3.132 \(\int \frac{1}{x^2 \sqrt{-3-4 x-x^2} \left (3+4 x+2 x^2\right )} \, dx\)

Optimal. Leaf size=151 \[ \frac{\sqrt{-x^2-4 x-3}}{9 x}+\frac{2 \tan ^{-1}\left (\frac{2 x+3}{\sqrt{3} \sqrt{-x^2-4 x-3}}\right )}{3 \sqrt{3}}+\frac{2}{27} \sqrt{2} \tan ^{-1}\left (\frac{1-\frac{x+3}{\sqrt{-x^2-4 x-3}}}{\sqrt{2}}\right )-\frac{2}{27} \sqrt{2} \tan ^{-1}\left (\frac{\frac{x+3}{\sqrt{-x^2-4 x-3}}+1}{\sqrt{2}}\right )+\frac{10}{27} \tanh ^{-1}\left (\frac{x}{\sqrt{-x^2-4 x-3}}\right ) \]

[Out]

Sqrt[-3 - 4*x - x^2]/(9*x) + (2*ArcTan[(3 + 2*x)/(Sqrt[3]*Sqrt[-3 - 4*x - x^2])]
)/(3*Sqrt[3]) + (2*Sqrt[2]*ArcTan[(1 - (3 + x)/Sqrt[-3 - 4*x - x^2])/Sqrt[2]])/2
7 - (2*Sqrt[2]*ArcTan[(1 + (3 + x)/Sqrt[-3 - 4*x - x^2])/Sqrt[2]])/27 + (10*ArcT
anh[x/Sqrt[-3 - 4*x - x^2]])/27

_______________________________________________________________________________________

Rubi [A]  time = 1.00088, antiderivative size = 151, normalized size of antiderivative = 1., number of steps used = 20, number of rules used = 12, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4 \[ \frac{\sqrt{-x^2-4 x-3}}{9 x}+\frac{2 \tan ^{-1}\left (\frac{2 x+3}{\sqrt{3} \sqrt{-x^2-4 x-3}}\right )}{3 \sqrt{3}}+\frac{2}{27} \sqrt{2} \tan ^{-1}\left (\frac{1-\frac{x+3}{\sqrt{-x^2-4 x-3}}}{\sqrt{2}}\right )-\frac{2}{27} \sqrt{2} \tan ^{-1}\left (\frac{\frac{x+3}{\sqrt{-x^2-4 x-3}}+1}{\sqrt{2}}\right )+\frac{10}{27} \tanh ^{-1}\left (\frac{x}{\sqrt{-x^2-4 x-3}}\right ) \]

Antiderivative was successfully verified.

[In]  Int[1/(x^2*Sqrt[-3 - 4*x - x^2]*(3 + 4*x + 2*x^2)),x]

[Out]

Sqrt[-3 - 4*x - x^2]/(9*x) + (2*ArcTan[(3 + 2*x)/(Sqrt[3]*Sqrt[-3 - 4*x - x^2])]
)/(3*Sqrt[3]) + (2*Sqrt[2]*ArcTan[(1 - (3 + x)/Sqrt[-3 - 4*x - x^2])/Sqrt[2]])/2
7 - (2*Sqrt[2]*ArcTan[(1 + (3 + x)/Sqrt[-3 - 4*x - x^2])/Sqrt[2]])/27 + (10*ArcT
anh[x/Sqrt[-3 - 4*x - x^2]])/27

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 118.371, size = 150, normalized size = 0.99 \[ - \frac{2 \sqrt{2} \operatorname{atan}{\left (\sqrt{2} \left (\frac{3 \left (\frac{x}{3} + 1\right )}{2 \sqrt{- x^{2} - 4 x - 3}} - \frac{1}{2}\right ) \right )}}{27} - \frac{2 \sqrt{2} \operatorname{atan}{\left (\sqrt{2} \left (\frac{3 \left (\frac{x}{3} + 1\right )}{2 \sqrt{- x^{2} - 4 x - 3}} + \frac{1}{2}\right ) \right )}}{27} + \frac{2 \sqrt{3} \operatorname{atan}{\left (- \frac{\sqrt{3} \left (- 4 x - 6\right )}{6 \sqrt{- x^{2} - 4 x - 3}} \right )}}{9} + \frac{10 \operatorname{atanh}{\left (\frac{x}{\sqrt{- x^{2} - 4 x - 3}} \right )}}{27} + \frac{\sqrt{- x^{2} - 4 x - 3}}{9 x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**2/(2*x**2+4*x+3)/(-x**2-4*x-3)**(1/2),x)

[Out]

