3.24.10 \(\int \frac {1+x}{(-1+x^3) \sqrt [8]{256-256 x^2+96 x^4-16 x^6+x^8}} \, dx\)

Optimal. Leaf size=177 \[ \frac {\left (\left (x^2-4\right )^4\right )^{7/8} \left (\frac {4 \tan ^{-1}\left (\frac {\sqrt {x^2-4}}{\sqrt {3}}-\frac {x}{\sqrt {3}}+\frac {1}{\sqrt {3}}\right )}{3 \sqrt {3}}-\frac {2}{3} \text {RootSum}\left [\text {$\#$1}^4-2 \text {$\#$1}^3+12 \text {$\#$1}^2-8 \text {$\#$1}+16\& ,\frac {\text {$\#$1}^2 \log \left (-\text {$\#$1}+\sqrt {x^2-4}-x\right )-\text {$\#$1} \log \left (-\text {$\#$1}+\sqrt {x^2-4}-x\right )+4 \log \left (-\text {$\#$1}+\sqrt {x^2-4}-x\right )}{2 \text {$\#$1}^3-3 \text {$\#$1}^2+12 \text {$\#$1}-4}\& \right ]\right )}{\left (x^2-4\right )^{7/2}} \]

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Rubi [A]  time = 0.72, antiderivative size = 218, normalized size of antiderivative = 1.23, number of steps used = 10, number of rules used = 6, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.171, Rules used = {6688, 6720, 6725, 725, 204, 206} \begin {gather*} -\frac {2 \sqrt {x^2-4} \tan ^{-1}\left (\frac {4-x}{\sqrt {3} \sqrt {x^2-4}}\right )}{3 \sqrt {3} \sqrt [8]{\left (x^2-4\right )^4}}+\frac {\left (1-\sqrt [3]{-1}\right ) \sqrt {x^2-4} \tanh ^{-1}\left (\frac {(-1)^{2/3} \left (\sqrt [3]{-1} x+4\right )}{\sqrt {1+4 \sqrt [3]{-1}} \sqrt {x^2-4}}\right )}{3 \sqrt {1+4 \sqrt [3]{-1}} \sqrt [8]{\left (x^2-4\right )^4}}-\frac {\left (1+(-1)^{2/3}\right ) \sqrt {x^2-4} \tanh ^{-1}\left (\frac {\sqrt [3]{-1} \left (4-(-1)^{2/3} x\right )}{\sqrt {1-4 (-1)^{2/3}} \sqrt {x^2-4}}\right )}{3 \sqrt {1-4 (-1)^{2/3}} \sqrt [8]{\left (x^2-4\right )^4}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 + x)/((-1 + x^3)*(256 - 256*x^2 + 96*x^4 - 16*x^6 + x^8)^(1/8)),x]

[Out]

(-2*Sqrt[-4 + x^2]*ArcTan[(4 - x)/(Sqrt[3]*Sqrt[-4 + x^2])])/(3*Sqrt[3]*((-4 + x^2)^4)^(1/8)) + ((1 - (-1)^(1/
3))*Sqrt[-4 + x^2]*ArcTanh[((-1)^(2/3)*(4 + (-1)^(1/3)*x))/(Sqrt[1 + 4*(-1)^(1/3)]*Sqrt[-4 + x^2])])/(3*Sqrt[1
+ 4*(-1)^(1/3)]*((-4 + x^2)^4)^(1/8)) - ((1 + (-1)^(2/3))*Sqrt[-4 + x^2]*ArcTanh[((-1)^(1/3)*(4 - (-1)^(2/3)*
x))/(Sqrt[1 - 4*(-1)^(2/3)]*Sqrt[-4 + x^2])])/(3*Sqrt[1 - 4*(-1)^(2/3)]*((-4 + x^2)^4)^(1/8))

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 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 6688

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, 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])

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]

