3.25.34 \(\int \frac {1-x^4+x^8}{x^2 (-1+x^4)^{3/4} (-1-x^4+x^8)} \, dx\)

Optimal. Leaf size=197 \[ -\frac {\sqrt [4]{x^4-1}}{x}+\sqrt {\frac {1}{10} \left (\sqrt {5}-1\right )} \tan ^{-1}\left (\frac {\sqrt {\frac {\sqrt {5}}{2}-\frac {1}{2}} x}{\sqrt [4]{x^4-1}}\right )-\sqrt {\frac {1}{10} \left (1+\sqrt {5}\right )} \tan ^{-1}\left (\frac {\sqrt {\frac {1}{2}+\frac {\sqrt {5}}{2}} x}{\sqrt [4]{x^4-1}}\right )-\sqrt {\frac {1}{10} \left (\sqrt {5}-1\right )} \tanh ^{-1}\left (\frac {\sqrt {\frac {\sqrt {5}}{2}-\frac {1}{2}} x}{\sqrt [4]{x^4-1}}\right )+\sqrt {\frac {1}{10} \left (1+\sqrt {5}\right )} \tanh ^{-1}\left (\frac {\sqrt {\frac {1}{2}+\frac {\sqrt {5}}{2}} x}{\sqrt [4]{x^4-1}}\right ) \]

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Rubi [C]  time = 0.55, antiderivative size = 157, normalized size of antiderivative = 0.80, number of steps used = 9, number of rules used = 5, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {6728, 264, 1528, 511, 510} \begin {gather*} \frac {4 \sqrt [4]{x^4-1} x^3 F_1\left (\frac {3}{4};-\frac {1}{4},1;\frac {7}{4};x^4,\frac {2 x^4}{1-\sqrt {5}}\right )}{3 \sqrt {5} \left (1-\sqrt {5}\right ) \sqrt [4]{1-x^4}}-\frac {4 \sqrt [4]{x^4-1} x^3 F_1\left (\frac {3}{4};1,-\frac {1}{4};\frac {7}{4};\frac {2 x^4}{1+\sqrt {5}},x^4\right )}{3 \sqrt {5} \left (1+\sqrt {5}\right ) \sqrt [4]{1-x^4}}-\frac {\sqrt [4]{x^4-1}}{x} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Int[(1 - x^4 + x^8)/(x^2*(-1 + x^4)^(3/4)*(-1 - x^4 + x^8)),x]

[Out]

-((-1 + x^4)^(1/4)/x) + (4*x^3*(-1 + x^4)^(1/4)*AppellF1[3/4, -1/4, 1, 7/4, x^4, (2*x^4)/(1 - Sqrt[5])])/(3*Sq
rt[5]*(1 - Sqrt[5])*(1 - x^4)^(1/4)) - (4*x^3*(-1 + x^4)^(1/4)*AppellF1[3/4, 1, -1/4, 7/4, (2*x^4)/(1 + Sqrt[5
]), x^4])/(3*Sqrt[5]*(1 + Sqrt[5])*(1 - x^4)^(1/4))

Rule 264

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a
*c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[(m + 1)/n + p + 1, 0] && NeQ[m, -1]

Rule 510

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(a^p*c^q
*(e*x)^(m + 1)*AppellF1[(m + 1)/n, -p, -q, 1 + (m + 1)/n, -((b*x^n)/a), -((d*x^n)/c)])/(e*(m + 1)), x] /; Free
Q[{a, b, c, d, e, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1] && NeQ[m, n - 1] && (IntegerQ[p] || GtQ[a
, 0]) && (IntegerQ[q] || GtQ[c, 0])

Rule 511

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Dist[(a^IntPa
rt[p]*(a + b*x^n)^FracPart[p])/(1 + (b*x^n)/a)^FracPart[p], Int[(e*x)^m*(1 + (b*x^n)/a)^p*(c + d*x^n)^q, x], x
] /; FreeQ[{a, b, c, d, e, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1] && NeQ[m, n - 1] &&  !(IntegerQ[
p] || GtQ[a, 0])

Rule 1528

Int[(((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(n_))^(q_))/((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_)), x_Symbol]
:> Int[ExpandIntegrand[(d + e*x^n)^q, (f*x)^m/(a + b*x^n + c*x^(2*n)), x], x] /; FreeQ[{a, b, c, d, e, f, q,
n}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] &&  !IntegerQ[q] && IntegerQ[m]

