3.11.69 \(\int \frac {\sqrt [4]{-1+x^4} (-1+x^4+2 x^8)}{x^6 (1-x^4+x^8)} \, dx\)

Optimal. Leaf size=80 \[ \frac {3}{4} \text {RootSum}\left [\text {$\#$1}^8-\text {$\#$1}^4+1\& ,\frac {\text {$\#$1} \log \left (\sqrt [4]{x^4-1}-\text {$\#$1} x\right )-\text {$\#$1} \log (x)}{2 \text {$\#$1}^4-1}\& \right ]+\frac {\sqrt [4]{x^4-1} \left (1-x^4\right )}{5 x^5} \]

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Rubi [C]  time = 0.56, antiderivative size = 165, normalized size of antiderivative = 2.06, 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 {2 \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-i \sqrt {3}}\right )}{\sqrt {3} \left (\sqrt {3}+i\right ) \sqrt [4]{1-x^4}}-\frac {2 \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+i \sqrt {3}},x^4\right )}{\sqrt {3} \left (-\sqrt {3}+i\right ) \sqrt [4]{1-x^4}}-\frac {\left (x^4-1\right )^{5/4}}{5 x^5} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Int[((-1 + x^4)^(1/4)*(-1 + x^4 + 2*x^8))/(x^6*(1 - x^4 + x^8)),x]

[Out]

-1/5*(-1 + x^4)^(5/4)/x^5 + (2*x^3*(-1 + x^4)^(1/4)*AppellF1[3/4, -1/4, 1, 7/4, x^4, (2*x^4)/(1 - I*Sqrt[3])])
/(Sqrt[3]*(I + Sqrt[3])*(1 - x^4)^(1/4)) - (2*x^3*(-1 + x^4)^(1/4)*AppellF1[3/4, 1, -1/4, 7/4, (2*x^4)/(1 + I*
Sqrt[3]), x^4])/(Sqrt[3]*(I - Sqrt[3])*(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 {\sqrt [4]{-1+x^4} \left (-1+x^4+2 x^8\right )}{x^6 \left (1-x^4+x^8\right )} \, dx &=\int \left (-\frac {\sqrt [4]{-1+x^4}}{x^6}+\frac {3 x^2 \sqrt [4]{-1+x^4}}{1-x^4+x^8}\right ) \, dx\\ &=3 \int \frac {x^2 \sqrt [4]{-1+x^4}}{1-x^4+x^8} \, dx-\int \frac {\sqrt [4]{-1+x^4}}{x^6} \, dx\\ &=-\frac {\left (-1+x^4\right )^{5/4}}{5 x^5}+3 \int \left (\frac {2 i x^2 \sqrt [4]{-1+x^4}}{\sqrt {3} \left (1+i \sqrt {3}-2 x^4\right )}+\frac {2 i x^2 \sqrt [4]{-1+x^4}}{\sqrt {3} \left (-1+i \sqrt {3}+2 x^4\right )}\right ) \, dx\\ &=-\frac {\left (-1+x^4\right )^{5/4}}{5 x^5}+\left (2 i \sqrt {3}\right ) \int \frac {x^2 \sqrt [4]{-1+x^4}}{1+i \sqrt {3}-2 x^4} \, dx+\left (2 i \sqrt {3}\right ) \int \frac {x^2 \sqrt [4]{-1+x^4}}{-1+i \sqrt {3}+2 x^4} \, dx\\ &=-\frac {\left (-1+x^4\right )^{5/4}}{5 x^5}+\frac {\left (2 i \sqrt {3} \sqrt [4]{-1+x^4}\right ) \int \frac {x^2 \sqrt [4]{1-x^4}}{1+i \sqrt {3}-2 x^4} \, dx}{\sqrt [4]{1-x^4}}+\frac {\left (2 i \sqrt {3} \sqrt [4]{-1+x^4}\right ) \int \frac {x^2 \sqrt [4]{1-x^4}}{-1+i \sqrt {3}+2 x^4} \, dx}{\sqrt [4]{1-x^4}}\\ &=-\frac {\left (-1+x^4\right )^{5/4}}{5 x^5}+\frac {2 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-i \sqrt {3}}\right )}{\sqrt {3} \left (i+\sqrt {3}\right ) \sqrt [4]{1-x^4}}-\frac {2 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+i \sqrt {3}},x^4\right )}{\sqrt {3} \left (i-\sqrt {3}\right ) \sqrt [4]{1-x^4}}\\ \end {align*}

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Mathematica [F]  time = 6.69, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt [4]{-1+x^4} \left (-1+x^4+2 x^8\right )}{x^6 \left (1-x^4+x^8\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[((-1 + x^4)^(1/4)*(-1 + x^4 + 2*x^8))/(x^6*(1 - x^4 + x^8)),x]

[Out]

Integrate[((-1 + x^4)^(1/4)*(-1 + x^4 + 2*x^8))/(x^6*(1 - x^4 + x^8)), x]

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IntegrateAlgebraic [A]  time = 0.29, size = 80, normalized size = 1.00 \begin {gather*} \frac {\left (1-x^4\right ) \sqrt [4]{-1+x^4}}{5 x^5}+\frac {3}{4} \text {RootSum}\left [1-\text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-\log (x) \text {$\#$1}+\log \left (\sqrt [4]{-1+x^4}-x \text {$\#$1}\right ) \text {$\#$1}}{-1+2 \text {$\#$1}^4}\&\right ] \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((-1 + x^4)^(1/4)*(-1 + x^4 + 2*x^8))/(x^6*(1 - x^4 + x^8)),x]

[Out]

((1 - x^4)*(-1 + x^4)^(1/4))/(5*x^5) + (3*RootSum[1 - #1^4 + #1^8 & , (-(Log[x]*#1) + Log[(-1 + x^4)^(1/4) - x
*#1]*#1)/(-1 + 2*#1^4) & ])/4

