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

Optimal. Leaf size=24 \[ -2 \tanh ^{-1}\left (\frac {\sqrt {x^3+x^2+x}}{x^2+x+1}\right ) \]

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Rubi [C]  time = 1.03, antiderivative size = 320, normalized size of antiderivative = 13.33, number of steps used = 17, number of rules used = 8, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {2056, 6725, 716, 1103, 934, 169, 538, 537} \begin {gather*} \frac {\sqrt {x} (x+1) \sqrt {\frac {x^2+x+1}{(x+1)^2}} F\left (2 \tan ^{-1}\left (\sqrt {x}\right )|\frac {1}{4}\right )}{\sqrt {x^3+x^2+x}}-\frac {4 \sqrt {x} \sqrt {1+\frac {2 x}{1-i \sqrt {3}}} \sqrt {1+\frac {2 x}{1+i \sqrt {3}}} \Pi \left (\frac {1}{2} \left (-i-\sqrt {3}\right );\sin ^{-1}\left (\frac {1}{2} \left (1-i \sqrt {3}\right ) \sqrt {x}\right )|\frac {i+\sqrt {3}}{i-\sqrt {3}}\right )}{\left (1-i \sqrt {3}\right ) \sqrt {x^3+x^2+x}}-\frac {4 \sqrt {x} \sqrt {1+\frac {2 x}{1-i \sqrt {3}}} \sqrt {1+\frac {2 x}{1+i \sqrt {3}}} \Pi \left (\frac {1}{2} \left (i+\sqrt {3}\right );\sin ^{-1}\left (\frac {1}{2} \left (1-i \sqrt {3}\right ) \sqrt {x}\right )|\frac {i+\sqrt {3}}{i-\sqrt {3}}\right )}{\left (1-i \sqrt {3}\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]*(1 + x)*Sqrt[(1 + x + x^2)/(1 + x)^2]*EllipticF[2*ArcTan[Sqrt[x]], 1/4])/Sqrt[x + x^2 + x^3] - (4*Sqr
t[x]*Sqrt[1 + (2*x)/(1 - I*Sqrt[3])]*Sqrt[1 + (2*x)/(1 + I*Sqrt[3])]*EllipticPi[(-I - Sqrt[3])/2, ArcSin[((1 -
I*Sqrt[3])*Sqrt[x])/2], (I + Sqrt[3])/(I - Sqrt[3])])/((1 - I*Sqrt[3])*Sqrt[x + x^2 + x^3]) - (4*Sqrt[x]*Sqrt
[1 + (2*x)/(1 - I*Sqrt[3])]*Sqrt[1 + (2*x)/(1 + I*Sqrt[3])]*EllipticPi[(I + Sqrt[3])/2, ArcSin[((1 - I*Sqrt[3]
)*Sqrt[x])/2], (I + Sqrt[3])/(I - Sqrt[3])])/((1 - I*Sqrt[3])*Sqrt[x + x^2 + x^3])

Rule 169

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] &&  !SimplerQ[e
+ f*x, c + d*x] &&  !SimplerQ[g + h*x, c + d*x]

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 1103

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

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

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Mathematica [C]  time = 0.53, size = 136, normalized size = 5.67 \begin {gather*} -\frac {2 (-1)^{2/3} \sqrt {\frac {\sqrt [3]{-1}}{x}+1} \sqrt {1-\frac {(-1)^{2/3}}{x}} x^{3/2} \left (-F\left (i \sinh ^{-1}\left (\frac {(-1)^{5/6}}{\sqrt {x}}\right )|(-1)^{2/3}\right )+\Pi \left (-(-1)^{5/6};i \sinh ^{-1}\left (\frac {(-1)^{5/6}}{\sqrt {x}}\right )|(-1)^{2/3}\right )+\Pi \left ((-1)^{5/6};i \sinh ^{-1}\left (\frac {(-1)^{5/6}}{\sqrt {x}}\right )|(-1)^{2/3}\right )\right )}{\sqrt {x \left (x^2+x+1\right )}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(-1)^(2/3)*Sqrt[1 + (-1)^(1/3)/x]*Sqrt[1 - (-1)^(2/3)/x]*x^(3/2)*(-EllipticF[I*ArcSinh[(-1)^(5/6)/Sqrt[x]]
, (-1)^(2/3)] + EllipticPi[-(-1)^(5/6), I*ArcSinh[(-1)^(5/6)/Sqrt[x]], (-1)^(2/3)] + EllipticPi[(-1)^(5/6), I*
ArcSinh[(-1)^(5/6)/Sqrt[x]], (-1)^(2/3)]))/Sqrt[x*(1 + x + x^2)]