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-x^4+x^8}{x^2 \left (-1+x^4\right )^{3/4} \left (-1-x^4+x^8\right )} \, dx &=\int \left (-\frac {1}{x^2 \left (-1+x^4\right )^{3/4}}+\frac {2 x^2 \sqrt [4]{-1+x^4}}{-1-x^4+x^8}\right ) \, dx\\ &=2 \int \frac {x^2 \sqrt [4]{-1+x^4}}{-1-x^4+x^8} \, dx-\int \frac {1}{x^2 \left (-1+x^4\right )^{3/4}} \, dx\\ &=-\frac {\sqrt [4]{-1+x^4}}{x}+2 \int \left (-\frac {2 x^2 \sqrt [4]{-1+x^4}}{\sqrt {5} \left (1+\sqrt {5}-2 x^4\right )}-\frac {2 x^2 \sqrt [4]{-1+x^4}}{\sqrt {5} \left (-1+\sqrt {5}+2 x^4\right )}\right ) \, dx\\ &=-\frac {\sqrt [4]{-1+x^4}}{x}-\frac {4 \int \frac {x^2 \sqrt [4]{-1+x^4}}{1+\sqrt {5}-2 x^4} \, dx}{\sqrt {5}}-\frac {4 \int \frac {x^2 \sqrt [4]{-1+x^4}}{-1+\sqrt {5}+2 x^4} \, dx}{\sqrt {5}}\\ &=-\frac {\sqrt [4]{-1+x^4}}{x}-\frac {\left (4 \sqrt [4]{-1+x^4}\right ) \int \frac {x^2 \sqrt [4]{1-x^4}}{1+\sqrt {5}-2 x^4} \, dx}{\sqrt {5} \sqrt [4]{1-x^4}}-\frac {\left (4 \sqrt [4]{-1+x^4}\right ) \int \frac {x^2 \sqrt [4]{1-x^4}}{-1+\sqrt {5}+2 x^4} \, dx}{\sqrt {5} \sqrt [4]{1-x^4}}\\ &=-\frac {\sqrt [4]{-1+x^4}}{x}+\frac {4 x^3 \sqrt [4]{-1+x^4} F_1\left (\frac {3}{4};-\frac {1}{4},1;\frac {7}{4};x^4,\frac {2 x^4}{1-\sqrt {5}}\right )}{3 \sqrt {5} \left (1-\sqrt {5}\right ) \sqrt [4]{1-x^4}}-\frac {4 x^3 \sqrt [4]{-1+x^4} F_1\left (\frac {3}{4};1,-\frac {1}{4};\frac {7}{4};\frac {2 x^4}{1+\sqrt {5}},x^4\right )}{3 \sqrt {5} \left (1+\sqrt {5}\right ) \sqrt [4]{1-x^4}}\\ \end {align*}

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Mathematica [F]  time = 7.22, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1-x^4+x^8}{x^2 \left (-1+x^4\right )^{3/4} \left (-1-x^4+x^8\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[(1 - x^4 + x^8)/(x^2*(-1 + x^4)^(3/4)*(-1 - x^4 + x^8)),x]

[Out]

Integrate[(1 - x^4 + x^8)/(x^2*(-1 + x^4)^(3/4)*(-1 - x^4 + x^8)), x]

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IntegrateAlgebraic [A]  time = 0.88, size = 197, normalized size = 1.00 \begin {gather*} -\frac {\sqrt [4]{-1+x^4}}{x}+\sqrt {\frac {1}{10} \left (-1+\sqrt {5}\right )} \tan ^{-1}\left (\frac {\sqrt {-\frac {1}{2}+\frac {\sqrt {5}}{2}} x}{\sqrt [4]{-1+x^4}}\right )-\sqrt {\frac {1}{10} \left (1+\sqrt {5}\right )} \tan ^{-1}\left (\frac {\sqrt {\frac {1}{2}+\frac {\sqrt {5}}{2}} x}{\sqrt [4]{-1+x^4}}\right )-\sqrt {\frac {1}{10} \left (-1+\sqrt {5}\right )} \tanh ^{-1}\left (\frac {\sqrt {-\frac {1}{2}+\frac {\sqrt {5}}{2}} x}{\sqrt [4]{-1+x^4}}\right )+\sqrt {\frac {1}{10} \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 - x^4 + x^8)/(x^2*(-1 + x^4)^(3/4)*(-1 - x^4 + x^8)),x]

