3.14.46 \(\int \frac {-1+x^2}{(1+x^2) \sqrt {-x-x^2+x^3}} \, dx\)

Optimal. Leaf size=97 \[ -\sqrt {\frac {1}{5}+\frac {2 i}{5}} \tan ^{-1}\left (\frac {\sqrt {1-2 i} \sqrt {x^3-x^2-x}}{x^2-x-1}\right )-\sqrt {\frac {1}{5}-\frac {2 i}{5}} \tan ^{-1}\left (\frac {\sqrt {1+2 i} \sqrt {x^3-x^2-x}}{x^2-x-1}\right ) \]

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Rubi [C]  time = 1.17, antiderivative size = 375, normalized size of antiderivative = 3.87, number of steps used = 15, number of rules used = 8, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.276, Rules used = {2056, 6725, 716, 1098, 934, 168, 538, 537} \begin {gather*} \frac {\sqrt {x} \sqrt {-\left (\left (1-\sqrt {5}\right ) x\right )-2} \sqrt {\frac {\left (1+\sqrt {5}\right ) x+2}{\left (1-\sqrt {5}\right ) x+2}} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{5} \sqrt {x}}{\sqrt {-\left (\left (1-\sqrt {5}\right ) x\right )-2}}\right )|\frac {1}{10} \left (5-\sqrt {5}\right )\right )}{\sqrt [4]{5} \sqrt {\frac {1}{\left (1-\sqrt {5}\right ) x+2}} \sqrt {x^3-x^2-x}}-\frac {\sqrt {3+\sqrt {5}} \sqrt {x} \sqrt {2 x+\sqrt {5}-1} \sqrt {1-\frac {2 x}{1+\sqrt {5}}} \Pi \left (-\frac {1}{2} i \left (1+\sqrt {5}\right );\sin ^{-1}\left (\sqrt {\frac {2}{1+\sqrt {5}}} \sqrt {x}\right )|\frac {1}{2} \left (-3-\sqrt {5}\right )\right )}{\sqrt {x^3-x^2-x}}-\frac {\sqrt {3+\sqrt {5}} \sqrt {x} \sqrt {2 x+\sqrt {5}-1} \sqrt {1-\frac {2 x}{1+\sqrt {5}}} \Pi \left (\frac {1}{2} i \left (1+\sqrt {5}\right );\sin ^{-1}\left (\sqrt {\frac {2}{1+\sqrt {5}}} \sqrt {x}\right )|\frac {1}{2} \left (-3-\sqrt {5}\right )\right )}{\sqrt {x^3-x^2-x}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(Sqrt[x]*Sqrt[-2 - (1 - Sqrt[5])*x]*Sqrt[(2 + (1 + Sqrt[5])*x)/(2 + (1 - Sqrt[5])*x)]*EllipticF[ArcSin[(Sqrt[2
]*5^(1/4)*Sqrt[x])/Sqrt[-2 - (1 - Sqrt[5])*x]], (5 - Sqrt[5])/10])/(5^(1/4)*Sqrt[(2 + (1 - Sqrt[5])*x)^(-1)]*S
qrt[-x - x^2 + x^3]) - (Sqrt[3 + Sqrt[5]]*Sqrt[x]*Sqrt[-1 + Sqrt[5] + 2*x]*Sqrt[1 - (2*x)/(1 + Sqrt[5])]*Ellip
ticPi[(-1/2*I)*(1 + Sqrt[5]), ArcSin[Sqrt[2/(1 + Sqrt[5])]*Sqrt[x]], (-3 - Sqrt[5])/2])/Sqrt[-x - x^2 + x^3] -
(Sqrt[3 + Sqrt[5]]*Sqrt[x]*Sqrt[-1 + Sqrt[5] + 2*x]*Sqrt[1 - (2*x)/(1 + Sqrt[5])]*EllipticPi[(I/2)*(1 + Sqrt[
5]), ArcSin[Sqrt[2/(1 + Sqrt[5])]*Sqrt[x]], (-3 - Sqrt[5])/2])/Sqrt[-x - x^2 + x^3]

Rule 168

Int[1/(((a_.) + (b_.)*(x_))*Sqrt[(c_.) + (d_.)*(x_)]*Sqrt[(e_.) + (f_.)*(x_)]*Sqrt[(g_.) + (h_.)*(x_)]), x_Sym
bol] :> Dist[-2, Subst[Int[1/(Simp[b*c - a*d - b*x^2, x]*Sqrt[Simp[(d*e - c*f)/d + (f*x^2)/d, x]]*Sqrt[Simp[(d
*g - c*h)/d + (h*x^2)/d, x]]), x], x, Sqrt[c + d*x]], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x] && GtQ[(d*e - c
*f)/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 716

Int[(x_)^(m_)/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[2, Subst[Int[x^(2*m + 1)/Sqrt[a + b*x^
2 + c*x^4], x], x, Sqrt[x]], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[m^2, 1/4]

Rule 934

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(f_.) + (g_.)*(x_)]*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :> Wi
th[{q = Rt[b^2 - 4*a*c, 2]}, Dist[(Sqrt[b - q + 2*c*x]*Sqrt[b + q + 2*c*x])/Sqrt[a + b*x + c*x^2], Int[1/((d +
e*x)*Sqrt[f + g*x]*Sqrt[b - q + 2*c*x]*Sqrt[b + q + 2*c*x]), x], x]] /; FreeQ[{a, b, c, d, e, f, g}, x] && Ne
Q[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0]

Rule 1098

Int[1/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[(Sqrt[(2*a +
(b - q)*x^2)/(2*a + (b + q)*x^2)]*Sqrt[(2*a + (b + q)*x^2)/q]*EllipticF[ArcSin[x/Sqrt[(2*a + (b + q)*x^2)/(2*q
