Integrand size = 27, antiderivative size = 16 \[ \int \frac {2-2 x-x^2}{\left (2+x^2\right ) \sqrt {1+x^3}} \, dx=2 \arctan \left (\frac {1+x}{\sqrt {1+x^3}}\right ) \] Output:
2*arctan((1+x)/(x^3+1)^(1/2))
Time = 1.27 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.44 \[ \int \frac {2-2 x-x^2}{\left (2+x^2\right ) \sqrt {1+x^3}} \, dx=2 \arctan \left (\frac {\sqrt {1+x^3}}{1-x+x^2}\right ) \] Input:
Integrate[(2 - 2*x - x^2)/((2 + x^2)*Sqrt[1 + x^3]),x]
Output:
2*ArcTan[Sqrt[1 + x^3]/(1 - x + x^2)]
Time = 0.35 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {2571, 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.
\(\displaystyle \int \frac {-x^2-2 x+2}{\left (x^2+2\right ) \sqrt {x^3+1}} \, dx\) |
\(\Big \downarrow \) 2571 |
\(\displaystyle 2 \int \frac {1}{\frac {(x+1)^2}{x^3+1}+1}d\frac {x+1}{\sqrt {x^3+1}}\) |
\(\Big \downarrow \) 216 |
\(\displaystyle 2 \arctan \left (\frac {x+1}{\sqrt {x^3+1}}\right )\) |
Input:
Int[(2 - 2*x - x^2)/((2 + x^2)*Sqrt[1 + x^3]),x]
Output:
2*ArcTan[(1 + x)/Sqrt[1 + x^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[((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]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.60 (sec) , antiderivative size = 46, normalized size of antiderivative = 2.88
method | result | size |
trager | \(\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{2}-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x +2 \sqrt {x^{3}+1}}{x^{2}+2}\right )\) | \(46\) |
default | \(\text {Expression too large to display}\) | \(1640\) |
elliptic | \(\text {Expression too large to display}\) | \(1845\) |
Input:
int((-x^2-2*x+2)/(x^2+2)/(x^3+1)^(1/2),x,method=_RETURNVERBOSE)
Output:
RootOf(_Z^2+1)*ln((RootOf(_Z^2+1)*x^2-2*RootOf(_Z^2+1)*x+2*(x^3+1)^(1/2))/ (x^2+2))
Time = 0.15 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.19 \[ \int \frac {2-2 x-x^2}{\left (2+x^2\right ) \sqrt {1+x^3}} \, dx=-\arctan \left (\frac {x^{2} - 2 \, x}{2 \, \sqrt {x^{3} + 1}}\right ) \] Input:
integrate((-x^2-2*x+2)/(x^2+2)/(x^3+1)^(1/2),x, algorithm="fricas")
Output:
-arctan(1/2*(x^2 - 2*x)/sqrt(x^3 + 1))
\[ \int \frac {2-2 x-x^2}{\left (2+x^2\right ) \sqrt {1+x^3}} \, dx=- \int \frac {2 x}{x^{2} \sqrt {x^{3} + 1} + 2 \sqrt {x^{3} + 1}}\, dx - \int \frac {x^{2}}{x^{2} \sqrt {x^{3} + 1} + 2 \sqrt {x^{3} + 1}}\, dx - \int \left (- \frac {2}{x^{2} \sqrt {x^{3} + 1} + 2 \sqrt {x^{3} + 1}}\right )\, dx \] 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 = 0.10 (sec) , antiderivative size = 273, normalized size of antiderivative = 17.06 \[ \int \frac {2-2 x-x^2}{\left (2+x^2\right ) \sqrt {1+x^3}} \, dx=\frac {\left (3+\sqrt {3}\,1{}\mathrm {i}\right )\,\sqrt {\frac {x-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {\frac {x+1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\sqrt {\frac {\frac {1}{2}-x+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\,\left (-\mathrm {F}\left (\mathrm {asin}\left (\sqrt {\frac {x+1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\right )\middle |-\frac {\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}\right )+\Pi \left (\frac {\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{1+\sqrt {2}\,1{}\mathrm {i}};\mathrm {asin}\left (\sqrt {\frac {x+1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\right )\middle |-\frac {\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}\right )+\Pi \left (-\frac {\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-1+\sqrt {2}\,1{}\mathrm {i}};\mathrm {asin}\left (\sqrt {\frac {x+1}{\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}}\right )\middle |-\frac {\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}{-\frac {3}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}}\right )\right )}{\sqrt {x^3+\left (-\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )-1\right )\,x-\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}} \] Input:
int(-(2*x + x^2 - 2)/((x^2 + 2)*(x^3 + 1)^(1/2)),x)
Output:
((3^(1/2)*1i + 3)*((x + (3^(1/2)*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 - 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)
\[ \int \frac {2-2 x-x^2}{\left (2+x^2\right ) \sqrt {1+x^3}} \, dx=2 \left (\int \frac {\sqrt {x^{3}+1}}{x^{5}+2 x^{3}+x^{2}+2}d x \right )-\left (\int \frac {\sqrt {x^{3}+1}\, x^{2}}{x^{5}+2 x^{3}+x^{2}+2}d x \right )-2 \left (\int \frac {\sqrt {x^{3}+1}\, x}{x^{5}+2 x^{3}+x^{2}+2}d x \right ) \] Input:
int((-x^2-2*x+2)/(x^2+2)/(x^3+1)^(1/2),x)
Output:
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)