3.22.26 \(\int \frac {\sqrt [8]{256-256 x^2+96 x^4-16 x^6+x^8}}{-1+x^3} \, dx\)
Optimal. Leaf size=154 \[ \frac {\sqrt [8]{\left (x^2-4\right )^4} \left (\frac {2 \tan ^{-1}\left (\frac {\sqrt {x^2-4}}{\sqrt {3} (x-2)}\right )}{\sqrt {3}}-\frac {1}{3} \sqrt {2 \left (-3-5 i \sqrt {3}\right )} \tan ^{-1}\left (\frac {\sqrt {1-\frac {2 i}{\sqrt {3}}} \sqrt {x^2-4}}{x-2}\right )-\frac {1}{3} \sqrt {2 \left (-3+5 i \sqrt {3}\right )} \tan ^{-1}\left (\frac {\sqrt {1+\frac {2 i}{\sqrt {3}}} \sqrt {x^2-4}}{x-2}\right )\right )}{\sqrt {x^2-4}} \]
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Rubi [A] time = 0.84, antiderivative size = 212, normalized size of antiderivative = 1.38,
number of steps used = 19, number of rules used = 14, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.438, Rules
used = {2074, 6720, 735, 844, 217, 206, 725, 204, 1020, 1080, 1037, 1031, 207, 203} \begin {gather*} \frac {\sqrt [8]{\left (x^2-4\right )^4} \tan ^{-1}\left (\frac {4-x}{\sqrt {3} \sqrt {x^2-4}}\right )}{\sqrt {3} \sqrt {x^2-4}}-\frac {\sqrt {2 \sqrt {21}-3} \sqrt [8]{\left (x^2-4\right )^4} \tan ^{-1}\left (\frac {\left (21-4 \sqrt {21}\right ) x+\sqrt {21}+21}{\sqrt {21 \left (2 \sqrt {21}-3\right )} \sqrt {x^2-4}}\right )}{3 \sqrt {x^2-4}}+\frac {\sqrt {3+2 \sqrt {21}} \sqrt [8]{\left (x^2-4\right )^4} \tanh ^{-1}\left (\frac {\left (21+4 \sqrt {21}\right ) x-\sqrt {21}+21}{\sqrt {21 \left (3+2 \sqrt {21}\right )} \sqrt {x^2-4}}\right )}{3 \sqrt {x^2-4}} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[(256 - 256*x^2 + 96*x^4 - 16*x^6 + x^8)^(1/8)/(-1 + x^3),x]
[Out]
(((-4 + x^2)^4)^(1/8)*ArcTan[(4 - x)/(Sqrt[3]*Sqrt[-4 + x^2])])/(Sqrt[3]*Sqrt[-4 + x^2]) - (Sqrt[-3 + 2*Sqrt[2
1]]*((-4 + x^2)^4)^(1/8)*ArcTan[(21 + Sqrt[21] + (21 - 4*Sqrt[21])*x)/(Sqrt[21*(-3 + 2*Sqrt[21])]*Sqrt[-4 + x^
2])])/(3*Sqrt[-4 + x^2]) + (Sqrt[3 + 2*Sqrt[21]]*((-4 + x^2)^4)^(1/8)*ArcTanh[(21 - Sqrt[21] + (21 + 4*Sqrt[21
])*x)/(Sqrt[21*(3 + 2*Sqrt[21])]*Sqrt[-4 + x^2])])/(3*Sqrt[-4 + x^2])
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 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 207
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTanh[(Rt[b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[a, 0] || GtQ[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 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 1020
