Integrand size = 22, antiderivative size = 25 \[ \int \frac {1+x}{(2-x) \sqrt {-1-x^3}} \, dx=\frac {2}{3} \arctan \left (\frac {(1+x)^2}{3 \sqrt {-1-x^3}}\right ) \] Output:
2/3*arctan(1/3*(1+x)^2/(-x^3-1)^(1/2))
Time = 1.16 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.92 \[ \int \frac {1+x}{(2-x) \sqrt {-1-x^3}} \, dx=-\frac {2}{3} \arctan \left (\frac {3 \sqrt {-1-x^3}}{(1+x)^2}\right ) \] Input:
Integrate[(1 + x)/((2 - x)*Sqrt[-1 - x^3]),x]
Output:
(-2*ArcTan[(3*Sqrt[-1 - x^3])/(1 + x)^2])/3
Time = 0.34 (sec) , antiderivative size = 25, 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, 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.
| \(\displaystyle \int \frac {x+1}{(2-x) \sqrt {-x^3-1}} \, dx\) |
| \(\Big \downarrow \) 2563 |
| \(\displaystyle 2 \int \frac {1}{\frac {(x+1)^4}{-x^3-1}+9}d\frac {(x+1)^2}{\sqrt {-x^3-1}}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle \frac {2}{3} \arctan \left (\frac {(x+1)^2}{3 \sqrt {-x^3-1}}\right )\) |
Input:
Int[(1 + x)/((2 - x)*Sqrt[-1 - x^3]),x]
Output:
(2*ArcTan[(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]))*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 = 0.62 (sec) , antiderivative size = 79, normalized size of antiderivative = 3.16
| method | result | size |
| trager | \(\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{3}+12 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{2}-6 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x -6 \sqrt {-x^{3}-1}\, x +10 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )-6 \sqrt {-x^{3}-1}}{\left (x -2\right )^{3}}\right )}{3}\) | \(79\) |
| 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((x+1)/(2-x)/(-x^3-1)^(1/2),x,method=_RETURNVERBOSE)
Output:
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)
Time = 0.14 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.52 \[ \int \frac {1+x}{(2-x) \sqrt {-1-x^3}} \, dx=-\frac {1}{3} \, \arctan \left (\frac {{\left (x^{3} + 12 \, x^{2} - 6 \, x + 10\right )} \sqrt {-x^{3} - 1}}{6 \, {\left (x^{4} + x^{3} + x + 1\right )}}\right ) \] Input:
integrate((1+x)/(2-x)/(-x^3-1)^(1/2),x, algorithm="fricas")
Output:
-1/3*arctan(1/6*(x^3 + 12*x^2 - 6*x + 10)*sqrt(-x^3 - 1)/(x^4 + x^3 + x + 1))
\[ \int \frac {1+x}{(2-x) \sqrt {-1-x^3}} \, dx=- \int \frac {x}{x \sqrt {- x^{3} - 1} - 2 \sqrt {- x^{3} - 1}}\, dx - \int \frac {1}{x \sqrt {- x^{3} - 1} - 2 \sqrt {- x^{3} - 1}}\, dx \] Input:
integrate((1+x)/(2-x)/(-x**3-1)**(1/2),x)
Output:
-Integral(x/(x*sqrt(-x**3 - 1) - 2*sqrt(-x**3 - 1)), x) - Integral(1/(x*sq rt(-x**3 - 1) - 2*sqrt(-x**3 - 1)), 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 = 0.07 (sec) , antiderivative size = 221, normalized size of antiderivative = 8.84 \[ \int \frac {1+x}{(2-x) \sqrt {-1-x^3}} \, dx=-\frac {\left (3+\sqrt {3}\,1{}\mathrm {i}\right )\,\sqrt {x^3+1}\,\sqrt {\frac {x-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\left (\mathrm {F}\left (\mathrm {asin}\left (\sqrt {\frac {x+1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\right )\middle |-\frac {\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}\right )-\Pi \left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6};\mathrm {asin}\left (\sqrt {\frac {x+1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\right )\middle |-\frac {\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}\right )\right )\,\sqrt {\frac {x+1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {\frac {\frac {1}{2}-x+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}}{\sqrt {-x^3-1}\,\sqrt {x^3+\left (-\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )-1\right )\,x-\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}} \] Input:
int(-(x + 1)/((- x^3 - 1)^(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)*(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 - x + 1/2)/((3^(1/2)*1i)/2 + 3/2))^(1/2))/((- x^3 - 1)^(1/2) *(x^3 - x*(((3^(1/2)*1i)/2 - 1/2)*((3^(1/2)*1i)/2 + 1/2) + 1) - ((3^(1/2)* 1i)/2 - 1/2)*((3^(1/2)*1i)/2 + 1/2))^(1/2))
\[ \int \frac {1+x}{(2-x) \sqrt {-1-x^3}} \, dx=\left (\int \frac {\sqrt {x^{3}+1}}{x^{3}-3 x^{2}+3 x -2}d x \right ) i \] 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)*i