3.22.1 \(\int \frac {(-1+x^4)^{3/4}}{-1+2 x^4+x^8} \, dx\)

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

________________________________________________________________________________________

Rubi [A]  time = 0.22, antiderivative size = 282, normalized size of antiderivative = 1.86, number of steps used = 25, number of rules used = 13, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.591, Rules used = {1428, 408, 240, 212, 206, 203, 377, 211, 1165, 628, 1162, 617, 204} \begin {gather*} \frac {\tan ^{-1}\left (\frac {\sqrt [8]{2} x}{\sqrt [4]{x^4-1}}\right )}{4 \sqrt [8]{2}}+\frac {\left (1-\sqrt {2}\right ) \tan ^{-1}\left (1-\frac {2^{5/8} x}{\sqrt [4]{x^4-1}}\right )}{4 \sqrt [8]{2} \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 \sqrt [8]{2} \left (2-\sqrt {2}\right )}+\frac {\tanh ^{-1}\left (\frac {\sqrt [8]{2} x}{\sqrt [4]{x^4-1}}\right )}{4 \sqrt [8]{2}}+\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 \sqrt [8]{2} \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 \sqrt [8]{2} \left (2-\sqrt {2}\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

ArcTan[(2^(1/8)*x)/(-1 + x^4)^(1/4)]/(4*2^(1/8)) + ((1 - Sqrt[2])*ArcTan[1 - (2^(5/8)*x)/(-1 + x^4)^(1/4)])/(4
*2^(1/8)*(2 - Sqrt[2])) - ((1 - Sqrt[2])*ArcTan[1 + (2^(5/8)*x)/(-1 + x^4)^(1/4)])/(4*2^(1/8)*(2 - Sqrt[2])) +
ArcTanh[(2^(1/8)*x)/(-1 + x^4)^(1/4)]/(4*2^(1/8)) + ((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^(1/8)*(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^(1/8)*(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 240

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[a^(p + 1/n), Subst[Int[1/(1 - b*x^n)^(p + 1/n + 1), x], x
, x/(a + b*x^n)^(1/n)], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[-1, p, 0] && NeQ[p, -2^(-1)] && IntegerQ[p
+ 1/n]

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 408

Int[((a_) + (b_.)*(x_)^4)^(p_)/((c_) + (d_.)*(x_)^4), x_Symbol] :> Dist[b/d, Int[(a + b*x^4)^(p - 1), x], x] -
Dist[(b*c - a*d)/d, Int[(a + b*x^4)^(p - 1)/(c + d*x^4), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0
] && (EqQ[p, 3/4] || EqQ[p, 5/4])

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 1428

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

Rubi steps

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

________________________________________________________________________________________

Mathematica [F]  time = 0.05, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (-1+x^4\right )^{3/4}}{-1+2 x^4+x^8} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 0.45, size = 152, normalized size = 1.00 \begin {gather*} \frac {\tan ^{-1}\left (\frac {\sqrt [8]{2} x}{\sqrt [4]{-1+x^4}}\right )}{4 \sqrt [8]{2}}-\frac {\tan ^{-1}\left (\frac {2^{5/8} x \sqrt [4]{-1+x^4}}{\sqrt [4]{2} x^2-\sqrt {-1+x^4}}\right )}{4\ 2^{5/8}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [8]{2} x}{\sqrt [4]{-1+x^4}}\right )}{4 \sqrt [8]{2}}+\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 )}{4\ 2^{5/8}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

