3.17.91 \(\int \frac {x^4 \sqrt [4]{x^2+x^4}}{-1+x^8} \, dx\)
Optimal. Leaf size=113 \[ -\frac {1}{8} \text {RootSum}\left [\text {$\#$1}^8-2 \text {$\#$1}^4+2\& ,\frac {\text {$\#$1} \log \left (\sqrt [4]{x^4+x^2}-\text {$\#$1} x\right )-\text {$\#$1} \log (x)}{\text {$\#$1}^4-1}\& \right ]+\frac {\tan ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^4+x^2}}\right )}{2\ 2^{3/4}}-\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^4+x^2}}\right )}{2\ 2^{3/4}} \]
________________________________________________________________________________________
Rubi [C] time = 0.96, antiderivative size = 207, normalized size of antiderivative = 1.83,
number of steps used = 43, number of rules used = 20, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.909, Rules used
= {2056, 1586, 6733, 6742, 1240, 410, 237, 335, 275, 231, 407, 409, 1213, 537, 494, 298, 203, 206,
1529, 510} \begin {gather*} \frac {x \sqrt [4]{x^4+x^2} F_1\left (\frac {3}{4};1,-\frac {1}{4};\frac {7}{4};-i x^2,-x^2\right )}{6 \sqrt [4]{x^2+1}}+\frac {x \sqrt [4]{x^4+x^2} F_1\left (\frac {3}{4};1,-\frac {1}{4};\frac {7}{4};i x^2,-x^2\right )}{6 \sqrt [4]{x^2+1}}+\frac {\sqrt [4]{x^4+x^2} \tan ^{-1}\left (\frac {\sqrt [4]{2} \sqrt {x}}{\sqrt [4]{x^2+1}}\right )}{2\ 2^{3/4} \sqrt {x} \sqrt [4]{x^2+1}}-\frac {\sqrt [4]{x^4+x^2} \tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt {x}}{\sqrt [4]{x^2+1}}\right )}{2\ 2^{3/4} \sqrt {x} \sqrt [4]{x^2+1}} \end {gather*}
Warning: Unable to verify antiderivative.
[In]
Int[(x^4*(x^2 + x^4)^(1/4))/(-1 + x^8),x]
[Out]
(x*(x^2 + x^4)^(1/4)*AppellF1[3/4, 1, -1/4, 7/4, (-I)*x^2, -x^2])/(6*(1 + x^2)^(1/4)) + (x*(x^2 + x^4)^(1/4)*A
ppellF1[3/4, 1, -1/4, 7/4, I*x^2, -x^2])/(6*(1 + x^2)^(1/4)) + ((x^2 + x^4)^(1/4)*ArcTan[(2^(1/4)*Sqrt[x])/(1
+ x^2)^(1/4)])/(2*2^(3/4)*Sqrt[x]*(1 + x^2)^(1/4)) - ((x^2 + x^4)^(1/4)*ArcTanh[(2^(1/4)*Sqrt[x])/(1 + x^2)^(1
/4)])/(2*2^(3/4)*Sqrt[x]*(1 + x^2)^(1/4))
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 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 231
Int[((a_) + (b_.)*(x_)^2)^(-3/4), x_Symbol] :> Simp[(2*EllipticF[(1*ArcTan[Rt[b/a, 2]*x])/2, 2])/(a^(3/4)*Rt[b
/a, 2]), x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && PosQ[b/a]
Rule 237
Int[((a_) + (b_.)*(x_)^4)^(-3/4), x_Symbol] :> Dist[(x^3*(1 + a/(b*x^4))^(3/4))/(a + b*x^4)^(3/4), Int[1/(x^3*
(1 + a/(b*x^4))^(3/4)), x], x] /; FreeQ[{a, b}, x]
Rule 275
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = GCD[m + 1, n]}, Dist[1/k, Subst[Int[x^((m
+ 1)/k - 1)*(a + b*x^(n/k))^p, x], x, x^k], x] /; k != 1] /; FreeQ[{a, b, p}, x] && IGtQ[n, 0] && IntegerQ[m]
Rule 298
Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[-(a/b), 2]], s = Denominator[Rt[-(a/b),
2]]}, Dist[s/(2*b), Int[1/(r + s*x^2), x], x] - Dist[s/(2*b), Int[1/(r - s*x^2), x], x]] /; FreeQ[{a, b}, x] &
& !GtQ[a/b, 0]
Rule 335
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Subst[Int[(a + b/x^n)^p/x^(m + 2), x], x, 1/x] /;
FreeQ[{a, b, p}, x] && ILtQ[n, 0] && IntegerQ[m]
Rule 407
Int[((a_) + (b_.)*(x_)^4)^(1/4)/((c_) + (d_.)*(x_)^4), x_Symbol] :> Dist[Sqrt[a + b*x^4]*Sqrt[a/(a + b*x^4)],
Subst[Int[1/(Sqrt[1 - b*x^4]*(c - (b*c - a*d)*x^4)), x], x, x/(a + b*x^4)^(1/4)], 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 410
