\(\int \frac {-2-2 x+x^2}{(3-x+x^2) \sqrt {-1+x^3}} \, dx\) [373]

Optimal result

Integrand size = 28, antiderivative size = 31 \[ \int \frac {-2-2 x+x^2}{\left (3-x+x^2\right ) \sqrt {-1+x^3}} \, dx=-\sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {-1+x^3}}{1+x+x^2}\right ) \]

[Out]

-2^(1/2)*arctanh(2^(1/2)*(x^3-1)^(1/2)/(x^2+x+1))

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.87, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {2170, 212} \[ \int \frac {-2-2 x+x^2}{\left (3-x+x^2\right ) \sqrt {-1+x^3}} \, dx=\sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} (1-x)}{\sqrt {x^3-1}}\right ) \]

[In]

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

[Out]

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

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 2170

Int[((f_) + (g_.)*(x_) + (h_.)*(x_)^2)/(((c_) + (d_.)*(x_) + (e_.)*(x_)^2)*Sqrt[(a_) + (b_.)*(x_)^3]), x_Symbo
l] :> Dist[-2*g*h, Subst[Int[1/(2*e*h - (b*d*f - 2*a*e*h)*x^2), x], x, (1 + 2*h*(x/g))/Sqrt[a + b*x^3]], x] /;
FreeQ[{a, b, c, d, e, f, g, h}, x] && NeQ[b*d*f - 2*a*e*h, 0] && EqQ[b*g^3 - 8*a*h^3, 0] && EqQ[g^2 + 2*f*h,
0] && EqQ[b*d*f + b*c*g - 4*a*e*h, 0]

Rubi steps \begin{align*} \text {integral}& = 4 \text {Subst}\left (\int \frac {1}{2-4 x^2} \, dx,x,\frac {1-x}{\sqrt {-1+x^3}}\right ) \\ & = \sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} (1-x)}{\sqrt {-1+x^3}}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 1.21 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00 \[ \int \frac {-2-2 x+x^2}{\left (3-x+x^2\right ) \sqrt {-1+x^3}} \, dx=-\sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {-1+x^3}}{1+x+x^2}\right ) \]

[In]

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

[Out]

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

Maple [C] (verified)

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

Time = 6.19 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.84

method	result	size
trager	\(\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) \ln \left (\frac {-\operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) x^{2}-3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) x +4 \sqrt {x^{3}-1}+\operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right )}{x^{2}-x +3}\right )}{2}\)	\(57\)
default	\(\frac {2 \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {x -1}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x -1}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \sqrt {\frac {\frac {3}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {x^{3}-1}}-\frac {2 \left (\frac {1}{2}-\frac {i \sqrt {11}}{2}\right ) \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {x -1}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \left (\frac {1}{2}+\frac {i \sqrt {11}}{2}\right ) \operatorname {EllipticPi}\left (\sqrt {\frac {x -1}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \frac {1}{4}+\frac {i \sqrt {11}}{4}+\frac {i \left (\frac {1}{2}+\frac {i \sqrt {11}}{2}\right ) \sqrt {3}}{6}, \sqrt {\frac {\frac {3}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{3 \sqrt {x^{3}-1}}-\frac {2 \left (\frac {1}{2}+\frac {i \sqrt {11}}{2}\right ) \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {x -1}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \left (\frac {1}{2}-\frac {i \sqrt {11}}{2}\right ) \operatorname {EllipticPi}\left (\sqrt {\frac {x -1}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \frac {1}{4}-\frac {i \sqrt {11}}{4}+\frac {i \left (\frac {1}{2}-\frac {i \sqrt {11}}{2}\right ) \sqrt {3}}{6}, \sqrt {\frac {\frac {3}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{3 \sqrt {x^{3}-1}}\)	\(423\)
elliptic	\(\frac {2 \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {x -1}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x -1}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \sqrt {\frac {\frac {3}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {x^{3}-1}}+\frac {2 \left (-\frac {1}{2}+\frac {i \sqrt {11}}{2}\right ) \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {x -1}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \left (\frac {1}{2}+\frac {i \sqrt {11}}{2}\right ) \operatorname {EllipticPi}\left (\sqrt {\frac {x -1}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \frac {1}{4}+\frac {i \sqrt {11}}{4}+\frac {i \left (\frac {1}{2}+\frac {i \sqrt {11}}{2}\right ) \sqrt {3}}{6}, \sqrt {\frac {\frac {3}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{3 \sqrt {x^{3}-1}}+\frac {2 \left (-\frac {1}{2}-\frac {i \sqrt {11}}{2}\right ) \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {x -1}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \left (\frac {1}{2}-\frac {i \sqrt {11}}{2}\right ) \operatorname {EllipticPi}\left (\sqrt {\frac {x -1}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \frac {1}{4}-\frac {i \sqrt {11}}{4}+\frac {i \left (\frac {1}{2}-\frac {i \sqrt {11}}{2}\right ) \sqrt {3}}{6}, \sqrt {\frac {\frac {3}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{3 \sqrt {x^{3}-1}}\)	\(423\)

