Integrand size = 21, antiderivative size = 87 \[ \int \frac {1}{1+2 x+\sqrt {-3-2 x+4 x^2}} \, dx=\frac {x}{3}-\frac {1}{6} \sqrt {-3-2 x+4 x^2}-\frac {11}{36} \text {arctanh}\left (\frac {1-4 x}{2 \sqrt {-3-2 x+4 x^2}}\right )-\frac {1}{18} \text {arctanh}\left (\frac {7+11 x}{\sqrt {-3-2 x+4 x^2}}\right )-\frac {1}{18} \log (2+3 x) \] Output:
1/3*x-1/6*(4*x^2-2*x-3)^(1/2)-11/36*arctanh(1/2*(1-4*x)/(4*x^2-2*x-3)^(1/2 ))-1/18*arctanh((7+11*x)/(4*x^2-2*x-3)^(1/2))-1/18*ln(2+3*x)
Time = 0.20 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.83 \[ \int \frac {1}{1+2 x+\sqrt {-3-2 x+4 x^2}} \, dx=\frac {1}{36} \left (12 x-6 \sqrt {-3-2 x+4 x^2}-9 \log \left (1-4 x+2 \sqrt {-3-2 x+4 x^2}\right )-4 \log \left (-5-6 x+3 \sqrt {-3-2 x+4 x^2}\right )\right ) \] Input:
Integrate[(1 + 2*x + Sqrt[-3 - 2*x + 4*x^2])^(-1),x]
Output:
(12*x - 6*Sqrt[-3 - 2*x + 4*x^2] - 9*Log[1 - 4*x + 2*Sqrt[-3 - 2*x + 4*x^2 ]] - 4*Log[-5 - 6*x + 3*Sqrt[-3 - 2*x + 4*x^2]])/36
Time = 0.23 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.94, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {2541, 27, 1195, 2009}
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.
\(\displaystyle \int \frac {1}{\sqrt {4 x^2-2 x-3}+2 x+1} \, dx\) |
\(\Big \downarrow \) 2541 |
\(\displaystyle 2 \int -\frac {-\left (2 x+\sqrt {4 x^2-2 x-3}+1\right )^2+3 \left (2 x+\sqrt {4 x^2-2 x-3}+1\right )+1}{2 \left (2 x+\sqrt {4 x^2-2 x-3}+1\right ) \left (3-2 \left (2 x+\sqrt {4 x^2-2 x-3}+1\right )\right )^2}d\left (2 x+\sqrt {4 x^2-2 x-3}+1\right )\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\int \frac {-\left (2 x+\sqrt {4 x^2-2 x-3}+1\right )^2+3 \left (2 x+\sqrt {4 x^2-2 x-3}+1\right )+1}{\left (2 x+\sqrt {4 x^2-2 x-3}+1\right ) \left (3-2 \left (2 x+\sqrt {4 x^2-2 x-3}+1\right )\right )^2}d\left (2 x+\sqrt {4 x^2-2 x-3}+1\right )\) |
\(\Big \downarrow \) 1195 |
\(\displaystyle -\int \left (-\frac {13}{18 \left (2 \left (2 x+\sqrt {4 x^2-2 x-3}+1\right )-3\right )}+\frac {13}{6 \left (2 \left (2 x+\sqrt {4 x^2-2 x-3}+1\right )-3\right )^2}+\frac {1}{9 \left (2 x+\sqrt {4 x^2-2 x-3}+1\right )}\right )d\left (2 x+\sqrt {4 x^2-2 x-3}+1\right )\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {13}{12 \left (3-2 \left (\sqrt {4 x^2-2 x-3}+2 x+1\right )\right )}-\frac {1}{9} \log \left (\sqrt {4 x^2-2 x-3}+2 x+1\right )+\frac {13}{36} \log \left (3-2 \left (\sqrt {4 x^2-2 x-3}+2 x+1\right )\right )\) |
Input:
Int[(1 + 2*x + Sqrt[-3 - 2*x + 4*x^2])^(-1),x]
Output:
-13/(12*(3 - 2*(1 + 2*x + Sqrt[-3 - 2*x + 4*x^2]))) - Log[1 + 2*x + Sqrt[- 3 - 2*x + 4*x^2]]/9 + (13*Log[3 - 2*(1 + 2*x + Sqrt[-3 - 2*x + 4*x^2])])/3 6