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fricas [B]  time = 8.30, size = 3713, normalized size = 46.41

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^4-1)^(1/4)*(2*x^8+x^4-1)/x^6/(x^8-x^4+1),x, algorithm="fricas")

[Out]

-1/320*(20*sqrt(6)*sqrt(2)*x^5*sqrt(-4*sqrt(3) + 8)*arctan(-1/36*(7560*x^16 - 15120*x^12 + 15048*x^8 - 7488*x^
4 + 12*sqrt(6)*(x^4 - 1)^(3/4)*(sqrt(3)*sqrt(2)*(30*x^13 + 75*x^9 - 467*x^5 - 7*x) + 3*sqrt(2)*(15*x^13 - 75*x
^9 - 149*x^5 - 4*x))*sqrt(-4*sqrt(3) + 8) - 12*sqrt(6)*(x^4 - 1)^(1/4)*(sqrt(3)*sqrt(2)*(30*x^15 - 165*x^11 -
227*x^7 + 369*x^3) + 3*sqrt(2)*(15*x^15 + 30*x^11 - 254*x^7 + 213*x^3))*sqrt(-4*sqrt(3) + 8) + sqrt(6)*(2*sqrt
(6)*sqrt(x^4 - 1)*(sqrt(3)*sqrt(2)*(105*x^14 + 240*x^10 - 352*x^6 - 47*x^2) - 3*sqrt(2)*(60*x^14 + 135*x^10 -
191*x^6 + 27*x^2))*sqrt(-4*sqrt(3) + 8) + sqrt(6)*(2*sqrt(3)*sqrt(2)*(120*x^16 + 150*x^12 - 653*x^8 + 381*x^4
+ 1) - 3*sqrt(2)*(105*x^16 + 240*x^12 - 353*x^8 + 10*x^4 - 1))*sqrt(-4*sqrt(3) + 8) + 24*(240*x^13 + 435*x^9 -
883*x^5 - 2*sqrt(3)*(60*x^13 + 135*x^9 - 135*x^5 - x) + 3*x)*(x^4 - 1)^(3/4) + 24*(210*x^15 + 360*x^11 - 752*
x^7 + 178*x^3 - sqrt(3)*(105*x^15 + 240*x^11 - 240*x^7 - 103*x^3))*(x^4 - 1)^(1/4))*sqrt((42*x^8 - 42*x^4 + sq
rt(6)*(3*sqrt(2)*x^5 - sqrt(3)*sqrt(2)*(2*x^5 - x))*(x^4 - 1)^(3/4)*sqrt(-4*sqrt(3) + 8) - sqrt(6)*(x^4 - 1)^(
1/4)*(sqrt(3)*sqrt(2)*(2*x^7 - x^3) - 3*sqrt(2)*(x^7 - x^3))*sqrt(-4*sqrt(3) + 8) - 24*sqrt(3)*(x^8 - x^4) + 1
2*(4*x^6 - 2*x^2 - sqrt(3)*(2*x^6 - x^2))*sqrt(x^4 - 1) + 6)/(x^8 - x^4 + 1)) - 36*sqrt(3)*(105*x^16 - 210*x^1
2 + 435*x^8 - 330*x^4 + 1) + 144*(60*x^14 - 90*x^10 - 22*x^6 + 26*x^2 - 15*sqrt(3)*(2*x^14 - 3*x^10 + 3*x^6 -
x^2))*sqrt(x^4 - 1) - 72)/(225*x^16 - 450*x^12 - 2685*x^8 + 2910*x^4 + 1)) - 20*sqrt(6)*sqrt(2)*x^5*sqrt(-4*sq
rt(3) + 8)*arctan(-1/36*(7560*x^16 - 15120*x^12 + 15048*x^8 - 7488*x^4 - 12*sqrt(6)*(x^4 - 1)^(3/4)*(sqrt(3)*s
qrt(2)*(30*x^13 + 75*x^9 - 467*x^5 - 7*x) + 3*sqrt(2)*(15*x^13 - 75*x^9 - 149*x^5 - 4*x))*sqrt(-4*sqrt(3) + 8)
+ 12*sqrt(6)*(x^4 - 1)^(1/4)*(sqrt(3)*sqrt(2)*(30*x^15 - 165*x^11 - 227*x^7 + 369*x^3) + 3*sqrt(2)*(15*x^15 +
30*x^11 - 254*x^7 + 213*x^3))*sqrt(-4*sqrt(3) + 8) - sqrt(6)*(2*sqrt(6)*sqrt(x^4 - 1)*(sqrt(3)*sqrt(2)*(105*x
^14 + 240*x^10 - 352*x^6 - 47*x^2) - 3*sqrt(2)*(60*x^14 + 135*x^10 - 191*x^6 + 27*x^2))*sqrt(-4*sqrt(3) + 8) +
sqrt(6)*(2*sqrt(3)*sqrt(2)*(120*x^16 + 150*x^12 - 653*x^8 + 381*x^4 + 1) - 3*sqrt(2)*(105*x^16 + 240*x^12 - 3
53*x^8 + 10*x^4 - 1))*sqrt(-4*sqrt(3) + 8) - 24*(240*x^13 + 435*x^9 - 883*x^5 - 2*sqrt(3)*(60*x^13 + 135*x^9 -
135*x^5 - x) + 3*x)*(x^4 - 1)^(3/4) - 24*(210*x^15 + 360*x^11 - 752*x^7 + 178*x^3 - sqrt(3)*(105*x^15 + 240*x
^11 - 240*x^7 - 103*x^3))*(x^4 - 1)^(1/4))*sqrt((42*x^8 - 42*x^4 - sqrt(6)*(3*sqrt(2)*x^5 - sqrt(3)*sqrt(2)*(2
*x^5 - x))*(x^4 - 1)^(3/4)*sqrt(-4*sqrt(3) + 8) + sqrt(6)*(x^4 - 1)^(1/4)*(sqrt(3)*sqrt(2)*(2*x^7 - x^3) - 3*s
qrt(2)*(x^7 - x^3))*sqrt(-4*sqrt(3) + 8) - 24*sqrt(3)*(x^8 - x^4) + 12*(4*x^6 - 2*x^2 - sqrt(3)*(2*x^6 - x^2))
*sqrt(x^4 - 1) + 6)/(x^8 - x^4 + 1)) - 36*sqrt(3)*(105*x^16 - 210*x^12 + 435*x^8 - 330*x^4 + 1) + 144*(60*x^14
- 90*x^10 - 22*x^6 + 26*x^2 - 15*sqrt(3)*(2*x^14 - 3*x^10 + 3*x^6 - x^2))*sqrt(x^4 - 1) - 72)/(225*x^16 - 450
*x^12 - 2685*x^8 + 2910*x^4 + 1)) + 40*sqrt(6)*sqrt(2)*x^5*sqrt(sqrt(3) + 2)*arctan(-1/9*(1890*x^16 - 3780*x^1
2 + 3762*x^8 - 1872*x^4 + sqrt(3)*(12*(240*x^13 + 435*x^9 - 883*x^5 + 2*sqrt(3)*(60*x^13 + 135*x^9 - 135*x^5 -
x) + 3*x)*(x^4 - 1)^(3/4) - (2*sqrt(6)*sqrt(x^4 - 1)*(sqrt(3)*sqrt(2)*(105*x^14 + 240*x^10 - 352*x^6 - 47*x^2
) + 3*sqrt(2)*(60*x^14 + 135*x^10 - 191*x^6 + 27*x^2)) + sqrt(6)*(2*sqrt(3)*sqrt(2)*(120*x^16 + 150*x^12 - 653
*x^8 + 381*x^4 + 1) + 3*sqrt(2)*(105*x^16 + 240*x^12 - 353*x^8 + 10*x^4 - 1)))*sqrt(sqrt(3) + 2) + 12*(210*x^1
5 + 360*x^11 - 752*x^7 + 178*x^3 + sqrt(3)*(105*x^15 + 240*x^11 - 240*x^7 - 103*x^3))*(x^4 - 1)^(1/4))*sqrt((2
