Integrand size = 22, antiderivative size = 168 \[ \int \frac {-3+x}{\sqrt [3]{1-x^2} \left (3+x^2\right )} \, dx=-\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{1-x^2}}{-2^{2/3}-2^{2/3} x+\sqrt [3]{1-x^2}}\right )}{2^{2/3}}-\frac {\log \left (2^{2/3}+2^{2/3} x+2 \sqrt [3]{1-x^2}\right )}{2^{2/3}}+\frac {\log \left (-\sqrt [3]{2}-2 \sqrt [3]{2} x-\sqrt [3]{2} x^2+\left (2^{2/3}+2^{2/3} x\right ) \sqrt [3]{1-x^2}-2 \left (1-x^2\right )^{2/3}\right )}{2\ 2^{2/3}} \] Output:
-1/2*3^(1/2)*arctan(3^(1/2)*(-x^2+1)^(1/3)/(-2^(2/3)-2^(2/3)*x+(-x^2+1)^(1 /3)))*2^(1/3)-1/2*ln(2^(2/3)+2^(2/3)*x+2*(-x^2+1)^(1/3))*2^(1/3)+1/4*ln(-2 ^(1/3)-2*2^(1/3)*x-2^(1/3)*x^2+(2^(2/3)+2^(2/3)*x)*(-x^2+1)^(1/3)-2*(-x^2+ 1)^(2/3))*2^(1/3)
Time = 0.22 (sec) , antiderivative size = 153, normalized size of antiderivative = 0.91 \[ \int \frac {-3+x}{\sqrt [3]{1-x^2} \left (3+x^2\right )} \, dx=\frac {-2 \sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{1-x^2}}{-2^{2/3}-2^{2/3} x+\sqrt [3]{1-x^2}}\right )-2 \log \left (2^{2/3}+2^{2/3} x+2 \sqrt [3]{1-x^2}\right )+\log \left (-\sqrt [3]{2}-2 \sqrt [3]{2} x-\sqrt [3]{2} x^2+2^{2/3} (1+x) \sqrt [3]{1-x^2}-2 \left (1-x^2\right )^{2/3}\right )}{2\ 2^{2/3}} \] Input:
Integrate[(-3 + x)/((1 - x^2)^(1/3)*(3 + x^2)),x]
Output:
(-2*Sqrt[3]*ArcTan[(Sqrt[3]*(1 - x^2)^(1/3))/(-2^(2/3) - 2^(2/3)*x + (1 - x^2)^(1/3))] - 2*Log[2^(2/3) + 2^(2/3)*x + 2*(1 - x^2)^(1/3)] + Log[-2^(1/ 3) - 2*2^(1/3)*x - 2^(1/3)*x^2 + 2^(2/3)*(1 + x)*(1 - x^2)^(1/3) - 2*(1 - x^2)^(2/3)])/(2*2^(2/3))
Time = 0.27 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.57, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {1341}
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.
\(\displaystyle \int \frac {x-3}{\sqrt [3]{1-x^2} \left (x^2+3\right )} \, dx\) |
\(\Big \downarrow \) 1341 |
\(\displaystyle \frac {\sqrt {3} \arctan \left (\frac {1}{\sqrt {3}}-\frac {2^{2/3} (x+1)^{2/3}}{\sqrt {3} \sqrt [3]{1-x}}\right )}{2^{2/3}}+\frac {\log \left (x^2+3\right )}{2\ 2^{2/3}}-\frac {3 \log \left ((x+1)^{2/3}+\sqrt [3]{2} \sqrt [3]{1-x}\right )}{2\ 2^{2/3}}\) |
Input:
Int[(-3 + x)/((1 - x^2)^(1/3)*(3 + x^2)),x]
Output:
(Sqrt[3]*ArcTan[1/Sqrt[3] - (2^(2/3)*(1 + x)^(2/3))/(Sqrt[3]*(1 - x)^(1/3) )])/2^(2/3) + Log[3 + x^2]/(2*2^(2/3)) - (3*Log[2^(1/3)*(1 - x)^(1/3) + (1 + x)^(2/3)])/(2*2^(2/3))
Int[((g_) + (h_.)*(x_))/(((a_) + (c_.)*(x_)^2)^(1/3)*((d_) + (f_.)*(x_)^2)) , x_Symbol] :> Simp[Sqrt[3]*h*(ArcTan[1/Sqrt[3] - 2^(2/3)*((1 - 3*h*(x/g))^ (2/3)/(Sqrt[3]*(1 + 3*h*(x/g))^(1/3)))]/(2^(2/3)*a^(1/3)*f)), x] + (-Simp[3 *h*(Log[(1 - 3*h*(x/g))^(2/3) + 2^(1/3)*(1 + 3*h*(x/g))^(1/3)]/(2^(5/3)*a^( 1/3)*f)), x] + Simp[h*(Log[d + f*x^2]/(2^(5/3)*a^(1/3)*f)), x]) /; FreeQ[{a , c, d, f, g, h}, x] && EqQ[c*d + 3*a*f, 0] && EqQ[c*g^2 + 9*a*h^2, 0] && G tQ[a, 0]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 6.18 (sec) , antiderivative size = 1042, normalized size of antiderivative = 6.20
