Integrand size = 29, antiderivative size = 18 \[ \int \frac {2-2 x-x^2}{\left (2+x^2\right ) \sqrt {-1-x^3}} \, dx=2 \text {arctanh}\left (\frac {1+x}{\sqrt {-1-x^3}}\right ) \] Output:
2*arctanh((1+x)/(-x^3-1)^(1/2))
Time = 1.28 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.39 \[ \int \frac {2-2 x-x^2}{\left (2+x^2\right ) \sqrt {-1-x^3}} \, dx=-2 \text {arctanh}\left (\frac {\sqrt {-1-x^3}}{1-x+x^2}\right ) \] Input:
Integrate[(2 - 2*x - x^2)/((2 + x^2)*Sqrt[-1 - x^3]),x]
Output:
-2*ArcTanh[Sqrt[-1 - x^3]/(1 - x + x^2)]
Time = 0.35 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {2571, 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.
\(\displaystyle \int \frac {-x^2-2 x+2}{\left (x^2+2\right ) \sqrt {-x^3-1}} \, dx\) |
\(\Big \downarrow \) 2571 |
\(\displaystyle 2 \int \frac {1}{1-\frac {(x+1)^2}{-x^3-1}}d\frac {x+1}{\sqrt {-x^3-1}}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle 2 \text {arctanh}\left (\frac {x+1}{\sqrt {-x^3-1}}\right )\) |
Input:
Int[(2 - 2*x - x^2)/((2 + x^2)*Sqrt[-1 - x^3]),x]
Output:
2*ArcTanh[(1 + x)/Sqrt[-1 - x^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[((f_) + (g_.)*(x_) + (h_.)*(x_)^2)/(((c_) + (e_.)*(x_)^2)*Sqrt[(a_) + ( b_.)*(x_)^3]), x_Symbol] :> Simp[-g/e Subst[Int[1/(1 + a*x^2), x], x, (1 + 2*h*(x/g))/Sqrt[a + b*x^3]], x] /; FreeQ[{a, b, c, e, f, g, h}, x] && EqQ [b*g^3 - 8*a*h^3, 0] && EqQ[g^2 + 2*f*h, 0] && EqQ[b*c*g - 4*a*e*h, 0]
Time = 0.46 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.67
method | result | size |
trager | \(\ln \left (\frac {-x^{2}+2 \sqrt {-x^{3}-1}+2 x}{x^{2}+2}\right )\) | \(30\) |
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 \sqrt {2}\, \sqrt {3}\, \sqrt {i x \sqrt {3}-\frac {i \sqrt {3}}{2}+\frac {3}{2}}\, \sqrt {\frac {x}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}+\frac {1}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {-i x \sqrt {3}+\frac {i \sqrt {3}}{2}+\frac {3}{2}}\, \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 {1}{2}+\frac {i \sqrt {3}}{2}-i \sqrt {2}}, \sqrt {\frac {i \sqrt {3}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\right )}{3 \sqrt {-x^{3}-1}\, \left (\frac {1}{2}+\frac {i \sqrt {3}}{2}-i \sqrt {2}\right )}+\frac {2 i \sqrt {3}\, \sqrt {i x \sqrt {3}-\frac {i \sqrt {3}}{2}+\frac {3}{2}}\, \sqrt {\frac {x}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}+\frac {1}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {-i x \sqrt {3}+\frac {i \sqrt {3}}{2}+\frac {3}{2}}\, \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 {1}{2}+\frac {i \sqrt {3}}{2}-i \sqrt {2}}, \sqrt {\frac {i \sqrt {3}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\right )}{3 \sqrt {-x^{3}-1}\, \left (\frac {1}{2}+\frac {i \sqrt {3}}{2}-i \sqrt {2}\right )}+\frac {2 i \sqrt {3}\, \sqrt {i x \sqrt {3}-\frac {i \sqrt {3}}{2}+\frac {3}{2}}\, \sqrt {\frac {x}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}+\frac {1}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {-i x \sqrt {3}+\frac {i \sqrt {3}}{2}+\frac {3}{2}}\, \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 {1}{2}+\frac {i \sqrt {3}}{2}+i \sqrt {2}}, \sqrt {\frac {i \sqrt {3}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\right )}{3 \sqrt {-x^{3}-1}\, \left (\frac {1}{2}+\frac {i \sqrt {3}}{2}+i \sqrt {2}\right )}+\frac {2 \sqrt {2}\, \sqrt {3}\, \sqrt {i x \sqrt {3}-\frac {i \sqrt {3}}{2}+\frac {3}{2}}\, \sqrt {\frac {x}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}+\frac {1}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {-i x \sqrt {3}+\frac {i \sqrt {3}}{2}+\frac {3}{2}}\, \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 {1}{2}+\frac {i \sqrt {3}}{2}+i \sqrt {2}}, \sqrt {\frac {i \sqrt {3}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\right )}{3 \sqrt {-x^{3}-1}\, \left (\frac {1}{2}+\frac {i \sqrt {3}}{2}+i \sqrt {2}\right )}\) | \(724\) |
