3.760 \(\int \frac {1}{(x+\sqrt {-3-4 x-x^2})^2} \, dx\)
Optimal. Leaf size=87 \[ \frac {1-\frac {\sqrt {-x-1}}{\sqrt {x+3}}}{-\frac {3 (x+1)}{x+3}-\frac {2 \sqrt {-x-1}}{\sqrt {x+3}}+1}+\frac {\tan ^{-1}\left (\frac {1-\frac {3 \sqrt {-x-1}}{\sqrt {x+3}}}{\sqrt {2}}\right )}{\sqrt {2}} \]
[Out]
1/2*arctan(1/2*(1-3*(-1-x)^(1/2)/(3+x)^(1/2))*2^(1/2))*2^(1/2)+(1-(-1-x)^(1/2)/(3+x)^(1/2))/(1-3*(1+x)/(3+x)-2
*(-1-x)^(1/2)/(3+x)^(1/2))
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Rubi [A] time = 0.07, antiderivative size = 87, normalized size of antiderivative = 1.00,
number of steps used = 5, number of rules used = 4, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used =
{12, 638, 618, 204} \[ \frac {1-\frac {\sqrt {-x-1}}{\sqrt {x+3}}}{-\frac {3 (x+1)}{x+3}-\frac {2 \sqrt {-x-1}}{\sqrt {x+3}}+1}+\frac {\tan ^{-1}\left (\frac {1-\frac {3 \sqrt {-x-1}}{\sqrt {x+3}}}{\sqrt {2}}\right )}{\sqrt {2}} \]
Antiderivative was successfully verified.
[In]
Int[(x + Sqrt[-3 - 4*x - x^2])^(-2),x]
[Out]
(1 - Sqrt[-1 - x]/Sqrt[3 + x])/(1 - (3*(1 + x))/(3 + x) - (2*Sqrt[-1 - x])/Sqrt[3 + x]) + ArcTan[(1 - (3*Sqrt[
-1 - x])/Sqrt[3 + x])/Sqrt[2]]/Sqrt[2]
Rule 12
Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] && !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]
Rule 204
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])
Rule 618
Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]
Rule 638
Int[((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((b*d - 2*a*e + (2*c*d -
b*e)*x)*(a + b*x + c*x^2)^(p + 1))/((p + 1)*(b^2 - 4*a*c)), x] - Dist[((2*p + 3)*(2*c*d - b*e))/((p + 1)*(b^2
- 4*a*c)), Int[(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[2*c*d - b*e, 0] && NeQ[b^
2 - 4*a*c, 0] && LtQ[p, -1] && NeQ[p, -3/2]
Rubi steps
\begin {align*} \int \frac {1}{\left (x+\sqrt {-3-4 x-x^2}\right )^2} \, dx &=2 \operatorname {Subst}\left (\int -\frac {2 x}{\left (1-2 x+3 x^2\right )^2} \, dx,x,\frac {\sqrt {-1-x}}{\sqrt {3+x}}\right )\\ &=-\left (4 \operatorname {Subst}\left (\int \frac {x}{\left (1-2 x+3 x^2\right )^2} \, dx,x,\frac {\sqrt {-1-x}}{\sqrt {3+x}}\right )\right )\\ &=\frac {1-\frac {\sqrt {-1-x}}{\sqrt {3+x}}}{1-\frac {3 (1+x)}{3+x}-\frac {2 \sqrt {-1-x}}{\sqrt {3+x}}}-\operatorname {Subst}\left (\int \frac {1}{1-2 x+3 x^2} \, dx,x,\frac {\sqrt {-1-x}}{\sqrt {3+x}}\right )\\ &=\frac {1-\frac {\sqrt {-1-x}}{\sqrt {3+x}}}{1-\frac {3 (1+x)}{3+x}-\frac {2 \sqrt {-1-x}}{\sqrt {3+x}}}+2 \operatorname {Subst}\left (\int \frac {1}{-8-x^2} \, dx,x,-2+\frac {6 \sqrt {-1-x}}{\sqrt {3+x}}\right )\\ &=\frac {1-\frac {\sqrt {-1-x}}{\sqrt {3+x}}}{1-\frac {3 (1+x)}{3+x}-\frac {2 \sqrt {-1-x}}{\sqrt {3+x}}}+\frac {\tan ^{-1}\left (\frac {1-\frac {3 \sqrt {-1-x}}{\sqrt {3+x}}}{\sqrt {2}}\right )}{\sqrt {2}}\\ \end {align*}