Rubi steps

\begin {align*} \int \frac {1+x}{\left (-1+x^3\right ) \sqrt [8]{256-256 x^2+96 x^4-16 x^6+x^8}} \, dx &=\int \frac {-1-x}{\sqrt [8]{\left (-4+x^2\right )^4} \left (1-x^3\right )} \, dx\\ &=\frac {\sqrt {-4+x^2} \int \frac {-1-x}{\sqrt {-4+x^2} \left (1-x^3\right )} \, dx}{\sqrt [8]{\left (-4+x^2\right )^4}}\\ &=\frac {\sqrt {-4+x^2} \int \left (-\frac {2}{3 (1-x) \sqrt {-4+x^2}}+\frac {-1-(-1)^{2/3}}{3 \left (1+\sqrt [3]{-1} x\right ) \sqrt {-4+x^2}}+\frac {-1+\sqrt [3]{-1}}{3 \left (1-(-1)^{2/3} x\right ) \sqrt {-4+x^2}}\right ) \, dx}{\sqrt [8]{\left (-4+x^2\right )^4}}\\ &=-\frac {\left (2 \sqrt {-4+x^2}\right ) \int \frac {1}{(1-x) \sqrt {-4+x^2}} \, dx}{3 \sqrt [8]{\left (-4+x^2\right )^4}}+\frac {\left (\left (-1+\sqrt [3]{-1}\right ) \sqrt {-4+x^2}\right ) \int \frac {1}{\left (1-(-1)^{2/3} x\right ) \sqrt {-4+x^2}} \, dx}{3 \sqrt [8]{\left (-4+x^2\right )^4}}+\frac {\left (\left (-1-(-1)^{2/3}\right ) \sqrt {-4+x^2}\right ) \int \frac {1}{\left (1+\sqrt [3]{-1} x\right ) \sqrt {-4+x^2}} \, dx}{3 \sqrt [8]{\left (-4+x^2\right )^4}}\\ &=\frac {\left (2 \sqrt {-4+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,\frac {4-x}{\sqrt {-4+x^2}}\right )}{3 \sqrt [8]{\left (-4+x^2\right )^4}}-\frac {\left (\left (-1+\sqrt [3]{-1}\right ) \sqrt {-4+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{1+4 \sqrt [3]{-1}-x^2} \, dx,x,\frac {4 (-1)^{2/3}-x}{\sqrt {-4+x^2}}\right )}{3 \sqrt [8]{\left (-4+x^2\right )^4}}-\frac {\left (\left (-1-(-1)^{2/3}\right ) \sqrt {-4+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{1-4 (-1)^{2/3}-x^2} \, dx,x,\frac {-4 \sqrt [3]{-1}-x}{\sqrt {-4+x^2}}\right )}{3 \sqrt [8]{\left (-4+x^2\right )^4}}\\ &=-\frac {2 \sqrt {-4+x^2} \tan ^{-1}\left (\frac {4-x}{\sqrt {3} \sqrt {-4+x^2}}\right )}{3 \sqrt {3} \sqrt [8]{\left (-4+x^2\right )^4}}+\frac {\left (1-\sqrt [3]{-1}\right ) \sqrt {-4+x^2} \tanh ^{-1}\left (\frac {(-1)^{2/3} \left (4+\sqrt [3]{-1} x\right )}{\sqrt {1+4 \sqrt [3]{-1}} \sqrt {-4+x^2}}\right )}{3 \sqrt {1+4 \sqrt [3]{-1}} \sqrt [8]{\left (-4+x^2\right )^4}}-\frac {\left (1+(-1)^{2/3}\right ) \sqrt {-4+x^2} \tanh ^{-1}\left (\frac {\sqrt [3]{-1} \left (4-(-1)^{2/3} x\right )}{\sqrt {1-4 (-1)^{2/3}} \sqrt {-4+x^2}}\right )}{3 \sqrt {1-4 (-1)^{2/3}} \sqrt [8]{\left (-4+x^2\right )^4}}\\ \end {align*}