Int[1/(((a_) + (b_.)*(x_)^4)^(3/4)*((c_) + (d_.)*(x_)^4)), x_Symbol] :> Dist[b/(b*c - a*d), Int[1/(a + b*x^4)^
(3/4), x], x] - Dist[d/(b*c - a*d), Int[(a + b*x^4)^(1/4)/(c + d*x^4), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ
[b*c - a*d, 0]
Rule 494
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> With[{k = Denominato
r[p]}, Dist[(k*a^(p + (m + 1)/n))/n, Subst[Int[(x^((k*(m + 1))/n - 1)*(c - (b*c - a*d)*x^k)^q)/(1 - b*x^k)^(p
+ q + (m + 1)/n + 1), x], x, x^(n/k)/(a + b*x^n)^(1/k)], x]] /; FreeQ[{a, b, c, d}, x] && IGtQ[n, 0] && Ration
alQ[m, p] && IntegersQ[p + (m + 1)/n, q] && LtQ[-1, p, 0]
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 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 1213
Int[1/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]), x_Symbol] :> With[{q = Rt[-(a*c), 2]}, Dist[Sqrt[-c],
Int[1/((d + e*x^2)*Sqrt[q + c*x^2]*Sqrt[q - c*x^2]), x], x]] /; FreeQ[{a, c, d, e}, x] && GtQ[a, 0] && LtQ[c,
0]
Rule 1240
Int[((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (c_.)*(x_)^4)^(p_), x_Symbol] :> Int[ExpandIntegrand[(a + c*x^4)^p, (d/
(d^2 - e^2*x^4) - (e*x^2)/(d^2 - e^2*x^4))^(-q), x], x] /; FreeQ[{a, c, d, e, p}, x] && NeQ[c*d^2 + a*e^2, 0]
&& !IntegerQ[p] && ILtQ[q, 0]
Rule 1529
Int[(((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(n_))^(q_))/((a_) + (c_.)*(x_)^(n2_.)), x_Symbol] :> Int[ExpandInte
grand[(d + e*x^n)^q, (f*x)^m/(a + c*x^(2*n)), x], x] /; FreeQ[{a, c, d, e, f, q, n}, x] && EqQ[n2, 2*n] && IGt
Q[n, 0] && !IntegerQ[q] && IntegerQ[m]
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 2056
Int[(u_.)*(P_)^(p_.), x_Symbol] :> With[{m = MinimumMonomialExponent[P, x]}, Dist[P^FracPart[p]/(x^(m*FracPart
[p])*Distrib[1/x^m, P]^FracPart[p]), Int[u*x^(m*p)*Distrib[1/x^m, P]^p, x], x]] /; FreeQ[p, x] && !IntegerQ[p
] && SumQ[P] && EveryQ[BinomialQ[#1, x] & , P] && !PolyQ[P, x, 2]
Rule 6733
Int[(u_)*(x_)^(m_), x_Symbol] :> With[{k = Denominator[m]}, Dist[k, Subst[Int[x^(k*(m + 1) - 1)*(u /. x -> x^k
), x], x, x^(1/k)], x]] /; FractionQ[m]
Rule 6742
Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]
Rubi steps
\begin {align*} \int \frac {x^4 \sqrt [4]{x^2+x^4}}{-1+x^8} \, dx &=\frac {\sqrt [4]{x^2+x^4} \int \frac {x^{9/2} \sqrt [4]{1+x^2}}{-1+x^8} \, dx}{\sqrt {x} \sqrt [4]{1+x^2}}\\ &=\frac {\sqrt [4]{x^2+x^4} \int \frac {x^{9/2}}{\left (1+x^2\right )^{3/4} \left (-1+x^2-x^4+x^6\right )} \, dx}{\sqrt {x} \sqrt [4]{1+x^2}}\\ &=\frac {\left (2 \sqrt [4]{x^2+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^{10}}{\left (1+x^4\right )^{3/4} \left (-1+x^4-x^8+x^{12}\right )} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt [4]{1+x^2}}\\ &=\frac {\left (2 \sqrt [4]{x^2+x^4}\right ) \operatorname {Subst}\left (\int \left (\frac {1}{4 \left (-1+x^2\right ) \left (1+x^4\right )^{3/4}}+\frac {1}{4 \left (1+x^2\right ) \left (1+x^4\right )^{3/4}}+\frac {x^2 \sqrt [4]{1+x^4}}{2 \left (1+x^8\right )}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt [4]{1+x^2}}\\ &=\frac {\sqrt [4]{x^2+x^4} \operatorname {Subst}\left (\int \frac {1}{\left (-1+x^2\right ) \left (1+x^4\right )^{3/4}} \, dx,x,\sqrt {x}\right )}{2 \sqrt {x} \sqrt [4]{1+x^2}}+\frac {\sqrt [4]{x^2+x^4} \operatorname {Subst}\left (\int \frac {1}{\left (1+x^2\right ) \left (1+x^4\right )^{3/4}} \, dx,x,\sqrt {x}\right )}{2 \sqrt {x} \sqrt [4]{1+x^2}}+\frac {\sqrt [4]{x^2+x^4} \operatorname {Subst}\left (\int \frac {x^2 \sqrt [4]{1+x^4}}{1+x^8} \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt [4]{1+x^2}}\\ &=\frac {\sqrt [4]{x^2+x^4} \operatorname {Subst}\left (\int \left (\frac {1}{\left (1-x^4\right ) \left (1+x^4\right )^{3/4}}+\frac {x^2}{\left (-1+x^4\right ) \left (1+x^4\right )^{3/4}}\right ) \, dx,x,\sqrt {x}\right )}{2 \sqrt {x} \sqrt [4]{1+x^2}}+\frac {\sqrt [4]{x^2+x^4} \operatorname {Subst}\left (\int \left (\frac {1}{\left (-1+x^4\right ) \left (1+x^4\right )^{3/4}}+\frac {x^2}{\left (-1+x^4\right ) \left (1+x^4\right )^{3/4}}\right ) \, dx,x,\sqrt {x}\right )}{2 \sqrt {x} \sqrt [4]{1+x^2}}+\frac {\sqrt [4]{x^2+x^4} \operatorname {Subst}\left (\int \left (-\frac {i x^2 \sqrt [4]{1+x^4}}{2 \left (-i+x^4\right )}+\frac {i x^2 \sqrt [4]{1+x^4}}{2 \left (i+x^4\right )}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt {x} \sqrt [4]{1+x^2}}\\ &=-\frac {\left (i \sqrt [4]{x^2+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2 \sqrt [4]{1+x^4}}{-i+x^4} \, dx,x,\sqrt {x}\right )}{2 \sqrt {x} \sqrt [4]{1+x^2}}+\frac {\left (i \sqrt [4]{x^2+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^2 \sqrt [4]{1+x^4}}{i+x^4} \, dx,x,\sqrt {x}\right )}{2 \sqrt {x} \sqrt [4]{1+x^2}}+\frac {\sqrt [4]{x^2+x^4} \operatorname {Subst}\left (\int \frac {1}{\left (1-x^4\right ) \left (1+x^4\right )^{3/4}} \, dx,x,\sqrt {x}\right )}{2 \sqrt {x} \sqrt [4]{1+x^2}}+\frac {\sqrt [4]{x^2+x^4} \operatorname {Subst}\left (\int \frac {1}{\left (-1+x^4\right ) \left (1+x^4\right )^{3/4}} \, dx,x,\sqrt {x}\right )}{2 \sqrt {x} \sqrt [4]{1+x^2}}+2 \frac {\sqrt [4]{x^2+x^4} \operatorname {Subst}\left (\int \frac {x^2}{\left (-1+x^4\right ) \left (1+x^4\right )^{3/4}} \, dx,x,\sqrt {x}\right )}{2 \sqrt {x} \sqrt [4]{1+x^2}}\\ &=\frac {x \sqrt [4]{x^2+x^4} F_1\left (\frac {3}{4};1,-\frac {1}{4};\frac {7}{4};-i x^2,-x^2\right )}{6 \sqrt [4]{1+x^2}}+\frac {x \sqrt [4]{x^2+x^4} F_1\left (\frac {3}{4};1,-\frac {1}{4};\frac {7}{4};i x^2,-x^2\right )}{6 \sqrt [4]{1+x^2}}+\frac {\sqrt [4]{x^2+x^4} \operatorname {Subst}\left (\int \frac {\sqrt [4]{1+x^4}}{1-x^4} \, dx,x,\sqrt {x}\right )}{4 \sqrt {x} \sqrt [4]{1+x^2}}+\frac {\sqrt [4]{x^2+x^4} \operatorname {Subst}\left (\int \frac {\sqrt [4]{1+x^4}}{-1+x^4} \, dx,x,\sqrt {x}\right )}{4 \sqrt {x} \sqrt [4]{1+x^2}}+2 \frac {\sqrt [4]{x^2+x^4} \operatorname {Subst}\left (\int \frac {x^2}{-1+2 x^4} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{1+x^2}}\right )}{2 \sqrt {x} \sqrt [4]{1+x^2}}\\ &=\frac {x \sqrt [4]{x^2+x^4} F_1\left (\frac {3}{4};1,-\frac {1}{4};\frac {7}{4};-i x^2,-x^2\right )}{6 \sqrt [4]{1+x^2}}+\frac {x \sqrt [4]{x^2+x^4} F_1\left (\frac {3}{4};1,-\frac {1}{4};\frac {7}{4};i x^2,-x^2\right )}{6 \sqrt [4]{1+x^2}}+2 \left (-\frac {\sqrt [4]{x^2+x^4} \operatorname {Subst}\left (\int \frac {1}{1-\sqrt {2} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{1+x^2}}\right )}{4 \sqrt {2} \sqrt {x} \sqrt [4]{1+x^2}}+\frac {\sqrt [4]{x^2+x^4} \operatorname {Subst}\left (\int \frac {1}{1+\sqrt {2} x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{1+x^2}}\right )}{4 \sqrt {2} \sqrt {x} \sqrt [4]{1+x^2}}\right )+\frac {\left (\sqrt {\frac {1}{1+x^2}} \sqrt [4]{1+x^2} \sqrt [4]{x^2+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1-2 x^4\right ) \sqrt {1-x^4}} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{1+x^2}}\right )}{4 \sqrt {x}}+\frac {\left (\sqrt {\frac {1}{1+x^2}} \sqrt [4]{1+x^2} \sqrt [4]{x^2+x^4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-x^4} \left (-1+2 x^4\right )} \, dx,x,\frac {\sqrt {x}}{\sqrt [4]{1+x^2}}\right )}{4 \sqrt {x}}\\ &=\frac {x \sqrt [4]{x^2+x^4} F_1\left (\frac {3}{4};1,-\frac {1}{4};\frac {7}{4};-i x^2,-x^2\right )}{6 \sqrt [4]{1+x^2}}+\frac {x \sqrt [4]{x^2+x^4} F_1\left (\frac {3}{4};1,-\frac {1}{4};\frac {7}{4};i x^2,-x^2\right )}{6 \sqrt [4]{1+x^2}}+2 \left (\frac {\sqrt [4]{x^2+x^4} \tan ^{-1}\left (\frac {\sqrt [4]{2} \sqrt {x}}{\sqrt [4]{1+x^2}}\right )}{4\ 2^{3/4} \sqrt {x} \sqrt [4]{1+x^2}}-\frac {\sqrt [4]{x^2+x^4} \tanh ^{-1}\left (\frac {\sqrt [4]{2} \sqrt {x}}{\sqrt [4]{1+x^2}}\right )}{4\ 2^{3/4} \sqrt {x} \sqrt [4]{1+x^2}}\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [F] time = 0.80, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^4 \sqrt [4]{x^2+x^4}}{-1+x^8} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
Integrate[(x^4*(x^2 + x^4)^(1/4))/(-1 + x^8),x]
[Out]
Integrate[(x^4*(x^2 + x^4)^(1/4))/(-1 + x^8), x]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 0.00, size = 113, normalized size = 1.00 \begin {gather*} \frac {\tan ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^2+x^4}}\right )}{2\ 2^{3/4}}-\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^2+x^4}}\right )}{2\ 2^{3/4}}-\frac {1}{8} \text {RootSum}\left [2-2 \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-\log (x) \text {$\#$1}+\log \left (\sqrt [4]{x^2+x^4}-x \text {$\#$1}\right ) \text {$\#$1}}{-1+\text {$\#$1}^4}\&\right ] \end {gather*}
Antiderivative was successfully verified.
[In]
IntegrateAlgebraic[(x^4*(x^2 + x^4)^(1/4))/(-1 + x^8),x]
[Out]
ArcTan[(2^(1/4)*x)/(x^2 + x^4)^(1/4)]/(2*2^(3/4)) - ArcTanh[(2^(1/4)*x)/(x^2 + x^4)^(1/4)]/(2*2^(3/4)) - RootS
um[2 - 2*#1^4 + #1^8 & , (-(Log[x]*#1) + Log[(x^2 + x^4)^(1/4) - x*#1]*#1)/(-1 + #1^4) & ]/8
________________________________________________________________________________________
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^4*(x^4+x^2)^(1/4)/(x^8-1),x, algorithm="fricas")
[Out]
Timed out
________________________________________________________________________________________
giac [B] time = 0.23, size = 218, normalized size = 1.93 \begin {gather*} -\frac {1}{4} \cdot 2^{\frac {1}{4}} \arctan \left (\frac {1}{2} \cdot 2^{\frac {3}{4}} {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}}\right ) - 8 i \, \left (-\frac {1}{16777216} i + \frac {1}{16777216}\right )^{\frac {1}{4}} \log \left (\left (-4722366482869645213696 i + 4722366482869645213696\right )^{\frac {1}{4}} - 262144 \, {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}}\right ) + 8 i \, \left (-\frac {1}{16777216} i + \frac {1}{16777216}\right )^{\frac {1}{4}} \log \left (-\left (-4722366482869645213696 i + 4722366482869645213696\right )^{\frac {1}{4}} - 262144 \, {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}}\right ) + i \, \left (\frac {1}{4096} i + \frac {1}{4096}\right )^{\frac {1}{4}} \log \left (\left (68719476736 i + 68719476736\right )^{\frac {1}{4}} - 512 \, {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}}\right ) - \left (\frac {1}{4096} i + \frac {1}{4096}\right )^{\frac {1}{4}} \log \left (i \, \left (68719476736 i + 68719476736\right )^{\frac {1}{4}} - 512 \, {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}}\right ) + \left (\frac {1}{4096} i + \frac {1}{4096}\right )^{\frac {1}{4}} \log \left (-i \, \left (68719476736 i + 68719476736\right )^{\frac {1}{4}} - 512 \, {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}}\right ) - i \, \left (\frac {1}{4096} i + \frac {1}{4096}\right )^{\frac {1}{4}} \log \left (-\left (68719476736 i + 68719476736\right )^{\frac {1}{4}} - 512 \, {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}}\right ) + \left (-\frac {1}{4096} i + \frac {1}{4096}\right )^{\frac {1}{4}} \log \left (i \, \left (-68719476736 i + 68719476736\right )^{\frac {1}{4}} - 512 \, {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}}\right ) - \left (-\frac {1}{4096} i + \frac {1}{4096}\right )^{\frac {1}{4}} \log \left (-i \, \left (-68719476736 i + 68719476736\right )^{\frac {1}{4}} - 512 \, {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}}\right ) - \frac {1}{8} \cdot 2^{\frac {1}{4}} \log \left (2^{\frac {1}{4}} + {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}}\right ) + \frac {1}{8} \cdot 2^{\frac {1}{4}} \log \left ({\left | -2^{\frac {1}{4}} + {\left (\frac {1}{x^{2}} + 1\right )}^{\frac {1}{4}} \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^4*(x^4+x^2)^(1/4)/(x^8-1),x, algorithm="giac")
[Out]
-1/4*2^(1/4)*arctan(1/2*2^(3/4)*(1/x^2 + 1)^(1/4)) - 8*I*(-1/16777216*I + 1/16777216)^(1/4)*log((-472236648286
9645213696*I + 4722366482869645213696)^(1/4) - 262144*(1/x^2 + 1)^(1/4)) + 8*I*(-1/16777216*I + 1/16777216)^(1
/4)*log(-(-4722366482869645213696*I + 4722366482869645213696)^(1/4) - 262144*(1/x^2 + 1)^(1/4)) + I*(1/4096*I
+ 1/4096)^(1/4)*log((68719476736*I + 68719476736)^(1/4) - 512*(1/x^2 + 1)^(1/4)) - (1/4096*I + 1/4096)^(1/4)*l
og(I*(68719476736*I + 68719476736)^(1/4) - 512*(1/x^2 + 1)^(1/4)) + (1/4096*I + 1/4096)^(1/4)*log(-I*(68719476
736*I + 68719476736)^(1/4) - 512*(1/x^2 + 1)^(1/4)) - I*(1/4096*I + 1/4096)^(1/4)*log(-(68719476736*I + 687194
76736)^(1/4) - 512*(1/x^2 + 1)^(1/4)) + (-1/4096*I + 1/4096)^(1/4)*log(I*(-68719476736*I + 68719476736)^(1/4)
- 512*(1/x^2 + 1)^(1/4)) - (-1/4096*I + 1/4096)^(1/4)*log(-I*(-68719476736*I + 68719476736)^(1/4) - 512*(1/x^2
+ 1)^(1/4)) - 1/8*2^(1/4)*log(2^(1/4) + (1/x^2 + 1)^(1/4)) + 1/8*2^(1/4)*log(abs(-2^(1/4) + (1/x^2 + 1)^(1/4)
))
________________________________________________________________________________________
maple [B] time = 64.33, size = 3624, normalized size = 32.07
| | |
| method |
result |
size |
| | |
| trager |
\(\text {Expression too large to display}\) |
\(3624\) |
| | |
|
|
|
|
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(x^4*(x^4+x^2)^(1/4)/(x^8-1),x,method=_RETURNVERBOSE)
[Out]
