\(\int \frac {1}{x \sqrt {-4-12 x-9 x^2}} \, dx\) [2422]

Optimal result

Integrand size = 18, antiderivative size = 27 \[ \int \frac {1}{x \sqrt {-4-12 x-9 x^2}} \, dx=-\frac {(2+3 x) \text {arctanh}(1+3 x)}{\sqrt {-4-12 x-9 x^2}} \]

[Out]

-(2+3*x)*arctanh(1+3*x)/(-(2+3*x)^2)^(1/2)

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(55\) vs. \(2(27)=54\).

Time = 0.01 (sec) , antiderivative size = 55, normalized size of antiderivative = 2.04, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {660, 36, 31, 29} \[ \int \frac {1}{x \sqrt {-4-12 x-9 x^2}} \, dx=\frac {(3 x+2) \log (x)}{2 \sqrt {-9 x^2-12 x-4}}-\frac {(3 x+2) \log (3 x+2)}{2 \sqrt {-9 x^2-12 x-4}} \]

[In]

Int[1/(x*Sqrt[-4 - 12*x - 9*x^2]),x]

[Out]

((2 + 3*x)*Log[x])/(2*Sqrt[-4 - 12*x - 9*x^2]) - ((2 + 3*x)*Log[2 + 3*x])/(2*Sqrt[-4 - 12*x - 9*x^2])

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 36

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Dist[b/(b*c - a*d), Int[1/(a + b*x), x], x] -
Dist[d/(b*c - a*d), Int[1/(c + d*x), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

Rule 660

Int[((d_.) + (e_.)*(x_))^(m_)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[(a + b*x + c*x^2)^Fra
cPart[p]/(c^IntPart[p]*(b/2 + c*x)^(2*FracPart[p])), Int[(d + e*x)^m*(b/2 + c*x)^(2*p), x], x] /; FreeQ[{a, b,
c, d, e, m, p}, x] && EqQ[b^2 - 4*a*c, 0] &&  !IntegerQ[p] && NeQ[2*c*d - b*e, 0]

Rubi steps \begin{align*} \text {integral}& = -\left (-\frac {(-6-9 x) \int \frac {1}{(-6-9 x) x} \, dx}{\sqrt {-4-12 x-9 x^2}}\right ) \\ & = -\frac {(3 (-6-9 x)) \int \frac {1}{-6-9 x} \, dx}{2 \sqrt {-4-12 x-9 x^2}}+-\frac {(-6-9 x) \int \frac {1}{x} \, dx}{6 \sqrt {-4-12 x-9 x^2}} \\ & = \frac {(2+3 x) \log (x)}{2 \sqrt {-4-12 x-9 x^2}}-\frac {(2+3 x) \log (2+3 x)}{2 \sqrt {-4-12 x-9 x^2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.01 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.22 \[ \int \frac {1}{x \sqrt {-4-12 x-9 x^2}} \, dx=\frac {(2+3 x) (\log (x)-\log (2+3 x))}{2 \sqrt {-(2+3 x)^2}} \]

[In]

Integrate[1/(x*Sqrt[-4 - 12*x - 9*x^2]),x]

[Out]

((2 + 3*x)*(Log[x] - Log[2 + 3*x]))/(2*Sqrt[-(2 + 3*x)^2])

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 0.33 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.78

method	result	size
meijerg	\(-\frac {i \left (\ln \left (x \right )+\ln \left (3\right )-\ln \left (2\right )-\ln \left (1+\frac {3 x}{2}\right )\right )}{2}\)	\(21\)
default	\(\frac {\left (2+3 x \right ) \left (\ln \left (x \right )-\ln \left (2+3 x \right )\right )}{2 \sqrt {-\left (2+3 x \right )^{2}}}\)	\(30\)
risch	\(\frac {\left (2+3 x \right ) \ln \left (x \right )}{2 \sqrt {-\left (2+3 x \right )^{2}}}-\frac {\left (2+3 x \right ) \ln \left (2+3 x \right )}{2 \sqrt {-\left (2+3 x \right )^{2}}}\)	\(46\)

[In]

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

[Out]

-1/2*I*(ln(x)+ln(3)-ln(2)-ln(1+3/2*x))

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.30 (sec) , antiderivative size = 11, normalized size of antiderivative = 0.41 \[ \int \frac {1}{x \sqrt {-4-12 x-9 x^2}} \, dx=\frac {1}{2} i \, \log \left (x + \frac {2}{3}\right ) - \frac {1}{2} i \, \log \left (x\right ) \]

[In]

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

[Out]

1/2*I*log(x + 2/3) - 1/2*I*log(x)

Sympy [F]

\[ \int \frac {1}{x \sqrt {-4-12 x-9 x^2}} \, dx=\int \frac {1}{x \sqrt {- \left (3 x + 2\right )^{2}}}\, dx \]

[In]

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

[Out]

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

Maxima [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.27 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.89 \[ \int \frac {1}{x \sqrt {-4-12 x-9 x^2}} \, dx=-\frac {1}{2} i \, \left (-1\right )^{12 \, x + 8} \log \left (\frac {12 \, x}{{\left | x \right |}} + \frac {8}{{\left | x \right |}}\right ) \]

[In]

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

[Out]

-1/2*I*(-1)^(12*x + 8)*log(12*x/abs(x) + 8/abs(x))

Giac [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.28 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.15 \[ \int \frac {1}{x \sqrt {-4-12 x-9 x^2}} \, dx=-\frac {i \, \log \left ({\left | 3 \, x + 2 \right |}\right )}{2 \, \mathrm {sgn}\left (-3 \, x - 2\right )} + \frac {i \, \log \left ({\left | x \right |}\right )}{2 \, \mathrm {sgn}\left (-3 \, x - 2\right )} \]

[In]

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

[Out]

-1/2*I*log(abs(3*x + 2))/sgn(-3*x - 2) + 1/2*I*log(abs(x))/sgn(-3*x - 2)

Mupad [B] (verification not implemented)

Time = 10.11 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x \sqrt {-4-12 x-9 x^2}} \, dx=\frac {\ln \left (\frac {6\,x+4-\sqrt {-{\left (3\,x+2\right )}^2}\,2{}\mathrm {i}}{x}\right )\,1{}\mathrm {i}}{2} \]

[In]

int(1/(x*(-(3*x + 2)^2)^(1/2)),x)

[Out]

(log((6*x - (-(3*x + 2)^2)^(1/2)*2i + 4)/x)*1i)/2