### 3.301 $$\int \frac{\log (x)}{\sqrt{a+b \log (x)}} \, dx$$

Optimal. Leaf size=60 $\frac{x \sqrt{a+b \log (x)}}{b}-\frac{\sqrt{\pi } (2 a+b) e^{-\frac{a}{b}} \text{Erfi}\left (\frac{\sqrt{a+b \log (x)}}{\sqrt{b}}\right )}{2 b^{3/2}}$

[Out]

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

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Rubi [A]  time = 0.0692622, antiderivative size = 60, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 13, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.308, Rules used = {2294, 2299, 2180, 2204} $\frac{x \sqrt{a+b \log (x)}}{b}-\frac{\sqrt{\pi } (2 a+b) e^{-\frac{a}{b}} \text{Erfi}\left (\frac{\sqrt{a+b \log (x)}}{\sqrt{b}}\right )}{2 b^{3/2}}$

Antiderivative was successfully veriﬁed.

[In]

Int[Log[x]/Sqrt[a + b*Log[x]],x]

[Out]

-((2*a + b)*Sqrt[Pi]*Erfi[Sqrt[a + b*Log[x]]/Sqrt[b]])/(2*b^(3/2)*E^(a/b)) + (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 2204

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[(F^a*Sqrt[Pi]*Erfi[(c + d*x)*Rt[b*Log[F], 2
]])/(2*d*Rt[b*Log[F], 2]), x] /; FreeQ[{F, a, b, c, d}, x] && PosQ[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^{-\frac{a}{b}} \sqrt{\pi } \text{erfi}\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 = 0.109793, size = 72, normalized size = 1.2 $\frac{2 x (a+b \log (x))-(2 a+b) e^{-\frac{a}{b}} \sqrt{-\frac{a+b \log (x)}{b}} \text{Gamma}\left (\frac{1}{2},-\frac{a+b \log (x)}{b}\right )}{2 b \sqrt{a+b \log (x)}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

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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)

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Maxima [B]  time = 1.22265, size = 146, normalized size = 2.43 \begin{align*} -\frac{\frac{2 \, \sqrt{\pi } a \operatorname{erf}\left (\sqrt{b \log \left (x\right ) + a} \sqrt{-\frac{1}{b}}\right ) e^{\left (-\frac{a}{b}\right )}}{\sqrt{-\frac{1}{b}}} + \frac{\sqrt{\pi } b \operatorname{erf}\left (\sqrt{b \log \left (x\right ) + a} \sqrt{-\frac{1}{b}}\right ) e^{\left (-\frac{a}{b}\right )}}{\sqrt{-\frac{1}{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*erf(sqrt(b*log(x) + a)*sqrt(-1/b))*e^(-a/b)/sqrt(-1/b) + sqrt(pi)*b*erf(sqrt(b*log(x) + a)*
sqrt(-1/b))*e^(-a/b)/sqrt(-1/b) - 2*sqrt(b*log(x) + a)*b*e^((b*log(x) + a)/b - a/b))/b^2

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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

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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)

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Giac [A]  time = 1.35694, size = 120, normalized size = 2. \begin{align*} \frac{\sqrt{\pi } \operatorname{erf}\left (-\frac{\sqrt{b \log \left (x\right ) + a} \sqrt{-b}}{b}\right ) e^{\left (-\frac{a}{b}\right )}}{2 \, \sqrt{-b}} + \frac{\sqrt{\pi } a \operatorname{erf}\left (-\frac{\sqrt{b \log \left (x\right ) + a} \sqrt{-b}}{b}\right ) e^{\left (-\frac{a}{b}\right )}}{\sqrt{-b} 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]

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