1*x^8 - 21*x^4 + 12*sqrt(3)*(x^8 - x^4) + 6*(4*x^6 - 2*x^2 + sqrt(3)*(2*x^6 - x^2))*sqrt(x^4 - 1) + (sqrt(6)*(
3*sqrt(2)*x^5 + sqrt(3)*sqrt(2)*(2*x^5 - x))*(x^4 - 1)^(3/4) + sqrt(6)*(x^4 - 1)^(1/4)*(sqrt(3)*sqrt(2)*(2*x^7
- x^3) + 3*sqrt(2)*(x^7 - x^3)))*sqrt(sqrt(3) + 2) + 3)/(x^8 - x^4 + 1)) + 9*sqrt(3)*(105*x^16 - 210*x^12 + 4
35*x^8 - 330*x^4 + 1) + 36*(60*x^14 - 90*x^10 - 22*x^6 + 26*x^2 + 15*sqrt(3)*(2*x^14 - 3*x^10 + 3*x^6 - x^2))*
sqrt(x^4 - 1) - 6*(sqrt(6)*(x^4 - 1)^(3/4)*(sqrt(3)*sqrt(2)*(30*x^13 + 75*x^9 - 467*x^5 - 7*x) - 3*sqrt(2)*(15
*x^13 - 75*x^9 - 149*x^5 - 4*x)) - sqrt(6)*(x^4 - 1)^(1/4)*(sqrt(3)*sqrt(2)*(30*x^15 - 165*x^11 - 227*x^7 + 36
9*x^3) - 3*sqrt(2)*(15*x^15 + 30*x^11 - 254*x^7 + 213*x^3)))*sqrt(sqrt(3) + 2) - 18)/(225*x^16 - 450*x^12 - 26
85*x^8 + 2910*x^4 + 1)) + 40*sqrt(6)*sqrt(2)*x^5*sqrt(sqrt(3) + 2)*arctan(1/9*(1890*x^16 - 3780*x^12 + 3762*x^
8 - 1872*x^4 + sqrt(3)*(12*(240*x^13 + 435*x^9 - 883*x^5 + 2*sqrt(3)*(60*x^13 + 135*x^9 - 135*x^5 - x) + 3*x)*
(x^4 - 1)^(3/4) + (2*sqrt(6)*sqrt(x^4 - 1)*(sqrt(3)*sqrt(2)*(105*x^14 + 240*x^10 - 352*x^6 - 47*x^2) + 3*sqrt(
2)*(60*x^14 + 135*x^10 - 191*x^6 + 27*x^2)) + sqrt(6)*(2*sqrt(3)*sqrt(2)*(120*x^16 + 150*x^12 - 653*x^8 + 381*
x^4 + 1) + 3*sqrt(2)*(105*x^16 + 240*x^12 - 353*x^8 + 10*x^4 - 1)))*sqrt(sqrt(3) + 2) + 12*(210*x^15 + 360*x^1
1 - 752*x^7 + 178*x^3 + sqrt(3)*(105*x^15 + 240*x^11 - 240*x^7 - 103*x^3))*(x^4 - 1)^(1/4))*sqrt((21*x^8 - 21*
x^4 + 12*sqrt(3)*(x^8 - x^4) + 6*(4*x^6 - 2*x^2 + sqrt(3)*(2*x^6 - x^2))*sqrt(x^4 - 1) - (sqrt(6)*(3*sqrt(2)*x
^5 + sqrt(3)*sqrt(2)*(2*x^5 - x))*(x^4 - 1)^(3/4) + sqrt(6)*(x^4 - 1)^(1/4)*(sqrt(3)*sqrt(2)*(2*x^7 - x^3) + 3
*sqrt(2)*(x^7 - x^3)))*sqrt(sqrt(3) + 2) + 3)/(x^8 - x^4 + 1)) + 9*sqrt(3)*(105*x^16 - 210*x^12 + 435*x^8 - 33
0*x^4 + 1) + 36*(60*x^14 - 90*x^10 - 22*x^6 + 26*x^2 + 15*sqrt(3)*(2*x^14 - 3*x^10 + 3*x^6 - x^2))*sqrt(x^4 -
1) + 6*(sqrt(6)*(x^4 - 1)^(3/4)*(sqrt(3)*sqrt(2)*(30*x^13 + 75*x^9 - 467*x^5 - 7*x) - 3*sqrt(2)*(15*x^13 - 75*
x^9 - 149*x^5 - 4*x)) - sqrt(6)*(x^4 - 1)^(1/4)*(sqrt(3)*sqrt(2)*(30*x^15 - 165*x^11 - 227*x^7 + 369*x^3) - 3*
sqrt(2)*(15*x^15 + 30*x^11 - 254*x^7 + 213*x^3)))*sqrt(sqrt(3) + 2) - 18)/(225*x^16 - 450*x^12 - 2685*x^8 + 29
10*x^4 + 1)) + 5*sqrt(6)*(sqrt(3)*sqrt(2)*x^5 + 2*sqrt(2)*x^5)*sqrt(-4*sqrt(3) + 8)*log(216*(42*x^8 - 42*x^4 +
sqrt(6)*(3*sqrt(2)*x^5 - sqrt(3)*sqrt(2)*(2*x^5 - x))*(x^4 - 1)^(3/4)*sqrt(-4*sqrt(3) + 8) - sqrt(6)*(x^4 - 1
)^(1/4)*(sqrt(3)*sqrt(2)*(2*x^7 - x^3) - 3*sqrt(2)*(x^7 - x^3))*sqrt(-4*sqrt(3) + 8) - 24*sqrt(3)*(x^8 - x^4)
+ 12*(4*x^6 - 2*x^2 - sqrt(3)*(2*x^6 - x^2))*sqrt(x^4 - 1) + 6)/(x^8 - x^4 + 1)) - 5*sqrt(6)*(sqrt(3)*sqrt(2)*
x^5 + 2*sqrt(2)*x^5)*sqrt(-4*sqrt(3) + 8)*log(216*(42*x^8 - 42*x^4 - sqrt(6)*(3*sqrt(2)*x^5 - sqrt(3)*sqrt(2)*
(2*x^5 - x))*(x^4 - 1)^(3/4)*sqrt(-4*sqrt(3) + 8) + sqrt(6)*(x^4 - 1)^(1/4)*(sqrt(3)*sqrt(2)*(2*x^7 - x^3) - 3
*sqrt(2)*(x^7 - x^3))*sqrt(-4*sqrt(3) + 8) - 24*sqrt(3)*(x^8 - x^4) + 12*(4*x^6 - 2*x^2 - sqrt(3)*(2*x^6 - x^2
))*sqrt(x^4 - 1) + 6)/(x^8 - x^4 + 1)) - 10*sqrt(6)*(sqrt(3)*sqrt(2)*x^5 - 2*sqrt(2)*x^5)*sqrt(sqrt(3) + 2)*lo
g(432*(21*x^8 - 21*x^4 + 12*sqrt(3)*(x^8 - x^4) + 6*(4*x^6 - 2*x^2 + sqrt(3)*(2*x^6 - x^2))*sqrt(x^4 - 1) + (s
qrt(6)*(3*sqrt(2)*x^5 + sqrt(3)*sqrt(2)*(2*x^5 - x))*(x^4 - 1)^(3/4) + sqrt(6)*(x^4 - 1)^(1/4)*(sqrt(3)*sqrt(2
)*(2*x^7 - x^3) + 3*sqrt(2)*(x^7 - x^3)))*sqrt(sqrt(3) + 2) + 3)/(x^8 - x^4 + 1)) + 10*sqrt(6)*(sqrt(3)*sqrt(2
)*x^5 - 2*sqrt(2)*x^5)*sqrt(sqrt(3) + 2)*log(432*(21*x^8 - 21*x^4 + 12*sqrt(3)*(x^8 - x^4) + 6*(4*x^6 - 2*x^2
+ sqrt(3)*(2*x^6 - x^2))*sqrt(x^4 - 1) - (sqrt(6)*(3*sqrt(2)*x^5 + sqrt(3)*sqrt(2)*(2*x^5 - x))*(x^4 - 1)^(3/4
) + sqrt(6)*(x^4 - 1)^(1/4)*(sqrt(3)*sqrt(2)*(2*x^7 - x^3) + 3*sqrt(2)*(x^7 - x^3)))*sqrt(sqrt(3) + 2) + 3)/(x
^8 - x^4 + 1)) + 64*(x^4 - 1)^(5/4))/x^5

