\(\int \frac {\text {erf}(d (a+b \log (c x^n)))}{x^2} \, dx\) [44]

Optimal result

Integrand size = 17, antiderivative size = 92 \[ \int \frac {\text {erf}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^2} \, dx=-\frac {\text {erf}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x}+\frac {e^{\frac {1}{4 b^2 d^2 n^2}+\frac {a}{b n}} \left (c x^n\right )^{\frac {1}{n}} \text {erf}\left (\frac {2 a b d^2+\frac {1}{n}+2 b^2 d^2 \log \left (c x^n\right )}{2 b d}\right )}{x} \]

[Out]

-erf(d*(a+b*ln(c*x^n)))/x+exp(1/4/b^2/d^2/n^2+a/b/n)*(c*x^n)^(1/n)*erf(1/2*(2*a*b*d^2+1/n+2*b^2*d^2*ln(c*x^n))
/b/d)/x

Rubi [A] (verified)

Time = 0.13 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.294, Rules used = {6536, 2314, 2308, 2266, 2236} \[ \int \frac {\text {erf}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^2} \, dx=\frac {\left (c x^n\right )^{\frac {1}{n}} e^{\frac {a}{b n}+\frac {1}{4 b^2 d^2 n^2}} \text {erf}\left (\frac {2 a b d^2+2 b^2 d^2 \log \left (c x^n\right )+\frac {1}{n}}{2 b d}\right )}{x}-\frac {\text {erf}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x} \]

[In]

Int[Erf[d*(a + b*Log[c*x^n])]/x^2,x]

[Out]

-(Erf[d*(a + b*Log[c*x^n])]/x) + (E^(1/(4*b^2*d^2*n^2) + a/(b*n))*(c*x^n)^n^(-1)*Erf[(2*a*b*d^2 + n^(-1) + 2*b
^2*d^2*Log[c*x^n])/(2*b*d)])/x

Rule 2236

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[F^a*Sqrt[Pi]*(Erf[(c + d*x)*Rt[(-b)*Log[F],
2]]/(2*d*Rt[(-b)*Log[F], 2])), x] /; FreeQ[{F, a, b, c, d}, x] && NegQ[b]

Rule 2266

Int[(F_)^((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[F^(a - b^2/(4*c)), Int[F^((b + 2*c*x)^2/(4*c))
, x], x] /; FreeQ[{F, a, b, c}, x]

Rule 2308

Int[(F_)^(((a_.) + Log[(c_.)*((d_.) + (e_.)*(x_))^(n_.)]^2*(b_.))*(f_.))*((g_.) + (h_.)*(x_))^(m_.), x_Symbol]
:> Dist[(g + h*x)^(m + 1)/(h*n*(c*(d + e*x)^n)^((m + 1)/n)), Subst[Int[E^(a*f*Log[F] + ((m + 1)*x)/n + b*f*Lo
g[F]*x^2), x], x, Log[c*(d + e*x)^n]], x] /; FreeQ[{F, a, b, c, d, e, f, g, h, m, n}, x] && EqQ[e*g - d*h, 0]

Rule 2314

Int[(F_)^(((a_.) + Log[(c_.)*((d_.) + (e_.)*(x_))^(n_.)]*(b_.))^2*(f_.))*((g_.) + (h_.)*(x_))^(m_.), x_Symbol]
:> Dist[(g + h*x)^m*((c*(d + e*x)^n)^(2*a*b*f*Log[F])/(d + e*x)^(m + 2*a*b*f*n*Log[F])), Int[(d + e*x)^(m + 2
*a*b*f*n*Log[F])*F^(a^2*f + b^2*f*Log[c*(d + e*x)^n]^2), x], x] /; FreeQ[{F, a, b, c, d, e, f, g, h, m, n}, x]
&& EqQ[e*g - d*h, 0]

Rule 6536

Int[Erf[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(d_.)]*((e_.)*(x_))^(m_.), x_Symbol] :> Simp[(e*x)^(m + 1)*(Erf[
d*(a + b*Log[c*x^n])]/(e*(m + 1))), x] - Dist[2*b*d*(n/(Sqrt[Pi]*(m + 1))), Int[(e*x)^m/E^(d*(a + b*Log[c*x^n]
))^2, x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && NeQ[m, -1]