Int[((g_.) + (h_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_)*((d_) + (e_.)*(x_) + (f_.)*(x_)^2)^(q_), x_Symbol] :> Simp
[(h*(a + c*x^2)^p*(d + e*x + f*x^2)^(q + 1))/(2*f*(p + q + 1)), x] + Dist[1/(2*f*(p + q + 1)), Int[(a + c*x^2)
^(p - 1)*(d + e*x + f*x^2)^q*Simp[a*h*e*p - a*(h*e - 2*g*f)*(p + q + 1) - 2*h*p*(c*d - a*f)*x - (h*c*e*p + c*(
h*e - 2*g*f)*(p + q + 1))*x^2, x], x], x] /; FreeQ[{a, c, d, e, f, g, h, q}, x] && NeQ[e^2 - 4*d*f, 0] && GtQ[
p, 0] && NeQ[p + q + 1, 0]
Rule 1031
Int[((g_) + (h_.)*(x_))/(((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)*Sqrt[(d_) + (f_.)*(x_)^2]), x_Symbol] :> Dist[-2*
g*(g*b - 2*a*h), Subst[Int[1/Simp[g*(g*b - 2*a*h)*(b^2 - 4*a*c) - b*d*x^2, x], x], x, Simp[g*b - 2*a*h - (b*h
- 2*g*c)*x, x]/Sqrt[d + f*x^2]], x] /; FreeQ[{a, b, c, d, f, g, h}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[b*h^2*d -
2*g*h*(c*d - a*f) - g^2*b*f, 0]
Rule 1037
Int[((g_.) + (h_.)*(x_))/(((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)*Sqrt[(d_) + (f_.)*(x_)^2]), x_Symbol] :> With[{q
= Rt[(c*d - a*f)^2 + b^2*d*f, 2]}, Dist[1/(2*q), Int[Simp[h*b*d - g*(c*d - a*f - q) + (h*(c*d - a*f + q) + g*
b*f)*x, x]/((a + b*x + c*x^2)*Sqrt[d + f*x^2]), x], x] - Dist[1/(2*q), Int[Simp[h*b*d - g*(c*d - a*f + q) + (h
*(c*d - a*f - q) + g*b*f)*x, x]/((a + b*x + c*x^2)*Sqrt[d + f*x^2]), x], x]] /; FreeQ[{a, b, c, d, f, g, h}, x
] && NeQ[b^2 - 4*a*c, 0] && NegQ[b^2 - 4*a*c]
Rule 1080
Int[((A_.) + (B_.)*(x_) + (C_.)*(x_)^2)/(((a_) + (b_.)*(x_) + (c_.)*(x_)^2)*Sqrt[(d_.) + (f_.)*(x_)^2]), x_Sym
bol] :> Dist[C/c, Int[1/Sqrt[d + f*x^2], x], x] + Dist[1/c, Int[(A*c - a*C + (B*c - b*C)*x)/((a + b*x + c*x^2)
*Sqrt[d + f*x^2]), x], x] /; FreeQ[{a, b, c, d, f, A, B, C}, x] && NeQ[b^2 - 4*a*c, 0]
Rule 2074
Int[(P_)^(p_)*(Q_)^(q_.), x_Symbol] :> With[{PP = Factor[P]}, Int[ExpandIntegrand[PP^p*Q^q, x], x] /; !SumQ[N
onfreeFactors[PP, x]]] /; FreeQ[q, x] && PolyQ[P, x] && PolyQ[Q, x] && IntegerQ[p] && NeQ[P, x]
Rule 6720
Int[(u_.)*((a_.)*(v_)^(m_.))^(p_), x_Symbol] :> Dist[(a^IntPart[p]*(a*v^m)^FracPart[p])/v^(m*FracPart[p]), Int
[u*v^(m*p), x], x] /; FreeQ[{a, m, p}, x] && !IntegerQ[p] && !FreeQ[v, x] && !(EqQ[a, 1] && EqQ[m, 1]) &&
!(EqQ[v, x] && EqQ[m, 1])
Rubi steps