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Mathematica [A]  time = 0.32, size = 168, normalized size = 0.95 \begin {gather*} -\frac {\sqrt {x^2-4} \left (2 \sqrt {7} \tan ^{-1}\left (\frac {4-x}{\sqrt {3} \sqrt {x^2-4}}\right )+\left (\sqrt [3]{-1}-1\right ) \sqrt {1-4 (-1)^{2/3}} \tanh ^{-1}\left (\frac {4 (-1)^{2/3}-x}{\sqrt {1+4 \sqrt [3]{-1}} \sqrt {x^2-4}}\right )+\sqrt {1+4 \sqrt [3]{-1}} \left (1+(-1)^{2/3}\right ) \tanh ^{-1}\left (\frac {x+4 \sqrt [3]{-1}}{\sqrt {1-4 (-1)^{2/3}} \sqrt {x^2-4}}\right )\right )}{3 \sqrt {21} \sqrt [8]{\left (x^2-4\right )^4}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1 + x)/((-1 + x^3)*(256 - 256*x^2 + 96*x^4 - 16*x^6 + x^8)^(1/8)),x]

[Out]

-1/3*(Sqrt[-4 + x^2]*(2*Sqrt[7]*ArcTan[(4 - x)/(Sqrt[3]*Sqrt[-4 + x^2])] + (-1 + (-1)^(1/3))*Sqrt[1 - 4*(-1)^(
2/3)]*ArcTanh[(4*(-1)^(2/3) - x)/(Sqrt[1 + 4*(-1)^(1/3)]*Sqrt[-4 + x^2])] + Sqrt[1 + 4*(-1)^(1/3)]*(1 + (-1)^(
2/3))*ArcTanh[(4*(-1)^(1/3) + x)/(Sqrt[1 - 4*(-1)^(2/3)]*Sqrt[-4 + x^2])]))/(Sqrt[21]*((-4 + x^2)^4)^(1/8))

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IntegrateAlgebraic [A]  time = 19.51, size = 177, normalized size = 1.00 \begin {gather*} \frac {\left (\left (-4+x^2\right )^4\right )^{7/8} \left (\frac {4 \tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {x}{\sqrt {3}}+\frac {\sqrt {-4+x^2}}{\sqrt {3}}\right )}{3 \sqrt {3}}-\frac {2}{3} \text {RootSum}\left [16-8 \text {$\#$1}+12 \text {$\#$1}^2-2 \text {$\#$1}^3+\text {$\#$1}^4\&,\frac {4 \log \left (-x+\sqrt {-4+x^2}-\text {$\#$1}\right )-\log \left (-x+\sqrt {-4+x^2}-\text {$\#$1}\right ) \text {$\#$1}+\log \left (-x+\sqrt {-4+x^2}-\text {$\#$1}\right ) \text {$\#$1}^2}{-4+12 \text {$\#$1}-3 \text {$\#$1}^2+2 \text {$\#$1}^3}\&\right ]\right )}{\left (-4+x^2\right )^{7/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(1 + x)/((-1 + x^3)*(256 - 256*x^2 + 96*x^4 - 16*x^6 + x^8)^(1/8)),x]

[Out]

(((-4 + x^2)^4)^(7/8)*((4*ArcTan[1/Sqrt[3] - x/Sqrt[3] + Sqrt[-4 + x^2]/Sqrt[3]])/(3*Sqrt[3]) - (2*RootSum[16
- 8*#1 + 12*#1^2 - 2*#1^3 + #1^4 & , (4*Log[-x + Sqrt[-4 + x^2] - #1] - Log[-x + Sqrt[-4 + x^2] - #1]*#1 + Log
[-x + Sqrt[-4 + x^2] - #1]*#1^2)/(-4 + 12*#1 - 3*#1^2 + 2*#1^3) & ])/3))/(-4 + x^2)^(7/2)

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fricas [B]  time = 0.81, size = 1289, normalized size = 7.28

