\(\int \frac {-3+x}{\sqrt [3]{1-x^2} (3+x^2)} \, dx\) [2249]

Optimal result

Integrand size = 22, antiderivative size = 168 \[ \int \frac {-3+x}{\sqrt [3]{1-x^2} \left (3+x^2\right )} \, dx=-\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{1-x^2}}{-2^{2/3}-2^{2/3} x+\sqrt [3]{1-x^2}}\right )}{2^{2/3}}-\frac {\log \left (2^{2/3}+2^{2/3} x+2 \sqrt [3]{1-x^2}\right )}{2^{2/3}}+\frac {\log \left (-\sqrt [3]{2}-2 \sqrt [3]{2} x-\sqrt [3]{2} x^2+\left (2^{2/3}+2^{2/3} x\right ) \sqrt [3]{1-x^2}-2 \left (1-x^2\right )^{2/3}\right )}{2\ 2^{2/3}} \]

[Out]

-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)

Rubi [A] (verified)

Time = 0.01 (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 = {1022} \[ \int \frac {-3+x}{\sqrt [3]{1-x^2} \left (3+x^2\right )} \, dx=\frac {\sqrt {3} \arctan \left (\frac {1}{\sqrt {3}}-\frac {2^{2/3} (x+1)^{2/3}}{\sqrt {3} \sqrt [3]{1-x}}\right )}{2^{2/3}}+\frac {\log \left (x^2+3\right )}{2\ 2^{2/3}}-\frac {3 \log \left ((x+1)^{2/3}+\sqrt [3]{2} \sqrt [3]{1-x}\right )}{2\ 2^{2/3}} \]

[In]

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

[Out]

(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))

Rule 1022

Int[((g_) + (h_.)*(x_))/(((a_) + (c_.)*(x_)^2)^(1/3)*((d_) + (f_.)*(x_)^2)), x_Symbol] :> Simp[Sqrt[3]*h*(ArcT
an[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] + (-S
imp[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] && GtQ[a, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {3} \arctan \left (\frac {1}{\sqrt {3}}-\frac {2^{2/3} (1+x)^{2/3}}{\sqrt {3} \sqrt [3]{1-x}}\right )}{2^{2/3}}+\frac {\log \left (3+x^2\right )}{2\ 2^{2/3}}-\frac {3 \log \left (\sqrt [3]{2} \sqrt [3]{1-x}+(1+x)^{2/3}\right )}{2\ 2^{2/3}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.21 (sec) , antiderivative size = 153, normalized size of antiderivative = 0.91 \[ \int \frac {-3+x}{\sqrt [3]{1-x^2} \left (3+x^2\right )} \, dx=\frac {-2 \sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{1-x^2}}{-2^{2/3}-2^{2/3} x+\sqrt [3]{1-x^2}}\right )-2 \log \left (2^{2/3}+2^{2/3} x+2 \sqrt [3]{1-x^2}\right )+\log \left (-\sqrt [3]{2}-2 \sqrt [3]{2} x-\sqrt [3]{2} x^2+2^{2/3} (1+x) \sqrt [3]{1-x^2}-2 \left (1-x^2\right )^{2/3}\right )}{2\ 2^{2/3}} \]

[In]

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

[Out]

(-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))

Maple [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 7.07 (sec) , antiderivative size = 1553, normalized size of antiderivative = 9.24

method	result	size
trager	\(\text {Expression too large to display}\)	\(1553\)

[In]

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

[Out]

