3.12 \(\int \frac{1}{\sqrt{\log (c (d+e x))}} \, dx\)

Optimal. Leaf size=25 \[ \frac{\sqrt{\pi } \text{Erfi}\left (\sqrt{\log (c (d+e x))}\right )}{c e} \]

[Out]

(Sqrt[Pi]*Erfi[Sqrt[Log[c*(d + e*x)]]])/(c*e)

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Rubi [A]  time = 0.0219619, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {2389, 2299, 2180, 2204} \[ \frac{\sqrt{\pi } \text{Erfi}\left (\sqrt{\log (c (d+e x))}\right )}{c e} \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[Log[c*(d + e*x)]],x]

[Out]

(Sqrt[Pi]*Erfi[Sqrt[Log[c*(d + e*x)]]])/(c*e)

Rule 2389

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.), x_Symbol] :> Dist[1/e, Subst[Int[(a + b*Log[c*
x^n])^p, x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, n, p}, 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{1}{\sqrt{\log (c (d+e x))}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{\sqrt{\log (c x)}} \, dx,x,d+e x\right )}{e}\\ &=\frac{\operatorname{Subst}\left (\int \frac{e^x}{\sqrt{x}} \, dx,x,\log (c (d+e x))\right )}{c e}\\ &=\frac{2 \operatorname{Subst}\left (\int e^{x^2} \, dx,x,\sqrt{\log (c (d+e x))}\right )}{c e}\\ &=\frac{\sqrt{\pi } \text{erfi}\left (\sqrt{\log (c (d+e x))}\right )}{c e}\\ \end{align*}

Mathematica [A]  time = 0.0019845, size = 25, normalized size = 1. \[ \frac{\sqrt{\pi } \text{Erfi}\left (\sqrt{\log (c (d+e x))}\right )}{c e} \]

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[Log[c*(d + e*x)]],x]

[Out]

(Sqrt[Pi]*Erfi[Sqrt[Log[c*(d + e*x)]]])/(c*e)

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Maple [F]  time = 0.229, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{\sqrt{\ln \left ( c \left ( ex+d \right ) \right ) }}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/ln(c*(e*x+d))^(1/2),x)

[Out]

int(1/ln(c*(e*x+d))^(1/2),x)

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Maxima [C]  time = 1.22635, size = 34, normalized size = 1.36 \begin{align*} -\frac{i \, \sqrt{\pi } \operatorname{erf}\left (i \, \sqrt{\log \left (c e x + c d\right )}\right )}{c e} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/log(c*(e*x+d))^(1/2),x, algorithm="maxima")

[Out]

-I*sqrt(pi)*erf(I*sqrt(log(c*e*x + c*d)))/(c*e)

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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(1/log(c*(e*x+d))^(1/2),x, algorithm="fricas")

[Out]

Exception raised: UnboundLocalError

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Sympy [A]  time = 5.08629, size = 63, normalized size = 2.52 \begin{align*} \begin{cases} 0 & \text{for}\: c = 0 \\\frac{x}{\sqrt{\log{\left (c d \right )}}} & \text{for}\: e = 0 \\\frac{\sqrt{\pi } \sqrt{- \log{\left (c d + c e x \right )}} \operatorname{erfc}{\left (\sqrt{- \log{\left (c d + c e x \right )}} \right )}}{c e \sqrt{\log{\left (c d + c e x \right )}}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/ln(c*(e*x+d))**(1/2),x)

[Out]

Piecewise((0, Eq(c, 0)), (x/sqrt(log(c*d)), Eq(e, 0)), (sqrt(pi)*sqrt(-log(c*d + c*e*x))*erfc(sqrt(-log(c*d +
c*e*x)))/(c*e*sqrt(log(c*d + c*e*x))), True))

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Giac [A]  time = 1.31681, size = 35, normalized size = 1.4 \begin{align*} \frac{\sqrt{\pi } i \operatorname{erf}\left (-i \sqrt{\log \left (c x e + c d\right )}\right ) e^{\left (-1\right )}}{c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/log(c*(e*x+d))^(1/2),x, algorithm="giac")

[Out]

sqrt(pi)*i*erf(-i*sqrt(log(c*x*e + c*d)))*e^(-1)/c