-4096*RootOf(8388608*_Z^8-4096*_Z^4+1)^5*ln((62277025792*RootOf(8388608*_Z^8-4096*_Z^4+1)^11*x^3-249108103168*
RootOf(8388608*_Z^8-4096*_Z^4+1)^11*x+180879360*x^3*RootOf(8388608*_Z^8-4096*_Z^4+1)^7-104595456*(x^4+x^2)^(1/
4)*RootOf(8388608*_Z^8-4096*_Z^4+1)^6*x^2+751304704*x*RootOf(8388608*_Z^8-4096*_Z^4+1)^7-13074432*(x^4+x^2)^(1
/2)*RootOf(8388608*_Z^8-4096*_Z^4+1)^5*x-1634304*(x^4+x^2)^(3/4)*RootOf(8388608*_Z^8-4096*_Z^4+1)^4-709632*x^3
*RootOf(8388608*_Z^8-4096*_Z^4+1)^3-76864*(x^4+x^2)^(1/4)*RootOf(8388608*_Z^8-4096*_Z^4+1)^2*x^2-439296*RootOf
(8388608*_Z^8-4096*_Z^4+1)^3*x-9608*(x^4+x^2)^(1/2)*RootOf(8388608*_Z^8-4096*_Z^4+1)*x-1201*(x^4+x^2)^(3/4))/x
/(4096*RootOf(8388608*_Z^8-4096*_Z^4+1)^4*x^2-16384*RootOf(8388608*_Z^8-4096*_Z^4+1)^4-5*x^2+3))+RootOf(838860
8*_Z^8-4096*_Z^4+1)*ln((62277025792*RootOf(8388608*_Z^8-4096*_Z^4+1)^11*x^3-249108103168*RootOf(8388608*_Z^8-4
096*_Z^4+1)^11*x+180879360*x^3*RootOf(8388608*_Z^8-4096*_Z^4+1)^7-104595456*(x^4+x^2)^(1/4)*RootOf(8388608*_Z^
8-4096*_Z^4+1)^6*x^2+751304704*x*RootOf(8388608*_Z^8-4096*_Z^4+1)^7-13074432*(x^4+x^2)^(1/2)*RootOf(8388608*_Z
^8-4096*_Z^4+1)^5*x-1634304*(x^4+x^2)^(3/4)*RootOf(8388608*_Z^8-4096*_Z^4+1)^4-709632*x^3*RootOf(8388608*_Z^8-
4096*_Z^4+1)^3-76864*(x^4+x^2)^(1/4)*RootOf(8388608*_Z^8-4096*_Z^4+1)^2*x^2-439296*RootOf(8388608*_Z^8-4096*_Z
^4+1)^3*x-9608*(x^4+x^2)^(1/2)*RootOf(8388608*_Z^8-4096*_Z^4+1)*x-1201*(x^4+x^2)^(3/4))/x/(4096*RootOf(8388608
*_Z^8-4096*_Z^4+1)^4*x^2-16384*RootOf(8388608*_Z^8-4096*_Z^4+1)^4-5*x^2+3))+1/32*RootOf(_Z^4+1048576*RootOf(83
88608*_Z^8-4096*_Z^4+1)^4-512)*ln((492830720*RootOf(_Z^4+1048576*RootOf(8388608*_Z^8-4096*_Z^4+1)^4-512)^3*Roo
tOf(8388608*_Z^8-4096*_Z^4+1)^8*x^3-1971322880*RootOf(_Z^4+1048576*RootOf(8388608*_Z^8-4096*_Z^4+1)^4-512)^3*R
ootOf(8388608*_Z^8-4096*_Z^4+1)^8*x-2902528*RootOf(_Z^4+1048576*RootOf(8388608*_Z^8-4096*_Z^4+1)^4-512)^3*Root
Of(8388608*_Z^8-4096*_Z^4+1)^4*x^3-3268608*RootOf(_Z^4+1048576*RootOf(8388608*_Z^8-4096*_Z^4+1)^4-512)^2*(x^4+
x^2)^(1/4)*RootOf(8388608*_Z^8-4096*_Z^4+1)^4*x^2-60928*RootOf(_Z^4+1048576*RootOf(8388608*_Z^8-4096*_Z^4+1)^4
-512)^3*RootOf(8388608*_Z^8-4096*_Z^4+1)^4*x+52428800*(x^4+x^2)^(1/2)*RootOf(8388608*_Z^8-4096*_Z^4+1)^4*RootO
f(_Z^4+1048576*RootOf(8388608*_Z^8-4096*_Z^4+1)^4-512)*x+52297728*(x^4+x^2)^(3/4)*RootOf(8388608*_Z^8-4096*_Z^
4+1)^4+759*RootOf(_Z^4+1048576*RootOf(8388608*_Z^8-4096*_Z^4+1)^4-512)^3*x^3+3998*RootOf(_Z^4+1048576*RootOf(8
388608*_Z^8-4096*_Z^4+1)^4-512)^2*(x^4+x^2)^(1/4)*x^2+165*RootOf(_Z^4+1048576*RootOf(8388608*_Z^8-4096*_Z^4+1)
^4-512)^3*x-9608*(x^4+x^2)^(1/2)*RootOf(_Z^4+1048576*RootOf(8388608*_Z^8-4096*_Z^4+1)^4-512)*x-63968*(x^4+x^2)
^(3/4))/x/(4096*RootOf(8388608*_Z^8-4096*_Z^4+1)^4*x^2-16384*RootOf(8388608*_Z^8-4096*_Z^4+1)^4+3*x^2+5))+128*
ln((243269632*RootOf(_Z^4+1048576*RootOf(8388608*_Z^8-4096*_Z^4+1)^4-512)^3*RootOf(8388608*_Z^8-4096*_Z^4+1)^8
*x^3-973078528*RootOf(_Z^4+1048576*RootOf(8388608*_Z^8-4096*_Z^4+1)^4-512)^3*RootOf(8388608*_Z^8-4096*_Z^4+1)^
8*x-944128*RootOf(_Z^4+1048576*RootOf(8388608*_Z^8-4096*_Z^4+1)^4-512)^3*RootOf(8388608*_Z^8-4096*_Z^4+1)^4*x^
3+13074432*RootOf(_Z^4+1048576*RootOf(8388608*_Z^8-4096*_Z^4+1)^4-512)^2*(x^4+x^2)^(1/4)*RootOf(8388608*_Z^8-4