-1/2*ln((2*RootOf(RootOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2)*RootOf(_Z^3+2)^3*x^2-8*RootOf(RootOf(_Z^3+2)^2+
2*_Z*RootOf(_Z^3+2)+4*_Z^2)^2*RootOf(_Z^3+2)^2*x^2+6*RootOf(RootOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2)*RootO
f(_Z^3+2)^3*x-24*RootOf(RootOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2)^2*RootOf(_Z^3+2)^2*x-18*(-x^2+1)^(2/3)*Ro
otOf(RootOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2)*RootOf(_Z^3+2)^2-6*(-x^2+1)^(1/3)*RootOf(_Z^3+2)^2*x-18*(-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-6*(-x^2+1)^(1/3)*RootOf(_Z^3+2
)^2-18*(-x^2+1)^(1/3)*RootOf(RootOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2)*RootOf(_Z^3+2)-RootOf(_Z^3+2)*x^2+4*
RootOf(RootOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2)*x^2+12*(-x^2+1)^(2/3)-3*RootOf(_Z^3+2)+12*RootOf(RootOf(_Z
^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2))/(2*RootOf(RootOf(_Z^3+2)^2+2*_Z*RootOf(_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))*RootOf(_Z^3+2)-ln((2*Roo
tOf(RootOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2)*RootOf(_Z^3+2)^3*x^2-8*RootOf(RootOf(_Z^3+2)^2+2*_Z*RootOf(_Z
^3+2)+4*_Z^2)^2*RootOf(_Z^3+2)^2*x^2+6*RootOf(RootOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2)*RootOf(_Z^3+2)^3*x-
24*RootOf(RootOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2)^2*RootOf(_Z^3+2)^2*x-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*(-x^2+1)^(1/3)*RootOf(_Z^3+2)^2*x-18*(-x^2+1)^(1/3)*Roo
tOf(RootOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2)*RootOf(_Z^3+2)*x-6*(-x^2+1)^(1/3)*RootOf(_Z^3+2)^2-18*(-x^2+1
)^(1/3)*RootOf(RootOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2)*RootOf(_Z^3+2)-RootOf(_Z^3+2)*x^2+4*RootOf(RootOf(
_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2)*x^2+12*(-x^2+1)^(2/3)-3*RootOf(_Z^3+2)+12*RootOf(RootOf(_Z^3+2)^2+2*_Z*R
ootOf(_Z^3+2)+4*_Z^2))/(2*RootOf(RootOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2)*RootOf(_Z^3+2)^2*x-x+3)/(2*RootO
f(RootOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2)*RootOf(_Z^3+2)^2*x-x-3))*RootOf(RootOf(_Z^3+2)^2+2*_Z*RootOf(_Z
^3+2)+4*_Z^2)+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+8*RootOf(RootOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2)^2*RootOf(_Z^3+2)^2*x
^2+18*RootOf(RootOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2)*RootOf(_Z^3+2)^3*x+24*RootOf(RootOf(_Z^3+2)^2+2*_Z*R
ootOf(_Z^3+2)+4*_Z^2)^2*RootOf(_Z^3+2)^2*x-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)^2-3*(-x^2+1)^(1/3)*RootOf(_Z^3+2)^2*x-18*(-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-3*(-x^2+1)^(1/3)*RootOf(_Z^3+2)^2-18*(-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^2-4*RootOf(RootOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4
*_Z^2)*x^2-18*RootOf(_Z^3+2)*x-24*RootOf(RootOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2)*x+6*(-x^2+1)^(2/3)+9*Roo
tOf(_Z^3+2)+12*RootOf(RootOf(_Z^3+2)^2+2*_Z*RootOf(_Z^3+2)+4*_Z^2))/(2*RootOf(RootOf(_Z^3+2)^2+2*_Z*RootOf(_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))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 315 vs. \(2 (128) = 256\).

