3.199 \(\int \frac{2+2 x-x^2}{(2+x^2) \sqrt{1-x^3}} \, dx\)
Optimal. Leaf size=20 \[ -2 \tan ^{-1}\left (\frac{1-x}{\sqrt{1-x^3}}\right ) \]
[Out]
-2*ArcTan[(1 - x)/Sqrt[1 - x^3]]
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Rubi [A] time = 0.0850639, antiderivative size = 20, normalized size of antiderivative = 1.,
number of steps used = 2, number of rules used = 2, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.069, Rules used =
{2146, 203} \[ -2 \tan ^{-1}\left (\frac{1-x}{\sqrt{1-x^3}}\right ) \]
Antiderivative was successfully verified.
[In]
Int[(2 + 2*x - x^2)/((2 + x^2)*Sqrt[1 - x^3]),x]
[Out]
-2*ArcTan[(1 - x)/Sqrt[1 - x^3]]
Rule 2146
Int[((f_) + (g_.)*(x_) + (h_.)*(x_)^2)/(((c_) + (e_.)*(x_)^2)*Sqrt[(a_) + (b_.)*(x_)^3]), x_Symbol] :> -Dist[g
/e, Subst[Int[1/(1 + a*x^2), x], x, (1 + (2*h*x)/g)/Sqrt[a + b*x^3]], x] /; FreeQ[{a, b, c, e, f, g, h}, x] &&
EqQ[b*g^3 - 8*a*h^3, 0] && EqQ[g^2 + 2*f*h, 0] && EqQ[b*c*g - 4*a*e*h, 0]
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])
Rubi steps
\begin{align*} \int \frac{2+2 x-x^2}{\left (2+x^2\right ) \sqrt{1-x^3}} \, dx &=-\left (2 \operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\frac{1-x}{\sqrt{1-x^3}}\right )\right )\\ &=-2 \tan ^{-1}\left (\frac{1-x}{\sqrt{1-x^3}}\right )\\ \end{align*}
Mathematica [C] time = 0.639948, size = 280, normalized size = 14. \[ \frac{2 \sqrt{\frac{1-x}{1+\sqrt [3]{-1}}} \sqrt{x^2+x+1} \left (\frac{\sqrt{3} \left (1+\sqrt [3]{-1}\right ) \left (x+\sqrt [3]{-1}\right ) F\left (\sin ^{-1}\left (\sqrt{\frac{1-(-1)^{2/3} x}{1+\sqrt [3]{-1}}}\right )|\sqrt [3]{-1}\right )}{(-1)^{2/3} x-1}+\frac{6 \left (1+i \sqrt{2}\right ) \Pi \left (\frac{2 \sqrt{3}}{-i-2 \sqrt{2}+\sqrt{3}};\sin ^{-1}\left (\sqrt{\frac{1-(-1)^{2/3} x}{1+\sqrt [3]{-1}}}\right )|\sqrt [3]{-1}\right )}{i+2 \sqrt{2}-\sqrt{3}}+\frac{3 \left (1-i \sqrt{2}\right ) \Pi \left (\frac{2 \sqrt{3}}{-i+2 \sqrt{2}+\sqrt{3}};\sin ^{-1}\left (\sqrt{\frac{1-(-1)^{2/3} x}{1+\sqrt [3]{-1}}}\right )|\sqrt [3]{-1}\right )}{(-1)^{5/6}-\sqrt{2}}\right )}{3 \sqrt{1-x^3}} \]
Antiderivative was successfully verified.
[In]
Integrate[(2 + 2*x - x^2)/((2 + x^2)*Sqrt[1 - x^3]),x]
[Out]
(2*Sqrt[(1 - x)/(1 + (-1)^(1/3))]*Sqrt[1 + x + x^2]*((Sqrt[3]*(1 + (-1)^(1/3))*((-1)^(1/3) + x)*EllipticF[ArcS
in[Sqrt[(1 - (-1)^(2/3)*x)/(1 + (-1)^(1/3))]], (-1)^(1/3)])/(-1 + (-1)^(2/3)*x) + (6*(1 + I*Sqrt[2])*EllipticP
i[(2*Sqrt[3])/(-I - 2*Sqrt[2] + Sqrt[3]), ArcSin[Sqrt[(1 - (-1)^(2/3)*x)/(1 + (-1)^(1/3))]], (-1)^(1/3)])/(I +
2*Sqrt[2] - Sqrt[3]) + (3*(1 - I*Sqrt[2])*EllipticPi[(2*Sqrt[3])/(-I + 2*Sqrt[2] + Sqrt[3]), ArcSin[Sqrt[(1 -
(-1)^(2/3)*x)/(1 + (-1)^(1/3))]], (-1)^(1/3)])/((-1)^(5/6) - Sqrt[2])))/(3*Sqrt[1 - x^3])
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Maple [C] time = 0.039, size = 732, normalized size = 36.6 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((-x^2+2*x+2)/(x^2+2)/(-x^3+1)^(1/2),x)
[Out]
2/3*I*3^(1/2)*(I*(x+1/2-1/2*I*3^(1/2))*3^(1/2))^(1/2)*((x-1)/(-3/2+1/2*I*3^(1/2)))^(1/2)*(-I*(x+1/2+1/2*I*3^(1
