\(\int \frac {1}{x^3 (1+2 x+\sqrt {-3-2 x+4 x^2})} \, dx\) [6]

Optimal result

Integrand size = 25, antiderivative size = 131 \[ \int \frac {1}{x^3 \left (1+2 x+\sqrt {-3-2 x+4 x^2}\right )} \, dx=-\frac {1}{8 x^2}-\frac {1}{8 x}+\frac {\sqrt {-3-2 x+4 x^2}}{8 x^2}-\frac {\sqrt {-3-2 x+4 x^2}}{3 x}-\frac {37 \arctan \left (\frac {3+x}{\sqrt {3} \sqrt {-3-2 x+4 x^2}}\right )}{48 \sqrt {3}}+\frac {3}{16} \text {arctanh}\left (\frac {7+11 x}{\sqrt {-3-2 x+4 x^2}}\right )-\frac {3 \log (x)}{16}+\frac {3}{16} \log (2+3 x) \] Output:

-1/8/x^2-1/8/x+1/8*(4*x^2-2*x-3)^(1/2)/x^2-1/3*(4*x^2-2*x-3)^(1/2)/x-37/14 
4*3^(1/2)*arctan(1/3*(3+x)*3^(1/2)/(4*x^2-2*x-3)^(1/2))+3/16*arctanh((7+11 
*x)/(4*x^2-2*x-3)^(1/2))-3/16*ln(x)+3/16*ln(2+3*x)
 

Mathematica [A] (verified)

Time = 0.42 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.91 \[ \int \frac {1}{x^3 \left (1+2 x+\sqrt {-3-2 x+4 x^2}\right )} \, dx=\frac {1}{144} \left (-\frac {18 (1+x)}{x^2}+\frac {6 (3-8 x) \sqrt {-3-2 x+4 x^2}}{x^2}+74 \sqrt {3} \arctan \left (\frac {-2 x+\sqrt {-3-2 x+4 x^2}}{\sqrt {3}}\right )-27 \log \left (x \left (-1+4 x-2 \sqrt {-3-2 x+4 x^2}\right )\right )+54 \log \left (-5-6 x+3 \sqrt {-3-2 x+4 x^2}\right )\right ) \] Input:

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

Output:

((-18*(1 + x))/x^2 + (6*(3 - 8*x)*Sqrt[-3 - 2*x + 4*x^2])/x^2 + 74*Sqrt[3] 
*ArcTan[(-2*x + Sqrt[-3 - 2*x + 4*x^2])/Sqrt[3]] - 27*Log[x*(-1 + 4*x - 2* 
Sqrt[-3 - 2*x + 4*x^2])] + 54*Log[-5 - 6*x + 3*Sqrt[-3 - 2*x + 4*x^2]])/14 
4
 

Rubi [A] (verified)

Time = 0.50 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.27, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, 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.

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

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {1}{4 x^3}-\frac {9 \sqrt {4 x^2-2 x-3}}{16 x}+\frac {27 \sqrt {4 x^2-2 x-3}}{16 (3 x+2)}+\frac {3 \sqrt {4 x^2-2 x-3}}{8 x^2}+\frac {1}{8 x^2}-\frac {\sqrt {4 x^2-2 x-3}}{4 x^3}-\frac {3}{16 x}+\frac {9}{16 (3 x+2)}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {7}{16} \sqrt {3} \arctan \left (\frac {x+3}{\sqrt {3} \sqrt {4 x^2-2 x-3}}\right )+\frac {13 \arctan \left (\frac {x+3}{\sqrt {3} \sqrt {4 x^2-2 x-3}}\right )}{24 \sqrt {3}}+\frac {3}{16} \text {arctanh}\left (\frac {11 x+7}{\sqrt {4 x^2-2 x-3}}\right )+\frac {\sqrt {4 x^2-2 x-3} (x+3)}{24 x^2}-\frac {3 \sqrt {4 x^2-2 x-3}}{8 x}-\frac {1}{8 x^2}-\frac {1}{8 x}-\frac {3 \log (x)}{16}+\frac {3}{16} \log (3 x+2)\)

Input:

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

Output:

-1/8*1/x^2 - 1/(8*x) - (3*Sqrt[-3 - 2*x + 4*x^2])/(8*x) + ((3 + x)*Sqrt[-3 
 - 2*x + 4*x^2])/(24*x^2) + (13*ArcTan[(3 + x)/(Sqrt[3]*Sqrt[-3 - 2*x + 4* 
x^2])])/(24*Sqrt[3]) - (7*Sqrt[3]*ArcTan[(3 + x)/(Sqrt[3]*Sqrt[-3 - 2*x + 
4*x^2])])/16 + (3*ArcTanh[(7 + 11*x)/Sqrt[-3 - 2*x + 4*x^2]])/16 - (3*Log[ 
x])/16 + (3*Log[2 + 3*x])/16
 

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. \(218\) vs. \(2(104)=208\).

Time = 0.18 (sec) , antiderivative size = 219, normalized size of antiderivative = 1.67

method	result	size
default	\(-\frac {1}{8 x}-\frac {3 \ln \left (x \right )}{16}+\frac {3 \ln \left (2+3 x \right )}{16}-\frac {1}{8 x^{2}}-\frac {\left (4 x^{2}-2 x -3\right )^{\frac {3}{2}}}{24 x^{2}}+\frac {5 \left (4 x^{2}-2 x -3\right )^{\frac {3}{2}}}{36 x}-\frac {37 \sqrt {4 x^{2}-2 x -3}}{144}+\frac {37 \sqrt {3}\, \arctan \left (\frac {\left (-6-2 x \right ) \sqrt {3}}{6 \sqrt {4 x^{2}-2 x -3}}\right )}{144}-\frac {5 \left (8 x -2\right ) \sqrt {4 x^{2}-2 x -3}}{72}+\frac {33 \ln \left (\frac {\left (4 x -1\right ) \sqrt {4}}{4}+\sqrt {4 x^{2}-2 x -3}\right ) \sqrt {4}}{64}+\frac {3 \sqrt {36 \left (x +\frac {2}{3}\right )^{2}-66 x -43}}{16}-\frac {33 \ln \left (\frac {\left (4 x -1\right ) \sqrt {4}}{4}+\sqrt {4 \left (x +\frac {2}{3}\right )^{2}-\frac {22 x}{3}-\frac {43}{9}}\right ) \sqrt {4}}{64}-\frac {3 \,\operatorname {arctanh}\left (\frac {-21-33 x}{\sqrt {36 \left (x +\frac {2}{3}\right )^{2}-66 x -43}}\right )}{16}\)	\(219\)
trager	\(\frac {\left (x -1\right ) \left (2 x +1\right )}{8 x^{2}}-\frac {\left (8 x -3\right ) \sqrt {4 x^{2}-2 x -3}}{24 x^{2}}+\frac {\operatorname {RootOf}\left (3 \textit {\_Z}^{2}+27 \textit {\_Z} +403\right ) \ln \left (-\frac {3 \operatorname {RootOf}\left (3 \textit {\_Z}^{2}+27 \textit {\_Z} +403\right )^{2} x -3 \operatorname {RootOf}\left (3 \textit {\_Z}^{2}+27 \textit {\_Z} +403\right )^{2}+1369 \operatorname {RootOf}\left (3 \textit {\_Z}^{2}+27 \textit {\_Z} +403\right ) \sqrt {4 x^{2}-2 x -3}+175 \operatorname {RootOf}\left (3 \textit {\_Z}^{2}+27 \textit {\_Z} +403\right ) x +528 \operatorname {RootOf}\left (3 \textit {\_Z}^{2}+27 \textit {\_Z} +403\right )+9583 \sqrt {4 x^{2}-2 x -3}-38632 x -23232}{2+3 x}\right )}{24}-\frac {\ln \left (-\frac {3 \operatorname {RootOf}\left (3 \textit {\_Z}^{2}+27 \textit {\_Z} +403\right )^{2} x -3 \operatorname {RootOf}\left (3 \textit {\_Z}^{2}+27 \textit {\_Z} +403\right )^{2}-1369 \operatorname {RootOf}\left (3 \textit {\_Z}^{2}+27 \textit {\_Z} +403\right ) \sqrt {4 x^{2}-2 x -3}-121 \operatorname {RootOf}\left (3 \textit {\_Z}^{2}+27 \textit {\_Z} +403\right ) x -582 \operatorname {RootOf}\left (3 \textit {\_Z}^{2}+27 \textit {\_Z} +403\right )-2738 \sqrt {4 x^{2}-2 x -3}-39964 x -28227}{2+3 x}\right ) \operatorname {RootOf}\left (3 \textit {\_Z}^{2}+27 \textit {\_Z} +403\right )}{24}-\frac {3 \ln \left (-\frac {3 \operatorname {RootOf}\left (3 \textit {\_Z}^{2}+27 \textit {\_Z} +403\right )^{2} x -3 \operatorname {RootOf}\left (3 \textit {\_Z}^{2}+27 \textit {\_Z} +403\right )^{2}-1369 \operatorname {RootOf}\left (3 \textit {\_Z}^{2}+27 \textit {\_Z} +403\right ) \sqrt {4 x^{2}-2 x -3}-121 \operatorname {RootOf}\left (3 \textit {\_Z}^{2}+27 \textit {\_Z} +403\right ) x -582 \operatorname {RootOf}\left (3 \textit {\_Z}^{2}+27 \textit {\_Z} +403\right )-2738 \sqrt {4 x^{2}-2 x -3}-39964 x -28227}{2+3 x}\right )}{8}\)	\(401\)

