3.21.100 \(\int \frac {x^4}{\sqrt [4]{-1+x^4} (-1+2 x^4+x^8)} \, dx\)

Optimal. Leaf size=152 \[ \frac {\tan ^{-1}\left (\frac {\sqrt [8]{2} x}{\sqrt [4]{x^4-1}}\right )}{4\ 2^{5/8}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [8]{2} x}{\sqrt [4]{x^4-1}}\right )}{4\ 2^{5/8}}+\frac {\tan ^{-1}\left (\frac {2^{5/8} x \sqrt [4]{x^4-1}}{\sqrt [4]{2} x^2-\sqrt {x^4-1}}\right )}{8 \sqrt [8]{2}}-\frac {\tanh ^{-1}\left (\frac {2\ 2^{3/8} x \sqrt [4]{x^4-1}}{2^{3/4} \sqrt {x^4-1}+2 x^2}\right )}{8 \sqrt [8]{2}} \]

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Rubi [A]  time = 0.26, antiderivative size = 282, normalized size of antiderivative = 1.86, number of steps used = 16, number of rules used = 11, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.440, Rules used = {1528, 377, 211, 1165, 628, 1162, 617, 204, 212, 206, 203} \begin {gather*} \frac {\tan ^{-1}\left (\frac {\sqrt [8]{2} x}{\sqrt [4]{x^4-1}}\right )}{4\ 2^{5/8}}-\frac {\left (1-\sqrt {2}\right ) \tan ^{-1}\left (1-\frac {2^{5/8} x}{\sqrt [4]{x^4-1}}\right )}{4\ 2^{5/8} \left (2-\sqrt {2}\right )}+\frac {\left (1-\sqrt {2}\right ) \tan ^{-1}\left (\frac {2^{5/8} x}{\sqrt [4]{x^4-1}}+1\right )}{4\ 2^{5/8} \left (2-\sqrt {2}\right )}+\frac {\tanh ^{-1}\left (\frac {\sqrt [8]{2} x}{\sqrt [4]{x^4-1}}\right )}{4\ 2^{5/8}}-\frac {\left (1-\sqrt {2}\right ) \log \left (-\frac {2 x}{\sqrt [4]{x^4-1}}+\frac {2^{5/8} x^2}{\sqrt {x^4-1}}+2^{3/8}\right )}{8\ 2^{5/8} \left (2-\sqrt {2}\right )}+\frac {\left (1-\sqrt {2}\right ) \log \left (\frac {2^{5/8} x}{\sqrt [4]{x^4-1}}+\frac {\sqrt [4]{2} x^2}{\sqrt {x^4-1}}+1\right )}{8\ 2^{5/8} \left (2-\sqrt {2}\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^4/((-1 + x^4)^(1/4)*(-1 + 2*x^4 + x^8)),x]

[Out]

ArcTan[(2^(1/8)*x)/(-1 + x^4)^(1/4)]/(4*2^(5/8)) - ((1 - Sqrt[2])*ArcTan[1 - (2^(5/8)*x)/(-1 + x^4)^(1/4)])/(4
*2^(5/8)*(2 - Sqrt[2])) + ((1 - Sqrt[2])*ArcTan[1 + (2^(5/8)*x)/(-1 + x^4)^(1/4)])/(4*2^(5/8)*(2 - Sqrt[2])) +
ArcTanh[(2^(1/8)*x)/(-1 + x^4)^(1/4)]/(4*2^(5/8)) - ((1 - Sqrt[2])*Log[2^(3/8) + (2^(5/8)*x^2)/Sqrt[-1 + x^4]
- (2*x)/(-1 + x^4)^(1/4)])/(8*2^(5/8)*(2 - Sqrt[2])) + ((1 - Sqrt[2])*Log[1 + (2^(1/4)*x^2)/Sqrt[-1 + x^4] +
(2^(5/8)*x)/(-1 + x^4)^(1/4)])/(8*2^(5/8)*(2 - Sqrt[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 211

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2]], s = Denominator[Rt[a/b, 2]]}, Di
st[1/(2*r), Int[(r - s*x^2)/(a + b*x^4), x], x] + Dist[1/(2*r), Int[(r + s*x^2)/(a + b*x^4), x], x]] /; FreeQ[
{a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] && AtomQ[SplitProduct[SumBaseQ, b
]]))

Rule 212

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[-(a/b), 2]], s = Denominator[Rt[-(a/b), 2]
]}, Dist[r/(2*a), Int[1/(r - s*x^2), x], x] + Dist[r/(2*a), Int[1/(r + s*x^2), x], x]] /; FreeQ[{a, b}, x] &&
!GtQ[a/b, 0]

