3.6.87 \(\int \frac {-1+x^{16}}{\sqrt {-1+x^4} (1-x^8+x^{16})} \, dx\)

Optimal. Leaf size=45 \[ \frac {1}{8} \text {RootSum}\left [\text {$\#$1}^8+4 \text {$\#$1}^4+1\& ,\frac {\log \left (\sqrt {x^4-1}-\text {$\#$1} x\right )-\log (x)}{\text {$\#$1}}\& \right ] \]

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Rubi [C]  time = 10.75, antiderivative size = 2783, normalized size of antiderivative = 61.84, number of steps used = 319, number of rules used = 14, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.518, Rules used = {1586, 6728, 1429, 406, 222, 409, 1215, 1457, 540, 253, 538, 537, 1519, 6725}

result too large to display

Warning: Unable to verify antiderivative.

[In]

Int[(-1 + x^16)/(Sqrt[-1 + x^4]*(1 - x^8 + x^16)),x]

[Out]

-1/2*((I - Sqrt[3])*Sqrt[-1 + x^2]*Sqrt[1 + x^2]*EllipticF[ArcSin[(Sqrt[2]*x)/Sqrt[-1 + x^2]], 1/2])/(Sqrt[6]*
Sqrt[-1 + x^4]) + ((I + Sqrt[3])*Sqrt[-1 + x^2]*Sqrt[1 + x^2]*EllipticF[ArcSin[(Sqrt[2]*x)/Sqrt[-1 + x^2]], 1/
2])/(2*Sqrt[6]*Sqrt[-1 + x^4]) - ((I/4)*(1 + 1/Sqrt[(1 - I*Sqrt[3])/2])*Sqrt[1 - x^2]*Sqrt[1 + x^2]*EllipticPi
[-Sqrt[(-I - Sqrt[3])/2], ArcSin[x], -1])/(Sqrt[3]*Sqrt[-1 + x^4]) - ((3 + I*Sqrt[3])*(1 + 1/Sqrt[(1 - I*Sqrt[
3])/2])*Sqrt[1 - x^2]*Sqrt[1 + x^2]*EllipticPi[-Sqrt[(-I - Sqrt[3])/2], ArcSin[x], -1])/(24*Sqrt[-1 + x^4]) +
((I/8)*(2 + Sqrt[2 - (2*I)*Sqrt[3]])*Sqrt[1 - x^2]*Sqrt[1 + x^2]*EllipticPi[-Sqrt[(-I - Sqrt[3])/2], ArcSin[x]
, -1])/(Sqrt[3]*Sqrt[-1 + x^4]) - ((2 + Sqrt[2 - (2*I)*Sqrt[3]])*Sqrt[1 - x^2]*Sqrt[1 + x^2]*EllipticPi[-Sqrt[
(-I - Sqrt[3])/2], ArcSin[x], -1])/(4*Sqrt[3]*(I + Sqrt[3])*Sqrt[-1 + x^4]) - ((I/4)*(1 + 1/Sqrt[(1 - I*Sqrt[3
])/2])*Sqrt[1 - x^2]*Sqrt[1 + x^2]*EllipticPi[Sqrt[(-I - Sqrt[3])/2], ArcSin[x], -1])/(Sqrt[3]*Sqrt[-1 + x^4])
- ((3 + I*Sqrt[3])*(1 + 1/Sqrt[(1 - I*Sqrt[3])/2])*Sqrt[1 - x^2]*Sqrt[1 + x^2]*EllipticPi[Sqrt[(-I - Sqrt[3])
