\(\int \frac {\text {erf}(d (a+b \log (c x^n)))}{x} \, dx\) [43]

Optimal result

Integrand size = 17, antiderivative size = 65 \[ \int \frac {\text {erf}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x} \, dx=\frac {e^{-d^2 \left (a+b \log \left (c x^n\right )\right )^2}}{b d n \sqrt {\pi }}+\frac {\text {erf}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \left (a+b \log \left (c x^n\right )\right )}{b n} \]

[Out]

erf(d*(a+b*ln(c*x^n)))*(a+b*ln(c*x^n))/b/n+1/b/d/exp(d^2*(a+b*ln(c*x^n))^2)/n/Pi^(1/2)

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {6484} \[ \int \frac {\text {erf}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x} \, dx=\frac {e^{-d^2 \left (a+b \log \left (c x^n\right )\right )^2}}{\sqrt {\pi } b d n}+\frac {\left (a+b \log \left (c x^n\right )\right ) \text {erf}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{b n} \]

[In]

Int[Erf[d*(a + b*Log[c*x^n])]/x,x]

[Out]

1/(b*d*E^(d^2*(a + b*Log[c*x^n])^2)*n*Sqrt[Pi]) + (Erf[d*(a + b*Log[c*x^n])]*(a + b*Log[c*x^n]))/(b*n)

Rule 6484

Int[Erf[(a_.) + (b_.)*(x_)], x_Symbol] :> Simp[(a + b*x)*(Erf[a + b*x]/b), x] + Simp[1/(b*Sqrt[Pi]*E^(a + b*x)
^2), x] /; FreeQ[{a, b}, x]

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \text {erf}(d (a+b x)) \, dx,x,\log \left (c x^n\right )\right )}{n} \\ & = \frac {\text {Subst}\left (\int \text {erf}(x) \, dx,x,a d+b d \log \left (c x^n\right )\right )}{b d n} \\ & = \frac {e^{-\left (a d+b d \log \left (c x^n\right )\right )^2}}{b d n \sqrt {\pi }}+\frac {\text {erf}\left (a d+b d \log \left (c x^n\right )\right ) \left (a+b \log \left (c x^n\right )\right )}{b n} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.22 \[ \int \frac {\text {erf}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x} \, dx=\frac {\frac {e^{-d^2 \left (a^2+b^2 \log ^2\left (c x^n\right )\right )} \left (c x^n\right )^{-2 a b d^2}}{b d \sqrt {\pi }}+\text {erf}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \left (\frac {a}{b}+\log \left (c x^n\right )\right )}{n} \]

[In]

Integrate[Erf[d*(a + b*Log[c*x^n])]/x,x]

[Out]

(1/(b*d*E^(d^2*(a^2 + b^2*Log[c*x^n]^2))*Sqrt[Pi]*(c*x^n)^(2*a*b*d^2)) + Erf[d*(a + b*Log[c*x^n])]*(a/b + Log[
c*x^n]))/n

Maple [A] (verified)

Time = 1.21 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.95

method	result	size
derivativedivides	\(\frac {\operatorname {erf}\left (a d +b d \ln \left (c \,x^{n}\right )\right ) \left (a d +b d \ln \left (c \,x^{n}\right )\right )+\frac {{\mathrm e}^{-{\left (a d +b d \ln \left (c \,x^{n}\right )\right )}^{2}}}{\sqrt {\pi }}}{n d b}\)	\(62\)
default	\(\frac {\operatorname {erf}\left (a d +b d \ln \left (c \,x^{n}\right )\right ) \left (a d +b d \ln \left (c \,x^{n}\right )\right )+\frac {{\mathrm e}^{-{\left (a d +b d \ln \left (c \,x^{n}\right )\right )}^{2}}}{\sqrt {\pi }}}{n d b}\)	\(62\)
parts	\(\ln \left (x \right ) \operatorname {erf}\left (d \left (a +b \ln \left (c \,x^{n}\right )\right )\right )-\frac {2 d b n \left (-\frac {{\mathrm e}^{-\ln \left (x \right )^{2} b^{2} d^{2} n^{2}-2 d^{2} \left (b \left (\ln \left (c \,x^{n}\right )-n \ln \left (x \right )\right )+a \right ) n b \ln \left (x \right )-d^{2} {\left (b \left (\ln \left (c \,x^{n}\right )-n \ln \left (x \right )\right )+a \right )}^{2}}}{2 b^{2} d^{2} n^{2}}-\frac {\left (b \left (\ln \left (c \,x^{n}\right )-n \ln \left (x \right )\right )+a \right ) \sqrt {\pi }\, \operatorname {erf}\left (d b n \ln \left (x \right )+d \left (b \left (\ln \left (c \,x^{n}\right )-n \ln \left (x \right )\right )+a \right )\right )}{2 d \,n^{2} b^{2}}\right )}{\sqrt {\pi }}\)	\(159\)