Rule 377

Int[((a_) + (b_.)*(x_)^(n_))^(p_)/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Subst[Int[1/(c - (b*c - a*d)*x^n), x]
, x, x/(a + b*x^n)^(1/n)] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && EqQ[n*p + 1, 0] && IntegerQ[n]

Rule 617

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[(a*c)/b^2]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + (2*c*x)/b], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 1162

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

Rule 1165

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

Rule 1528

Int[(((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(n_))^(q_))/((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_)), x_Symbol]
:> Int[ExpandIntegrand[(d + e*x^n)^q, (f*x)^m/(a + b*x^n + c*x^(2*n)), x], x] /; FreeQ[{a, b, c, d, e, f, q,
n}, x] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] &&  !IntegerQ[q] && IntegerQ[m]

Rubi steps

\begin {align*} \int \frac {x^4}{\sqrt [4]{-1+x^4} \left (-1+2 x^4+x^8\right )} \, dx &=\int \left (\frac {1-\frac {1}{\sqrt {2}}}{\sqrt [4]{-1+x^4} \left (2-2 \sqrt {2}+2 x^4\right )}+\frac {1+\frac {1}{\sqrt {2}}}{\sqrt [4]{-1+x^4} \left (2+2 \sqrt {2}+2 x^4\right )}\right ) \, dx\\ &=\frac {1}{2} \left (2-\sqrt {2}\right ) \int \frac {1}{\sqrt [4]{-1+x^4} \left (2-2 \sqrt {2}+2 x^4\right )} \, dx+\frac {1}{2} \left (2+\sqrt {2}\right ) \int \frac {1}{\sqrt [4]{-1+x^4} \left (2+2 \sqrt {2}+2 x^4\right )} \, dx\\ &=\frac {1}{2} \left (2-\sqrt {2}\right ) \operatorname {Subst}\left (\int \frac {1}{2-2 \sqrt {2}-\left (4-2 \sqrt {2}\right ) x^4} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right )+\frac {1}{2} \left (2+\sqrt {2}\right ) \operatorname {Subst}\left (\int \frac {1}{2+2 \sqrt {2}-\left (4+2 \sqrt {2}\right ) x^4} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right )\\ &=\frac {\operatorname {Subst}\left (\int \frac {1}{1-\sqrt [4]{2} x^2} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right )}{4 \sqrt {2}}+\frac {\operatorname {Subst}\left (\int \frac {1}{1+\sqrt [4]{2} x^2} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right )}{4 \sqrt {2}}+\frac {1}{4} \left (2-\sqrt {2}\right ) \operatorname {Subst}\left (\int \frac {1-\sqrt [4]{2} x^2}{2-2 \sqrt {2}+\left (-4+2 \sqrt {2}\right ) x^4} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right )+\frac {1}{4} \left (2-\sqrt {2}\right ) \operatorname {Subst}\left (\int \frac {1+\sqrt [4]{2} x^2}{2-2 \sqrt {2}+\left (-4+2 \sqrt {2}\right ) x^4} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right )\\ &=\frac {\tan ^{-1}\left (\frac {\sqrt [8]{2} x}{\sqrt [4]{-1+x^4}}\right )}{4\ 2^{5/8}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [8]{2} x}{\sqrt [4]{-1+x^4}}\right )}{4\ 2^{5/8}}+\frac {\operatorname {Subst}\left (\int \frac {2^{3/8}+2 x}{-\frac {1}{\sqrt [4]{2}}-2^{3/8} x-x^2} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right )}{16 \sqrt [8]{2}}+\frac {\operatorname {Subst}\left (\int \frac {2^{3/8}-2 x}{-\frac {1}{\sqrt [4]{2}}+2^{3/8} x-x^2} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right )}{16 \sqrt [8]{2}}+\frac {\left (1-\sqrt {2}\right ) \operatorname {Subst}\left (\int \frac {1}{\frac {1}{\sqrt [4]{2}}-2^{3/8} x+x^2} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right )}{8 \sqrt [4]{2} \left (2-\sqrt {2}\right )}+\frac {\left (1-\sqrt {2}\right ) \operatorname {Subst}\left (\int \frac {1}{\frac {1}{\sqrt [4]{2}}+2^{3/8} x+x^2} \, dx,x,\frac {x}{\sqrt [4]{-1+x^4}}\right )}{8 \sqrt [4]{2} \left (2-\sqrt {2}\right )}\\ &=\frac {\tan ^{-1}\left (\frac {\sqrt [8]{2} x}{\sqrt [4]{-1+x^4}}\right )}{4\ 2^{5/8}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [8]{2} x}{\sqrt [4]{-1+x^4}}\right )}{4\ 2^{5/8}}+\frac {\log \left (2^{3/8}+\frac {2^{5/8} x^2}{\sqrt {-1+x^4}}-\frac {2 x}{\sqrt [4]{-1+x^4}}\right )}{16 \sqrt [8]{2}}-\frac {\log \left (1+\frac {\sqrt [4]{2} x^2}{\sqrt {-1+x^4}}+\frac {2^{5/8} x}{\sqrt [4]{-1+x^4}}\right )}{16 \sqrt [8]{2}}+\frac {\left (1-\sqrt {2}\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {2^{5/8} x}{\sqrt [4]{-1+x^4}}\right )}{4\ 2^{5/8} \left (2-\sqrt {2}\right )}-\frac {\left (1-\sqrt {2}\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {2^{5/8} x}{\sqrt [4]{-1+x^4}}\right )}{4\ 2^{5/8} \left (2-\sqrt {2}\right )}\\ &=\frac {\tan ^{-1}\left (\frac {\sqrt [8]{2} x}{\sqrt [4]{-1+x^4}}\right )}{4\ 2^{5/8}}-\frac {\left (1-\sqrt {2}\right ) \tan ^{-1}\left (1-\frac {2^{5/8} x}{\sqrt [4]{-1+x^4}}\right )}{4\ 2^{5/8} \left (2-\sqrt {2}\right )}+\frac {\left (1-\sqrt {2}\right ) \tan ^{-1}\left (1+\frac {2^{5/8} x}{\sqrt [4]{-1+x^4}}\right )}{4\ 2^{5/8} \left (2-\sqrt {2}\right )}+\frac {\tanh ^{-1}\left (\frac {\sqrt [8]{2} x}{\sqrt [4]{-1+x^4}}\right )}{4\ 2^{5/8}}+\frac {\log \left (2^{3/8}+\frac {2^{5/8} x^2}{\sqrt {-1+x^4}}-\frac {2 x}{\sqrt [4]{-1+x^4}}\right )}{16 \sqrt [8]{2}}-\frac {\log \left (1+\frac {\sqrt [4]{2} x^2}{\sqrt {-1+x^4}}+\frac {2^{5/8} x}{\sqrt [4]{-1+x^4}}\right )}{16 \sqrt [8]{2}}\\ \end {align*}

