3.12.38 \(\int \frac {-1+x^4}{\sqrt {-x+x^3} (1+x^4)} \, dx\)

Optimal. Leaf size=85 \[ \frac {\tan ^{-1}\left (\frac {2^{3/4} \sqrt {x^3-x}}{-x^2+\sqrt {2} x+1}\right )}{2^{3/4}}-\frac {\tanh ^{-1}\left (\frac {\frac {x^2}{2^{3/4}}+\frac {x}{\sqrt [4]{2}}-\frac {1}{2^{3/4}}}{\sqrt {x^3-x}}\right )}{2^{3/4}} \]

________________________________________________________________________________________

Rubi [C]  time = 1.66, antiderivative size = 654, normalized size of antiderivative = 7.69, number of steps used = 43, number of rules used = 13, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.542, Rules used = {2056, 1586, 6715, 6725, 406, 222, 409, 1215, 1457, 540, 253, 538, 537} \begin {gather*} \frac {2 \sqrt {x-1} \sqrt {x} \sqrt {x+1} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt {x}}{\sqrt {x-1}}\right )|\frac {1}{2}\right )}{\left (2 \sqrt {2}+(2+2 i)\right ) \sqrt {x^3-x}}+\frac {2 \sqrt {x-1} \sqrt {x} \sqrt {x+1} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt {x}}{\sqrt {x-1}}\right )|\frac {1}{2}\right )}{\left (2 \sqrt {2}+(2-2 i)\right ) \sqrt {x^3-x}}-\frac {\sqrt {x-1} \sqrt {x} \sqrt {x+1} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt {x}}{\sqrt {x-1}}\right )|\frac {1}{2}\right )}{\left (\sqrt {2}+(1+i)\right ) \sqrt {x^3-x}}-\frac {\sqrt {x-1} \sqrt {x} \sqrt {x+1} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt {x}}{\sqrt {x-1}}\right )|\frac {1}{2}\right )}{\left (\sqrt {2}+(1-i)\right ) \sqrt {x^3-x}}-\frac {\sqrt {x-1} \sqrt {x} \sqrt {x+1} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt {x}}{\sqrt {x-1}}\right )|\frac {1}{2}\right )}{\left (\sqrt {2}+(-1-i)\right ) \sqrt {x^3-x}}-\frac {2 \sqrt {x-1} \sqrt {x} \sqrt {x+1} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt {x}}{\sqrt {x-1}}\right )|\frac {1}{2}\right )}{\left (-2 \sqrt {2}+(2+2 i)\right ) \sqrt {x^3-x}}+\frac {\sqrt {2} \sqrt {x-1} \sqrt {x} \sqrt {x+1} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt {x}}{\sqrt {x-1}}\right )|\frac {1}{2}\right )}{\sqrt {x^3-x}}-\frac {\sqrt {1-x} \sqrt {x} \sqrt {x+1} \Pi \left (-\sqrt [4]{-1};\left .\sin ^{-1}\left (\sqrt {x}\right )\right |-1\right )}{\sqrt {x^3-x}}-\frac {\sqrt {1-x} \sqrt {x} \sqrt {x+1} \Pi \left (\sqrt [4]{-1};\left .\sin ^{-1}\left (\sqrt {x}\right )\right |-1\right )}{\sqrt {x^3-x}}-\frac {\sqrt {1-x} \sqrt {x} \sqrt {x+1} \Pi \left (-(-1)^{3/4};\left .\sin ^{-1}\left (\sqrt {x}\right )\right |-1\right )}{\sqrt {x^3-x}}-\frac {\sqrt {1-x} \sqrt {x} \sqrt {x+1} \Pi \left ((-1)^{3/4};\left .\sin ^{-1}\left (\sqrt {x}\right )\right |-1\right )}{\sqrt {x^3-x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[2]*Sqrt[-1 + x]*Sqrt[x]*Sqrt[1 + x]*EllipticF[ArcSin[(Sqrt[2]*Sqrt[x])/Sqrt[-1 + x]], 1/2])/Sqrt[-x + x^
3] - (2*Sqrt[-1 + x]*Sqrt[x]*Sqrt[1 + x]*EllipticF[ArcSin[(Sqrt[2]*Sqrt[x])/Sqrt[-1 + x]], 1/2])/(((2 + 2*I) -
2*Sqrt[2])*Sqrt[-x + x^3]) - (Sqrt[-1 + x]*Sqrt[x]*Sqrt[1 + x]*EllipticF[ArcSin[(Sqrt[2]*Sqrt[x])/Sqrt[-1 + x
]], 1/2])/(((-1 - I) + Sqrt[2])*Sqrt[-x + x^3]) - (Sqrt[-1 + x]*Sqrt[x]*Sqrt[1 + x]*EllipticF[ArcSin[(Sqrt[2]*
Sqrt[x])/Sqrt[-1 + x]], 1/2])/(((1 - I) + Sqrt[2])*Sqrt[-x + x^3]) - (Sqrt[-1 + x]*Sqrt[x]*Sqrt[1 + x]*Ellipti
cF[ArcSin[(Sqrt[2]*Sqrt[x])/Sqrt[-1 + x]], 1/2])/(((1 + I) + Sqrt[2])*Sqrt[-x + x^3]) + (2*Sqrt[-1 + x]*Sqrt[x
]*Sqrt[1 + x]*EllipticF[ArcSin[(Sqrt[2]*Sqrt[x])/Sqrt[-1 + x]], 1/2])/(((2 - 2*I) + 2*Sqrt[2])*Sqrt[-x + x^3])
+ (2*Sqrt[-1 + x]*Sqrt[x]*Sqrt[1 + x]*EllipticF[ArcSin[(Sqrt[2]*Sqrt[x])/Sqrt[-1 + x]], 1/2])/(((2 + 2*I) + 2
*Sqrt[2])*Sqrt[-x + x^3]) - (Sqrt[1 - x]*Sqrt[x]*Sqrt[1 + x]*EllipticPi[-(-1)^(1/4), ArcSin[Sqrt[x]], -1])/Sqr
t[-x + x^3] - (Sqrt[1 - x]*Sqrt[x]*Sqrt[1 + x]*EllipticPi[(-1)^(1/4), ArcSin[Sqrt[x]], -1])/Sqrt[-x + x^3] - (
Sqrt[1 - x]*Sqrt[x]*Sqrt[1 + x]*EllipticPi[-(-1)^(3/4), ArcSin[Sqrt[x]], -1])/Sqrt[-x + x^3] - (Sqrt[1 - x]*Sq
rt[x]*Sqrt[1 + x]*EllipticPi[(-1)^(3/4), ArcSin[Sqrt[x]], -1])/Sqrt[-x + x^3]