\begin {align*} \int \frac {\sqrt [8]{256-256 x^2+96 x^4-16 x^6+x^8}}{-1+x^3} \, dx &=\int \left (\frac {\sqrt [8]{\left (-4+x^2\right )^4}}{3 (-1+x)}+\frac {(-2-x) \sqrt [8]{\left (-4+x^2\right )^4}}{3 \left (1+x+x^2\right )}\right ) \, dx\\ &=\frac {1}{3} \int \frac {\sqrt [8]{\left (-4+x^2\right )^4}}{-1+x} \, dx+\frac {1}{3} \int \frac {(-2-x) \sqrt [8]{\left (-4+x^2\right )^4}}{1+x+x^2} \, dx\\ &=\frac {\sqrt [8]{\left (-4+x^2\right )^4} \int \frac {\sqrt {-4+x^2}}{-1+x} \, dx}{3 \sqrt {-4+x^2}}+\frac {\sqrt [8]{\left (-4+x^2\right )^4} \int \frac {(-2-x) \sqrt {-4+x^2}}{1+x+x^2} \, dx}{3 \sqrt {-4+x^2}}\\ &=\frac {\sqrt [8]{\left (-4+x^2\right )^4} \int \frac {-4+x}{(-1+x) \sqrt {-4+x^2}} \, dx}{3 \sqrt {-4+x^2}}+\frac {\sqrt [8]{\left (-4+x^2\right )^4} \int \frac {8+5 x-x^2}{\sqrt {-4+x^2} \left (1+x+x^2\right )} \, dx}{3 \sqrt {-4+x^2}}\\ &=\frac {\sqrt [8]{\left (-4+x^2\right )^4} \int \frac {9+6 x}{\sqrt {-4+x^2} \left (1+x+x^2\right )} \, dx}{3 \sqrt {-4+x^2}}-\frac {\sqrt [8]{\left (-4+x^2\right )^4} \int \frac {1}{(-1+x) \sqrt {-4+x^2}} \, dx}{\sqrt {-4+x^2}}\\ &=\frac {\sqrt [8]{\left (-4+x^2\right )^4} \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,\frac {-4+x}{\sqrt {-4+x^2}}\right )}{\sqrt {-4+x^2}}+\frac {\sqrt [8]{\left (-4+x^2\right )^4} \int \frac {3 \left (7+3 \sqrt {21}\right )-3 \left (7-2 \sqrt {21}\right ) x}{\sqrt {-4+x^2} \left (1+x+x^2\right )} \, dx}{6 \sqrt {21} \sqrt {-4+x^2}}-\frac {\sqrt [8]{\left (-4+x^2\right )^4} \int \frac {3 \left (7-3 \sqrt {21}\right )-3 \left (7+2 \sqrt {21}\right ) x}{\sqrt {-4+x^2} \left (1+x+x^2\right )} \, dx}{6 \sqrt {21} \sqrt {-4+x^2}}\\ &=\frac {\sqrt [8]{\left (-4+x^2\right )^4} \tan ^{-1}\left (\frac {4-x}{\sqrt {3} \sqrt {-4+x^2}}\right )}{\sqrt {3} \sqrt {-4+x^2}}+\frac {\left (4 \sqrt {21} \left (3-2 \sqrt {21}\right ) \sqrt [8]{\left (-4+x^2\right )^4}\right ) \operatorname {Subst}\left (\int \frac {1}{-756 \left (3-2 \sqrt {21}\right )+4 x^2} \, dx,x,\frac {3 \left (21+\sqrt {21}\right )+3 \left (21-4 \sqrt {21}\right ) x}{\sqrt {-4+x^2}}\right )}{\sqrt {-4+x^2}}-\frac {\left (4 \sqrt {21} \left (3+2 \sqrt {21}\right ) \sqrt [8]{\left (-4+x^2\right )^4}\right ) \operatorname {Subst}\left (\int \frac {1}{-756 \left (3+2 \sqrt {21}\right )+4 x^2} \, dx,x,\frac {3 \left (21-\sqrt {21}\right )+3 \left (21+4 \sqrt {21}\right ) x}{\sqrt {-4+x^2}}\right )}{\sqrt {-4+x^2}}\\ &=\frac {\sqrt [8]{\left (-4+x^2\right )^4} \tan ^{-1}\left (\frac {4-x}{\sqrt {3} \sqrt {-4+x^2}}\right )}{\sqrt {3} \sqrt {-4+x^2}}-\frac {\sqrt {-3+2 \sqrt {21}} \sqrt [8]{\left (-4+x^2\right )^4} \tan ^{-1}\left (\frac {21+\sqrt {21}+\left (21-4 \sqrt {21}\right ) x}{\sqrt {21 \left (-3+2 \sqrt {21}\right )} \sqrt {-4+x^2}}\right )}{3 \sqrt {-4+x^2}}+\frac {\sqrt {3+2 \sqrt {21}} \sqrt [8]{\left (-4+x^2\right )^4} \tanh ^{-1}\left (\frac {21-\sqrt {21}+\left (21+4 \sqrt {21}\right ) x}{\sqrt {21 \left (3+2 \sqrt {21}\right )} \sqrt {-4+x^2}}\right )}{3 \sqrt {-4+x^2}}\\ \end {align*}
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Mathematica [A] time = 0.30, size = 157, normalized size = 1.02 \begin {gather*} \frac {\sqrt [8]{\left (x^2-4\right )^4} \left (-\sqrt {3} \tan ^{-1}\left (\frac {x-4}{\sqrt {3} \sqrt {x^2-4}}\right )+(-1)^{2/3} \sqrt {1+4 \sqrt [3]{-1}} \tanh ^{-1}\left (\frac {4 (-1)^{2/3}-x}{\sqrt {1+4 \sqrt [3]{-1}} \sqrt {x^2-4}}\right )+\sqrt [3]{-1} \sqrt {1-4 (-1)^{2/3}} \tanh ^{-1}\left (\frac {x+4 \sqrt [3]{-1}}{\sqrt {1-4 (-1)^{2/3}} \sqrt {x^2-4}}\right )\right )}{3 \sqrt {x^2-4}} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[(256 - 256*x^2 + 96*x^4 - 16*x^6 + x^8)^(1/8)/(-1 + x^3),x]
[Out]
(((-4 + x^2)^4)^(1/8)*(-(Sqrt[3]*ArcTan[(-4 + x)/(Sqrt[3]*Sqrt[-4 + x^2])]) + (-1)^(2/3)*Sqrt[1 + 4*(-1)^(1/3)
]*ArcTanh[(4*(-1)^(2/3) - x)/(Sqrt[1 + 4*(-1)^(1/3)]*Sqrt[-4 + x^2])] + (-1)^(1/3)*Sqrt[1 - 4*(-1)^(2/3)]*ArcT
anh[(4*(-1)^(1/3) + x)/(Sqrt[1 - 4*(-1)^(2/3)]*Sqrt[-4 + x^2])]))/(3*Sqrt[-4 + x^2])
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IntegrateAlgebraic [A] time = 20.56, size = 154, normalized size = 1.00 \begin {gather*} \frac {\sqrt [8]{\left (-4+x^2\right )^4} \left (\frac {2 \tan ^{-1}\left (\frac {\sqrt {-4+x^2}}{\sqrt {3} (-2+x)}\right )}{\sqrt {3}}-\frac {1}{3} \sqrt {2 \left (-3-5 i \sqrt {3}\right )} \tan ^{-1}\left (\frac {\sqrt {1-\frac {2 i}{\sqrt {3}}} \sqrt {-4+x^2}}{-2+x}\right )-\frac {1}{3} \sqrt {2 \left (-3+5 i \sqrt {3}\right )} \tan ^{-1}\left (\frac {\sqrt {1+\frac {2 i}{\sqrt {3}}} \sqrt {-4+x^2}}{-2+x}\right )\right )}{\sqrt {-4+x^2}} \end {gather*}
Antiderivative was successfully verified.