/2], ArcSin[x], -1])/(24*Sqrt[-1 + x^4]) + ((I/8)*(2 + Sqrt[2 - (2*I)*Sqrt[3]])*Sqrt[1 - x^2]*Sqrt[1 + x^2]*El
lipticPi[Sqrt[(-I - Sqrt[3])/2], ArcSin[x], -1])/(Sqrt[3]*Sqrt[-1 + x^4]) - ((2 + Sqrt[2 - (2*I)*Sqrt[3]])*Sqr
t[1 - x^2]*Sqrt[1 + x^2]*EllipticPi[Sqrt[(-I - Sqrt[3])/2], ArcSin[x], -1])/(4*Sqrt[3]*(I + Sqrt[3])*Sqrt[-1 +
x^4]) + ((I/4)*(1 + 1/Sqrt[(1 + I*Sqrt[3])/2])*Sqrt[1 - x^2]*Sqrt[1 + x^2]*EllipticPi[-Sqrt[(I - Sqrt[3])/2],
ArcSin[x], -1])/(Sqrt[3]*Sqrt[-1 + x^4]) - ((3 - I*Sqrt[3])*(1 + 1/Sqrt[(1 + I*Sqrt[3])/2])*Sqrt[1 - x^2]*Sqr
t[1 + x^2]*EllipticPi[-Sqrt[(I - Sqrt[3])/2], ArcSin[x], -1])/(24*Sqrt[-1 + x^4]) - ((I/8)*(2 + Sqrt[2 + (2*I)
*Sqrt[3]])*Sqrt[1 - x^2]*Sqrt[1 + x^2]*EllipticPi[-Sqrt[(I - Sqrt[3])/2], ArcSin[x], -1])/(Sqrt[3]*Sqrt[-1 + x
^4]) + ((2 + Sqrt[2 + (2*I)*Sqrt[3]])*Sqrt[1 - x^2]*Sqrt[1 + x^2]*EllipticPi[-Sqrt[(I - Sqrt[3])/2], ArcSin[x]
, -1])/(4*Sqrt[3]*(I - Sqrt[3])*Sqrt[-1 + x^4]) + ((I/4)*(1 + 1/Sqrt[(1 + I*Sqrt[3])/2])*Sqrt[1 - x^2]*Sqrt[1
+ x^2]*EllipticPi[Sqrt[(I - Sqrt[3])/2], ArcSin[x], -1])/(Sqrt[3]*Sqrt[-1 + x^4]) - ((3 - I*Sqrt[3])*(1 + 1/Sq
rt[(1 + I*Sqrt[3])/2])*Sqrt[1 - x^2]*Sqrt[1 + x^2]*EllipticPi[Sqrt[(I - Sqrt[3])/2], ArcSin[x], -1])/(24*Sqrt[
-1 + x^4]) - ((I/8)*(2 + Sqrt[2 + (2*I)*Sqrt[3]])*Sqrt[1 - x^2]*Sqrt[1 + x^2]*EllipticPi[Sqrt[(I - Sqrt[3])/2]
, ArcSin[x], -1])/(Sqrt[3]*Sqrt[-1 + x^4]) + ((2 + Sqrt[2 + (2*I)*Sqrt[3]])*Sqrt[1 - x^2]*Sqrt[1 + x^2]*Ellipt
icPi[Sqrt[(I - Sqrt[3])/2], ArcSin[x], -1])/(4*Sqrt[3]*(I - Sqrt[3])*Sqrt[-1 + x^4]) + ((I/24)*(3 - (2 - I)*Sq
rt[3])*Sqrt[1 - x^2]*Sqrt[1 + x^2]*EllipticPi[-((1 - I*Sqrt[3])/2)^(-1/4), ArcSin[x], -1])/Sqrt[-1 + x^4] - ((
1/24 - I/24)*(3 - Sqrt[3])*Sqrt[1 - x^2]*Sqrt[1 + x^2]*EllipticPi[-((1 - I*Sqrt[3])/2)^(-1/4), ArcSin[x], -1])
/Sqrt[-1 + x^4] + ((I/8)*(2 - Sqrt[2 - (2*I)*Sqrt[3]])*Sqrt[1 - x^2]*Sqrt[1 + x^2]*EllipticPi[-((1 - I*Sqrt[3]