)]], (b + q)/(2*q)])/(2*Sqrt[a + b*x^2 + c*x^4]*Sqrt[a/(2*a + (b + q)*x^2)]), x]] /; FreeQ[{a, b, c}, x] && Gt
Q[b^2 - 4*a*c, 0] && LtQ[a, 0] && GtQ[c, 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 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^2}{\left (1+x^2\right ) \sqrt {-x-x^2+x^3}} \, dx &=\frac {\left (\sqrt {x} \sqrt {-1-x+x^2}\right ) \int \frac {-1+x^2}{\sqrt {x} \left (1+x^2\right ) \sqrt {-1-x+x^2}} \, dx}{\sqrt {-x-x^2+x^3}}\\ &=\frac {\left (\sqrt {x} \sqrt {-1-x+x^2}\right ) \int \left (\frac {1}{\sqrt {x} \sqrt {-1-x+x^2}}-\frac {2}{\sqrt {x} \left (1+x^2\right ) \sqrt {-1-x+x^2}}\right ) \, dx}{\sqrt {-x-x^2+x^3}}\\ &=\frac {\left (\sqrt {x} \sqrt {-1-x+x^2}\right ) \int \frac {1}{\sqrt {x} \sqrt {-1-x+x^2}} \, dx}{\sqrt {-x-x^2+x^3}}-\frac {\left (2 \sqrt {x} \sqrt {-1-x+x^2}\right ) \int \frac {1}{\sqrt {x} \left (1+x^2\right ) \sqrt {-1-x+x^2}} \, dx}{\sqrt {-x-x^2+x^3}}\\ &=-\frac {\left (2 \sqrt {x} \sqrt {-1-x+x^2}\right ) \int \left (\frac {i}{2 (i-x) \sqrt {x} \sqrt {-1-x+x^2}}+\frac {i}{2 \sqrt {x} (i+x) \sqrt {-1-x+x^2}}\right ) \, dx}{\sqrt {-x-x^2+x^3}}+\frac {\left (2 \sqrt {x} \sqrt {-1-x+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1-x^2+x^4}} \, dx,x,\sqrt {x}\right )}{\sqrt {-x-x^2+x^3}}\\ &=\frac {\sqrt {x} \sqrt {-2-\left (1-\sqrt {5}\right ) x} \sqrt {\frac {2+\left (1+\sqrt {5}\right ) x}{2+\left (1-\sqrt {5}\right ) x}} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{5} \sqrt {x}}{\sqrt {-2-\left (1-\sqrt {5}\right ) x}}\right )|\frac {1}{10} \left (5-\sqrt {5}\right )\right )}{\sqrt [4]{5} \sqrt {\frac {1}{2+\left (1-\sqrt {5}\right ) x}} \sqrt {-x-x^2+x^3}}-\frac {\left (i \sqrt {x} \sqrt {-1-x+x^2}\right ) \int \frac {1}{(i-x) \sqrt {x} \sqrt {-1-x+x^2}} \, dx}{\sqrt {-x-x^2+x^3}}-\frac {\left (i \sqrt {x} \sqrt {-1-x+x^2}\right ) \int \frac {1}{\sqrt {x} (i+x) \sqrt {-1-x+x^2}} \, dx}{\sqrt {-x-x^2+x^3}}\\ &=\frac {\sqrt {x} \sqrt {-2-\left (1-\sqrt {5}\right ) x} \sqrt {\frac {2+\left (1+\sqrt {5}\right ) x}{2+\left (1-\sqrt {5}\right ) x}} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{5} \sqrt {x}}{\sqrt {-2-\left (1-\sqrt {5}\right ) x}}\right )|\frac {1}{10} \left (5-\sqrt {5}\right )\right )}{\sqrt [4]{5} \sqrt {\frac {1}{2+\left (1-\sqrt {5}\right ) x}} \sqrt {-x-x^2+x^3}}-\frac {\left (i \sqrt {x} \sqrt {-1-\sqrt {5}+2 x} \sqrt {-1+\sqrt {5}+2 x}\right ) \int \frac {1}{(i-x) \sqrt {x} \sqrt {-1-\sqrt {5}+2 x} \sqrt {-1+\sqrt {5}+2 x}} \, dx}{\sqrt {-x-x^2+x^3}}-\frac {\left (i \sqrt {x} \sqrt {-1-\sqrt {5}+2 x} \sqrt {-1+\sqrt {5}+2 x}\right ) \int \frac {1}{\sqrt {x} (i+x) \sqrt {-1-\sqrt {5}+2 x} \sqrt {-1+\sqrt {5}+2 x}} \, dx}{\sqrt {-x-x^2+x^3}}\\ &=\frac {\sqrt {x} \sqrt {-2-\left (1-\sqrt {5}\right ) x} \sqrt {\frac {2+\left (1+\sqrt {5}\right ) x}{2+\left (1-\sqrt {5}\right ) x}} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{5} \sqrt {x}}{\sqrt {-2-\left (1-\sqrt {5}\right ) x}}\right )|\frac {1}{10} \left (5-\sqrt {5}\right )\right )}{\sqrt [4]{5} \sqrt {\frac {1}{2+\left (1-\sqrt {5}\right ) x}} \sqrt {-x-x^2+x^3}}+\frac {\left (2 i \sqrt {x} \sqrt {-1-\sqrt {5}+2 x} \sqrt {-1+\sqrt {5}+2 x}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-i-x^2\right ) \sqrt {-1-\sqrt {5}+2 x^2} \sqrt {-1+\sqrt {5}+2 x^2}} \, dx,x,\sqrt {x}\right )}{\sqrt {-x-x^2+x^3}}+\frac {\left (2 i \sqrt {x} \sqrt {-1-\sqrt {5}+2 x} \sqrt {-1+\sqrt {5}+2 x}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-i+x^2\right ) \sqrt {-1-\sqrt {5}+2 x^2} \sqrt {-1+\sqrt {5}+2 x^2}} \, dx,x,\sqrt {x}\right )}{\sqrt {-x-x^2+x^3}}\\ &=\frac {\sqrt {x} \sqrt {-2-\left (1-\sqrt {5}\right ) x} \sqrt {\frac {2+\left (1+\sqrt {5}\right ) x}{2+\left (1-\sqrt {5}\right ) x}} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{5} \sqrt {x}}{\sqrt {-2-\left (1-\sqrt {5}\right ) x}}\right )|\frac {1}{10} \left (5-\sqrt {5}\right )\right )}{\sqrt [4]{5} \sqrt {\frac {1}{2+\left (1-\sqrt {5}\right ) x}} \sqrt {-x-x^2+x^3}}+\frac {\left (2 i \sqrt {x} \sqrt {-1+\sqrt {5}+2 x} \sqrt {1-\frac {2 x}{1+\sqrt {5}}}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-i-x^2\right ) \sqrt {-1+\sqrt {5}+2 x^2} \sqrt {1+\frac {2 x^2}{-1-\sqrt {5}}}} \, dx,x,\sqrt {x}\right )}{\sqrt {-x-x^2+x^3}}+\frac {\left (2 i \sqrt {x} \sqrt {-1+\sqrt {5}+2 x} \sqrt {1-\frac {2 x}{1+\sqrt {5}}}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (-i+x^2\right ) \sqrt {-1+\sqrt {5}+2 x^2} \sqrt {1+\frac {2 x^2}{-1-\sqrt {5}}}} \, dx,x,\sqrt {x}\right )}{\sqrt {-x-x^2+x^3}}\\ &=\frac {\sqrt {x} \sqrt {-2-\left (1-\sqrt {5}\right ) x} \sqrt {\frac {2+\left (1+\sqrt {5}\right ) x}{2+\left (1-\sqrt {5}\right ) x}} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{5} \sqrt {x}}{\sqrt {-2-\left (1-\sqrt {5}\right ) x}}\right )|\frac {1}{10} \left (5-\sqrt {5}\right )\right )}{\sqrt [4]{5} \sqrt {\frac {1}{2+\left (1-\sqrt {5}\right ) x}} \sqrt {-x-x^2+x^3}}-\frac {\sqrt {3+\sqrt {5}} \sqrt {x} \sqrt {-1+\sqrt {5}+2 x} \sqrt {1-\frac {2 x}{1+\sqrt {5}}} \Pi \left (-\frac {1}{2} i \left (1+\sqrt {5}\right );\sin ^{-1}\left (\sqrt {\frac {2}{1+\sqrt {5}}} \sqrt {x}\right )|\frac {1}{2} \left (-3-\sqrt {5}\right )\right )}{\sqrt {-x-x^2+x^3}}-\frac {\sqrt {3+\sqrt {5}} \sqrt {x} \sqrt {-1+\sqrt {5}+2 x} \sqrt {1-\frac {2 x}{1+\sqrt {5}}} \Pi \left (\frac {1}{2} i \left (1+\sqrt {5}\right );\sin ^{-1}\left (\sqrt {\frac {2}{1+\sqrt {5}}} \sqrt {x}\right )|\frac {1}{2} \left (-3-\sqrt {5}\right )\right )}{\sqrt {-x-x^2+x^3}}\\ \end {align*}

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Mathematica [C]  time = 0.66, size = 205, normalized size = 2.11 \begin {gather*} -\frac {2 i \sqrt {\frac {2}{\sqrt {5}-1}} \sqrt {-\frac {1}{x^2}-\frac {1}{x}+1} x^{3/2} \left (F\left (i \sinh ^{-1}\left (\frac {\sqrt {\frac {2}{1+\sqrt {5}}}}{\sqrt {x}}\right )|-\frac {3}{2}-\frac {\sqrt {5}}{2}\right )-\Pi \left (-\frac {1}{2} i \left (1+\sqrt {5}\right );i \sinh ^{-1}\left (\frac {\sqrt {\frac {2}{1+\sqrt {5}}}}{\sqrt {x}}\right )|\frac {1}{2} \left (-3-\sqrt {5}\right )\right )-\Pi \left (\frac {1}{2} i \left (1+\sqrt {5}\right );i \sinh ^{-1}\left (\frac {\sqrt {\frac {2}{1+\sqrt {5}}}}{\sqrt {x}}\right )|\frac {1}{2} \left (-3-\sqrt {5}\right )\right )\right )}{\sqrt {x \left (x^2-x-1\right )}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((-2*I)*Sqrt[2/(-1 + Sqrt[5])]*Sqrt[1 - x^(-2) - x^(-1)]*x^(3/2)*(EllipticF[I*ArcSinh[Sqrt[2/(1 + Sqrt[5])]/Sq
rt[x]], -3/2 - Sqrt[5]/2] - EllipticPi[(-1/2*I)*(1 + Sqrt[5]), I*ArcSinh[Sqrt[2/(1 + Sqrt[5])]/Sqrt[x]], (-3 -
Sqrt[5])/2] - EllipticPi[(I/2)*(1 + Sqrt[5]), I*ArcSinh[Sqrt[2/(1 + Sqrt[5])]/Sqrt[x]], (-3 - Sqrt[5])/2]))/S
qrt[x*(-1 - x + x^2)]

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IntegrateAlgebraic [A]  time = 0.29, size = 97, normalized size = 1.00 \begin {gather*} -\sqrt {\frac {1}{5}+\frac {2 i}{5}} \tan ^{-1}\left (\frac {\sqrt {1-2 i} \sqrt {-x-x^2+x^3}}{-1-x+x^2}\right )-\sqrt {\frac {1}{5}-\frac {2 i}{5}} \tan ^{-1}\left (\frac {\sqrt {1+2 i} \sqrt {-x-x^2+x^3}}{-1-x+x^2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-(Sqrt[1/5 + (2*I)/5]*ArcTan[(Sqrt[1 - 2*I]*Sqrt[-x - x^2 + x^3])/(-1 - x + x^2)]) - Sqrt[1/5 - (2*I)/5]*ArcTa
n[(Sqrt[1 + 2*I]*Sqrt[-x - x^2 + x^3])/(-1 - x + x^2)]

