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

Optimal. Leaf size=111 \[ -\frac {1}{2} \sqrt {\frac {1}{3} \left (3+2 \sqrt {3}\right )} \tan ^{-1}\left (\frac {\sqrt {3+2 \sqrt {3}} \sqrt {x^3+1}}{x^2-x+1}\right )-\frac {1}{2} \sqrt {\frac {1}{3} \left (2 \sqrt {3}-3\right )} \tanh ^{-1}\left (\frac {\sqrt {2 \sqrt {3}-3} \sqrt {x^3+1}}{x^2-x+1}\right ) \]

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Rubi [C]  time = 0.92, antiderivative size = 412, normalized size of antiderivative = 3.71, number of steps used = 13, number of rules used = 6, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {6728, 218, 2135, 2140, 206, 203} \begin {gather*} -\frac {1}{2} \sqrt {\frac {1}{3} \left (3+2 \sqrt {3}\right )} \tan ^{-1}\left (\frac {\sqrt {3+2 \sqrt {3}} (x+1)}{\sqrt {x^3+1}}\right )-\frac {1}{2} \sqrt {\frac {1}{3} \left (2 \sqrt {3}-3\right )} \tanh ^{-1}\left (\frac {\sqrt {2 \sqrt {3}-3} (x+1)}{\sqrt {x^3+1}}\right )-\frac {\left (2+\sqrt {3}\right )^{3/2} (x+1) \sqrt {\frac {x^2-x+1}{\left (x+\sqrt {3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac {x-\sqrt {3}+1}{x+\sqrt {3}+1}\right )|-7-4 \sqrt {3}\right )}{2 \sqrt [4]{3} \sqrt {\frac {x+1}{\left (x+\sqrt {3}+1\right )^2}} \sqrt {x^3+1}}+\frac {2 \sqrt {2+\sqrt {3}} (x+1) \sqrt {\frac {x^2-x+1}{\left (x+\sqrt {3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac {x-\sqrt {3}+1}{x+\sqrt {3}+1}\right )|-7-4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {\frac {x+1}{\left (x+\sqrt {3}+1\right )^2}} \sqrt {x^3+1}}-\frac {\sqrt {2-\sqrt {3}} (x+1) \sqrt {\frac {x^2-x+1}{\left (x+\sqrt {3}+1\right )^2}} F\left (\sin ^{-1}\left (\frac {x-\sqrt {3}+1}{x+\sqrt {3}+1}\right )|-7-4 \sqrt {3}\right )}{2 \sqrt [4]{3} \sqrt {\frac {x+1}{\left (x+\sqrt {3}+1\right )^2}} \sqrt {x^3+1}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/2*(Sqrt[(3 + 2*Sqrt[3])/3]*ArcTan[(Sqrt[3 + 2*Sqrt[3]]*(1 + x))/Sqrt[1 + x^3]]) - (Sqrt[(-3 + 2*Sqrt[3])/3]
*ArcTanh[(Sqrt[-3 + 2*Sqrt[3]]*(1 + x))/Sqrt[1 + x^3]])/2 - (Sqrt[2 - Sqrt[3]]*(1 + x)*Sqrt[(1 - x + x^2)/(1 +
Sqrt[3] + x)^2]*EllipticF[ArcSin[(1 - Sqrt[3] + x)/(1 + Sqrt[3] + x)], -7 - 4*Sqrt[3]])/(2*3^(1/4)*Sqrt[(1 +
x)/(1 + Sqrt[3] + x)^2]*Sqrt[1 + x^3]) + (2*Sqrt[2 + Sqrt[3]]*(1 + x)*Sqrt[(1 - x + x^2)/(1 + Sqrt[3] + x)^2]*
EllipticF[ArcSin[(1 - Sqrt[3] + x)/(1 + Sqrt[3] + x)], -7 - 4*Sqrt[3]])/(3^(1/4)*Sqrt[(1 + x)/(1 + Sqrt[3] + x
)^2]*Sqrt[1 + x^3]) - ((2 + Sqrt[3])^(3/2)*(1 + x)*Sqrt[(1 - x + x^2)/(1 + Sqrt[3] + x)^2]*EllipticF[ArcSin[(1
- Sqrt[3] + x)/(1 + Sqrt[3] + x)], -7 - 4*Sqrt[3]])/(2*3^(1/4)*Sqrt[(1 + x)/(1 + Sqrt[3] + x)^2]*Sqrt[1 + x^3
])

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 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 218

Int[1/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], s = Denom[Rt[b/a, 3]]}, Simp[(2*Sqr
t[2 + Sqrt[3]]*(s + r*x)*Sqrt[(s^2 - r*s*x + r^2*x^2)/((1 + Sqrt[3])*s + r*x)^2]*EllipticF[ArcSin[((1 - Sqrt[3
])*s + r*x)/((1 + Sqrt[3])*s + r*x)], -7 - 4*Sqrt[3]])/(3^(1/4)*r*Sqrt[a + b*x^3]*Sqrt[(s*(s + r*x))/((1 + Sqr
t[3])*s + r*x)^2]), x]] /; FreeQ[{a, b}, x] && PosQ[a]

Rule 2135

Int[1/(((c_) + (d_.)*(x_))*Sqrt[(a_) + (b_.)*(x_)^3]), x_Symbol] :> Dist[(-6*a*d^3)/(c*(b*c^3 - 28*a*d^3)), In
t[1/Sqrt[a + b*x^3], x], x] + Dist[1/(c*(b*c^3 - 28*a*d^3)), Int[Simp[c*(b*c^3 - 22*a*d^3) + 6*a*d^4*x, x]/((c
+ d*x)*Sqrt[a + b*x^3]), x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[b^2*c^6 - 20*a*b*c^3*d^3 - 8*a^2*d^6, 0]

Rule 2140

Int[((e_) + (f_.)*(x_))/(((c_) + (d_.)*(x_))*Sqrt[(a_) + (b_.)*(x_)^3]), x_Symbol] :> With[{k = Simplify[(d*e
+ 2*c*f)/(c*f)]}, Dist[((1 + k)*e)/d, Subst[Int[1/(1 + (3 + 2*k)*a*x^2), x], x, (1 + ((1 + k)*d*x)/c)/Sqrt[a +
b*x^3]], x]] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[d*e - c*f, 0] && EqQ[b^2*c^6 - 20*a*b*c^3*d^3 - 8*a^2*d^6
, 0] && EqQ[6*a*d^4*e - c*f*(b*c^3 - 22*a*d^3), 0]

Rule 6728

Int[(u_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a +