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_.) + (b_.)*(x _) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(f + g*x)^n*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n}, x ] && IGtQ[p, 0]
Int[((g_.) + (h_.)*((d_.) + (e_.)*(x_) + (f_.)*Sqrt[(a_.) + (b_.)*(x_) + (c _.)*(x_)^2])^(n_))^(p_.), x_Symbol] :> Simp[2 Subst[Int[(g + h*x^n)^p*((d ^2*e - (b*d - a*e)*f^2 - (2*d*e - b*f^2)*x + e*x^2)/(-2*d*e + b*f^2 + 2*e*x )^2), x], x, d + e*x + f*Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c, d, e , f, g, h, n}, x] && EqQ[e^2 - c*f^2, 0] && IntegerQ[p]
Time = 0.14 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.06
method | result | size |
default | \(-\frac {\sqrt {36 \left (x +\frac {2}{3}\right )^{2}-66 x -43}}{18}+\frac {11 \ln \left (\frac {\left (4 x -1\right ) \sqrt {4}}{4}+\sqrt {4 \left (x +\frac {2}{3}\right )^{2}-\frac {22 x}{3}-\frac {43}{9}}\right ) \sqrt {4}}{72}+\frac {\operatorname {arctanh}\left (\frac {-21-33 x}{\sqrt {36 \left (x +\frac {2}{3}\right )^{2}-66 x -43}}\right )}{18}+\frac {\ln \left (-2-3 x \right )}{6}+\frac {x}{3}-\frac {2 \ln \left (2+3 x \right )}{9}\) | \(92\) |
trager | \(\frac {x}{3}-\frac {\sqrt {4 x^{2}-2 x -3}}{6}+\frac {\ln \left (\frac {35511830+832470756 x -3099091026 x^{2} \sqrt {4 x^{2}-2 x -3}-376883380224 x^{12}+780593528832 x^{10}+962873786368 x^{9}-23414043276 x^{3}-447475041 x^{2}-19279420992 x^{4}+347892350976 x^{13}-572102688768 x^{7}+176217689856 x^{5}-871542816768 x^{11}-593255989248 x^{8}+188914372608 x^{6}+34568346 \sqrt {4 x^{2}-2 x -3}-156400554 \sqrt {4 x^{2}-2 x -3}\, x +28960489472 \sqrt {4 x^{2}-2 x -3}\, x^{5}+39948350336 \sqrt {4 x^{2}-2 x -3}\, x^{4}-474702144 \sqrt {4 x^{2}-2 x -3}\, x^{3}+173946175488 \sqrt {4 x^{2}-2 x -3}\, x^{12}-144955146240 \sqrt {4 x^{2}-2 x -3}\, x^{11}-401344561152 \sqrt {4 x^{2}-2 x -3}\, x^{10}+248739004416 \sqrt {4 x^{2}-2 x -3}\, x^{9}+384623968256 \sqrt {4 x^{2}-2 x -3}\, x^{8}-143955853312 \sqrt {4 x^{2}-2 x -3}\, x^{7}-182234701824 \sqrt {4 x^{2}-2 x -3}\, x^{6}}{\left (2+3 x \right )^{4}}\right )}{36}\) | \(311\) |
Input:
int(1/(1+2*x+(4*x^2-2*x-3)^(1/2)),x,method=_RETURNVERBOSE)
Output:
-1/18*(36*(x+2/3)^2-66*x-43)^(1/2)+11/72*ln(1/4*(4*x-1)*4^(1/2)+(4*(x+2/3) ^2-22/3*x-43/9)^(1/2))*4^(1/2)+1/18*arctanh(9/2*(-14/3-22/3*x)/(36*(x+2/3) ^2-66*x-43)^(1/2))+1/6*ln(-2-3*x)+1/3*x-2/9*ln(2+3*x)
Time = 0.09 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.03 \[ \int \frac {1}{1+2 x+\sqrt {-3-2 x+4 x^2}} \, dx=\frac {1}{3} \, x - \frac {1}{6} \, \sqrt {4 \, x^{2} - 2 \, x - 3} - \frac {1}{18} \, \log \left (3 \, x + 2\right ) + \frac {1}{18} \, \log \left (-2 \, x + \sqrt {4 \, x^{2} - 2 \, x - 3} - 1\right ) - \frac {11}{36} \, \log \left (-4 \, x + 2 \, \sqrt {4 \, x^{2} - 2 \, x - 3} + 1\right ) - \frac {1}{18} \, \log \left (-6 \, x + 3 \, \sqrt {4 \, x^{2} - 2 \, x - 3} - 5\right ) \] Input:
integrate(1/(1+2*x+(4*x^2-2*x-3)^(1/2)),x, algorithm="fricas")
Output:
1/3*x - 1/6*sqrt(4*x^2 - 2*x - 3) - 1/18*log(3*x + 2) + 1/18*log(-2*x + sq rt(4*x^2 - 2*x - 3) - 1) - 11/36*log(-4*x + 2*sqrt(4*x^2 - 2*x - 3) + 1) - 1/18*log(-6*x + 3*sqrt(4*x^2 - 2*x - 3) - 5)
\[ \int \frac {1}{1+2 x+\sqrt {-3-2 x+4 x^2}} \, dx=\int \frac {1}{2 x + \sqrt {4 x^{2} - 2 x - 3} + 1}\, dx \] Input:
integrate(1/(1+2*x+(4*x**2-2*x-3)**(1/2)),x)
Output:
Integral(1/(2*x + sqrt(4*x**2 - 2*x - 3) + 1), x)
\[ \int \frac {1}{1+2 x+\sqrt {-3-2 x+4 x^2}} \, dx=\int { \frac {1}{2 \, x + \sqrt {4 \, x^{2} - 2 \, x - 3} + 1} \,d x } \] Input:
integrate(1/(1+2*x+(4*x^2-2*x-3)^(1/2)),x, algorithm="maxima")
Output:
integrate(1/(2*x + sqrt(4*x^2 - 2*x - 3) + 1), x)
Time = 0.14 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.08 \[ \int \frac {1}{1+2 x+\sqrt {-3-2 x+4 x^2}} \, dx=\frac {1}{3} \, x - \frac {1}{6} \, \sqrt {4 \, x^{2} - 2 \, x - 3} - \frac {1}{18} \, \log \left ({\left | 3 \, x + 2 \right |}\right ) + \frac {1}{18} \, \log \left ({\left | -2 \, x + \sqrt {4 \, x^{2} - 2 \, x - 3} - 1 \right |}\right ) - \frac {11}{36} \, \log \left ({\left | -4 \, x + 2 \, \sqrt {4 \, x^{2} - 2 \, x - 3} + 1 \right |}\right ) - \frac {1}{18} \, \log \left ({\left | -6 \, x + 3 \, \sqrt {4 \, x^{2} - 2 \, x - 3} - 5 \right |}\right ) \] Input:
integrate(1/(1+2*x+(4*x^2-2*x-3)^(1/2)),x, algorithm="giac")
Output:
1/3*x - 1/6*sqrt(4*x^2 - 2*x - 3) - 1/18*log(abs(3*x + 2)) + 1/18*log(abs( -2*x + sqrt(4*x^2 - 2*x - 3) - 1)) - 11/36*log(abs(-4*x + 2*sqrt(4*x^2 - 2 *x - 3) + 1)) - 1/18*log(abs(-6*x + 3*sqrt(4*x^2 - 2*x - 3) - 5))
Timed out. \[ \int \frac {1}{1+2 x+\sqrt {-3-2 x+4 x^2}} \, dx=\frac {x}{3}-\frac {\ln \left (x+\frac {2}{3}\right )}{18}-\int \frac {\sqrt {4\,x^2-2\,x-3}}{6\,x+4} \,d x \] Input:
int(1/(2*x + (4*x^2 - 2*x - 3)^(1/2) + 1),x)
Output:
x/3 - log(x + 2/3)/18 - int((4*x^2 - 2*x - 3)^(1/2)/(6*x + 4), x)
Time = 0.16 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.80 \[ \int \frac {1}{1+2 x+\sqrt {-3-2 x+4 x^2}} \, dx=-\frac {\sqrt {4 x^{2}-2 x -3}}{6}-\frac {\mathrm {log}\left (\frac {26 \sqrt {4 x^{2}-2 x -3}+52 x +26}{\sqrt {13}}\right )}{9}+\frac {13 \,\mathrm {log}\left (\frac {2 \sqrt {4 x^{2}-2 x -3}+4 x -1}{\sqrt {13}}\right )}{36}+\frac {x}{3}-\frac {1}{12} \] Input:
int(1/(1+2*x+(4*x^2-2*x-3)^(1/2)),x)
Output:
( - 6*sqrt(4*x**2 - 2*x - 3) - 4*log((26*sqrt(4*x**2 - 2*x - 3) + 52*x + 2 6)/sqrt(13)) + 13*log((2*sqrt(4*x**2 - 2*x - 3) + 4*x - 1)/sqrt(13)) + 12* x - 3)/36