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giac [B]  time = 0.42, size = 372, normalized size = 4.65 \begin {gather*} -\frac {1}{8} \, {\left (\sqrt {6} - 3 \, \sqrt {2}\right )} \arctan \left (\frac {\sqrt {6} - \sqrt {2} + \frac {4 \, {\left (x^{4} - 1\right )}^{\frac {1}{4}}}{x}}{\sqrt {6} + \sqrt {2}}\right ) - \frac {1}{8} \, {\left (\sqrt {6} - 3 \, \sqrt {2}\right )} \arctan \left (-\frac {\sqrt {6} - \sqrt {2} - \frac {4 \, {\left (x^{4} - 1\right )}^{\frac {1}{4}}}{x}}{\sqrt {6} + \sqrt {2}}\right ) - \frac {1}{8} \, {\left (\sqrt {6} + 3 \, \sqrt {2}\right )} \arctan \left (\frac {\sqrt {6} + \sqrt {2} + \frac {4 \, {\left (x^{4} - 1\right )}^{\frac {1}{4}}}{x}}{\sqrt {6} - \sqrt {2}}\right ) - \frac {1}{8} \, {\left (\sqrt {6} + 3 \, \sqrt {2}\right )} \arctan \left (-\frac {\sqrt {6} + \sqrt {2} - \frac {4 \, {\left (x^{4} - 1\right )}^{\frac {1}{4}}}{x}}{\sqrt {6} - \sqrt {2}}\right ) - \frac {1}{16} \, {\left (\sqrt {6} - 3 \, \sqrt {2}\right )} \log \left (\frac {{\left (x^{4} - 1\right )}^{\frac {1}{4}} {\left (\sqrt {6} + \sqrt {2}\right )}}{2 \, x} + \frac {\sqrt {x^{4} - 1}}{x^{2}} + 1\right ) + \frac {1}{16} \, {\left (\sqrt {6} - 3 \, \sqrt {2}\right )} \log \left (-\frac {{\left (x^{4} - 1\right )}^{\frac {1}{4}} {\left (\sqrt {6} + \sqrt {2}\right )}}{2 \, x} + \frac {\sqrt {x^{4} - 1}}{x^{2}} + 1\right ) - \frac {1}{16} \, {\left (\sqrt {6} + 3 \, \sqrt {2}\right )} \log \left (\frac {{\left (x^{4} - 1\right )}^{\frac {1}{4}} {\left (\sqrt {6} - \sqrt {2}\right )}}{2 \, x} + \frac {\sqrt {x^{4} - 1}}{x^{2}} + 1\right ) + \frac {1}{16} \, {\left (\sqrt {6} + 3 \, \sqrt {2}\right )} \log \left (-\frac {{\left (x^{4} - 1\right )}^{\frac {1}{4}} {\left (\sqrt {6} - \sqrt {2}\right )}}{2 \, x} + \frac {\sqrt {x^{4} - 1}}{x^{2}} + 1\right ) - \frac {{\left (x^{4} - 1\right )}^{\frac {1}{4}} {\left (\frac {1}{x^{4}} - 1\right )}}{5 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^4-1)^(1/4)*(2*x^8+x^4-1)/x^6/(x^8-x^4+1),x, algorithm="giac")

