Integrand size = 18, antiderivative size = 21 \[ \int \frac {-1+x}{(2+x) \sqrt {-1+x^3}} \, dx=-\frac {2}{3} \arctan \left (\frac {3 \sqrt {-1+x^3}}{(-1+x)^2}\right ) \]
Time = 0.93 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00 \[ \int \frac {-1+x}{(2+x) \sqrt {-1+x^3}} \, dx=-\frac {2}{3} \arctan \left (\frac {3 \sqrt {-1+x^3}}{(-1+x)^2}\right ) \]
Time = 0.20 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.19, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {2563, 216}
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.
\(\displaystyle \int \frac {x-1}{(x+2) \sqrt {x^3-1}} \, dx\) |
\(\Big \downarrow \) 2563 |
\(\displaystyle 2 \int \frac {1}{\frac {(1-x)^4}{x^3-1}+9}d\frac {(1-x)^2}{\sqrt {x^3-1}}\) |
\(\Big \downarrow \) 216 |
\(\displaystyle \frac {2}{3} \arctan \left (\frac {(1-x)^2}{3 \sqrt {x^3-1}}\right )\) |
3.2.91.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[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]
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 | \(\frac {2 \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {x -1}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x -1}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \sqrt {\frac {\frac {3}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {x^{3}-1}}-\frac {2 \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {x -1}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \operatorname {EllipticPi}\left (\sqrt {\frac {x -1}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \frac {i \sqrt {3}}{6}+\frac {1}{2}, \sqrt {\frac {\frac {3}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {x^{3}-1}}\) | \(240\) |
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)
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 \[ \int \frac {-1+x}{(2+x) \sqrt {-1+x^3}} \, dx=\frac {1}{3} \, \arctan \left (\frac {{\left (x^{3} - 12 \, x^{2} - 6 \, x - 10\right )} \sqrt {x^{3} - 1}}{6 \, {\left (x^{4} - x^{3} - x + 1\right )}}\right ) \]
\[ \int \frac {-1+x}{(2+x) \sqrt {-1+x^3}} \, dx=\int \frac {x - 1}{\sqrt {\left (x - 1\right ) \left (x^{2} + x + 1\right )} \left (x + 2\right )}\, dx \]
\[ \int \frac {-1+x}{(2+x) \sqrt {-1+x^3}} \, dx=\int { \frac {x - 1}{\sqrt {x^{3} - 1} {\left (x + 2\right )}} \,d x } \]
\[ \int \frac {-1+x}{(2+x) \sqrt {-1+x^3}} \, dx=\int { \frac {x - 1}{\sqrt {x^{3} - 1} {\left (x + 2\right )}} \,d x } \]
Time = 5.54 (sec) , antiderivative size = 206, normalized size of antiderivative = 9.81 \[ \int \frac {-1+x}{(2+x) \sqrt {-1+x^3}} \, dx=-\frac {\left (3+\sqrt {3}\,1{}\mathrm {i}\right )\,\sqrt {-\frac {x+\frac {1}{2}-\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {\frac {x+\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\left (\mathrm {F}\left (\mathrm {asin}\left (\sqrt {-\frac {x-1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\right )\middle |-\frac {\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}\right )-\Pi \left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6};\mathrm {asin}\left (\sqrt {-\frac {x-1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\right )\middle |-\frac {\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}\right )\right )\,\sqrt {-\frac {x-1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}}{\sqrt {x^3+\left (-\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )-1\right )\,x+\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}} \]
-((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(a sin((-(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)