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

Optimal. Leaf size=239 \[ \frac {\tan ^{-1}\left (\frac {\frac {x^4}{\sqrt {2}}-\frac {x^2}{\sqrt {2}}-\frac {1}{\sqrt {2}}}{x \sqrt {x^4-1}}\right )}{4 \sqrt {2}}+\frac {\tan ^{-1}\left (\frac {\frac {x^4}{\sqrt {2} \sqrt [4]{3}}-\frac {\sqrt [4]{3} x^2}{\sqrt {2}}-\frac {1}{\sqrt {2} \sqrt [4]{3}}}{x \sqrt {x^4-1}}\right )}{4 \sqrt {2} \sqrt [4]{3}}-\frac {\tanh ^{-1}\left (\frac {\frac {x^4}{\sqrt {2}}+\frac {x^2}{\sqrt {2}}-\frac {1}{\sqrt {2}}}{x \sqrt {x^4-1}}\right )}{4 \sqrt {2}}-\frac {\tanh ^{-1}\left (\frac {\frac {x^4}{\sqrt {2} \sqrt [4]{3}}+\frac {\sqrt [4]{3} x^2}{\sqrt {2}}-\frac {1}{\sqrt {2} \sqrt [4]{3}}}{x \sqrt {x^4-1}}\right )}{4 \sqrt {2} \sqrt [4]{3}} \]