fricas [B]  time = 12.40, size = 1933, normalized size = 12.72

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8*2^(7/8)*arctan(1/2*(4*(2^(5/8)*x^5 + 2^(1/8)*(x^5 - x))*(x^4 - 1)^(3/4) + (2^(5/8)*(5*x^8 - 4*x^4 + 1) +
2*sqrt(x^4 - 1)*(2^(7/8)*(2*x^6 - x^2) + 2^(3/8)*(3*x^6 - x^2)) + 2^(1/8)*(7*x^8 - 6*x^4 + 1))*sqrt(6*2^(3/4)
- 8*2^(1/4)) + 4*(2^(7/8)*x^7 + 2^(3/8)*(x^7 - x^3))*(x^4 - 1)^(1/4))/(x^8 + 2*x^4 - 1)) + 1/32*2^(7/8)*log((2
^(7/8)*(x^8 - 2*x^4 - 1) + 4*(x^5 - sqrt(2)*x + x)*(x^4 - 1)^(3/4) - 4*sqrt(x^4 - 1)*(2^(5/8)*x^2 - 2^(1/8)*(x
^6 + x^2)) + 2*2^(3/8)*(x^8 + 1) - 4*(x^4 - 1)^(1/4)*(2^(3/4)*x^3 - 2^(1/4)*(x^7 + x^3)))/(x^8 + 2*x^4 - 1)) -
1/32*2^(7/8)*log(-(2^(7/8)*(x^8 - 2*x^4 - 1) - 4*(x^5 - sqrt(2)*x + x)*(x^4 - 1)^(3/4) - 4*sqrt(x^4 - 1)*(2^(
5/8)*x^2 - 2^(1/8)*(x^6 + x^2)) + 2*2^(3/8)*(x^8 + 1) + 4*(x^4 - 1)^(1/4)*(2^(3/4)*x^3 - 2^(1/4)*(x^7 + x^3)))
/(x^8 + 2*x^4 - 1)) + 1/8*2^(3/8)*arctan(-1/2*(130*x^16 - 248*x^12 + 132*x^8 - 8*x^4 - sqrt(2)*(16*(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)) + 2^(5/8)*(151*x^16 - 392*x^12 +
254*x^8 - 8*x^4 - 1) + 2*sqrt(x^4 - 1)*(2^(7/8)*(91*x^14 - 123*x^10 + 19*x^6 + x^2) - 2*2^(3/8)*(86*x^14 - 101
*x^10 + 8*x^6 + x^2)) + 4*(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) - 2*2^(1/8)*(189*x^16 - 418*x^12 + 236*x^8 - 2*x^4 - 1))*sqrt((3*2^(3/4)*(x^8 + 2*x^4 - 1) + 4*(x^
4 - 1)^(3/4)*(2^(7/8)*(x^5 + 2*x) + 2^(3/8)*(x^5 + 3*x)) + 8*(x^6 + 3*x^2 + sqrt(2)*(x^6 + 2*x^2))*sqrt(x^4 -
1) + 4*2^(1/4)*(x^8 + 2*x^4 - 1) + 4*(x^4 - 1)^(1/4)*(2^(5/8)*(x^7 + 3*x^3) + 2*2^(1/8)*(x^7 + 2*x^3)))/(x^8 +
2*x^4 - 1)) - 4*(x^4 - 1)^(3/4)*(2^(5/8)*(81*x^13 - 79*x^9 - 3*x^5 + x) - 2*2^(1/8)*(5*x^13 - 22*x^9 + 11*x^5
)) + 32*sqrt(2)*(3*x^16 - 5*x^12 + 3*x^8 - x^4) + 8*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)*(2^(7/8)*(19*x^15 - 13*x^11 - 9*x^7 + 3*x^3) - 2*2^
(3/8)*(39*x^15 - 82*x^11 + 41*x^7)) + 2)/(383*x^16 - 772*x^12 + 382*x^8 + 4*x^4 - 1)) - 1/8*2^(3/8)*arctan(-1/
2*(130*x^16 - 248*x^12 + 132*x^8 - 8*x^4 - sqrt(2)*(16*(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)) - 2^(5/8)*(151*x^16 - 392*x^12 + 254*x^8 - 8*x^4 - 1) - 2*sqrt(x^4 - 1)*(2^(7
/8)*(91*x^14 - 123*x^10 + 19*x^6 + x^2) - 2*2^(3/8)*(86*x^14 - 101*x^10 + 8*x^6 + x^2)) + 4*(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) + 2*2^(1/8)*(189*x^16 - 418*x^12
+ 236*x^8 - 2*x^4 - 1))*sqrt((3*2^(3/4)*(x^8 + 2*x^4 - 1) - 4*(x^4 - 1)^(3/4)*(2^(7/8)*(x^5 + 2*x) + 2^(3/8)*
(x^5 + 3*x)) + 8*(x^6 + 3*x^2 + sqrt(2)*(x^6 + 2*x^2))*sqrt(x^4 - 1) + 4*2^(1/4)*(x^8 + 2*x^4 - 1) - 4*(x^4 -
1)^(1/4)*(2^(5/8)*(x^7 + 3*x^3) + 2*2^(1/8)*(x^7 + 2*x^3)))/(x^8 + 2*x^4 - 1)) + 4*(x^4 - 1)^(3/4)*(2^(5/8)*(8
1*x^13 - 79*x^9 - 3*x^5 + x) - 2*2^(1/8)*(5*x^13 - 22*x^9 + 11*x^5)) + 32*sqrt(2)*(3*x^16 - 5*x^12 + 3*x^8 - x
^4) + 8*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)*(2^(7/8)*(19*x^15 - 13*x^11 - 9*x^7 + 3*x^3) - 2*2^(3/8)*(39*x^15 - 82*x^11 + 41*x^7)) + 2)/(38
3*x^16 - 772*x^12 + 382*x^8 + 4*x^4 - 1)) + 1/32*2^(3/8)*log(8*(3*2^(3/4)*(x^8 + 2*x^4 - 1) + 4*(x^4 - 1)^(3/4
)*(2^(7/8)*(x^5 + 2*x) + 2^(3/8)*(x^5 + 3*x)) + 8*(x^6 + 3*x^2 + sqrt(2)*(x^6 + 2*x^2))*sqrt(x^4 - 1) + 4*2^(1
/4)*(x^8 + 2*x^4 - 1) + 4*(x^4 - 1)^(1/4)*(2^(5/8)*(x^7 + 3*x^3) + 2*2^(1/8)*(x^7 + 2*x^3)))/(x^8 + 2*x^4 - 1)
) - 1/32*2^(3/8)*log(8*(3*2^(3/4)*(x^8 + 2*x^4 - 1) - 4*(x^4 - 1)^(3/4)*(2^(7/8)*(x^5 + 2*x) + 2^(3/8)*(x^5 +
3*x)) + 8*(x^6 + 3*x^2 + sqrt(2)*(x^6 + 2*x^2))*sqrt(x^4 - 1) + 4*2^(1/4)*(x^8 + 2*x^4 - 1) - 4*(x^4 - 1)^(1/4
)*(2^(5/8)*(x^7 + 3*x^3) + 2*2^(1/8)*(x^7 + 2*x^3)))/(x^8 + 2*x^4 - 1))

