Optimal. Leaf size=149 \[ -\frac{13-\frac{27 \sqrt{-x-1}}{\sqrt{x+3}}}{18 \left (-\frac{3 (x+1)}{x+3}-\frac{2 \sqrt{-x-1}}{\sqrt{x+3}}+1\right )}-\frac{2 \left (2-\frac{\sqrt{-x-1}}{\sqrt{x+3}}\right )}{9 \left (-\frac{3 (x+1)}{x+3}-\frac{2 \sqrt{-x-1}}{\sqrt{x+3}}+1\right )^2}-\frac{3 \tan ^{-1}\left (\frac{1-\frac{3 \sqrt{-x-1}}{\sqrt{x+3}}}{\sqrt{2}}\right )}{2 \sqrt{2}} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.186621, antiderivative size = 149, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.278 \[ -\frac{13-\frac{27 \sqrt{-x-1}}{\sqrt{x+3}}}{18 \left (-\frac{3 (x+1)}{x+3}-\frac{2 \sqrt{-x-1}}{\sqrt{x+3}}+1\right )}-\frac{2 \left (2-\frac{\sqrt{-x-1}}{\sqrt{x+3}}\right )}{9 \left (-\frac{3 (x+1)}{x+3}-\frac{2 \sqrt{-x-1}}{\sqrt{x+3}}+1\right )^2}-\frac{3 \tan ^{-1}\left (\frac{1-\frac{3 \sqrt{-x-1}}{\sqrt{x+3}}}{\sqrt{2}}\right )}{2 \sqrt{2}} \]
Antiderivative was successfully verified.
[In] Int[(x + Sqrt[-3 - 4*x - x^2])^(-3),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 10.823, size = 214, normalized size = 1.44 \[ - \frac{2 - \frac{6 \sqrt{- x^{2} - 4 x - 3}}{x + 3}}{12 \left (1 - \frac{2 \sqrt{- x^{2} - 4 x - 3}}{x + 3} + \frac{3 \left (- x^{2} - 4 x - 3\right )}{\left (x + 3\right )^{2}}\right )} - \frac{8 - \frac{4 \sqrt{- x^{2} - 4 x - 3}}{x + 3}}{18 \left (1 - \frac{2 \sqrt{- x^{2} - 4 x - 3}}{x + 3} + \frac{3 \left (- x^{2} - 4 x - 3\right )}{\left (x + 3\right )^{2}}\right )^{2}} - \frac{10 - \frac{18 \sqrt{- x^{2} - 4 x - 3}}{x + 3}}{18 \left (1 - \frac{2 \sqrt{- x^{2} - 4 x - 3}}{x + 3} + \frac{3 \left (- x^{2} - 4 x - 3\right )}{\left (x + 3\right )^{2}}\right )} + \frac{3 \sqrt{2} \operatorname{atan}{\left (\sqrt{2} \left (- \frac{1}{2} + \frac{3 \sqrt{- x^{2} - 4 x - 3}}{2 \left (x + 3\right )}\right ) \right )}}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(x+(-x**2-4*x-3)**(1/2))**3,x)
[Out]
_______________________________________________________________________________________
Mathematica [C] time = 6.11845, size = 914, normalized size = 6.13 \[ \frac{1}{32} \left (\frac{8 (2 x-3)}{\left (2 x^2+4 x+3\right )^2}-\frac{8 \sqrt{-x^2-4 x-3} \left (8 x^3+22 x^2+26 x+15\right )}{\left (2 x^2+4 x+3\right )^2}-12 \sqrt{2} \tan ^{-1}\left (\sqrt{2} (x+1)\right )+\frac{6 \left (2+i \sqrt{2}\right ) \tan ^{-1}\left (\frac{(x+2) \left (2 \left (9+2 i \sqrt{2}\right ) x^2+16 \left (2+i \sqrt{2}\right ) x+3 \left (5+4 i \sqrt{2}\right )\right )}{\left (8 i+6 \sqrt{2}\right ) x^3+\left (-6 \sqrt{1+2 i \sqrt{2}} \sqrt{-x^2-4 x-3}+8 \sqrt{2}+36 i\right ) x^2+\left (-12 \sqrt{1+2 i \sqrt{2}} \sqrt{-x^2-4 x-3}-5 \sqrt{2}+40 i\right ) x-9 \sqrt{1+2 i \sqrt{2}} \sqrt{-x^2-4 x-3}-6 \sqrt{2}+12 i}\right )}{\sqrt{1+2 i \sqrt{2}}}-\frac{6 \left (2 i+\sqrt{2}\right ) \tanh ^{-1}\left (\frac{(x+2) \left (2 \left (9 i+2 \sqrt{2}\right ) x^2+16 \left (2 i+\sqrt{2}\right ) x+3 \left (5 i+4 \sqrt{2}\right )\right )}{\left (-8 i+6 \sqrt{2}\right ) x^3+\left (-6 \sqrt{1-2 i \sqrt{2}} \sqrt{-x^2-4 x-3}+8 \sqrt{2}-36 i\right ) x^2-12 \sqrt{1-2 i \sqrt{2}} \sqrt{-x^2-4 x-3} x-5 \left (8 i+\sqrt{2}\right ) x-3 \left (3 \sqrt{1-2 i \sqrt{2}} \sqrt{-x^2-4 x-3}+2 \sqrt{2}+4 i\right )}\right )}{\sqrt{1-2 i \sqrt{2}}}+\frac{3 \left (2 i+\sqrt{2}\right ) \log \left (4 \left (2 x^2+4 x+3\right )^2\right )}{\sqrt{1-2 i \sqrt{2}}}+\frac{3 \left (-2 i+\sqrt{2}\right ) \log \left (4 \left (2 x^2+4 x+3\right )^2\right )}{\sqrt{1+2 i \sqrt{2}}}-\frac{3 \left (2 i+\sqrt{2}\right ) \log \left (\left (2 x^2+4 x+3\right ) \left (\left (2+2 i \sqrt{2}\right ) x^2+\left (-2 \sqrt{2-4 i \sqrt{2}} \sqrt{-x^2-4 x-3}+8 i \sqrt{2}+4\right ) x-2 \sqrt{2-4 i \sqrt{2}} \sqrt{-x^2-4 x-3}+6 i \sqrt{2}+3\right )\right )}{\sqrt{1-2 i \sqrt{2}}}-\frac{3 \left (-2 i+\sqrt{2}\right ) \log \left (\left (2 x^2+4 x+3\right ) \left (\left (2-2 i \sqrt{2}\right ) x^2-2 \left (\sqrt{2+4 i \sqrt{2}} \sqrt{-x^2-4 x-3}+4 i \sqrt{2}-2\right ) x-2 \sqrt{2+4 i \sqrt{2}} \sqrt{-x^2-4 x-3}-6 i \sqrt{2}+3\right )\right )}{\sqrt{1+2 i \sqrt{2}}}-\frac{8 (3 x+2)}{2 x^2+4 x+3}\right ) \]
Antiderivative was successfully verified.
