\(\int \frac {1}{1+3 x+\sqrt {-3-2 x+4 x^2}} \, dx\) [34]

Optimal result

Integrand size = 21, antiderivative size = 105 \[ \int \frac {1}{1+3 x+\sqrt {-3-2 x+4 x^2}} \, dx=\frac {7}{5} \arctan \left (\frac {1}{2} \left (1+5 \left (2 x+\sqrt {-3-2 x+4 x^2}\right )\right )\right )+\frac {3}{5} \log \left (7+3 x-20 x^2-\sqrt {-3-2 x+4 x^2}-10 x \sqrt {-3-2 x+4 x^2}\right )-\log \left (1-2 \left (2 x+\sqrt {-3-2 x+4 x^2}\right )\right ) \] Output:

7/5*arctan(1/2+5*x+5/2*(4*x^2-2*x-3)^(1/2))+3/5*ln(7+3*x-20*x^2-(4*x^2-2*x 
-3)^(1/2)-10*x*(4*x^2-2*x-3)^(1/2))-ln(1-4*x-2*(4*x^2-2*x-3)^(1/2))
 

Mathematica [A] (verified)

Time = 0.30 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.86 \[ \int \frac {1}{1+3 x+\sqrt {-3-2 x+4 x^2}} \, dx=-\frac {7}{5} \arctan \left (\frac {3}{2}+x-\frac {1}{2} \sqrt {-3-2 x+4 x^2}\right )-\frac {1}{5} \log \left (1-4 x+2 \sqrt {-3-2 x+4 x^2}\right )+\frac {3}{5} \log \left (-5-5 x-4 x^2+(3+2 x) \sqrt {-3-2 x+4 x^2}\right ) \] Input:

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

Output:

(-7*ArcTan[3/2 + x - Sqrt[-3 - 2*x + 4*x^2]/2])/5 - Log[1 - 4*x + 2*Sqrt[- 
3 - 2*x + 4*x^2]]/5 + (3*Log[-5 - 5*x - 4*x^2 + (3 + 2*x)*Sqrt[-3 - 2*x + 
4*x^2]])/5
 

Rubi [A] (verified)

Time = 0.36 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.04, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {7293, 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.

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

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {3 x+1}{5 x^2+8 x+4}-\frac {\sqrt {4 x^2-2 x-3}}{5 x^2+8 x+4}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {7}{10} \arctan \left (\frac {7 x+8}{2 \sqrt {4 x^2-2 x-3}}\right )-\frac {7}{10} \arctan \left (\frac {5 x}{2}+2\right )+\frac {2}{5} \text {arctanh}\left (\frac {1-4 x}{2 \sqrt {4 x^2-2 x-3}}\right )+\frac {3}{5} \text {arctanh}\left (\frac {3 x+1}{\sqrt {4 x^2-2 x-3}}\right )+\frac {3}{10} \log \left (5 x^2+8 x+4\right )\)

Input:

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

Output:

(-7*ArcTan[2 + (5*x)/2])/10 - (7*ArcTan[(8 + 7*x)/(2*Sqrt[-3 - 2*x + 4*x^2 
])])/10 + (2*ArcTanh[(1 - 4*x)/(2*Sqrt[-3 - 2*x + 4*x^2])])/5 + (3*ArcTanh 
[(1 + 3*x)/Sqrt[-3 - 2*x + 4*x^2]])/5 + (3*Log[4 + 8*x + 5*x^2])/10
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 7293

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v] 
]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(484\) vs. \(2(87)=174\).

Time = 0.93 (sec) , antiderivative size = 485, normalized size of antiderivative = 4.62

