((256*RootOf(_Z^2+16*Roo
tOf(256*_Z^4-16*_Z^2-1)^2-1)*RootOf(256*_Z^4-16*_Z^2-1)^4*x^4-96*(x^4-1)^(1/2)*RootOf(_Z^2+16*RootOf(256*_Z^4-
16*_Z^2-1)^2-1)*RootOf(256*_Z^4-16*_Z^2-1)^2*x^2-96*RootOf(_Z^2+16*RootOf(256*_Z^4-16*_Z^2-1)^2-1)*RootOf(256*
_Z^4-16*_Z^2-1)^2*x^4-96*RootOf(256*_Z^4-16*_Z^2-1)^2*(x^4-1)^(3/4)*x-128*(x^4-1)^(1/4)*RootOf(256*_Z^4-16*_Z^
2-1)^2*x^3+8*(x^4-1)^(1/2)*RootOf(_Z^2+16*RootOf(256*_Z^4-16*_Z^2-1)^2-1)*x^2+9*RootOf(_Z^2+16*RootOf(256*_Z^4
-16*_Z^2-1)^2-1)*x^4+8*(x^4-1)^(3/4)*x+14*x^3*(x^4-1)^(1/4)+16*RootOf(256*_Z^4-16*_Z^2-1)^2*RootOf(_Z^2+16*Roo
tOf(256*_Z^4-16*_Z^2-1)^2-1)-3*RootOf(_Z^2+16*RootOf(256*_Z^4-16*_Z^2-1)^2-1))/(64*RootOf(256*_Z^4-16*_Z^2-1)^
3*x^2-4*RootOf(256*_Z^4-16*_Z^2-1)*x^2-1)/(64*RootOf(256*_Z^4-16*_Z^2-1)^3*x^2-4*RootOf(256*_Z^4-16*_Z^2-1)*x^
2+1))+4*RootOf(256*_Z^4-16*_Z^2-1)^2*RootOf(_Z^2+16*RootOf(256*_Z^4-16*_Z^2-1)^2-1)*ln((768*RootOf(_Z^2+16*Roo
tOf(256*_Z^4-16*_Z^2-1)^2-1)*RootOf(256*_Z^4-16*_Z^2-1)^4*x^4-128*(x^4-1)^(1/2)*RootOf(_Z^2+16*RootOf(256*_Z^4
-16*_Z^2-1)^2-1)*RootOf(256*_Z^4-16*_Z^2-1)^2*x^2+112*RootOf(_Z^2+16*RootOf(256*_Z^4-16*_Z^2-1)^2-1)*RootOf(25
6*_Z^4-16*_Z^2-1)^2*x^4+96*RootOf(256*_Z^4-16*_Z^2-1)^2*(x^4-1)^(3/4)*x-128*(x^4-1)^(1/4)*RootOf(256*_Z^4-16*_
Z^2-1)^2*x^3-6*(x^4-1)^(1/2)*RootOf(_Z^2+16*RootOf(256*_Z^4-16*_Z^2-1)^2-1)*x^2+2*RootOf(_Z^2+16*RootOf(256*_Z
^4-16*_Z^2-1)^2-1)*x^4+2*(x^4-1)^(3/4)*x-6*x^3*(x^4-1)^(1/4)-48*RootOf(256*_Z^4-16*_Z^2-1)^2*RootOf(_Z^2+16*Ro
otOf(256*_Z^4-16*_Z^2-1)^2-1)-RootOf(_Z^2+16*RootOf(256*_Z^4-16*_Z^2-1)^2-1))/(16*RootOf(256*_Z^4-16*_Z^2-1)^2
*x^4+1))-16*RootOf(256*_Z^4-16*_Z^2-1)^3*ln((2048*RootOf(256*_Z^4-16*_Z^2-1)^5*x^4+448*RootOf(256*_Z^4-16*_Z^2
-1)^3*(x^4-1)^(1/2)*x^2-608*x^4*RootOf(256*_Z^4-16*_Z^2-1)^3-64*RootOf(256*_Z^4-16*_Z^2-1)^2*(x^4-1)^(3/4)*x+1
12*(x^4-1)^(1/4)*RootOf(256*_Z^4-16*_Z^2-1)^2*x^3-44*RootOf(256*_Z^4-16*_Z^2-1)*(x^4-1)^(1/2)*x^2+42*x^4*RootO
f(256*_Z^4-16*_Z^2-1)+7*(x^4-1)^(3/4)*x-11*x^3*(x^4-1)^(1/4)+128*RootOf(256*_Z^4-16*_Z^2-1)^3-14*RootOf(256*_Z
^4-16*_Z^2-1))/(64*RootOf(256*_Z^4-16*_Z^2-1)^3*x^2-4*RootOf(256*_Z^4-16*_Z^2-1)*x^2-1)/(64*RootOf(256*_Z^4-16
*_Z^2-1)^3*x^2-4*RootOf(256*_Z^4-16*_Z^2-1)*x^2+1))+RootOf(256*_Z^4-16*_Z^2-1)*ln((2048*RootOf(256*_Z^4-16*_Z^
2-1)^5*x^4+448*RootOf(256*_Z^4-16*_Z^2-1)^3*(x^4-1)^(1/2)*x^2-608*x^4*RootOf(256*_Z^4-16*_Z^2-1)^3-64*RootOf(2
56*_Z^4-16*_Z^2-1)^2*(x^4-1)^(3/4)*x+112*(x^4-1)^(1/4)*RootOf(256*_Z^4-16*_Z^2-1)^2*x^3-44*RootOf(256*_Z^4-16*
_Z^2-1)*(x^4-1)^(1/2)*x^2+42*x^4*RootOf(256*_Z^4-16*_Z^2-1)+7*(x^4-1)^(3/4)*x-11*x^3*(x^4-1)^(1/4)+128*RootOf(
256*_Z^4-16*_Z^2-1)^3-14*RootOf(256*_Z^4-16*_Z^2-1))/(64*RootOf(256*_Z^4-16*_Z^2-1)^3*x^2-4*RootOf(256*_Z^4-16
*_Z^2-1)*x^2-1)/(64*RootOf(256*_Z^4-16*_Z^2-1)^3*x^2-4*RootOf(256*_Z^4-16*_Z^2-1)*x^2+1))
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {2 \, x^{4} - 1}{{\left (x^{8} - x^{4} - 1\right )} {\left (x^{4} - 1\right )}^{\frac {1}{4}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((2*x^4-1)/(x^4-1)^(1/4)/(x^8-x^4-1),x, algorithm="maxima")
[Out]
integrate((2*x^4 - 1)/((x^8 - x^4 - 1)*(x^4 - 1)^(1/4)), x)
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} -\int \frac {2\,x^4-1}{{\left (x^4-1\right )}^{1/4}\,\left (-x^8+x^4+1\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(-(2*x^4 - 1)/((x^4 - 1)^(1/4)*(x^4 - x^8 + 1)),x)
[Out]
-int((2*x^4 - 1)/((x^4 - 1)^(1/4)*(x^4 - x^8 + 1)), x)
________________________________________________________________________________________
sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((2*x**4-1)/(x**4-1)**(1/4)/(x**8-x**4-1),x)
[Out]
Timed out
________________________________________________________________________________________