b*x^n + c*x^(2*n)), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]

Rubi steps

\begin {align*} \int \frac {1-x+x^2}{\left (-2+2 x+x^2\right ) \sqrt {1+x^3}} \, dx &=\int \left (\frac {1}{\sqrt {1+x^3}}+\frac {3 (1-x)}{\left (-2+2 x+x^2\right ) \sqrt {1+x^3}}\right ) \, dx\\ &=3 \int \frac {1-x}{\left (-2+2 x+x^2\right ) \sqrt {1+x^3}} \, dx+\int \frac {1}{\sqrt {1+x^3}} \, dx\\ &=\frac {2 \sqrt {2+\sqrt {3}} (1+x) \sqrt {\frac {1-x+x^2}{\left (1+\sqrt {3}+x\right )^2}} F\left (\sin ^{-1}\left (\frac {1-\sqrt {3}+x}{1+\sqrt {3}+x}\right )|-7-4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {\frac {1+x}{\left (1+\sqrt {3}+x\right )^2}} \sqrt {1+x^3}}+3 \int \left (\frac {-1+\frac {2}{\sqrt {3}}}{\left (2-2 \sqrt {3}+2 x\right ) \sqrt {1+x^3}}+\frac {-1-\frac {2}{\sqrt {3}}}{\left (2+2 \sqrt {3}+2 x\right ) \sqrt {1+x^3}}\right ) \, dx\\ &=\frac {2 \sqrt {2+\sqrt {3}} (1+x) \sqrt {\frac {1-x+x^2}{\left (1+\sqrt {3}+x\right )^2}} F\left (\sin ^{-1}\left (\frac {1-\sqrt {3}+x}{1+\sqrt {3}+x}\right )|-7-4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {\frac {1+x}{\left (1+\sqrt {3}+x\right )^2}} \sqrt {1+x^3}}+\left (-3+2 \sqrt {3}\right ) \int \frac {1}{\left (2-2 \sqrt {3}+2 x\right ) \sqrt {1+x^3}} \, dx-\left (3+2 \sqrt {3}\right ) \int \frac {1}{\left (2+2 \sqrt {3}+2 x\right ) \sqrt {1+x^3}} \, dx\\ &=\frac {2 \sqrt {2+\sqrt {3}} (1+x) \sqrt {\frac {1-x+x^2}{\left (1+\sqrt {3}+x\right )^2}} F\left (\sin ^{-1}\left (\frac {1-\sqrt {3}+x}{1+\sqrt {3}+x}\right )|-7-4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {\frac {1+x}{\left (1+\sqrt {3}+x\right )^2}} \sqrt {1+x^3}}+\frac {1}{192} \left (2-\sqrt {3}\right ) \int \frac {96 \left (1+\sqrt {3}\right )+96 x}{\left (2-2 \sqrt {3}+2 x\right ) \sqrt {1+x^3}} \, dx+\frac {1}{4} \left (-2+\sqrt {3}\right ) \int \frac {1}{\sqrt {1+x^3}} \, dx+\frac {1}{192} \left (2+\sqrt {3}\right ) \int \frac {96 \left (1-\sqrt {3}\right )+96 x}{\left (2+2 \sqrt {3}+2 x\right ) \sqrt {1+x^3}} \, dx-\frac {1}{4} \left (2+\sqrt {3}\right ) \int \frac {1}{\sqrt {1+x^3}} \, dx\\ &=-\frac {\sqrt {2-\sqrt {3}} (1+x) \sqrt {\frac {1-x+x^2}{\left (1+\sqrt {3}+x\right )^2}} F\left (\sin ^{-1}\left (\frac {1-\sqrt {3}+x}{1+\sqrt {3}+x}\right )|-7-4 \sqrt {3}\right )}{2 \sqrt [4]{3} \sqrt {\frac {1+x}{\left (1+\sqrt {3}+x\right )^2}} \sqrt {1+x^3}}+\frac {2 \sqrt {2+\sqrt {3}} (1+x) \sqrt {\frac {1-x+x^2}{\left (1+\sqrt {3}+x\right )^2}} F\left (\sin ^{-1}\left (\frac {1-\sqrt {3}+x}{1+\sqrt {3}+x}\right )|-7-4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {\frac {1+x}{\left (1+\sqrt {3}+x\right )^2}} \sqrt {1+x^3}}-\frac {\left (2+\sqrt {3}\right )^{3/2} (1+x) \sqrt {\frac {1-x+x^2}{\left (1+\sqrt {3}+x\right )^2}} F\left (\sin ^{-1}\left (\frac {1-\sqrt {3}+x}{1+\sqrt {3}+x}\right )|-7-4 \sqrt {3}\right )}{2 \sqrt [4]{3} \sqrt {\frac {1+x}{\left (1+\sqrt {3}+x\right )^2}} \sqrt {1+x^3}}+\frac {1}{2} \left (-2-\sqrt {3}\right ) \operatorname {Subst}\left (\int \frac {1}{1+\left (3+2 \sqrt {3}\right ) x^2} \, dx,x,\frac {1+\frac {2 \left (1+\sqrt {3}\right ) x}{2+2 \sqrt {3}}}{\sqrt {1+x^3}}\right )+\frac {1}{2} \left (-2+\sqrt {3}\right ) \operatorname {Subst}\left (\int \frac {1}{1+\left (3-2 \sqrt {3}\right ) x^2} \, dx,x,\frac {1+\frac {2 \left (1-\sqrt {3}\right ) x}{2-2 \sqrt {3}}}{\sqrt {1+x^3}}\right )\\ &=-\frac {1}{2} \sqrt {1+\frac {2}{\sqrt {3}}} \tan ^{-1}\left (\frac {\sqrt {3+2 \sqrt {3}} (1+x)}{\sqrt {1+x^3}}\right )-\frac {1}{2} \sqrt {-1+\frac {2}{\sqrt {3}}} \tanh ^{-1}\left (\frac {\sqrt {-3+2 \sqrt {3}} (1+x)}{\sqrt {1+x^3}}\right )-\frac {\sqrt {2-\sqrt {3}} (1+x) \sqrt {\frac {1-x+x^2}{\left (1+\sqrt {3}+x\right )^2}} F\left (\sin ^{-1}\left (\frac {1-\sqrt {3}+x}{1+\sqrt {3}+x}\right )|-7-4 \sqrt {3}\right )}{2 \sqrt [4]{3} \sqrt {\frac {1+x}{\left (1+\sqrt {3}+x\right )^2}} \sqrt {1+x^3}}+\frac {2 \sqrt {2+\sqrt {3}} (1+x) \sqrt {\frac {1-x+x^2}{\left (1+\sqrt {3}+x\right )^2}} F\left (\sin ^{-1}\left (\frac {1-\sqrt {3}+x}{1+\sqrt {3}+x}\right )|-7-4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {\frac {1+x}{\left (1+\sqrt {3}+x\right )^2}} \sqrt {1+x^3}}-\frac {\left (2+\sqrt {3}\right )^{3/2} (1+x) \sqrt {\frac {1-x+x^2}{\left (1+\sqrt {3}+x\right )^2}} F\left (\sin ^{-1}\left (\frac {1-\sqrt {3}+x}{1+\sqrt {3}+x}\right )|-7-4 \sqrt {3}\right )}{2 \sqrt [4]{3} \sqrt {\frac {1+x}{\left (1+\sqrt {3}+x\right )^2}} \sqrt {1+x^3}}\\ \end {align*}

