Integrand size = 29, antiderivative size = 20 \[ \int \frac {2+2 x-x^2}{\left (2+x^2\right ) \sqrt {1-x^3}} \, dx=-2 \arctan \left (\frac {1-x}{\sqrt {1-x^3}}\right ) \]
Time = 1.11 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.15 \[ \int \frac {2+2 x-x^2}{\left (2+x^2\right ) \sqrt {1-x^3}} \, dx=-2 \arctan \left (\frac {\sqrt {1-x^3}}{1+x+x^2}\right ) \]
Time = 0.21 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {2571, 216}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {-x^2+2 x+2}{\left (x^2+2\right ) \sqrt {1-x^3}} \, dx\) |
| \(\Big \downarrow \) 2571 |
| \(\displaystyle -2 \int \frac {1}{\frac {(1-x)^2}{1-x^3}+1}d\frac {1-x}{\sqrt {1-x^3}}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle -2 \arctan \left (\frac {1-x}{\sqrt {1-x^3}}\right )\) |
3.2.99.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
Int[((f_) + (g_.)*(x_) + (h_.)*(x_)^2)/(((c_) + (e_.)*(x_)^2)*Sqrt[(a_) + ( b_.)*(x_)^3]), x_Symbol] :> Simp[-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]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 1.56 (sec) , antiderivative size = 49, normalized size of antiderivative = 2.45
| method | result | size |
| trager | \(\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{2}+2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x -2 \sqrt {-x^{3}+1}}{x^{2}+2}\right )\) | \(49\) |
| default | \(\frac {2 i \sqrt {3}\, \sqrt {i \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {\frac {x -1}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {-i \left (x +\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, F\left (\frac {\sqrt {3}\, \sqrt {i \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \sqrt {\frac {i \sqrt {3}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\right )}{3 \sqrt {-x^{3}+1}}-\frac {2 i \sqrt {3}\, \sqrt {i \sqrt {3}\, x +\frac {i \sqrt {3}}{2}+\frac {3}{2}}\, \sqrt {\frac {x}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}-\frac {1}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {-i \sqrt {3}\, x -\frac {i \sqrt {3}}{2}+\frac {3}{2}}\, \Pi \left (\frac {\sqrt {3}\, \sqrt {i \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \frac {i \sqrt {3}}{-i \sqrt {2}-\frac {1}{2}+\frac {i \sqrt {3}}{2}}, \sqrt {\frac {i \sqrt {3}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\right )}{3 \sqrt {-x^{3}+1}\, \left (-i \sqrt {2}-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right )}-\frac {2 \sqrt {2}\, \sqrt {3}\, \sqrt {i \sqrt {3}\, x +\frac {i \sqrt {3}}{2}+\frac {3}{2}}\, \sqrt {\frac {x}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}-\frac {1}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {-i \sqrt {3}\, x -\frac {i \sqrt {3}}{2}+\frac {3}{2}}\, \Pi \left (\frac {\sqrt {3}\, \sqrt {i \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \frac {i \sqrt {3}}{-i \sqrt {2}-\frac {1}{2}+\frac {i \sqrt {3}}{2}}, \sqrt {\frac {i \sqrt {3}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\right )}{3 \sqrt {-x^{3}+1}\, \left (-i \sqrt {2}-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right )}+\frac {2 \sqrt {2}\, \sqrt {3}\, \sqrt {i \sqrt {3}\, x +\frac {i \sqrt {3}}{2}+\frac {3}{2}}\, \sqrt {\frac {x}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}-\frac {1}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {-i \sqrt {3}\, x -\frac {i \sqrt {3}}{2}+\frac {3}{2}}\, \Pi \left (\frac {\sqrt {3}\, \sqrt {i \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \frac {i \sqrt {3}}{i \sqrt {2}-\frac {1}{2}+\frac {i \sqrt {3}}{2}}, \sqrt {\frac {i \sqrt {3}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\right )}{3 \sqrt {-x^{3}+1}\, \left (i \sqrt {2}-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right )}-\frac {2 i \sqrt {3}\, \sqrt {i \sqrt {3}\, x +\frac {i \sqrt {3}}{2}+\frac {3}{2}}\, \sqrt {\frac {x}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}-\frac {1}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {-i \sqrt {3}\, x -\frac {i \sqrt {3}}{2}+\frac {3}{2}}\, \Pi \left (\frac {\sqrt {3}\, \sqrt {i \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \frac {i \sqrt {3}}{i \sqrt {2}-\frac {1}{2}+\frac {i \sqrt {3}}{2}}, \sqrt {\frac {i \sqrt {3}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\right )}{3 \sqrt {-x^{3}+1}\, \left (i \sqrt {2}-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right )}\) | \(732\) |
| elliptic | \(\frac {2 i \sqrt {3}\, \sqrt {i \sqrt {3}\, x +\frac {i \sqrt {3}}{2}+\frac {3}{2}}\, \sqrt {\frac {x}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}-\frac {1}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {-i \sqrt {3}\, x -\frac {i \sqrt {3}}{2}+\frac {3}{2}}\, F\left (\frac {\sqrt {3}\, \sqrt {i \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \sqrt {\frac {i \sqrt {3}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\right )}{3 \sqrt {-x^{3}+1}}-\frac {2 \sqrt {2}\, \sqrt {3}\, \sqrt {i \sqrt {3}\, x +\frac {i \sqrt {3}}{2}+\frac {3}{2}}\, \sqrt {\frac {x}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}-\frac {1}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {-i \sqrt {3}\, x -\frac {i \sqrt {3}}{2}+\frac {3}{2}}\, \Pi \left (\frac {\sqrt {3}\, \sqrt {i \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \frac {i \sqrt {3}}{-i \sqrt {2}-\frac {1}{2}+\frac {i \sqrt {3}}{2}}, \sqrt {\frac {i \sqrt {3}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\right )}{3 \sqrt {-x^{3}+1}\, \left (-i \sqrt {2}-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right )}-\frac {2 i \sqrt {3}\, \sqrt {i \sqrt {3}\, x +\frac {i \sqrt {3}}{2}+\frac {3}{2}}\, \sqrt {\frac {x}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}-\frac {1}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {-i \sqrt {3}\, x -\frac {i \sqrt {3}}{2}+\frac {3}{2}}\, \Pi \left (\frac {\sqrt {3}\, \sqrt {i \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \frac {i \sqrt {3}}{-i \sqrt {2}-\frac {1}{2}+\frac {i \sqrt {3}}{2}}, \sqrt {\frac {i \sqrt {3}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\right )}{3 \sqrt {-x^{3}+1}\, \left (-i \sqrt {2}-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right )}+\frac {2 \sqrt {2}\, \sqrt {3}\, \sqrt {i \sqrt {3}\, x +\frac {i \sqrt {3}}{2}+\frac {3}{2}}\, \sqrt {\frac {x}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}-\frac {1}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {-i \sqrt {3}\, x -\frac {i \sqrt {3}}{2}+\frac {3}{2}}\, \Pi \left (\frac {\sqrt {3}\, \sqrt {i \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \frac {i \sqrt {3}}{i \sqrt {2}-\frac {1}{2}+\frac {i \sqrt {3}}{2}}, \sqrt {\frac {i \sqrt {3}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\right )}{3 \sqrt {-x^{3}+1}\, \left (i \sqrt {2}-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right )}-\frac {2 i \sqrt {3}\, \sqrt {i \sqrt {3}\, x +\frac {i \sqrt {3}}{2}+\frac {3}{2}}\, \sqrt {\frac {x}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}-\frac {1}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {-i \sqrt {3}\, x -\frac {i \sqrt {3}}{2}+\frac {3}{2}}\, \Pi \left (\frac {\sqrt {3}\, \sqrt {i \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \frac {i \sqrt {3}}{i \sqrt {2}-\frac {1}{2}+\frac {i \sqrt {3}}{2}}, \sqrt {\frac {i \sqrt {3}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\right )}{3 \sqrt {-x^{3}+1}\, \left (i \sqrt {2}-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right )}\) | \(743\) |
Time = 0.27 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.40 \[ \int \frac {2+2 x-x^2}{\left (2+x^2\right ) \sqrt {1-x^3}} \, dx=-\arctan \left (\frac {\sqrt {-x^{3} + 1} {\left (x^{2} + 2 \, x\right )}}{2 \, {\left (x^{3} - 1\right )}}\right ) \]
\[ \int \frac {2+2 x-x^2}{\left (2+x^2\right ) \sqrt {1-x^3}} \, dx=- \int \left (- \frac {2 x}{x^{2} \sqrt {1 - x^{3}} + 2 \sqrt {1 - x^{3}}}\right )\, dx - \int \frac {x^{2}}{x^{2} \sqrt {1 - x^{3}} + 2 \sqrt {1 - x^{3}}}\, dx - \int \left (- \frac {2}{x^{2} \sqrt {1 - x^{3}} + 2 \sqrt {1 - x^{3}}}\right )\, dx \]
-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)
\[ \int \frac {2+2 x-x^2}{\left (2+x^2\right ) \sqrt {1-x^3}} \, dx=\int { -\frac {x^{2} - 2 \, x - 2}{\sqrt {-x^{3} + 1} {\left (x^{2} + 2\right )}} \,d x } \]
\[ \int \frac {2+2 x-x^2}{\left (2+x^2\right ) \sqrt {1-x^3}} \, dx=\int { -\frac {x^{2} - 2 \, x - 2}{\sqrt {-x^{3} + 1} {\left (x^{2} + 2\right )}} \,d x } \]
Time = 19.53 (sec) , antiderivative size = 292, normalized size of antiderivative = 14.60 \[ \int \frac {2+2 x-x^2}{\left (2+x^2\right ) \sqrt {1-x^3}} \, dx=-\frac {\left (3+\sqrt {3}\,1{}\mathrm {i}\right )\,\sqrt {x^3-1}\,\sqrt {-\frac {x+\frac {1}{2}-\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {\frac {x+\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {-\frac {x-1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\left (-\mathrm {F}\left (\mathrm {asin}\left (\sqrt {-\frac {x-1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\right )\middle |-\frac {\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}\right )+\Pi \left (\frac {\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{1+\sqrt {2}\,1{}\mathrm {i}};\mathrm {asin}\left (\sqrt {-\frac {x-1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\right )\middle |-\frac {\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}\right )+\Pi \left (-\frac {\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-1+\sqrt {2}\,1{}\mathrm {i}};\mathrm {asin}\left (\sqrt {-\frac {x-1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\right )\middle |-\frac {\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}\right )\right )}{\sqrt {1-x^3}\,\sqrt {x^3+\left (-\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )-1\right )\,x+\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}} \]
-((3^(1/2)*1i + 3)*(x^3 - 1)^(1/2)*(-(x - (3^(1/2)*1i)/2 + 1/2)/((3^(1/2)* 1i)/2 - 3/2))^(1/2)*((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)*(ellipticPi(((3^(1/2)*1i)/2 + 3/2)/(2^(1/2)*1i + 1), 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)) - 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 )) + ellipticPi(-((3^(1/2)*1i)/2 + 3/2)/(2^(1/2)*1i - 1), 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))))/((1 - x^3)^(1/2)*(((3^(1/2)*1i)/2 - 1/2)*((3^(1/2)*1i)/2 + 1/2) - x *(((3^(1/2)*1i)/2 - 1/2)*((3^(1/2)*1i)/2 + 1/2) + 1) + x^3)^(1/2))