)/2)^(-1/4), ArcSin[x], -1])/(Sqrt[3]*Sqrt[-1 + x^4]) - ((2 - Sqrt[2 - (2*I)*Sqrt[3]])*Sqrt[1 - x^2]*Sqrt[1 +
x^2]*EllipticPi[-((1 - I*Sqrt[3])/2)^(-1/4), ArcSin[x], -1])/(4*Sqrt[3]*(I + Sqrt[3])*Sqrt[-1 + x^4]) + ((I/24
)*(3 - (2 - I)*Sqrt[3])*Sqrt[1 - x^2]*Sqrt[1 + x^2]*EllipticPi[((1 - I*Sqrt[3])/2)^(-1/4), ArcSin[x], -1])/Sqr
t[-1 + x^4] - ((1/24 - I/24)*(3 - Sqrt[3])*Sqrt[1 - x^2]*Sqrt[1 + x^2]*EllipticPi[((1 - I*Sqrt[3])/2)^(-1/4),
ArcSin[x], -1])/Sqrt[-1 + x^4] + ((I/8)*(2 - Sqrt[2 - (2*I)*Sqrt[3]])*Sqrt[1 - x^2]*Sqrt[1 + x^2]*EllipticPi[(
(1 - I*Sqrt[3])/2)^(-1/4), ArcSin[x], -1])/(Sqrt[3]*Sqrt[-1 + x^4]) - ((2 - Sqrt[2 - (2*I)*Sqrt[3]])*Sqrt[1 -
x^2]*Sqrt[1 + x^2]*EllipticPi[((1 - I*Sqrt[3])/2)^(-1/4), ArcSin[x], -1])/(4*Sqrt[3]*(I + Sqrt[3])*Sqrt[-1 + x
^4]) - ((I/24)*(3 - (2 + I)*Sqrt[3])*Sqrt[1 - x^2]*Sqrt[1 + x^2]*EllipticPi[-((1 + I*Sqrt[3])/2)^(-1/4), ArcSi
n[x], -1])/Sqrt[-1 + x^4] - ((1/24 + I/24)*(3 - Sqrt[3])*Sqrt[1 - x^2]*Sqrt[1 + x^2]*EllipticPi[-((1 + I*Sqrt[
3])/2)^(-1/4), ArcSin[x], -1])/Sqrt[-1 + x^4] - ((I/8)*(2 - Sqrt[2 + (2*I)*Sqrt[3]])*Sqrt[1 - x^2]*Sqrt[1 + x^
2]*EllipticPi[-((1 + I*Sqrt[3])/2)^(-1/4), ArcSin[x], -1])/(Sqrt[3]*Sqrt[-1 + x^4]) + ((2 - Sqrt[2 + (2*I)*Sqr
t[3]])*Sqrt[1 - x^2]*Sqrt[1 + x^2]*EllipticPi[-((1 + I*Sqrt[3])/2)^(-1/4), ArcSin[x], -1])/(4*Sqrt[3]*(I - Sqr
t[3])*Sqrt[-1 + x^4]) - ((I/24)*(3 - (2 + I)*Sqrt[3])*Sqrt[1 - x^2]*Sqrt[1 + x^2]*EllipticPi[((1 + I*Sqrt[3])/
2)^(-1/4), ArcSin[x], -1])/Sqrt[-1 + x^4] - ((1/24 + I/24)*(3 - Sqrt[3])*Sqrt[1 - x^2]*Sqrt[1 + x^2]*EllipticP
i[((1 + I*Sqrt[3])/2)^(-1/4), ArcSin[x], -1])/Sqrt[-1 + x^4] - ((I/8)*(2 - Sqrt[2 + (2*I)*Sqrt[3]])*Sqrt[1 - x
^2]*Sqrt[1 + x^2]*EllipticPi[((1 + I*Sqrt[3])/2)^(-1/4), ArcSin[x], -1])/(Sqrt[3]*Sqrt[-1 + x^4]) + ((2 - Sqrt
[2 + (2*I)*Sqrt[3]])*Sqrt[1 - x^2]*Sqrt[1 + x^2]*EllipticPi[((1 + I*Sqrt[3])/2)^(-1/4), ArcSin[x], -1])/(4*Sqr
t[3]*(I - Sqrt[3])*Sqrt[-1 + x^4])