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Mathematica [C]  time = 0.91, size = 278, normalized size = 2.50 \begin {gather*} \frac {\sqrt {\frac {x+1}{1+\sqrt [3]{-1}}} \left (i \sqrt {x^2-x+1} \left (\left (\sqrt {3}+(-2+i)\right ) \Pi \left (\frac {2 \sqrt {3}}{-3 i+(1+2 i) \sqrt {3}};\sin ^{-1}\left (\sqrt {\frac {(-1)^{2/3} x+1}{1+\sqrt [3]{-1}}}\right )|\sqrt [3]{-1}\right )+\left (\sqrt {3}+(2+i)\right ) \Pi \left (\frac {2 i \sqrt {3}}{3+(2+i) \sqrt {3}};\sin ^{-1}\left (\sqrt {\frac {(-1)^{2/3} x+1}{1+\sqrt [3]{-1}}}\right )|\sqrt [3]{-1}\right )\right )+\frac {2 \sqrt {\frac {\sqrt [3]{-1}-(-1)^{2/3} x}{1+\sqrt [3]{-1}}} \left (\left (\sqrt {3}+i\right ) x-2 i\right ) F\left (\sin ^{-1}\left (\sqrt {\frac {(-1)^{2/3} x+1}{1+\sqrt [3]{-1}}}\right )|\sqrt [3]{-1}\right )}{\sqrt {\frac {(-1)^{2/3} x+1}{1+\sqrt [3]{-1}}}}\right )}{\left (\sqrt {3}+i\right ) \sqrt {x^3+1}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 1.05, size = 111, normalized size = 1.00 \begin {gather*} -\frac {1}{2} \sqrt {\frac {1}{3} \left (3+2 \sqrt {3}\right )} \tan ^{-1}\left (\frac {\sqrt {3+2 \sqrt {3}} \sqrt {1+x^3}}{1-x+x^2}\right )-\frac {1}{2} \sqrt {\frac {1}{3} \left (-3+2 \sqrt {3}\right )} \tanh ^{-1}\left (\frac {\sqrt {-3+2 \sqrt {3}} \sqrt {1+x^3}}{1-x+x^2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [B]  time = 0.71, size = 229, normalized size = 2.06 \begin {gather*} -\frac {1}{6} \, \sqrt {3} \sqrt {2 \, \sqrt {3} + 3} \arctan \left (\frac {\sqrt {x^{3} + 1} \sqrt {2 \, \sqrt {3} + 3}}{x^{2} - x + 1}\right ) - \frac {1}{24} \, \sqrt {3} \sqrt {2 \, \sqrt {3} - 3} \log \left (\frac {x^{4} - 2 \, x^{3} + 6 \, x^{2} + 2 \, \sqrt {x^{3} + 1} {\left (x^{2} + 2 \, \sqrt {3} {\left (x + 1\right )} + 2 \, x + 4\right )} \sqrt {2 \, \sqrt {3} - 3} + 4 \, \sqrt {3} {\left (x^{3} + 1\right )} + 4 \, x + 4}{x^{4} + 4 \, x^{3} - 8 \, x + 4}\right ) + \frac {1}{24} \, \sqrt {3} \sqrt {2 \, \sqrt {3} - 3} \log \left (\frac {x^{4} - 2 \, x^{3} + 6 \, x^{2} - 2 \, \sqrt {x^{3} + 1} {\left (x^{2} + 2 \, \sqrt {3} {\left (x + 1\right )} + 2 \, x + 4\right )} \sqrt {2 \, \sqrt {3} - 3} + 4 \, \sqrt {3} {\left (x^{3} + 1\right )} + 4 \, x + 4}{x^{4} + 4 \, x^{3} - 8 \, x + 4}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*sqrt(3)*sqrt(2*sqrt(3) + 3)*arctan(sqrt(x^3 + 1)*sqrt(2*sqrt(3) + 3)/(x^2 - x + 1)) - 1/24*sqrt(3)*sqrt(2
*sqrt(3) - 3)*log((x^4 - 2*x^3 + 6*x^2 + 2*sqrt(x^3 + 1)*(x^2 + 2*sqrt(3)*(x + 1) + 2*x + 4)*sqrt(2*sqrt(3) -
3) + 4*sqrt(3)*(x^3 + 1) + 4*x + 4)/(x^4 + 4*x^3 - 8*x + 4)) + 1/24*sqrt(3)*sqrt(2*sqrt(3) - 3)*log((x^4 - 2*x
^3 + 6*x^2 - 2*sqrt(x^3 + 1)*(x^2 + 2*sqrt(3)*(x + 1) + 2*x + 4)*sqrt(2*sqrt(3) - 3) + 4*sqrt(3)*(x^3 + 1) + 4
*x + 4)/(x^4 + 4*x^3 - 8*x + 4))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C]  time = 2.71, size = 582, normalized size = 5.24

method	result	size