[In]

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

[Out]

1/2*RootOf(_Z^2-2)*ln((-RootOf(_Z^2-2)*x^2-3*RootOf(_Z^2-2)*x+4*(x^3-1)^(1/2)+RootOf(_Z^2-2))/(x^2-x+3))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 65 vs. \(2 (25) = 50\).

Time = 0.25 (sec) , antiderivative size = 65, normalized size of antiderivative = 2.10 \[ \int \frac {-2-2 x+x^2}{\left (3-x+x^2\right ) \sqrt {-1+x^3}} \, dx=\frac {1}{4} \, \sqrt {2} \log \left (\frac {x^{4} + 14 \, x^{3} - 4 \, \sqrt {2} \sqrt {x^{3} - 1} {\left (x^{2} + 3 \, x - 1\right )} + 7 \, x^{2} - 6 \, x - 7}{x^{4} - 2 \, x^{3} + 7 \, x^{2} - 6 \, x + 9}\right ) \]

[In]

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

[Out]

1/4*sqrt(2)*log((x^4 + 14*x^3 - 4*sqrt(2)*sqrt(x^3 - 1)*(x^2 + 3*x - 1) + 7*x^2 - 6*x - 7)/(x^4 - 2*x^3 + 7*x^
2 - 6*x + 9))

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 5.58 (sec) , antiderivative size = 275, normalized size of antiderivative = 8.87 \[ \int \frac {-2-2 x+x^2}{\left (3-x+x^2\right ) \sqrt {-1+x^3}} \, dx=\frac {\left (3+\sqrt {3}\,1{}\mathrm {i}\right )\,\sqrt {-\frac {x+\frac {1}{2}-\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {\frac {x+\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {-\frac {x-1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\left (-\mathrm {F}\left (\mathrm {asin}\left (\sqrt {-\frac {x-1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\right )\middle |-\frac {\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}\right )+\Pi \left (\frac {\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{\frac {1}{2}+\frac {\sqrt {11}\,1{}\mathrm {i}}{2}};\mathrm {asin}\left (\sqrt {-\frac {x-1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\right )\middle |-\frac {\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}\right )+\Pi \left (-\frac {\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {1}{2}+\frac {\sqrt {11}\,1{}\mathrm {i}}{2}};\mathrm {asin}\left (\sqrt {-\frac {x-1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\right )\middle |-\frac {\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}\right )\right )}{\sqrt {x^3+\left (-\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )-1\right )\,x+\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}} \]

[In]

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

[Out]

((3^(1/2)*1i + 3)*(-(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)/((11^(1/2)
*1i)/2 + 1/2), 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)) + e
llipticPi(-((3^(1/2)*1i)/2 + 3/2)/((11^(1/2)*1i)/2 - 1/2), 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^(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)