3.13 \(\int \frac{1}{\log ^{\frac{3}{2}}(c (d+e x))} \, dx\)

Optimal. Leaf size=49 \[ \frac{2 \sqrt{\pi } \text{Erfi}\left (\sqrt{\log (c (d+e x))}\right )}{c e}-\frac{2 (d+e x)}{e \sqrt{\log (c (d+e x))}} \]

[Out]

(2*Sqrt[Pi]*Erfi[Sqrt[Log[c*(d + e*x)]]])/(c*e) - (2*(d + e*x))/(e*Sqrt[Log[c*(d + e*x)]])

________________________________________________________________________________________

Rubi [A]  time = 0.0301805, antiderivative size = 49, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.417, Rules used = {2389, 2297, 2299, 2180, 2204} \[ \frac{2 \sqrt{\pi } \text{Erfi}\left (\sqrt{\log (c (d+e x))}\right )}{c e}-\frac{2 (d+e x)}{e \sqrt{\log (c (d+e x))}} \]

Antiderivative was successfully verified.

[In]

Int[Log[c*(d + e*x)]^(-3/2),x]

[Out]

(2*Sqrt[Pi]*Erfi[Sqrt[Log[c*(d + e*x)]]])/(c*e) - (2*(d + e*x))/(e*Sqrt[Log[c*(d + e*x)]])

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 2297

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_), x_Symbol] :> Simp[(x*(a + b*Log[c*x^n])^(p + 1))/(b*n*(p + 1))
, x] - Dist[1/(b*n*(p + 1)), Int[(a + b*Log[c*x^n])^(p + 1), x], x] /; FreeQ[{a, b, c, n}, x] && LtQ[p, -1] &&
IntegerQ[2*p]

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}{\log ^{\frac{3}{2}}(c (d+e x))} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{\log ^{\frac{3}{2}}(c x)} \, dx,x,d+e x\right )}{e}\\ &=-\frac{2 (d+e x)}{e \sqrt{\log (c (d+e x))}}+\frac{2 \operatorname{Subst}\left (\int \frac{1}{\sqrt{\log (c x)}} \, dx,x,d+e x\right )}{e}\\ &=-\frac{2 (d+e x)}{e \sqrt{\log (c (d+e x))}}+\frac{2 \operatorname{Subst}\left (\int \frac{e^x}{\sqrt{x}} \, dx,x,\log (c (d+e x))\right )}{c e}\\ &=-\frac{2 (d+e x)}{e \sqrt{\log (c (d+e x))}}+\frac{4 \operatorname{Subst}\left (\int e^{x^2} \, dx,x,\sqrt{\log (c (d+e x))}\right )}{c e}\\ &=\frac{2 \sqrt{\pi } \text{erfi}\left (\sqrt{\log (c (d+e x))}\right )}{c e}-\frac{2 (d+e x)}{e \sqrt{\log (c (d+e x))}}\\ \end{align*}

Mathematica [A]  time = 0.024192, size = 58, normalized size = 1.18 \[ \frac{2 \sqrt{-\log (c (d+e x))} \text{Gamma}\left (\frac{1}{2},-\log (c (d+e x))\right )-2 c (d+e x)}{c e \sqrt{\log (c (d+e x))}} \]

Antiderivative was successfully verified.

[In]

Integrate[Log[c*(d + e*x)]^(-3/2),x]

[Out]

(-2*c*(d + e*x) + 2*Gamma[1/2, -Log[c*(d + e*x)]]*Sqrt[-Log[c*(d + e*x)]])/(c*e*Sqrt[Log[c*(d + e*x)]])

________________________________________________________________________________________

Maple [F]  time = 0.259, size = 0, normalized size = 0. \begin{align*} \int \left ( \ln \left ( c \left ( ex+d \right ) \right ) \right ) ^{-{\frac{3}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/ln(c*(e*x+d))^(3/2),x)

[Out]

int(1/ln(c*(e*x+d))^(3/2),x)

________________________________________________________________________________________

Maxima [A]  time = 1.25852, size = 61, normalized size = 1.24 \begin{align*} -\frac{\sqrt{-\log \left (c e x + c d\right )} \Gamma \left (-\frac{1}{2}, -\log \left (c e x + c d\right )\right )}{c e \sqrt{\log \left (c e x + c d\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/log(c*(e*x+d))^(3/2),x, algorithm="maxima")

[Out]

-sqrt(-log(c*e*x + c*d))*gamma(-1/2, -log(c*e*x + c*d))/(c*e*sqrt(log(c*e*x + c*d)))

________________________________________________________________________________________

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))^(3/2),x, algorithm="fricas")

[Out]

Exception raised: UnboundLocalError

________________________________________________________________________________________

Sympy [A]  time = 143.548, size = 92, normalized size = 1.88 \begin{align*} \begin{cases} 0 & \text{for}\: c = 0 \\\frac{x}{\log{\left (c d \right )}^{\frac{3}{2}}} & \text{for}\: e = 0 \\\frac{\left (- \log{\left (c d + c e x \right )}\right )^{\frac{3}{2}} \left (- 2 \sqrt{\pi } \operatorname{erfc}{\left (\sqrt{- \log{\left (c d + c e x \right )}} \right )} + \frac{2 \left (c d + c e x\right )}{\sqrt{- \log{\left (c d + c e x \right )}}}\right )}{c e \log{\left (c d + c e x \right )}^{\frac{3}{2}}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/ln(c*(e*x+d))**(3/2),x)

[Out]

Piecewise((0, Eq(c, 0)), (x/log(c*d)**(3/2), Eq(e, 0)), ((-log(c*d + c*e*x))**(3/2)*(-2*sqrt(pi)*erfc(sqrt(-lo
g(c*d + c*e*x))) + 2*(c*d + c*e*x)/sqrt(-log(c*d + c*e*x)))/(c*e*log(c*d + c*e*x)**(3/2)), True))

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\log \left ({\left (e x + d\right )} c\right )^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/log(c*(e*x+d))^(3/2),x, algorithm="giac")

[Out]

integrate(log((e*x + d)*c)^(-3/2), x)