3.43 \(\int \frac{\text{Erf}(d (a+b \log (c x^n)))}{x} \, dx\)

Optimal. Leaf size=65 \[ \frac{e^{-d^2 \left (a+b \log \left (c x^n\right )\right )^2}}{\sqrt{\pi } b d n}+\frac{\left (a+b \log \left (c x^n\right )\right ) \text{Erf}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{b n} \]

[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)

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Rubi [A]  time = 0.0463315, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 1, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.059, Rules used = {6349} \[ \frac{e^{-d^2 \left (a+b \log \left (c x^n\right )\right )^2}}{\sqrt{\pi } b d n}+\frac{\left (a+b \log \left (c x^n\right )\right ) \text{Erf}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{b n} \]

Antiderivative was successfully verified.

[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 6349

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*} \int \frac{\text{erf}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x} \, dx &=\frac{\operatorname{Subst}\left (\int \text{erf}(d (a+b x)) \, dx,x,\log \left (c x^n\right )\right )}{n}\\ &=\frac{\operatorname{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]  time = 0.139583, size = 79, normalized size = 1.22 \[ \frac{\frac{\left (c x^n\right )^{-2 a b d^2} e^{-d^2 \left (a^2+b^2 \log ^2\left (c x^n\right )\right )}}{\sqrt{\pi } b d}+\left (\frac{a}{b}+\log \left (c x^n\right )\right ) \text{Erf}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{n} \]

Antiderivative was successfully verified.

[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

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Maple [A]  time = 0.108, size = 79, normalized size = 1.2 \begin{align*}{\frac{\ln \left ( c{x}^{n} \right ){\it Erf} \left ( ad+bd\ln \left ( c{x}^{n} \right ) \right ) }{n}}+{\frac{{\it Erf} \left ( ad+bd\ln \left ( c{x}^{n} \right ) \right ) a}{bn}}+{\frac{{{\rm e}^{- \left ( ad+bd\ln \left ( c{x}^{n} \right ) \right ) ^{2}}}}{bdn\sqrt{\pi }}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.03546, size = 78, normalized size = 1.2 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [A]  time = 2.81078, size = 284, normalized size = 4.37 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{erf}{\left (a d + b d \log{\left (c x^{n} \right )} \right )}}{x}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Giac [A]  time = 1.31471, size = 90, normalized size = 1.38 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)