-2*sqrt(2)*atan(sqrt(2)*(3*(x/3 + 1)/(2*sqrt(-x**2 - 4*x - 3)) - 1/2))/27 - 2*sq
rt(2)*atan(sqrt(2)*(3*(x/3 + 1)/(2*sqrt(-x**2 - 4*x - 3)) + 1/2))/27 + 2*sqrt(3)
*atan(-sqrt(3)*(-4*x - 6)/(6*sqrt(-x**2 - 4*x - 3)))/9 + 10*atanh(x/sqrt(-x**2 -
4*x - 3))/27 + sqrt(-x**2 - 4*x - 3)/(9*x)

_______________________________________________________________________________________

Mathematica [C]  time = 6.29769, size = 1135, normalized size = 7.52 \[ -\frac{2 \tan ^{-1}\left (\frac{(2 x+3) \sqrt{-x^2-4 x-3}}{\sqrt{3} \left (x^2+4 x+3\right )}\right )}{3 \sqrt{3}}-\frac{i \left (-i+2 \sqrt{2}\right ) \tan ^{-1}\left (\frac{22 i \sqrt{2} x^4+16 x^4+18 i \sqrt{1-2 i \sqrt{2}} \sqrt{-x^2-4 x-3} x^3+100 i \sqrt{2} x^3+124 x^3+72 i \sqrt{1-2 i \sqrt{2}} \sqrt{-x^2-4 x-3} x^2+137 i \sqrt{2} x^2+324 x^2+99 i \sqrt{1-2 i \sqrt{2}} \sqrt{-x^2-4 x-3} x+48 i \sqrt{2} x+324 x+54 i \sqrt{1-2 i \sqrt{2}} \sqrt{-x^2-4 x-3}-9 i \sqrt{2}+108}{32 \sqrt{2} x^4+34 i x^4+176 \sqrt{2} x^3+112 i x^3+306 \sqrt{2} x^2+125 i x^2+216 \sqrt{2} x+84 i x+54 \sqrt{2}+45 i}\right )}{9 \sqrt{1-2 i \sqrt{2}}}+\frac{\left (i+2 \sqrt{2}\right ) \tanh ^{-1}\left (\frac{22 \sqrt{2} x^4+16 i x^4+18 \sqrt{1+2 i \sqrt{2}} \sqrt{-x^2-4 x-3} x^3+100 \sqrt{2} x^3+124 i x^3+72 \sqrt{1+2 i \sqrt{2}} \sqrt{-x^2-4 x-3} x^2+137 \sqrt{2} x^2+324 i x^2+99 \sqrt{1+2 i \sqrt{2}} \sqrt{-x^2-4 x-3} x+48 \sqrt{2} x+324 i x+54 \sqrt{1+2 i \sqrt{2}} \sqrt{-x^2-4 x-3}-9 \sqrt{2}+108 i}{32 \sqrt{2} x^4-34 i x^4+176 \sqrt{2} x^3-112 i x^3+306 \sqrt{2} x^2-125 i x^2+216 \sqrt{2} x-84 i x+54 \sqrt{2}-45 i}\right )}{9 \sqrt{1+2 i \sqrt{2}}}+\frac{\left (i+2 \sqrt{2}\right ) \log \left (\left (-2 i x+\sqrt{2}-2 i\right )^2 \left (2 i x+\sqrt{2}+2 i\right )^2\right )}{18 \sqrt{1+2 i \sqrt{2}}}+\frac{\left (-i+2 \sqrt{2}\right ) \log \left (\left (-2 i x+\sqrt{2}-2 i\right )^2 \left (2 i x+\sqrt{2}+2 i\right )^2\right )}{18 \sqrt{1-2 i \sqrt{2}}}-\frac{\left (-i+2 \sqrt{2}\right ) \log \left (\left (2 x^2+4 x+3\right ) \left (2 i \sqrt{2} x^2+2 x^2-2 \sqrt{2 \left (1-2 i \sqrt{2}\right )} \sqrt{-x^2-4 x-3} x+8 i \sqrt{2} x+4 x-2 \sqrt{2 \left (1-2 i \sqrt{2}\right )} \sqrt{-x^2-4 x-3}+6 i \sqrt{2}+3\right )\right )}{18 \sqrt{1-2 i \sqrt{2}}}-\frac{\left (i+2 \sqrt{2}\right ) \log \left (\left (2 x^2+4 x+3\right ) \left (-2 i \sqrt{2} x^2+2 x^2-2 \sqrt{2 \left (1+2 i \sqrt{2}\right )} \sqrt{-x^2-4 x-3} x-8 i \sqrt{2} x+4 x-2 \sqrt{2 \left (1+2 i \sqrt{2}\right )} \sqrt{-x^2-4 x-3}-6 i \sqrt{2}+3\right )\right )}{18 \sqrt{1+2 i \sqrt{2}}}+\frac{\sqrt{-x^2-4 x-3}}{9 x} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^2*Sqrt[-3 - 4*x - x^2]*(3 + 4*x + 2*x^2)),x]