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Rubi [C]  time = 4.44, antiderivative size = 853, normalized size of antiderivative = 3.57, number of steps used = 153, number of rules used = 22, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.880, Rules used = {1586, 6728, 1729, 1209, 1188, 222, 1185, 1215, 1457, 540, 253, 538, 537, 1248, 735, 844, 217, 206, 725, 204, 406, 409} \begin {gather*} -\frac {1}{32} \sqrt {\frac {1}{6} \left (3+i \sqrt {3}\right )} \left (i+\sqrt {3}\right ) \tan ^{-1}\left (\frac {\left (1-i \sqrt {3}\right ) x^2+2}{\sqrt {2 \left (3+i \sqrt {3}\right )} \sqrt {x^4-1}}\right )+\frac {1}{32} \sqrt {\frac {1}{6} \left (3+i \sqrt {3}\right )} \left (i+\sqrt {3}\right ) \tan ^{-1}\left (\frac {4-\left (1+i \sqrt {3}\right )^2 x^2}{2 \sqrt {2 \left (3+i \sqrt {3}\right )} \sqrt {x^4-1}}\right )+\frac {\left (i+\sqrt {3}\right ) \sqrt {x^2-1} \sqrt {x^2+1} F\left (\sin ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {x^2-1}}\right )|\frac {1}{2}\right )}{8 \sqrt {6} \sqrt {x^4-1}}+\frac {3 \left (1+i \sqrt {3}\right ) \sqrt {x^2-1} \sqrt {x^2+1} F\left (\sin ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {x^2-1}}\right )|\frac {1}{2}\right )}{8 \sqrt {2} \sqrt {x^4-1}}+\frac {3 \left (1-i \sqrt {3}\right ) \sqrt {x^2-1} \sqrt {x^2+1} F\left (\sin ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {x^2-1}}\right )|\frac {1}{2}\right )}{8 \sqrt {2} \sqrt {x^4-1}}-\frac {\left (i-\sqrt {3}\right ) \sqrt {x^2-1} \sqrt {x^2+1} F\left (\sin ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {x^2-1}}\right )|\frac {1}{2}\right )}{8 \sqrt {6} \sqrt {x^4-1}}-\frac {\sqrt {1-x^2} \sqrt {x^2+1} \Pi \left (\frac {1}{2} \left (-i-\sqrt {3}\right );\left .\sin ^{-1}(x)\right |-1\right )}{4 \sqrt {x^4-1}}-\frac {\sqrt {1-x^2} \sqrt {x^2+1} \Pi \left (-\frac {4}{\left (i-\sqrt {3}\right )^2};\left .\sin ^{-1}(x)\right |-1\right )}{4 \sqrt {x^4-1}}-\frac {\sqrt {1-x^2} \sqrt {x^2+1} \Pi \left (\frac {1}{2} \left (i-\sqrt {3}\right );\left .\sin ^{-1}(x)\right |-1\right )}{4 \sqrt {x^4-1}}-\frac {\sqrt {1-x^2} \sqrt {x^2+1} \Pi \left (\frac {1}{\sqrt {\frac {1}{2} \left (1-i \sqrt {3}\right )}};\left .\sin ^{-1}(x)\right |-1\right )}{4 \sqrt {x^4-1}}-\frac {\sqrt {1-x^2} \sqrt {x^2+1} \Pi \left (\frac {2}{1-i \sqrt {3}};\left .\sin ^{-1}(x)\right |-1\right )}{4 \sqrt {x^4-1}}-\frac {\sqrt {1-x^2} \sqrt {x^2+1} \Pi \left (\frac {1}{\sqrt {\frac {1}{2} \left (1+i \sqrt {3}\right )}};\left .\sin ^{-1}(x)\right |-1\right )}{4 \sqrt {x^4-1}}-\frac {\sqrt {1-x^2} \sqrt {x^2+1} \Pi \left (\frac {2}{1+i \sqrt {3}};\left .\sin ^{-1}(x)\right |-1\right )}{4 \sqrt {x^4-1}}-\frac {\sqrt {1-x^2} \sqrt {x^2+1} \Pi \left (-\frac {4}{\left (i+\sqrt {3}\right )^2};\left .\sin ^{-1}(x)\right |-1\right )}{4 \sqrt {x^4-1}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-1/32*(Sqrt[(3 + I*Sqrt[3])/6]*(I + Sqrt[3])*ArcTan[(2 + (1 - I*Sqrt[3])*x^2)/(Sqrt[2*(3 + I*Sqrt[3])]*Sqrt[-1
+ x^4])]) + (Sqrt[(3 + I*Sqrt[3])/6]*(I + Sqrt[3])*ArcTan[(4 - (1 + I*Sqrt[3])^2*x^2)/(2*Sqrt[2*(3 + I*Sqrt[3
])]*Sqrt[-1 + x^4])])/32 - ((I - Sqrt[3])*Sqrt[-1 + x^2]*Sqrt[1 + x^2]*EllipticF[ArcSin[(Sqrt[2]*x)/Sqrt[-1 +
x^2]], 1/2])/(8*Sqrt[6]*Sqrt[-1 + x^4]) + (3*(1 - I*Sqrt[3])*Sqrt[-1 + x^2]*Sqrt[1 + x^2]*EllipticF[ArcSin[(Sq
rt[2]*x)/Sqrt[-1 + x^2]], 1/2])/(8*Sqrt[2]*Sqrt[-1 + x^4]) + (3*(1 + I*Sqrt[3])*Sqrt[-1 + x^2]*Sqrt[1 + x^2]*E
llipticF[ArcSin[(Sqrt[2]*x)/Sqrt[-1 + x^2]], 1/2])/(8*Sqrt[2]*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])/(8*Sqrt[6]*Sqrt[-1 + x^4]) - (Sqrt[1 - x^2]*
Sqrt[1 + x^2]*EllipticPi[(-I - Sqrt[3])/2, ArcSin[x], -1])/(4*Sqrt[-1 + x^4]) - (Sqrt[1 - x^2]*Sqrt[1 + x^2]*E
llipticPi[-4/(I - Sqrt[3])^2, ArcSin[x], -1])/(4*Sqrt[-1 + x^4]) - (Sqrt[1 - x^2]*Sqrt[1 + x^2]*EllipticPi[(I
- Sqrt[3])/2, ArcSin[x], -1])/(4*Sqrt[-1 + x^4]) - (Sqrt[1 - x^2]*Sqrt[1 + x^2]*EllipticPi[1/Sqrt[(1 - I*Sqrt[
3])/2], ArcSin[x], -1])/(4*Sqrt[-1 + x^4]) - (Sqrt[1 - x^2]*Sqrt[1 + x^2]*EllipticPi[2/(1 - I*Sqrt[3]), ArcSin
[x], -1])/(4*Sqrt[-1 + x^4]) - (Sqrt[1 - x^2]*Sqrt[1 + x^2]*EllipticPi[1/Sqrt[(1 + I*Sqrt[3])/2], ArcSin[x], -
1])/(4*Sqrt[-1 + x^4]) - (Sqrt[1 - x^2]*Sqrt[1 + x^2]*EllipticPi[2/(1 + I*Sqrt[3]), ArcSin[x], -1])/(4*Sqrt[-1
+ x^4]) - (Sqrt[1 - x^2]*Sqrt[1 + x^2]*EllipticPi[-4/(I + Sqrt[3])^2, ArcSin[x], -1])/(4*Sqrt[-1 + x^4])

