\(\int \frac {1}{\log ^{\frac {3}{2}}(c (d+e x))} \, dx\) [13]

Optimal result

Integrand size = 12, antiderivative size = 49 \[ \int \frac {1}{\log ^{\frac {3}{2}}(c (d+e x))} \, dx=\frac {2 \sqrt {\pi } \text {erfi}\left (\sqrt {\log (c (d+e x))}\right )}{c e}-\frac {2 (d+e x)}{e \sqrt {\log (c (d+e x))}} \]

[Out]

2*erfi(ln(c*(e*x+d))^(1/2))*Pi^(1/2)/c/e-2*(e*x+d)/e/ln(c*(e*x+d))^(1/2)

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {2436, 2334, 2336, 2211, 2235} \[ \int \frac {1}{\log ^{\frac {3}{2}}(c (d+e x))} \, dx=\frac {2 \sqrt {\pi } \text {erfi}\left (\sqrt {\log (c (d+e x))}\right )}{c e}-\frac {2 (d+e x)}{e \sqrt {\log (c (d+e x))}} \]

[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 2211

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] &&  !TrueQ[$UseGamma]

Rule 2235

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]

Rule 2334

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 2336

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 2436

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]

Rubi steps \begin{align*} \text {integral}& = \frac {\text {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 \text {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 \text {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 \text {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] (verified)

Time = 0.07 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.18 \[ \int \frac {1}{\log ^{\frac {3}{2}}(c (d+e x))} \, dx=\frac {-2 c (d+e x)+2 \Gamma \left (\frac {1}{2},-\log (c (d+e x))\right ) \sqrt {-\log (c (d+e x))}}{c e \sqrt {\log (c (d+e x))}} \]

[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]

\[\int \frac {1}{\ln \left (c \left (e x +d \right )\right )^{\frac {3}{2}}}d x\]

[In]

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

[Out]

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

Fricas [F(-2)]

Exception generated. \[ \int \frac {1}{\log ^{\frac {3}{2}}(c (d+e x))} \, dx=\text {Exception raised: TypeError} \]

[In]

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

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (co
nstant residues)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 92 vs. \(2 (44) = 88\).

Time = 13.49 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.88 \[ \int \frac {1}{\log ^{\frac {3}{2}}(c (d+e x))} \, dx=\begin {cases} 0 & \text {for}\: c = 0 \\\frac {x}{\log {\left (c d \right )}^{\frac {3}{2}}} & \text {for}\: e = 0 \\\frac {\left (- \log {\left (c d + c e x \right )}\right )^{\frac {3}{2}} \left (- 2 \sqrt {\pi } \operatorname {erfc}{\left (\sqrt {- \log {\left (c d + c e x \right )}} \right )} + \frac {2 \left (c d + c e x\right )}{\sqrt {- \log {\left (c d + c e x \right )}}}\right )}{c e \log {\left (c d + c e x \right )}^{\frac {3}{2}}} & \text {otherwise} \end {cases} \]

[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))

Maxima [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.92 \[ \int \frac {1}{\log ^{\frac {3}{2}}(c (d+e x))} \, dx=-\frac {\sqrt {-\log \left (c e x + c d\right )} \Gamma \left (-\frac {1}{2}, -\log \left (c e x + c d\right )\right )}{c e \sqrt {\log \left (c e x + c d\right )}} \]

[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)))

Giac [F]

\[ \int \frac {1}{\log ^{\frac {3}{2}}(c (d+e x))} \, dx=\int { \frac {1}{\log \left ({\left (e x + d\right )} c\right )^{\frac {3}{2}}} \,d x } \]

[In]

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

[Out]

integrate(log((e*x + d)*c)^(-3/2), x)

Mupad [B] (verification not implemented)

Time = 1.29 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.37 \[ \int \frac {1}{\log ^{\frac {3}{2}}(c (d+e x))} \, dx=-\frac {2\,\left (d+e\,x\right )}{e\,\sqrt {\ln \left (c\,\left (d+e\,x\right )\right )}}-\frac {2\,\sqrt {\pi }\,{\left (-\ln \left (c\,\left (d+e\,x\right )\right )\right )}^{3/2}\,\mathrm {erfc}\left (\sqrt {-\ln \left (c\,\left (d+e\,x\right )\right )}\right )}{c\,e\,{\ln \left (c\,\left (d+e\,x\right )\right )}^{3/2}} \]

[In]

int(1/log(c*(d + e*x))^(3/2),x)

[Out]

- (2*(d + e*x))/(e*log(c*(d + e*x))^(1/2)) - (2*pi^(1/2)*(-log(c*(d + e*x)))^(3/2)*erfc((-log(c*(d + e*x)))^(1
/2)))/(c*e*log(c*(d + e*x))^(3/2))