\(\int \text {erf}(d (a+b \log (c x^n))) \, dx\) [42]

Optimal result

Integrand size = 13, antiderivative size = 93 \[ \int \text {erf}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=x \text {erf}\left (d \left (a+b \log \left (c x^n\right )\right )\right )-e^{\frac {1-4 a b d^2 n}{4 b^2 d^2 n^2}} x \left (c x^n\right )^{-1/n} \text {erf}\left (\frac {2 a b d^2-\frac {1}{n}+2 b^2 d^2 \log \left (c x^n\right )}{2 b d}\right ) \] Output:

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

Mathematica [A] (verified)

Time = 0.17 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.86 \[ \int \text {erf}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=x \text {erf}\left (d \left (a+b \log \left (c x^n\right )\right )\right )-e^{-\frac {\frac {-\frac {1}{d^2}+4 a b n}{b^2}+4 n \log \left (c x^n\right )}{4 n^2}} x \text {erf}\left (a d-\frac {1}{2 b d n}+b d \log \left (c x^n\right )\right ) \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.45 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {6951, 2710, 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.

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

\(\Big \downarrow \) 6951

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

\(\Big \downarrow \) 2710

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

\(\Big \downarrow \) 2706

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

\(\Big \downarrow \) 2664

\(\displaystyle x \text {erf}\left (d \left (a+b \log \left (c x^n\right )\right )\right )-\frac {2 b d x \left (c x^n\right )^{-1/n} e^{\frac {1-4 a b d^2 n}{4 b^2 d^2 n^2}} \int \exp \left (-\frac {\left (2 a b d^2+2 b^2 \log \left (c x^n\right ) d^2-\frac {1}{n}\right )^2}{4 b^2 d^2}\right )d\log \left (c x^n\right )}{\sqrt {\pi }}\)

\(\Big \downarrow \) 2634

\(\displaystyle x \text {erf}\left (d \left (a+b \log \left (c x^n\right )\right )\right )-x \left (c x^n\right )^{-1/n} e^{\frac {1-4 a b d^2 n}{4 b^2 d^2 n^2}} \text {erf}\left (\frac {2 a b d^2+2 b^2 d^2 \log \left (c x^n\right )-\frac {1}{n}}{2 b d}\right )\)

Input:

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

Output:

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

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 2710

Int[(F_)^(((a_.) + Log[(c_.)*((d_.) + (e_.)*(x_))^(n_.)]*(b_.))^2*(f_.)), x 
_Symbol] :> Simp[((c*(d + e*x)^n)^(2*a*b*f*Log[F])/(d + e*x)^(2*a*b*f*n*Log 
[F]))*Int[(d + e*x)^(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, n}, x] &&  !IntegerQ[2*a*b*f*Log[ 
F]]
 

rule 6951

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

Maple [F]

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

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.31 \[ \int \text {erf}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=-\sqrt {b^{2} d^{2} n^{2}} \operatorname {erf}\left (\frac {{\left (2 \, b^{2} d^{2} n^{2} \log \left (x\right ) + 2 \, b^{2} d^{2} n \log \left (c\right ) + 2 \, a b d^{2} n - 1\right )} \sqrt {b^{2} d^{2} n^{2}}}{2 \, b^{2} d^{2} n^{2}}\right ) e^{\left (-\frac {4 \, b^{2} d^{2} n \log \left (c\right ) + 4 \, a b d^{2} n - 1}{4 \, b^{2} d^{2} n^{2}}\right )} + x \operatorname {erf}\left (b d \log \left (c x^{n}\right ) + a d\right ) \] Input:

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [A] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.85 \[ \int \text {erf}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=x \operatorname {erf}\left (b d n \log \left (x\right ) + b d \log \left (c\right ) + a d\right ) + \frac {\operatorname {erf}\left (-b d n \log \left (x\right ) - b d \log \left (c\right ) - a d + \frac {1}{2 \, b d n}\right ) e^{\left (-\frac {a}{b n} + \frac {1}{4 \, b^{2} d^{2} n^{2}}\right )}}{c^{\left (\frac {1}{n}\right )}} \] Input:

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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