Input:

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

Output:

-1/8/x-3/16*ln(x)+3/16*ln(2+3*x)-1/8/x^2-1/24/x^2*(4*x^2-2*x-3)^(3/2)+5/36 
/x*(4*x^2-2*x-3)^(3/2)-37/144*(4*x^2-2*x-3)^(1/2)+37/144*3^(1/2)*arctan(1/ 
6*(-6-2*x)*3^(1/2)/(4*x^2-2*x-3)^(1/2))-5/72*(8*x-2)*(4*x^2-2*x-3)^(1/2)+3 
3/64*ln(1/4*(4*x-1)*4^(1/2)+(4*x^2-2*x-3)^(1/2))*4^(1/2)+3/16*(36*(x+2/3)^ 
2-66*x-43)^(1/2)-33/64*ln(1/4*(4*x-1)*4^(1/2)+(4*(x+2/3)^2-22/3*x-43/9)^(1 
/2))*4^(1/2)-3/16*arctanh(9/2*(-14/3-22/3*x)/(36*(x+2/3)^2-66*x-43)^(1/2))
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.02 \[ \int \frac {1}{x^3 \left (1+2 x+\sqrt {-3-2 x+4 x^2}\right )} \, dx=\frac {74 \, \sqrt {3} x^{2} \arctan \left (-\frac {2}{3} \, \sqrt {3} x + \frac {1}{3} \, \sqrt {3} \sqrt {4 \, x^{2} - 2 \, x - 3}\right ) + 27 \, x^{2} \log \left (3 \, x + 2\right ) - 27 \, x^{2} \log \left (x\right ) - 27 \, x^{2} \log \left (-2 \, x + \sqrt {4 \, x^{2} - 2 \, x - 3} - 1\right ) + 27 \, x^{2} \log \left (-6 \, x + 3 \, \sqrt {4 \, x^{2} - 2 \, x - 3} - 5\right ) - 96 \, x^{2} - 6 \, \sqrt {4 \, x^{2} - 2 \, x - 3} {\left (8 \, x - 3\right )} - 18 \, x - 18}{144 \, x^{2}} \] Input:

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

Output:

1/144*(74*sqrt(3)*x^2*arctan(-2/3*sqrt(3)*x + 1/3*sqrt(3)*sqrt(4*x^2 - 2*x 
 - 3)) + 27*x^2*log(3*x + 2) - 27*x^2*log(x) - 27*x^2*log(-2*x + sqrt(4*x^ 
2 - 2*x - 3) - 1) + 27*x^2*log(-6*x + 3*sqrt(4*x^2 - 2*x - 3) - 5) - 96*x^ 
2 - 6*sqrt(4*x^2 - 2*x - 3)*(8*x - 3) - 18*x - 18)/x^2
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

integrate(1/x^3/(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^3), x)
 