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[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 217

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

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 725

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

Rule 735

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((d + e*x)^(m + 1)*(a + c*x^2)^p)/(
e*(m + 2*p + 1)), x] + Dist[(2*p)/(e*(m + 2*p + 1)), Int[(d + e*x)^m*Simp[a*e - c*d*x, x]*(a + c*x^2)^(p - 1),
x], x] /; FreeQ[{a, c, d, e, m}, x] && NeQ[c*d^2 + a*e^2, 0] && GtQ[p, 0] && NeQ[m + 2*p + 1, 0] && ( !Ration
alQ[m] || LtQ[m, 1]) &&  !ILtQ[m + 2*p, 0] && IntQuadraticQ[a, 0, c, d, e, m, p, x]

Rule 844

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[g/e, Int[(d
+ e*x)^(m + 1)*(a + c*x^2)^p, x], x] + Dist[(e*f - d*g)/e, Int[(d + e*x)^m*(a + c*x^2)^p, x], x] /; FreeQ[{a,
c, d, e, f, g, m, p}, x] && NeQ[c*d^2 + a*e^2, 0] &&  !IGtQ[m, 0]

Rule 1185

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

Rule 1188

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

Rule 1209

Int[((a_) + (c_.)*(x_)^4)^(p_)/((d_) + (e_.)*(x_)^2), x_Symbol] :> -Dist[(e^2)^(-1), Int[(c*d - c*e*x^2)*(a +
c*x^4)^(p - 1), x], x] + Dist[(c*d^2 + a*e^2)/e^2, Int[(a + c*x^4)^(p - 1)/(d + e*x^2), x], x] /; FreeQ[{a, c,
d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && IGtQ[p + 1/2, 0]

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 1248

Int[(x_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2, Subst[Int[(d + e*x)^q
*(a + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, c, d, e, p, q}, x]

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 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 1729

Int[((a_) + (c_.)*(x_)^4)^(p_.)/((d_) + (e_.)*(x_)), x_Symbol] :> Dist[d, Int[(a + c*x^4)^p/(d^2 - e^2*x^2), x
], x] - Dist[e, Int[(x*(a + c*x^4)^p)/(d^2 - e^2*x^2), x], x] /; FreeQ[{a, c, d, e}, x] && IntegerQ[p + 1/2]

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 {(1-x) \sqrt {-1+x^4}}{8 \left (1-x+x^2\right )}+\frac {(1+x) \sqrt {-1+x^4}}{8 \left (1+x+x^2\right )}+\frac {\sqrt {-1+x^4}}{4 \left (1-x^2+x^4\right )}+\frac {\sqrt {-1+x^4} \left (1+x^4\right )}{2 \left (1-x^4+x^8\right )}\right ) \, dx\\ &=\frac {1}{8} \int \frac {(1-x) \sqrt {-1+x^4}}{1-x+x^2} \, dx+\frac {1}{8} \int \frac {(1+x) \sqrt {-1+x^4}}{1+x+x^2} \, dx+\frac {1}{4} \int \frac {\sqrt {-1+x^4}}{1-x^2+x^4} \, dx+\frac {1}{2} \int \frac {\sqrt {-1+x^4} \left (1+x^4\right )}{1-x^4+x^8} \, dx\\ &=\text {rest of steps removed due to Latex formating problem} \end {align*}

