\(\int \frac {\text {erf}(d (a+b \log (c x^n)))}{x^3} \, dx\) [45]

Optimal result

Integrand size = 17, antiderivative size = 95 \[ \int \frac {\text {erf}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^3} \, dx=-\frac {\text {erf}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{2 x^2}+\frac {e^{\frac {1+2 a b d^2 n}{b^2 d^2 n^2}} \left (c x^n\right )^{2/n} \text {erf}\left (\frac {1+a b d^2 n+b^2 d^2 n \log \left (c x^n\right )}{b d n}\right )}{2 x^2} \] Output:

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

Mathematica [A] (verified)

Time = 0.17 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.81 \[ \int \frac {\text {erf}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^3} \, dx=\frac {-\text {erf}\left (d \left (a+b \log \left (c x^n\right )\right )\right )+e^{\frac {\frac {\frac {1}{d^2}+2 a b n}{b^2}+2 n \log \left (c x^n\right )}{n^2}} \text {erf}\left (a d+\frac {1}{b d n}+b d \log \left (c x^n\right )\right )}{2 x^2} \] Input:

Integrate[Erf[d*(a + b*Log[c*x^n])]/x^3,x]
 

Output:

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

Rubi [A] (verified)

Time = 0.49 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.16, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.294, Rules used = {6955, 2712, 2706, 2664, 2634}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {\text {erf}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^3} \, dx\)

\(\Big \downarrow \) 6955

\(\displaystyle \frac {b d n \int \frac {e^{-d^2 \left (a+b \log \left (c x^n\right )\right )^2}}{x^3}dx}{\sqrt {\pi }}-\frac {\text {erf}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{2 x^2}\)

\(\Big \downarrow \) 2712

\(\displaystyle \frac {b d n x^{2 a b d^2 n} \left (c x^n\right )^{-2 a b d^2} \int e^{-a^2 d^2-b^2 \log ^2\left (c x^n\right ) d^2} x^{-2 a b n d^2-3}dx}{\sqrt {\pi }}-\frac {\text {erf}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{2 x^2}\)

\(\Big \downarrow \) 2706

\(\displaystyle \frac {b d \left (c x^n\right )^{2 \left (a b d^2+\frac {1}{n}\right )-2 a b d^2} \int \exp \left (-a^2 d^2-b^2 \log ^2\left (c x^n\right ) d^2-\frac {2 \left (a b n d^2+1\right ) \log \left (c x^n\right )}{n}\right )d\log \left (c x^n\right )}{\sqrt {\pi } x^2}-\frac {\text {erf}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{2 x^2}\)

\(\Big \downarrow \) 2664

\(\displaystyle \frac {b d e^{\frac {2 a b d^2 n+1}{b^2 d^2 n^2}} \left (c x^n\right )^{2 \left (a b d^2+\frac {1}{n}\right )-2 a b d^2} \int \exp \left (-\frac {\left (a b n d^2+b^2 n \log \left (c x^n\right ) d^2+1\right )^2}{b^2 d^2 n^2}\right )d\log \left (c x^n\right )}{\sqrt {\pi } x^2}-\frac {\text {erf}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{2 x^2}\)

\(\Big \downarrow \) 2634

\(\displaystyle \frac {e^{\frac {2 a b d^2 n+1}{b^2 d^2 n^2}} \left (c x^n\right )^{2 \left (a b d^2+\frac {1}{n}\right )-2 a b d^2} \text {erf}\left (\frac {a b d^2 n+b^2 d^2 n \log \left (c x^n\right )+1}{b d n}\right )}{2 x^2}-\frac {\text {erf}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{2 x^2}\)

Input:

Int[Erf[d*(a + b*Log[c*x^n])]/x^3,x]
 

Output:

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

Defintions of rubi rules used

rule 2634

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] /; Fr 
eeQ[{F, a, b, c, d}, x] && NegQ[b]
 

rule 2664

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

rule 2706

Int[(F_)^(((a_.) + Log[(c_.)*((d_.) + (e_.)*(x_))^(n_.)]^2*(b_.))*(f_.))*(( 
g_.) + (h_.)*(x_))^(m_.), x_Symbol] :> Simp[(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*Log[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 2712

Int[(F_)^(((a_.) + Log[(c_.)*((d_.) + (e_.)*(x_))^(n_.)]*(b_.))^2*(f_.))*(( 
g_.) + (h_.)*(x_))^(m_.), x_Symbol] :> Simp[(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 6955

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] - 
 Simp[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]
 

Maple [F]

\[\int \frac {\operatorname {erf}\left (d \left (a +b \ln \left (c \,x^{n}\right )\right )\right )}{x^{3}}d x\]

Input:

int(erf(d*(a+b*ln(c*x^n)))/x^3,x)
 

Output:

int(erf(d*(a+b*ln(c*x^n)))/x^3,x)
 

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.31 \[ \int \frac {\text {erf}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^3} \, dx=\frac {\sqrt {b^{2} d^{2} n^{2}} x^{2} \operatorname {erf}\left (\frac {{\left (b^{2} d^{2} n^{2} \log \left (x\right ) + b^{2} d^{2} n \log \left (c\right ) + a b d^{2} n + 1\right )} \sqrt {b^{2} d^{2} n^{2}}}{b^{2} d^{2} n^{2}}\right ) e^{\left (\frac {2 \, b^{2} d^{2} n \log \left (c\right ) + 2 \, a b d^{2} n + 1}{b^{2} d^{2} n^{2}}\right )} - \operatorname {erf}\left (b d \log \left (c x^{n}\right ) + a d\right )}{2 \, x^{2}} \] Input:

integrate(erf(d*(a+b*log(c*x^n)))/x^3,x, algorithm="fricas")
 

Output:

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

Sympy [F]

\[ \int \frac {\text {erf}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^3} \, dx=\int \frac {\operatorname {erf}{\left (a d + b d \log {\left (c x^{n} \right )} \right )}}{x^{3}}\, dx \] Input:

integrate(erf(d*(a+b*ln(c*x**n)))/x**3,x)
 

Output:

Integral(erf(a*d + b*d*log(c*x**n))/x**3, x)
 

Maxima [F]

\[ \int \frac {\text {erf}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^3} \, dx=\int { \frac {\operatorname {erf}\left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )}{x^{3}} \,d x } \] Input:

integrate(erf(d*(a+b*log(c*x^n)))/x^3,x, algorithm="maxima")
 

Output:

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^3, x)/(sqrt(pi)*c^(2*a*b*d^2 
)) - 1/2*erf(b*d*log(x^n) + (b*log(c) + a)*d)/x^2
 

Giac [F]

\[ \int \frac {\text {erf}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^3} \, dx=\int { \frac {\operatorname {erf}\left ({\left (b \log \left (c x^{n}\right ) + a\right )} d\right )}{x^{3}} \,d x } \] Input:

integrate(erf(d*(a+b*log(c*x^n)))/x^3,x, algorithm="giac")
 

Output:

integrate(erf((b*log(c*x^n) + a)*d)/x^3, x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\text {erf}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^3} \, dx=\int \frac {\mathrm {erf}\left (d\,\left (a+b\,\ln \left (c\,x^n\right )\right )\right )}{x^3} \,d x \] Input:

int(erf(d*(a + b*log(c*x^n)))/x^3,x)
 

Output:

int(erf(d*(a + b*log(c*x^n)))/x^3, x)
 

Reduce [F]

\[ \int \frac {\text {erf}\left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^3} \, dx=\int \frac {\mathrm {erf}\left (\mathrm {log}\left (x^{n} c \right ) b d +a d \right )}{x^{3}}d x \] Input:

int(erf(d*(a+b*log(c*x^n)))/x^3,x)
 

Output:

int(erf(log(x**n*c)*b*d + a*d)/x**3,x)