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fricas [B]  time = 0.95, size = 2458, normalized size = 25.34

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/80*5^(1/4)*(sqrt(5)*sqrt(2) - sqrt(2))*sqrt(sqrt(5) + 5)*log(5*(5*x^4 - 20*x^3 + 5^(1/4)*sqrt(x^3 - x^2 - x)
*(sqrt(5)*sqrt(2)*(x^2 - 6*x - 1) - 5*sqrt(2)*(x^2 - 2*x - 1))*sqrt(sqrt(5) + 5) + 30*x^2 + 20*sqrt(5)*(x^3 -
x^2 - x) + 20*x + 5)/(x^4 + 2*x^2 + 1)) - 1/80*5^(1/4)*(sqrt(5)*sqrt(2) - sqrt(2))*sqrt(sqrt(5) + 5)*log(5*(5*
x^4 - 20*x^3 - 5^(1/4)*sqrt(x^3 - x^2 - x)*(sqrt(5)*sqrt(2)*(x^2 - 6*x - 1) - 5*sqrt(2)*(x^2 - 2*x - 1))*sqrt(
sqrt(5) + 5) + 30*x^2 + 20*sqrt(5)*(x^3 - x^2 - x) + 20*x + 5)/(x^4 + 2*x^2 + 1)) - 1/10*5^(1/4)*sqrt(2)*sqrt(
sqrt(5) + 5)*arctan(-1/200*(100*x^11 + 1300*x^10 - 6700*x^9 - 4400*x^8 + 28400*x^7 + 1400*x^6 - 28400*x^5 - 44
00*x^4 + 6700*x^3 + 1300*x^2 + 5*sqrt(x^3 - x^2 - x)*(5^(3/4)*(sqrt(5)*sqrt(2)*(x^10 - 4*x^9 - 17*x^8 + 56*x^7
+ 78*x^6 - 136*x^5 - 78*x^4 + 56*x^3 + 17*x^2 - 4*x - 1) - sqrt(2)*(x^10 + 8*x^9 - 57*x^8 - 24*x^7 + 294*x^6
+ 64*x^5 - 294*x^4 - 24*x^3 + 57*x^2 + 8*x - 1)) + 4*5^(1/4)*(sqrt(5)*sqrt(2)*(3*x^9 + 7*x^8 + 14*x^7 - 81*x^6
- 10*x^5 + 81*x^4 + 14*x^3 - 7*x^2 + 3*x) - 5*sqrt(2)*(x^9 + 9*x^8 - 18*x^7 - 15*x^6 + 26*x^5 + 15*x^4 - 18*x
^3 - 9*x^2 + x)))*sqrt(sqrt(5) + 5) - sqrt(5)*(240*x^10 + 160*x^9 - 1680*x^8 - 480*x^7 + 3200*x^6 + 480*x^5 -
1680*x^4 - 160*x^3 + 240*x^2 + sqrt(x^3 - x^2 - x)*(5^(3/4)*(sqrt(5)*sqrt(2)*(x^10 - 6*x^9 - 25*x^8 + 120*x^7
- 58*x^6 - 196*x^5 + 58*x^4 + 120*x^3 + 25*x^2 - 6*x - 1) - sqrt(2)*(x^10 - 2*x^9 - 17*x^8 + 216*x^7 - 306*x^6
- 396*x^5 + 306*x^4 + 216*x^3 + 17*x^2 - 2*x - 1)) + 4*5^(1/4)*(sqrt(5)*sqrt(2)*(3*x^9 - 3*x^8 - 56*x^7 + 69*
x^6 + 90*x^5 - 69*x^4 - 56*x^3 + 3*x^2 + 3*x) - 5*sqrt(2)*(x^9 + 3*x^8 - 28*x^7 + 11*x^6 + 54*x^5 - 11*x^4 - 2
8*x^3 - 3*x^2 + x)))*sqrt(sqrt(5) + 5) + 4*sqrt(5)*(5*x^11 - 25*x^10 - 105*x^9 + 440*x^8 + 50*x^7 - 830*x^6 -
50*x^5 + 440*x^4 + 105*x^3 - 25*x^2 - sqrt(5)*(x^11 + 3*x^10 - 37*x^9 + 96*x^8 - 6*x^7 - 166*x^6 + 6*x^5 + 96*
x^4 + 37*x^3 + 3*x^2 - x) - 5*x) - 80*sqrt(5)*(x^10 + 6*x^9 - 23*x^8 - 2*x^7 + 40*x^6 + 2*x^5 - 23*x^4 - 6*x^3
+ x^2))*sqrt((5*x^4 - 20*x^3 + 5^(1/4)*sqrt(x^3 - x^2 - x)*(sqrt(5)*sqrt(2)*(x^2 - 6*x - 1) - 5*sqrt(2)*(x^2
- 2*x - 1))*sqrt(sqrt(5) + 5) + 30*x^2 + 20*sqrt(5)*(x^3 - x^2 - x) + 20*x + 5)/(x^4 + 2*x^2 + 1)) + 20*sqrt(5
)*(5*x^11 - 15*x^10 - 15*x^9 + 20*x^8 - 20*x^7 + 70*x^6 + 20*x^5 + 20*x^4 + 15*x^3 - 15*x^2 - sqrt(5)*(x^11 +
13*x^10 - 67*x^9 - 44*x^8 + 284*x^7 + 14*x^6 - 284*x^5 - 44*x^4 + 67*x^3 + 13*x^2 - x) - 5*x) - 100*sqrt(5)*(x
^11 - 3*x^10 - 3*x^9 + 4*x^8 - 4*x^7 + 14*x^6 + 4*x^5 + 4*x^4 + 3*x^3 - 3*x^2 - x) - 100*x)/(x^11 - 9*x^10 - 4
5*x^9 + 180*x^8 + 18*x^7 - 326*x^6 - 18*x^5 + 180*x^4 + 45*x^3 - 9*x^2 - x)) - 1/10*5^(1/4)*sqrt(2)*sqrt(sqrt(
5) + 5)*arctan(1/200*(100*x^11 + 1300*x^10 - 6700*x^9 - 4400*x^8 + 28400*x^7 + 1400*x^6 - 28400*x^5 - 4400*x^4
+ 6700*x^3 + 1300*x^2 - 5*sqrt(x^3 - x^2 - x)*(5^(3/4)*(sqrt(5)*sqrt(2)*(x^10 - 4*x^9 - 17*x^8 + 56*x^7 + 78*
x^6 - 136*x^5 - 78*x^4 + 56*x^3 + 17*x^2 - 4*x - 1) - sqrt(2)*(x^10 + 8*x^9 - 57*x^8 - 24*x^7 + 294*x^6 + 64*x
^5 - 294*x^4 - 24*x^3 + 57*x^2 + 8*x - 1)) + 4*5^(1/4)*(sqrt(5)*sqrt(2)*(3*x^9 + 7*x^8 + 14*x^7 - 81*x^6 - 10*
x^5 + 81*x^4 + 14*x^3 - 7*x^2 + 3*x) - 5*sqrt(2)*(x^9 + 9*x^8 - 18*x^7 - 15*x^6 + 26*x^5 + 15*x^4 - 18*x^3 - 9
*x^2 + x)))*sqrt(sqrt(5) + 5) - sqrt(5)*(240*x^10 + 160*x^9 - 1680*x^8 - 480*x^7 + 3200*x^6 + 480*x^5 - 1680*x
^4 - 160*x^3 + 240*x^2 - sqrt(x^3 - x^2 - x)*(5^(3/4)*(sqrt(5)*sqrt(2)*(x^10 - 6*x^9 - 25*x^8 + 120*x^7 - 58*x