Rule 222

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[-(a*b), 2]}, Simp[(Sqrt[-a + q*x^2]*Sqrt[(a + q*x^2
)/q]*EllipticF[ArcSin[x/Sqrt[(a + q*x^2)/(2*q)]], 1/2])/(Sqrt[2]*Sqrt[-a]*Sqrt[a + b*x^4]), x] /; IntegerQ[q]]
/; FreeQ[{a, b}, x] && LtQ[a, 0] && GtQ[b, 0]

Rule 253

Int[((a1_.) + (b1_.)*(x_)^(n_))^(p_)*((a2_.) + (b2_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[((a1 + b1*x^n)^FracPa
rt[p]*(a2 + b2*x^n)^FracPart[p])/(a1*a2 + b1*b2*x^(2*n))^FracPart[p], Int[(a1*a2 + b1*b2*x^(2*n))^p, x], x] /;
FreeQ[{a1, b1, a2, b2, n, p}, x] && EqQ[a2*b1 + a1*b2, 0] &&  !IntegerQ[p]

Rule 406

Int[Sqrt[(a_) + (b_.)*(x_)^4]/((c_) + (d_.)*(x_)^4), x_Symbol] :> Dist[b/d, Int[1/Sqrt[a + b*x^4], x], x] - Di
st[(b*c - a*d)/d, Int[1/(Sqrt[a + b*x^4]*(c + d*x^4)), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

Rule 409

Int[1/(Sqrt[(a_) + (b_.)*(x_)^4]*((c_) + (d_.)*(x_)^4)), x_Symbol] :> Dist[1/(2*c), Int[1/(Sqrt[a + b*x^4]*(1
- Rt[-(d/c), 2]*x^2)), x], x] + Dist[1/(2*c), Int[1/(Sqrt[a + b*x^4]*(1 + Rt[-(d/c), 2]*x^2)), x], x] /; FreeQ
[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

Rule 537

Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x_)^2]), x_Symbol] :> Simp[(1*Ellipt
icPi[(b*c)/(a*d), ArcSin[Rt[-(d/c), 2]*x], (c*f)/(d*e)])/(a*Sqrt[c]*Sqrt[e]*Rt[-(d/c), 2]), x] /; FreeQ[{a, b,
c, d, e, f}, x] &&  !GtQ[d/c, 0] && GtQ[c, 0] && GtQ[e, 0] &&  !( !GtQ[f/e, 0] && SimplerSqrtQ[-(f/e), -(d/c)
])

Rule 538

Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x_)^2]), x_Symbol] :> Dist[Sqrt[1 +
(d*x^2)/c]/Sqrt[c + d*x^2], Int[1/((a + b*x^2)*Sqrt[1 + (d*x^2)/c]*Sqrt[e + f*x^2]), x], x] /; FreeQ[{a, b, c,
d, e, f}, x] &&  !GtQ[c, 0]

Rule 540

Int[Sqrt[(c_) + (d_.)*(x_)^2]/(((a_) + (b_.)*(x_)^2)*Sqrt[(e_) + (f_.)*(x_)^2]), x_Symbol] :> Dist[d/b, Int[1/
(Sqrt[c + d*x^2]*Sqrt[e + f*x^2]), x], x] + Dist[(b*c - a*d)/b, Int[1/((a + b*x^2)*Sqrt[c + d*x^2]*Sqrt[e + f*
x^2]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NegQ[d/c]

Rule 1215

Int[1/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]), x_Symbol] :> With[{q = Rt[-(a*c), 2]}, Dist[c/(c*d +
e*q), Int[1/Sqrt[a + c*x^4], x], x] + Dist[e/(c*d + e*q), Int[(q - c*x^2)/((d + e*x^2)*Sqrt[a + c*x^4]), x], x
]] /; FreeQ[{a, c, d, e}, x] && GtQ[-(a*c), 0] &&  !LtQ[c, 0]

Rule 1429

Int[((d_) + (e_.)*(x_)^(n_))^(q_)/((a_) + (c_.)*(x_)^(n2_)), x_Symbol] :> With[{r = Rt[-(a*c), 2]}, -Dist[c/(2
*r), Int[(d + e*x^n)^q/(r - c*x^n), x], x] - Dist[c/(2*r), Int[(d + e*x^n)^q/(r + c*x^n), x], x]] /; FreeQ[{a,
c, d, e, n, q}, x] && EqQ[n2, 2*n] && NeQ[c*d^2 + a*e^2, 0] &&  !IntegerQ[q]

Rule 1457

Int[((d_) + (e_.)*(x_)^(n_))^(q_.)*((f_) + (g_.)*(x_)^(n_))^(r_.)*((a_) + (c_.)*(x_)^(n2_))^(p_), x_Symbol] :>
Dist[(a + c*x^(2*n))^FracPart[p]/((d + e*x^n)^FracPart[p]*(a/d + (c*x^n)/e)^FracPart[p]), Int[(d + e*x^n)^(p
+ q)*(f + g*x^n)^r*(a/d + (c*x^n)/e)^p, x], x] /; FreeQ[{a, c, d, e, f, g, n, p, q, r}, x] && EqQ[n2, 2*n] &&
EqQ[c*d^2 + a*e^2, 0] &&  !IntegerQ[p]

