3.25.71 \(\int \frac {-1+2 x^4}{\sqrt [4]{1+x^4} (-1+x^4+x^8)} \, dx\)
Optimal. Leaf size=201 \[ \frac {1}{2} \sqrt {\frac {1}{10} \left (11+5 \sqrt {5}\right )} \tan ^{-1}\left (\frac {\sqrt {\frac {\sqrt {5}}{2}-\frac {1}{2}} x}{\sqrt [4]{x^4+1}}\right )-\frac {1}{2} \sqrt {\frac {1}{10} \left (5 \sqrt {5}-11\right )} \tan ^{-1}\left (\frac {\sqrt {\frac {1}{2}+\frac {\sqrt {5}}{2}} x}{\sqrt [4]{x^4+1}}\right )+\frac {1}{2} \sqrt {\frac {1}{10} \left (11+5 \sqrt {5}\right )} \tanh ^{-1}\left (\frac {\sqrt {\frac {\sqrt {5}}{2}-\frac {1}{2}} x}{\sqrt [4]{x^4+1}}\right )-\frac {1}{2} \sqrt {\frac {1}{10} \left (5 \sqrt {5}-11\right )} \tanh ^{-1}\left (\frac {\sqrt {\frac {1}{2}+\frac {\sqrt {5}}{2}} x}{\sqrt [4]{x^4+1}}\right ) \]
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Rubi [A] time = 0.40, antiderivative size = 213, normalized size of antiderivative = 1.06,
number of steps used = 10, number of rules used = 5, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used
= {6728, 377, 212, 206, 203} \begin {gather*} \frac {\sqrt [4]{\frac {1}{2} \left (123+55 \sqrt {5}\right )} \tan ^{-1}\left (\frac {\sqrt [4]{\frac {2}{3+\sqrt {5}}} x}{\sqrt [4]{x^4+1}}\right )}{2 \sqrt {5}}-\frac {\sqrt [4]{\frac {1}{2} \left (123-55 \sqrt {5}\right )} \tan ^{-1}\left (\frac {\sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} x}{\sqrt [4]{x^4+1}}\right )}{2 \sqrt {5}}+\frac {\sqrt [4]{\frac {1}{2} \left (123+55 \sqrt {5}\right )} \tanh ^{-1}\left (\frac {\sqrt [4]{\frac {2}{3+\sqrt {5}}} x}{\sqrt [4]{x^4+1}}\right )}{2 \sqrt {5}}-\frac {\sqrt [4]{\frac {1}{2} \left (123-55 \sqrt {5}\right )} \tanh ^{-1}\left (\frac {\sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} x}{\sqrt [4]{x^4+1}}\right )}{2 \sqrt {5}} \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]
(((123 + 55*Sqrt[5])/2)^(1/4)*ArcTan[((2/(3 + Sqrt[5]))^(1/4)*x)/(1 + x^4)^(1/4)])/(2*Sqrt[5]) - (((123 - 55*S
qrt[5])/2)^(1/4)*ArcTan[(((3 + Sqrt[5])/2)^(1/4)*x)/(1 + x^4)^(1/4)])/(2*Sqrt[5]) + (((123 + 55*Sqrt[5])/2)^(1
/4)*ArcTanh[((2/(3 + Sqrt[5]))^(1/4)*x)/(1 + x^4)^(1/4)])/(2*Sqrt[5]) - (((123 - 55*Sqrt[5])/2)^(1/4)*ArcTanh[
(((3 + Sqrt[5])/2)^(1/4)*x)/(1 + x^4)^(1/4)])/(2*Sqrt[5])