[Out]

-((-1 + x^4)^(1/4)/x) + Sqrt[(-1 + Sqrt[5])/10]*ArcTan[(Sqrt[-1/2 + Sqrt[5]/2]*x)/(-1 + x^4)^(1/4)] - Sqrt[(1
+ Sqrt[5])/10]*ArcTan[(Sqrt[1/2 + Sqrt[5]/2]*x)/(-1 + x^4)^(1/4)] - Sqrt[(-1 + Sqrt[5])/10]*ArcTanh[(Sqrt[-1/2
+ Sqrt[5]/2]*x)/(-1 + x^4)^(1/4)] + Sqrt[(1 + Sqrt[5])/10]*ArcTanh[(Sqrt[1/2 + Sqrt[5]/2]*x)/(-1 + x^4)^(1/4)
]

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fricas [B]  time = 14.83, size = 971, normalized size = 4.93

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^8-x^4+1)/x^2/(x^4-1)^(3/4)/(x^8-x^4-1),x, algorithm="fricas")

[Out]

-1/40*(4*sqrt(10)*x*sqrt(sqrt(5) + 1)*arctan(-1/40*(sqrt(2)*(2*sqrt(10)*(5*x^6 - sqrt(5)*(x^6 + 2*x^2))*sqrt(x
^4 - 1) - sqrt(10)*(5*x^8 + 5*x^4 - sqrt(5)*(5*x^8 - 3*x^4 - 1) - 5))*(sqrt(5) + 1) + 4*(sqrt(10)*(5*x^5 - sqr
t(5)*(x^5 + 2*x))*(x^4 - 1)^(3/4) - sqrt(10)*(x^4 - 1)^(1/4)*(5*x^3 - sqrt(5)*(2*x^7 - x^3)))*sqrt(sqrt(5) + 1
))/(x^8 - x^4 - 1)) + 4*sqrt(10)*x*sqrt(sqrt(5) - 1)*arctan(1/40*(sqrt(2)*(2*sqrt(10)*(5*x^6 + sqrt(5)*(x^6 +
2*x^2))*sqrt(x^4 - 1) + sqrt(10)*(5*x^8 + 5*x^4 + sqrt(5)*(5*x^8 - 3*x^4 - 1) - 5))*(sqrt(5) - 1) - 4*(sqrt(10
)*(5*x^5 + sqrt(5)*(x^5 + 2*x))*(x^4 - 1)^(3/4) + sqrt(10)*(x^4 - 1)^(1/4)*(5*x^3 + sqrt(5)*(2*x^7 - x^3)))*sq
rt(sqrt(5) - 1))/(x^8 - x^4 - 1)) + sqrt(10)*x*sqrt(sqrt(5) - 1)*log((10*(2*x^5 + sqrt(5)*x - x)*(x^4 - 1)^(3/
4) + (sqrt(10)*sqrt(x^4 - 1)*(5*x^2 + sqrt(5)*(2*x^6 - x^2)) + sqrt(10)*(5*x^8 - 5*x^4 + sqrt(5)*(2*x^4 - 1)))
*sqrt(sqrt(5) - 1) - 10*(x^7 - 3*x^3 - sqrt(5)*(x^7 - x^3))*(x^4 - 1)^(1/4))/(x^8 - x^4 - 1)) - sqrt(10)*x*sqr
t(sqrt(5) - 1)*log((10*(2*x^5 + sqrt(5)*x - x)*(x^4 - 1)^(3/4) - (sqrt(10)*sqrt(x^4 - 1)*(5*x^2 + sqrt(5)*(2*x
^6 - x^2)) + sqrt(10)*(5*x^8 - 5*x^4 + sqrt(5)*(2*x^4 - 1)))*sqrt(sqrt(5) - 1) - 10*(x^7 - 3*x^3 - sqrt(5)*(x^
7 - x^3))*(x^4 - 1)^(1/4))/(x^8 - x^4 - 1)) + sqrt(10)*x*sqrt(sqrt(5) + 1)*log((10*(2*x^5 - sqrt(5)*x - x)*(x^
4 - 1)^(3/4) + (sqrt(10)*sqrt(x^4 - 1)*(5*x^2 - sqrt(5)*(2*x^6 - x^2)) - sqrt(10)*(5*x^8 - 5*x^4 - sqrt(5)*(2*
x^4 - 1)))*sqrt(sqrt(5) + 1) + 10*(x^7 - 3*x^3 + sqrt(5)*(x^7 - x^3))*(x^4 - 1)^(1/4))/(x^8 - x^4 - 1)) - sqrt
(10)*x*sqrt(sqrt(5) + 1)*log((10*(2*x^5 - sqrt(5)*x - x)*(x^4 - 1)^(3/4) - (sqrt(10)*sqrt(x^4 - 1)*(5*x^2 - sq
rt(5)*(2*x^6 - x^2)) - sqrt(10)*(5*x^8 - 5*x^4 - sqrt(5)*(2*x^4 - 1)))*sqrt(sqrt(5) + 1) + 10*(x^7 - 3*x^3 + s
qrt(5)*(x^7 - x^3))*(x^4 - 1)^(1/4))/(x^8 - x^4 - 1)) + 40*(x^4 - 1)^(1/4))/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((x^8-x^4+1)/x^2/(x^4-1)^(3/4)/(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 = 14.08, size = 3552, normalized size = 18.03 \[\text {output too large to display}\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^8-x^4+1)/x^2/(x^4-1)^(3/4)/(x^8-x^4-1),x)