Rule 1519

Int[(((f_.)*(x_))^(m_.)*((d_.) + (e_.)*(x_)^(n_))^(q_))/((a_) + (c_.)*(x_)^(n2_.)), x_Symbol] :> Dist[(e*f^n)/
c, Int[(f*x)^(m - n)*(d + e*x^n)^(q - 1), x], x] - Dist[f^n/c, Int[((f*x)^(m - n)*(d + e*x^n)^(q - 1)*Simp[a*e
- c*d*x^n, x])/(a + c*x^(2*n)), x], x] /; FreeQ[{a, c, d, e, f}, x] && EqQ[n2, 2*n] && IGtQ[n, 0] &&  !Intege
rQ[q] && GtQ[q, 0] && GtQ[m, n - 1] && LeQ[m, 2*n - 1]

Rule 1586

Int[(u_.)*(Px_)^(p_.)*(Qx_)^(q_.), x_Symbol] :> Int[u*PolynomialQuotient[Px, Qx, x]^p*Qx^(p + q), x] /; FreeQ[
q, x] && PolyQ[Px, x] && PolyQ[Qx, x] && EqQ[PolynomialRemainder[Px, Qx, x], 0] && IntegerQ[p] && LtQ[p*q, 0]

Rule 6725

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]

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^{16}}{\sqrt {-1+x^4} \left (1-x^8+x^{16}\right )} \, dx &=\int \frac {\sqrt {-1+x^4} \left (1+x^4+x^8+x^{12}\right )}{1-x^8+x^{16}} \, dx\\ &=\int \left (\frac {\sqrt {-1+x^4}}{1-x^8+x^{16}}+\frac {x^4 \sqrt {-1+x^4}}{1-x^8+x^{16}}+\frac {x^8 \sqrt {-1+x^4}}{1-x^8+x^{16}}+\frac {x^{12} \sqrt {-1+x^4}}{1-x^8+x^{16}}\right ) \, dx\\ &=\int \frac {\sqrt {-1+x^4}}{1-x^8+x^{16}} \, dx+\int \frac {x^4 \sqrt {-1+x^4}}{1-x^8+x^{16}} \, dx+\int \frac {x^8 \sqrt {-1+x^4}}{1-x^8+x^{16}} \, dx+\int \frac {x^{12} \sqrt {-1+x^4}}{1-x^8+x^{16}} \, dx\\ &=\int \left (\frac {2 i \sqrt {-1+x^4}}{\sqrt {3} \left (1+i \sqrt {3}-2 x^8\right )}+\frac {2 i \sqrt {-1+x^4}}{\sqrt {3} \left (-1+i \sqrt {3}+2 x^8\right )}\right ) \, dx+\int \left (\frac {\left (1-\frac {i}{\sqrt {3}}\right ) \sqrt {-1+x^4}}{-1-i \sqrt {3}+2 x^8}+\frac {\left (1+\frac {i}{\sqrt {3}}\right ) \sqrt {-1+x^4}}{-1+i \sqrt {3}+2 x^8}\right ) \, dx+\int \left (\frac {2 i x^4 \sqrt {-1+x^4}}{\sqrt {3} \left (1+i \sqrt {3}-2 x^8\right )}+\frac {2 i x^4 \sqrt {-1+x^4}}{\sqrt {3} \left (-1+i \sqrt {3}+2 x^8\right )}\right ) \, dx+\int \left (\frac {i \left (1+i \sqrt {3}\right ) x^4 \sqrt {-1+x^4}}{\sqrt {3} \left (1+i \sqrt {3}-2 x^8\right )}-\frac {i \left (-1+i \sqrt {3}\right ) x^4 \sqrt {-1+x^4}}{\sqrt {3} \left (-1+i \sqrt {3}+2 x^8\right )}\right ) \, dx\\ \end {align*}

rest of steps removed due to Latex formating problem.