________________________________________________________________________________________

Mathematica [C]  time = 2.75, size = 195, normalized size = 0.82 \begin {gather*} \frac {\sqrt {1-x^4} \left (4 F\left (\left .\sin ^{-1}(x)\right |-1\right )-\Pi \left (-\frac {i}{2}-\frac {\sqrt {3}}{2};\left .\sin ^{-1}(x)\right |-1\right )-\Pi \left (\frac {i}{2}-\frac {\sqrt {3}}{2};\left .\sin ^{-1}(x)\right |-1\right )-\Pi \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2};\left .\sin ^{-1}(x)\right |-1\right )-\Pi \left (\frac {1}{2}-\frac {i \sqrt {3}}{2};\left .\sin ^{-1}(x)\right |-1\right )-\Pi \left (\frac {1}{2}+\frac {i \sqrt {3}}{2};\left .\sin ^{-1}(x)\right |-1\right )-\Pi \left (\frac {1}{2} \left (-i+\sqrt {3}\right );\left .\sin ^{-1}(x)\right |-1\right )-\Pi \left (\frac {1}{2} i \left (i+\sqrt {3}\right );\left .\sin ^{-1}(x)\right |-1\right )-\Pi \left (\frac {1}{2} \left (i+\sqrt {3}\right );\left .\sin ^{-1}(x)\right |-1\right )\right )}{4 \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]

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

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IntegrateAlgebraic [C]  time = 2.28, size = 278, normalized size = 1.16 \begin {gather*} \frac {\tan ^{-1}\left (\frac {\frac {x^4}{\sqrt {2}}-\frac {x^2}{\sqrt {2}}-\frac {1}{\sqrt {2}}}{x \sqrt {x^4-1}}\right )}{4 \sqrt {2}}+\frac {\tan ^{-1}\left (\frac {\frac {x^4}{\sqrt {2} \sqrt [4]{3}}+\frac {\left (2 i-\sqrt {-1+4 i \sqrt {3}}\right ) x^2}{\sqrt {2} \sqrt [4]{3}}-\frac {1}{\sqrt {2} \sqrt [4]{3}}}{x \sqrt {x^4-1}}\right )}{4 \sqrt {2} \sqrt [4]{3}}-\frac {\tanh ^{-1}\left (\frac {\frac {x^4}{\sqrt {2}}+\frac {x^2}{\sqrt {2}}-\frac {1}{\sqrt {2}}}{x \sqrt {x^4-1}}\right )}{4 \sqrt {2}}-\frac {\tanh ^{-1}\left (\frac {\frac {x^4}{\sqrt {2} \sqrt [4]{3}}+\frac {\left (2 i+\sqrt {-1-4 i \sqrt {3}}\right ) x^2}{\sqrt {2} \sqrt [4]{3}}-\frac {1}{\sqrt {2} \sqrt [4]{3}}}{x \sqrt {x^4-1}}\right )}{4 \sqrt {2} \sqrt [4]{3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

ArcTan[(-(1/Sqrt[2]) - x^2/Sqrt[2] + x^4/Sqrt[2])/(x*Sqrt[-1 + x^4])]/(4*Sqrt[2]) + ArcTan[(-(1/(Sqrt[2]*3^(1/
4))) + ((2*I - Sqrt[-1 + (4*I)*Sqrt[3]])*x^2)/(Sqrt[2]*3^(1/4)) + x^4/(Sqrt[2]*3^(1/4)))/(x*Sqrt[-1 + x^4])]/(
4*Sqrt[2]*3^(1/4)) - ArcTanh[(-(1/Sqrt[2]) + x^2/Sqrt[2] + x^4/Sqrt[2])/(x*Sqrt[-1 + x^4])]/(4*Sqrt[2]) - ArcT
anh[(-(1/(Sqrt[2]*3^(1/4))) + ((2*I + Sqrt[-1 - (4*I)*Sqrt[3]])*x^2)/(Sqrt[2]*3^(1/4)) + x^4/(Sqrt[2]*3^(1/4))
)/(x*Sqrt[-1 + x^4])]/(4*Sqrt[2]*3^(1/4))

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fricas [B]  time = 1.04, size = 1173, normalized size = 4.91

result too large to display

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]