[Out]

-(x^4-1)^(1/4)/x+(-1/10*RootOf(_Z^2+400*RootOf(6400*_Z^4-80*_Z^2-1)^2-5)*ln(-(32000*RootOf(6400*_Z^4-80*_Z^2-1
)^4*x^12-1600*RootOf(6400*_Z^4-80*_Z^2-1)^2*x^12+160*RootOf(_Z^2+400*RootOf(6400*_Z^4-80*_Z^2-1)^2-5)*RootOf(6
400*_Z^4-80*_Z^2-1)^2*(x^12-3*x^8+3*x^4-1)^(1/4)*x^9-64000*RootOf(6400*_Z^4-80*_Z^2-1)^4*x^8+15*x^12-6*RootOf(
_Z^2+400*RootOf(6400*_Z^4-80*_Z^2-1)^2-5)*(x^12-3*x^8+3*x^4-1)^(1/4)*x^9-800*(x^12-3*x^8+3*x^4-1)^(1/2)*RootOf
(6400*_Z^4-80*_Z^2-1)^2*x^6+3600*RootOf(6400*_Z^4-80*_Z^2-1)^2*x^8+480*RootOf(_Z^2+400*RootOf(6400*_Z^4-80*_Z^
2-1)^2-5)*RootOf(6400*_Z^4-80*_Z^2-1)^2*(x^12-3*x^8+3*x^4-1)^(3/4)*x^3-320*RootOf(_Z^2+400*RootOf(6400*_Z^4-80
*_Z^2-1)^2-5)*RootOf(6400*_Z^4-80*_Z^2-1)^2*(x^12-3*x^8+3*x^4-1)^(1/4)*x^5+32000*RootOf(6400*_Z^4-80*_Z^2-1)^4
*x^4+20*(x^12-3*x^8+3*x^4-1)^(1/2)*x^6-35*x^8-8*RootOf(_Z^2+400*RootOf(6400*_Z^4-80*_Z^2-1)^2-5)*(x^12-3*x^8+3
*x^4-1)^(3/4)*x^3+12*RootOf(_Z^2+400*RootOf(6400*_Z^4-80*_Z^2-1)^2-5)*(x^12-3*x^8+3*x^4-1)^(1/4)*x^5+800*(x^12
-3*x^8+3*x^4-1)^(1/2)*RootOf(6400*_Z^4-80*_Z^2-1)^2*x^2-2400*RootOf(6400*_Z^4-80*_Z^2-1)^2*x^4+160*RootOf(_Z^2
+400*RootOf(6400*_Z^4-80*_Z^2-1)^2-5)*RootOf(6400*_Z^4-80*_Z^2-1)^2*(x^12-3*x^8+3*x^4-1)^(1/4)*x-20*(x^12-3*x^
8+3*x^4-1)^(1/2)*x^2+25*x^4-6*RootOf(_Z^2+400*RootOf(6400*_Z^4-80*_Z^2-1)^2-5)*(x^12-3*x^8+3*x^4-1)^(1/4)*x+40
0*RootOf(6400*_Z^4-80*_Z^2-1)^2-5)/(320*RootOf(6400*_Z^4-80*_Z^2-1)^3*x^2-12*RootOf(6400*_Z^4-80*_Z^2-1)*x^2+1
)/(320*RootOf(6400*_Z^4-80*_Z^2-1)^3*x^2-12*RootOf(6400*_Z^4-80*_Z^2-1)*x^2-1)/(-1+x)^2/(1+x)^2/(x^2+1)^2)-2*R
ootOf(6400*_Z^4-80*_Z^2-1)*ln(-(3200*RootOf(6400*_Z^4-80*_Z^2-1)^4*x^12+80*RootOf(6400*_Z^4-80*_Z^2-1)^2*x^12-
320*RootOf(6400*_Z^4-80*_Z^2-1)^3*(x^12-3*x^8+3*x^4-1)^(1/4)*x^9-6400*RootOf(6400*_Z^4-80*_Z^2-1)^4*x^8-8*Root