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Mathematica [A]  time = 0.25, size = 175, normalized size = 1.15 \begin {gather*} \frac {4 \tan ^{-1}\left (\frac {\sqrt [8]{2} x}{\sqrt [4]{x^4-1}}\right )+4 \tanh ^{-1}\left (\frac {\sqrt [8]{2} x}{\sqrt [4]{x^4-1}}\right )+\sqrt {2} \left (2 \tan ^{-1}\left (1-\frac {2^{5/8} x}{\sqrt [4]{x^4-1}}\right )-2 \tan ^{-1}\left (\frac {2^{5/8} x}{\sqrt [4]{x^4-1}}+1\right )+\log \left (-\frac {2^{5/8} x}{\sqrt [4]{x^4-1}}+\frac {\sqrt [4]{2} x^2}{\sqrt {x^4-1}}+1\right )-\log \left (\frac {2^{5/8} x}{\sqrt [4]{x^4-1}}+\frac {\sqrt [4]{2} x^2}{\sqrt {x^4-1}}+1\right )\right )}{16\ 2^{5/8}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^4/((-1 + x^4)^(1/4)*(-1 + 2*x^4 + x^8)),x]

[Out]

(4*ArcTan[(2^(1/8)*x)/(-1 + x^4)^(1/4)] + 4*ArcTanh[(2^(1/8)*x)/(-1 + x^4)^(1/4)] + Sqrt[2]*(2*ArcTan[1 - (2^(
5/8)*x)/(-1 + x^4)^(1/4)] - 2*ArcTan[1 + (2^(5/8)*x)/(-1 + x^4)^(1/4)] + Log[1 + (2^(1/4)*x^2)/Sqrt[-1 + x^4]
- (2^(5/8)*x)/(-1 + x^4)^(1/4)] - Log[1 + (2^(1/4)*x^2)/Sqrt[-1 + x^4] + (2^(5/8)*x)/(-1 + x^4)^(1/4)]))/(16*2
^(5/8))