________________________________________________________________________________________

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^4-1)^(3/4)/(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

________________________________________________________________________________________

maple [C]  time = 16.55, size = 1570, normalized size = 10.33

method	result	size
trager	\(\text {Expression too large to display}\)	\(1570\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^4-1)^(3/4)/(x^8+2*x^4-1),x,method=_RETURNVERBOSE)

[Out]

-1/128*ln((RootOf(_Z^8-128)^9*x^4-64*(x^4-1)^(1/2)*RootOf(-2*RootOf(_Z^8-128)^5*_Z+RootOf(_Z^8-128)^2+256*_Z^2
)*RootOf(_Z^8-128)^6*x^2-8*(x^4-1)^(1/4)*RootOf(_Z^8-128)^6*x^3-8*RootOf(_Z^8-128)^5*x^4-128*RootOf(-2*RootOf(
_Z^8-128)^5*_Z+RootOf(_Z^8-128)^2+256*_Z^2)*RootOf(_Z^8-128)^4*x^4+2048*(x^4-1)^(1/4)*RootOf(-2*RootOf(_Z^8-12
8)^5*_Z+RootOf(_Z^8-128)^2+256*_Z^2)*RootOf(_Z^8-128)*x^3+8*RootOf(_Z^8-128)^5+1024*RootOf(-2*RootOf(_Z^8-128)
^5*_Z+RootOf(_Z^8-128)^2+256*_Z^2)*x^4+256*(x^4-1)^(3/4)*x-1024*RootOf(-2*RootOf(_Z^8-128)^5*_Z+RootOf(_Z^8-12
8)^2+256*_Z^2))/(RootOf(_Z^8-128)^4*x^4+8*x^4-8))*RootOf(_Z^8-128)^5+ln((RootOf(_Z^8-128)^9*x^4-64*(x^4-1)^(1/
2)*RootOf(-2*RootOf(_Z^8-128)^5*_Z+RootOf(_Z^8-128)^2+256*_Z^2)*RootOf(_Z^8-128)^6*x^2-8*(x^4-1)^(1/4)*RootOf(
_Z^8-128)^6*x^3-8*RootOf(_Z^8-128)^5*x^4-128*RootOf(-2*RootOf(_Z^8-128)^5*_Z+RootOf(_Z^8-128)^2+256*_Z^2)*Root
Of(_Z^8-128)^4*x^4+2048*(x^4-1)^(1/4)*RootOf(-2*RootOf(_Z^8-128)^5*_Z+RootOf(_Z^8-128)^2+256*_Z^2)*RootOf(_Z^8
-128)*x^3+8*RootOf(_Z^8-128)^5+1024*RootOf(-2*RootOf(_Z^8-128)^5*_Z+RootOf(_Z^8-128)^2+256*_Z^2)*x^4+256*(x^4-
1)^(3/4)*x-1024*RootOf(-2*RootOf(_Z^8-128)^5*_Z+RootOf(_Z^8-128)^2+256*_Z^2))/(RootOf(_Z^8-128)^4*x^4+8*x^4-8)
)*RootOf(-2*RootOf(_Z^8-128)^5*_Z+RootOf(_Z^8-128)^2+256*_Z^2)+1/8*ln((4*RootOf(-2*RootOf(_Z^8-128)^5*_Z+RootO
f(_Z^8-128)^2+256*_Z^2)*RootOf(_Z^8-128)^8*x^4+(x^4-1)^(1/2)*RootOf(_Z^8-128)^7*x^2-2*(x^4-1)^(1/4)*RootOf(_Z^
8-128)^6*x^3-2*RootOf(_Z^8-128)^5*x^4+32*RootOf(-2*RootOf(_Z^8-128)^5*_Z+RootOf(_Z^8-128)^2+256*_Z^2)*RootOf(_
Z^8-128)^4*x^4-256*(x^4-1)^(1/2)*RootOf(-2*RootOf(_Z^8-128)^5*_Z+RootOf(_Z^8-128)^2+256*_Z^2)*RootOf(_Z^8-128)