[In]
IntegrateAlgebraic[(256 - 256*x^2 + 96*x^4 - 16*x^6 + x^8)^(1/8)/(-1 + x^3),x]
[Out]
(((-4 + x^2)^4)^(1/8)*((2*ArcTan[Sqrt[-4 + x^2]/(Sqrt[3]*(-2 + x))])/Sqrt[3] - (Sqrt[2*(-3 - (5*I)*Sqrt[3])]*A
rcTan[(Sqrt[1 - (2*I)/Sqrt[3]]*Sqrt[-4 + x^2])/(-2 + x)])/3 - (Sqrt[2*(-3 + (5*I)*Sqrt[3])]*ArcTan[(Sqrt[1 + (
2*I)/Sqrt[3]]*Sqrt[-4 + x^2])/(-2 + x)])/3))/Sqrt[-4 + x^2]
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fricas [B] time = 0.54, size = 1214, normalized size = 7.88
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((x^8-16*x^6+96*x^4-256*x^2+256)^(1/8)/(x^3-1),x, algorithm="fricas")
[Out]
-1/63*21^(3/4)*sqrt(3)*sqrt(-4*sqrt(21) + 56)*arctan(1/1764000*sqrt(2)*sqrt(70*x^2 - 21^(1/4)*(sqrt(21)*(2*x +
3) - 7*x + 7)*sqrt(-4*sqrt(21) + 56) + (x^8 - 16*x^6 + 96*x^4 - 256*x^2 + 256)^(1/8)*(21^(1/4)*(2*sqrt(21) -
7)*sqrt(-4*sqrt(21) + 56) - 140*x - 70) + 70*x + 70*sqrt(21) + 70*(x^8 - 16*x^6 + 96*x^4 - 256*x^2 + 256)^(1/4
) + 70)*(sqrt(35)*(21^(3/4)*(13*sqrt(21)*sqrt(3) + 57*sqrt(3)) + 3*21^(1/4)*(11*sqrt(21)*sqrt(3) - 21*sqrt(3))
)*sqrt(-4*sqrt(21) + 56) + 30*sqrt(35)*(sqrt(21)*(3*sqrt(21)*sqrt(3) + 7*sqrt(3)) + 63*sqrt(21)*sqrt(3) - 273*
sqrt(3))) + 1/120*sqrt(21)*sqrt(3)*(9*x + 4) + 1/840*sqrt(21)*(sqrt(21)*sqrt(3)*(3*x + 8) + 7*sqrt(3)*(x - 4))
- 1/40*sqrt(3)*(13*x + 8) + 1/25200*(21^(3/4)*(sqrt(21)*sqrt(3)*(13*x + 8) + 3*sqrt(3)*(19*x + 4)) + 3*21^(1/
4)*(sqrt(21)*sqrt(3)*(11*x + 76) - 21*sqrt(3)*(x + 16)))*sqrt(-4*sqrt(21) + 56) - 1/25200*(x^8 - 16*x^6 + 96*x
^4 - 256*x^2 + 256)^(1/8)*(30*sqrt(21)*(3*sqrt(21)*sqrt(3) + 7*sqrt(3)) + (21^(3/4)*(13*sqrt(21)*sqrt(3) + 57*
sqrt(3)) + 3*21^(1/4)*(11*sqrt(21)*sqrt(3) - 21*sqrt(3)))*sqrt(-4*sqrt(21) + 56) + 1890*sqrt(21)*sqrt(3) - 819
0*sqrt(3))) - 1/63*21^(3/4)*sqrt(3)*sqrt(-4*sqrt(21) + 56)*arctan(1/1764000*sqrt(2)*sqrt(70*x^2 + 21^(1/4)*(sq
rt(21)*(2*x + 3) - 7*x + 7)*sqrt(-4*sqrt(21) + 56) - (x^8 - 16*x^6 + 96*x^4 - 256*x^2 + 256)^(1/8)*(21^(1/4)*(
2*sqrt(21) - 7)*sqrt(-4*sqrt(21) + 56) + 140*x + 70) + 70*x + 70*sqrt(21) + 70*(x^8 - 16*x^6 + 96*x^4 - 256*x^
2 + 256)^(1/4) + 70)*(sqrt(35)*(21^(3/4)*(13*sqrt(21)*sqrt(3) + 57*sqrt(3)) + 3*21^(1/4)*(11*sqrt(21)*sqrt(3)
- 21*sqrt(3)))*sqrt(-4*sqrt(21) + 56) - 30*sqrt(35)*(sqrt(21)*(3*sqrt(21)*sqrt(3) + 7*sqrt(3)) + 63*sqrt(21)*s
qrt(3) - 273*sqrt(3))) - 1/120*sqrt(21)*sqrt(3)*(9*x + 4) - 1/840*sqrt(21)*(sqrt(21)*sqrt(3)*(3*x + 8) + 7*sqr
t(3)*(x - 4)) + 1/40*sqrt(3)*(13*x + 8) + 1/25200*(21^(3/4)*(sqrt(21)*sqrt(3)*(13*x + 8) + 3*sqrt(3)*(19*x + 4
)) + 3*21^(1/4)*(sqrt(21)*sqrt(3)*(11*x + 76) - 21*sqrt(3)*(x + 16)))*sqrt(-4*sqrt(21) + 56) + 1/25200*(x^8 -
16*x^6 + 96*x^4 - 256*x^2 + 256)^(1/8)*(30*sqrt(21)*(3*sqrt(21)*sqrt(3) + 7*sqrt(3)) - (21^(3/4)*(13*sqrt(21)*