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IntegrateAlgebraic [A]  time = 0.50, size = 152, normalized size = 1.00 \begin {gather*} \frac {\tan ^{-1}\left (\frac {\sqrt [8]{2} x}{\sqrt [4]{-1+x^4}}\right )}{4\ 2^{5/8}}+\frac {\tan ^{-1}\left (\frac {2^{5/8} x \sqrt [4]{-1+x^4}}{\sqrt [4]{2} x^2-\sqrt {-1+x^4}}\right )}{8 \sqrt [8]{2}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [8]{2} x}{\sqrt [4]{-1+x^4}}\right )}{4\ 2^{5/8}}-\frac {\tanh ^{-1}\left (\frac {2\ 2^{3/8} x \sqrt [4]{-1+x^4}}{2 x^2+2^{3/4} \sqrt {-1+x^4}}\right )}{8 \sqrt [8]{2}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[x^4/((-1 + x^4)^(1/4)*(-1 + 2*x^4 + x^8)),x]

[Out]

ArcTan[(2^(1/8)*x)/(-1 + x^4)^(1/4)]/(4*2^(5/8)) + ArcTan[(2^(5/8)*x*(-1 + x^4)^(1/4))/(2^(1/4)*x^2 - Sqrt[-1
+ x^4])]/(8*2^(1/8)) + ArcTanh[(2^(1/8)*x)/(-1 + x^4)^(1/4)]/(4*2^(5/8)) - ArcTanh[(2*2^(3/8)*x*(-1 + x^4)^(1/
4))/(2*x^2 + 2^(3/4)*Sqrt[-1 + x^4])]/(8*2^(1/8))

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fricas [B]  time = 13.50, size = 2047, normalized size = 13.47