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Mathematica [C] time = 1.67, size = 881, normalized size = 10.13 \[ \frac {1}{16} \left (\frac {8 (x+3)}{2 x^2+4 x+3}+4 \sqrt {2} \tan ^{-1}\left (\sqrt {2} (x+1)\right )-\frac {2 i \left (-2 i+\sqrt {2}\right ) \tan ^{-1}\left (\frac {(x+2) \left (2 \left (9+2 i \sqrt {2}\right ) x^2+16 \left (2+i \sqrt {2}\right ) x+3 \left (5+4 i \sqrt {2}\right )\right )}{\left (8 i+6 \sqrt {2}\right ) x^3+\left (-6 \sqrt {1+2 i \sqrt {2}} \sqrt {-x^2-4 x-3}+8 \sqrt {2}+36 i\right ) x^2+\left (-12 \sqrt {1+2 i \sqrt {2}} \sqrt {-x^2-4 x-3}-5 \sqrt {2}+40 i\right ) x-9 \sqrt {1+2 i \sqrt {2}} \sqrt {-x^2-4 x-3}-6 \sqrt {2}+12 i}\right )}{\sqrt {1+2 i \sqrt {2}}}+\frac {2 \left (2 i+\sqrt {2}\right ) \tanh ^{-1}\left (\frac {(x+2) \left (2 \left (9 i+2 \sqrt {2}\right ) x^2+16 \left (2 i+\sqrt {2}\right ) x+3 \left (5 i+4 \sqrt {2}\right )\right )}{\left (-8 i+6 \sqrt {2}\right ) x^3+\left (-6 \sqrt {1-2 i \sqrt {2}} \sqrt {-x^2-4 x-3}+8 \sqrt {2}-36 i\right ) x^2-12 \sqrt {1-2 i \sqrt {2}} \sqrt {-x^2-4 x-3} x-5 \left (8 i+\sqrt {2}\right ) x-3 \left (3 \sqrt {1-2 i \sqrt {2}} \sqrt {-x^2-4 x-3}+2 \sqrt {2}+4 i\right )}\right )}{\sqrt {1-2 i \sqrt {2}}}-\frac {\left (2 i+\sqrt {2}\right ) \log \left (4 \left (2 x^2+4 x+3\right )^2\right )}{\sqrt {1-2 i \sqrt {2}}}-\frac {\left (-2 i+\sqrt {2}\right ) \log \left (4 \left (2 x^2+4 x+3\right )^2\right )}{\sqrt {1+2 i \sqrt {2}}}+\frac {\left (2 i+\sqrt {2}\right ) \log \left (\left (2 x^2+4 x+3\right ) \left (\left (2+2 i \sqrt {2}\right ) x^2+\left (-2 \sqrt {2-4 i \sqrt {2}} \sqrt {-x^2-4 x-3}+8 i \sqrt {2}+4\right ) x-2 \sqrt {2-4 i \sqrt {2}} \sqrt {-x^2-4 x-3}+6 i \sqrt {2}+3\right )\right )}{\sqrt {1-2 i \sqrt {2}}}+\frac {\left (-2 i+\sqrt {2}\right ) \log \left (\left (2 x^2+4 x+3\right ) \left (\left (2-2 i \sqrt {2}\right ) x^2-2 \left (\sqrt {2+4 i \sqrt {2}} \sqrt {-x^2-4 x-3}+4 i \sqrt {2}-2\right ) x-2 \sqrt {2+4 i \sqrt {2}} \sqrt {-x^2-4 x-3}-6 i \sqrt {2}+3\right )\right )}{\sqrt {1+2 i \sqrt {2}}}+\frac {8 (2 x+3) \sqrt {-x^2-4 x-3}}{2 x^2+4 x+3}\right ) \]
Antiderivative was successfully verified.