Giac [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 185, normalized size of antiderivative = 1.41 \[ \int \frac {1}{x^3 \left (1+2 x+\sqrt {-3-2 x+4 x^2}\right )} \, dx=\frac {37}{72} \, \sqrt {3} \arctan \left (-\frac {1}{3} \, \sqrt {3} {\left (2 \, x - \sqrt {4 \, x^{2} - 2 \, x - 3}\right )}\right ) - \frac {10 \, {\left (2 \, x - \sqrt {4 \, x^{2} - 2 \, x - 3}\right )}^{3} + 15 \, {\left (2 \, x - \sqrt {4 \, x^{2} - 2 \, x - 3}\right )}^{2} - 6 \, x + 3 \, \sqrt {4 \, x^{2} - 2 \, x - 3} + 81}{6 \, {\left ({\left (2 \, x - \sqrt {4 \, x^{2} - 2 \, x - 3}\right )}^{2} + 3\right )}^{2}} - \frac {x + 1}{8 \, x^{2}} + \frac {3}{16} \, \log \left ({\left | 3 \, x + 2 \right |}\right ) - \frac {3}{16} \, \log \left ({\left | x \right |}\right ) - \frac {3}{16} \, \log \left ({\left | -2 \, x + \sqrt {4 \, x^{2} - 2 \, x - 3} - 1 \right |}\right ) + \frac {3}{16} \, \log \left ({\left | -6 \, x + 3 \, \sqrt {4 \, x^{2} - 2 \, x - 3} - 5 \right |}\right ) \] Input:

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

Output:

37/72*sqrt(3)*arctan(-1/3*sqrt(3)*(2*x - sqrt(4*x^2 - 2*x - 3))) - 1/6*(10 
*(2*x - sqrt(4*x^2 - 2*x - 3))^3 + 15*(2*x - sqrt(4*x^2 - 2*x - 3))^2 - 6* 
x + 3*sqrt(4*x^2 - 2*x - 3) + 81)/((2*x - sqrt(4*x^2 - 2*x - 3))^2 + 3)^2 
- 1/8*(x + 1)/x^2 + 3/16*log(abs(3*x + 2)) - 3/16*log(abs(x)) - 3/16*log(a 
bs(-2*x + sqrt(4*x^2 - 2*x - 3) - 1)) + 3/16*log(abs(-6*x + 3*sqrt(4*x^2 - 
 2*x - 3) - 5))
 

Mupad [F(-1)]

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

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

Output:

- (atan(x*3i + 1i)*3i)/8 - int((4*x^2 - 2*x - 3)^(1/2)/(2*(2*x^3 + 3*x^4)) 
, x) - (x/8 + 1/8)/x^2
 

Reduce [B] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.11 \[ \int \frac {1}{x^3 \left (1+2 x+\sqrt {-3-2 x+4 x^2}\right )} \, dx=\frac {4070 \sqrt {3}\, \mathit {atan} \left (\frac {\sqrt {4 x^{2}-2 x -3}+2 x}{\sqrt {3}}\right ) x^{2}-2640 \sqrt {4 x^{2}-2 x -3}\, x +990 \sqrt {4 x^{2}-2 x -3}+1485 \,\mathrm {log}\left (3 x +2\right ) x^{2}+1485 \,\mathrm {log}\left (\frac {26 \sqrt {4 x^{2}-2 x -3}+52 x +26}{\sqrt {13}}\right ) x^{2}-1485 \,\mathrm {log}\left (\frac {6 \sqrt {4 x^{2}-2 x -3}+12 x +10}{\sqrt {13}}\right ) x^{2}-1485 \,\mathrm {log}\left (x \right ) x^{2}+3184 x^{2}-990 x -990}{7920 x^{2}} \] Input:

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

Output:

(4070*sqrt(3)*atan((sqrt(4*x**2 - 2*x - 3) + 2*x)/sqrt(3))*x**2 - 2640*sqr 
t(4*x**2 - 2*x - 3)*x + 990*sqrt(4*x**2 - 2*x - 3) + 1485*log(3*x + 2)*x** 
2 + 1485*log((26*sqrt(4*x**2 - 2*x - 3) + 52*x + 26)/sqrt(13))*x**2 - 1485 
*log((6*sqrt(4*x**2 - 2*x - 3) + 12*x + 10)/sqrt(13))*x**2 - 1485*log(x)*x 
**2 + 3184*x**2 - 990*x - 990)/(7920*x**2)