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-\frac {4}{\sqrt {5}}}{\sqrt [4]{1+x^4} \left (1-\sqrt {5}+2 x^4\right )}+\frac {2+\frac {4}{\sqrt {5}}}{\sqrt [4]{1+x^4} \left (1+\sqrt {5}+2 x^4\right )}\right ) \, dx\\ &=\frac {1}{5} \left (2 \left (5-2 \sqrt {5}\right )\right ) \int \frac {1}{\sqrt [4]{1+x^4} \left (1-\sqrt {5}+2 x^4\right )} \, dx+\frac {1}{5} \left (2 \left (5+2 \sqrt {5}\right )\right ) \int \frac {1}{\sqrt [4]{1+x^4} \left (1+\sqrt {5}+2 x^4\right )} \, dx\\ &=\frac {1}{5} \left (2 \left (5-2 \sqrt {5}\right )\right ) \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 {1}{5} \left (2 \left (5+2 \sqrt {5}\right )\right ) \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 {\left (2-\sqrt {5}\right ) \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 {10}}+\frac {\left (2-\sqrt {5}\right ) \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 {10}}+\frac {\left (2+\sqrt {5}\right ) \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 {10}}+\frac {\left (2+\sqrt {5}\right ) \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 {10}}\\ &=\frac {\sqrt [4]{\frac {1}{2} \left (123+55 \sqrt {5}\right )} \tan ^{-1}\left (\frac {\sqrt [4]{\frac {2}{3+\sqrt {5}}} x}{\sqrt [4]{1+x^4}}\right )}{2 \sqrt {5}}-\frac {\sqrt [4]{\frac {1}{2} \left (123-55 \sqrt {5}\right )} \tan ^{-1}\left (\frac {\sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} x}{\sqrt [4]{1+x^4}}\right )}{2 \sqrt {5}}+\frac {\sqrt [4]{\frac {1}{2} \left (123+55 \sqrt {5}\right )} \tanh ^{-1}\left (\frac {\sqrt [4]{\frac {2}{3+\sqrt {5}}} x}{\sqrt [4]{1+x^4}}\right )}{2 \sqrt {5}}-\frac {\sqrt [4]{\frac {1}{2} \left (123-55 \sqrt {5}\right )} \tanh ^{-1}\left (\frac {\sqrt [4]{\frac {1}{2} \left (3+\sqrt {5}\right )} x}{\sqrt [4]{1+x^4}}\right )}{2 \sqrt {5}}\\ \end {align*}
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Mathematica [A] time = 0.28, size = 181, normalized size = 0.90 \begin {gather*} \frac {\sqrt [4]{123+55 \sqrt {5}} \tan ^{-1}\left (\frac {\sqrt [4]{\frac {2}{3+\sqrt {5}}} x}{\sqrt [4]{x^4+1}}\right )-\sqrt [4]{123-55 \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]{123+55 \sqrt {5}} \tanh ^{-1}\left (\frac {\sqrt [4]{\frac {2}{3+\sqrt {5}}} x}{\sqrt [4]{x^4+1}}\right )-\sqrt [4]{123-55 \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} \sqrt {5}} \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]
((123 + 55*Sqrt[5])^(1/4)*ArcTan[((2/(3 + Sqrt[5]))^(1/4)*x)/(1 + x^4)^(1/4)] - (123 - 55*Sqrt[5])^(1/4)*ArcTa
n[(((3 + Sqrt[5])/2)^(1/4)*x)/(1 + x^4)^(1/4)] + (123 + 55*Sqrt[5])^(1/4)*ArcTanh[((2/(3 + Sqrt[5]))^(1/4)*x)/