[Out]

Sqrt[-3 - 4*x - x^2]/(9*x) - (2*ArcTan[((3 + 2*x)*Sqrt[-3 - 4*x - x^2])/(Sqrt[3]
*(3 + 4*x + x^2))])/(3*Sqrt[3]) - ((I/9)*(-I + 2*Sqrt[2])*ArcTan[(108 - (9*I)*Sq
rt[2] + 324*x + (48*I)*Sqrt[2]*x + 324*x^2 + (137*I)*Sqrt[2]*x^2 + 124*x^3 + (10
0*I)*Sqrt[2]*x^3 + 16*x^4 + (22*I)*Sqrt[2]*x^4 + (54*I)*Sqrt[1 - (2*I)*Sqrt[2]]*
Sqrt[-3 - 4*x - x^2] + (99*I)*Sqrt[1 - (2*I)*Sqrt[2]]*x*Sqrt[-3 - 4*x - x^2] + (
72*I)*Sqrt[1 - (2*I)*Sqrt[2]]*x^2*Sqrt[-3 - 4*x - x^2] + (18*I)*Sqrt[1 - (2*I)*S
qrt[2]]*x^3*Sqrt[-3 - 4*x - x^2])/(45*I + 54*Sqrt[2] + (84*I)*x + 216*Sqrt[2]*x
+ (125*I)*x^2 + 306*Sqrt[2]*x^2 + (112*I)*x^3 + 176*Sqrt[2]*x^3 + (34*I)*x^4 + 3
2*Sqrt[2]*x^4)])/Sqrt[1 - (2*I)*Sqrt[2]] + ((I + 2*Sqrt[2])*ArcTanh[(108*I - 9*S
qrt[2] + (324*I)*x + 48*Sqrt[2]*x + (324*I)*x^2 + 137*Sqrt[2]*x^2 + (124*I)*x^3
+ 100*Sqrt[2]*x^3 + (16*I)*x^4 + 22*Sqrt[2]*x^4 + 54*Sqrt[1 + (2*I)*Sqrt[2]]*Sqr
t[-3 - 4*x - x^2] + 99*Sqrt[1 + (2*I)*Sqrt[2]]*x*Sqrt[-3 - 4*x - x^2] + 72*Sqrt[
1 + (2*I)*Sqrt[2]]*x^2*Sqrt[-3 - 4*x - x^2] + 18*Sqrt[1 + (2*I)*Sqrt[2]]*x^3*Sqr
t[-3 - 4*x - x^2])/(-45*I + 54*Sqrt[2] - (84*I)*x + 216*Sqrt[2]*x - (125*I)*x^2
+ 306*Sqrt[2]*x^2 - (112*I)*x^3 + 176*Sqrt[2]*x^3 - (34*I)*x^4 + 32*Sqrt[2]*x^4)
])/(9*Sqrt[1 + (2*I)*Sqrt[2]]) + ((-I + 2*Sqrt[2])*Log[(-2*I + Sqrt[2] - (2*I)*x
)^2*(2*I + Sqrt[2] + (2*I)*x)^2])/(18*Sqrt[1 - (2*I)*Sqrt[2]]) + ((I + 2*Sqrt[2]
)*Log[(-2*I + Sqrt[2] - (2*I)*x)^2*(2*I + Sqrt[2] + (2*I)*x)^2])/(18*Sqrt[1 + (2
*I)*Sqrt[2]]) - ((-I + 2*Sqrt[2])*Log[(3 + 4*x + 2*x^2)*(3 + (6*I)*Sqrt[2] + 4*x
+ (8*I)*Sqrt[2]*x + 2*x^2 + (2*I)*Sqrt[2]*x^2 - 2*Sqrt[2*(1 - (2*I)*Sqrt[2])]*S
qrt[-3 - 4*x - x^2] - 2*Sqrt[2*(1 - (2*I)*Sqrt[2])]*x*Sqrt[-3 - 4*x - x^2])])/(1
8*Sqrt[1 - (2*I)*Sqrt[2]]) - ((I + 2*Sqrt[2])*Log[(3 + 4*x + 2*x^2)*(3 - (6*I)*S
qrt[2] + 4*x - (8*I)*Sqrt[2]*x + 2*x^2 - (2*I)*Sqrt[2]*x^2 - 2*Sqrt[2*(1 + (2*I)
*Sqrt[2])]*Sqrt[-3 - 4*x - x^2] - 2*Sqrt[2*(1 + (2*I)*Sqrt[2])]*x*Sqrt[-3 - 4*x
- x^2])])/(18*Sqrt[1 + (2*I)*Sqrt[2]])

