\(\int \frac {-1+x}{(2+x) \sqrt {-1+x^3}} \, dx\) [191]

Optimal result

Integrand size = 18, antiderivative size = 21 \[ \int \frac {-1+x}{(2+x) \sqrt {-1+x^3}} \, dx=-\frac {2}{3} \arctan \left (\frac {3 \sqrt {-1+x^3}}{(-1+x)^2}\right ) \]

[Out]

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

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.19, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {2163, 209} \[ \int \frac {-1+x}{(2+x) \sqrt {-1+x^3}} \, dx=\frac {2}{3} \arctan \left (\frac {(1-x)^2}{3 \sqrt {x^3-1}}\right ) \]

[In]

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

[Out]

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

Rule 209

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

Rule 2163

Int[((e_) + (f_.)*(x_))/(((c_) + (d_.)*(x_))*Sqrt[(a_) + (b_.)*(x_)^3]), x_Symbol] :> Dist[-2*(e/d), Subst[Int
[1/(9 - a*x^2), x], x, (1 + f*(x/e))^2/Sqrt[a + b*x^3]], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[d*e - c*f,
0] && EqQ[b*c^3 + 8*a*d^3, 0] && EqQ[2*d*e + c*f, 0]

Rubi steps \begin{align*} \text {integral}& = 2 \text {Subst}\left (\int \frac {1}{9+x^2} \, dx,x,\frac {(1-x)^2}{\sqrt {-1+x^3}}\right ) \\ & = \frac {2}{3} \arctan \left (\frac {(1-x)^2}{3 \sqrt {-1+x^3}}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.93 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00 \[ \int \frac {-1+x}{(2+x) \sqrt {-1+x^3}} \, dx=-\frac {2}{3} \arctan \left (\frac {3 \sqrt {-1+x^3}}{(-1+x)^2}\right ) \]

[In]

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

[Out]

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

Maple [C] (verified)

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

Time = 2.83 (sec) , antiderivative size = 74, normalized size of antiderivative = 3.52

method	result	size
trager	\(\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{3}-12 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{2}-6 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x -6 \sqrt {x^{3}-1}\, x -10 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )+6 \sqrt {x^{3}-1}}{\left (x +2\right )^{3}}\right )}{3}\)	\(74\)
default	\(\frac {2 \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {x -1}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x -1}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \sqrt {\frac {\frac {3}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {x^{3}-1}}-\frac {2 \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {x -1}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \operatorname {EllipticPi}\left (\sqrt {\frac {x -1}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \frac {i \sqrt {3}}{6}+\frac {1}{2}, \sqrt {\frac {\frac {3}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {x^{3}-1}}\)	\(240\)
elliptic	\(\frac {2 \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {x -1}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x -1}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \sqrt {\frac {\frac {3}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {x^{3}-1}}-\frac {2 \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {x -1}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \operatorname {EllipticPi}\left (\sqrt {\frac {x -1}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \frac {i \sqrt {3}}{6}+\frac {1}{2}, \sqrt {\frac {\frac {3}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {x^{3}-1}}\)	\(240\)

[In]

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

[Out]

1/3*RootOf(_Z^2+1)*ln((RootOf(_Z^2+1)*x^3-12*RootOf(_Z^2+1)*x^2-6*RootOf(_Z^2+1)*x-6*(x^3-1)^(1/2)*x-10*RootOf
(_Z^2+1)+6*(x^3-1)^(1/2))/(x+2)^3)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 40 vs. \(2 (17) = 34\).

Time = 0.26 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.90 \[ \int \frac {-1+x}{(2+x) \sqrt {-1+x^3}} \, dx=\frac {1}{3} \, \arctan \left (\frac {{\left (x^{3} - 12 \, x^{2} - 6 \, x - 10\right )} \sqrt {x^{3} - 1}}{6 \, {\left (x^{4} - x^{3} - x + 1\right )}}\right ) \]

[In]

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

[Out]

1/3*arctan(1/6*(x^3 - 12*x^2 - 6*x - 10)*sqrt(x^3 - 1)/(x^4 - x^3 - x + 1))

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 5.54 (sec) , antiderivative size = 206, normalized size of antiderivative = 9.81 \[ \int \frac {-1+x}{(2+x) \sqrt {-1+x^3}} \, dx=-\frac {\left (3+\sqrt {3}\,1{}\mathrm {i}\right )\,\sqrt {-\frac {x+\frac {1}{2}-\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {\frac {x+\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\left (\mathrm {F}\left (\mathrm {asin}\left (\sqrt {-\frac {x-1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\right )\middle |-\frac {\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}\right )-\Pi \left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6};\mathrm {asin}\left (\sqrt {-\frac {x-1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\right )\middle |-\frac {\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}\right )\right )\,\sqrt {-\frac {x-1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}}{\sqrt {x^3+\left (-\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )-1\right )\,x+\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}} \]

[In]

int((x - 1)/((x^3 - 1)^(1/2)*(x + 2)),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)*(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)/6 + 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)))*(-(x - 1)/((3^(1/2)*1i)/2 + 3/2))^(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)