[In]
Integrate[(x + Sqrt[-3 - 4*x - x^2])^(-2),x]
[Out]
((8*(3 + x))/(3 + 4*x + 2*x^2) + (8*(3 + 2*x)*Sqrt[-3 - 4*x - x^2])/(3 + 4*x + 2*x^2) + 4*Sqrt[2]*ArcTan[Sqrt[
2]*(1 + x)] - ((2*I)*(-2*I + Sqrt[2])*ArcTan[((2 + x)*(3*(5 + (4*I)*Sqrt[2]) + 16*(2 + I*Sqrt[2])*x + 2*(9 + (
2*I)*Sqrt[2])*x^2))/(12*I - 6*Sqrt[2] + (8*I + 6*Sqrt[2])*x^3 - 9*Sqrt[1 + (2*I)*Sqrt[2]]*Sqrt[-3 - 4*x - x^2]
+ x*(40*I - 5*Sqrt[2] - 12*Sqrt[1 + (2*I)*Sqrt[2]]*Sqrt[-3 - 4*x - x^2]) + x^2*(36*I + 8*Sqrt[2] - 6*Sqrt[1 +
(2*I)*Sqrt[2]]*Sqrt[-3 - 4*x - x^2]))])/Sqrt[1 + (2*I)*Sqrt[2]] + (2*(2*I + Sqrt[2])*ArcTanh[((2 + x)*(3*(5*I
+ 4*Sqrt[2]) + 16*(2*I + Sqrt[2])*x + 2*(9*I + 2*Sqrt[2])*x^2))/(-5*(8*I + Sqrt[2])*x + (-8*I + 6*Sqrt[2])*x^
3 - 12*Sqrt[1 - (2*I)*Sqrt[2]]*x*Sqrt[-3 - 4*x - x^2] + x^2*(-36*I + 8*Sqrt[2] - 6*Sqrt[1 - (2*I)*Sqrt[2]]*Sqr
t[-3 - 4*x - x^2]) - 3*(4*I + 2*Sqrt[2] + 3*Sqrt[1 - (2*I)*Sqrt[2]]*Sqrt[-3 - 4*x - x^2]))])/Sqrt[1 - (2*I)*Sq
rt[2]] - ((-2*I + Sqrt[2])*Log[4*(3 + 4*x + 2*x^2)^2])/Sqrt[1 + (2*I)*Sqrt[2]] - ((2*I + Sqrt[2])*Log[4*(3 + 4
*x + 2*x^2)^2])/Sqrt[1 - (2*I)*Sqrt[2]] + ((2*I + Sqrt[2])*Log[(3 + 4*x + 2*x^2)*(3 + (6*I)*Sqrt[2] + (2 + (2*
I)*Sqrt[2])*x^2 - 2*Sqrt[2 - (4*I)*Sqrt[2]]*Sqrt[-3 - 4*x - x^2] + x*(4 + (8*I)*Sqrt[2] - 2*Sqrt[2 - (4*I)*Sqr
t[2]]*Sqrt[-3 - 4*x - x^2]))])/Sqrt[1 - (2*I)*Sqrt[2]] + ((-2*I + Sqrt[2])*Log[(3 + 4*x + 2*x^2)*(3 - (6*I)*Sq
rt[2] + (2 - (2*I)*Sqrt[2])*x^2 - 2*Sqrt[2 + (4*I)*Sqrt[2]]*Sqrt[-3 - 4*x - x^2] - 2*x*(-2 + (4*I)*Sqrt[2] + S
qrt[2 + (4*I)*Sqrt[2]]*Sqrt[-3 - 4*x - x^2]))])/Sqrt[1 + (2*I)*Sqrt[2]])/16
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fricas [A] time = 0.42, size = 121, normalized size = 1.39 \[ \frac {2 \, \sqrt {2} {\left (2 \, x^{2} + 4 \, x + 3\right )} \arctan \left (\sqrt {2} {\left (x + 1\right )}\right ) - \sqrt {2} {\left (2 \, x^{2} + 4 \, x + 3\right )} \arctan \left (\frac {\sqrt {2} {\left (6 \, x^{2} + 20 \, x + 15\right )} \sqrt {-x^{2} - 4 \, x - 3}}{4 \, {\left (2 \, x^{3} + 11 \, x^{2} + 18 \, x + 9\right )}}\right ) + 4 \, \sqrt {-x^{2} - 4 \, x - 3} {\left (2 \, x + 3\right )} + 4 \, x + 12}{8 \, {\left (2 \, x^{2} + 4 \, x + 3\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(x+(-x^2-4*x-3)^(1/2))^2,x, algorithm="fricas")
[Out]
1/8*(2*sqrt(2)*(2*x^2 + 4*x + 3)*arctan(sqrt(2)*(x + 1)) - sqrt(2)*(2*x^2 + 4*x + 3)*arctan(1/4*sqrt(2)*(6*x^2
+ 20*x + 15)*sqrt(-x^2 - 4*x - 3)/(2*x^3 + 11*x^2 + 18*x + 9)) + 4*sqrt(-x^2 - 4*x - 3)*(2*x + 3) + 4*x + 12)
/(2*x^2 + 4*x + 3)
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giac [B] time = 0.46, size = 263, normalized size = 3.02 \[ \frac {1}{4} \, \sqrt {2} \arctan \left (\sqrt {2} {\left (x + 1\right )}\right ) - \frac {1}{4} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\frac {3 \, {\left (\sqrt {-x^{2} - 4 \, x - 3} - 1\right )}}{x + 2} + 1\right )}\right ) - \frac {1}{4} \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\frac {\sqrt {-x^{2} - 4 \, x - 3} - 1}{x + 2} + 1\right )}\right ) + \frac {x + 3}{2 \, {\left (2 \, x^{2} + 4 \, x + 3\right )}} - \frac {\frac {10 \, {\left (\sqrt {-x^{2} - 4 \, x - 3} - 1\right )}}{x + 2} + \frac {7 \, {\left (\sqrt {-x^{2} - 4 \, x - 3} - 1\right )}^{2}}{{\left (x + 2\right )}^{2}} - \frac {2 \, {\left (\sqrt {-x^{2} - 4 \, x - 3} - 1\right )}^{3}}{{\left (x + 2\right )}^{3}} + 3}{3 \, {\left (\frac {8 \, {\left (\sqrt {-x^{2} - 4 \, x - 3} - 1\right )}}{x + 2} + \frac {14 \, {\left (\sqrt {-x^{2} - 4 \, x - 3} - 1\right )}^{2}}{{\left (x + 2\right )}^{2}} + \frac {8 \, {\left (\sqrt {-x^{2} - 4 \, x - 3} - 1\right )}^{3}}{{\left (x + 2\right )}^{3}} + \frac {3 \, {\left (\sqrt {-x^{2} - 4 \, x - 3} - 1\right )}^{4}}{{\left (x + 2\right )}^{4}} + 3\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(x+(-x^2-4*x-3)^(1/2))^2,x, algorithm="giac")