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/256*32^(7/8)*sqrt(2)*arctan(-1/32*(2080*x^16 - 3968*x^12 + 2112*x^8 - 128*x^4 - sqrt(2)*(32^(5/8)*sqrt(2)*(
151*x^16 - 392*x^12 + 254*x^8 - 8*x^4 - 1) + 128*(x^4 - 1)^(3/4)*(2^(3/4)*(25*x^13 - 26*x^9 - x^5) - 2*2^(1/4)
*(11*x^13 - 12*x^9 - x^5)) - 8*32^(1/8)*sqrt(2)*(189*x^16 - 418*x^12 + 236*x^8 - 2*x^4 - 1) + sqrt(x^4 - 1)*(3
2^(7/8)*sqrt(2)*(91*x^14 - 123*x^10 + 19*x^6 + x^2) - 8*32^(3/8)*sqrt(2)*(86*x^14 - 101*x^10 + 8*x^6 + x^2)) +
32*(28*x^15 - 6*x^11 - 24*x^7 - 2*x^3 + sqrt(2)*(3*x^15 - 27*x^11 + 27*x^7 + x^3))*(x^4 - 1)^(1/4))*sqrt((12*
2^(3/4)*(x^8 + 2*x^4 - 1) + (x^4 - 1)^(3/4)*(32^(7/8)*sqrt(2)*(x^5 + 2*x) + 4*32^(3/8)*sqrt(2)*(x^5 + 3*x)) +
32*(x^6 + 3*x^2 + sqrt(2)*(x^6 + 2*x^2))*sqrt(x^4 - 1) + 16*2^(1/4)*(x^8 + 2*x^4 - 1) + 2*(x^4 - 1)^(1/4)*(32^
(5/8)*sqrt(2)*(x^7 + 3*x^3) + 8*32^(1/8)*sqrt(2)*(x^7 + 2*x^3)))/(x^8 + 2*x^4 - 1)) - 8*(x^4 - 1)^(3/4)*(32^(5
/8)*sqrt(2)*(81*x^13 - 79*x^9 - 3*x^5 + x) - 8*32^(1/8)*sqrt(2)*(5*x^13 - 22*x^9 + 11*x^5)) + 512*sqrt(2)*(3*x
^16 - 5*x^12 + 3*x^8 - x^4) + 128*sqrt(x^4 - 1)*(2^(3/4)*(17*x^14 - 30*x^10 + 15*x^6) + 2^(1/4)*(31*x^14 - 33*
x^10 + 3*x^6 - x^2)) - 4*(x^4 - 1)^(1/4)*(32^(7/8)*sqrt(2)*(19*x^15 - 13*x^11 - 9*x^7 + 3*x^3) - 8*32^(3/8)*sq
rt(2)*(39*x^15 - 82*x^11 + 41*x^7)) + 32)/(383*x^16 - 772*x^12 + 382*x^8 + 4*x^4 - 1)) + 1/256*32^(7/8)*sqrt(2
)*arctan(-1/32*(2080*x^16 - 3968*x^12 + 2112*x^8 - 128*x^4 + sqrt(2)*(32^(5/8)*sqrt(2)*(151*x^16 - 392*x^12 +
254*x^8 - 8*x^4 - 1) - 128*(x^4 - 1)^(3/4)*(2^(3/4)*(25*x^13 - 26*x^9 - x^5) - 2*2^(1/4)*(11*x^13 - 12*x^9 - x
^5)) - 8*32^(1/8)*sqrt(2)*(189*x^16 - 418*x^12 + 236*x^8 - 2*x^4 - 1) + sqrt(x^4 - 1)*(32^(7/8)*sqrt(2)*(91*x^
14 - 123*x^10 + 19*x^6 + x^2) - 8*32^(3/8)*sqrt(2)*(86*x^14 - 101*x^10 + 8*x^6 + x^2)) - 32*(28*x^15 - 6*x^11
- 24*x^7 - 2*x^3 + sqrt(2)*(3*x^15 - 27*x^11 + 27*x^7 + x^3))*(x^4 - 1)^(1/4))*sqrt((12*2^(3/4)*(x^8 + 2*x^4 -
1) - (x^4 - 1)^(3/4)*(32^(7/8)*sqrt(2)*(x^5 + 2*x) + 4*32^(3/8)*sqrt(2)*(x^5 + 3*x)) + 32*(x^6 + 3*x^2 + sqrt