elliptic | \(\frac {2 i \sqrt {3}\, \sqrt {i x \sqrt {3}-\frac {i \sqrt {3}}{2}+\frac {3}{2}}\, \sqrt {\frac {x}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}+\frac {1}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {-i x \sqrt {3}+\frac {i \sqrt {3}}{2}+\frac {3}{2}}\, \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 x \sqrt {3}-\frac {i \sqrt {3}}{2}+\frac {3}{2}}\, \sqrt {\frac {x}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}+\frac {1}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {-i x \sqrt {3}+\frac {i \sqrt {3}}{2}+\frac {3}{2}}\, \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 {1}{2}+\frac {i \sqrt {3}}{2}-i \sqrt {2}}, \sqrt {\frac {i \sqrt {3}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\right )}{3 \sqrt {-x^{3}-1}\, \left (\frac {1}{2}+\frac {i \sqrt {3}}{2}-i \sqrt {2}\right )}-\frac {2 \sqrt {2}\, \sqrt {3}\, \sqrt {i x \sqrt {3}-\frac {i \sqrt {3}}{2}+\frac {3}{2}}\, \sqrt {\frac {x}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}+\frac {1}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {-i x \sqrt {3}+\frac {i \sqrt {3}}{2}+\frac {3}{2}}\, \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 {1}{2}+\frac {i \sqrt {3}}{2}-i \sqrt {2}}, \sqrt {\frac {i \sqrt {3}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\right )}{3 \sqrt {-x^{3}-1}\, \left (\frac {1}{2}+\frac {i \sqrt {3}}{2}-i \sqrt {2}\right )}+\frac {2 i \sqrt {3}\, \sqrt {i x \sqrt {3}-\frac {i \sqrt {3}}{2}+\frac {3}{2}}\, \sqrt {\frac {x}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}+\frac {1}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {-i x \sqrt {3}+\frac {i \sqrt {3}}{2}+\frac {3}{2}}\, \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 {1}{2}+\frac {i \sqrt {3}}{2}+i \sqrt {2}}, \sqrt {\frac {i \sqrt {3}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\right )}{3 \sqrt {-x^{3}-1}\, \left (\frac {1}{2}+\frac {i \sqrt {3}}{2}+i \sqrt {2}\right )}+\frac {2 \sqrt {2}\, \sqrt {3}\, \sqrt {i x \sqrt {3}-\frac {i \sqrt {3}}{2}+\frac {3}{2}}\, \sqrt {\frac {x}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}+\frac {1}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \sqrt {-i x \sqrt {3}+\frac {i \sqrt {3}}{2}+\frac {3}{2}}\, \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 {1}{2}+\frac {i \sqrt {3}}{2}+i \sqrt {2}}, \sqrt {\frac {i \sqrt {3}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\right )}{3 \sqrt {-x^{3}-1}\, \left (\frac {1}{2}+\frac {i \sqrt {3}}{2}+i \sqrt {2}\right )}\) | \(733\) |
Input:
int((-x^2-2*x+2)/(x^2+2)/(-x^3-1)^(1/2),x,method=_RETURNVERBOSE)
Output:
ln((-x^2+2*(-x^3-1)^(1/2)+2*x)/(x^2+2))
Time = 0.10 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.56 \[ \int \frac {2-2 x-x^2}{\left (2+x^2\right ) \sqrt {-1-x^3}} \, dx=\log \left (-\frac {x^{2} - 2 \, x - 2 \, \sqrt {-x^{3} - 1}}{x^{2} + 2}\right ) \] Input:
integrate((-x^2-2*x+2)/(x^2+2)/(-x^3-1)^(1/2),x, algorithm="fricas")
Output:
log(-(x^2 - 2*x - 2*sqrt(-x^3 - 1))/(x^2 + 2))