096*_Z^4+1)^4*x^2-1984512*RootOf(_Z^4+1048576*RootOf(8388608*_Z^8-4096*_Z^4+1)^4-512)^3*RootOf(8388608*_Z^8-40
96*_Z^4+1)^4*x+52297728*(x^4+x^2)^(1/2)*RootOf(8388608*_Z^8-4096*_Z^4+1)^4*RootOf(_Z^4+1048576*RootOf(8388608*
_Z^8-4096*_Z^4+1)^4-512)*x+209190912*(x^4+x^2)^(3/4)*RootOf(8388608*_Z^8-4096*_Z^4+1)^4-2369*RootOf(_Z^4+10485
76*RootOf(8388608*_Z^8-4096*_Z^4+1)^4-512)^3*x^3-15992*RootOf(_Z^4+1048576*RootOf(8388608*_Z^8-4096*_Z^4+1)^4-
512)^2*(x^4+x^2)^(1/4)*x^2-515*RootOf(_Z^4+1048576*RootOf(8388608*_Z^8-4096*_Z^4+1)^4-512)^3*x-63968*(x^4+x^2)
^(1/2)*RootOf(_Z^4+1048576*RootOf(8388608*_Z^8-4096*_Z^4+1)^4-512)*x-255872*(x^4+x^2)^(3/4))/x/(4096*RootOf(83
88608*_Z^8-4096*_Z^4+1)^4*x^2-16384*RootOf(8388608*_Z^8-4096*_Z^4+1)^4+3*x^2+5))*RootOf(8388608*_Z^8-4096*_Z^4
+1)^4*RootOf(_Z^4+1048576*RootOf(8388608*_Z^8-4096*_Z^4+1)^4-512)-1/32*ln((243269632*RootOf(_Z^4+1048576*RootO
f(8388608*_Z^8-4096*_Z^4+1)^4-512)^3*RootOf(8388608*_Z^8-4096*_Z^4+1)^8*x^3-973078528*RootOf(_Z^4+1048576*Root
Of(8388608*_Z^8-4096*_Z^4+1)^4-512)^3*RootOf(8388608*_Z^8-4096*_Z^4+1)^8*x-944128*RootOf(_Z^4+1048576*RootOf(8
388608*_Z^8-4096*_Z^4+1)^4-512)^3*RootOf(8388608*_Z^8-4096*_Z^4+1)^4*x^3+13074432*RootOf(_Z^4+1048576*RootOf(8
388608*_Z^8-4096*_Z^4+1)^4-512)^2*(x^4+x^2)^(1/4)*RootOf(8388608*_Z^8-4096*_Z^4+1)^4*x^2-1984512*RootOf(_Z^4+1
048576*RootOf(8388608*_Z^8-4096*_Z^4+1)^4-512)^3*RootOf(8388608*_Z^8-4096*_Z^4+1)^4*x+52297728*(x^4+x^2)^(1/2)
*RootOf(8388608*_Z^8-4096*_Z^4+1)^4*RootOf(_Z^4+1048576*RootOf(8388608*_Z^8-4096*_Z^4+1)^4-512)*x+209190912*(x
^4+x^2)^(3/4)*RootOf(8388608*_Z^8-4096*_Z^4+1)^4-2369*RootOf(_Z^4+1048576*RootOf(8388608*_Z^8-4096*_Z^4+1)^4-5
12)^3*x^3-15992*RootOf(_Z^4+1048576*RootOf(8388608*_Z^8-4096*_Z^4+1)^4-512)^2*(x^4+x^2)^(1/4)*x^2-515*RootOf(_
Z^4+1048576*RootOf(8388608*_Z^8-4096*_Z^4+1)^4-512)^3*x-63968*(x^4+x^2)^(1/2)*RootOf(_Z^4+1048576*RootOf(83886
08*_Z^8-4096*_Z^4+1)^4-512)*x-255872*(x^4+x^2)^(3/4))/x/(4096*RootOf(8388608*_Z^8-4096*_Z^4+1)^4*x^2-16384*Roo
tOf(8388608*_Z^8-4096*_Z^4+1)^4+3*x^2+5))*RootOf(_Z^4+1048576*RootOf(8388608*_Z^8-4096*_Z^4+1)^4-512)-RootOf(8
388608*_Z^8-4096*_Z^4+1)*ln(-(504658657280*RootOf(8388608*_Z^8-4096*_Z^4+1)^11*x^3-2018634629120*RootOf(838860
8*_Z^8-4096*_Z^4+1)^11*x+2479357952*x^3*RootOf(8388608*_Z^8-4096*_Z^4+1)^7-104595456*(x^4+x^2)^(1/4)*RootOf(83
88608*_Z^8-4096*_Z^4+1)^6*x^2+2033713152*x*RootOf(8388608*_Z^8-4096*_Z^4+1)^7-52428800*(x^4+x^2)^(1/2)*RootOf(
8388608*_Z^8-4096*_Z^4+1)^5*x+1634304*(x^4+x^2)^(3/4)*RootOf(8388608*_Z^8-4096*_Z^4+1)^4-553728*x^3*RootOf(838
8608*_Z^8-4096*_Z^4+1)^3-76864*(x^4+x^2)^(1/4)*RootOf(8388608*_Z^8-4096*_Z^4+1)^2*x^2-342784*RootOf(8388608*_Z
^8-4096*_Z^4+1)^3*x+15992*(x^4+x^2)^(1/2)*RootOf(8388608*_Z^8-4096*_Z^4+1)*x+1201*(x^4+x^2)^(3/4))/x/(4096*Roo
tOf(8388608*_Z^8-4096*_Z^4+1)^4*x^2-16384*RootOf(8388608*_Z^8-4096*_Z^4+1)^4-5*x^2+3))-1024*ln(-(24576*RootOf(
_Z^4+1048576*RootOf(8388608*_Z^8-4096*_Z^4+1)^4-512)^3*RootOf(8388608*_Z^8-4096*_Z^4+1)^7*x^3+8192*RootOf(_Z^4
+1048576*RootOf(8388608*_Z^8-4096*_Z^4+1)^4-512)^3*RootOf(8388608*_Z^8-4096*_Z^4+1)^7*x-6*RootOf(8388608*_Z^8-
4096*_Z^4+1)^3*RootOf(_Z^4+1048576*RootOf(8388608*_Z^8-4096*_Z^4+1)^4-512)^3*x^3-8192*(x^4+x^2)^(1/2)*RootOf(_
