\(\int x \text {erfc}(d (a+b \log (c x^n))) \, dx\) [144]

Optimal result

Integrand size = 15, antiderivative size = 94 \[ \int x \text {erfc}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\frac {1}{2} e^{\frac {1-2 a b d^2 n}{b^2 d^2 n^2}} x^2 \left (c x^n\right )^{-2/n} \text {erf}\left (\frac {a b d^2-\frac {1}{n}+b^2 d^2 \log \left (c x^n\right )}{b d}\right )+\frac {1}{2} x^2 \text {erfc}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \]

[Out]

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

Rubi [A] (verified)

Time = 0.09 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6537, 2314, 2308, 2266, 2236} \[ \int x \text {erfc}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\frac {1}{2} x^2 \left (c x^n\right )^{-2/n} e^{\frac {1-2 a b d^2 n}{b^2 d^2 n^2}} \text {erf}\left (\frac {a b d^2+b^2 d^2 \log \left (c x^n\right )-\frac {1}{n}}{b d}\right )+\frac {1}{2} x^2 \text {erfc}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \]

[In]

Int[x*Erfc[d*(a + b*Log[c*x^n])],x]

[Out]

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

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 6537

Int[Erfc[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(d_.)]*((e_.)*(x_))^(m_.), x_Symbol] :> Simp[(e*x)^(m + 1)*(Erf
c[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 {1}{2} x^2 \text {erfc}\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} x \, dx}{\sqrt {\pi }} \\ & = \frac {1}{2} x^2 \text {erfc}\left (d \left (a+b \log \left (c x^n\right )\right )\right )+\frac {\left (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^{1-2 a b d^2 n} \, dx}{\sqrt {\pi }} \\ & = \frac {1}{2} x^2 \text {erfc}\left (d \left (a+b \log \left (c x^n\right )\right )\right )+\frac {\left (b d x^2 \left (c x^n\right )^{-2 a b d^2-\frac {2-2 a b d^2 n}{n}}\right ) \text {Subst}\left (\int \exp \left (-a^2 d^2+\frac {\left (2-2 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 }} \\ & = \frac {1}{2} x^2 \text {erfc}\left (d \left (a+b \log \left (c x^n\right )\right )\right )+\frac {\left (b d e^{\frac {1-2 a b d^2 n}{b^2 d^2 n^2}} x^2 \left (c x^n\right )^{-2 a b d^2-\frac {2-2 a b d^2 n}{n}}\right ) \text {Subst}\left (\int \exp \left (-\frac {\left (\frac {2-2 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 }} \\ & = \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 {erf}\left (\frac {a b d^2-\frac {1}{n}+b^2 d^2 \log \left (c x^n\right )}{b d}\right )+\frac {1}{2} x^2 \text {erfc}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.20 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.85 \[ \int x \text {erfc}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=\frac {1}{2} \left (e^{\frac {\frac {\frac {1}{d^2}-2 a b n}{b^2}-2 n \log \left (c x^n\right )}{n^2}} x^2 \text {erf}\left (a d-\frac {1}{b d n}+b d \log \left (c x^n\right )\right )+x^2 \text {erfc}\left (d \left (a+b \log \left (c x^n\right )\right )\right )\right ) \]

[In]

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

[Out]

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

Maple [F]

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

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.34 \[ \int x \text {erfc}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=-\frac {1}{2} \, x^{2} \operatorname {erf}\left (b d \log \left (c x^{n}\right ) + a d\right ) + \frac {1}{2} \, \sqrt {b^{2} d^{2} n^{2}} \operatorname {erf}\left (\frac {{\left (b^{2} d^{2} n^{2} \log \left (x\right ) + b^{2} d^{2} n \log \left (c\right ) + a b d^{2} n - 1\right )} \sqrt {b^{2} d^{2} n^{2}}}{b^{2} d^{2} n^{2}}\right ) e^{\left (-\frac {2 \, b^{2} d^{2} n \log \left (c\right ) + 2 \, a b d^{2} n - 1}{b^{2} d^{2} n^{2}}\right )} + \frac {1}{2} \, x^{2} \]

[In]

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

[Out]

-1/2*x^2*erf(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)) + 1/2*x^2

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.36 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.94 \[ \int x \text {erfc}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \, dx=-\frac {1}{2} \, x^{2} \operatorname {erf}\left (b d n \log \left (x\right ) + b d \log \left (c\right ) + a d\right ) + \frac {1}{2} \, x^{2} - \frac {\operatorname {erf}\left (-b d n \log \left (x\right ) - b d \log \left (c\right ) - a d + \frac {1}{b d n}\right ) e^{\left (-\frac {2 \, a}{b n} + \frac {1}{b^{2} d^{2} n^{2}}\right )}}{2 \, c^{\frac {2}{n}}} \]

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

int(x*erfc(d*(a + b*log(c*x^n))),x)

[Out]

int(x*erfc(d*(a + b*log(c*x^n))), x)