_______________________________________________________________________________________

Maple [A]  time = 0.023, size = 169, normalized size = 1.1 \[{\frac{1}{9\,x}\sqrt{-{x}^{2}-4\,x-3}}-{\frac{2\,\sqrt{3}}{9}\arctan \left ({\frac{ \left ( -6-4\,x \right ) \sqrt{3}}{6}{\frac{1}{\sqrt{-{x}^{2}-4\,x-3}}}} \right ) }+{\frac{\sqrt{3}\sqrt{4}}{81}\sqrt{3\,{\frac{{x}^{2}}{ \left ( -3/2-x \right ) ^{2}}}-12} \left ( \sqrt{2}\arctan \left ({\frac{\sqrt{2}}{6}\sqrt{3\,{\frac{{x}^{2}}{ \left ( -3/2-x \right ) ^{2}}}-12}} \right ) -5\,{\it Artanh} \left ( 3\,{\frac{x}{-3/2-x}{\frac{1}{\sqrt{3\,{\frac{{x}^{2}}{ \left ( -3/2-x \right ) ^{2}}}-12}}}} \right ) \right ){\frac{1}{\sqrt{{1 \left ({{x}^{2} \left ( -{\frac{3}{2}}-x \right ) ^{-2}}-4 \right ) \left ( 1+{x \left ( -{\frac{3}{2}}-x \right ) ^{-1}} \right ) ^{-2}}}}} \left ( 1+{x \left ( -{\frac{3}{2}}-x \right ) ^{-1}} \right ) ^{-1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^2/(2*x^2+4*x+3)/(-x^2-4*x-3)^(1/2),x)

[Out]

1/9*(-x^2-4*x-3)^(1/2)/x-2/9*3^(1/2)*arctan(1/6*(-6-4*x)*3^(1/2)/(-x^2-4*x-3)^(1
/2))+1/81*3^(1/2)*4^(1/2)*(3*x^2/(-3/2-x)^2-12)^(1/2)*(2^(1/2)*arctan(1/6*(3*x^2
/(-3/2-x)^2-12)^(1/2)*2^(1/2))-5*arctanh(3*x/(-3/2-x)/(3*x^2/(-3/2-x)^2-12)^(1/2
)))/((x^2/(-3/2-x)^2-4)/(1+x/(-3/2-x))^2)^(1/2)/(1+x/(-3/2-x))

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (2 \, x^{2} + 4 \, x + 3\right )} \sqrt{-x^{2} - 4 \, x - 3} x^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((2*x^2 + 4*x + 3)*sqrt(-x^2 - 4*x - 3)*x^2),x, algorithm="maxima")

[Out]

integrate(1/((2*x^2 + 4*x + 3)*sqrt(-x^2 - 4*x - 3)*x^2), x)

_______________________________________________________________________________________

Fricas [A]  time = 0.305584, size = 258, normalized size = 1.71 \[ \frac{\sqrt{3}{\left (2 \, \sqrt{3} \sqrt{2} x \arctan \left (\frac{\sqrt{2}{\left (x + 3 \, \sqrt{-x^{2} - 4 \, x - 3}\right )}}{2 \,{\left (2 \, x + 3\right )}}\right ) + 2 \, \sqrt{3} \sqrt{2} x \arctan \left (-\frac{\sqrt{2}{\left (x - 3 \, \sqrt{-x^{2} - 4 \, x - 3}\right )}}{2 \,{\left (2 \, x + 3\right )}}\right ) - 5 \, \sqrt{3} x \log \left (-\frac{2 \, \sqrt{-x^{2} - 4 \, x - 3} x + 4 \, x + 3}{x^{2}}\right ) + 5 \, \sqrt{3} x \log \left (\frac{2 \, \sqrt{-x^{2} - 4 \, x - 3} x - 4 \, x - 3}{x^{2}}\right ) + 36 \, x \arctan \left (\frac{\sqrt{3}{\left (2 \, x + 3\right )}}{3 \, \sqrt{-x^{2} - 4 \, x - 3}}\right ) + 6 \, \sqrt{3} \sqrt{-x^{2} - 4 \, x - 3}\right )}}{162 \, x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((2*x^2 + 4*x + 3)*sqrt(-x^2 - 4*x - 3)*x^2),x, algorithm="fricas")