________________________________________________________________________________________

Mathematica [C]  time = 22.23, size = 127, normalized size = 2.82 \begin {gather*} \frac {(-1)^{5/6} \sqrt {1-x^4} \left (-4 F\left (\left .\sin ^{-1}(x)\right |-1\right )+\Pi \left (-\sqrt [12]{-1};\left .\sin ^{-1}(x)\right |-1\right )+\Pi \left (\sqrt [12]{-1};\left .\sin ^{-1}(x)\right |-1\right )+\Pi \left (-(-1)^{5/12};\left .\sin ^{-1}(x)\right |-1\right )+\Pi \left ((-1)^{5/12};\left .\sin ^{-1}(x)\right |-1\right )+\Pi \left (-(-1)^{7/12};\left .\sin ^{-1}(x)\right |-1\right )+\Pi \left ((-1)^{7/12};\left .\sin ^{-1}(x)\right |-1\right )+\Pi \left (-(-1)^{11/12};\left .\sin ^{-1}(x)\right |-1\right )+\Pi \left ((-1)^{11/12};\left .\sin ^{-1}(x)\right |-1\right )\right )}{2 \left (\sqrt {3}-i\right ) \sqrt {x^4-1}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-1 + x^16)/(Sqrt[-1 + x^4]*(1 - x^8 + x^16)),x]

[Out]

((-1)^(5/6)*Sqrt[1 - x^4]*(-4*EllipticF[ArcSin[x], -1] + EllipticPi[-(-1)^(1/12), ArcSin[x], -1] + EllipticPi[
(-1)^(1/12), ArcSin[x], -1] + EllipticPi[-(-1)^(5/12), ArcSin[x], -1] + EllipticPi[(-1)^(5/12), ArcSin[x], -1]
+ EllipticPi[-(-1)^(7/12), ArcSin[x], -1] + EllipticPi[(-1)^(7/12), ArcSin[x], -1] + EllipticPi[-(-1)^(11/12)
, ArcSin[x], -1] + EllipticPi[(-1)^(11/12), ArcSin[x], -1]))/(2*(-I + Sqrt[3])*Sqrt[-1 + x^4])

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IntegrateAlgebraic [A]  time = 0.35, size = 45, normalized size = 1.00 \begin {gather*} \frac {1}{8} \text {RootSum}\left [1+4 \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-\log (x)+\log \left (\sqrt {-1+x^4}-x \text {$\#$1}\right )}{\text {$\#$1}}\&\right ] \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(-1 + x^16)/(Sqrt[-1 + x^4]*(1 - x^8 + x^16)),x]

[Out]

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

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fricas [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^16-1)/(x^4-1)^(1/2)/(x^16-x^8+1),x, algorithm="fricas")

[Out]

Timed out

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{16} - 1}{{\left (x^{16} - x^{8} + 1\right )} \sqrt {x^{4} - 1}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^16-1)/(x^4-1)^(1/2)/(x^16-x^8+1),x, algorithm="giac")

[Out]

integrate((x^16 - 1)/((x^16 - x^8 + 1)*sqrt(x^4 - 1)), x)

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maple [B]  time = 6.63, size = 63, normalized size = 1.40