[Out]
1/4*sqrt(2)*arctan(sqrt(2)*(x + 1)) - 1/4*sqrt(2)*arctan(1/2*sqrt(2)*(3*(sqrt(-x^2 - 4*x - 3) - 1)/(x + 2) + 1
)) - 1/4*sqrt(2)*arctan(1/2*sqrt(2)*((sqrt(-x^2 - 4*x - 3) - 1)/(x + 2) + 1)) + 1/2*(x + 3)/(2*x^2 + 4*x + 3)
- 1/3*(10*(sqrt(-x^2 - 4*x - 3) - 1)/(x + 2) + 7*(sqrt(-x^2 - 4*x - 3) - 1)^2/(x + 2)^2 - 2*(sqrt(-x^2 - 4*x -
3) - 1)^3/(x + 2)^3 + 3)/(8*(sqrt(-x^2 - 4*x - 3) - 1)/(x + 2) + 14*(sqrt(-x^2 - 4*x - 3) - 1)^2/(x + 2)^2 +
8*(sqrt(-x^2 - 4*x - 3) - 1)^3/(x + 2)^3 + 3*(sqrt(-x^2 - 4*x - 3) - 1)^4/(x + 2)^4 + 3)
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maple [B] time = 0.09, size = 2407, normalized size = 27.67 \[ \text {Expression too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/(x+(-x^2-4*x-3)^(1/2))^2,x)
[Out]
-3/8*(4*x+4)/(2*x^2+4*x+3)+1/4*2^(1/2)*arctan(1/4*(4*x+4)*2^(1/2))-1/2*(-4*x-6)/(2*x^2+4*x+3)+1/72*3^(1/2)*4^(
1/2)*(3/(-x-3/2)^2*x^2-12)^(1/2)*(2^(1/2)*arctan(1/6*(3/(-x-3/2)^2*x^2-12)^(1/2)*2^(1/2))*x^2/(-x-3/2)^2-8*arc
tanh(3/(-x-3/2)/(3/(-x-3/2)^2*x^2-12)^(1/2)*x)*x^2/(-x-3/2)^2+2*2^(1/2)*arctan(1/6*(3/(-x-3/2)^2*x^2-12)^(1/2)
*2^(1/2))-6*(3/(-x-3/2)^2*x^2-12)^(1/2)-16*arctanh(3/(-x-3/2)/(3/(-x-3/2)^2*x^2-12)^(1/2)*x))/((1/(-x-3/2)^2*x
^2-4)/(1/(-x-3/2)*x+1)^2)^(1/2)/(1/(-x-3/2)*x+1)/(1/(-x-3/2)^2*x^2+2)+1/36*3^(1/2)*4^(1/2)*(3/(-x-3/2)^2*x^2-1
2)^(1/2)*(7*2^(1/2)*arctan(1/6*(3/(-x-3/2)^2*x^2-12)^(1/2)*2^(1/2))+4*arctanh(3/(-x-3/2)/(3/(-x-3/2)^2*x^2-12)
^(1/2)*x))/(1/(-x-3/2)*x+1)/((1/(-x-3/2)^2*x^2-4)/(1/(-x-3/2)*x+1)^2)^(1/2)-2/9*3^(1/2)*4^(1/2)*(3/(-x-3/2)^2*
x^2-12)^(1/2)*(3*2^(1/2)*arctan(1/6*(3/(-x-3/2)^2*x^2-12)^(1/2)*2^(1/2))*x^6/(-x-3/2)^6+2*ln(((3/(-x-3/2)^2*x^
2-12)^(1/2)*x/(-x-3/2)-1/(-x-3/2)^2*x^2+4)/(1/(-x-3/2)^2*x^2-4))*x^6/(-x-3/2)^6-2*ln(((3/(-x-3/2)^2*x^2-12)^(1
/2)*x/(-x-3/2)+1/(-x-3/2)^2*x^2-4)/(1/(-x-3/2)^2*x^2-4))*x^6/(-x-3/2)^6+4*arctanh(3/(-x-3/2)/(3/(-x-3/2)^2*x^2