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x)/(x^3-1)/(x^8-16*x^6+96*x^4-256*x^2+256)^(1/8),x, algorithm="fricas")

[Out]

-1/126*21^(1/4)*sqrt(3*sqrt(21) + 14)*(2*sqrt(21) - 9)*log(7056*x^2 + 1008*(21^(1/4)*(x^8 - 16*x^6 + 96*x^4 -
256*x^2 + 256)^(1/8)*(3*sqrt(21) - 14) - 21^(1/4)*(sqrt(21)*(3*x + 1) - 14*x - 7))*sqrt(3*sqrt(21) + 14) - 705
6*(x^8 - 16*x^6 + 96*x^4 - 256*x^2 + 256)^(1/8)*(2*x + 1) + 7056*x + 7056*sqrt(21) + 7056*(x^8 - 16*x^6 + 96*x
^4 - 256*x^2 + 256)^(1/4) + 7056) + 1/126*21^(1/4)*sqrt(3*sqrt(21) + 14)*(2*sqrt(21) - 9)*log(7056*x^2 - 1008*
(21^(1/4)*(x^8 - 16*x^6 + 96*x^4 - 256*x^2 + 256)^(1/8)*(3*sqrt(21) - 14) - 21^(1/4)*(sqrt(21)*(3*x + 1) - 14*
x - 7))*sqrt(3*sqrt(21) + 14) - 7056*(x^8 - 16*x^6 + 96*x^4 - 256*x^2 + 256)^(1/8)*(2*x + 1) + 7056*x + 7056*s
qrt(21) + 7056*(x^8 - 16*x^6 + 96*x^4 - 256*x^2 + 256)^(1/4) + 7056) + 2/63*21^(1/4)*sqrt(3)*sqrt(3*sqrt(21) +
14)*arctan(1/17640*sqrt(7)*sqrt(7*x^2 - (21^(1/4)*(x^8 - 16*x^6 + 96*x^4 - 256*x^2 + 256)^(1/8)*(3*sqrt(21) -
14) - 21^(1/4)*(sqrt(21)*(3*x + 1) - 14*x - 7))*sqrt(3*sqrt(21) + 14) - 7*(x^8 - 16*x^6 + 96*x^4 - 256*x^2 +
256)^(1/8)*(2*x + 1) + 7*x + 7*sqrt(21) + 7*(x^8 - 16*x^6 + 96*x^4 - 256*x^2 + 256)^(1/4) + 7)*(3*sqrt(21)*(3*
sqrt(21)*sqrt(3) + 7*sqrt(3)) + (21^(3/4)*(sqrt(21)*sqrt(3) + 9*sqrt(3)) - 3*21^(1/4)*(13*sqrt(21)*sqrt(3) - 6
3*sqrt(3)))*sqrt(3*sqrt(21) + 14) + 189*sqrt(21)*sqrt(3) - 819*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) + 1/2520*(21^(3/4)*(s
qrt(21)*sqrt(3)*(x - 4) + 3*sqrt(3)*(3*x + 8)) - 3*21^(1/4)*(sqrt(21)*sqrt(3)*(13*x + 128) - 21*sqrt(3)*(3*x +
28)) - (x^8 - 16*x^6 + 96*x^4 - 256*x^2 + 256)^(1/8)*(21^(3/4)*(sqrt(21)*sqrt(3) + 9*sqrt(3)) - 3*21^(1/4)*(1
3*sqrt(21)*sqrt(3) - 63*sqrt(3))))*sqrt(3*sqrt(21) + 14) - 1/840*(x^8 - 16*x^6 + 96*x^4 - 256*x^2 + 256)^(1/8)
*(sqrt(21)*(3*sqrt(21)*sqrt(3) + 7*sqrt(3)) + 63*sqrt(21)*sqrt(3) - 273*sqrt(3))) + 2/63*21^(1/4)*sqrt(3)*sqrt
(3*sqrt(21) + 14)*arctan(-1/17640*sqrt(7)*sqrt(7*x^2 + (21^(1/4)*(x^8 - 16*x^6 + 96*x^4 - 256*x^2 + 256)^(1/8)
*(3*sqrt(21) - 14) - 21^(1/4)*(sqrt(21)*(3*x + 1) - 14*x - 7))*sqrt(3*sqrt(21) + 14) - 7*(x^8 - 16*x^6 + 96*x^
4 - 256*x^2 + 256)^(1/8)*(2*x + 1) + 7*x + 7*sqrt(21) + 7*(x^8 - 16*x^6 + 96*x^4 - 256*x^2 + 256)^(1/4) + 7)*(
3*sqrt(21)*(3*sqrt(21)*sqrt(3) + 7*sqrt(3)) - (21^(3/4)*(sqrt(21)*sqrt(3) + 9*sqrt(3)) - 3*21^(1/4)*(13*sqrt(2
1)*sqrt(3) - 63*sqrt(3)))*sqrt(3*sqrt(21) + 14) + 189*sqrt(21)*sqrt(3) - 819*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) + 1/252
0*(21^(3/4)*(sqrt(21)*sqrt(3)*(x - 4) + 3*sqrt(3)*(3*x + 8)) - 3*21^(1/4)*(sqrt(21)*sqrt(3)*(13*x + 128) - 21*
sqrt(3)*(3*x + 28)) - (x^8 - 16*x^6 + 96*x^4 - 256*x^2 + 256)^(1/8)*(21^(3/4)*(sqrt(21)*sqrt(3) + 9*sqrt(3)) -
3*21^(1/4)*(13*sqrt(21)*sqrt(3) - 63*sqrt(3))))*sqrt(3*sqrt(21) + 14) + 1/840*(x^8 - 16*x^6 + 96*x^4 - 256*x^
2 + 256)^(1/8)*(sqrt(21)*(3*sqrt(21)*sqrt(3) + 7*sqrt(3)) + 63*sqrt(21)*sqrt(3) - 273*sqrt(3))) + 4/9*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 {x + 1}{{\left (x^{8} - 16 \, x^{6} + 96 \, x^{4} - 256 \, x^{2} + 256\right )}^{\frac {1}{8}} {\left (x^{3} - 1\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x)/(x^3-1)/(x^8-16*x^6+96*x^4-256*x^2+256)^(1/8),x, algorithm="giac")