method	result	size
default	\(-\frac {\ln \left (\frac {\left (4 x -1\right ) \sqrt {4}}{4}+\sqrt {4 x^{2}-2 x -3}\right ) \sqrt {4}}{5}-\frac {8 \sqrt {-\frac {833 \left (\frac {8}{7}+x \right )^{2}}{\left (-\frac {1}{3}-x \right )^{2}}+1989}\, \left (2 \,\operatorname {arctanh}\left (\frac {\sqrt {-\frac {833 \left (\frac {8}{7}+x \right )^{2}}{\left (-\frac {1}{3}-x \right )^{2}}+1989}}{51}\right )-9 \arctan \left (\frac {7 \sqrt {-\frac {833 \left (\frac {8}{7}+x \right )^{2}}{\left (-\frac {1}{3}-x \right )^{2}}+1989}\, \left (\frac {8}{7}+x \right )}{2 \left (\frac {49 \left (\frac {8}{7}+x \right )^{2}}{\left (-\frac {1}{3}-x \right )^{2}}-117\right ) \left (-\frac {1}{3}-x \right )}\right )\right )}{85 \sqrt {-\frac {17 \left (\frac {49 \left (\frac {8}{7}+x \right )^{2}}{\left (-\frac {1}{3}-x \right )^{2}}-117\right )}{\left (\frac {\frac {8}{7}+x}{-\frac {1}{3}-x}+1\right )^{2}}}\, \left (\frac {\frac {8}{7}+x}{-\frac {1}{3}-x}+1\right )}+\frac {2 \sqrt {-\frac {833 \left (\frac {8}{7}+x \right )^{2}}{\left (-\frac {1}{3}-x \right )^{2}}+1989}\, \left (\operatorname {arctanh}\left (\frac {\sqrt {-\frac {833 \left (\frac {8}{7}+x \right )^{2}}{\left (-\frac {1}{3}-x \right )^{2}}+1989}}{51}\right )+4 \arctan \left (\frac {7 \sqrt {-\frac {833 \left (\frac {8}{7}+x \right )^{2}}{\left (-\frac {1}{3}-x \right )^{2}}+1989}\, \left (\frac {8}{7}+x \right )}{2 \left (\frac {49 \left (\frac {8}{7}+x \right )^{2}}{\left (-\frac {1}{3}-x \right )^{2}}-117\right ) \left (-\frac {1}{3}-x \right )}\right )\right )}{17 \sqrt {-\frac {17 \left (\frac {49 \left (\frac {8}{7}+x \right )^{2}}{\left (-\frac {1}{3}-x \right )^{2}}-117\right )}{\left (\frac {\frac {8}{7}+x}{-\frac {1}{3}-x}+1\right )^{2}}}\, \left (\frac {\frac {8}{7}+x}{-\frac {1}{3}-x}+1\right )}-\frac {3 \sqrt {-\frac {833 \left (\frac {8}{7}+x \right )^{2}}{\left (-\frac {1}{3}-x \right )^{2}}+1989}\, \left (6 \,\operatorname {arctanh}\left (\frac {\sqrt {-\frac {833 \left (\frac {8}{7}+x \right )^{2}}{\left (-\frac {1}{3}-x \right )^{2}}+1989}}{51}\right )+7 \arctan \left (\frac {7 \sqrt {-\frac {833 \left (\frac {8}{7}+x \right )^{2}}{\left (-\frac {1}{3}-x \right )^{2}}+1989}\, \left (\frac {8}{7}+x \right )}{2 \left (\frac {49 \left (\frac {8}{7}+x \right )^{2}}{\left (-\frac {1}{3}-x \right )^{2}}-117\right ) \left (-\frac {1}{3}-x \right )}\right )\right )}{34 \sqrt {-\frac {17 \left (\frac {49 \left (\frac {8}{7}+x \right )^{2}}{\left (-\frac {1}{3}-x \right )^{2}}-117\right )}{\left (\frac {\frac {8}{7}+x}{-\frac {1}{3}-x}+1\right )^{2}}}\, \left (\frac {\frac {8}{7}+x}{-\frac {1}{3}-x}+1\right )}-\frac {7 \arctan \left (\frac {5 x}{2}+2\right )}{10}+\frac {3 \ln \left (5 x^{2}+8 x +4\right )}{10}\)	\(485\)
trager	\(\text {Expression too large to display}\)	\(719\)

Input:

int(1/(1+3*x+(4*x^2-2*x-3)^(1/2)),x,method=_RETURNVERBOSE)
 

Output:

-1/5*ln(1/4*(4*x-1)*4^(1/2)+(4*x^2-2*x-3)^(1/2))*4^(1/2)-8/85*(-833*(8/7+x 
)^2/(-1/3-x)^2+1989)^(1/2)*(2*arctanh(1/51*(-833*(8/7+x)^2/(-1/3-x)^2+1989 
)^(1/2))-9*arctan(7/2/(49*(8/7+x)^2/(-1/3-x)^2-117)*(-833*(8/7+x)^2/(-1/3- 
x)^2+1989)^(1/2)*(8/7+x)/(-1/3-x)))/(-17*(49*(8/7+x)^2/(-1/3-x)^2-117)/((8 
/7+x)/(-1/3-x)+1)^2)^(1/2)/((8/7+x)/(-1/3-x)+1)+2/17*(-833*(8/7+x)^2/(-1/3 
-x)^2+1989)^(1/2)*(arctanh(1/51*(-833*(8/7+x)^2/(-1/3-x)^2+1989)^(1/2))+4* 
arctan(7/2/(49*(8/7+x)^2/(-1/3-x)^2-117)*(-833*(8/7+x)^2/(-1/3-x)^2+1989)^ 
(1/2)*(8/7+x)/(-1/3-x)))/(-17*(49*(8/7+x)^2/(-1/3-x)^2-117)/((8/7+x)/(-1/3 
-x)+1)^2)^(1/2)/((8/7+x)/(-1/3-x)+1)-3/34*(-833*(8/7+x)^2/(-1/3-x)^2+1989) 
^(1/2)*(6*arctanh(1/51*(-833*(8/7+x)^2/(-1/3-x)^2+1989)^(1/2))+7*arctan(7/ 
2/(49*(8/7+x)^2/(-1/3-x)^2-117)*(-833*(8/7+x)^2/(-1/3-x)^2+1989)^(1/2)*(8/ 
7+x)/(-1/3-x)))/(-17*(49*(8/7+x)^2/(-1/3-x)^2-117)/((8/7+x)/(-1/3-x)+1)^2) 
^(1/2)/((8/7+x)/(-1/3-x)+1)-7/10*arctan(5/2*x+2)+3/10*ln(5*x^2+8*x+4)
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.45 \[ \int \frac {1}{1+3 x+\sqrt {-3-2 x+4 x^2}} \, dx=-\frac {7}{10} \, \arctan \left (\frac {5}{2} \, x + 2\right ) + \frac {7}{10} \, \arctan \left (-x + \frac {1}{2} \, \sqrt {4 \, x^{2} - 2 \, x - 3} - \frac {3}{2}\right ) + \frac {7}{10} \, \arctan \left (-5 \, x + \frac {5}{2} \, \sqrt {4 \, x^{2} - 2 \, x - 3} - \frac {1}{2}\right ) - \frac {3}{10} \, \log \left (20 \, x^{2} - \sqrt {4 \, x^{2} - 2 \, x - 3} {\left (10 \, x + 1\right )} - 3 \, x - 7\right ) + \frac {3}{10} \, \log \left (5 \, x^{2} + 8 \, x + 4\right ) + \frac {3}{10} \, \log \left (4 \, x^{2} - \sqrt {4 \, x^{2} - 2 \, x - 3} {\left (2 \, x + 3\right )} + 5 \, x + 5\right ) + \frac {2}{5} \, \log \left (-4 \, x + 2 \, \sqrt {4 \, x^{2} - 2 \, x - 3} + 1\right ) \] Input:

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

Output:

-7/10*arctan(5/2*x + 2) + 7/10*arctan(-x + 1/2*sqrt(4*x^2 - 2*x - 3) - 3/2 
) + 7/10*arctan(-5*x + 5/2*sqrt(4*x^2 - 2*x - 3) - 1/2) - 3/10*log(20*x^2 
- sqrt(4*x^2 - 2*x - 3)*(10*x + 1) - 3*x - 7) + 3/10*log(5*x^2 + 8*x + 4) 
+ 3/10*log(4*x^2 - sqrt(4*x^2 - 2*x - 3)*(2*x + 3) + 5*x + 5) + 2/5*log(-4 
*x + 2*sqrt(4*x^2 - 2*x - 3) + 1)
 

Sympy [F]

\[ \int \frac {1}{1+3 x+\sqrt {-3-2 x+4 x^2}} \, dx=\int \frac {1}{3 x + \sqrt {4 x^{2} - 2 x - 3} + 1}\, dx \] Input:

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

Output:

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

Maxima [F]

\[ \int \frac {1}{1+3 x+\sqrt {-3-2 x+4 x^2}} \, dx=\int { \frac {1}{3 \, x + \sqrt {4 \, x^{2} - 2 \, x - 3} + 1} \,d x } \] Input:

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

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 175 vs. \(2 (87) = 174\).