Of(6400*_Z^4-80*_Z^2-1)*(x^12-3*x^8+3*x^4-1)^(1/4)*x^9+80*(x^12-3*x^8+3*x^4-1)^(1/2)*RootOf(6400*_Z^4-80*_Z^2-
1)^2*x^6-200*RootOf(6400*_Z^4-80*_Z^2-1)^2*x^8-960*RootOf(6400*_Z^4-80*_Z^2-1)^3*(x^12-3*x^8+3*x^4-1)^(3/4)*x^
3+640*RootOf(6400*_Z^4-80*_Z^2-1)^3*(x^12-3*x^8+3*x^4-1)^(1/4)*x^5+3200*RootOf(6400*_Z^4-80*_Z^2-1)^4*x^4+(x^1
2-3*x^8+3*x^4-1)^(1/2)*x^6-4*RootOf(6400*_Z^4-80*_Z^2-1)*(x^12-3*x^8+3*x^4-1)^(3/4)*x^3+16*RootOf(6400*_Z^4-80
*_Z^2-1)*(x^12-3*x^8+3*x^4-1)^(1/4)*x^5-80*(x^12-3*x^8+3*x^4-1)^(1/2)*RootOf(6400*_Z^4-80*_Z^2-1)^2*x^2+160*Ro
otOf(6400*_Z^4-80*_Z^2-1)^2*x^4-320*RootOf(6400*_Z^4-80*_Z^2-1)^3*(x^12-3*x^8+3*x^4-1)^(1/4)*x-(x^12-3*x^8+3*x
^4-1)^(1/2)*x^2-8*RootOf(6400*_Z^4-80*_Z^2-1)*(x^12-3*x^8+3*x^4-1)^(1/4)*x-40*RootOf(6400*_Z^4-80*_Z^2-1)^2)/(
80*RootOf(6400*_Z^4-80*_Z^2-1)^2*x^4+1)/(-1+x)^2/(1+x)^2/(x^2+1)^2)-8*RootOf(6400*_Z^4-80*_Z^2-1)^2*RootOf(_Z^
2+400*RootOf(6400*_Z^4-80*_Z^2-1)^2-5)*ln((16000*RootOf(6400*_Z^4-80*_Z^2-1)^4*x^12+400*RootOf(6400*_Z^4-80*_Z
^2-1)^2*x^12-240*RootOf(_Z^2+400*RootOf(6400*_Z^4-80*_Z^2-1)^2-5)*RootOf(6400*_Z^4-80*_Z^2-1)^2*(x^12-3*x^8+3*
x^4-1)^(1/4)*x^9-32000*RootOf(6400*_Z^4-80*_Z^2-1)^4*x^8-RootOf(_Z^2+400*RootOf(6400*_Z^4-80*_Z^2-1)^2-5)*(x^1
2-3*x^8+3*x^4-1)^(1/4)*x^9-400*(x^12-3*x^8+3*x^4-1)^(1/2)*RootOf(6400*_Z^4-80*_Z^2-1)^2*x^6-1000*RootOf(6400*_
Z^4-80*_Z^2-1)^2*x^8+320*RootOf(_Z^2+400*RootOf(6400*_Z^4-80*_Z^2-1)^2-5)*RootOf(6400*_Z^4-80*_Z^2-1)^2*(x^12-
3*x^8+3*x^4-1)^(3/4)*x^3+480*RootOf(_Z^2+400*RootOf(6400*_Z^4-80*_Z^2-1)^2-5)*RootOf(6400*_Z^4-80*_Z^2-1)^2*(x
^12-3*x^8+3*x^4-1)^(1/4)*x^5+16000*RootOf(6400*_Z^4-80*_Z^2-1)^4*x^4-5*(x^12-3*x^8+3*x^4-1)^(1/2)*x^6+3*RootOf
(_Z^2+400*RootOf(6400*_Z^4-80*_Z^2-1)^2-5)*(x^12-3*x^8+3*x^4-1)^(3/4)*x^3+2*RootOf(_Z^2+400*RootOf(6400*_Z^4-8
0*_Z^2-1)^2-5)*(x^12-3*x^8+3*x^4-1)^(1/4)*x^5+400*(x^12-3*x^8+3*x^4-1)^(1/2)*RootOf(6400*_Z^4-80*_Z^2-1)^2*x^2