Time = 3.24 (sec) , antiderivative size = 315, normalized size of antiderivative = 1.88 \[ \int \frac {-3+x}{\sqrt [3]{1-x^2} \left (3+x^2\right )} \, dx=-\frac {1}{6} \cdot 4^{\frac {1}{6}} \sqrt {3} \left (-1\right )^{\frac {1}{3}} \arctan \left (\frac {4^{\frac {1}{6}} \sqrt {3} {\left (12 \cdot 4^{\frac {2}{3}} \left (-1\right )^{\frac {2}{3}} {\left (x^{4} + 3 \, x^{3} + 3 \, x^{2} + 9 \, x\right )} {\left (-x^{2} + 1\right )}^{\frac {2}{3}} - 12 \, \left (-1\right )^{\frac {1}{3}} {\left (x^{5} + 19 \, x^{4} + 42 \, x^{3} + 6 \, x^{2} - 27 \, x - 9\right )} {\left (-x^{2} + 1\right )}^{\frac {1}{3}} + 4^{\frac {1}{3}} {\left (x^{6} - 18 \, x^{5} - 117 \, x^{4} - 36 \, x^{3} + 207 \, x^{2} + 54 \, x - 27\right )}\right )}}{6 \, {\left (x^{6} + 54 \, x^{5} + 171 \, x^{4} + 108 \, x^{3} - 81 \, x^{2} - 162 \, x - 27\right )}}\right ) - \frac {1}{24} \cdot 4^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} \log \left (-\frac {6 \cdot 4^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} {\left (x^{2} + 3 \, x\right )} {\left (-x^{2} + 1\right )}^{\frac {2}{3}} - 4^{\frac {1}{3}} \left (-1\right )^{\frac {2}{3}} {\left (x^{4} + 18 \, x^{3} + 24 \, x^{2} - 18 \, x - 9\right )} + 6 \, {\left (x^{3} + 7 \, x^{2} + 3 \, x - 3\right )} {\left (-x^{2} + 1\right )}^{\frac {1}{3}}}{x^{4} + 6 \, x^{2} + 9}\right ) + \frac {1}{12} \cdot 4^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} \log \left (\frac {6 \cdot 4^{\frac {1}{3}} \left (-1\right )^{\frac {2}{3}} {\left (-x^{2} + 1\right )}^{\frac {1}{3}} {\left (x + 1\right )} - 4^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} {\left (x^{2} + 3\right )} + 12 \, {\left (-x^{2} + 1\right )}^{\frac {2}{3}}}{x^{2} + 3}\right ) \]

[In]

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

[Out]

-1/6*4^(1/6)*sqrt(3)*(-1)^(1/3)*arctan(1/6*4^(1/6)*sqrt(3)*(12*4^(2/3)*(-1)^(2/3)*(x^4 + 3*x^3 + 3*x^2 + 9*x)*
(-x^2 + 1)^(2/3) - 12*(-1)^(1/3)*(x^5 + 19*x^4 + 42*x^3 + 6*x^2 - 27*x - 9)*(-x^2 + 1)^(1/3) + 4^(1/3)*(x^6 -
18*x^5 - 117*x^4 - 36*x^3 + 207*x^2 + 54*x - 27))/(x^6 + 54*x^5 + 171*x^4 + 108*x^3 - 81*x^2 - 162*x - 27)) -
1/24*4^(2/3)*(-1)^(1/3)*log(-(6*4^(2/3)*(-1)^(1/3)*(x^2 + 3*x)*(-x^2 + 1)^(2/3) - 4^(1/3)*(-1)^(2/3)*(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)*(-1)
^(1/3)*log((6*4^(1/3)*(-1)^(2/3)*(-x^2 + 1)^(1/3)*(x + 1) - 4^(2/3)*(-1)^(1/3)*(x^2 + 3) + 12*(-x^2 + 1)^(2/3)
)/(x^2 + 3))

Sympy [F]

\[ \int \frac {-3+x}{\sqrt [3]{1-x^2} \left (3+x^2\right )} \, dx=\int \frac {x - 3}{\sqrt [3]{- \left (x - 1\right ) \left (x + 1\right )} \left (x^{2} + 3\right )}\, dx \]

[In]

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

[Out]

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

Maxima [F]

\[ \int \frac {-3+x}{\sqrt [3]{1-x^2} \left (3+x^2\right )} \, dx=\int { \frac {x - 3}{{\left (x^{2} + 3\right )} {\left (-x^{2} + 1\right )}^{\frac {1}{3}}} \,d x } \]

[In]

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

[Out]

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

Giac [F]

\[ \int \frac {-3+x}{\sqrt [3]{1-x^2} \left (3+x^2\right )} \, dx=\int { \frac {x - 3}{{\left (x^{2} + 3\right )} {\left (-x^{2} + 1\right )}^{\frac {1}{3}}} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {-3+x}{\sqrt [3]{1-x^2} \left (3+x^2\right )} \, dx=\int \frac {x-3}{{\left (1-x^2\right )}^{1/3}\,\left (x^2+3\right )} \,d x \]

[In]

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

[Out]

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