trager	\(\frac {\RootOf \left (48 \textit {\_Z}^{4}+24 \textit {\_Z}^{2}-1\right ) \ln \left (-\frac {-240 \RootOf \left (48 \textit {\_Z}^{4}+24 \textit {\_Z}^{2}-1\right )^{5} x^{2}+480 \RootOf \left (48 \textit {\_Z}^{4}+24 \textit {\_Z}^{2}-1\right )^{5} x -232 \RootOf \left (48 \textit {\_Z}^{4}+24 \textit {\_Z}^{2}-1\right )^{3} x^{2}+304 \RootOf \left (48 \textit {\_Z}^{4}+24 \textit {\_Z}^{2}-1\right )^{3} x +32 \sqrt {x^{3}+1}\, \RootOf \left (48 \textit {\_Z}^{4}+24 \textit {\_Z}^{2}-1\right )^{2}-160 \RootOf \left (48 \textit {\_Z}^{4}+24 \textit {\_Z}^{2}-1\right )^{3}-55 \RootOf \left (48 \textit {\_Z}^{4}+24 \textit {\_Z}^{2}-1\right ) x^{2}+22 \RootOf \left (48 \textit {\_Z}^{4}+24 \textit {\_Z}^{2}-1\right ) x +16 \sqrt {x^{3}+1}-88 \RootOf \left (48 \textit {\_Z}^{4}+24 \textit {\_Z}^{2}-1\right )}{\left (12 \RootOf \left (48 \textit {\_Z}^{4}+24 \textit {\_Z}^{2}-1\right )^{2} x +5 x -4\right )^{2}}\right )}{2}-\frac {\RootOf \left (\textit {\_Z}^{2}+4 \RootOf \left (48 \textit {\_Z}^{4}+24 \textit {\_Z}^{2}-1\right )^{2}+2\right ) \ln \left (\frac {-240 \RootOf \left (\textit {\_Z}^{2}+4 \RootOf \left (48 \textit {\_Z}^{4}+24 \textit {\_Z}^{2}-1\right )^{2}+2\right ) \RootOf \left (48 \textit {\_Z}^{4}+24 \textit {\_Z}^{2}-1\right )^{4} x^{2}+480 \RootOf \left (\textit {\_Z}^{2}+4 \RootOf \left (48 \textit {\_Z}^{4}+24 \textit {\_Z}^{2}-1\right )^{2}+2\right ) \RootOf \left (48 \textit {\_Z}^{4}+24 \textit {\_Z}^{2}-1\right )^{4} x -8 \RootOf \left (\textit {\_Z}^{2}+4 \RootOf \left (48 \textit {\_Z}^{4}+24 \textit {\_Z}^{2}-1\right )^{2}+2\right ) \RootOf \left (48 \textit {\_Z}^{4}+24 \textit {\_Z}^{2}-1\right )^{2} x^{2}+176 \RootOf \left (48 \textit {\_Z}^{4}+24 \textit {\_Z}^{2}-1\right )^{2} \RootOf \left (\textit {\_Z}^{2}+4 \RootOf \left (48 \textit {\_Z}^{4}+24 \textit {\_Z}^{2}-1\right )^{2}+2\right ) x +64 \sqrt {x^{3}+1}\, \RootOf \left (48 \textit {\_Z}^{4}+24 \textit {\_Z}^{2}-1\right )^{2}+160 \RootOf \left (48 \textit {\_Z}^{4}+24 \textit {\_Z}^{2}-1\right )^{2} \RootOf \left (\textit {\_Z}^{2}+4 \RootOf \left (48 \textit {\_Z}^{4}+24 \textit {\_Z}^{2}-1\right )^{2}+2\right )+\RootOf \left (\textit {\_Z}^{2}+4 \RootOf \left (48 \textit {\_Z}^{4}+24 \textit {\_Z}^{2}-1\right )^{2}+2\right ) x^{2}-10 \RootOf \left (\textit {\_Z}^{2}+4 \RootOf \left (48 \textit {\_Z}^{4}+24 \textit {\_Z}^{2}-1\right )^{2}+2\right ) x -8 \RootOf \left (\textit {\_Z}^{2}+4 \RootOf \left (48 \textit {\_Z}^{4}+24 \textit {\_Z}^{2}-1\right )^{2}+2\right )}{\left (12 \RootOf \left (48 \textit {\_Z}^{4}+24 \textit {\_Z}^{2}-1\right )^{2} x +x +4\right )^{2}}\right )}{4}\)	\(582\)
default	\(\frac {2 \left (\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}-\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x -\frac {1}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \EllipticF \left (\sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \sqrt {\frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {x^{3}+1}}+\frac {3 \sqrt {\frac {1}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}+\frac {x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}-\frac {1}{2 \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}-\frac {i \sqrt {3}}{2 \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\, \sqrt {\frac {x}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}-\frac {1}{2 \left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right )}+\frac {i \sqrt {3}}{-3+i \sqrt {3}}}\, \EllipticPi \left (\sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, -\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}{3}, \sqrt {\frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right ) \sqrt {3}}{2 \sqrt {x^{3}+1}}-\frac {3 \sqrt {\frac {1}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}+\frac {x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}-\frac {1}{2 \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}-\frac {i \sqrt {3}}{2 \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\, \sqrt {\frac {x}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}-\frac {1}{2 \left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right )}+\frac {i \sqrt {3}}{-3+i \sqrt {3}}}\, \EllipticPi \left (\sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, -\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}{3}, \sqrt {\frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {x^{3}+1}}-\frac {3 i \sqrt {\frac {1}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}+\frac {x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}-\frac {1}{2 \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}-\frac {i \sqrt {3}}{2 \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\, \sqrt {\frac {x}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}-\frac {1}{2 \left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right )}+\frac {i \sqrt {3}}{-3+i \sqrt {3}}}\, \EllipticPi \left (\sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, -\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}{3}, \sqrt {\frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{2 \sqrt {x^{3}+1}}+\frac {i \sqrt {\frac {1}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}+\frac {x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}-\frac {1}{2 \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}-\frac {i \sqrt {3}}{2 \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\, \sqrt {\frac {x}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}-\frac {1}{2 \left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right )}+\frac {i \sqrt {3}}{-3+i \sqrt {3}}}\, \EllipticPi \left (\sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, -\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}{3}, \sqrt {\frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right ) \sqrt {3}}{\sqrt {x^{3}+1}}-\frac {3 \sqrt {\frac {1}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}+\frac {x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}-\frac {1}{2 \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}-\frac {i \sqrt {3}}{2 \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\, \sqrt {\frac {x}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}-\frac {1}{2 \left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right )}+\frac {i \sqrt {3}}{-3+i \sqrt {3}}}\, \EllipticPi \left (\sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}{3}, \sqrt {\frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {x^{3}+1}}+\frac {3 i \sqrt {\frac {1}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}+\frac {x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}-\frac {1}{2 \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}-\frac {i \sqrt {3}}{2 \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\, \sqrt {\frac {x}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}-\frac {1}{2 \left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right )}+\frac {i \sqrt {3}}{-3+i \sqrt {3}}}\, \EllipticPi \left (\sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}{3}, \sqrt {\frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{2 \sqrt {x^{3}+1}}+\frac {i \sqrt {\frac {1}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}+\frac {x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}-\frac {1}{2 \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}-\frac {i \sqrt {3}}{2 \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\, \sqrt {\frac {x}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}-\frac {1}{2 \left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right )}+\frac {i \sqrt {3}}{-3+i \sqrt {3}}}\, \EllipticPi \left (\sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}{3}, \sqrt {\frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right ) \sqrt {3}}{\sqrt {x^{3}+1}}-\frac {3 \sqrt {\frac {1}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}+\frac {x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}-\frac {1}{2 \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}-\frac {i \sqrt {3}}{2 \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\, \sqrt {\frac {x}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}-\frac {1}{2 \left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right )}+\frac {i \sqrt {3}}{-3+i \sqrt {3}}}\, \EllipticPi \left (\sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}{3}, \sqrt {\frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right ) \sqrt {3}}{2 \sqrt {x^{3}+1}}\)	\(1501\)