-12)^(1/2)*x)*x^6/(-x-3/2)^6+(3/(-x-3/2)^2*x^2-12)^(1/2)*x^5/(-x-3/2)^5-(3/(-x-3/2)^2*x^2-12)^(3/2)*x^2/(-x-3/
2)^2+(3/(-x-3/2)^2*x^2-12)^(1/2)*x^4/(-x-3/2)^4-36*2^(1/2)*arctan(1/6*(3/(-x-3/2)^2*x^2-12)^(1/2)*2^(1/2))*x^2
/(-x-3/2)^2-2*(3/(-x-3/2)^2*x^2-12)^(1/2)*x^3/(-x-3/2)^3-8*(3/(-x-3/2)^2*x^2-12)^(1/2)*x^2/(-x-3/2)^2-24*ln(((
3/(-x-3/2)^2*x^2-12)^(1/2)*x/(-x-3/2)-1/(-x-3/2)^2*x^2+4)/(1/(-x-3/2)^2*x^2-4))*x^2/(-x-3/2)^2+24*ln(((3/(-x-3
/2)^2*x^2-12)^(1/2)*x/(-x-3/2)+1/(-x-3/2)^2*x^2-4)/(1/(-x-3/2)^2*x^2-4))*x^2/(-x-3/2)^2-48*arctanh(3/(-x-3/2)/
(3/(-x-3/2)^2*x^2-12)^(1/2)*x)*x^2/(-x-3/2)^2-48*2^(1/2)*arctan(1/6*(3/(-x-3/2)^2*x^2-12)^(1/2)*2^(1/2))-8*(3/
(-x-3/2)^2*x^2-12)^(1/2)*x/(-x-3/2)+16*(3/(-x-3/2)^2*x^2-12)^(1/2)-32*ln(((3/(-x-3/2)^2*x^2-12)^(1/2)*x/(-x-3/
2)-1/(-x-3/2)^2*x^2+4)/(1/(-x-3/2)^2*x^2-4))+32*ln(((3/(-x-3/2)^2*x^2-12)^(1/2)*x/(-x-3/2)+1/(-x-3/2)^2*x^2-4)
/(1/(-x-3/2)^2*x^2-4))-64*arctanh(3/(-x-3/2)/(3/(-x-3/2)^2*x^2-12)^(1/2)*x))/((1/(-x-3/2)^2*x^2-4)/(1/(-x-3/2)
*x+1)^2)^(1/2)/(1/(-x-3/2)*x+1)/(1/(-x-3/2)^2*x^2+2)/((3/(-x-3/2)^2*x^2-12)^(1/2)*x/(-x-3/2)-1/(-x-3/2)^2*x^2+
4)/((3/(-x-3/2)^2*x^2-12)^(1/2)*x/(-x-3/2)+1/(-x-3/2)^2*x^2-4)-2/9*3^(1/2)*4^(1/2)*(3/(-x-3/2)^2*x^2-12)^(1/2)
*(2^(1/2)*arctan(1/6*(3/(-x-3/2)^2*x^2-12)^(1/2)*2^(1/2))+arctanh(3/(-x-3/2)/(3/(-x-3/2)^2*x^2-12)^(1/2)*x))/(
(1/(-x-3/2)^2*x^2-4)/(1/(-x-3/2)*x+1)^2)^(1/2)/(1/(-x-3/2)*x+1)+1/18*3^(1/2)*4^(1/2)*(3/(-x-3/2)^2*x^2-12)^(1/
2)*(11*2^(1/2)*arctan(1/6*(3/(-x-3/2)^2*x^2-12)^(1/2)*2^(1/2))*x^6/(-x-3/2)^6+8*ln(((3/(-x-3/2)^2*x^2-12)^(1/2
)*x/(-x-3/2)-1/(-x-3/2)^2*x^2+4)/(1/(-x-3/2)^2*x^2-4))*x^6/(-x-3/2)^6-8*ln(((3/(-x-3/2)^2*x^2-12)^(1/2)*x/(-x-
3/2)+1/(-x-3/2)^2*x^2-4)/(1/(-x-3/2)^2*x^2-4))*x^6/(-x-3/2)^6+24*arctanh(3/(-x-3/2)/(3/(-x-3/2)^2*x^2-12)^(1/2
