\(\int \frac {1}{\sqrt {b x^2}} \, dx\) [90]

Optimal result

Integrand size = 9, antiderivative size = 13 \[ \int \frac {1}{\sqrt {b x^2}} \, dx=\frac {x \log (x)}{\sqrt {b x^2}} \]

[Out]

x*ln(x)/(b*x^2)^(1/2)

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {15, 29} \[ \int \frac {1}{\sqrt {b x^2}} \, dx=\frac {x \log (x)}{\sqrt {b x^2}} \]

[In]

Int[1/Sqrt[b*x^2],x]

[Out]

(x*Log[x])/Sqrt[b*x^2]

Rule 15

Int[(u_.)*((a_.)*(x_)^(n_))^(m_), x_Symbol] :> Dist[a^IntPart[m]*((a*x^n)^FracPart[m]/x^(n*FracPart[m])), Int[
u*x^(m*n), x], x] /; FreeQ[{a, m, n}, x] &&  !IntegerQ[m]

Rule 29

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

Rubi steps \begin{align*} \text {integral}& = \frac {x \int \frac {1}{x} \, dx}{\sqrt {b x^2}} \\ & = \frac {x \log (x)}{\sqrt {b x^2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\sqrt {b x^2}} \, dx=\frac {x \log (x)}{\sqrt {b x^2}} \]

[In]

Integrate[1/Sqrt[b*x^2],x]

[Out]

(x*Log[x])/Sqrt[b*x^2]

Maple [A] (verified)

Time = 0.04 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.92

method	result	size
default	\(\frac {x \ln \left (x \right )}{\sqrt {b \,x^{2}}}\)	\(12\)
risch	\(\frac {x \ln \left (x \right )}{\sqrt {b \,x^{2}}}\)	\(12\)

[In]

int(1/(b*x^2)^(1/2),x,method=_RETURNVERBOSE)

[Out]

x*ln(x)/(b*x^2)^(1/2)

Fricas [A] (verification not implemented)

none

Time = 0.34 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.23 \[ \int \frac {1}{\sqrt {b x^2}} \, dx=\frac {\sqrt {b x^{2}} \log \left (x\right )}{b x} \]

[In]

integrate(1/(b*x^2)^(1/2),x, algorithm="fricas")

[Out]

sqrt(b*x^2)*log(x)/(b*x)

Sympy [A] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.92 \[ \int \frac {1}{\sqrt {b x^2}} \, dx=\frac {x \log {\left (x \right )}}{\sqrt {b x^{2}}} \]

[In]

integrate(1/(b*x**2)**(1/2),x)

[Out]

x*log(x)/sqrt(b*x**2)

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 6, normalized size of antiderivative = 0.46 \[ \int \frac {1}{\sqrt {b x^2}} \, dx=\frac {\log \left (x\right )}{\sqrt {b}} \]

[In]

integrate(1/(b*x^2)^(1/2),x, algorithm="maxima")

[Out]

log(x)/sqrt(b)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.46 \[ \int \frac {1}{\sqrt {b x^2}} \, dx=\frac {\log \left (\sqrt {{\left | b \right |}} {\left | x \right |} {\left | \mathrm {sgn}\left (x\right ) \right |}\right )}{\sqrt {b} \mathrm {sgn}\left (x\right )} \]

[In]

integrate(1/(b*x^2)^(1/2),x, algorithm="giac")

[Out]

log(sqrt(abs(b))*abs(x)*abs(sgn(x)))/(sqrt(b)*sgn(x))

Mupad [B] (verification not implemented)

Time = 0.00 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.77 \[ \int \frac {1}{\sqrt {b x^2}} \, dx=\frac {\ln \left (b\,x\right )\,\mathrm {sign}\left (x\right )}{\sqrt {b}} \]

[In]

int(1/(b*x^2)^(1/2),x)

[Out]

(log(b*x)*sign(x))/b^(1/2)