Integrand size = 22, antiderivative size = 27 \[ \int \frac {1-x}{(2+x) \sqrt {1-x^3}} \, dx=-\frac {2}{3} \text {arctanh}\left (\frac {(1-x)^2}{3 \sqrt {1-x^3}}\right ) \] Output:
-2/3*arctanh(1/3*(1-x)^2/(-x^3+1)^(1/2))
Time = 1.12 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.22 \[ \int \frac {1-x}{(2+x) \sqrt {1-x^3}} \, dx=-\frac {2}{3} \text {arctanh}\left (\frac {\frac {1}{3}-\frac {2 x}{3}+\frac {x^2}{3}}{\sqrt {1-x^3}}\right ) \] Input:
Integrate[(1 - x)/((2 + x)*Sqrt[1 - x^3]),x]
Output:
(-2*ArcTanh[(1/3 - (2*x)/3 + x^2/3)/Sqrt[1 - x^3]])/3
Time = 0.34 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {2563, 219}
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.
\(\displaystyle \int \frac {1-x}{(x+2) \sqrt {1-x^3}} \, dx\) |
\(\Big \downarrow \) 2563 |
\(\displaystyle -2 \int \frac {1}{9-\frac {(1-x)^4}{1-x^3}}d\frac {(1-x)^2}{\sqrt {1-x^3}}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle -\frac {2}{3} \text {arctanh}\left (\frac {(1-x)^2}{3 \sqrt {1-x^3}}\right )\) |
Input:
Int[(1 - x)/((2 + x)*Sqrt[1 - x^3]),x]
Output:
(-2*ArcTanh[(1 - x)^2/(3*Sqrt[1 - x^3])])/3
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] && (Gt Q[a, 0] || LtQ[b, 0])
Int[((e_) + (f_.)*(x_))/(((c_) + (d_.)*(x_))*Sqrt[(a_) + (b_.)*(x_)^3]), x_ Symbol] :> Simp[-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]
Leaf count of result is larger than twice the leaf count of optimal. \(48\) vs. \(2(21)=42\).
Time = 0.46 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.81
method | result | size |
trager | \(\frac {\ln \left (-\frac {-x^{3}+6 \sqrt {-x^{3}+1}\, x +12 x^{2}-6 \sqrt {-x^{3}+1}+6 x +10}{\left (2+x \right )^{3}}\right )}{3}\) | \(49\) |
default | \(\frac {2 i \sqrt {3}\, \sqrt {i \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {\frac {x -1}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {-i \left (x +\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \operatorname {EllipticF}\left (\frac {\sqrt {3}\, \sqrt {i \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \sqrt {\frac {i \sqrt {3}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\right )}{3 \sqrt {-x^{3}+1}}-\frac {2 i \sqrt {3}\, \sqrt {i \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {\frac {x -1}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {-i \left (x +\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \operatorname {EllipticPi}\left (\frac {\sqrt {3}\, \sqrt {i \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \frac {i \sqrt {3}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}, \sqrt {\frac {i \sqrt {3}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {-x^{3}+1}\, \left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right )}\) | \(240\) |
elliptic | \(\frac {2 i \sqrt {3}\, \sqrt {i \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {\frac {x -1}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {-i \left (x +\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \operatorname {EllipticF}\left (\frac {\sqrt {3}\, \sqrt {i \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \sqrt {\frac {i \sqrt {3}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\right )}{3 \sqrt {-x^{3}+1}}-\frac {2 i \sqrt {3}\, \sqrt {i \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \sqrt {\frac {x -1}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {-i \left (x +\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}\, \operatorname {EllipticPi}\left (\frac {\sqrt {3}\, \sqrt {i \left (x +\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {3}}}{3}, \frac {i \sqrt {3}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}, \sqrt {\frac {i \sqrt {3}}{-\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {-x^{3}+1}\, \left (\frac {3}{2}+\frac {i \sqrt {3}}{2}\right )}\) | \(240\) |
Input:
int((1-x)/(2+x)/(-x^3+1)^(1/2),x,method=_RETURNVERBOSE)
Output:
1/3*ln(-(-x^3+6*(-x^3+1)^(1/2)*x+12*x^2-6*(-x^3+1)^(1/2)+6*x+10)/(2+x)^3)
Leaf count of result is larger than twice the leaf count of optimal. 47 vs. \(2 (19) = 38\).