-1/24*3^(3/4)*sqrt(2)*arctan(1/3*(3*x^16 + 6*x^12 + 9*x^8 + 6*x^4 + sqrt(3)*(2*3^(3/4)*sqrt(2)*(x^14 - 12*x^10
+ 12*x^6 - x^2) + 3^(1/4)*sqrt(2)*(x^16 - 28*x^12 + 45*x^8 - 28*x^4 + 1) + 12*(x^11 + x^7 + x^3 + 4*sqrt(3)*(
x^9 - x^5))*sqrt(x^4 - 1))*sqrt((12*x^6 - 12*x^2 + sqrt(3)*(x^8 + x^4 + 1) + 2*sqrt(x^4 - 1)*(3*3^(1/4)*sqrt(2
)*x^3 + 3^(3/4)*sqrt(2)*(x^5 - x)))/(x^8 + x^4 + 1)) + 12*sqrt(3)*(x^14 - x^2) + 6*sqrt(x^4 - 1)*(3*3^(3/4)*sq
rt(2)*(x^11 - 3*x^7 + x^3) + 3^(1/4)*sqrt(2)*(x^13 - 12*x^9 + 12*x^5 - x)) + 3)/(x^16 - 46*x^12 + 99*x^8 - 46*
x^4 + 1)) + 1/24*3^(3/4)*sqrt(2)*arctan(1/3*(3*x^16 + 6*x^12 + 9*x^8 + 6*x^4 - sqrt(3)*(2*3^(3/4)*sqrt(2)*(x^1
4 - 12*x^10 + 12*x^6 - x^2) + 3^(1/4)*sqrt(2)*(x^16 - 28*x^12 + 45*x^8 - 28*x^4 + 1) - 12*(x^11 + x^7 + x^3 +
4*sqrt(3)*(x^9 - x^5))*sqrt(x^4 - 1))*sqrt((12*x^6 - 12*x^2 + sqrt(3)*(x^8 + x^4 + 1) - 2*sqrt(x^4 - 1)*(3*3^(
1/4)*sqrt(2)*x^3 + 3^(3/4)*sqrt(2)*(x^5 - x)))/(x^8 + x^4 + 1)) + 12*sqrt(3)*(x^14 - x^2) - 6*sqrt(x^4 - 1)*(3
*3^(3/4)*sqrt(2)*(x^11 - 3*x^7 + x^3) + 3^(1/4)*sqrt(2)*(x^13 - 12*x^9 + 12*x^5 - x)) + 3)/(x^16 - 46*x^12 + 9
9*x^8 - 46*x^4 + 1)) - 1/96*3^(3/4)*sqrt(2)*log(12*(12*x^6 - 12*x^2 + sqrt(3)*(x^8 + x^4 + 1) + 2*sqrt(x^4 - 1
)*(3*3^(1/4)*sqrt(2)*x^3 + 3^(3/4)*sqrt(2)*(x^5 - x)))/(x^8 + x^4 + 1)) + 1/96*3^(3/4)*sqrt(2)*log(12*(12*x^6
- 12*x^2 + sqrt(3)*(x^8 + x^4 + 1) - 2*sqrt(x^4 - 1)*(3*3^(1/4)*sqrt(2)*x^3 + 3^(3/4)*sqrt(2)*(x^5 - x)))/(x^8
+ x^4 + 1)) - 1/8*sqrt(2)*arctan((x^8 - x^4 + 2*sqrt(2)*(x^5 - x^3 - x)*sqrt(x^4 - 1) + (4*sqrt(x^4 - 1)*x^3
+ sqrt(2)*(x^8 - 2*x^6 - 3*x^4 + 2*x^2 + 1))*sqrt((x^8 + 4*x^6 - x^4 + 2*sqrt(2)*(x^5 + x^3 - x)*sqrt(x^4 - 1)
- 4*x^2 + 1)/(x^8 - x^4 + 1)) + 1)/(x^8 - 4*x^6 - x^4 + 4*x^2 + 1)) + 1/8*sqrt(2)*arctan((x^8 - x^4 - 2*sqrt(
2)*(x^5 - x^3 - x)*sqrt(x^4 - 1) + (4*sqrt(x^4 - 1)*x^3 - sqrt(2)*(x^8 - 2*x^6 - 3*x^4 + 2*x^2 + 1))*sqrt((x^8
+ 4*x^6 - x^4 - 2*sqrt(2)*(x^5 + x^3 - x)*sqrt(x^4 - 1) - 4*x^2 + 1)/(x^8 - x^4 + 1)) + 1)/(x^8 - 4*x^6 - x^4
+ 4*x^2 + 1)) - 1/32*sqrt(2)*log(4*(x^8 + 4*x^6 - x^4 + 2*sqrt(2)*(x^5 + x^3 - x)*sqrt(x^4 - 1) - 4*x^2 + 1)/
(x^8 - x^4 + 1)) + 1/32*sqrt(2)*log(4*(x^8 + 4*x^6 - x^4 - 2*sqrt(2)*(x^5 + x^3 - x)*sqrt(x^4 - 1) - 4*x^2 + 1
)/(x^8 - x^4 + 1))