Rule 222

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[-(a*b), 2]}, Simp[(Sqrt[-a + q*x^2]*Sqrt[(a + q*x^2
)/q]*EllipticF[ArcSin[x/Sqrt[(a + q*x^2)/(2*q)]], 1/2])/(Sqrt[2]*Sqrt[-a]*Sqrt[a + b*x^4]), x] /; IntegerQ[q]]
/; FreeQ[{a, b}, x] && LtQ[a, 0] && GtQ[b, 0]

Rule 253

Int[((a1_.) + (b1_.)*(x_)^(n_))^(p_)*((a2_.) + (b2_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[((a1 + b1*x^n)^FracPa
rt[p]*(a2 + b2*x^n)^FracPart[p])/(a1*a2 + b1*b2*x^(2*n))^FracPart[p], Int[(a1*a2 + b1*b2*x^(2*n))^p, x], x] /;
FreeQ[{a1, b1, a2, b2, n, p}, x] && EqQ[a2*b1 + a1*b2, 0] &&  !IntegerQ[p]

Rule 406

Int[Sqrt[(a_) + (b_.)*(x_)^4]/((c_) + (d_.)*(x_)^4), x_Symbol] :> Dist[b/d, Int[1/Sqrt[a + b*x^4], x], x] - Di
st[(b*c - a*d)/d, Int[1/(Sqrt[a + b*x^4]*(c + d*x^4)), x], 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 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 538

Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x_)^2]), x_Symbol] :> Dist[Sqrt[1 +
(d*x^2)/c]/Sqrt[c + d*x^2], Int[1/((a + b*x^2)*Sqrt[1 + (d*x^2)/c]*Sqrt[e + f*x^2]), x], x] /; FreeQ[{a, b, c,
d, e, f}, x] &&  !GtQ[c, 0]

Rule 540

Int[Sqrt[(c_) + (d_.)*(x_)^2]/(((a_) + (b_.)*(x_)^2)*Sqrt[(e_) + (f_.)*(x_)^2]), x_Symbol] :> Dist[d/b, Int[1/
(Sqrt[c + d*x^2]*Sqrt[e + f*x^2]), x], x] + Dist[(b*c - a*d)/b, Int[1/((a + b*x^2)*Sqrt[c + d*x^2]*Sqrt[e + f*
x^2]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NegQ[d/c]

Rule 1215

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

Rule 1457

Int[((d_) + (e_.)*(x_)^(n_))^(q_.)*((f_) + (g_.)*(x_)^(n_))^(r_.)*((a_) + (c_.)*(x_)^(n2_))^(p_), x_Symbol] :>
Dist[(a + c*x^(2*n))^FracPart[p]/((d + e*x^n)^FracPart[p]*(a/d + (c*x^n)/e)^FracPart[p]), Int[(d + e*x^n)^(p
+ q)*(f + g*x^n)^r*(a/d + (c*x^n)/e)^p, x], x] /; FreeQ[{a, c, d, e, f, g, n, p, q, r}, x] && EqQ[n2, 2*n] &&
EqQ[c*d^2 + a*e^2, 0] &&  !IntegerQ[p]

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 6715

Int[(u_)*(x_)^(m_.), x_Symbol] :> Dist[1/(m + 1), Subst[Int[SubstFor[x^(m + 1), u, x], x], x, x^(m + 1)], x] /
; FreeQ[m, x] && NeQ[m, -1] && FunctionOfQ[x^(m + 1), u, x]