Time = 0.12 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.74 \[ \int \frac {1-x}{(2+x) \sqrt {1-x^3}} \, dx=\frac {1}{3} \, \log \left (-\frac {x^{3} - 12 \, x^{2} - 6 \, \sqrt {-x^{3} + 1} {\left (x - 1\right )} - 6 \, x - 10}{x^{3} + 6 \, x^{2} + 12 \, x + 8}\right ) \] Input:
integrate((1-x)/(2+x)/(-x^3+1)^(1/2),x, algorithm="fricas")
Output:
1/3*log(-(x^3 - 12*x^2 - 6*sqrt(-x^3 + 1)*(x - 1) - 6*x - 10)/(x^3 + 6*x^2 + 12*x + 8))
\[ \int \frac {1-x}{(2+x) \sqrt {1-x^3}} \, dx=- \int \frac {x}{x \sqrt {1 - x^{3}} + 2 \sqrt {1 - x^{3}}}\, dx - \int \left (- \frac {1}{x \sqrt {1 - x^{3}} + 2 \sqrt {1 - x^{3}}}\right )\, dx \] Input:
integrate((1-x)/(2+x)/(-x**3+1)**(1/2),x)
Output:
-Integral(x/(x*sqrt(1 - x**3) + 2*sqrt(1 - x**3)), x) - Integral(-1/(x*sqr t(1 - x**3) + 2*sqrt(1 - x**3)), x)
\[ \int \frac {1-x}{(2+x) \sqrt {1-x^3}} \, dx=\int { -\frac {x - 1}{\sqrt {-x^{3} + 1} {\left (x + 2\right )}} \,d x } \] Input:
integrate((1-x)/(2+x)/(-x^3+1)^(1/2),x, algorithm="maxima")
Output:
-integrate((x - 1)/(sqrt(-x^3 + 1)*(x + 2)), x)
\[ \int \frac {1-x}{(2+x) \sqrt {1-x^3}} \, dx=\int { -\frac {x - 1}{\sqrt {-x^{3} + 1} {\left (x + 2\right )}} \,d x } \] Input:
integrate((1-x)/(2+x)/(-x^3+1)^(1/2),x, algorithm="giac")
Output:
integrate(-(x - 1)/(sqrt(-x^3 + 1)*(x + 2)), x)
Time = 21.87 (sec) , antiderivative size = 221, normalized size of antiderivative = 8.19 \[ \int \frac {1-x}{(2+x) \sqrt {1-x^3}} \, dx=\frac {\left (3+\sqrt {3}\,1{}\mathrm {i}\right )\,\sqrt {x^3-1}\,\sqrt {-\frac {x+\frac {1}{2}-\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {\frac {x+\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\left (\mathrm {F}\left (\mathrm {asin}\left (\sqrt {-\frac {x-1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\right )\middle |-\frac {\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}\right )-\Pi \left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6};\mathrm {asin}\left (\sqrt {-\frac {x-1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\right )\middle |-\frac {\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}\right )\right )\,\sqrt {-\frac {x-1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}}{\sqrt {1-x^3}\,\sqrt {x^3+\left (-\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )-1\right )\,x+\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}} \] Input:
int(-(x - 1)/((1 - x^3)^(1/2)*(x + 2)),x)
Output:
((3^(1/2)*1i + 3)*(x^3 - 1)^(1/2)*(-(x - (3^(1/2)*1i)/2 + 1/2)/((3^(1/2)*1 i)/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))/((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))
\[ \int \frac {1-x}{(2+x) \sqrt {1-x^3}} \, dx=\int \frac {\sqrt {-x^{3}+1}}{x^{3}+3 x^{2}+3 x +2}d x \] Input:
int((1-x)/(2+x)/(-x^3+1)^(1/2),x)
Output:
int(sqrt( - x**3 + 1)/(x**3 + 3*x**2 + 3*x + 2),x)