Rubi steps \begin{align*} \text {integral}& = -\frac {\text {erf}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x}+\frac {(2 b d n) \int \frac {e^{-d^2 \left (a+b \log \left (c x^n\right )\right )^2}}{x^2} \, dx}{\sqrt {\pi }} \\ & = -\frac {\text {erf}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x}+\frac {\left (2 b d n x^{2 a b d^2 n} \left (c x^n\right )^{-2 a b d^2}\right ) \int e^{-a^2 d^2-b^2 d^2 \log ^2\left (c x^n\right )} x^{-2-2 a b d^2 n} \, dx}{\sqrt {\pi }} \\ & = -\frac {\text {erf}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x}+\frac {\left (2 b d \left (c x^n\right )^{-2 a b d^2-\frac {-1-2 a b d^2 n}{n}}\right ) \text {Subst}\left (\int \exp \left (-a^2 d^2+\frac {\left (-1-2 a b d^2 n\right ) x}{n}-b^2 d^2 x^2\right ) \, dx,x,\log \left (c x^n\right )\right )}{\sqrt {\pi } x} \\ & = -\frac {\text {erf}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x}+\frac {\left (2 b d e^{\frac {1}{4 b^2 d^2 n^2}+\frac {a}{b n}} \left (c x^n\right )^{-2 a b d^2-\frac {-1-2 a b d^2 n}{n}}\right ) \text {Subst}\left (\int \exp \left (-\frac {\left (\frac {-1-2 a b d^2 n}{n}-2 b^2 d^2 x\right )^2}{4 b^2 d^2}\right ) \, dx,x,\log \left (c x^n\right )\right )}{\sqrt {\pi } x} \\ & = -\frac {\text {erf}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x}+\frac {e^{\frac {1}{4 b^2 d^2 n^2}+\frac {a}{b n}} \left (c x^n\right )^{\frac {1}{n}} \text {erf}\left (\frac {2 a b d^2+\frac {1}{n}+2 b^2 d^2 \log \left (c x^n\right )}{2 b d}\right )}{x} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.17 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.87 \[ \int \frac {\text {erf}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^2} \, dx=\frac {-\text {erf}\left (d \left (a+b \log \left (c x^n\right )\right )\right )+e^{\frac {\frac {\frac {1}{d^2}+4 a b n}{b^2}+4 n \log \left (c x^n\right )}{4 n^2}} \text {erf}\left (a d+\frac {1}{2 b d n}+b d \log \left (c x^n\right )\right )}{x} \]

[In]

Integrate[Erf[d*(a + b*Log[c*x^n])]/x^2,x]

[Out]

(-Erf[d*(a + b*Log[c*x^n])] + E^(((d^(-2) + 4*a*b*n)/b^2 + 4*n*Log[c*x^n])/(4*n^2))*Erf[a*d + 1/(2*b*d*n) + b*
d*Log[c*x^n]])/x

Maple [F]

\[\int \frac {\operatorname {erf}\left (d \left (a +b \ln \left (c \,x^{n}\right )\right )\right )}{x^{2}}d x\]

[In]

int(erf(d*(a+b*ln(c*x^n)))/x^2,x)

[Out]

int(erf(d*(a+b*ln(c*x^n)))/x^2,x)

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.37 \[ \int \frac {\text {erf}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^2} \, dx=\frac {\sqrt {b^{2} d^{2} n^{2}} x \operatorname {erf}\left (\frac {{\left (2 \, b^{2} d^{2} n^{2} \log \left (x\right ) + 2 \, b^{2} d^{2} n \log \left (c\right ) + 2 \, a b d^{2} n + 1\right )} \sqrt {b^{2} d^{2} n^{2}}}{2 \, b^{2} d^{2} n^{2}}\right ) e^{\left (\frac {4 \, b^{2} d^{2} n \log \left (c\right ) + 4 \, a b d^{2} n + 1}{4 \, b^{2} d^{2} n^{2}}\right )} - \operatorname {erf}\left (b d \log \left (c x^{n}\right ) + a d\right )}{x} \]

[In]

integrate(erf(d*(a+b*log(c*x^n)))/x^2,x, algorithm="fricas")

[Out]

(sqrt(b^2*d^2*n^2)*x*erf(1/2*(2*b^2*d^2*n^2*log(x) + 2*b^2*d^2*n*log(c) + 2*a*b*d^2*n + 1)*sqrt(b^2*d^2*n^2)/(
b^2*d^2*n^2))*e^(1/4*(4*b^2*d^2*n*log(c) + 4*a*b*d^2*n + 1)/(b^2*d^2*n^2)) - erf(b*d*log(c*x^n) + a*d))/x

Sympy [F]

\[ \int \frac {\text {erf}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^2} \, dx=\int \frac {\operatorname {erf}{\left (a d + b d \log {\left (c x^{n} \right )} \right )}}{x^{2}}\, dx \]

[In]

integrate(erf(d*(a+b*ln(c*x**n)))/x**2,x)

[Out]

Integral(erf(a*d + b*d*log(c*x**n))/x**2, x)

Maxima [F]

\[ \int \frac {\text {erf}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^2} \, dx=\int { \frac {\operatorname {erf}\left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )}{x^{2}} \,d x } \]

[In]

integrate(erf(d*(a+b*log(c*x^n)))/x^2,x, algorithm="maxima")

[Out]

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

Giac [F]

\[ \int \frac {\text {erf}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^2} \, dx=\int { \frac {\operatorname {erf}\left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )}{x^{2}} \,d x } \]

[In]

integrate(erf(d*(a+b*log(c*x^n)))/x^2,x, algorithm="giac")

[Out]

integrate(erf((b*log(c*x^n) + a)*d)/x^2, x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\text {erf}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^2} \, dx=\int \frac {\mathrm {erf}\left (d\,\left (a+b\,\ln \left (c\,x^n\right )\right )\right )}{x^2} \,d x \]

[In]

int(erf(d*(a + b*log(c*x^n)))/x^2,x)

[Out]

int(erf(d*(a + b*log(c*x^n)))/x^2, x)