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IntegrateAlgebraic [A]  time = 0.07, size = 24, normalized size = 1.00 \begin {gather*} -2 \tanh ^{-1}\left (\frac {\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]

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

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fricas [A]  time = 0.45, size = 29, normalized size = 1.21 \begin {gather*} \log \left (\frac {x^{2} + 2 \, x - 2 \, \sqrt {x^{3} + x^{2} + x} + 1}{x^{2} + 1}\right ) \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="fricas")

[Out]

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

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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 [A]  time = 0.18, size = 32, normalized size = 1.33

method	result	size
trager	\(\ln \left (\frac {-x^{2}+2 \sqrt {x^{3}+x^{2}+x}-2 x -1}{x^{2}+1}\right )\)	\(32\)
default	\(\frac {2 \left (\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {1}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {3}\, \sqrt {i \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {\frac {x}{-\frac {1}{2}-\frac {i \sqrt {3}}{2}}}\, \EllipticF \left (\sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {1}{2}+\frac {i \sqrt {3}}{2}}}, \frac {\sqrt {3}\, \sqrt {i \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}\right )}{3 \sqrt {x^{3}+x^{2}+x}}+\frac {i \sqrt {\frac {x}{\frac {1}{2}+\frac {i \sqrt {3}}{2}}+\frac {1}{1+i \sqrt {3}}+\frac {i \sqrt {3}}{1+i \sqrt {3}}}\, \sqrt {3}\, \sqrt {i \sqrt {3}\, x +\frac {i \sqrt {3}}{2}+\frac {3}{2}}\, \sqrt {\frac {x}{-\frac {1}{2}-\frac {i \sqrt {3}}{2}}}\, \EllipticPi \left (\sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {1}{2}+\frac {i \sqrt {3}}{2}}}, \frac {-\frac {1}{2}-\frac {i \sqrt {3}}{2}}{-\frac {1}{2}-i-\frac {i \sqrt {3}}{2}}, \frac {\sqrt {3}\, \sqrt {i \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}\right )}{3 \sqrt {x^{3}+x^{2}+x}\, \left (-\frac {1}{2}-i-\frac {i \sqrt {3}}{2}\right )}-\frac {\sqrt {\frac {x}{\frac {1}{2}+\frac {i \sqrt {3}}{2}}+\frac {1}{1+i \sqrt {3}}+\frac {i \sqrt {3}}{1+i \sqrt {3}}}\, \sqrt {i \sqrt {3}\, x +\frac {i \sqrt {3}}{2}+\frac {3}{2}}\, \sqrt {\frac {x}{-\frac {1}{2}-\frac {i \sqrt {3}}{2}}}\, \EllipticPi \left (\sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {1}{2}+\frac {i \sqrt {3}}{2}}}, \frac {-\frac {1}{2}-\frac {i \sqrt {3}}{2}}{-\frac {1}{2}-i-\frac {i \sqrt {3}}{2}}, \frac {\sqrt {3}\, \sqrt {i \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}\right )}{\sqrt {x^{3}+x^{2}+x}\, \left (-\frac {1}{2}-i-\frac {i \sqrt {3}}{2}\right )}-\frac {i \sqrt {\frac {x}{\frac {1}{2}+\frac {i \sqrt {3}}{2}}+\frac {1}{1+i \sqrt {3}}+\frac {i \sqrt {3}}{1+i \sqrt {3}}}\, \sqrt {3}\, \sqrt {i \sqrt {3}\, x +\frac {i \sqrt {3}}{2}+\frac {3}{2}}\, \sqrt {\frac {x}{-\frac {1}{2}-\frac {i \sqrt {3}}{2}}}\, \EllipticPi \left (\sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {1}{2}+\frac {i \sqrt {3}}{2}}}, \frac {-\frac {1}{2}-\frac {i \sqrt {3}}{2}}{-\frac {1}{2}+i-\frac {i \sqrt {3}}{2}}, \frac {\sqrt {3}\, \sqrt {i \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}\right )}{3 \sqrt {x^{3}+x^{2}+x}\, \left (-\frac {1}{2}+i-\frac {i \sqrt {3}}{2}\right )}+\frac {\sqrt {\frac {x}{\frac {1}{2}+\frac {i \sqrt {3}}{2}}+\frac {1}{1+i \sqrt {3}}+\frac {i \sqrt {3}}{1+i \sqrt {3}}}\, \sqrt {i \sqrt {3}\, x +\frac {i \sqrt {3}}{2}+\frac {3}{2}}\, \sqrt {\frac {x}{-\frac {1}{2}-\frac {i \sqrt {3}}{2}}}\, \EllipticPi \left (\sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {1}{2}+\frac {i \sqrt {3}}{2}}}, \frac {-\frac {1}{2}-\frac {i \sqrt {3}}{2}}{-\frac {1}{2}+i-\frac {i \sqrt {3}}{2}}, \frac {\sqrt {3}\, \sqrt {i \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}\right )}{\sqrt {x^{3}+x^{2}+x}\, \left (-\frac {1}{2}+i-\frac {i \sqrt {3}}{2}\right )}\)	\(781\)
elliptic	\(\frac {\sqrt {\frac {x}{\frac {1}{2}+\frac {i \sqrt {3}}{2}}+\frac {1}{1+i \sqrt {3}}+\frac {i \sqrt {3}}{1+i \sqrt {3}}}\, \sqrt {3}\, \sqrt {i \sqrt {3}\, x +\frac {i \sqrt {3}}{2}+\frac {3}{2}}\, \sqrt {\frac {x}{-\frac {1}{2}-\frac {i \sqrt {3}}{2}}}\, \EllipticF \left (\sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {1}{2}+\frac {i \sqrt {3}}{2}}}, \frac {\sqrt {3}\, \sqrt {i \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}\right )}{3 \sqrt {x^{3}+x^{2}+x}}+\frac {i \sqrt {\frac {x}{\frac {1}{2}+\frac {i \sqrt {3}}{2}}+\frac {1}{1+i \sqrt {3}}+\frac {i \sqrt {3}}{1+i \sqrt {3}}}\, \sqrt {i \sqrt {3}\, x +\frac {i \sqrt {3}}{2}+\frac {3}{2}}\, \sqrt {\frac {x}{-\frac {1}{2}-\frac {i \sqrt {3}}{2}}}\, \EllipticF \left (\sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {1}{2}+\frac {i \sqrt {3}}{2}}}, \frac {\sqrt {3}\, \sqrt {i \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}\right )}{\sqrt {x^{3}+x^{2}+x}}+\frac {i \sqrt {\frac {x}{\frac {1}{2}+\frac {i \sqrt {3}}{2}}+\frac {1}{1+i \sqrt {3}}+\frac {i \sqrt {3}}{1+i \sqrt {3}}}\, \sqrt {3}\, \sqrt {i \sqrt {3}\, x +\frac {i \sqrt {3}}{2}+\frac {3}{2}}\, \sqrt {\frac {x}{-\frac {1}{2}-\frac {i \sqrt {3}}{2}}}\, \EllipticPi \left (\sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {1}{2}+\frac {i \sqrt {3}}{2}}}, \frac {-\frac {1}{2}-\frac {i \sqrt {3}}{2}}{-\frac {1}{2}-i-\frac {i \sqrt {3}}{2}}, \frac {\sqrt {3}\, \sqrt {i \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}\right )}{3 \sqrt {x^{3}+x^{2}+x}\, \left (-\frac {1}{2}-i-\frac {i \sqrt {3}}{2}\right )}-\frac {\sqrt {\frac {x}{\frac {1}{2}+\frac {i \sqrt {3}}{2}}+\frac {1}{1+i \sqrt {3}}+\frac {i \sqrt {3}}{1+i \sqrt {3}}}\, \sqrt {i \sqrt {3}\, x +\frac {i \sqrt {3}}{2}+\frac {3}{2}}\, \sqrt {\frac {x}{-\frac {1}{2}-\frac {i \sqrt {3}}{2}}}\, \EllipticPi \left (\sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {1}{2}+\frac {i \sqrt {3}}{2}}}, \frac {-\frac {1}{2}-\frac {i \sqrt {3}}{2}}{-\frac {1}{2}-i-\frac {i \sqrt {3}}{2}}, \frac {\sqrt {3}\, \sqrt {i \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}\right )}{\sqrt {x^{3}+x^{2}+x}\, \left (-\frac {1}{2}-i-\frac {i \sqrt {3}}{2}\right )}-\frac {i \sqrt {\frac {x}{\frac {1}{2}+\frac {i \sqrt {3}}{2}}+\frac {1}{1+i \sqrt {3}}+\frac {i \sqrt {3}}{1+i \sqrt {3}}}\, \sqrt {3}\, \sqrt {i \sqrt {3}\, x +\frac {i \sqrt {3}}{2}+\frac {3}{2}}\, \sqrt {\frac {x}{-\frac {1}{2}-\frac {i \sqrt {3}}{2}}}\, \EllipticPi \left (\sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {1}{2}+\frac {i \sqrt {3}}{2}}}, \frac {-\frac {1}{2}-\frac {i \sqrt {3}}{2}}{-\frac {1}{2}+i-\frac {i \sqrt {3}}{2}}, \frac {\sqrt {3}\, \sqrt {i \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}\right )}{3 \sqrt {x^{3}+x^{2}+x}\, \left (-\frac {1}{2}+i-\frac {i \sqrt {3}}{2}\right )}+\frac {\sqrt {\frac {x}{\frac {1}{2}+\frac {i \sqrt {3}}{2}}+\frac {1}{1+i \sqrt {3}}+\frac {i \sqrt {3}}{1+i \sqrt {3}}}\, \sqrt {i \sqrt {3}\, x +\frac {i \sqrt {3}}{2}+\frac {3}{2}}\, \sqrt {\frac {x}{-\frac {1}{2}-\frac {i \sqrt {3}}{2}}}\, \EllipticPi \left (\sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {1}{2}+\frac {i \sqrt {3}}{2}}}, \frac {-\frac {1}{2}-\frac {i \sqrt {3}}{2}}{-\frac {1}{2}+i-\frac {i \sqrt {3}}{2}}, \frac {\sqrt {3}\, \sqrt {i \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}\right )}{\sqrt {x^{3}+x^{2}+x}\, \left (-\frac {1}{2}+i-\frac {i \sqrt {3}}{2}\right )}\)	\(925\)

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]

ln((-x^2+2*(x^3+x^2+x)^(1/2)-2*x-1)/(x^2+1))

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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 = 0.15, size = 223, normalized size = 9.29 \begin {gather*} -\frac {\sqrt {\frac {x}{-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {-\frac {x+\frac {1}{2}-\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {\frac {x+\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\left (\sqrt {3}+1{}\mathrm {i}\right )\,\left (-\mathrm {F}\left (\mathrm {asin}\left (\sqrt {\frac {x}{-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\right )\middle |-\frac {-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}\right )+\Pi \left (-\frac {\sqrt {3}}{2}-\frac {1}{2}{}\mathrm {i};\mathrm {asin}\left (\sqrt {\frac {x}{-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\right )\middle |-\frac {-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}\right )+\Pi \left (\frac {\sqrt {3}}{2}+\frac {1}{2}{}\mathrm {i};\mathrm {asin}\left (\sqrt {\frac {x}{-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\right )\middle |-\frac {-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}\right )\right )\,1{}\mathrm {i}}{\sqrt {x^3+x^2-\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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