(2)*(x^6 + 2*x^2))*sqrt(x^4 - 1) + 16*2^(1/4)*(x^8 + 2*x^4 - 1) - 2*(x^4 - 1)^(1/4)*(32^(5/8)*sqrt(2)*(x^7 + 3
*x^3) + 8*32^(1/8)*sqrt(2)*(x^7 + 2*x^3)))/(x^8 + 2*x^4 - 1)) + 8*(x^4 - 1)^(3/4)*(32^(5/8)*sqrt(2)*(81*x^13 -
79*x^9 - 3*x^5 + x) - 8*32^(1/8)*sqrt(2)*(5*x^13 - 22*x^9 + 11*x^5)) + 512*sqrt(2)*(3*x^16 - 5*x^12 + 3*x^8 -
x^4) + 128*sqrt(x^4 - 1)*(2^(3/4)*(17*x^14 - 30*x^10 + 15*x^6) + 2^(1/4)*(31*x^14 - 33*x^10 + 3*x^6 - x^2)) +
4*(x^4 - 1)^(1/4)*(32^(7/8)*sqrt(2)*(19*x^15 - 13*x^11 - 9*x^7 + 3*x^3) - 8*32^(3/8)*sqrt(2)*(39*x^15 - 82*x^
11 + 41*x^7)) + 32)/(383*x^16 - 772*x^12 + 382*x^8 + 4*x^4 - 1)) - 1/1024*32^(7/8)*sqrt(2)*log(128*(12*2^(3/4)
*(x^8 + 2*x^4 - 1) + (x^4 - 1)^(3/4)*(32^(7/8)*sqrt(2)*(x^5 + 2*x) + 4*32^(3/8)*sqrt(2)*(x^5 + 3*x)) + 32*(x^6
+ 3*x^2 + sqrt(2)*(x^6 + 2*x^2))*sqrt(x^4 - 1) + 16*2^(1/4)*(x^8 + 2*x^4 - 1) + 2*(x^4 - 1)^(1/4)*(32^(5/8)*s
qrt(2)*(x^7 + 3*x^3) + 8*32^(1/8)*sqrt(2)*(x^7 + 2*x^3)))/(x^8 + 2*x^4 - 1)) + 1/1024*32^(7/8)*sqrt(2)*log(128
*(12*2^(3/4)*(x^8 + 2*x^4 - 1) - (x^4 - 1)^(3/4)*(32^(7/8)*sqrt(2)*(x^5 + 2*x) + 4*32^(3/8)*sqrt(2)*(x^5 + 3*x
)) + 32*(x^6 + 3*x^2 + sqrt(2)*(x^6 + 2*x^2))*sqrt(x^4 - 1) + 16*2^(1/4)*(x^8 + 2*x^4 - 1) - 2*(x^4 - 1)^(1/4)
*(32^(5/8)*sqrt(2)*(x^7 + 3*x^3) + 8*32^(1/8)*sqrt(2)*(x^7 + 2*x^3)))/(x^8 + 2*x^4 - 1)) - 1/128*32^(7/8)*arct
an(1/16*(sqrt(2)*(32^(5/8)*(7*x^8 - 6*x^4 + 1) + sqrt(x^4 - 1)*(32^(7/8)*(3*x^6 - x^2) + 8*32^(3/8)*(2*x^6 - x
^2)) + 8*32^(1/8)*(5*x^8 - 4*x^4 + 1))*sqrt(3*2^(3/4) - 4*2^(1/4)) + 4*(8*32^(1/8)*x^5 + 32^(5/8)*(x^5 - x))*(
x^4 - 1)^(3/4) + 2*(8*32^(3/8)*x^7 + 32^(7/8)*(x^7 - x^3))*(x^4 - 1)^(1/4))/(x^8 + 2*x^4 - 1)) + 1/512*32^(7/8
)*log((32^(7/8)*(x^8 + 1) + 32*(x^5 - sqrt(2)*x + x)*(x^4 - 1)^(3/4) - 4*sqrt(x^4 - 1)*(8*32^(1/8)*x^2 - 32^(5
/8)*(x^6 + x^2)) + 4*32^(3/8)*(x^8 - 2*x^4 - 1) - 32*(x^4 - 1)^(1/4)*(2^(3/4)*x^3 - 2^(1/4)*(x^7 + x^3)))/(x^8
+ 2*x^4 - 1)) - 1/512*32^(7/8)*log(-(32^(7/8)*(x^8 + 1) - 32*(x^5 - sqrt(2)*x + x)*(x^4 - 1)^(3/4) - 4*sqrt(x
^4 - 1)*(8*32^(1/8)*x^2 - 32^(5/8)*(x^6 + x^2)) + 4*32^(3/8)*(x^8 - 2*x^4 - 1) + 32*(x^4 - 1)^(1/4)*(2^(3/4)*x
^3 - 2^(1/4)*(x^7 + x^3)))/(x^8 + 2*x^4 - 1))