elliptic	\(\frac {3 \sqrt {\frac {1}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}+\frac {x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}-\frac {1}{2 \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}-\frac {i \sqrt {3}}{2 \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\, \sqrt {\frac {x}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}-\frac {1}{2 \left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right )}+\frac {i \sqrt {3}}{-3+i \sqrt {3}}}\, \EllipticF \left (\sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \sqrt {\frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {x^{3}+1}}-\frac {i \sqrt {\frac {1}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}+\frac {x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}-\frac {1}{2 \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}-\frac {i \sqrt {3}}{2 \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\, \sqrt {\frac {x}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}-\frac {1}{2 \left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right )}+\frac {i \sqrt {3}}{-3+i \sqrt {3}}}\, \EllipticF \left (\sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \sqrt {\frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right ) \sqrt {3}}{\sqrt {x^{3}+1}}+\frac {3 \sqrt {\frac {1}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}+\frac {x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}-\frac {1}{2 \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}-\frac {i \sqrt {3}}{2 \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\, \sqrt {\frac {x}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}-\frac {1}{2 \left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right )}+\frac {i \sqrt {3}}{-3+i \sqrt {3}}}\, \EllipticPi \left (\sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, -\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}{3}, \sqrt {\frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right ) \sqrt {3}}{2 \sqrt {x^{3}+1}}-\frac {3 \sqrt {\frac {1}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}+\frac {x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}-\frac {1}{2 \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}-\frac {i \sqrt {3}}{2 \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\, \sqrt {\frac {x}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}-\frac {1}{2 \left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right )}+\frac {i \sqrt {3}}{-3+i \sqrt {3}}}\, \EllipticPi \left (\sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, -\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}{3}, \sqrt {\frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {x^{3}+1}}-\frac {3 i \sqrt {\frac {1}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}+\frac {x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}-\frac {1}{2 \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}-\frac {i \sqrt {3}}{2 \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\, \sqrt {\frac {x}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}-\frac {1}{2 \left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right )}+\frac {i \sqrt {3}}{-3+i \sqrt {3}}}\, \EllipticPi \left (\sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, -\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}{3}, \sqrt {\frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{2 \sqrt {x^{3}+1}}+\frac {i \sqrt {\frac {1}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}+\frac {x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}-\frac {1}{2 \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}-\frac {i \sqrt {3}}{2 \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\, \sqrt {\frac {x}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}-\frac {1}{2 \left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right )}+\frac {i \sqrt {3}}{-3+i \sqrt {3}}}\, \EllipticPi \left (\sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, -\frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}{3}, \sqrt {\frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right ) \sqrt {3}}{\sqrt {x^{3}+1}}-\frac {3 \sqrt {\frac {1}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}+\frac {x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}-\frac {1}{2 \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}-\frac {i \sqrt {3}}{2 \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\, \sqrt {\frac {x}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}-\frac {1}{2 \left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right )}+\frac {i \sqrt {3}}{-3+i \sqrt {3}}}\, \EllipticPi \left (\sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}{3}, \sqrt {\frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {x^{3}+1}}+\frac {3 i \sqrt {\frac {1}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}+\frac {x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}-\frac {1}{2 \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}-\frac {i \sqrt {3}}{2 \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\, \sqrt {\frac {x}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}-\frac {1}{2 \left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right )}+\frac {i \sqrt {3}}{-3+i \sqrt {3}}}\, \EllipticPi \left (\sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}{3}, \sqrt {\frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{2 \sqrt {x^{3}+1}}+\frac {i \sqrt {\frac {1}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}+\frac {x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}-\frac {1}{2 \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}-\frac {i \sqrt {3}}{2 \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\, \sqrt {\frac {x}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}-\frac {1}{2 \left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right )}+\frac {i \sqrt {3}}{-3+i \sqrt {3}}}\, \EllipticPi \left (\sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}{3}, \sqrt {\frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right ) \sqrt {3}}{\sqrt {x^{3}+1}}-\frac {3 \sqrt {\frac {1}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}+\frac {x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}-\frac {1}{2 \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}-\frac {i \sqrt {3}}{2 \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right )}}\, \sqrt {\frac {x}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}-\frac {1}{2 \left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right )}+\frac {i \sqrt {3}}{-3+i \sqrt {3}}}\, \EllipticPi \left (\sqrt {\frac {1+x}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \frac {\left (-\frac {3}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}{3}, \sqrt {\frac {-\frac {3}{2}+\frac {i \sqrt {3}}{2}}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right ) \sqrt {3}}{2 \sqrt {x^{3}+1}}\)	\(1706\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*RootOf(48*_Z^4+24*_Z^2-1)*ln(-(-240*RootOf(48*_Z^4+24*_Z^2-1)^5*x^2+480*RootOf(48*_Z^4+24*_Z^2-1)^5*x-232*
RootOf(48*_Z^4+24*_Z^2-1)^3*x^2+304*RootOf(48*_Z^4+24*_Z^2-1)^3*x+32*(x^3+1)^(1/2)*RootOf(48*_Z^4+24*_Z^2-1)^2
-160*RootOf(48*_Z^4+24*_Z^2-1)^3-55*RootOf(48*_Z^4+24*_Z^2-1)*x^2+22*RootOf(48*_Z^4+24*_Z^2-1)*x+16*(x^3+1)^(1
/2)-88*RootOf(48*_Z^4+24*_Z^2-1))/(12*RootOf(48*_Z^4+24*_Z^2-1)^2*x+5*x-4)^2)-1/4*RootOf(_Z^2+4*RootOf(48*_Z^4
+24*_Z^2-1)^2+2)*ln((-240*RootOf(_Z^2+4*RootOf(48*_Z^4+24*_Z^2-1)^2+2)*RootOf(48*_Z^4+24*_Z^2-1)^4*x^2+480*Roo
tOf(_Z^2+4*RootOf(48*_Z^4+24*_Z^2-1)^2+2)*RootOf(48*_Z^4+24*_Z^2-1)^4*x-8*RootOf(_Z^2+4*RootOf(48*_Z^4+24*_Z^2
-1)^2+2)*RootOf(48*_Z^4+24*_Z^2-1)^2*x^2+176*RootOf(48*_Z^4+24*_Z^2-1)^2*RootOf(_Z^2+4*RootOf(48*_Z^4+24*_Z^2-
1)^2+2)*x+64*(x^3+1)^(1/2)*RootOf(48*_Z^4+24*_Z^2-1)^2+160*RootOf(48*_Z^4+24*_Z^2-1)^2*RootOf(_Z^2+4*RootOf(48