^6 - 196*x^5 + 58*x^4 + 120*x^3 + 25*x^2 - 6*x - 1) - sqrt(2)*(x^10 - 2*x^9 - 17*x^8 + 216*x^7 - 306*x^6 - 396
*x^5 + 306*x^4 + 216*x^3 + 17*x^2 - 2*x - 1)) + 4*5^(1/4)*(sqrt(5)*sqrt(2)*(3*x^9 - 3*x^8 - 56*x^7 + 69*x^6 +
90*x^5 - 69*x^4 - 56*x^3 + 3*x^2 + 3*x) - 5*sqrt(2)*(x^9 + 3*x^8 - 28*x^7 + 11*x^6 + 54*x^5 - 11*x^4 - 28*x^3
- 3*x^2 + x)))*sqrt(sqrt(5) + 5) + 4*sqrt(5)*(5*x^11 - 25*x^10 - 105*x^9 + 440*x^8 + 50*x^7 - 830*x^6 - 50*x^5
+ 440*x^4 + 105*x^3 - 25*x^2 - sqrt(5)*(x^11 + 3*x^10 - 37*x^9 + 96*x^8 - 6*x^7 - 166*x^6 + 6*x^5 + 96*x^4 +
37*x^3 + 3*x^2 - x) - 5*x) - 80*sqrt(5)*(x^10 + 6*x^9 - 23*x^8 - 2*x^7 + 40*x^6 + 2*x^5 - 23*x^4 - 6*x^3 + x^2
))*sqrt((5*x^4 - 20*x^3 - 5^(1/4)*sqrt(x^3 - x^2 - x)*(sqrt(5)*sqrt(2)*(x^2 - 6*x - 1) - 5*sqrt(2)*(x^2 - 2*x
- 1))*sqrt(sqrt(5) + 5) + 30*x^2 + 20*sqrt(5)*(x^3 - x^2 - x) + 20*x + 5)/(x^4 + 2*x^2 + 1)) + 20*sqrt(5)*(5*x
^11 - 15*x^10 - 15*x^9 + 20*x^8 - 20*x^7 + 70*x^6 + 20*x^5 + 20*x^4 + 15*x^3 - 15*x^2 - sqrt(5)*(x^11 + 13*x^1
0 - 67*x^9 - 44*x^8 + 284*x^7 + 14*x^6 - 284*x^5 - 44*x^4 + 67*x^3 + 13*x^2 - x) - 5*x) - 100*sqrt(5)*(x^11 -
3*x^10 - 3*x^9 + 4*x^8 - 4*x^7 + 14*x^6 + 4*x^5 + 4*x^4 + 3*x^3 - 3*x^2 - x) - 100*x)/(x^11 - 9*x^10 - 45*x^9
+ 180*x^8 + 18*x^7 - 326*x^6 - 18*x^5 + 180*x^4 + 45*x^3 - 9*x^2 - x))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C]  time = 2.80, size = 654, normalized size = 6.74

method	result	size
trager	\(-\frac {\RootOf \left (5 \textit {\_Z}^{4}+2 \textit {\_Z}^{2}+1\right ) \ln \left (-\frac {-325 \RootOf \left (5 \textit {\_Z}^{4}+2 \textit {\_Z}^{2}+1\right )^{5} x^{2}+325 \RootOf \left (5 \textit {\_Z}^{4}+2 \textit {\_Z}^{2}+1\right )^{5}-90 \RootOf \left (5 \textit {\_Z}^{4}+2 \textit {\_Z}^{2}+1\right )^{3} x^{2}-520 x \RootOf \left (5 \textit {\_Z}^{4}+2 \textit {\_Z}^{2}+1\right )^{3}+80 \RootOf \left (5 \textit {\_Z}^{4}+2 \textit {\_Z}^{2}+1\right )^{2} \sqrt {x^{3}-x^{2}-x}+90 \RootOf \left (5 \textit {\_Z}^{4}+2 \textit {\_Z}^{2}+1\right )^{3}+31 \RootOf \left (5 \textit {\_Z}^{4}+2 \textit {\_Z}^{2}+1\right ) x^{2}-248 \RootOf \left (5 \textit {\_Z}^{4}+2 \textit {\_Z}^{2}+1\right ) x +192 \sqrt {x^{3}-x^{2}-x}-31 \RootOf \left (5 \textit {\_Z}^{4}+2 \textit {\_Z}^{2}+1\right )}{\left (5 \RootOf \left (5 \textit {\_Z}^{4}+2 \textit {\_Z}^{2}+1\right )^{2} x -5 \RootOf \left (5 \textit {\_Z}^{4}+2 \textit {\_Z}^{2}+1\right )^{2}-x -3\right )^{2}}\right )}{2}-\frac {\RootOf \left (\textit {\_Z}^{2}+25 \RootOf \left (5 \textit {\_Z}^{4}+2 \textit {\_Z}^{2}+1\right )^{2}+10\right ) \ln \left (\frac {65 \RootOf \left (\textit {\_Z}^{2}+25 \RootOf \left (5 \textit {\_Z}^{4}+2 \textit {\_Z}^{2}+1\right )^{2}+10\right ) \RootOf \left (5 \textit {\_Z}^{4}+2 \textit {\_Z}^{2}+1\right )^{4} x^{2}-65 \RootOf \left (\textit {\_Z}^{2}+25 \RootOf \left (5 \textit {\_Z}^{4}+2 \textit {\_Z}^{2}+1\right )^{2}+10\right ) \RootOf \left (5 \textit {\_Z}^{4}+2 \textit {\_Z}^{2}+1\right )^{4}+34 \RootOf \left (5 \textit {\_Z}^{4}+2 \textit {\_Z}^{2}+1\right )^{2} \RootOf \left (\textit {\_Z}^{2}+25 \RootOf \left (5 \textit {\_Z}^{4}+2 \textit {\_Z}^{2}+1\right )^{2}+10\right ) x^{2}-104 \RootOf \left (5 \textit {\_Z}^{4}+2 \textit {\_Z}^{2}+1\right )^{2} \RootOf \left (\textit {\_Z}^{2}+25 \RootOf \left (5 \textit {\_Z}^{4}+2 \textit {\_Z}^{2}+1\right )^{2}+10\right ) x +80 \RootOf \left (5 \textit {\_Z}^{4}+2 \textit {\_Z}^{2}+1\right )^{2} \sqrt {x^{3}-x^{2}-x}-34 \RootOf \left (5 \textit {\_Z}^{4}+2 \textit {\_Z}^{2}+1\right )^{2} \RootOf \left (\textit {\_Z}^{2}+25 \RootOf \left (5 \textit {\_Z}^{4}+2 \textit {\_Z}^{2}+1\right )^{2}+10\right )-3 \RootOf \left (\textit {\_Z}^{2}+25 \RootOf \left (5 \textit {\_Z}^{4}+2 \textit {\_Z}^{2}+1\right )^{2}+10\right ) x^{2}+8 \RootOf \left (\textit {\_Z}^{2}+25 \RootOf \left (5 \textit {\_Z}^{4}+2 \textit {\_Z}^{2}+1\right )^{2}+10\right ) x -160 \sqrt {x^{3}-x^{2}-x}+3 \RootOf \left (\textit {\_Z}^{2}+25 \RootOf \left (5 \textit {\_Z}^{4}+2 \textit {\_Z}^{2}+1\right )^{2}+10\right )}{\left (5 \RootOf \left (5 \textit {\_Z}^{4}+2 \textit {\_Z}^{2}+1\right )^{2} x -5 \RootOf \left (5 \textit {\_Z}^{4}+2 \textit {\_Z}^{2}+1\right )^{2}+3 x +1\right )^{2}}\right )}{10}\)	\(654\)
default	\(\frac {2 \left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right ) \sqrt {\frac {x -\frac {1}{2}+\frac {\sqrt {5}}{2}}{\frac {\sqrt {5}}{2}-\frac {1}{2}}}\, \sqrt {-5 \left (x -\frac {1}{2}-\frac {\sqrt {5}}{2}\right ) \sqrt {5}}\, \sqrt {-\frac {x}{\frac {\sqrt {5}}{2}-\frac {1}{2}}}\, \EllipticF \left (\sqrt {\frac {x -\frac {1}{2}+\frac {\sqrt {5}}{2}}{\frac {\sqrt {5}}{2}-\frac {1}{2}}}, \frac {\sqrt {5}\, \sqrt {\left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right ) \sqrt {5}}}{5}\right )}{5 \sqrt {x^{3}-x^{2}-x}}+\frac {i \sqrt {\frac {x}{\frac {\sqrt {5}}{2}-\frac {1}{2}}-\frac {1}{2 \left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right )}+\frac {\sqrt {5}}{\sqrt {5}-1}}\, \sqrt {-5 x \sqrt {5}+\frac {5 \sqrt {5}}{2}+\frac {25}{2}}\, \sqrt {-\frac {x}{\frac {\sqrt {5}}{2}-\frac {1}{2}}}\, \EllipticPi \left (\sqrt {\frac {x -\frac {1}{2}+\frac {\sqrt {5}}{2}}{\frac {\sqrt {5}}{2}-\frac {1}{2}}}, \frac {\frac {1}{2}-\frac {\sqrt {5}}{2}}{\frac {1}{2}-i-\frac {\sqrt {5}}{2}}, \frac {\sqrt {5}\, \sqrt {\left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right ) \sqrt {5}}}{5}\right ) \sqrt {5}}{5 \sqrt {x^{3}-x^{2}-x}\, \left (\frac {1}{2}-i-\frac {\sqrt {5}}{2}\right )}-\frac {i \sqrt {\frac {x}{\frac {\sqrt {5}}{2}-\frac {1}{2}}-\frac {1}{2 \left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right )}+\frac {\sqrt {5}}{\sqrt {5}-1}}\, \sqrt {-5 x \sqrt {5}+\frac {5 \sqrt {5}}{2}+\frac {25}{2}}\, \sqrt {-\frac {x}{\frac {\sqrt {5}}{2}-\frac {1}{2}}}\, \EllipticPi \left (\sqrt {\frac {x -\frac {1}{2}+\frac {\sqrt {5}}{2}}{\frac {\sqrt {5}}{2}-\frac {1}{2}}}, \frac {\frac {1}{2}-\frac {\sqrt {5}}{2}}{\frac {1}{2}-i-\frac {\sqrt {5}}{2}}, \frac {\sqrt {5}\, \sqrt {\left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right ) \sqrt {5}}}{5}\right )}{5 \sqrt {x^{3}-x^{2}-x}\, \left (\frac {1}{2}-i-\frac {\sqrt {5}}{2}\right )}-\frac {i \sqrt {\frac {x}{\frac {\sqrt {5}}{2}-\frac {1}{2}}-\frac {1}{2 \left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right )}+\frac {\sqrt {5}}{\sqrt {5}-1}}\, \sqrt {-5 x \sqrt {5}+\frac {5 \sqrt {5}}{2}+\frac {25}{2}}\, \sqrt {-\frac {x}{\frac {\sqrt {5}}{2}-\frac {1}{2}}}\, \EllipticPi \left (\sqrt {\frac {x -\frac {1}{2}+\frac {\sqrt {5}}{2}}{\frac {\sqrt {5}}{2}-\frac {1}{2}}}, \frac {\frac {1}{2}-\frac {\sqrt {5}}{2}}{\frac {1}{2}+i-\frac {\sqrt {5}}{2}}, \frac {\sqrt {5}\, \sqrt {\left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right ) \sqrt {5}}}{5}\right ) \sqrt {5}}{5 \sqrt {x^{3}-x^{2}-x}\, \left (\frac {1}{2}+i-\frac {\sqrt {5}}{2}\right )}+\frac {i \sqrt {\frac {x}{\frac {\sqrt {5}}{2}-\frac {1}{2}}-\frac {1}{2 \left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right )}+\frac {\sqrt {5}}{\sqrt {5}-1}}\, \sqrt {-5 x \sqrt {5}+\frac {5 \sqrt {5}}{2}+\frac {25}{2}}\, \sqrt {-\frac {x}{\frac {\sqrt {5}}{2}-\frac {1}{2}}}\, \EllipticPi \left (\sqrt {\frac {x -\frac {1}{2}+\frac {\sqrt {5}}{2}}{\frac {\sqrt {5}}{2}-\frac {1}{2}}}, \frac {\frac {1}{2}-\frac {\sqrt {5}}{2}}{\frac {1}{2}+i-\frac {\sqrt {5}}{2}}, \frac {\sqrt {5}\, \sqrt {\left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right ) \sqrt {5}}}{5}\right )}{5 \sqrt {x^{3}-x^{2}-x}\, \left (\frac {1}{2}+i-\frac {\sqrt {5}}{2}\right )}\)	\(735\)
