### 3.302 $$\int \frac{\log (x)}{\sqrt{a-b \log (x)}} \, dx$$

Optimal. Leaf size=64 $-\frac{\sqrt{\pi } (2 a-b) e^{a/b} \text{Erf}\left (\frac{\sqrt{a-b \log (x)}}{\sqrt{b}}\right )}{2 b^{3/2}}-\frac{x \sqrt{a-b \log (x)}}{b}$

[Out]

-((2*a - b)*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a - b*Log[x]]/Sqrt[b]])/(2*b^(3/2)) - (x*Sqrt[a - b*Log[x]])/b

________________________________________________________________________________________

Rubi [A]  time = 0.0677266, antiderivative size = 64, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 14, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.286, Rules used = {2294, 2299, 2180, 2205} $-\frac{\sqrt{\pi } (2 a-b) e^{a/b} \text{Erf}\left (\frac{\sqrt{a-b \log (x)}}{\sqrt{b}}\right )}{2 b^{3/2}}-\frac{x \sqrt{a-b \log (x)}}{b}$

Antiderivative was successfully veriﬁed.

[In]

Int[Log[x]/Sqrt[a - b*Log[x]],x]

[Out]

-((2*a - b)*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a - b*Log[x]]/Sqrt[b]])/(2*b^(3/2)) - (x*Sqrt[a - b*Log[x]])/b

Rule 2294

Int[((A_.) + Log[(c_.)*((d_.) + (e_.)*(x_))^(n_.)]*(B_.))/Sqrt[Log[(c_.)*((d_.) + (e_.)*(x_))^(n_.)]*(b_.) + (
a_)], x_Symbol] :> Simp[(B*(d + e*x)*Sqrt[a + b*Log[c*(d + e*x)^n]])/(b*e), x] + Dist[(2*A*b - B*(2*a + b*n))/
(2*b), Int[1/Sqrt[a + b*Log[c*(d + e*x)^n]], x], x] /; FreeQ[{a, b, c, d, e, A, B, n}, x]

Rule 2299

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_), x_Symbol] :> Dist[1/(n*c^(1/n)), Subst[Int[E^(x/n)*(a + b*x)^p
, x], x, Log[c*x^n]], x] /; FreeQ[{a, b, c, p}, x] && IntegerQ[1/n]

Rule 2180

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Dist[2/d, Subst[Int[F^(g*(e - (c*
f)/d) + (f*g*x^2)/d), x], x, Sqrt[c + d*x]], x] /; FreeQ[{F, c, d, e, f, g}, x] &&  !\$UseGamma === True

Rule 2205

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[(F^a*Sqrt[Pi]*Erf[(c + d*x)*Rt[-(b*Log[F]),
2]])/(2*d*Rt[-(b*Log[F]), 2]), x] /; FreeQ[{F, a, b, c, d}, x] && NegQ[b]

Rubi steps

\begin{align*} \int \frac{\log (x)}{\sqrt{a-b \log (x)}} \, dx &=-\frac{x \sqrt{a-b \log (x)}}{b}-\frac{(-2 a+b) \int \frac{1}{\sqrt{a-b \log (x)}} \, dx}{2 b}\\ &=-\frac{x \sqrt{a-b \log (x)}}{b}-\frac{(-2 a+b) \operatorname{Subst}\left (\int \frac{e^x}{\sqrt{a-b x}} \, dx,x,\log (x)\right )}{2 b}\\ &=-\frac{x \sqrt{a-b \log (x)}}{b}-\frac{(2 a-b) \operatorname{Subst}\left (\int e^{\frac{a}{b}-\frac{x^2}{b}} \, dx,x,\sqrt{a-b \log (x)}\right )}{b^2}\\ &=-\frac{(2 a-b) e^{a/b} \sqrt{\pi } \text{erf}\left (\frac{\sqrt{a-b \log (x)}}{\sqrt{b}}\right )}{2 b^{3/2}}-\frac{x \sqrt{a-b \log (x)}}{b}\\ \end{align*}

Mathematica [A]  time = 0.0908824, size = 71, normalized size = 1.11 $\frac{-(b-2 a) e^{a/b} \sqrt{\frac{a}{b}-\log (x)} \text{Gamma}\left (\frac{1}{2},\frac{a}{b}-\log (x)\right )-2 x (a-b \log (x))}{2 b \sqrt{a-b \log (x)}}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[Log[x]/Sqrt[a - b*Log[x]],x]

[Out]

(-((-2*a + b)*E^(a/b)*Gamma[1/2, a/b - Log[x]]*Sqrt[a/b - Log[x]]) - 2*x*(a - b*Log[x]))/(2*b*Sqrt[a - b*Log[x
]])

________________________________________________________________________________________

Maple [F]  time = 0.02, size = 0, normalized size = 0. \begin{align*} \int{\ln \left ( x \right ){\frac{1}{\sqrt{a-b\ln \left ( x \right ) }}}}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(ln(x)/(a-b*ln(x))^(1/2),x)

[Out]

int(ln(x)/(a-b*ln(x))^(1/2),x)

________________________________________________________________________________________

Maxima [A]  time = 1.21922, size = 127, normalized size = 1.98 \begin{align*} -\frac{2 \, \sqrt{\pi } a \sqrt{b} \operatorname{erf}\left (\frac{\sqrt{-b \log \left (x\right ) + a}}{\sqrt{b}}\right ) e^{\frac{a}{b}} - \sqrt{\pi } b^{\frac{3}{2}} \operatorname{erf}\left (\frac{\sqrt{-b \log \left (x\right ) + a}}{\sqrt{b}}\right ) e^{\frac{a}{b}} + 2 \, \sqrt{-b \log \left (x\right ) + a} b e^{\left (\frac{b \log \left (x\right ) - a}{b} + \frac{a}{b}\right )}}{2 \, b^{2}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(2*sqrt(pi)*a*sqrt(b)*erf(sqrt(-b*log(x) + a)/sqrt(b))*e^(a/b) - sqrt(pi)*b^(3/2)*erf(sqrt(-b*log(x) + a)
/sqrt(b))*e^(a/b) + 2*sqrt(-b*log(x) + a)*b*e^((b*log(x) - a)/b + a/b))/b^2

________________________________________________________________________________________

Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: UnboundLocalError

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\log{\left (x \right )}}{\sqrt{a - b \log{\left (x \right )}}}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(ln(x)/(a-b*ln(x))**(1/2),x)

[Out]

Integral(log(x)/sqrt(a - b*log(x)), x)

________________________________________________________________________________________

Giac [A]  time = 1.26626, size = 100, normalized size = 1.56 \begin{align*} \frac{\sqrt{\pi } a \operatorname{erf}\left (-\frac{\sqrt{-b \log \left (x\right ) + a}}{\sqrt{b}}\right ) e^{\frac{a}{b}}}{b^{\frac{3}{2}}} - \frac{\sqrt{\pi } \operatorname{erf}\left (-\frac{\sqrt{-b \log \left (x\right ) + a}}{\sqrt{b}}\right ) e^{\frac{a}{b}}}{2 \, \sqrt{b}} - \frac{\sqrt{-b \log \left (x\right ) + a} x}{b} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(pi)*a*erf(-sqrt(-b*log(x) + a)/sqrt(b))*e^(a/b)/b^(3/2) - 1/2*sqrt(pi)*erf(-sqrt(-b*log(x) + a)/sqrt(b))*
e^(a/b)/sqrt(b) - sqrt(-b*log(x) + a)*x/b