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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 [C]  time = 0.11, size = 685, normalized size = 2.87

result too large to display

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]

-I*(x^2+1)^(1/2)*(-x^2+1)^(1/2)/(x^4-1)^(1/2)*EllipticF(I*x,I)+1/8*(-1/2+1/2*I*3^(1/2))*(1/2/(-3/2+1/2*I*3^(1/
2))^(1/2)*arctanh((1/2+1/2*I*3^(1/2))*(x^2+1/2-1/2*I*3^(1/2))/(-3/2+1/2*I*3^(1/2))^(1/2)/(x^4-1)^(1/2))+I*(-1/
2-1/2*I*3^(1/2))*(x^2+1)^(1/2)*(-x^2+1)^(1/2)/(x^4-1)^(1/2)*EllipticPi(I*x,1/2-1/2*I*3^(1/2),I))+1/8*(-1/2-1/2
*I*3^(1/2))*(1/2/(-3/2-1/2*I*3^(1/2))^(1/2)*arctanh((1/2-1/2*I*3^(1/2))*(x^2+1/2+1/2*I*3^(1/2))/(-3/2-1/2*I*3^
(1/2))^(1/2)/(x^4-1)^(1/2))+I*(-1/2+1/2*I*3^(1/2))*(x^2+1)^(1/2)*(-x^2+1)^(1/2)/(x^4-1)^(1/2)*EllipticPi(I*x,1
/2+1/2*I*3^(1/2),I))+1/16*sum(_alpha*(-1/(_alpha^2-2)^(1/2)*arctanh(_alpha^2*(_alpha^2+x^2-1)/(_alpha^2-2)^(1/
2)/(x^4-1)^(1/2))+2*I*(-_alpha^3+_alpha)*(x^2+1)^(1/2)*(-x^2+1)^(1/2)/(x^4-1)^(1/2)*EllipticPi(I*x,_alpha^2-1,
I)),_alpha=RootOf(_Z^4-_Z^2+1))+1/8*(1/2+1/2*I*3^(1/2))*(-1/2/(-3/2-1/2*I*3^(1/2))^(1/2)*arctanh((-1/2+1/2*I*3
^(1/2))*(x^2+1/2+1/2*I*3^(1/2))/(-3/2-1/2*I*3^(1/2))^(1/2)/(x^4-1)^(1/2))+I*(1/2-1/2*I*3^(1/2))*(x^2+1)^(1/2)*
(-x^2+1)^(1/2)/(x^4-1)^(1/2)*EllipticPi(I*x,1/2+1/2*I*3^(1/2),I))+1/8*(1/2-1/2*I*3^(1/2))*(-1/2/(-3/2+1/2*I*3^
(1/2))^(1/2)*arctanh((-1/2-1/2*I*3^(1/2))*(x^2+1/2-1/2*I*3^(1/2))/(-3/2+1/2*I*3^(1/2))^(1/2)/(x^4-1)^(1/2))+I*
(1/2+1/2*I*3^(1/2))*(x^2+1)^(1/2)*(-x^2+1)^(1/2)/(x^4-1)^(1/2)*EllipticPi(I*x,1/2-1/2*I*3^(1/2),I))+1/16*sum(_
alpha*(-1/(_alpha^4-1)^(1/2)*arctanh(_alpha^2*(_alpha^6-_alpha^2+x^2)/(_alpha^4-1)^(1/2)/(x^4-1)^(1/2))+2*I*(-
_alpha^7+_alpha^3)*(x^2+1)^(1/2)*(-x^2+1)^(1/2)/(x^4-1)^(1/2)*EllipticPi(I*x,_alpha^6-_alpha^2,I)),_alpha=Root
Of(_Z^8-_Z^4+1))

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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 = -0.00 \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^8 + x^16 + 1)),x)

[Out]

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

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

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

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