[In] Integrate[(x + Sqrt[-3 - 4*x - x^2])^(-3),x]
[Out]
_______________________________________________________________________________________
Maple [B] time = 0.301, size = 14545, normalized size = 97.6 \[ \text{output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(x+(-x^2-4*x-3)^(1/2))^3,x)
[Out]
_______________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (x + \sqrt{-x^{2} - 4 \, x - 3}\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((x + sqrt(-x^2 - 4*x - 3))^(-3),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.277454, size = 225, normalized size = 1.51 \[ -\frac{\sqrt{2}{\left (2 \, \sqrt{2}{\left (8 \, x^{3} + 22 \, x^{2} + 26 \, x + 15\right )} \sqrt{-x^{2} - 4 \, x - 3} + 6 \,{\left (4 \, x^{4} + 16 \, x^{3} + 28 \, x^{2} + 24 \, x + 9\right )} \arctan \left (\sqrt{2}{\left (x + 1\right )}\right ) + 3 \,{\left (4 \, x^{4} + 16 \, x^{3} + 28 \, x^{2} + 24 \, x + 9\right )} \arctan \left (\frac{\sqrt{2}{\left (6 \, x^{2} + 20 \, x + 15\right )}}{4 \, \sqrt{-x^{2} - 4 \, x - 3}{\left (2 \, x + 3\right )}}\right ) + 2 \, \sqrt{2}{\left (6 \, x^{3} + 16 \, x^{2} + 15 \, x + 9\right )}\right )}}{16 \,{\left (4 \, x^{4} + 16 \, x^{3} + 28 \, x^{2} + 24 \, x + 9\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((x + sqrt(-x^2 - 4*x - 3))^(-3),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (x + \sqrt{- x^{2} - 4 x - 3}\right )^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(x+(-x**2-4*x-3)**(1/2))**3,x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.282357, size = 495, normalized size = 3.32 \[ -\frac{3}{8} \, \sqrt{2} \arctan \left (\sqrt{2}{\left (x + 1\right )}\right ) + \frac{3}{8} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (\frac{3 \,{\left (\sqrt{-x^{2} - 4 \, x - 3} - 1\right )}}{x + 2} + 1\right )}\right ) + \frac{3}{8} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (\frac{\sqrt{-x^{2} - 4 \, x - 3} - 1}{x + 2} + 1\right )}\right ) - \frac{6 \, x^{3} + 16 \, x^{2} + 15 \, x + 9}{4 \,{\left (2 \, x^{2} + 4 \, x + 3\right )}^{2}} + \frac{\frac{618 \,{\left (\sqrt{-x^{2} - 4 \, x - 3} - 1\right )}}{x + 2} + \frac{1547 \,{\left (\sqrt{-x^{2} - 4 \, x - 3} - 1\right )}^{2}}{{\left (x + 2\right )}^{2}} + \frac{2362 \,{\left (\sqrt{-x^{2} - 4 \, x - 3} - 1\right )}^{3}}{{\left (x + 2\right )}^{3}} + \frac{2223 \,{\left (\sqrt{-x^{2} - 4 \, x - 3} - 1\right )}^{4}}{{\left (x + 2\right )}^{4}} + \frac{1174 \,{\left (\sqrt{-x^{2} - 4 \, x - 3} - 1\right )}^{5}}{{\left (x + 2\right )}^{5}} + \frac{377 \,{\left (\sqrt{-x^{2} - 4 \, x - 3} - 1\right )}^{6}}{{\left (x + 2\right )}^{6}} + \frac{6 \,{\left (\sqrt{-x^{2} - 4 \, x - 3} - 1\right )}^{7}}{{\left (x + 2\right )}^{7}} + 117}{18 \,{\left (\frac{8 \,{\left (\sqrt{-x^{2} - 4 \, x - 3} - 1\right )}}{x + 2} + \frac{14 \,{\left (\sqrt{-x^{2} - 4 \, x - 3} - 1\right )}^{2}}{{\left (x + 2\right )}^{2}} + \frac{8 \,{\left (\sqrt{-x^{2} - 4 \, x - 3} - 1\right )}^{3}}{{\left (x + 2\right )}^{3}} + \frac{3 \,{\left (\sqrt{-x^{2} - 4 \, x - 3} - 1\right )}^{4}}{{\left (x + 2\right )}^{4}} + 3\right )}^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((x + sqrt(-x^2 - 4*x - 3))^(-3),x, algorithm="giac")
[Out]