Input:
int((-3+x)/(-x^2+1)^(1/3)/(x^2+3),x,method=_RETURNVERBOSE)
Output:
RootOf(RootOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2)*ln(-(6*RootOf(RootOf(_ Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2)*RootOf(_Z^3+2)^3*x^2+4*RootOf(RootOf( _Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2)^2*RootOf(_Z^3+2)^2*x^2+18*(-x^2+1)^( 2/3)*RootOf(RootOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2)*RootOf(_Z^3+2)^2+ 18*RootOf(RootOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2)*RootOf(_Z^3+2)^3*x+ 12*RootOf(RootOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2)^2*RootOf(_Z^3+2)^2* x-9*(-x^2+1)^(1/3)*RootOf(_Z^3+2)^2*x-6*(-x^2+1)^(1/3)*RootOf(RootOf(_Z^3+ 2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2)*RootOf(_Z^3+2)*x-9*(-x^2+1)^(1/3)*RootOf( _Z^3+2)^2-6*(-x^2+1)^(1/3)*RootOf(RootOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_ Z^2)*RootOf(_Z^3+2)-3*RootOf(_Z^3+2)*x^2-2*RootOf(RootOf(_Z^3+2)^2+2*_Z*Ro otOf(_Z^3+2)+4*_Z^2)*x^2-12*(-x^2+1)^(2/3)-9*RootOf(_Z^3+2)-6*RootOf(RootO f(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2))/(2*RootOf(RootOf(_Z^3+2)^2+2*_Z*R ootOf(_Z^3+2)+4*_Z^2)*RootOf(_Z^3+2)^2*x-x-3)/(2*RootOf(RootOf(_Z^3+2)^2+2 *_Z*RootOf(_Z^3+2)+4*_Z^2)*RootOf(_Z^3+2)^2*x-x+3))+1/2*RootOf(_Z^3+2)*ln( (2*RootOf(RootOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2)*RootOf(_Z^3+2)^3*x^ 2+12*RootOf(RootOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2)^2*RootOf(_Z^3+2)^ 2*x^2+18*(-x^2+1)^(2/3)*RootOf(RootOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2 )*RootOf(_Z^3+2)^2+6*RootOf(RootOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2)*R ootOf(_Z^3+2)^3*x+36*RootOf(RootOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2)^2 *RootOf(_Z^3+2)^2*x-9*(-x^2+1)^(1/3)*RootOf(_Z^3+2)^2*x-12*(-x^2+1)^(1/...
Leaf count of result is larger than twice the leaf count of optimal. 285 vs. \(2 (128) = 256\).
Time = 3.09 (sec) , antiderivative size = 285, normalized size of antiderivative = 1.70 \[ \int \frac {-3+x}{\sqrt [3]{1-x^2} \left (3+x^2\right )} \, dx=\frac {1}{6} \cdot 4^{\frac {1}{6}} \sqrt {3} \arctan \left (\frac {4^{\frac {1}{6}} \sqrt {3} {\left (12 \cdot 4^{\frac {2}{3}} {\left (x^{4} + 3 \, x^{3} + 3 \, x^{2} + 9 \, x\right )} {\left (-x^{2} + 1\right )}^{\frac {2}{3}} + 4^{\frac {1}{3}} {\left (x^{6} - 18 \, x^{5} - 117 \, x^{4} - 36 \, x^{3} + 207 \, x^{2} + 54 \, x - 27\right )} + 12 \, {\left (x^{5} + 19 \, x^{4} + 42 \, x^{3} + 6 \, x^{2} - 27 \, x - 9\right )} {\left (-x^{2} + 1\right )}^{\frac {1}{3}}\right )}}{6 \, {\left (x^{6} + 54 \, x^{5} + 171 \, x^{4} + 108 \, x^{3} - 81 \, x^{2} - 162 \, x - 27\right )}}\right ) + \frac {1}{24} \cdot 4^{\frac {2}{3}} \log \left (\frac {6 \cdot 4^{\frac {2}{3}} {\left (x^{2} + 3 \, x\right )} {\left (-x^{2} + 1\right )}^{\frac {2}{3}} + 4^{\frac {1}{3}} {\left (x^{4} + 18 \, x^{3} + 24 \, x^{2} - 18 \, x - 9\right )} - 6 \, {\left (x^{3} + 7 \, x^{2} + 3 \, x - 3\right )} {\left (-x^{2} + 1\right )}^{\frac {1}{3}}}{x^{4} + 6 \, x^{2} + 9}\right ) - \frac {1}{12} \cdot 4^{\frac {2}{3}} \log \left (\frac {4^{\frac {2}{3}} {\left (x^{2} + 3\right )} + 6 \cdot 4^{\frac {1}{3}} {\left (-x^{2} + 1\right )}^{\frac {1}{3}} {\left (x + 1\right )} + 12 \, {\left (-x^{2} + 1\right )}^{\frac {2}{3}}}{x^{2} + 3}\right ) \] Input:
integrate((-3+x)/(-x^2+1)^(1/3)/(x^2+3),x, algorithm="fricas")
Output:
1/6*4^(1/6)*sqrt(3)*arctan(1/6*4^(1/6)*sqrt(3)*(12*4^(2/3)*(x^4 + 3*x^3 + 3*x^2 + 9*x)*(-x^2 + 1)^(2/3) + 4^(1/3)*(x^6 - 18*x^5 - 117*x^4 - 36*x^3 + 207*x^2 + 54*x - 27) + 12*(x^5 + 19*x^4 + 42*x^3 + 6*x^2 - 27*x - 9)*(-x^ 2 + 1)^(1/3))/(x^6 + 54*x^5 + 171*x^4 + 108*x^3 - 81*x^2 - 162*x - 27)) + 1/24*4^(2/3)*log((6*4^(2/3)*(x^2 + 3*x)*(-x^2 + 1)^(2/3) + 4^(1/3)*(x^4 + 18*x^3 + 24*x^2 - 18*x - 9) - 6*(x^3 + 7*x^2 + 3*x - 3)*(-x^2 + 1)^(1/3))/ (x^4 + 6*x^2 + 9)) - 1/12*4^(2/3)*log((4^(2/3)*(x^2 + 3) + 6*4^(1/3)*(-x^2 + 1)^(1/3)*(x + 1) + 12*(-x^2 + 1)^(2/3))/(x^2 + 3))
\[ \int \frac {-3+x}{\sqrt [3]{1-x^2} \left (3+x^2\right )} \, dx=\int \frac {x - 3}{\sqrt [3]{- \left (x - 1\right ) \left (x + 1\right )} \left (x^{2} + 3\right )}\, dx \] Input:
integrate((-3+x)/(-x**2+1)**(1/3)/(x**2+3),x)
Output:
Integral((x - 3)/((-(x - 1)*(x + 1))**(1/3)*(x**2 + 3)), x)
\[ \int \frac {-3+x}{\sqrt [3]{1-x^2} \left (3+x^2\right )} \, dx=\int { \frac {x - 3}{{\left (x^{2} + 3\right )} {\left (-x^{2} + 1\right )}^{\frac {1}{3}}} \,d x } \] Input:
integrate((-3+x)/(-x^2+1)^(1/3)/(x^2+3),x, algorithm="maxima")
Output:
integrate((x - 3)/((x^2 + 3)*(-x^2 + 1)^(1/3)), x)
\[ \int \frac {-3+x}{\sqrt [3]{1-x^2} \left (3+x^2\right )} \, dx=\int { \frac {x - 3}{{\left (x^{2} + 3\right )} {\left (-x^{2} + 1\right )}^{\frac {1}{3}}} \,d x } \] Input:
integrate((-3+x)/(-x^2+1)^(1/3)/(x^2+3),x, algorithm="giac")
Output:
integrate((x - 3)/((x^2 + 3)*(-x^2 + 1)^(1/3)), x)
Timed out. \[ \int \frac {-3+x}{\sqrt [3]{1-x^2} \left (3+x^2\right )} \, dx=\int \frac {x-3}{{\left (1-x^2\right )}^{1/3}\,\left (x^2+3\right )} \,d x \] Input:
int((x - 3)/((1 - x^2)^(1/3)*(x^2 + 3)),x)
Output:
int((x - 3)/((1 - x^2)^(1/3)*(x^2 + 3)), x)
\[ \int \frac {-3+x}{\sqrt [3]{1-x^2} \left (3+x^2\right )} \, dx=\int \frac {x}{\left (-x^{2}+1\right )^{\frac {1}{3}} x^{2}+3 \left (-x^{2}+1\right )^{\frac {1}{3}}}d x -3 \left (\int \frac {1}{\left (-x^{2}+1\right )^{\frac {1}{3}} x^{2}+3 \left (-x^{2}+1\right )^{\frac {1}{3}}}d x \right ) \] Input:
int((-3+x)/(-x^2+1)^(1/3)/(x^2+3),x)
Output:
int(x/(( - x**2 + 1)**(1/3)*x**2 + 3*( - x**2 + 1)**(1/3)),x) - 3*int(1/(( - x**2 + 1)**(1/3)*x**2 + 3*( - x**2 + 1)**(1/3)),x)