[Out]

-1/8*(sqrt(6) - 3*sqrt(2))*arctan((sqrt(6) - sqrt(2) + 4*(x^4 - 1)^(1/4)/x)/(sqrt(6) + sqrt(2))) - 1/8*(sqrt(6
) - 3*sqrt(2))*arctan(-(sqrt(6) - sqrt(2) - 4*(x^4 - 1)^(1/4)/x)/(sqrt(6) + sqrt(2))) - 1/8*(sqrt(6) + 3*sqrt(
2))*arctan((sqrt(6) + sqrt(2) + 4*(x^4 - 1)^(1/4)/x)/(sqrt(6) - sqrt(2))) - 1/8*(sqrt(6) + 3*sqrt(2))*arctan(-
(sqrt(6) + sqrt(2) - 4*(x^4 - 1)^(1/4)/x)/(sqrt(6) - sqrt(2))) - 1/16*(sqrt(6) - 3*sqrt(2))*log(1/2*(x^4 - 1)^
(1/4)*(sqrt(6) + sqrt(2))/x + sqrt(x^4 - 1)/x^2 + 1) + 1/16*(sqrt(6) - 3*sqrt(2))*log(-1/2*(x^4 - 1)^(1/4)*(sq
rt(6) + sqrt(2))/x + sqrt(x^4 - 1)/x^2 + 1) - 1/16*(sqrt(6) + 3*sqrt(2))*log(1/2*(x^4 - 1)^(1/4)*(sqrt(6) - sq
rt(2))/x + sqrt(x^4 - 1)/x^2 + 1) + 1/16*(sqrt(6) + 3*sqrt(2))*log(-1/2*(x^4 - 1)^(1/4)*(sqrt(6) - sqrt(2))/x
+ sqrt(x^4 - 1)/x^2 + 1) - 1/5*(x^4 - 1)^(1/4)*(1/x^4 - 1)/x

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maple [B]  time = 20.91, size = 3975, normalized size = 49.69 \[\text {output too large to display}\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^4-1)^(1/4)*(2*x^8+x^4-1)/x^6/(x^8-x^4+1),x)

[Out]