)*x)*x^6/(-x-3/2)^6+4*(3/(-x-3/2)^2*x^2-12)^(1/2)*x^5/(-x-3/2)^5-(3/(-x-3/2)^2*x^2-12)^(3/2)*x^2/(-x-3/2)^2+(3
/(-x-3/2)^2*x^2-12)^(1/2)*x^4/(-x-3/2)^4-132*2^(1/2)*arctan(1/6*(3/(-x-3/2)^2*x^2-12)^(1/2)*2^(1/2))*x^2/(-x-3
/2)^2-8*(3/(-x-3/2)^2*x^2-12)^(1/2)*x^3/(-x-3/2)^3-8*(3/(-x-3/2)^2*x^2-12)^(1/2)*x^2/(-x-3/2)^2-96*ln(((3/(-x-
3/2)^2*x^2-12)^(1/2)*x/(-x-3/2)-1/(-x-3/2)^2*x^2+4)/(1/(-x-3/2)^2*x^2-4))*x^2/(-x-3/2)^2+96*ln(((3/(-x-3/2)^2*
x^2-12)^(1/2)*x/(-x-3/2)+1/(-x-3/2)^2*x^2-4)/(1/(-x-3/2)^2*x^2-4))*x^2/(-x-3/2)^2-288*arctanh(3/(-x-3/2)/(3/(-
x-3/2)^2*x^2-12)^(1/2)*x)*x^2/(-x-3/2)^2-176*2^(1/2)*arctan(1/6*(3/(-x-3/2)^2*x^2-12)^(1/2)*2^(1/2))-32*(3/(-x
-3/2)^2*x^2-12)^(1/2)*x/(-x-3/2)+16*(3/(-x-3/2)^2*x^2-12)^(1/2)-128*ln(((3/(-x-3/2)^2*x^2-12)^(1/2)*x/(-x-3/2)
-1/(-x-3/2)^2*x^2+4)/(1/(-x-3/2)^2*x^2-4))+128*ln(((3/(-x-3/2)^2*x^2-12)^(1/2)*x/(-x-3/2)+1/(-x-3/2)^2*x^2-4)/
(1/(-x-3/2)^2*x^2-4))-384*arctanh(3/(-x-3/2)/(3/(-x-3/2)^2*x^2-12)^(1/2)*x))/((1/(-x-3/2)^2*x^2-4)/(1/(-x-3/2)
*x+1)^2)^(1/2)/(1/(-x-3/2)*x+1)/(1/(-x-3/2)^2*x^2+2)/((3/(-x-3/2)^2*x^2-12)^(1/2)*x/(-x-3/2)-1/(-x-3/2)^2*x^2+
4)/((3/(-x-3/2)^2*x^2-12)^(1/2)*x/(-x-3/2)+1/(-x-3/2)^2*x^2-4)
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (x + \sqrt {-x^{2} - 4 \, x - 3}\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(x+(-x^2-4*x-3)^(1/2))^2,x, algorithm="maxima")
[Out]
integrate((x + sqrt(-x^2 - 4*x - 3))^(-2), x)
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{{\left (x+\sqrt {-x^2-4\,x-3}\right )}^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/(x + (- 4*x - x^2 - 3)^(1/2))^2,x)
[Out]
int(1/(x + (- 4*x - x^2 - 3)^(1/2))^2, x)
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (x + \sqrt {- x^{2} - 4 x - 3}\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(x+(-x**2-4*x-3)**(1/2))**2,x)
[Out]
Integral((x + sqrt(-x**2 - 4*x - 3))**(-2), x)
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