Z^4+1048576*RootOf(8388608*_Z^8-4096*_Z^4+1)^4-512)*RootOf(8388608*_Z^8-4096*_Z^4+1)^5*x-2*RootOf(8388608*_Z^8
-4096*_Z^4+1)^3*RootOf(_Z^4+1048576*RootOf(8388608*_Z^8-4096*_Z^4+1)^4-512)^3*x+4*RootOf(_Z^4+1048576*RootOf(8
388608*_Z^8-4096*_Z^4+1)^4-512)^2*(x^4+x^2)^(1/4)*RootOf(8388608*_Z^8-4096*_Z^4+1)^2*x^2+2*RootOf(8388608*_Z^8
-4096*_Z^4+1)*RootOf(_Z^4+1048576*RootOf(8388608*_Z^8-4096*_Z^4+1)^4-512)*(x^4+x^2)^(1/2)*x-(x^4+x^2)^(3/4))/(
1+x)/x/(-1+x))*RootOf(8388608*_Z^8-4096*_Z^4+1)^5*RootOf(_Z^4+1048576*RootOf(8388608*_Z^8-4096*_Z^4+1)^4-512)+
1/4*ln(-(24576*RootOf(_Z^4+1048576*RootOf(8388608*_Z^8-4096*_Z^4+1)^4-512)^3*RootOf(8388608*_Z^8-4096*_Z^4+1)^
7*x^3+8192*RootOf(_Z^4+1048576*RootOf(8388608*_Z^8-4096*_Z^4+1)^4-512)^3*RootOf(8388608*_Z^8-4096*_Z^4+1)^7*x-
6*RootOf(8388608*_Z^8-4096*_Z^4+1)^3*RootOf(_Z^4+1048576*RootOf(8388608*_Z^8-4096*_Z^4+1)^4-512)^3*x^3-8192*(x
^4+x^2)^(1/2)*RootOf(_Z^4+1048576*RootOf(8388608*_Z^8-4096*_Z^4+1)^4-512)*RootOf(8388608*_Z^8-4096*_Z^4+1)^5*x
-2*RootOf(8388608*_Z^8-4096*_Z^4+1)^3*RootOf(_Z^4+1048576*RootOf(8388608*_Z^8-4096*_Z^4+1)^4-512)^3*x+4*RootOf
(_Z^4+1048576*RootOf(8388608*_Z^8-4096*_Z^4+1)^4-512)^2*(x^4+x^2)^(1/4)*RootOf(8388608*_Z^8-4096*_Z^4+1)^2*x^2
+2*RootOf(8388608*_Z^8-4096*_Z^4+1)*RootOf(_Z^4+1048576*RootOf(8388608*_Z^8-4096*_Z^4+1)^4-512)*(x^4+x^2)^(1/2
)*x-(x^4+x^2)^(3/4))/(1+x)/x/(-1+x))*RootOf(8388608*_Z^8-4096*_Z^4+1)*RootOf(_Z^4+1048576*RootOf(8388608*_Z^8-
4096*_Z^4+1)^4-512)-1/4*RootOf(8388608*_Z^8-4096*_Z^4+1)*RootOf(_Z^4+1048576*RootOf(8388608*_Z^8-4096*_Z^4+1)^
4-512)*ln((6*RootOf(8388608*_Z^8-4096*_Z^4+1)^3*RootOf(_Z^4+1048576*RootOf(8388608*_Z^8-4096*_Z^4+1)^4-512)^3*
x^3+2*RootOf(8388608*_Z^8-4096*_Z^4+1)^3*RootOf(_Z^4+1048576*RootOf(8388608*_Z^8-4096*_Z^4+1)^4-512)^3*x+4*Roo
tOf(_Z^4+1048576*RootOf(8388608*_Z^8-4096*_Z^4+1)^4-512)^2*(x^4+x^2)^(1/4)*RootOf(8388608*_Z^8-4096*_Z^4+1)^2*
x^2+2*RootOf(8388608*_Z^8-4096*_Z^4+1)*RootOf(_Z^4+1048576*RootOf(8388608*_Z^8-4096*_Z^4+1)^4-512)*(x^4+x^2)^(
1/2)*x+(x^4+x^2)^(3/4))/(1+x)/x/(-1+x))
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} -\frac {2 \, {\left (x^{3} + x\right )} {\left (x^{2} + 1\right )}^{\frac {1}{4}} x^{\frac {9}{2}}}{5 \, {\left (x^{8} - 1\right )}} - \int \frac {16 \, {\left (x^{2} + 1\right )}^{\frac {5}{4}} x^{\frac {9}{2}}}{5 \, {\left (x^{16} - 2 \, x^{8} + 1\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^4*(x^4+x^2)^(1/4)/(x^8-1),x, algorithm="maxima")
[Out]
-2/5*(x^3 + x)*(x^2 + 1)^(1/4)*x^(9/2)/(x^8 - 1) - integrate(16/5*(x^2 + 1)^(5/4)*x^(9/2)/(x^16 - 2*x^8 + 1),
x)
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x^4\,{\left (x^4+x^2\right )}^{1/4}}{x^8-1} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((x^4*(x^2 + x^4)^(1/4))/(x^8 - 1),x)
[Out]
int((x^4*(x^2 + x^4)^(1/4))/(x^8 - 1), x)
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{4} \sqrt [4]{x^{2} \left (x^{2} + 1\right )}}{\left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right ) \left (x^{4} + 1\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x**4*(x**4+x**2)**(1/4)/(x**8-1),x)
[Out]
Integral(x**4*(x**2*(x**2 + 1))**(1/4)/((x - 1)*(x + 1)*(x**2 + 1)*(x**4 + 1)), x)
________________________________________________________________________________________