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^4}{\sqrt {-x+x^3} \left (1+x^4\right )} \, dx &=\frac {\left (\sqrt {x} \sqrt {-1+x^2}\right ) \int \frac {-1+x^4}{\sqrt {x} \sqrt {-1+x^2} \left (1+x^4\right )} \, dx}{\sqrt {-x+x^3}}\\ &=\frac {\left (\sqrt {x} \sqrt {-1+x^2}\right ) \int \frac {\sqrt {-1+x^2} \left (1+x^2\right )}{\sqrt {x} \left (1+x^4\right )} \, dx}{\sqrt {-x+x^3}}\\ &=\frac {\left (2 \sqrt {x} \sqrt {-1+x^2}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {-1+x^4} \left (1+x^4\right )}{1+x^8} \, dx,x,\sqrt {x}\right )}{\sqrt {-x+x^3}}\\ &=\frac {\left (2 \sqrt {x} \sqrt {-1+x^2}\right ) \operatorname {Subst}\left (\int \left (-\frac {\left (\frac {1}{2}-\frac {i}{2}\right ) \sqrt {-1+x^4}}{i-x^4}+\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {-1+x^4}}{i+x^4}\right ) \, dx,x,\sqrt {x}\right )}{\sqrt {-x+x^3}}\\ &=-\frac {\left ((1-i) \sqrt {x} \sqrt {-1+x^2}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {-1+x^4}}{i-x^4} \, dx,x,\sqrt {x}\right )}{\sqrt {-x+x^3}}+\frac {\left ((1+i) \sqrt {x} \sqrt {-1+x^2}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {-1+x^4}}{i+x^4} \, dx,x,\sqrt {x}\right )}{\sqrt {-x+x^3}}\\ &=-\frac {\left (2 i \sqrt {x} \sqrt {-1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (i-x^4\right ) \sqrt {-1+x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt {-x+x^3}}-\frac {\left (2 i \sqrt {x} \sqrt {-1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1+x^4} \left (i+x^4\right )} \, dx,x,\sqrt {x}\right )}{\sqrt {-x+x^3}}+\frac {\left ((1-i) \sqrt {x} \sqrt {-1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1+x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt {-x+x^3}}+\frac {\left ((1+i) \sqrt {x} \sqrt {-1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1+x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt {-x+x^3}}\\ &=\frac {\sqrt {2} \sqrt {-1+x} \sqrt {x} \sqrt {1+x} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt {x}}{\sqrt {-1+x}}\right )|\frac {1}{2}\right )}{\sqrt {-x+x^3}}-\frac {\left (\sqrt {x} \sqrt {-1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1-\sqrt [4]{-1} x^2\right ) \sqrt {-1+x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt {-x+x^3}}-\frac {\left (\sqrt {x} \sqrt {-1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1+\sqrt [4]{-1} x^2\right ) \sqrt {-1+x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt {-x+x^3}}-\frac {\left (\sqrt {x} \sqrt {-1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1-(-1)^{3/4} x^2\right ) \sqrt {-1+x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt {-x+x^3}}-\frac {\left (\sqrt {x} \sqrt {-1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1+(-1)^{3/4} x^2\right ) \sqrt {-1+x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt {-x+x^3}}\\ &=\frac {\sqrt {2} \sqrt {-1+x} \sqrt {x} \sqrt {1+x} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt {x}}{\sqrt {-1+x}}\right )|\frac {1}{2}\right )}{\sqrt {-x+x^3}}-\frac {\left (\sqrt {x} \sqrt {-1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1+x^4}} \, dx,x,\sqrt {x}\right )}{\left (1-\sqrt [4]{-1}\right ) \sqrt {-x+x^3}}+\frac {\left (\sqrt [4]{-1} \sqrt {x} \sqrt {-1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1-x^2}{\left (1-\sqrt [4]{-1} x^2\right ) \sqrt {-1+x^4}} \, dx,x,\sqrt {x}\right )}{\left (1-\sqrt [4]{-1}\right ) \sqrt {-x+x^3}}-\frac {\left (\sqrt {x} \sqrt {-1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1+x^4}} \, dx,x,\sqrt {x}\right )}{\left (1+\sqrt [4]{-1}\right ) \sqrt {-x+x^3}}-\frac {\left (\sqrt [4]{-1} \sqrt {x} \sqrt {-1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1-x^2}{\left (1+\sqrt [4]{-1} x^2\right ) \sqrt {-1+x^4}} \, dx,x,\sqrt {x}\right )}{\left (1+\sqrt [4]{-1}\right ) \sqrt {-x+x^3}}-\frac {\left (\sqrt {x} \sqrt {-1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1+x^4}} \, dx,x,\sqrt {x}\right )}{\left (1-(-1)^{3/4}\right ) \sqrt {-x+x^3}}-\frac {\left (\sqrt {x} \sqrt {-1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1+x^4}} \, dx,x,\sqrt {x}\right )}{\left (1+(-1)^{3/4}\right ) \sqrt {-x+x^3}}-\frac {\left ((-1)^{3/4} \sqrt {x} \sqrt {-1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1-x^2}{\left (1+(-1)^{3/4} x^2\right ) \sqrt {-1+x^4}} \, dx,x,\sqrt {x}\right )}{\left (1+(-1)^{3/4}\right ) \sqrt {-x+x^3}}-\frac {\left (\sqrt {2} \sqrt {x} \sqrt {-1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1-x^2}{\left (1-(-1)^{3/4} x^2\right ) \sqrt {-1+x^4}} \, dx,x,\sqrt {x}\right )}{\left ((1+i)+\sqrt {2}\right ) \sqrt {-x+x^3}}\\ &=\frac {\sqrt {2} \sqrt {-1+x} \sqrt {x} \sqrt {1+x} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt {x}}{\sqrt {-1+x}}\right )|\frac {1}{2}\right )}{\sqrt {-x+x^3}}-\frac {\sqrt {-1+x} \sqrt {x} \sqrt {1+x} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt {x}}{\sqrt {-1+x}}\right )|\frac {1}{2}\right )}{\sqrt {2} \left (1-\sqrt [4]{-1}\right ) \sqrt {-x+x^3}}-\frac {\sqrt {-1+x} \sqrt {x} \sqrt {1+x} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt {x}}{\sqrt {-1+x}}\right )|\frac {1}{2}\right )}{\sqrt {2} \left (1+\sqrt [4]{-1}\right ) \sqrt {-x+x^3}}-\frac {\sqrt {-1+x} \sqrt {x} \sqrt {1+x} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt {x}}{\sqrt {-1+x}}\right )|\frac {1}{2}\right )}{\sqrt {2} \left (1-(-1)^{3/4}\right ) \sqrt {-x+x^3}}-\frac {\sqrt {-1+x} \sqrt {x} \sqrt {1+x} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt {x}}{\sqrt {-1+x}}\right )|\frac {1}{2}\right )}{\sqrt {2} \left (1+(-1)^{3/4}\right ) \sqrt {-x+x^3}}+\frac {\left (\sqrt [4]{-1} \sqrt {-1-x} \sqrt {1-x} \sqrt {x}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {1-x^2}}{\sqrt {-1-x^2} \left (1-\sqrt [4]{-1} x^2\right )} \, dx,x,\sqrt {x}\right )}{\left (1-\sqrt [4]{-1}\right ) \sqrt {-x+x^3}}-\frac {\left (\sqrt [4]{-1} \sqrt {-1-x} \sqrt {1-x} \sqrt {x}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {1-x^2}}{\sqrt {-1-x^2} \left (1+\sqrt [4]{-1} x^2\right )} \, dx,x,\sqrt {x}\right )}{\left (1+\sqrt [4]{-1}\right ) \sqrt {-x+x^3}}-\frac {\left ((-1)^{3/4} \sqrt {-1-x} \sqrt {1-x} \sqrt {x}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {1-x^2}}{\sqrt {-1-x^2} \left (1+(-1)^{3/4} x^2\right )} \, dx,x,\sqrt {x}\right )}{\left (1+(-1)^{3/4}\right ) \sqrt {-x+x^3}}-\frac {\left (\sqrt {2} \sqrt {-1-x} \sqrt {1-x} \sqrt {x}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {1-x^2}}{\sqrt {-1-x^2} \left (1-(-1)^{3/4} x^2\right )} \, dx,x,\sqrt {x}\right )}{\left ((1+i)+\sqrt {2}\right ) \sqrt {-x+x^3}}\\ &=\frac {\sqrt {2} \sqrt {-1+x} \sqrt {x} \sqrt {1+x} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt {x}}{\sqrt {-1+x}}\right )|\frac {1}{2}\right )}{\sqrt {-x+x^3}}-\frac {\sqrt {-1+x} \sqrt {x} \sqrt {1+x} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt {x}}{\sqrt {-1+x}}\right )|\frac {1}{2}\right )}{\sqrt {2} \left (1-\sqrt [4]{-1}\right ) \sqrt {-x+x^3}}-\frac {\sqrt {-1+x} \sqrt {x} \sqrt {1+x} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt {x}}{\sqrt {-1+x}}\right )|\frac {1}{2}\right )}{\sqrt {2} \left (1+\sqrt [4]{-1}\right ) \sqrt {-x+x^3}}-\frac {\sqrt {-1+x} \sqrt {x} \sqrt {1+x} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt {x}}{\sqrt {-1+x}}\right )|\frac {1}{2}\right )}{\sqrt {2} \left (1-(-1)^{3/4}\right ) \sqrt {-x+x^3}}-\frac {\sqrt {-1+x} \sqrt {x} \sqrt {1+x} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt {x}}{\sqrt {-1+x}}\right )|\frac {1}{2}\right )}{\sqrt {2} \left (1+(-1)^{3/4}\right ) \sqrt {-x+x^3}}+\frac {\left (\sqrt {-1-x} \sqrt {1-x} \sqrt {x}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1-x^2} \sqrt {1-x^2}} \, dx,x,\sqrt {x}\right )}{\left (1-\sqrt [4]{-1}\right ) \sqrt {-x+x^3}}+\frac {\left (\sqrt {-1-x} \sqrt {1-x} \sqrt {x}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1-x^2} \sqrt {1-x^2}} \, dx,x,\sqrt {x}\right )}{\left (1+\sqrt [4]{-1}\right ) \sqrt {-x+x^3}}+\frac {\left (\sqrt [4]{-1} \left (-1+(-1)^{3/4}\right ) \sqrt {-1-x} \sqrt {1-x} \sqrt {x}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1-x^2} \sqrt {1-x^2} \left (1+\sqrt [4]{-1} x^2\right )} \, dx,x,\sqrt {x}\right )}{\left (1+\sqrt [4]{-1}\right ) \sqrt {-x+x^3}}+\frac {\left (\sqrt {-1-x} \sqrt {1-x} \sqrt {x}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1-x^2} \sqrt {1-x^2}} \, dx,x,\sqrt {x}\right )}{\left (1+(-1)^{3/4}\right ) \sqrt {-x+x^3}}+\frac {\left ((-1)^{3/4} \left (-1+\sqrt [4]{-1}\right ) \sqrt {-1-x} \sqrt {1-x} \sqrt {x}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1-x^2} \sqrt {1-x^2} \left (1+(-1)^{3/4} x^2\right )} \, dx,x,\sqrt {x}\right )}{\left (1+(-1)^{3/4}\right ) \sqrt {-x+x^3}}+\frac {\left (\sqrt [4]{-1} \left (1+(-1)^{3/4}\right ) \sqrt {-1-x} \sqrt {1-x} \sqrt {x}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1-x^2} \sqrt {1-x^2} \left (1-\sqrt [4]{-1} x^2\right )} \, dx,x,\sqrt {x}\right )}{\left (1-\sqrt [4]{-1}\right ) \sqrt {-x+x^3}}+\frac {\left (\sqrt [4]{-1} \sqrt {2} \sqrt {-1-x} \sqrt {1-x} \sqrt {x}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1-x^2} \sqrt {1-x^2}} \, dx,x,\sqrt {x}\right )}{\left ((1+i)+\sqrt {2}\right ) \sqrt {-x+x^3}}-\frac {\left (\sqrt {2} \left (1+\sqrt [4]{-1}\right ) \sqrt {-1-x} \sqrt {1-x} \sqrt {x}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1-x^2} \sqrt {1-x^2} \left (1-(-1)^{3/4} x^2\right )} \, dx,x,\sqrt {x}\right )}{\left ((1+i)+\sqrt {2}\right ) \sqrt {-x+x^3}}\\ &=\frac {\sqrt {2} \sqrt {-1+x} \sqrt {x} \sqrt {1+x} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt {x}}{\sqrt {-1+x}}\right )|\frac {1}{2}\right )}{\sqrt {-x+x^3}}-\frac {\sqrt {-1+x} \sqrt {x} \sqrt {1+x} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt {x}}{\sqrt {-1+x}}\right )|\frac {1}{2}\right )}{\sqrt {2} \left (1-\sqrt [4]{-1}\right ) \sqrt {-x+x^3}}-\frac {\sqrt {-1+x} \sqrt {x} \sqrt {1+x} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt {x}}{\sqrt {-1+x}}\right )|\frac {1}{2}\right )}{\sqrt {2} \left (1+\sqrt [4]{-1}\right ) \sqrt {-x+x^3}}-\frac {\sqrt {-1+x} \sqrt {x} \sqrt {1+x} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt {x}}{\sqrt {-1+x}}\right )|\frac {1}{2}\right )}{\sqrt {2} \left (1-(-1)^{3/4}\right ) \sqrt {-x+x^3}}-\frac {\sqrt {-1+x} \sqrt {x} \sqrt {1+x} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt {x}}{\sqrt {-1+x}}\right )|\frac {1}{2}\right )}{\sqrt {2} \left (1+(-1)^{3/4}\right ) \sqrt {-x+x^3}}+\frac {\left (\sqrt [4]{-1} \left (-1+(-1)^{3/4}\right ) \sqrt {1-x} \sqrt {x} \sqrt {1+x}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \sqrt {1+x^2} \left (1+\sqrt [4]{-1} x^2\right )} \, dx,x,\sqrt {x}\right )}{\left (1+\sqrt [4]{-1}\right ) \sqrt {-x+x^3}}+\frac {\left ((-1)^{3/4} \left (-1+\sqrt [4]{-1}\right ) \sqrt {1-x} \sqrt {x} \sqrt {1+x}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \sqrt {1+x^2} \left (1+(-1)^{3/4} x^2\right )} \, dx,x,\sqrt {x}\right )}{\left (1+(-1)^{3/4}\right ) \sqrt {-x+x^3}}+\frac {\left (\sqrt [4]{-1} \left (1+(-1)^{3/4}\right ) \sqrt {1-x} \sqrt {x} \sqrt {1+x}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \sqrt {1+x^2} \left (1-\sqrt [4]{-1} x^2\right )} \, dx,x,\sqrt {x}\right )}{\left (1-\sqrt [4]{-1}\right ) \sqrt {-x+x^3}}-\frac {\left (\sqrt {2} \left (1+\sqrt [4]{-1}\right ) \sqrt {1-x} \sqrt {x} \sqrt {1+x}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \sqrt {1+x^2} \left (1-(-1)^{3/4} x^2\right )} \, dx,x,\sqrt {x}\right )}{\left ((1+i)+\sqrt {2}\right ) \sqrt {-x+x^3}}+\frac {\left (\sqrt {x} \sqrt {-1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1+x^4}} \, dx,x,\sqrt {x}\right )}{\left (1-\sqrt [4]{-1}\right ) \sqrt {-x+x^3}}+\frac {\left (\sqrt {x} \sqrt {-1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1+x^4}} \, dx,x,\sqrt {x}\right )}{\left (1+\sqrt [4]{-1}\right ) \sqrt {-x+x^3}}+\frac {\left (\sqrt {x} \sqrt {-1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1+x^4}} \, dx,x,\sqrt {x}\right )}{\left (1+(-1)^{3/4}\right ) \sqrt {-x+x^3}}+\frac {\left (\sqrt [4]{-1} \sqrt {2} \sqrt {x} \sqrt {-1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1+x^4}} \, dx,x,\sqrt {x}\right )}{\left ((1+i)+\sqrt {2}\right ) \sqrt {-x+x^3}}\\ &=\frac {\sqrt {2} \sqrt {-1+x} \sqrt {x} \sqrt {1+x} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt {x}}{\sqrt {-1+x}}\right )|\frac {1}{2}\right )}{\sqrt {-x+x^3}}-\frac {\sqrt {-1+x} \sqrt {x} \sqrt {1+x} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt {x}}{\sqrt {-1+x}}\right )|\frac {1}{2}\right )}{\sqrt {2} \left (1-(-1)^{3/4}\right ) \sqrt {-x+x^3}}+\frac {\sqrt [4]{-1} \sqrt {-1+x} \sqrt {x} \sqrt {1+x} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt {x}}{\sqrt {-1+x}}\right )|\frac {1}{2}\right )}{\left ((1+i)+\sqrt {2}\right ) \sqrt {-x+x^3}}-\frac {\sqrt [4]{-1} \left (1-(-1)^{3/4}\right ) \sqrt {1-x} \sqrt {x} \sqrt {1+x} \Pi \left (-\sqrt [4]{-1};\left .\sin ^{-1}\left (\sqrt {x}\right )\right |-1\right )}{\left (1+\sqrt [4]{-1}\right ) \sqrt {-x+x^3}}-\frac {\sqrt {1-x} \sqrt {x} \sqrt {1+x} \Pi \left (\sqrt [4]{-1};\left .\sin ^{-1}\left (\sqrt {x}\right )\right |-1\right )}{\sqrt {-x+x^3}}-\frac {(-1)^{3/4} \left (1-\sqrt [4]{-1}\right ) \sqrt {1-x} \sqrt {x} \sqrt {1+x} \Pi \left (-(-1)^{3/4};\left .\sin ^{-1}\left (\sqrt {x}\right )\right |-1\right )}{\left (1+(-1)^{3/4}\right ) \sqrt {-x+x^3}}-\frac {\sqrt {1-x} \sqrt {x} \sqrt {1+x} \Pi \left ((-1)^{3/4};\left .\sin ^{-1}\left (\sqrt {x}\right )\right |-1\right )}{\sqrt {-x+x^3}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [C]  time = 0.72, size = 96, normalized size = 1.13 \begin {gather*} -\frac {\sqrt {1-\frac {1}{x^2}} x^{3/2} \left (-2 F\left (\left .\sin ^{-1}\left (\frac {1}{\sqrt {x}}\right )\right |-1\right )+\Pi \left (-\sqrt [4]{-1};\left .\sin ^{-1}\left (\frac {1}{\sqrt {x}}\right )\right |-1\right )+\Pi \left (\sqrt [4]{-1};\left .\sin ^{-1}\left (\frac {1}{\sqrt {x}}\right )\right |-1\right )+\Pi \left (-(-1)^{3/4};\left .\sin ^{-1}\left (\frac {1}{\sqrt {x}}\right )\right |-1\right )+\Pi \left ((-1)^{3/4};\left .\sin ^{-1}\left (\frac {1}{\sqrt {x}}\right )\right |-1\right )\right )}{\sqrt {x \left (x^2-1\right )}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-((Sqrt[1 - x^(-2)]*x^(3/2)*(-2*EllipticF[ArcSin[1/Sqrt[x]], -1] + EllipticPi[-(-1)^(1/4), ArcSin[1/Sqrt[x]],
-1] + EllipticPi[(-1)^(1/4), ArcSin[1/Sqrt[x]], -1] + EllipticPi[-(-1)^(3/4), ArcSin[1/Sqrt[x]], -1] + Ellipti
cPi[(-1)^(3/4), ArcSin[1/Sqrt[x]], -1]))/Sqrt[x*(-1 + x^2)])