*_Z^4+24*_Z^2-1)^2+2)+RootOf(_Z^2+4*RootOf(48*_Z^4+24*_Z^2-1)^2+2)*x^2-10*RootOf(_Z^2+4*RootOf(48*_Z^4+24*_Z^2
-1)^2+2)*x-8*RootOf(_Z^2+4*RootOf(48*_Z^4+24*_Z^2-1)^2+2))/(12*RootOf(48*_Z^4+24*_Z^2-1)^2*x+x+4)^2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.97, size = 509, normalized size = 4.59 \begin {gather*} \frac {2\,\left (\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\sqrt {\frac {x-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {\frac {x+1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {\frac {\frac {1}{2}-x+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\mathrm {F}\left (\mathrm {asin}\left (\sqrt {\frac {x+1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\right )\middle |-\frac {\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}\right )}{\sqrt {x^3+\left (-\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )-1\right )\,x-\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}}+\frac {\left (3\,\sqrt {3}-6\right )\,\left (\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\sqrt {\frac {x-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {\frac {x+1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {\frac {\frac {1}{2}-x+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\Pi \left (\frac {\sqrt {3}\,\left (\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{3};\mathrm {asin}\left (\sqrt {\frac {x+1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\right )\middle |-\frac {\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}\right )}{3\,\sqrt {x^3+\left (-\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )-1\right )\,x-\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}}-\frac {\left (3\,\sqrt {3}+6\right )\,\left (\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\sqrt {\frac {x-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {\frac {x+1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {\frac {\frac {1}{2}-x+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\Pi \left (-\frac {\sqrt {3}\,\left (\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{3};\mathrm {asin}\left (\sqrt {\frac {x+1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\right )\middle |-\frac {\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}\right )}{3\,\sqrt {x^3+\left (-\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )-1\right )\,x-\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2*((3^(1/2)*1i)/2 + 3/2)*((x + (3^(1/2)*1i)/2 - 1/2)/((3^(1/2)*1i)/2 - 3/2))^(1/2)*((x + 1)/((3^(1/2)*1i)/2 +
3/2))^(1/2)*(((3^(1/2)*1i)/2 - x + 1/2)/((3^(1/2)*1i)/2 + 3/2))^(1/2)*ellipticF(asin(((x + 1)/((3^(1/2)*1i)/2
+ 3/2))^(1/2)), -((3^(1/2)*1i)/2 + 3/2)/((3^(1/2)*1i)/2 - 3/2)))/(x^3 - x*(((3^(1/2)*1i)/2 - 1/2)*((3^(1/2)*1
i)/2 + 1/2) + 1) - ((3^(1/2)*1i)/2 - 1/2)*((3^(1/2)*1i)/2 + 1/2))^(1/2) + ((3*3^(1/2) - 6)*((3^(1/2)*1i)/2 + 3
/2)*((x + (3^(1/2)*1i)/2 - 1/2)/((3^(1/2)*1i)/2 - 3/2))^(1/2)*((x + 1)/((3^(1/2)*1i)/2 + 3/2))^(1/2)*(((3^(1/2
)*1i)/2 - x + 1/2)/((3^(1/2)*1i)/2 + 3/2))^(1/2)*ellipticPi((3^(1/2)*((3^(1/2)*1i)/2 + 3/2))/3, asin(((x + 1)/
((3^(1/2)*1i)/2 + 3/2))^(1/2)), -((3^(1/2)*1i)/2 + 3/2)/((3^(1/2)*1i)/2 - 3/2)))/(3*(x^3 - x*(((3^(1/2)*1i)/2
- 1/2)*((3^(1/2)*1i)/2 + 1/2) + 1) - ((3^(1/2)*1i)/2 - 1/2)*((3^(1/2)*1i)/2 + 1/2))^(1/2)) - ((3*3^(1/2) + 6)*
((3^(1/2)*1i)/2 + 3/2)*((x + (3^(1/2)*1i)/2 - 1/2)/((3^(1/2)*1i)/2 - 3/2))^(1/2)*((x + 1)/((3^(1/2)*1i)/2 + 3/
2))^(1/2)*(((3^(1/2)*1i)/2 - x + 1/2)/((3^(1/2)*1i)/2 + 3/2))^(1/2)*ellipticPi(-(3^(1/2)*((3^(1/2)*1i)/2 + 3/2
))/3, asin(((x + 1)/((3^(1/2)*1i)/2 + 3/2))^(1/2)), -((3^(1/2)*1i)/2 + 3/2)/((3^(1/2)*1i)/2 - 3/2)))/(3*(x^3 -
x*(((3^(1/2)*1i)/2 - 1/2)*((3^(1/2)*1i)/2 + 1/2) + 1) - ((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 {x^{2} - x + 1}{\sqrt {\left (x + 1\right ) \left (x^{2} - x + 1\right )} \left (x^{2} + 2 x - 2\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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