-1/5*(x^8-2*x^4+1)/x^5/(x^4-1)^(3/4)+(331776*RootOf(5308416*_Z^8-2304*_Z^4+1)^7*ln(-(127401984*RootOf(5308416*
_Z^8-2304*_Z^4+1)^10*x^12-254803968*RootOf(5308416*_Z^8-2304*_Z^4+1)^10*x^8-165888*x^12*RootOf(5308416*_Z^8-23
04*_Z^4+1)^6-9216*RootOf(5308416*_Z^8-2304*_Z^4+1)^5*(x^12-3*x^8+3*x^4-1)^(1/4)*x^9+127401984*RootOf(5308416*_
Z^8-2304*_Z^4+1)^10*x^4+387072*x^8*RootOf(5308416*_Z^8-2304*_Z^4+1)^6+48*RootOf(5308416*_Z^8-2304*_Z^4+1)^2*x^
12-442368*(x^12-3*x^8+3*x^4-1)^(3/4)*RootOf(5308416*_Z^8-2304*_Z^4+1)^7*x^3+2304*(x^12-3*x^8+3*x^4-1)^(1/2)*Ro
otOf(5308416*_Z^8-2304*_Z^4+1)^4*x^6+18432*RootOf(5308416*_Z^8-2304*_Z^4+1)^5*(x^12-3*x^8+3*x^4-1)^(1/4)*x^5-4
*RootOf(5308416*_Z^8-2304*_Z^4+1)*(x^12-3*x^8+3*x^4-1)^(1/4)*x^9-276480*x^4*RootOf(5308416*_Z^8-2304*_Z^4+1)^6
-120*RootOf(5308416*_Z^8-2304*_Z^4+1)^2*x^8+384*RootOf(5308416*_Z^8-2304*_Z^4+1)^3*(x^12-3*x^8+3*x^4-1)^(3/4)*
x^3-2304*(x^12-3*x^8+3*x^4-1)^(1/2)*RootOf(5308416*_Z^8-2304*_Z^4+1)^4*x^2-(x^12-3*x^8+3*x^4-1)^(1/2)*x^6-9216
*RootOf(5308416*_Z^8-2304*_Z^4+1)^5*(x^12-3*x^8+3*x^4-1)^(1/4)*x+8*RootOf(5308416*_Z^8-2304*_Z^4+1)*(x^12-3*x^
8+3*x^4-1)^(1/4)*x^5+55296*RootOf(5308416*_Z^8-2304*_Z^4+1)^6+96*x^4*RootOf(5308416*_Z^8-2304*_Z^4+1)^2+(x^12-
3*x^8+3*x^4-1)^(1/2)*x^2-4*RootOf(5308416*_Z^8-2304*_Z^4+1)*(x^12-3*x^8+3*x^4-1)^(1/4)*x-24*RootOf(5308416*_Z^
8-2304*_Z^4+1)^2)/(442368*RootOf(5308416*_Z^8-2304*_Z^4+1)^7*x-384*RootOf(5308416*_Z^8-2304*_Z^4+1)^3*x+1)/(18
432*RootOf(5308416*_Z^8-2304*_Z^4+1)^5*x-4*x*RootOf(5308416*_Z^8-2304*_Z^4+1)+1)/(442368*RootOf(5308416*_Z^8-2
304*_Z^4+1)^7*x-384*RootOf(5308416*_Z^8-2304*_Z^4+1)^3*x-1)/(18432*RootOf(5308416*_Z^8-2304*_Z^4+1)^5*x-4*x*Ro
otOf(5308416*_Z^8-2304*_Z^4+1)-1)/(110592*x*RootOf(5308416*_Z^8-2304*_Z^4+1)^6+1)^2/(-1+x)^2/(1+x)^2/(110592*x
*RootOf(5308416*_Z^8-2304*_Z^4+1)^6-1)^2)-331776*RootOf(5308416*_Z^8-2304*_Z^4+1)^7*ln((-13824*RootOf(5308416*
_Z^8-2304*_Z^4+1)^5*x^12+2304*(x^12-3*x^8+3*x^4-1)^(1/4)*RootOf(5308416*_Z^8-2304*_Z^4+1)^4*x^9+27648*RootOf(5
308416*_Z^8-2304*_Z^4+1)^5*x^8-6*RootOf(5308416*_Z^8-2304*_Z^4+1)*x^12+110592*(x^12-3*x^8+3*x^4-1)^(3/4)*RootO
f(5308416*_Z^8-2304*_Z^4+1)^6*x^3-576*(x^12-3*x^8+3*x^4-1)^(1/2)*RootOf(5308416*_Z^8-2304*_Z^4+1)^3*x^6-4608*(
x^12-3*x^8+3*x^4-1)^(1/4)*RootOf(5308416*_Z^8-2304*_Z^4+1)^4*x^5+(x^12-3*x^8+3*x^4-1)^(1/4)*x^9-13824*RootOf(5
308416*_Z^8-2304*_Z^4+1)^5*x^4+18*RootOf(5308416*_Z^8-2304*_Z^4+1)*x^8+48*RootOf(5308416*_Z^8-2304*_Z^4+1)^2*(
x^12-3*x^8+3*x^4-1)^(3/4)*x^3+576*(x^12-3*x^8+3*x^4-1)^(1/2)*RootOf(5308416*_Z^8-2304*_Z^4+1)^3*x^2+2304*(x^12
-3*x^8+3*x^4-1)^(1/4)*RootOf(5308416*_Z^8-2304*_Z^4+1)^4*x-2*(x^12-3*x^8+3*x^4-1)^(1/4)*x^5-18*RootOf(5308416*
_Z^8-2304*_Z^4+1)*x^4+(x^12-3*x^8+3*x^4-1)^(1/4)*x+6*RootOf(5308416*_Z^8-2304*_Z^4+1))/(442368*RootOf(5308416*
_Z^8-2304*_Z^4+1)^7*x+192*RootOf(5308416*_Z^8-2304*_Z^4+1)^3*x+1)/(9216*RootOf(5308416*_Z^8-2304*_Z^4+1)^5*x-8
*x*RootOf(5308416*_Z^8-2304*_Z^4+1)+1)/(9216*RootOf(5308416*_Z^8-2304*_Z^4+1)^5*x-8*x*RootOf(5308416*_Z^8-2304
*_Z^4+1)-1)/(442368*RootOf(5308416*_Z^8-2304*_Z^4+1)^7*x+192*RootOf(5308416*_Z^8-2304*_Z^4+1)^3*x-1)/(110592*x