\[ \int \frac {2-2 x-x^2}{\left (2+x^2\right ) \sqrt {-1-x^3}} \, dx=- \int \frac {2 x}{x^{2} \sqrt {- x^{3} - 1} + 2 \sqrt {- x^{3} - 1}}\, dx - \int \frac {x^{2}}{x^{2} \sqrt {- x^{3} - 1} + 2 \sqrt {- x^{3} - 1}}\, dx - \int \left (- \frac {2}{x^{2} \sqrt {- x^{3} - 1} + 2 \sqrt {- x^{3} - 1}}\right )\, dx \] Input:
integrate((-x**2-2*x+2)/(x**2+2)/(-x**3-1)**(1/2),x)
Output:
-Integral(2*x/(x**2*sqrt(-x**3 - 1) + 2*sqrt(-x**3 - 1)), x) - Integral(x* *2/(x**2*sqrt(-x**3 - 1) + 2*sqrt(-x**3 - 1)), x) - Integral(-2/(x**2*sqrt (-x**3 - 1) + 2*sqrt(-x**3 - 1)), x)
\[ \int \frac {2-2 x-x^2}{\left (2+x^2\right ) \sqrt {-1-x^3}} \, dx=\int { -\frac {x^{2} + 2 \, x - 2}{\sqrt {-x^{3} - 1} {\left (x^{2} + 2\right )}} \,d x } \] Input:
integrate((-x^2-2*x+2)/(x^2+2)/(-x^3-1)^(1/2),x, algorithm="maxima")
Output:
-integrate((x^2 + 2*x - 2)/(sqrt(-x^3 - 1)*(x^2 + 2)), x)
\[ \int \frac {2-2 x-x^2}{\left (2+x^2\right ) \sqrt {-1-x^3}} \, dx=\int { -\frac {x^{2} + 2 \, x - 2}{\sqrt {-x^{3} - 1} {\left (x^{2} + 2\right )}} \,d x } \] Input:
integrate((-x^2-2*x+2)/(x^2+2)/(-x^3-1)^(1/2),x, algorithm="giac")
Output:
integrate(-(x^2 + 2*x - 2)/(sqrt(-x^3 - 1)*(x^2 + 2)), x)
Time = 0.09 (sec) , antiderivative size = 289, normalized size of antiderivative = 16.06 \[ \int \frac {2-2 x-x^2}{\left (2+x^2\right ) \sqrt {-1-x^3}} \, dx=\frac {\left (3+\sqrt {3}\,1{}\mathrm {i}\right )\,\sqrt {x^3+1}\,\sqrt {\frac {x-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {\frac {x+1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {\frac {\frac {1}{2}-x+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\left (-\mathrm {F}\left (\mathrm {asin}\left (\sqrt {\frac {x+1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\right )\middle |-\frac {\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}\right )+\Pi \left (\frac {\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{1+\sqrt {2}\,1{}\mathrm {i}};\mathrm {asin}\left (\sqrt {\frac {x+1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\right )\middle |-\frac {\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}\right )+\Pi \left (-\frac {\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-1+\sqrt {2}\,1{}\mathrm {i}};\mathrm {asin}\left (\sqrt {\frac {x+1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\right )\middle |-\frac {\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}\right )\right )}{\sqrt {-x^3-1}\,\sqrt {x^3+\left (-\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )-1\right )\,x-\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}} \] Input:
int(-(2*x + x^2 - 2)/((x^2 + 2)*(- x^3 - 1)^(1/2)),x)
Output:
((3^(1/2)*1i + 3)*(x^3 + 1)^(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)*(((3^(1/2)*1i)/2 - x + 1/2)/((3^(1/2)*1i)/2 + 3/2))^(1/2)*(ellipticPi(((3^(1/2)*1i)/2 + 3/2 )/(2^(1/2)*1i + 1), 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)) + ellipticPi(-((3^(1/2)*1i)/2 + 3/2)/(2^(1/2)*1i - 1), 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^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))
\[ \int \frac {2-2 x-x^2}{\left (2+x^2\right ) \sqrt {-1-x^3}} \, dx=i \left (-2 \left (\int \frac {\sqrt {x^{3}+1}}{x^{5}+2 x^{3}+x^{2}+2}d x \right )+\int \frac {\sqrt {x^{3}+1}\, x^{2}}{x^{5}+2 x^{3}+x^{2}+2}d x +2 \left (\int \frac {\sqrt {x^{3}+1}\, x}{x^{5}+2 x^{3}+x^{2}+2}d x \right )\right ) \] Input:
int((-x^2-2*x+2)/(x^2+2)/(-x^3-1)^(1/2),x)
Output:
i*( - 2*int(sqrt(x**3 + 1)/(x**5 + 2*x**3 + x**2 + 2),x) + int((sqrt(x**3 + 1)*x**2)/(x**5 + 2*x**3 + x**2 + 2),x) + 2*int((sqrt(x**3 + 1)*x)/(x**5 + 2*x**3 + x**2 + 2),x))