method	result	size
elliptic	\(\frac {\left (\munderset {\textit {\_R} =\RootOf \left (16 \textit {\_Z}^{8}+16 \textit {\_Z}^{4}+1\right )}{\sum }\frac {\left (2 \textit {\_R}^{6}+\textit {\_R}^{2}\right ) \ln \left (\frac {\sqrt {x^{4}-1}\, \sqrt {2}}{2 x}-\textit {\_R} \right )}{2 \textit {\_R}^{7}+\textit {\_R}^{3}}\right ) \sqrt {2}}{16}\)	\(63\)
default	\(-\frac {i \sqrt {x^{2}+1}\, \sqrt {-x^{2}+1}\, \EllipticF \left (i x , i\right )}{\sqrt {x^{4}-1}}+\frac {\left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (\textit {\_Z}^{16}-\textit {\_Z}^{8}+1\right )}{\sum }\underline {\hspace {1.25 ex}}\alpha \left (-\frac {\arctanh \left (\frac {\underline {\hspace {1.25 ex}}\alpha ^{2} \left (\underline {\hspace {1.25 ex}}\alpha ^{14}-\underline {\hspace {1.25 ex}}\alpha ^{6}+x^{2}\right )}{\sqrt {\underline {\hspace {1.25 ex}}\alpha ^{4}-1}\, \sqrt {x^{4}-1}}\right )}{\sqrt {\underline {\hspace {1.25 ex}}\alpha ^{4}-1}}+\frac {2 i \left (-\underline {\hspace {1.25 ex}}\alpha ^{15}+\underline {\hspace {1.25 ex}}\alpha ^{7}\right ) \sqrt {x^{2}+1}\, \sqrt {-x^{2}+1}\, \EllipticPi \left (i x , \underline {\hspace {1.25 ex}}\alpha ^{14}-\underline {\hspace {1.25 ex}}\alpha ^{6}, i\right )}{\sqrt {x^{4}-1}}\right )\right )}{16}\)	\(145\)
trager	\(-262144 \ln \left (\frac {-64 \RootOf \left (16777216 \textit {\_Z}^{8}+16384 \textit {\_Z}^{4}+1\right )^{2} x^{2}-x^{4}+16 \RootOf \left (16777216 \textit {\_Z}^{8}+16384 \textit {\_Z}^{4}+1\right ) \sqrt {x^{4}-1}\, x +1}{64 \RootOf \left (16777216 \textit {\_Z}^{8}+16384 \textit {\_Z}^{4}+1\right )^{2} x^{2}-x^{4}+1}\right ) \RootOf \left (16777216 \textit {\_Z}^{8}+16384 \textit {\_Z}^{4}+1\right )^{7}-256 \ln \left (\frac {-64 \RootOf \left (16777216 \textit {\_Z}^{8}+16384 \textit {\_Z}^{4}+1\right )^{2} x^{2}-x^{4}+16 \RootOf \left (16777216 \textit {\_Z}^{8}+16384 \textit {\_Z}^{4}+1\right ) \sqrt {x^{4}-1}\, x +1}{64 \RootOf \left (16777216 \textit {\_Z}^{8}+16384 \textit {\_Z}^{4}+1\right )^{2} x^{2}-x^{4}+1}\right ) \RootOf \left (16777216 \textit {\_Z}^{8}+16384 \textit {\_Z}^{4}+1\right )^{3}+64 \RootOf \left (16777216 \textit {\_Z}^{8}+16384 \textit {\_Z}^{4}+1\right )^{2} \RootOf \left (-4096 \RootOf \left (16777216 \textit {\_Z}^{8}+16384 \textit {\_Z}^{4}+1\right )^{6}-4 \RootOf \left (16777216 \textit {\_Z}^{8}+16384 \textit {\_Z}^{4}+1\right )^{2}+\textit {\_Z}^{2}\right ) \ln \left (-\frac {-67108864 \RootOf \left (-4096 \RootOf \left (16777216 \textit {\_Z}^{8}+16384 \textit {\_Z}^{4}+1\right )^{6}-4 \RootOf \left (16777216 \textit {\_Z}^{8}+16384 \textit {\_Z}^{4}+1\right )^{2}+\textit {\_Z}^{2}\right ) \RootOf \left (16777216 \textit {\_Z}^{8}+16384 \textit {\_Z}^{4}+1\right )^{8} x^{2}-65536 \RootOf \left (-4096 \RootOf \left (16777216 \textit {\_Z}^{8}+16384 \textit {\_Z}^{4}+1\right )^{6}-4 \RootOf \left (16777216 \textit {\_Z}^{8}+16384 \textit {\_Z}^{4}+1\right )^{2}+\textit {\_Z}^{2}\right ) \RootOf \left (16777216 \textit {\_Z}^{8}+16384 \textit {\_Z}^{4}+1\right )^{4} x^{2}-256 \RootOf \left (-4096 \RootOf \left (16777216 \textit {\_Z}^{8}+16384 \textit {\_Z}^{4}+1\right )^{6}-4 \RootOf \left (16777216 \textit {\_Z}^{8}+16384 \textit {\_Z}^{4}+1\right )^{2}+\textit {\_Z}^{2}\right ) \RootOf \left (16777216 \textit {\_Z}^{8}+16384 \textit {\_Z}^{4}+1\right )^{2} x^{4}+256 \RootOf \left (16777216 \textit {\_Z}^{8}+16384 \textit {\_Z}^{4}+1\right )^{2} \RootOf \left (-4096 \RootOf \left (16777216 \textit {\_Z}^{8}+16384 \textit {\_Z}^{4}+1\right )^{6}-4 \RootOf \left (16777216 \textit {\_Z}^{8}+16384 \textit {\_Z}^{4}+1\right )^{2}+\textit {\_Z}^{2}\right )+x \sqrt {x^{4}-1}}{262144 \RootOf \left (16777216 \textit {\_Z}^{8}+16384 \textit {\_Z}^{4}+1\right )^{6} x^{2}+256 \RootOf \left (16777216 \textit {\_Z}^{8}+16384 \textit {\_Z}^{4}+1\right )^{2} x^{2}-x^{4}+1}\right )+\RootOf \left (-4096 \RootOf \left (16777216 \textit {\_Z}^{8}+16384 \textit {\_Z}^{4}+1\right )^{6}-4 \RootOf \left (16777216 \textit {\_Z}^{8}+16384 \textit {\_Z}^{4}+1\right )^{2}+\textit {\_Z}^{2}\right ) \ln \left (\frac {256 \RootOf \left (-4096 \RootOf \left (16777216 \textit {\_Z}^{8}+16384 \textit {\_Z}^{4}+1\right )^{6}-4 \RootOf \left (16777216 \textit {\_Z}^{8}+16384 \textit {\_Z}^{4}+1\right )^{2}+\textit {\_Z}^{2}\right ) \RootOf \left (16777216 \textit {\_Z}^{8}+16384 \textit {\_Z}^{4}+1\right )^{2} x^{2}-4 \RootOf \left (-4096 \RootOf \left (16777216 \textit {\_Z}^{8}+16384 \textit {\_Z}^{4}+1\right )^{6}-4 \RootOf \left (16777216 \textit {\_Z}^{8}+16384 \textit {\_Z}^{4}+1\right )^{2}+\textit {\_Z}^{2}\right ) x^{4}+x \sqrt {x^{4}-1}+4 \RootOf \left (-4096 \RootOf \left (16777216 \textit {\_Z}^{8}+16384 \textit {\_Z}^{4}+1\right )^{6}-4 \RootOf \left (16777216 \textit {\_Z}^{8}+16384 \textit {\_Z}^{4}+1\right )^{2}+\textit {\_Z}^{2}\right )}{64 \RootOf \left (16777216 \textit {\_Z}^{8}+16384 \textit {\_Z}^{4}+1\right )^{2} x^{2}+x^{4}-1}\right )+\RootOf \left (16777216 \textit {\_Z}^{8}+16384 \textit {\_Z}^{4}+1\right ) \ln \left (-\frac {1048576 \RootOf \left (16777216 \textit {\_Z}^{8}+16384 \textit {\_Z}^{4}+1\right )^{7} x^{2}+1024 \RootOf \left (16777216 \textit {\_Z}^{8}+16384 \textit {\_Z}^{4}+1\right )^{3} x^{2}-4 \RootOf \left (16777216 \textit {\_Z}^{8}+16384 \textit {\_Z}^{4}+1\right ) x^{4}+x \sqrt {x^{4}-1}+4 \RootOf \left (16777216 \textit {\_Z}^{8}+16384 \textit {\_Z}^{4}+1\right )}{262144 \RootOf \left (16777216 \textit {\_Z}^{8}+16384 \textit {\_Z}^{4}+1\right )^{6} x^{2}+256 \RootOf \left (16777216 \textit {\_Z}^{8}+16384 \textit {\_Z}^{4}+1\right )^{2} x^{2}+x^{4}-1}\right )\)	\(919\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^16-1)/(x^4-1)^(1/2)/(x^16-x^8+1),x,method=_RETURNVERBOSE)

[Out]

1/16*sum((2*_R^6+_R^2)/(2*_R^7+_R^3)*ln(1/2*(x^4-1)^(1/2)*2^(1/2)/x-_R),_R=RootOf(16*_Z^8+16*_Z^4+1))*2^(1/2)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{16} - 1}{{\left (x^{16} - x^{8} + 1\right )} \sqrt {x^{4} - 1}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^16-1)/(x^4-1)^(1/2)/(x^16-x^8+1),x, algorithm="maxima")

[Out]

integrate((x^16 - 1)/((x^16 - x^8 + 1)*sqrt(x^4 - 1)), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {x^{16}-1}{\sqrt {x^4-1}\,\left (x^{16}-x^8+1\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^16 - 1)/((x^4 - 1)^(1/2)*(x^16 - x^8 + 1)),x)

[Out]

int((x^16 - 1)/((x^4 - 1)^(1/2)*(x^16 - 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**16-1)/(x**4-1)**(1/2)/(x**16-x**8+1),x)

[Out]

Timed out

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