*RootOf(5308416*_Z^8-2304*_Z^4+1)^6+1)^2/(-1+x)^2/(1+x)^2/(110592*x*RootOf(5308416*_Z^8-2304*_Z^4+1)^6-1)^2)-6
912*RootOf(5308416*_Z^8-2304*_Z^4+1)^5*ln((127401984*RootOf(5308416*_Z^8-2304*_Z^4+1)^10*x^12-254803968*RootOf
(5308416*_Z^8-2304*_Z^4+1)^10*x^8-165888*x^12*RootOf(5308416*_Z^8-2304*_Z^4+1)^6-884736*RootOf(5308416*_Z^8-23
04*_Z^4+1)^7*(x^12-3*x^8+3*x^4-1)^(1/4)*x^9+127401984*RootOf(5308416*_Z^8-2304*_Z^4+1)^10*x^4+387072*x^8*RootO
f(5308416*_Z^8-2304*_Z^4+1)^6+48*RootOf(5308416*_Z^8-2304*_Z^4+1)^2*x^12+1769472*RootOf(5308416*_Z^8-2304*_Z^4
+1)^7*(x^12-3*x^8+3*x^4-1)^(1/4)*x^5+192*RootOf(5308416*_Z^8-2304*_Z^4+1)^3*(x^12-3*x^8+3*x^4-1)^(1/4)*x^9-230
4*(x^12-3*x^8+3*x^4-1)^(1/2)*RootOf(5308416*_Z^8-2304*_Z^4+1)^4*x^6-18432*RootOf(5308416*_Z^8-2304*_Z^4+1)^5*(
x^12-3*x^8+3*x^4-1)^(3/4)*x^3-276480*x^4*RootOf(5308416*_Z^8-2304*_Z^4+1)^6-120*RootOf(5308416*_Z^8-2304*_Z^4+
1)^2*x^8-884736*RootOf(5308416*_Z^8-2304*_Z^4+1)^7*(x^12-3*x^8+3*x^4-1)^(1/4)*x-384*RootOf(5308416*_Z^8-2304*_
Z^4+1)^3*(x^12-3*x^8+3*x^4-1)^(1/4)*x^5+2304*(x^12-3*x^8+3*x^4-1)^(1/2)*RootOf(5308416*_Z^8-2304*_Z^4+1)^4*x^2
+(x^12-3*x^8+3*x^4-1)^(1/2)*x^6+4*RootOf(5308416*_Z^8-2304*_Z^4+1)*(x^12-3*x^8+3*x^4-1)^(3/4)*x^3+55296*RootOf
(5308416*_Z^8-2304*_Z^4+1)^6+96*x^4*RootOf(5308416*_Z^8-2304*_Z^4+1)^2+192*RootOf(5308416*_Z^8-2304*_Z^4+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^2-24*RootOf(5308416*_Z^8-2304*_Z^4+1)^2)/(442368*Roo
tOf(5308416*_Z^8-2304*_Z^4+1)^7*x-384*RootOf(5308416*_Z^8-2304*_Z^4+1)^3*x+1)/(18432*RootOf(5308416*_Z^8-2304*
_Z^4+1)^5*x-4*x*RootOf(5308416*_Z^8-2304*_Z^4+1)+1)/(442368*RootOf(5308416*_Z^8-2304*_Z^4+1)^7*x-384*RootOf(53
08416*_Z^8-2304*_Z^4+1)^3*x-1)/(18432*RootOf(5308416*_Z^8-2304*_Z^4+1)^5*x-4*x*RootOf(5308416*_Z^8-2304*_Z^4+1
)-1)/(110592*x*RootOf(5308416*_Z^8-2304*_Z^4+1)^6+1)^2/(-1+x)^2/(1+x)^2/(110592*x*RootOf(5308416*_Z^8-2304*_Z^
4+1)^6-1)^2)-144*RootOf(5308416*_Z^8-2304*_Z^4+1)^3*ln(-(127401984*RootOf(5308416*_Z^8-2304*_Z^4+1)^10*x^12-25
4803968*RootOf(5308416*_Z^8-2304*_Z^4+1)^10*x^8-165888*x^12*RootOf(5308416*_Z^8-2304*_Z^4+1)^6-9216*RootOf(530
8416*_Z^8-2304*_Z^4+1)^5*(x^12-3*x^8+3*x^4-1)^(1/4)*x^9+127401984*RootOf(5308416*_Z^8-2304*_Z^4+1)^10*x^4+3870
72*x^8*RootOf(5308416*_Z^8-2304*_Z^4+1)^6+48*RootOf(5308416*_Z^8-2304*_Z^4+1)^2*x^12-442368*(x^12-3*x^8+3*x^4-
1)^(3/4)*RootOf(5308416*_Z^8-2304*_Z^4+1)^7*x^3+2304*(x^12-3*x^8+3*x^4-1)^(1/2)*RootOf(5308416*_Z^8-2304*_Z^4+
1)^4*x^6+18432*RootOf(5308416*_Z^8-2304*_Z^4+1)^5*(x^12-3*x^8+3*x^4-1)^(1/4)*x^5-4*RootOf(5308416*_Z^8-2304*_Z
^4+1)*(x^12-3*x^8+3*x^4-1)^(1/4)*x^9-276480*x^4*RootOf(5308416*_Z^8-2304*_Z^4+1)^6-120*RootOf(5308416*_Z^8-230
4*_Z^4+1)^2*x^8+384*RootOf(5308416*_Z^8-2304*_Z^4+1)^3*(x^12-3*x^8+3*x^4-1)^(3/4)*x^3-2304*(x^12-3*x^8+3*x^4-1
)^(1/2)*RootOf(5308416*_Z^8-2304*_Z^4+1)^4*x^2-(x^12-3*x^8+3*x^4-1)^(1/2)*x^6-9216*RootOf(5308416*_Z^8-2304*_Z
^4+1)^5*(x^12-3*x^8+3*x^4-1)^(1/4)*x+8*RootOf(5308416*_Z^8-2304*_Z^4+1)*(x^12-3*x^8+3*x^4-1)^(1/4)*x^5+55296*R
ootOf(5308416*_Z^8-2304*_Z^4+1)^6+96*x^4*RootOf(5308416*_Z^8-2304*_Z^4+1)^2+(x^12-3*x^8+3*x^4-1)^(1/2)*x^2-4*R
ootOf(5308416*_Z^8-2304*_Z^4+1)*(x^12-3*x^8+3*x^4-1)^(1/4)*x-24*RootOf(5308416*_Z^8-2304*_Z^4+1)^2)/(442368*Ro