(1 + x^4)^(1/4)] - (123 - 55*Sqrt[5])^(1/4)*ArcTanh[(((3 + Sqrt[5])/2)^(1/4)*x)/(1 + x^4)^(1/4)])/(2*2^(1/4)*S
qrt[5])
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IntegrateAlgebraic [A] time = 1.03, size = 201, normalized size = 1.00 \begin {gather*} \frac {1}{2} \sqrt {\frac {1}{10} \left (11+5 \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}{10} \left (-11+5 \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}{10} \left (11+5 \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}{10} \left (-11+5 \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]
(Sqrt[(11 + 5*Sqrt[5])/10]*ArcTan[(Sqrt[-1/2 + Sqrt[5]/2]*x)/(1 + x^4)^(1/4)])/2 - (Sqrt[(-11 + 5*Sqrt[5])/10]
*ArcTan[(Sqrt[1/2 + Sqrt[5]/2]*x)/(1 + x^4)^(1/4)])/2 + (Sqrt[(11 + 5*Sqrt[5])/10]*ArcTanh[(Sqrt[-1/2 + Sqrt[5
]/2]*x)/(1 + x^4)^(1/4)])/2 - (Sqrt[(-11 + 5*Sqrt[5])/10]*ArcTanh[(Sqrt[1/2 + Sqrt[5]/2]*x)/(1 + x^4)^(1/4)])/
2
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fricas [B] time = 28.10, size = 1049, normalized size = 5.22
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]
1/20*sqrt(10)*sqrt(5*sqrt(5) - 11)*arctan(1/40*(sqrt(2)*(2*sqrt(10)*(5*x^6 + 10*x^2 + sqrt(5)*(3*x^6 + 4*x^2))
*sqrt(x^4 + 1) + sqrt(10)*(15*x^8 + 25*x^4 + sqrt(5)*(5*x^8 + 11*x^4 + 3) + 5))*sqrt(5*sqrt(5) - 11)*sqrt(sqrt
(5) + 1) + 4*(sqrt(10)*(5*x^5 + sqrt(5)*(3*x^5 + 4*x) + 10*x)*(x^4 + 1)^(3/4) + sqrt(10)*(10*x^7 + 15*x^3 + sq
rt(5)*(4*x^7 + 7*x^3))*(x^4 + 1)^(1/4))*sqrt(5*sqrt(5) - 11))/(x^8 + x^4 - 1)) - 1/20*sqrt(10)*sqrt(5*sqrt(5)
+ 11)*arctan(-1/40*(sqrt(2)*(2*sqrt(10)*(5*x^6 + 10*x^2 - sqrt(5)*(3*x^6 + 4*x^2))*sqrt(x^4 + 1) - sqrt(10)*(1
5*x^8 + 25*x^4 - sqrt(5)*(5*x^8 + 11*x^4 + 3) + 5))*sqrt(5*sqrt(5) + 11)*sqrt(sqrt(5) - 1) - 4*(sqrt(10)*(5*x^
5 - sqrt(5)*(3*x^5 + 4*x) + 10*x)*(x^4 + 1)^(3/4) - sqrt(10)*(10*x^7 + 15*x^3 - sqrt(5)*(4*x^7 + 7*x^3))*(x^4
+ 1)^(1/4))*sqrt(5*sqrt(5) + 11))/(x^8 + x^4 - 1)) - 1/80*sqrt(10)*sqrt(5*sqrt(5) - 11)*log((10*(2*x^5 + sqrt(
5)*x + x)*(x^4 + 1)^(3/4) + (sqrt(10)*(10*x^6 + 15*x^2 + sqrt(5)*(4*x^6 + 7*x^2))*sqrt(x^4 + 1) + sqrt(10)*(10
*x^8 + 20*x^4 + sqrt(5)*(5*x^8 + 9*x^4 + 2) + 5))*sqrt(5*sqrt(5) - 11) + 10*(x^7 + 3*x^3 + sqrt(5)*(x^7 + x^3)
)*(x^4 + 1)^(1/4))/(x^8 + x^4 - 1)) + 1/80*sqrt(10)*sqrt(5*sqrt(5) - 11)*log((10*(2*x^5 + sqrt(5)*x + x)*(x^4
+ 1)^(3/4) - (sqrt(10)*(10*x^6 + 15*x^2 + sqrt(5)*(4*x^6 + 7*x^2))*sqrt(x^4 + 1) + sqrt(10)*(10*x^8 + 20*x^4 +