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 0.35, size = 85, normalized size = 1.00 \begin {gather*} \frac {\tan ^{-1}\left (\frac {2^{3/4} \sqrt {-x+x^3}}{1+\sqrt {2} x-x^2}\right )}{2^{3/4}}-\frac {\tanh ^{-1}\left (\frac {-\frac {1}{2^{3/4}}+\frac {x}{\sqrt [4]{2}}+\frac {x^2}{2^{3/4}}}{\sqrt {-x+x^3}}\right )}{2^{3/4}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

fricas [B]  time = 0.71, size = 385, normalized size = 4.53 \begin {gather*} -\frac {1}{2} \cdot 2^{\frac {1}{4}} \arctan \left (\frac {\sqrt {x^{3} - x} {\left (2^{\frac {3}{4}} x - 2^{\frac {1}{4}} {\left (x^{2} - 1\right )}\right )} - {\left (2 \, x^{3} - \sqrt {x^{3} - x} {\left (2^{\frac {3}{4}} x + 2^{\frac {1}{4}} {\left (x^{2} - 1\right )}\right )} - 2 \, x\right )} \sqrt {\frac {x^{4} + 4 \, \sqrt {2} {\left (x^{3} - x\right )} + 2 \, \sqrt {x^{3} - x} {\left (2^{\frac {3}{4}} {\left (x^{2} - 1\right )} + 2 \cdot 2^{\frac {1}{4}} x\right )} + 1}{x^{4} + 1}}}{2 \, {\left (x^{3} - x\right )}}\right ) - \frac {1}{2} \cdot 2^{\frac {1}{4}} \arctan \left (\frac {\sqrt {x^{3} - x} {\left (2^{\frac {3}{4}} x - 2^{\frac {1}{4}} {\left (x^{2} - 1\right )}\right )} + {\left (2 \, x^{3} + \sqrt {x^{3} - x} {\left (2^{\frac {3}{4}} x + 2^{\frac {1}{4}} {\left (x^{2} - 1\right )}\right )} - 2 \, x\right )} \sqrt {\frac {x^{4} + 4 \, \sqrt {2} {\left (x^{3} - x\right )} - 2 \, \sqrt {x^{3} - x} {\left (2^{\frac {3}{4}} {\left (x^{2} - 1\right )} + 2 \cdot 2^{\frac {1}{4}} x\right )} + 1}{x^{4} + 1}}}{2 \, {\left (x^{3} - x\right )}}\right ) - \frac {1}{8} \cdot 2^{\frac {1}{4}} \log \left (\frac {x^{4} + 4 \, \sqrt {2} {\left (x^{3} - x\right )} + 2 \, \sqrt {x^{3} - x} {\left (2^{\frac {3}{4}} {\left (x^{2} - 1\right )} + 2 \cdot 2^{\frac {1}{4}} x\right )} + 1}{x^{4} + 1}\right ) + \frac {1}{8} \cdot 2^{\frac {1}{4}} \log \left (\frac {x^{4} + 4 \, \sqrt {2} {\left (x^{3} - x\right )} - 2 \, \sqrt {x^{3} - x} {\left (2^{\frac {3}{4}} {\left (x^{2} - 1\right )} + 2 \cdot 2^{\frac {1}{4}} x\right )} + 1}{x^{4} + 1}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*2^(1/4)*arctan(1/2*(sqrt(x^3 - x)*(2^(3/4)*x - 2^(1/4)*(x^2 - 1)) - (2*x^3 - sqrt(x^3 - x)*(2^(3/4)*x + 2
^(1/4)*(x^2 - 1)) - 2*x)*sqrt((x^4 + 4*sqrt(2)*(x^3 - x) + 2*sqrt(x^3 - x)*(2^(3/4)*(x^2 - 1) + 2*2^(1/4)*x) +
1)/(x^4 + 1)))/(x^3 - x)) - 1/2*2^(1/4)*arctan(1/2*(sqrt(x^3 - x)*(2^(3/4)*x - 2^(1/4)*(x^2 - 1)) + (2*x^3 +
sqrt(x^3 - x)*(2^(3/4)*x + 2^(1/4)*(x^2 - 1)) - 2*x)*sqrt((x^4 + 4*sqrt(2)*(x^3 - x) - 2*sqrt(x^3 - x)*(2^(3/4
)*(x^2 - 1) + 2*2^(1/4)*x) + 1)/(x^4 + 1)))/(x^3 - x)) - 1/8*2^(1/4)*log((x^4 + 4*sqrt(2)*(x^3 - x) + 2*sqrt(x
^3 - x)*(2^(3/4)*(x^2 - 1) + 2*2^(1/4)*x) + 1)/(x^4 + 1)) + 1/8*2^(1/4)*log((x^4 + 4*sqrt(2)*(x^3 - x) - 2*sqr
t(x^3 - x)*(2^(3/4)*(x^2 - 1) + 2*2^(1/4)*x) + 1)/(x^4 + 1))