otOf(5308416*_Z^8-2304*_Z^4+1)^7*x-384*RootOf(5308416*_Z^8-2304*_Z^4+1)^3*x+1)/(18432*RootOf(5308416*_Z^8-2304
*_Z^4+1)^5*x-4*x*RootOf(5308416*_Z^8-2304*_Z^4+1)+1)/(442368*RootOf(5308416*_Z^8-2304*_Z^4+1)^7*x-384*RootOf(5
308416*_Z^8-2304*_Z^4+1)^3*x-1)/(18432*RootOf(5308416*_Z^8-2304*_Z^4+1)^5*x-4*x*RootOf(5308416*_Z^8-2304*_Z^4+
1)-1)/(110592*x*RootOf(5308416*_Z^8-2304*_Z^4+1)^6+1)^2/(-1+x)^2/(1+x)^2/(110592*x*RootOf(5308416*_Z^8-2304*_Z
^4+1)^6-1)^2)+3*RootOf(5308416*_Z^8-2304*_Z^4+1)*ln(-(-6912*RootOf(5308416*_Z^8-2304*_Z^4+1)^5*x^12+110592*(x^
12-3*x^8+3*x^4-1)^(1/4)*RootOf(5308416*_Z^8-2304*_Z^4+1)^6*x^9+13824*RootOf(5308416*_Z^8-2304*_Z^4+1)^5*x^8-3*
RootOf(5308416*_Z^8-2304*_Z^4+1)*x^12-221184*(x^12-3*x^8+3*x^4-1)^(1/4)*RootOf(5308416*_Z^8-2304*_Z^4+1)^6*x^5
-24*(x^12-3*x^8+3*x^4-1)^(1/4)*RootOf(5308416*_Z^8-2304*_Z^4+1)^2*x^9+288*(x^12-3*x^8+3*x^4-1)^(1/2)*RootOf(53
08416*_Z^8-2304*_Z^4+1)^3*x^6-1152*(x^12-3*x^8+3*x^4-1)^(3/4)*RootOf(5308416*_Z^8-2304*_Z^4+1)^4*x^3-6912*Root
Of(5308416*_Z^8-2304*_Z^4+1)^5*x^4+9*RootOf(5308416*_Z^8-2304*_Z^4+1)*x^8+110592*(x^12-3*x^8+3*x^4-1)^(1/4)*Ro
otOf(5308416*_Z^8-2304*_Z^4+1)^6*x+48*(x^12-3*x^8+3*x^4-1)^(1/4)*RootOf(5308416*_Z^8-2304*_Z^4+1)^2*x^5-288*(x
^12-3*x^8+3*x^4-1)^(1/2)*RootOf(5308416*_Z^8-2304*_Z^4+1)^3*x^2+(x^12-3*x^8+3*x^4-1)^(3/4)*x^3-9*RootOf(530841
6*_Z^8-2304*_Z^4+1)*x^4-24*(x^12-3*x^8+3*x^4-1)^(1/4)*RootOf(5308416*_Z^8-2304*_Z^4+1)^2*x+3*RootOf(5308416*_Z
^8-2304*_Z^4+1))/(442368*RootOf(5308416*_Z^8-2304*_Z^4+1)^7*x+192*RootOf(5308416*_Z^8-2304*_Z^4+1)^3*x+1)/(921
6*RootOf(5308416*_Z^8-2304*_Z^4+1)^5*x-8*x*RootOf(5308416*_Z^8-2304*_Z^4+1)+1)/(9216*RootOf(5308416*_Z^8-2304*
_Z^4+1)^5*x-8*x*RootOf(5308416*_Z^8-2304*_Z^4+1)-1)/(442368*RootOf(5308416*_Z^8-2304*_Z^4+1)^7*x+192*RootOf(53
08416*_Z^8-2304*_Z^4+1)^3*x-1)/(110592*x*RootOf(5308416*_Z^8-2304*_Z^4+1)^6+1)^2/(-1+x)^2/(1+x)^2/(110592*x*Ro
otOf(5308416*_Z^8-2304*_Z^4+1)^6-1)^2))/(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 {{\left (2 \, x^{8} + x^{4} - 1\right )} {\left (x^{4} - 1\right )}^{\frac {1}{4}}}{{\left (x^{8} - x^{4} + 1\right )} x^{6}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^4-1)^(1/4)*(2*x^8+x^4-1)/x^6/(x^8-x^4+1),x, algorithm="maxima")

[Out]

integrate((2*x^8 + x^4 - 1)*(x^4 - 1)^(1/4)/((x^8 - x^4 + 1)*x^6), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (x^4-1\right )}^{1/4}\,\left (2\,x^8+x^4-1\right )}{x^6\,\left (x^8-x^4+1\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((x^4 - 1)^(1/4)*(x^4 + 2*x^8 - 1))/(x^6*(x^8 - x^4 + 1)),x)

[Out]

int(((x^4 - 1)^(1/4)*(x^4 + 2*x^8 - 1))/(x^6*(x^8 - x^4 + 1)), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt [4]{\left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right )} \left (x^{4} + 1\right ) \left (2 x^{4} - 1\right )}{x^{6} \left (x^{8} - x^{4} + 1\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**4-1)**(1/4)*(2*x**8+x**4-1)/x**6/(x**8-x**4+1),x)

[Out]

Integral(((x - 1)*(x + 1)*(x**2 + 1))**(1/4)*(x**4 + 1)*(2*x**4 - 1)/(x**6*(x**8 - x**4 + 1)), x)

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