sqrt(3) + 57*sqrt(3)) + 3*21^(1/4)*(11*sqrt(21)*sqrt(3) - 21*sqrt(3)))*sqrt(-4*sqrt(21) + 56) + 1890*sqrt(21)*
sqrt(3) - 8190*sqrt(3))) + 1/420*21^(1/4)*(sqrt(21) + 14)*sqrt(-4*sqrt(21) + 56)*log(16*x^2 + 8/35*21^(1/4)*(s
qrt(21)*(2*x + 3) - 7*x + 7)*sqrt(-4*sqrt(21) + 56) - 8/35*(x^8 - 16*x^6 + 96*x^4 - 256*x^2 + 256)^(1/8)*(21^(
1/4)*(2*sqrt(21) - 7)*sqrt(-4*sqrt(21) + 56) + 140*x + 70) + 16*x + 16*sqrt(21) + 16*(x^8 - 16*x^6 + 96*x^4 -
256*x^2 + 256)^(1/4) + 16) - 1/420*21^(1/4)*(sqrt(21) + 14)*sqrt(-4*sqrt(21) + 56)*log(16*x^2 - 8/35*21^(1/4)*
(sqrt(21)*(2*x + 3) - 7*x + 7)*sqrt(-4*sqrt(21) + 56) + 8/35*(x^8 - 16*x^6 + 96*x^4 - 256*x^2 + 256)^(1/8)*(21
^(1/4)*(2*sqrt(21) - 7)*sqrt(-4*sqrt(21) + 56) - 140*x - 70) + 16*x + 16*sqrt(21) + 16*(x^8 - 16*x^6 + 96*x^4
- 256*x^2 + 256)^(1/4) + 16) - 2/3*sqrt(3)*arctan(-1/3*sqrt(3)*(x - 1) + 1/3*sqrt(3)*(x^8 - 16*x^6 + 96*x^4 -
256*x^2 + 256)^(1/8))
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (x^{8} - 16 \, x^{6} + 96 \, x^{4} - 256 \, x^{2} + 256\right )}^{\frac {1}{8}}}{x^{3} - 1}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((x^8-16*x^6+96*x^4-256*x^2+256)^(1/8)/(x^3-1),x, algorithm="giac")
[Out]
integrate((x^8 - 16*x^6 + 96*x^4 - 256*x^2 + 256)^(1/8)/(x^3 - 1), x)
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maple [F] time = 0.03, size = 0, normalized size = 0.00 \[\int \frac {\left (x^{8}-16 x^{6}+96 x^{4}-256 x^{2}+256\right )^{\frac {1}{8}}}{x^{3}-1}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((x^8-16*x^6+96*x^4-256*x^2+256)^(1/8)/(x^3-1),x)
[Out]
int((x^8-16*x^6+96*x^4-256*x^2+256)^(1/8)/(x^3-1),x)
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (x^{8} - 16 \, x^{6} + 96 \, x^{4} - 256 \, x^{2} + 256\right )}^{\frac {1}{8}}}{x^{3} - 1}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((x^8-16*x^6+96*x^4-256*x^2+256)^(1/8)/(x^3-1),x, algorithm="maxima")
[Out]
integrate((x^8 - 16*x^6 + 96*x^4 - 256*x^2 + 256)^(1/8)/(x^3 - 1), x)
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (x^8-16\,x^6+96\,x^4-256\,x^2+256\right )}^{1/8}}{x^3-1} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((96*x^4 - 256*x^2 - 16*x^6 + x^8 + 256)^(1/8)/(x^3 - 1),x)
[Out]
int((96*x^4 - 256*x^2 - 16*x^6 + x^8 + 256)^(1/8)/(x^3 - 1), x)
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt [8]{\left (x - 2\right )^{4} \left (x + 2\right )^{4}}}{\left (x - 1\right ) \left (x^{2} + x + 1\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((x**8-16*x**6+96*x**4-256*x**2+256)**(1/8)/(x**3-1),x)
[Out]
Integral(((x - 2)**4*(x + 2)**4)**(1/8)/((x - 1)*(x**2 + x + 1)), x)
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