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:Unab
le to convert to real 1/4 Error: Bad Argument ValueUnable to convert to real 1/4 Error: Bad Argument Value

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maple [C]  time = 10.86, size = 998, normalized size = 6.57

method	result	size
trager	\(-\frac {\RootOf \left (\textit {\_Z}^{8}-8\right ) \ln \left (-\frac {\RootOf \left (\textit {\_Z}^{8}-8\right )^{9} x^{4}-2 \RootOf \left (\textit {\_Z}^{8}-8\right )^{7} \sqrt {x^{4}-1}\, x^{2}-4 \RootOf \left (\textit {\_Z}^{8}-8\right )^{6} \left (x^{4}-1\right )^{\frac {1}{4}} x^{3}+4 \RootOf \left (\textit {\_Z}^{8}-8\right )^{4} \left (x^{4}-1\right )^{\frac {3}{4}} x +8 \RootOf \left (\textit {\_Z}^{8}-8\right )^{3} \sqrt {x^{4}-1}\, x^{2}+8 \RootOf \left (\textit {\_Z}^{8}-8\right )^{2} \left (x^{4}-1\right )^{\frac {1}{4}} x^{3}-2 \RootOf \left (\textit {\_Z}^{8}-8\right )^{5}-4 \RootOf \left (\textit {\_Z}^{8}-8\right ) x^{4}-16 \left (x^{4}-1\right )^{\frac {3}{4}} x +4 \RootOf \left (\textit {\_Z}^{8}-8\right )}{\RootOf \left (\textit {\_Z}^{8}-8\right )^{4} x^{4}-2 x^{4}+2}\right )}{16}+\frac {\ln \left (\frac {3 \RootOf \left (\textit {\_Z}^{8}-8\right )^{9} x^{4}-8 \RootOf \left (\textit {\_Z}^{8}-8\right )^{7} \sqrt {x^{4}-1}\, x^{2}+8 \RootOf \left (\textit {\_Z}^{8}-8\right )^{6} \left (x^{4}-1\right )^{\frac {1}{4}} x^{3}+2 \RootOf \left (\textit {\_Z}^{8}-8\right )^{5} x^{4}-24 \RootOf \left (\textit {\_Z}^{8}-8\right )^{4} \left (x^{4}-1\right )^{\frac {3}{4}} x +48 \RootOf \left (\textit {\_Z}^{8}-8\right )^{3} \sqrt {x^{4}-1}\, x^{2}-48 \RootOf \left (\textit {\_Z}^{8}-8\right )^{2} \left (x^{4}-1\right )^{\frac {1}{4}} x^{3}-6 \RootOf \left (\textit {\_Z}^{8}-8\right )^{5}-8 \RootOf \left (\textit {\_Z}^{8}-8\right ) x^{4}+32 \left (x^{4}-1\right )^{\frac {3}{4}} x +8 \RootOf \left (\textit {\_Z}^{8}-8\right )}{\RootOf \left (\textit {\_Z}^{8}-8\right )^{4} x^{4}+2 x^{4}-2}\right ) \RootOf \left (\textit {\_Z}^{8}-8\right )^{5}}{64}-\frac {\ln \left (\frac {3 \RootOf \left (\textit {\_Z}^{8}-8\right )^{9} x^{4}-8 \RootOf \left (\textit {\_Z}^{8}-8\right )^{7} \sqrt {x^{4}-1}\, x^{2}+8 \RootOf \left (\textit {\_Z}^{8}-8\right )^{6} \left (x^{4}-1\right )^{\frac {1}{4}} x^{3}+2 \RootOf \left (\textit {\_Z}^{8}-8\right )^{5} x^{4}-24 \RootOf \left (\textit {\_Z}^{8}-8\right )^{4} \left (x^{4}-1\right )^{\frac {3}{4}} x +48 \RootOf \left (\textit {\_Z}^{8}-8\right )^{3} \sqrt {x^{4}-1}\, x^{2}-48 \RootOf \left (\textit {\_Z}^{8}-8\right )^{2} \left (x^{4}-1\right )^{\frac {1}{4}} x^{3}-6 \RootOf \left (\textit {\_Z}^{8}-8\right )^{5}-8 \RootOf \left (\textit {\_Z}^{8}-8\right ) x^{4}+32 \left (x^{4}-1\right )^{\frac {3}{4}} x +8 \RootOf \left (\textit {\_Z}^{8}-8\right )}{\RootOf \left (\textit {\_Z}^{8}-8\right )^{4} x^{4}+2 x^{4}-2}\right ) \RootOf \left (\textit {\_Z}^{8}-8\right )^{4} \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{8}-8\right )^{2}\right )}{64}+\frac {\RootOf \left (\textit {\_Z}^{8}-8\right )^{4} \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{8}-8\right )^{2}\right ) \ln \left (\frac {\RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{8}-8\right )^{2}\right ) \RootOf \left (\textit {\_Z}^{8}-8\right )^{6} \sqrt {x^{4}-1}\, x^{2}-\RootOf \left (\textit {\_Z}^{8}-8\right )^{7} \sqrt {x^{4}-1}\, x^{2}-2 \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{8}-8\right )^{2}\right ) \RootOf \left (\textit {\_Z}^{8}-8\right )^{5} \left (x^{4}-1\right )^{\frac {1}{4}} x^{3}+\RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{8}-8\right )^{2}\right ) \RootOf \left (\textit {\_Z}^{8}-8\right )^{4} x^{4}+\RootOf \left (\textit {\_Z}^{8}-8\right )^{5} x^{4}-2 \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{8}-8\right )^{2}\right ) x^{4}-2 \RootOf \left (\textit {\_Z}^{8}-8\right ) x^{4}+8 \left (x^{4}-1\right )^{\frac {3}{4}} x +2 \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{8}-8\right )^{2}\right )+2 \RootOf \left (\textit {\_Z}^{8}-8\right )}{\RootOf \left (\textit {\_Z}^{8}-8\right )^{4} x^{4}+2 x^{4}-2}\right )}{32}+\frac {\RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{8}-8\right )^{2}\right ) \ln \left (\frac {\RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{8}-8\right )^{2}\right ) \RootOf \left (\textit {\_Z}^{8}-8\right )^{8} x^{4}-4 \RootOf \left (\textit {\_Z}^{8}-8\right )^{6} \left (x^{4}-1\right )^{\frac {1}{4}} x^{3}+2 \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{8}-8\right )^{2}\right ) \RootOf \left (\textit {\_Z}^{8}-8\right )^{4} x^{4}-8 \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{8}-8\right )^{2}\right ) \RootOf \left (\textit {\_Z}^{8}-8\right )^{2} \sqrt {x^{4}-1}\, x^{2}-2 \RootOf \left (\textit {\_Z}^{8}-8\right )^{4} \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{8}-8\right )^{2}\right )+16 \left (x^{4}-1\right )^{\frac {3}{4}} x}{\RootOf \left (\textit {\_Z}^{8}-8\right )^{4} x^{4}-2 x^{4}+2}\right )}{16}\)	\(998\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/16*RootOf(_Z^8-8)*ln(-(RootOf(_Z^8-8)^9*x^4-2*RootOf(_Z^8-8)^7*(x^4-1)^(1/2)*x^2-4*RootOf(_Z^8-8)^6*(x^4-1)
^(1/4)*x^3+4*RootOf(_Z^8-8)^4*(x^4-1)^(3/4)*x+8*RootOf(_Z^8-8)^3*(x^4-1)^(1/2)*x^2+8*RootOf(_Z^8-8)^2*(x^4-1)^
(1/4)*x^3-2*RootOf(_Z^8-8)^5-4*RootOf(_Z^8-8)*x^4-16*(x^4-1)^(3/4)*x+4*RootOf(_Z^8-8))/(RootOf(_Z^8-8)^4*x^4-2
*x^4+2))+1/64*ln((3*RootOf(_Z^8-8)^9*x^4-8*RootOf(_Z^8-8)^7*(x^4-1)^(1/2)*x^2+8*RootOf(_Z^8-8)^6*(x^4-1)^(1/4)
*x^3+2*RootOf(_Z^8-8)^5*x^4-24*RootOf(_Z^8-8)^4*(x^4-1)^(3/4)*x+48*RootOf(_Z^8-8)^3*(x^4-1)^(1/2)*x^2-48*RootO
f(_Z^8-8)^2*(x^4-1)^(1/4)*x^3-6*RootOf(_Z^8-8)^5-8*RootOf(_Z^8-8)*x^4+32*(x^4-1)^(3/4)*x+8*RootOf(_Z^8-8))/(Ro
otOf(_Z^8-8)^4*x^4+2*x^4-2))*RootOf(_Z^8-8)^5-1/64*ln((3*RootOf(_Z^8-8)^9*x^4-8*RootOf(_Z^8-8)^7*(x^4-1)^(1/2)
*x^2+8*RootOf(_Z^8-8)^6*(x^4-1)^(1/4)*x^3+2*RootOf(_Z^8-8)^5*x^4-24*RootOf(_Z^8-8)^4*(x^4-1)^(3/4)*x+48*RootOf
(_Z^8-8)^3*(x^4-1)^(1/2)*x^2-48*RootOf(_Z^8-8)^2*(x^4-1)^(1/4)*x^3-6*RootOf(_Z^8-8)^5-8*RootOf(_Z^8-8)*x^4+32*
(x^4-1)^(3/4)*x+8*RootOf(_Z^8-8))/(RootOf(_Z^8-8)^4*x^4+2*x^4-2))*RootOf(_Z^8-8)^4*RootOf(_Z^2+RootOf(_Z^8-8)^
2)+1/32*RootOf(_Z^8-8)^4*RootOf(_Z^2+RootOf(_Z^8-8)^2)*ln((RootOf(_Z^2+RootOf(_Z^8-8)^2)*RootOf(_Z^8-8)^6*(x^4
-1)^(1/2)*x^2-RootOf(_Z^8-8)^7*(x^4-1)^(1/2)*x^2-2*RootOf(_Z^2+RootOf(_Z^8-8)^2)*RootOf(_Z^8-8)^5*(x^4-1)^(1/4
)*x^3+RootOf(_Z^2+RootOf(_Z^8-8)^2)*RootOf(_Z^8-8)^4*x^4+RootOf(_Z^8-8)^5*x^4-2*RootOf(_Z^2+RootOf(_Z^8-8)^2)*
x^4-2*RootOf(_Z^8-8)*x^4+8*(x^4-1)^(3/4)*x+2*RootOf(_Z^2+RootOf(_Z^8-8)^2)+2*RootOf(_Z^8-8))/(RootOf(_Z^8-8)^4
*x^4+2*x^4-2))+1/16*RootOf(_Z^2+RootOf(_Z^8-8)^2)*ln((RootOf(_Z^2+RootOf(_Z^8-8)^2)*RootOf(_Z^8-8)^8*x^4-4*Roo
tOf(_Z^8-8)^6*(x^4-1)^(1/4)*x^3+2*RootOf(_Z^2+RootOf(_Z^8-8)^2)*RootOf(_Z^8-8)^4*x^4-8*RootOf(_Z^2+RootOf(_Z^8
-8)^2)*RootOf(_Z^8-8)^2*(x^4-1)^(1/2)*x^2-2*RootOf(_Z^8-8)^4*RootOf(_Z^2+RootOf(_Z^8-8)^2)+16*(x^4-1)^(3/4)*x)
/(RootOf(_Z^8-8)^4*x^4-2*x^4+2))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x^4}{{\left (x^4-1\right )}^{1/4}\,\left (x^8+2\,x^4-1\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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