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maple [C]  time = 2.66, size = 119, normalized size = 1.40

method	result	size
default	\(\frac {\sqrt {1+x}\, \sqrt {2-2 x}\, \sqrt {-x}\, \EllipticF \left (\sqrt {1+x}, \frac {\sqrt {2}}{2}\right )}{\sqrt {x^{3}-x}}+\frac {\sqrt {2}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (\textit {\_Z}^{4}+1\right )}{\sum }\frac {\underline {\hspace {1.25 ex}}\alpha \left (\underline {\hspace {1.25 ex}}\alpha ^{3}-\underline {\hspace {1.25 ex}}\alpha ^{2}+\underline {\hspace {1.25 ex}}\alpha -1\right ) \sqrt {1+x}\, \sqrt {1-x}\, \sqrt {-x}\, \EllipticPi \left (\sqrt {1+x}, -\frac {1}{2} \underline {\hspace {1.25 ex}}\alpha ^{3}+\frac {1}{2} \underline {\hspace {1.25 ex}}\alpha ^{2}-\frac {1}{2} \underline {\hspace {1.25 ex}}\alpha +\frac {1}{2}, \frac {\sqrt {2}}{2}\right )}{\sqrt {x \left (x^{2}-1\right )}}\right )}{4}\)	\(119\)
elliptic	\(\frac {\sqrt {1+x}\, \sqrt {2-2 x}\, \sqrt {-x}\, \EllipticF \left (\sqrt {1+x}, \frac {\sqrt {2}}{2}\right )}{\sqrt {x^{3}-x}}+\frac {\sqrt {2}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (\textit {\_Z}^{4}+1\right )}{\sum }\frac {\underline {\hspace {1.25 ex}}\alpha \left (\underline {\hspace {1.25 ex}}\alpha ^{3}-\underline {\hspace {1.25 ex}}\alpha ^{2}+\underline {\hspace {1.25 ex}}\alpha -1\right ) \sqrt {1+x}\, \sqrt {1-x}\, \sqrt {-x}\, \EllipticPi \left (\sqrt {1+x}, -\frac {1}{2} \underline {\hspace {1.25 ex}}\alpha ^{3}+\frac {1}{2} \underline {\hspace {1.25 ex}}\alpha ^{2}-\frac {1}{2} \underline {\hspace {1.25 ex}}\alpha +\frac {1}{2}, \frac {\sqrt {2}}{2}\right )}{\sqrt {x \left (x^{2}-1\right )}}\right )}{4}\)	\(119\)
trager	\(\frac {\RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+8\right )^{2}\right ) \ln \left (-\frac {8 \RootOf \left (\textit {\_Z}^{4}+8\right )^{4} x^{2} \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+8\right )^{2}\right )-12 \RootOf \left (\textit {\_Z}^{4}+8\right )^{4} x \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+8\right )^{2}\right )-8 \RootOf \left (\textit {\_Z}^{4}+8\right )^{4} \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+8\right )^{2}\right )-50 \RootOf \left (\textit {\_Z}^{4}+8\right )^{2} \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+8\right )^{2}\right ) x^{2}+7 \RootOf \left (\textit {\_Z}^{4}+8\right )^{2} \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+8\right )^{2}\right ) x +200 \sqrt {x^{3}-x}\, \RootOf \left (\textit {\_Z}^{4}+8\right )^{2}+50 \RootOf \left (\textit {\_Z}^{4}+8\right )^{2} \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+8\right )^{2}\right )+78 \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+8\right )^{2}\right ) x^{2}+104 \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+8\right )^{2}\right ) x -56 \sqrt {x^{3}-x}-78 \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}+8\right )^{2}\right )}{2 \RootOf \left (\textit {\_Z}^{4}+8\right )^{2} x^{2}-3 \RootOf \left (\textit {\_Z}^{4}+8\right )^{2} x -2 \RootOf \left (\textit {\_Z}^{4}+8\right )^{2}+6 x^{2}+8 x -6}\right )}{4}+\frac {\RootOf \left (\textit {\_Z}^{4}+8\right ) \ln \left (-\frac {8 \RootOf \left (\textit {\_Z}^{4}+8\right )^{5} x^{2}-12 \RootOf \left (\textit {\_Z}^{4}+8\right )^{5} x -8 \RootOf \left (\textit {\_Z}^{4}+8\right )^{5}+50 \RootOf \left (\textit {\_Z}^{4}+8\right )^{3} x^{2}-7 \RootOf \left (\textit {\_Z}^{4}+8\right )^{3} x -50 \RootOf \left (\textit {\_Z}^{4}+8\right )^{3}-200 \sqrt {x^{3}-x}\, \RootOf \left (\textit {\_Z}^{4}+8\right )^{2}+78 \RootOf \left (\textit {\_Z}^{4}+8\right ) x^{2}+104 \RootOf \left (\textit {\_Z}^{4}+8\right ) x -78 \RootOf \left (\textit {\_Z}^{4}+8\right )-56 \sqrt {x^{3}-x}}{2 \RootOf \left (\textit {\_Z}^{4}+8\right )^{2} x^{2}-3 \RootOf \left (\textit {\_Z}^{4}+8\right )^{2} x -2 \RootOf \left (\textit {\_Z}^{4}+8\right )^{2}-6 x^{2}-8 x +6}\right )}{4}\)	\(476\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(1+x)^(1/2)*(2-2*x)^(1/2)*(-x)^(1/2)/(x^3-x)^(1/2)*EllipticF((1+x)^(1/2),1/2*2^(1/2))+1/4*2^(1/2)*sum(_alpha*(
_alpha^3-_alpha^2+_alpha-1)*(1+x)^(1/2)*(1-x)^(1/2)*(-x)^(1/2)/(x*(x^2-1))^(1/2)*EllipticPi((1+x)^(1/2),-1/2*_
alpha^3+1/2*_alpha^2-1/2*_alpha+1/2,1/2*2^(1/2)),_alpha=RootOf(_Z^4+1))

