\(\int \sqrt {\log (c (d+e x))} \, dx\) [11]

Optimal result

Integrand size = 12, antiderivative size = 50 \[ \int \sqrt {\log (c (d+e x))} \, dx=-\frac {\sqrt {\pi } \text {erfi}\left (\sqrt {\log (c (d+e x))}\right )}{2 c e}+\frac {(d+e x) \sqrt {\log (c (d+e x))}}{e} \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 50, 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, 2333, 2336, 2211, 2235} \[ \int \sqrt {\log (c (d+e x))} \, dx=\frac {(d+e x) \sqrt {\log (c (d+e x))}}{e}-\frac {\sqrt {\pi } \text {erfi}\left (\sqrt {\log (c (d+e x))}\right )}{2 c e} \]

[In]

Int[Sqrt[Log[c*(d + e*x)]],x]

[Out]

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

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 2333

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.), x_Symbol] :> Simp[x*(a + b*Log[c*x^n])^p, x] - Dist[b*n*p, In
t[(a + b*Log[c*x^n])^(p - 1), x], x] /; FreeQ[{a, b, c, n}, x] && GtQ[p, 0] && 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 \sqrt {\log (c x)} \, dx,x,d+e x\right )}{e} \\ & = \frac {(d+e x) \sqrt {\log (c (d+e x))}}{e}-\frac {\text {Subst}\left (\int \frac {1}{\sqrt {\log (c x)}} \, dx,x,d+e x\right )}{2 e} \\ & = \frac {(d+e x) \sqrt {\log (c (d+e x))}}{e}-\frac {\text {Subst}\left (\int \frac {e^x}{\sqrt {x}} \, dx,x,\log (c (d+e x))\right )}{2 c e} \\ & = \frac {(d+e x) \sqrt {\log (c (d+e x))}}{e}-\frac {\text {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 )}{2 c e}+\frac {(d+e x) \sqrt {\log (c (d+e x))}}{e} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.00 \[ \int \sqrt {\log (c (d+e x))} \, dx=-\frac {\sqrt {\pi } \text {erfi}\left (\sqrt {\log (c (d+e x))}\right )}{2 c e}+\frac {(d+e x) \sqrt {\log (c (d+e x))}}{e} \]

[In]

Integrate[Sqrt[Log[c*(d + e*x)]],x]

[Out]

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

Maple [F]

\[\int \sqrt {\ln \left (c \left (e x +d \right )\right )}d x\]

[In]

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

[Out]

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

Fricas [F(-2)]

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

[In]

integrate(log(c*(e*x+d))^(1/2),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. 90 vs. \(2 (41) = 82\).

Time = 0.90 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.80 \[ \int \sqrt {\log (c (d+e x))} \, dx=\begin {cases} \tilde {\infty } x & \text {for}\: c = 0 \\x \sqrt {\log {\left (c d \right )}} & \text {for}\: e = 0 \\\frac {\left (\sqrt {- \log {\left (c d + c e x \right )}} \left (c d + c e x\right ) + \frac {\sqrt {\pi } \operatorname {erfc}{\left (\sqrt {- \log {\left (c d + c e x \right )}} \right )}}{2}\right ) \sqrt {\log {\left (c d + c e x \right )}}}{c e \sqrt {- \log {\left (c d + c e x \right )}}} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

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

Maxima [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.19 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.98 \[ \int \sqrt {\log (c (d+e x))} \, dx=-\frac {-i \, \sqrt {\pi } \operatorname {erf}\left (i \, \sqrt {\log \left (c e x + c d\right )}\right ) - 2 \, {\left (c e x + c d\right )} \sqrt {\log \left (c e x + c d\right )}}{2 \, c e} \]

[In]

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

[Out]

-1/2*(-I*sqrt(pi)*erf(I*sqrt(log(c*e*x + c*d))) - 2*(c*e*x + c*d)*sqrt(log(c*e*x + c*d)))/(c*e)

Giac [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.31 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.04 \[ \int \sqrt {\log (c (d+e x))} \, dx=-\frac {i \, \sqrt {\pi } \operatorname {erf}\left (-i \, \sqrt {\log \left (c e x + c d\right )}\right )}{2 \, c e} + \frac {{\left (c e x + c d\right )} \sqrt {\log \left (c e x + c d\right )}}{c e} \]

[In]

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

[Out]

-1/2*I*sqrt(pi)*erf(-I*sqrt(log(c*e*x + c*d)))/(c*e) + (c*e*x + c*d)*sqrt(log(c*e*x + c*d))/(c*e)

Mupad [B] (verification not implemented)

Time = 1.28 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.92 \[ \int \sqrt {\log (c (d+e x))} \, dx=\frac {\sqrt {\ln \left (c\,\left (d+e\,x\right )\right )}\,\left (d+e\,x\right )}{e}+\frac {\sqrt {\pi }\,\mathrm {erf}\left (\sqrt {\ln \left (c\,\left (d+e\,x\right )\right )}\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{2\,c\,e} \]

[In]

int(log(c*(d + e*x))^(1/2),x)

[Out]

(log(c*(d + e*x))^(1/2)*(d + e*x))/e + (pi^(1/2)*erf(log(c*(d + e*x))^(1/2)*1i)*1i)/(2*c*e)