elliptic	\(\frac {\sqrt {\frac {x}{\frac {\sqrt {5}}{2}-\frac {1}{2}}-\frac {1}{2 \left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right )}+\frac {\sqrt {5}}{\sqrt {5}-1}}\, \sqrt {-5 x \sqrt {5}+\frac {5 \sqrt {5}}{2}+\frac {25}{2}}\, \sqrt {-\frac {x}{\frac {\sqrt {5}}{2}-\frac {1}{2}}}\, \EllipticF \left (\sqrt {\frac {x -\frac {1}{2}+\frac {\sqrt {5}}{2}}{\frac {\sqrt {5}}{2}-\frac {1}{2}}}, \frac {\sqrt {5}\, \sqrt {\left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right ) \sqrt {5}}}{5}\right ) \sqrt {5}}{5 \sqrt {x^{3}-x^{2}-x}}-\frac {\sqrt {\frac {x}{\frac {\sqrt {5}}{2}-\frac {1}{2}}-\frac {1}{2 \left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right )}+\frac {\sqrt {5}}{\sqrt {5}-1}}\, \sqrt {-5 x \sqrt {5}+\frac {5 \sqrt {5}}{2}+\frac {25}{2}}\, \sqrt {-\frac {x}{\frac {\sqrt {5}}{2}-\frac {1}{2}}}\, \EllipticF \left (\sqrt {\frac {x -\frac {1}{2}+\frac {\sqrt {5}}{2}}{\frac {\sqrt {5}}{2}-\frac {1}{2}}}, \frac {\sqrt {5}\, \sqrt {\left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right ) \sqrt {5}}}{5}\right )}{5 \sqrt {x^{3}-x^{2}-x}}+\frac {i \sqrt {\frac {x}{\frac {\sqrt {5}}{2}-\frac {1}{2}}-\frac {1}{2 \left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right )}+\frac {\sqrt {5}}{\sqrt {5}-1}}\, \sqrt {-5 x \sqrt {5}+\frac {5 \sqrt {5}}{2}+\frac {25}{2}}\, \sqrt {-\frac {x}{\frac {\sqrt {5}}{2}-\frac {1}{2}}}\, \EllipticPi \left (\sqrt {\frac {x -\frac {1}{2}+\frac {\sqrt {5}}{2}}{\frac {\sqrt {5}}{2}-\frac {1}{2}}}, \frac {\frac {1}{2}-\frac {\sqrt {5}}{2}}{\frac {1}{2}-i-\frac {\sqrt {5}}{2}}, \frac {\sqrt {5}\, \sqrt {\left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right ) \sqrt {5}}}{5}\right ) \sqrt {5}}{5 \sqrt {x^{3}-x^{2}-x}\, \left (\frac {1}{2}-i-\frac {\sqrt {5}}{2}\right )}-\frac {i \sqrt {\frac {x}{\frac {\sqrt {5}}{2}-\frac {1}{2}}-\frac {1}{2 \left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right )}+\frac {\sqrt {5}}{\sqrt {5}-1}}\, \sqrt {-5 x \sqrt {5}+\frac {5 \sqrt {5}}{2}+\frac {25}{2}}\, \sqrt {-\frac {x}{\frac {\sqrt {5}}{2}-\frac {1}{2}}}\, \EllipticPi \left (\sqrt {\frac {x -\frac {1}{2}+\frac {\sqrt {5}}{2}}{\frac {\sqrt {5}}{2}-\frac {1}{2}}}, \frac {\frac {1}{2}-\frac {\sqrt {5}}{2}}{\frac {1}{2}-i-\frac {\sqrt {5}}{2}}, \frac {\sqrt {5}\, \sqrt {\left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right ) \sqrt {5}}}{5}\right )}{5 \sqrt {x^{3}-x^{2}-x}\, \left (\frac {1}{2}-i-\frac {\sqrt {5}}{2}\right )}-\frac {i \sqrt {\frac {x}{\frac {\sqrt {5}}{2}-\frac {1}{2}}-\frac {1}{2 \left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right )}+\frac {\sqrt {5}}{\sqrt {5}-1}}\, \sqrt {-5 x \sqrt {5}+\frac {5 \sqrt {5}}{2}+\frac {25}{2}}\, \sqrt {-\frac {x}{\frac {\sqrt {5}}{2}-\frac {1}{2}}}\, \EllipticPi \left (\sqrt {\frac {x -\frac {1}{2}+\frac {\sqrt {5}}{2}}{\frac {\sqrt {5}}{2}-\frac {1}{2}}}, \frac {\frac {1}{2}-\frac {\sqrt {5}}{2}}{\frac {1}{2}+i-\frac {\sqrt {5}}{2}}, \frac {\sqrt {5}\, \sqrt {\left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right ) \sqrt {5}}}{5}\right ) \sqrt {5}}{5 \sqrt {x^{3}-x^{2}-x}\, \left (\frac {1}{2}+i-\frac {\sqrt {5}}{2}\right )}+\frac {i \sqrt {\frac {x}{\frac {\sqrt {5}}{2}-\frac {1}{2}}-\frac {1}{2 \left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right )}+\frac {\sqrt {5}}{\sqrt {5}-1}}\, \sqrt {-5 x \sqrt {5}+\frac {5 \sqrt {5}}{2}+\frac {25}{2}}\, \sqrt {-\frac {x}{\frac {\sqrt {5}}{2}-\frac {1}{2}}}\, \EllipticPi \left (\sqrt {\frac {x -\frac {1}{2}+\frac {\sqrt {5}}{2}}{\frac {\sqrt {5}}{2}-\frac {1}{2}}}, \frac {\frac {1}{2}-\frac {\sqrt {5}}{2}}{\frac {1}{2}+i-\frac {\sqrt {5}}{2}}, \frac {\sqrt {5}\, \sqrt {\left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right ) \sqrt {5}}}{5}\right )}{5 \sqrt {x^{3}-x^{2}-x}\, \left (\frac {1}{2}+i-\frac {\sqrt {5}}{2}\right )}\)	\(873\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*RootOf(5*_Z^4+2*_Z^2+1)*ln(-(-325*RootOf(5*_Z^4+2*_Z^2+1)^5*x^2+325*RootOf(5*_Z^4+2*_Z^2+1)^5-90*RootOf(5
*_Z^4+2*_Z^2+1)^3*x^2-520*x*RootOf(5*_Z^4+2*_Z^2+1)^3+80*RootOf(5*_Z^4+2*_Z^2+1)^2*(x^3-x^2-x)^(1/2)+90*RootOf
(5*_Z^4+2*_Z^2+1)^3+31*RootOf(5*_Z^4+2*_Z^2+1)*x^2-248*RootOf(5*_Z^4+2*_Z^2+1)*x+192*(x^3-x^2-x)^(1/2)-31*Root
Of(5*_Z^4+2*_Z^2+1))/(5*RootOf(5*_Z^4+2*_Z^2+1)^2*x-5*RootOf(5*_Z^4+2*_Z^2+1)^2-x-3)^2)-1/10*RootOf(_Z^2+25*Ro
otOf(5*_Z^4+2*_Z^2+1)^2+10)*ln((65*RootOf(_Z^2+25*RootOf(5*_Z^4+2*_Z^2+1)^2+10)*RootOf(5*_Z^4+2*_Z^2+1)^4*x^2-
65*RootOf(_Z^2+25*RootOf(5*_Z^4+2*_Z^2+1)^2+10)*RootOf(5*_Z^4+2*_Z^2+1)^4+34*RootOf(5*_Z^4+2*_Z^2+1)^2*RootOf(
_Z^2+25*RootOf(5*_Z^4+2*_Z^2+1)^2+10)*x^2-104*RootOf(5*_Z^4+2*_Z^2+1)^2*RootOf(_Z^2+25*RootOf(5*_Z^4+2*_Z^2+1)
^2+10)*x+80*RootOf(5*_Z^4+2*_Z^2+1)^2*(x^3-x^2-x)^(1/2)-34*RootOf(5*_Z^4+2*_Z^2+1)^2*RootOf(_Z^2+25*RootOf(5*_
Z^4+2*_Z^2+1)^2+10)-3*RootOf(_Z^2+25*RootOf(5*_Z^4+2*_Z^2+1)^2+10)*x^2+8*RootOf(_Z^2+25*RootOf(5*_Z^4+2*_Z^2+1
)^2+10)*x-160*(x^3-x^2-x)^(1/2)+3*RootOf(_Z^2+25*RootOf(5*_Z^4+2*_Z^2+1)^2+10))/(5*RootOf(5*_Z^4+2*_Z^2+1)^2*x
-5*RootOf(5*_Z^4+2*_Z^2+1)^2+3*x+1)^2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.89, size = 210, normalized size = 2.16 \begin {gather*} -\frac {\sqrt {\frac {x}{\frac {\sqrt {5}}{2}+\frac {1}{2}}}\,\sqrt {\frac {x+\frac {\sqrt {5}}{2}-\frac {1}{2}}{\frac {\sqrt {5}}{2}-\frac {1}{2}}}\,\left (\sqrt {5}+1\right )\,\sqrt {\frac {\frac {\sqrt {5}}{2}-x+\frac {1}{2}}{\frac {\sqrt {5}}{2}+\frac {1}{2}}}\,\left (-\mathrm {F}\left (\mathrm {asin}\left (\sqrt {\frac {x}{\frac {\sqrt {5}}{2}+\frac {1}{2}}}\right )\middle |-\frac {\frac {\sqrt {5}}{2}+\frac {1}{2}}{\frac {\sqrt {5}}{2}-\frac {1}{2}}\right )+\Pi \left (-\frac {\sqrt {5}\,1{}\mathrm {i}}{2}-\frac {1}{2}{}\mathrm {i};\mathrm {asin}\left (\sqrt {\frac {x}{\frac {\sqrt {5}}{2}+\frac {1}{2}}}\right )\middle |-\frac {\frac {\sqrt {5}}{2}+\frac {1}{2}}{\frac {\sqrt {5}}{2}-\frac {1}{2}}\right )+\Pi \left (\frac {\sqrt {5}\,1{}\mathrm {i}}{2}+\frac {1}{2}{}\mathrm {i};\mathrm {asin}\left (\sqrt {\frac {x}{\frac {\sqrt {5}}{2}+\frac {1}{2}}}\right )\middle |-\frac {\frac {\sqrt {5}}{2}+\frac {1}{2}}{\frac {\sqrt {5}}{2}-\frac {1}{2}}\right )\right )}{\sqrt {x^3-x^2-\left (\frac {\sqrt {5}}{2}-\frac {1}{2}\right )\,\left (\frac {\sqrt {5}}{2}+\frac {1}{2}\right )\,x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-((x/(5^(1/2)/2 + 1/2))^(1/2)*((x + 5^(1/2)/2 - 1/2)/(5^(1/2)/2 - 1/2))^(1/2)*(5^(1/2) + 1)*((5^(1/2)/2 - x +
1/2)/(5^(1/2)/2 + 1/2))^(1/2)*(ellipticPi(- (5^(1/2)*1i)/2 - 1i/2, asin((x/(5^(1/2)/2 + 1/2))^(1/2)), -(5^(1/2
)/2 + 1/2)/(5^(1/2)/2 - 1/2)) - ellipticF(asin((x/(5^(1/2)/2 + 1/2))^(1/2)), -(5^(1/2)/2 + 1/2)/(5^(1/2)/2 - 1
/2)) + ellipticPi((5^(1/2)*1i)/2 + 1i/2, asin((x/(5^(1/2)/2 + 1/2))^(1/2)), -(5^(1/2)/2 + 1/2)/(5^(1/2)/2 - 1/
2))))/(x^3 - x^2 - x*(5^(1/2)/2 - 1/2)*(5^(1/2)/2 + 1/2))^(1/2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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