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

mupad [B]  time = 0.03, size = 205, normalized size = 2.41 \begin {gather*} \frac {\sqrt {-x}\,\sqrt {1-x}\,\sqrt {x+1}\,\Pi \left (\sqrt {2}\,\left (-\frac {1}{2}-\frac {1}{2}{}\mathrm {i}\right );\mathrm {asin}\left (\sqrt {-x}\right )\middle |-1\right )}{\sqrt {x^3-x}}+\frac {\sqrt {-x}\,\sqrt {1-x}\,\sqrt {x+1}\,\Pi \left (\sqrt {2}\,\left (-\frac {1}{2}+\frac {1}{2}{}\mathrm {i}\right );\mathrm {asin}\left (\sqrt {-x}\right )\middle |-1\right )}{\sqrt {x^3-x}}+\frac {\sqrt {-x}\,\sqrt {1-x}\,\sqrt {x+1}\,\Pi \left (\sqrt {2}\,\left (\frac {1}{2}-\frac {1}{2}{}\mathrm {i}\right );\mathrm {asin}\left (\sqrt {-x}\right )\middle |-1\right )}{\sqrt {x^3-x}}+\frac {\sqrt {-x}\,\sqrt {1-x}\,\sqrt {x+1}\,\Pi \left (\sqrt {2}\,\left (\frac {1}{2}+\frac {1}{2}{}\mathrm {i}\right );\mathrm {asin}\left (\sqrt {-x}\right )\middle |-1\right )}{\sqrt {x^3-x}}-\frac {2\,\sqrt {-x}\,\sqrt {1-x}\,\sqrt {x+1}\,\mathrm {F}\left (\mathrm {asin}\left (\sqrt {-x}\right )\middle |-1\right )}{\sqrt {x^3-x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((-x)^(1/2)*(1 - x)^(1/2)*(x + 1)^(1/2)*ellipticPi(2^(1/2)*(- 1/2 - 1i/2), asin((-x)^(1/2)), -1))/(x^3 - x)^(1
/2) + ((-x)^(1/2)*(1 - x)^(1/2)*(x + 1)^(1/2)*ellipticPi(2^(1/2)*(- 1/2 + 1i/2), asin((-x)^(1/2)), -1))/(x^3 -
x)^(1/2) + ((-x)^(1/2)*(1 - x)^(1/2)*(x + 1)^(1/2)*ellipticPi(2^(1/2)*(1/2 - 1i/2), asin((-x)^(1/2)), -1))/(x
^3 - x)^(1/2) + ((-x)^(1/2)*(1 - x)^(1/2)*(x + 1)^(1/2)*ellipticPi(2^(1/2)*(1/2 + 1i/2), asin((-x)^(1/2)), -1)
)/(x^3 - x)^(1/2) - (2*(-x)^(1/2)*(1 - x)^(1/2)*(x + 1)^(1/2)*ellipticF(asin((-x)^(1/2)), -1))/(x^3 - x)^(1/2)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________