sqrt(5)*(5*x^8 + 9*x^4 + 2) + 5))*sqrt(5*sqrt(5) - 11) + 10*(x^7 + 3*x^3 + sqrt(5)*(x^7 + x^3))*(x^4 + 1)^(1/
4))/(x^8 + x^4 - 1)) + 1/80*sqrt(10)*sqrt(5*sqrt(5) + 11)*log((10*(2*x^5 - sqrt(5)*x + x)*(x^4 + 1)^(3/4) + (s
qrt(10)*(10*x^6 + 15*x^2 - sqrt(5)*(4*x^6 + 7*x^2))*sqrt(x^4 + 1) - sqrt(10)*(10*x^8 + 20*x^4 - sqrt(5)*(5*x^8
+ 9*x^4 + 2) + 5))*sqrt(5*sqrt(5) + 11) - 10*(x^7 + 3*x^3 - sqrt(5)*(x^7 + x^3))*(x^4 + 1)^(1/4))/(x^8 + x^4
- 1)) - 1/80*sqrt(10)*sqrt(5*sqrt(5) + 11)*log((10*(2*x^5 - sqrt(5)*x + x)*(x^4 + 1)^(3/4) - (sqrt(10)*(10*x^6
+ 15*x^2 - sqrt(5)*(4*x^6 + 7*x^2))*sqrt(x^4 + 1) - sqrt(10)*(10*x^8 + 20*x^4 - sqrt(5)*(5*x^8 + 9*x^4 + 2) +
5))*sqrt(5*sqrt(5) + 11) - 10*(x^7 + 3*x^3 - sqrt(5)*(x^7 + x^3))*(x^4 + 1)^(1/4))/(x^8 + x^4 - 1))
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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((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
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maple [C] time = 13.67, size = 1608, normalized size = 8.00
| | |
| method |
result |
size |
| | |
| trager |
\(\text {Expression too large to display}\) |
\(1608\) |
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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(6400*_Z^4-880*_Z^2-1)*ln(-(-64000*RootOf(6400*_Z^4-880*_Z^2-1)^5*x^4-12800*RootOf(6400*_Z^4-880*_Z^2-1
)^3*(x^4+1)^(1/2)*x^2+23200*x^4*RootOf(6400*_Z^4-880*_Z^2-1)^3+320*(x^4+1)^(3/4)*RootOf(6400*_Z^4-880*_Z^2-1)^
2*x-560*(x^4+1)^(1/4)*RootOf(6400*_Z^4-880*_Z^2-1)^2*x^3+1780*RootOf(6400*_Z^4-880*_Z^2-1)*(x^4+1)^(1/2)*x^2-1
980*x^4*RootOf(6400*_Z^4-880*_Z^2-1)-47*(x^4+1)^(3/4)*x+76*x^3*(x^4+1)^(1/4)+4000*RootOf(6400*_Z^4-880*_Z^2-1)
^3-550*RootOf(6400*_Z^4-880*_Z^2-1))/(80*RootOf(6400*_Z^4-880*_Z^2-1)^2*x^4-8*x^4+5))+1/20*RootOf(_Z^2+400*Roo
tOf(6400*_Z^4-880*_Z^2-1)^2-55)*ln(5*(-3200*RootOf(_Z^2+400*RootOf(6400*_Z^4-880*_Z^2-1)^2-55)*RootOf(6400*_Z^
4-880*_Z^2-1)^4*x^4+640*(x^4+1)^(1/2)*RootOf(6400*_Z^4-880*_Z^2-1)^2*RootOf(_Z^2+400*RootOf(6400*_Z^4-880*_Z^2
-1)^2-55)*x^2-280*RootOf(_Z^2+400*RootOf(6400*_Z^4-880*_Z^2-1)^2-55)*RootOf(6400*_Z^4-880*_Z^2-1)^2*x^4+320*(x
^4+1)^(3/4)*RootOf(6400*_Z^4-880*_Z^2-1)^2*x-560*(x^4+1)^(1/4)*RootOf(6400*_Z^4-880*_Z^2-1)^2*x^3+(x^4+1)^(1/2
)*RootOf(_Z^2+400*RootOf(6400*_Z^4-880*_Z^2-1)^2-55)*x^2+3*(x^4+1)^(3/4)*x+x^3*(x^4+1)^(1/4)-200*RootOf(6400*_
Z^4-880*_Z^2-1)^2*RootOf(_Z^2+400*RootOf(6400*_Z^4-880*_Z^2-1)^2-55))/(960*RootOf(6400*_Z^4-880*_Z^2-1)^3*x^2-