+800*RootOf(6400*_Z^4-80*_Z^2-1)^2*x^4-240*RootOf(_Z^2+400*RootOf(6400*_Z^4-80*_Z^2-1)^2-5)*RootOf(6400*_Z^4-8
0*_Z^2-1)^2*(x^12-3*x^8+3*x^4-1)^(1/4)*x+5*(x^12-3*x^8+3*x^4-1)^(1/2)*x^2-RootOf(_Z^2+400*RootOf(6400*_Z^4-80*
_Z^2-1)^2-5)*(x^12-3*x^8+3*x^4-1)^(1/4)*x-200*RootOf(6400*_Z^4-80*_Z^2-1)^2)/(80*RootOf(6400*_Z^4-80*_Z^2-1)^2
*x^4+1)/(-1+x)^2/(1+x)^2/(x^2+1)^2)+160*ln((6400*RootOf(6400*_Z^4-80*_Z^2-1)^4*x^12-320*RootOf(6400*_Z^4-80*_Z
^2-1)^2*x^12+1920*RootOf(6400*_Z^4-80*_Z^2-1)^3*(x^12-3*x^8+3*x^4-1)^(1/4)*x^9-12800*RootOf(6400*_Z^4-80*_Z^2-
1)^4*x^8+3*x^12-32*RootOf(6400*_Z^4-80*_Z^2-1)*(x^12-3*x^8+3*x^4-1)^(1/4)*x^9+160*(x^12-3*x^8+3*x^4-1)^(1/2)*R
ootOf(6400*_Z^4-80*_Z^2-1)^2*x^6+720*RootOf(6400*_Z^4-80*_Z^2-1)^2*x^8-2560*RootOf(6400*_Z^4-80*_Z^2-1)^3*(x^1
2-3*x^8+3*x^4-1)^(3/4)*x^3-3840*RootOf(6400*_Z^4-80*_Z^2-1)^3*(x^12-3*x^8+3*x^4-1)^(1/4)*x^5+6400*RootOf(6400*
_Z^4-80*_Z^2-1)^4*x^4-4*(x^12-3*x^8+3*x^4-1)^(1/2)*x^6-7*x^8+56*RootOf(6400*_Z^4-80*_Z^2-1)*(x^12-3*x^8+3*x^4-
1)^(3/4)*x^3+64*RootOf(6400*_Z^4-80*_Z^2-1)*(x^12-3*x^8+3*x^4-1)^(1/4)*x^5-160*(x^12-3*x^8+3*x^4-1)^(1/2)*Root
Of(6400*_Z^4-80*_Z^2-1)^2*x^2-480*RootOf(6400*_Z^4-80*_Z^2-1)^2*x^4+1920*RootOf(6400*_Z^4-80*_Z^2-1)^3*(x^12-3
*x^8+3*x^4-1)^(1/4)*x+4*(x^12-3*x^8+3*x^4-1)^(1/2)*x^2+5*x^4-32*RootOf(6400*_Z^4-80*_Z^2-1)*(x^12-3*x^8+3*x^4-
1)^(1/4)*x+80*RootOf(6400*_Z^4-80*_Z^2-1)^2-1)/(320*RootOf(6400*_Z^4-80*_Z^2-1)^3*x^2-12*RootOf(6400*_Z^4-80*_
Z^2-1)*x^2+1)/(320*RootOf(6400*_Z^4-80*_Z^2-1)^3*x^2-12*RootOf(6400*_Z^4-80*_Z^2-1)*x^2-1)/(-1+x)^2/(1+x)^2/(x
^2+1)^2)*RootOf(6400*_Z^4-80*_Z^2-1)^3-2*ln((6400*RootOf(6400*_Z^4-80*_Z^2-1)^4*x^12-320*RootOf(6400*_Z^4-80*_
Z^2-1)^2*x^12+1920*RootOf(6400*_Z^4-80*_Z^2-1)^3*(x^12-3*x^8+3*x^4-1)^(1/4)*x^9-12800*RootOf(6400*_Z^4-80*_Z^2
