\(\int x \text {erfi}(d (a+b \log (c x^n))) \, dx\) [247]

Optimal result

Integrand size = 15, antiderivative size = 93 \[ \int x \text {erfi}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\frac {1}{2} x^2 \text {erfi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )-\frac {1}{2} e^{-\frac {1+2 a b d^2 n}{b^2 d^2 n^2}} x^2 \left (c x^n\right )^{-2/n} \text {erfi}\left (\frac {a b d^2+\frac {1}{n}+b^2 d^2 \log \left (c x^n\right )}{b d}\right ) \] Output:

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

Mathematica [A] (verified)

Time = 0.19 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.87 \[ \int x \text {erfi}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\frac {1}{2} \left (x^2 \text {erfi}\left (d \left (a+b \log \left (c x^n\right )\right )\right )-e^{-\frac {\frac {\frac {1}{d^2}+2 a b n}{b^2}+2 n \log \left (c x^n\right )}{n^2}} x^2 \text {erfi}\left (a d+\frac {1}{b d n}+b d \log \left (c x^n\right )\right )\right ) \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.47 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.18, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6957, 2712, 2706, 2664, 2633}

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.

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

\(\Big \downarrow \) 6957

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

\(\Big \downarrow \) 2712

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

\(\Big \downarrow \) 2706

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

\(\Big \downarrow \) 2664

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

\(\Big \downarrow \) 2633

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 2633

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[F^a*Sqrt 
[Pi]*(Erfi[(c + d*x)*Rt[b*Log[F], 2]]/(2*d*Rt[b*Log[F], 2])), x] /; FreeQ[{ 
F, a, b, c, d}, x] && PosQ[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 6957

Int[Erfi[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(d_.)]*((e_.)*(x_))^(m_.), x 
_Symbol] :> Simp[(e*x)^(m + 1)*(Erfi[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]

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

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.32 \[ \int x \text {erfi}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\frac {1}{2} \, x^{2} \operatorname {erfi}\left (b d \log \left (c x^{n}\right ) + a d\right ) + \frac {1}{2} \, \sqrt {-b^{2} d^{2} n^{2}} \operatorname {erf}\left (\frac {{\left (b^{2} d^{2} n^{2} \log \left (x\right ) + b^{2} d^{2} n \log \left (c\right ) + a b d^{2} n + 1\right )} \sqrt {-b^{2} d^{2} n^{2}}}{b^{2} d^{2} n^{2}}\right ) e^{\left (-\frac {2 \, b^{2} d^{2} n \log \left (c\right ) + 2 \, a b d^{2} n + 1}{b^{2} d^{2} n^{2}}\right )} \] Input:

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

integrate(x*erfi((b*log(c*x^n) + a)*d), x)
 

Giac [F]

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

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

Output:

integrate(x*erfi((b*log(c*x^n) + a)*d), x)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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