[Out]

integrate((x + 1)/((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 = 0.04, size = 0, normalized size = 0.00 \[\int \frac {1+x}{\left (x^{3}-1\right ) \left (x^{8}-16 x^{6}+96 x^{4}-256 x^{2}+256\right )^{\frac {1}{8}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1+x)/(x^3-1)/(x^8-16*x^6+96*x^4-256*x^2+256)^(1/8),x)

[Out]

int((1+x)/(x^3-1)/(x^8-16*x^6+96*x^4-256*x^2+256)^(1/8),x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x + 1}{{\left (x^{8} - 16 \, x^{6} + 96 \, x^{4} - 256 \, x^{2} + 256\right )}^{\frac {1}{8}} {\left (x^{3} - 1\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x)/(x^3-1)/(x^8-16*x^6+96*x^4-256*x^2+256)^(1/8),x, algorithm="maxima")

[Out]

integrate((x + 1)/((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 {x+1}{\left (x^3-1\right )\,{\left (x^8-16\,x^6+96\,x^4-256\,x^2+256\right )}^{1/8}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x + 1)/((x^3 - 1)*(96*x^4 - 256*x^2 - 16*x^6 + x^8 + 256)^(1/8)),x)

[Out]

int((x + 1)/((x^3 - 1)*(96*x^4 - 256*x^2 - 16*x^6 + x^8 + 256)^(1/8)), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x + 1}{\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((1+x)/(x**3-1)/(x**8-16*x**6+96*x**4-256*x**2+256)**(1/8),x)

[Out]

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

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