116*RootOf(6400*_Z^4-880*_Z^2-1)*x^2+5)/(960*RootOf(6400*_Z^4-880*_Z^2-1)^3*x^2-116*RootOf(6400*_Z^4-880*_Z^2-
1)*x^2-5))+4*RootOf(6400*_Z^4-880*_Z^2-1)^2*RootOf(_Z^2+400*RootOf(6400*_Z^4-880*_Z^2-1)^2-55)*ln((6400*RootOf
(_Z^2+400*RootOf(6400*_Z^4-880*_Z^2-1)^2-55)*RootOf(6400*_Z^4-880*_Z^2-1)^4*x^4-1680*RootOf(_Z^2+400*RootOf(64
00*_Z^4-880*_Z^2-1)^2-55)*RootOf(6400*_Z^4-880*_Z^2-1)^2*x^4-2400*(x^4+1)^(3/4)*RootOf(6400*_Z^4-880*_Z^2-1)^2
*x-3200*(x^4+1)^(1/4)*RootOf(6400*_Z^4-880*_Z^2-1)^2*x^3+50*(x^4+1)^(1/2)*RootOf(_Z^2+400*RootOf(6400*_Z^4-880
*_Z^2-1)^2-55)*x^2+54*RootOf(_Z^2+400*RootOf(6400*_Z^4-880*_Z^2-1)^2-55)*x^4+290*(x^4+1)^(3/4)*x+470*x^3*(x^4+
1)^(1/4)-400*RootOf(6400*_Z^4-880*_Z^2-1)^2*RootOf(_Z^2+400*RootOf(6400*_Z^4-880*_Z^2-1)^2-55)+15*RootOf(_Z^2+
400*RootOf(6400*_Z^4-880*_Z^2-1)^2-55))/(80*RootOf(6400*_Z^4-880*_Z^2-1)^2*x^4-8*x^4+5))+80*RootOf(6400*_Z^4-8
80*_Z^2-1)^3*ln(5*(-12800*RootOf(6400*_Z^4-880*_Z^2-1)^5*x^4+160*x^4*RootOf(6400*_Z^4-880*_Z^2-1)^3+240*(x^4+1
)^(3/4)*RootOf(6400*_Z^4-880*_Z^2-1)^2*x+320*(x^4+1)^(1/4)*RootOf(6400*_Z^4-880*_Z^2-1)^2*x^3-100*RootOf(6400*
_Z^4-880*_Z^2-1)*(x^4+1)^(1/2)*x^2+112*x^4*RootOf(6400*_Z^4-880*_Z^2-1)-4*(x^4+1)^(3/4)*x+3*x^3*(x^4+1)^(1/4)-
800*RootOf(6400*_Z^4-880*_Z^2-1)^3+80*RootOf(6400*_Z^4-880*_Z^2-1))/(960*RootOf(6400*_Z^4-880*_Z^2-1)^3*x^2-11
6*RootOf(6400*_Z^4-880*_Z^2-1)*x^2+5)/(960*RootOf(6400*_Z^4-880*_Z^2-1)^3*x^2-116*RootOf(6400*_Z^4-880*_Z^2-1)
*x^2-5))-11*RootOf(6400*_Z^4-880*_Z^2-1)*ln(5*(-12800*RootOf(6400*_Z^4-880*_Z^2-1)^5*x^4+160*x^4*RootOf(6400*_
Z^4-880*_Z^2-1)^3+240*(x^4+1)^(3/4)*RootOf(6400*_Z^4-880*_Z^2-1)^2*x+320*(x^4+1)^(1/4)*RootOf(6400*_Z^4-880*_Z
^2-1)^2*x^3-100*RootOf(6400*_Z^4-880*_Z^2-1)*(x^4+1)^(1/2)*x^2+112*x^4*RootOf(6400*_Z^4-880*_Z^2-1)-4*(x^4+1)^
(3/4)*x+3*x^3*(x^4+1)^(1/4)-800*RootOf(6400*_Z^4-880*_Z^2-1)^3+80*RootOf(6400*_Z^4-880*_Z^2-1))/(960*RootOf(64
00*_Z^4-880*_Z^2-1)^3*x^2-116*RootOf(6400*_Z^4-880*_Z^2-1)*x^2+5)/(960*RootOf(6400*_Z^4-880*_Z^2-1)^3*x^2-116*
RootOf(6400*_Z^4-880*_Z^2-1)*x^2-5))
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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)
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \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)
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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
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