[Out]

1/162*sqrt(3)*(2*sqrt(3)*sqrt(2)*x*arctan(1/2*sqrt(2)*(x + 3*sqrt(-x^2 - 4*x - 3
))/(2*x + 3)) + 2*sqrt(3)*sqrt(2)*x*arctan(-1/2*sqrt(2)*(x - 3*sqrt(-x^2 - 4*x -
3))/(2*x + 3)) - 5*sqrt(3)*x*log(-(2*sqrt(-x^2 - 4*x - 3)*x + 4*x + 3)/x^2) + 5
*sqrt(3)*x*log((2*sqrt(-x^2 - 4*x - 3)*x - 4*x - 3)/x^2) + 36*x*arctan(1/3*sqrt(
3)*(2*x + 3)/sqrt(-x^2 - 4*x - 3)) + 6*sqrt(3)*sqrt(-x^2 - 4*x - 3))/x

_______________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x^{2} \sqrt{- \left (x + 1\right ) \left (x + 3\right )} \left (2 x^{2} + 4 x + 3\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**2/(2*x**2+4*x+3)/(-x**2-4*x-3)**(1/2),x)

[Out]

Integral(1/(x**2*sqrt(-(x + 1)*(x + 3))*(2*x**2 + 4*x + 3)), x)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.273651, size = 363, normalized size = 2.4 \[ \frac{2}{27} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (\frac{3 \,{\left (\sqrt{-x^{2} - 4 \, x - 3} - 1\right )}}{x + 2} + 1\right )}\right ) - \frac{4}{9} \, \sqrt{3} \arctan \left (\frac{1}{3} \, \sqrt{3}{\left (\frac{2 \,{\left (\sqrt{-x^{2} - 4 \, x - 3} - 1\right )}}{x + 2} + 1\right )}\right ) + \frac{2}{27} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (\frac{\sqrt{-x^{2} - 4 \, x - 3} - 1}{x + 2} + 1\right )}\right ) - \frac{\frac{\sqrt{-x^{2} - 4 \, x - 3} - 1}{x + 2} + 2}{18 \,{\left (\frac{\sqrt{-x^{2} - 4 \, x - 3} - 1}{x + 2} + \frac{{\left (\sqrt{-x^{2} - 4 \, x - 3} - 1\right )}^{2}}{{\left (x + 2\right )}^{2}} + 1\right )}} + \frac{5}{27} \,{\rm ln}\left (\frac{2 \,{\left (\sqrt{-x^{2} - 4 \, x - 3} - 1\right )}}{x + 2} + \frac{3 \,{\left (\sqrt{-x^{2} - 4 \, x - 3} - 1\right )}^{2}}{{\left (x + 2\right )}^{2}} + 1\right ) - \frac{5}{27} \,{\rm ln}\left (\frac{2 \,{\left (\sqrt{-x^{2} - 4 \, x - 3} - 1\right )}}{x + 2} + \frac{{\left (\sqrt{-x^{2} - 4 \, x - 3} - 1\right )}^{2}}{{\left (x + 2\right )}^{2}} + 3\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((2*x^2 + 4*x + 3)*sqrt(-x^2 - 4*x - 3)*x^2),x, algorithm="giac")

[Out]

2/27*sqrt(2)*arctan(1/2*sqrt(2)*(3*(sqrt(-x^2 - 4*x - 3) - 1)/(x + 2) + 1)) - 4/
9*sqrt(3)*arctan(1/3*sqrt(3)*(2*(sqrt(-x^2 - 4*x - 3) - 1)/(x + 2) + 1)) + 2/27*
sqrt(2)*arctan(1/2*sqrt(2)*((sqrt(-x^2 - 4*x - 3) - 1)/(x + 2) + 1)) - 1/18*((sq
rt(-x^2 - 4*x - 3) - 1)/(x + 2) + 2)/((sqrt(-x^2 - 4*x - 3) - 1)/(x + 2) + (sqrt
(-x^2 - 4*x - 3) - 1)^2/(x + 2)^2 + 1) + 5/27*ln(2*(sqrt(-x^2 - 4*x - 3) - 1)/(x
+ 2) + 3*(sqrt(-x^2 - 4*x - 3) - 1)^2/(x + 2)^2 + 1) - 5/27*ln(2*(sqrt(-x^2 - 4
*x - 3) - 1)/(x + 2) + (sqrt(-x^2 - 4*x - 3) - 1)^2/(x + 2)^2 + 3)