-1)^4*x^8+3*x^12-32*RootOf(6400*_Z^4-80*_Z^2-1)*(x^12-3*x^8+3*x^4-1)^(1/4)*x^9+160*(x^12-3*x^8+3*x^4-1)^(1/2)*
RootOf(6400*_Z^4-80*_Z^2-1)^2*x^6+720*RootOf(6400*_Z^4-80*_Z^2-1)^2*x^8-2560*RootOf(6400*_Z^4-80*_Z^2-1)^3*(x^
12-3*x^8+3*x^4-1)^(3/4)*x^3-3840*RootOf(6400*_Z^4-80*_Z^2-1)^3*(x^12-3*x^8+3*x^4-1)^(1/4)*x^5+6400*RootOf(6400
*_Z^4-80*_Z^2-1)^4*x^4-4*(x^12-3*x^8+3*x^4-1)^(1/2)*x^6-7*x^8+56*RootOf(6400*_Z^4-80*_Z^2-1)*(x^12-3*x^8+3*x^4
-1)^(3/4)*x^3+64*RootOf(6400*_Z^4-80*_Z^2-1)*(x^12-3*x^8+3*x^4-1)^(1/4)*x^5-160*(x^12-3*x^8+3*x^4-1)^(1/2)*Roo
tOf(6400*_Z^4-80*_Z^2-1)^2*x^2-480*RootOf(6400*_Z^4-80*_Z^2-1)^2*x^4+1920*RootOf(6400*_Z^4-80*_Z^2-1)^3*(x^12-
3*x^8+3*x^4-1)^(1/4)*x+4*(x^12-3*x^8+3*x^4-1)^(1/2)*x^2+5*x^4-32*RootOf(6400*_Z^4-80*_Z^2-1)*(x^12-3*x^8+3*x^4
-1)^(1/4)*x+80*RootOf(6400*_Z^4-80*_Z^2-1)^2-1)/(320*RootOf(6400*_Z^4-80*_Z^2-1)^3*x^2-12*RootOf(6400*_Z^4-80*
_Z^2-1)*x^2+1)/(320*RootOf(6400*_Z^4-80*_Z^2-1)^3*x^2-12*RootOf(6400*_Z^4-80*_Z^2-1)*x^2-1)/(-1+x)^2/(1+x)^2/(
x^2+1)^2)*RootOf(6400*_Z^4-80*_Z^2-1))/(x^4-1)^(3/4)*((x^4-1)^3)^(1/4)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{8} - x^{4} + 1}{{\left (x^{8} - x^{4} - 1\right )} {\left (x^{4} - 1\right )}^{\frac {3}{4}} x^{2}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^8-x^4+1)/x^2/(x^4-1)^(3/4)/(x^8-x^4-1),x, algorithm="maxima")

[Out]

integrate((x^8 - x^4 + 1)/((x^8 - x^4 - 1)*(x^4 - 1)^(3/4)*x^2), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int -\frac {x^8-x^4+1}{x^2\,{\left (x^4-1\right )}^{3/4}\,\left (-x^8+x^4+1\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(x^8 - x^4 + 1)/(x^2*(x^4 - 1)^(3/4)*(x^4 - x^8 + 1)),x)

[Out]

int(-(x^8 - x^4 + 1)/(x^2*(x^4 - 1)^(3/4)*(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((x**8-x**4+1)/x**2/(x**4-1)**(3/4)/(x**8-x**4-1),x)

[Out]

Timed out

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