^2*x^2-32*RootOf(_Z^8-128)^4*RootOf(-2*RootOf(_Z^8-128)^5*_Z+RootOf(_Z^8-128)^2+256*_Z^2)-16*RootOf(_Z^8-128)*
x^4+64*(x^4-1)^(3/4)*x+16*RootOf(_Z^8-128))/(RootOf(_Z^8-128)^4*x^4-8*x^4+8))*RootOf(_Z^8-128)^4*RootOf(-2*Roo
tOf(_Z^8-128)^5*_Z+RootOf(_Z^8-128)^2+256*_Z^2)-1/16*ln((4*RootOf(-2*RootOf(_Z^8-128)^5*_Z+RootOf(_Z^8-128)^2+
256*_Z^2)*RootOf(_Z^8-128)^8*x^4+(x^4-1)^(1/2)*RootOf(_Z^8-128)^7*x^2-2*(x^4-1)^(1/4)*RootOf(_Z^8-128)^6*x^3-2
*RootOf(_Z^8-128)^5*x^4+32*RootOf(-2*RootOf(_Z^8-128)^5*_Z+RootOf(_Z^8-128)^2+256*_Z^2)*RootOf(_Z^8-128)^4*x^4
-256*(x^4-1)^(1/2)*RootOf(-2*RootOf(_Z^8-128)^5*_Z+RootOf(_Z^8-128)^2+256*_Z^2)*RootOf(_Z^8-128)^2*x^2-32*Root
Of(_Z^8-128)^4*RootOf(-2*RootOf(_Z^8-128)^5*_Z+RootOf(_Z^8-128)^2+256*_Z^2)-16*RootOf(_Z^8-128)*x^4+64*(x^4-1)
^(3/4)*x+16*RootOf(_Z^8-128))/(RootOf(_Z^8-128)^4*x^4-8*x^4+8))*RootOf(_Z^8-128)+RootOf(-2*RootOf(_Z^8-128)^5*
_Z+RootOf(_Z^8-128)^2+256*_Z^2)*ln(-(8*(x^4-1)^(1/2)*RootOf(-2*RootOf(_Z^8-128)^5*_Z+RootOf(_Z^8-128)^2+256*_Z
^2)*RootOf(_Z^8-128)^6*x^2-(x^4-1)^(1/4)*RootOf(_Z^8-128)^6*x^3+16*RootOf(-2*RootOf(_Z^8-128)^5*_Z+RootOf(_Z^8
-128)^2+256*_Z^2)*RootOf(_Z^8-128)^4*x^4-8*(x^4-1)^(1/2)*RootOf(_Z^8-128)^3*x^2+256*(x^4-1)^(1/4)*RootOf(-2*Ro
otOf(_Z^8-128)^5*_Z+RootOf(_Z^8-128)^2+256*_Z^2)*RootOf(_Z^8-128)*x^3-128*RootOf(-2*RootOf(_Z^8-128)^5*_Z+Root
Of(_Z^8-128)^2+256*_Z^2)*x^4-32*(x^4-1)^(3/4)*x+128*RootOf(-2*RootOf(_Z^8-128)^5*_Z+RootOf(_Z^8-128)^2+256*_Z^
2))/(RootOf(_Z^8-128)^4*x^4+8*x^4-8))+1/16*RootOf(_Z^8-128)*ln(((x^4-1)^(1/2)*RootOf(_Z^8-128)^7*x^2+2*(x^4-1)
^(1/4)*RootOf(_Z^8-128)^6*x^3+2*RootOf(_Z^8-128)^5*x^4+16*RootOf(_Z^8-128)*x^4+64*(x^4-1)^(3/4)*x-16*RootOf(_Z
^8-128))/(RootOf(_Z^8-128)^4*x^4-8*x^4+8))

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((x^4 - 1)^(3/4)/(x^8 + 2*x^4 - 1), x)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (x^4-1\right )}^{3/4}}{x^8+2\,x^4-1} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^4 - 1)^(3/4)/(2*x^4 + x^8 - 1),x)

[Out]

int((x^4 - 1)^(3/4)/(2*x^4 + x^8 - 1), x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**4-1)**(3/4)/(x**8+2*x**4-1),x)

[Out]

Integral(((x - 1)*(x + 1)*(x**2 + 1))**(3/4)/(x**8 + 2*x**4 - 1), x)

________________________________________________________________________________________