[In]

int(erf(d*(a+b*ln(c*x^n)))/x,x,method=_RETURNVERBOSE)

[Out]

1/n/d/b*(erf(a*d+b*d*ln(c*x^n))*(a*d+b*d*ln(c*x^n))+1/Pi^(1/2)*exp(-(a*d+b*d*ln(c*x^n))^2))

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.83 \[ \int \frac {\text {erf}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x} \, dx=\frac {{\left (\pi b d n \log \left (x\right ) + \pi b d \log \left (c\right ) + \pi a d\right )} \operatorname {erf}\left (b d \log \left (c x^{n}\right ) + a d\right ) + \sqrt {\pi } e^{\left (-b^{2} d^{2} n^{2} \log \left (x\right )^{2} - b^{2} d^{2} \log \left (c\right )^{2} - 2 \, a b d^{2} \log \left (c\right ) - a^{2} d^{2} - 2 \, {\left (b^{2} d^{2} n \log \left (c\right ) + a b d^{2} n\right )} \log \left (x\right )\right )}}{\pi b d n} \]

[In]

integrate(erf(d*(a+b*log(c*x^n)))/x,x, algorithm="fricas")

[Out]

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

Sympy [F]

\[ \int \frac {\text {erf}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x} \, dx=\int \frac {\operatorname {erf}{\left (a d + b d \log {\left (c x^{n} \right )} \right )}}{x}\, dx \]

[In]

integrate(erf(d*(a+b*ln(c*x**n)))/x,x)

[Out]

Integral(erf(a*d + b*d*log(c*x**n))/x, x)

Maxima [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.89 \[ \int \frac {\text {erf}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x} \, dx=\frac {{\left (b \log \left (c x^{n}\right ) + a\right )} d \operatorname {erf}\left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right ) + \frac {e^{\left (-{\left (b \log \left (c x^{n}\right ) + a\right )}^{2} d^{2}\right )}}{\sqrt {\pi }}}{b d n} \]

[In]

integrate(erf(d*(a+b*log(c*x^n)))/x,x, algorithm="maxima")

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.03 \[ \int \frac {\text {erf}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x} \, dx=\frac {{\left (b d n \log \left (x\right ) + b d \log \left (c\right ) + a d\right )} \operatorname {erf}\left (b d n \log \left (x\right ) + b d \log \left (c\right ) + a d\right ) + \frac {e^{\left (-{\left (b d n \log \left (x\right ) + b d \log \left (c\right ) + a d\right )}^{2}\right )}}{\sqrt {\pi }}}{b d n} \]

[In]

integrate(erf(d*(a+b*log(c*x^n)))/x,x, algorithm="giac")

[Out]

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

Mupad [B] (verification not implemented)

Time = 5.37 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.86 \[ \int \frac {\text {erf}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x} \, dx=\frac {\ln \left (c\,x^n\right )\,\mathrm {erf}\left (a\,d+b\,d\,\ln \left (c\,x^n\right )\right )}{n}+\frac {a\,d\,\mathrm {erfi}\left (a\,\sqrt {-d^2}+b\,\ln \left (c\,x^n\right )\,\sqrt {-d^2}\right )}{b\,n\,\sqrt {-d^2}}+\frac {{\mathrm {e}}^{-b^2\,d^2\,{\ln \left (c\,x^n\right )}^2}\,{\mathrm {e}}^{-a^2\,d^2}}{b\,d\,n\,\sqrt {\pi }\,{\left (c\,x^n\right )}^{2\,a\,b\,d^2}} \]

[In]

int(erf(d*(a + b*log(c*x^n)))/x,x)

[Out]

(log(c*x^n)*erf(a*d + b*d*log(c*x^n)))/n + (a*d*erfi(a*(-d^2)^(1/2) + b*log(c*x^n)*(-d^2)^(1/2)))/(b*n*(-d^2)^
(1/2)) + (exp(-b^2*d^2*log(c*x^n)^2)*exp(-a^2*d^2))/(b*d*n*pi^(1/2)*(c*x^n)^(2*a*b*d^2))