Time = 0.12 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.67 \[ \int \frac {1}{1+3 x+\sqrt {-3-2 x+4 x^2}} \, dx=-\frac {7}{10} \, \arctan \left (\frac {5}{2} \, x + 2\right ) + \frac {7}{10} \, \arctan \left (-x + \frac {1}{2} \, \sqrt {4 \, x^{2} - 2 \, x - 3} - \frac {3}{2}\right ) + \frac {7}{10} \, \arctan \left (-5 \, x + \frac {5}{2} \, \sqrt {4 \, x^{2} - 2 \, x - 3} - \frac {1}{2}\right ) - \frac {3}{10} \, \log \left (5 \, {\left (2 \, x - \sqrt {4 \, x^{2} - 2 \, x - 3}\right )}^{2} + 4 \, x - 2 \, \sqrt {4 \, x^{2} - 2 \, x - 3} + 1\right ) + \frac {3}{10} \, \log \left ({\left (2 \, x - \sqrt {4 \, x^{2} - 2 \, x - 3}\right )}^{2} + 12 \, x - 6 \, \sqrt {4 \, x^{2} - 2 \, x - 3} + 13\right ) + \frac {3}{10} \, \log \left (5 \, x^{2} + 8 \, x + 4\right ) + \frac {2}{5} \, \log \left ({\left | -4 \, x + 2 \, \sqrt {4 \, x^{2} - 2 \, x - 3} + 1 \right |}\right ) \] Input:

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

Output:

-7/10*arctan(5/2*x + 2) + 7/10*arctan(-x + 1/2*sqrt(4*x^2 - 2*x - 3) - 3/2 
) + 7/10*arctan(-5*x + 5/2*sqrt(4*x^2 - 2*x - 3) - 1/2) - 3/10*log(5*(2*x 
- sqrt(4*x^2 - 2*x - 3))^2 + 4*x - 2*sqrt(4*x^2 - 2*x - 3) + 1) + 3/10*log 
((2*x - sqrt(4*x^2 - 2*x - 3))^2 + 12*x - 6*sqrt(4*x^2 - 2*x - 3) + 13) + 
3/10*log(5*x^2 + 8*x + 4) + 2/5*log(abs(-4*x + 2*sqrt(4*x^2 - 2*x - 3) + 1 
))
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{1+3 x+\sqrt {-3-2 x+4 x^2}} \, dx=-\int \frac {\sqrt {4\,x^2-2\,x-3}}{5\,x^2+8\,x+4} \,d x+\ln \left (x+\frac {4}{5}-\frac {2}{5}{}\mathrm {i}\right )\,\left (\frac {3}{10}+\frac {7}{20}{}\mathrm {i}\right )+\ln \left (x+\frac {4}{5}+\frac {2}{5}{}\mathrm {i}\right )\,\left (\frac {3}{10}-\frac {7}{20}{}\mathrm {i}\right ) \] Input:

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

Output:

log(x + (4/5 - 2i/5))*(3/10 + 7i/20) + log(x + (4/5 + 2i/5))*(3/10 - 7i/20 
) - int((4*x^2 - 2*x - 3)^(1/2)/(8*x + 5*x^2 + 4), x)
 

Reduce [B] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.89 \[ \int \frac {1}{1+3 x+\sqrt {-3-2 x+4 x^2}} \, dx=\frac {7 \mathit {atan} \left (\frac {5 \sqrt {4 x^{2}-2 x -3}}{2}+5 x +\frac {1}{2}\right )}{5}+\frac {3 \,\mathrm {log}\left (\frac {80 \sqrt {4 x^{2}-2 x -3}\, x +8 \sqrt {4 x^{2}-2 x -3}+160 x^{2}-24 x -56}{\sqrt {13}}\right )}{5}-\mathrm {log}\left (\frac {2 \sqrt {4 x^{2}-2 x -3}+4 x -1}{\sqrt {13}}\right ) \] Input:

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

Output:

(7*atan((5*sqrt(4*x**2 - 2*x - 3) + 10*x + 1)/2) + 3*log((80*sqrt(4*x**2 - 
 2*x - 3)*x + 8*sqrt(4*x**2 - 2*x - 3) + 160*x**2 - 24*x - 56)/sqrt(13)) - 
 5*log((2*sqrt(4*x**2 - 2*x - 3) + 4*x - 1)/sqrt(13)))/5