/2))*3^(1/2))^(1/2)/(-x^3+1)^(1/2)*EllipticF(1/3*3^(1/2)*(I*(x+1/2-1/2*I*3^(1/2))*3^(1/2))^(1/2),(I*3^(1/2)/(-
3/2+1/2*I*3^(1/2)))^(1/2))-2/3*I*3^(1/2)*(I*3^(1/2)*x+1/2*I*3^(1/2)+3/2)^(1/2)*(1/(-3/2+1/2*I*3^(1/2))*x-1/(-3
/2+1/2*I*3^(1/2)))^(1/2)*(-I*3^(1/2)*x-1/2*I*3^(1/2)+3/2)^(1/2)/(-x^3+1)^(1/2)/(-1/2+1/2*I*3^(1/2)-I*2^(1/2))*
EllipticPi(1/3*3^(1/2)*(I*(x+1/2-1/2*I*3^(1/2))*3^(1/2))^(1/2),I*3^(1/2)/(-1/2+1/2*I*3^(1/2)-I*2^(1/2)),(I*3^(
1/2)/(-3/2+1/2*I*3^(1/2)))^(1/2))-2/3*2^(1/2)*3^(1/2)*(I*3^(1/2)*x+1/2*I*3^(1/2)+3/2)^(1/2)*(1/(-3/2+1/2*I*3^(
1/2))*x-1/(-3/2+1/2*I*3^(1/2)))^(1/2)*(-I*3^(1/2)*x-1/2*I*3^(1/2)+3/2)^(1/2)/(-x^3+1)^(1/2)/(-1/2+1/2*I*3^(1/2
)-I*2^(1/2))*EllipticPi(1/3*3^(1/2)*(I*(x+1/2-1/2*I*3^(1/2))*3^(1/2))^(1/2),I*3^(1/2)/(-1/2+1/2*I*3^(1/2)-I*2^
(1/2)),(I*3^(1/2)/(-3/2+1/2*I*3^(1/2)))^(1/2))-2/3*I*3^(1/2)*(I*3^(1/2)*x+1/2*I*3^(1/2)+3/2)^(1/2)*(1/(-3/2+1/
2*I*3^(1/2))*x-1/(-3/2+1/2*I*3^(1/2)))^(1/2)*(-I*3^(1/2)*x-1/2*I*3^(1/2)+3/2)^(1/2)/(-x^3+1)^(1/2)/(-1/2+1/2*I
*3^(1/2)+I*2^(1/2))*EllipticPi(1/3*3^(1/2)*(I*(x+1/2-1/2*I*3^(1/2))*3^(1/2))^(1/2),I*3^(1/2)/(-1/2+1/2*I*3^(1/
2)+I*2^(1/2)),(I*3^(1/2)/(-3/2+1/2*I*3^(1/2)))^(1/2))+2/3*2^(1/2)*3^(1/2)*(I*3^(1/2)*x+1/2*I*3^(1/2)+3/2)^(1/2
)*(1/(-3/2+1/2*I*3^(1/2))*x-1/(-3/2+1/2*I*3^(1/2)))^(1/2)*(-I*3^(1/2)*x-1/2*I*3^(1/2)+3/2)^(1/2)/(-x^3+1)^(1/2
)/(-1/2+1/2*I*3^(1/2)+I*2^(1/2))*EllipticPi(1/3*3^(1/2)*(I*(x+1/2-1/2*I*3^(1/2))*3^(1/2))^(1/2),I*3^(1/2)/(-1/
2+1/2*I*3^(1/2)+I*2^(1/2)),(I*3^(1/2)/(-3/2+1/2*I*3^(1/2)))^(1/2))
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} -\int \frac{x^{2} - 2 \, x - 2}{\sqrt{-x^{3} + 1}{\left (x^{2} + 2\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((-x^2+2*x+2)/(x^2+2)/(-x^3+1)^(1/2),x, algorithm="maxima")
[Out]
-integrate((x^2 - 2*x - 2)/(sqrt(-x^3 + 1)*(x^2 + 2)), x)
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Fricas [A] time = 1.3752, size = 69, normalized size = 3.45 \begin{align*} -\arctan \left (\frac{\sqrt{-x^{3} + 1}{\left (x^{2} + 2 \, x\right )}}{2 \,{\left (x^{3} - 1\right )}}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((-x^2+2*x+2)/(x^2+2)/(-x^3+1)^(1/2),x, algorithm="fricas")
[Out]
-arctan(1/2*sqrt(-x^3 + 1)*(x^2 + 2*x)/(x^3 - 1))
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} - \int - \frac{2 x}{x^{2} \sqrt{1 - x^{3}} + 2 \sqrt{1 - x^{3}}}\, dx - \int \frac{x^{2}}{x^{2} \sqrt{1 - x^{3}} + 2 \sqrt{1 - x^{3}}}\, dx - \int - \frac{2}{x^{2} \sqrt{1 - x^{3}} + 2 \sqrt{1 - x^{3}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((-x**2+2*x+2)/(x**2+2)/(-x**3+1)**(1/2),x)
[Out]
-Integral(-2*x/(x**2*sqrt(1 - x**3) + 2*sqrt(1 - x**3)), x) - Integral(x**2/(x**2*sqrt(1 - x**3) + 2*sqrt(1 -
x**3)), x) - Integral(-2/(x**2*sqrt(1 - x**3) + 2*sqrt(1 - x**3)), x)
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int -\frac{x^{2} - 2 \, x - 2}{\sqrt{-x^{3} + 1}{\left (x^{2} + 2\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((-x^2+2*x+2)/(x^2+2)/(-x^3+1)^(1/2),x, algorithm="giac")
[Out]
integrate(-(x^2 - 2*x - 2)/(sqrt(-x^3 + 1)*(x^2 + 2)), x)