\(\int \text {erfc}(b x)^2 \, dx\) [134]

Optimal result

Integrand size = 6, antiderivative size = 56 \[ \int \text {erfc}(b x)^2 \, dx=-\frac {\sqrt {\frac {2}{\pi }} \text {erf}\left (\sqrt {2} b x\right )}{b}-\frac {2 e^{-b^2 x^2} \text {erfc}(b x)}{b \sqrt {\pi }}+x \text {erfc}(b x)^2 \]

[Out]

x*erfc(b*x)^2-erf(b*x*2^(1/2))*2^(1/2)/Pi^(1/2)/b-2*erfc(b*x)/b/exp(b^2*x^2)/Pi^(1/2)

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {6488, 12, 6518, 2236} \[ \int \text {erfc}(b x)^2 \, dx=-\frac {2 e^{-b^2 x^2} \text {erfc}(b x)}{\sqrt {\pi } b}-\frac {\sqrt {\frac {2}{\pi }} \text {erf}\left (\sqrt {2} b x\right )}{b}+x \text {erfc}(b x)^2 \]

[In]

Int[Erfc[b*x]^2,x]

[Out]

-((Sqrt[2/Pi]*Erf[Sqrt[2]*b*x])/b) - (2*Erfc[b*x])/(b*E^(b^2*x^2)*Sqrt[Pi]) + x*Erfc[b*x]^2

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, 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 6488

Int[Erfc[(a_.) + (b_.)*(x_)]^2, x_Symbol] :> Simp[(a + b*x)*(Erfc[a + b*x]^2/b), x] + Dist[4/Sqrt[Pi], Int[(a
+ b*x)*(Erfc[a + b*x]/E^(a + b*x)^2), x], x] /; FreeQ[{a, b}, x]

Rule 6518

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

Rubi steps \begin{align*} \text {integral}& = x \text {erfc}(b x)^2+\frac {4 \int b e^{-b^2 x^2} x \text {erfc}(b x) \, dx}{\sqrt {\pi }} \\ & = x \text {erfc}(b x)^2+\frac {(4 b) \int e^{-b^2 x^2} x \text {erfc}(b x) \, dx}{\sqrt {\pi }} \\ & = -\frac {2 e^{-b^2 x^2} \text {erfc}(b x)}{b \sqrt {\pi }}+x \text {erfc}(b x)^2-\frac {4 \int e^{-2 b^2 x^2} \, dx}{\pi } \\ & = -\frac {\sqrt {\frac {2}{\pi }} \text {erf}\left (\sqrt {2} b x\right )}{b}-\frac {2 e^{-b^2 x^2} \text {erfc}(b x)}{b \sqrt {\pi }}+x \text {erfc}(b x)^2 \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.00 \[ \int \text {erfc}(b x)^2 \, dx=-\frac {\sqrt {\frac {2}{\pi }} \text {erf}\left (\sqrt {2} b x\right )}{b}-\frac {2 e^{-b^2 x^2} \text {erfc}(b x)}{b \sqrt {\pi }}+x \text {erfc}(b x)^2 \]

[In]

Integrate[Erfc[b*x]^2,x]

[Out]

-((Sqrt[2/Pi]*Erf[Sqrt[2]*b*x])/b) - (2*Erfc[b*x])/(b*E^(b^2*x^2)*Sqrt[Pi]) + x*Erfc[b*x]^2

Maple [A] (verified)

Time = 0.33 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.86

method	result	size
derivativedivides	\(\frac {\operatorname {erf}\left (b x \right )^{2} b x +\frac {2 \,\operatorname {erf}\left (b x \right ) {\mathrm e}^{-b^{2} x^{2}}}{\sqrt {\pi }}-\frac {\sqrt {2}\, \operatorname {erf}\left (b x \sqrt {2}\right )}{\sqrt {\pi }}}{b}\)	\(48\)
default	\(\frac {\operatorname {erf}\left (b x \right )^{2} b x +\frac {2 \,\operatorname {erf}\left (b x \right ) {\mathrm e}^{-b^{2} x^{2}}}{\sqrt {\pi }}-\frac {\sqrt {2}\, \operatorname {erf}\left (b x \sqrt {2}\right )}{\sqrt {\pi }}}{b}\)	\(48\)

[In]

int(erfc(b*x)^2,x,method=_RETURNVERBOSE)

[Out]

1/b*(erf(b*x)^2*b*x+2*erf(b*x)/Pi^(1/2)*exp(-b^2*x^2)-1/Pi^(1/2)*2^(1/2)*erf(b*x*2^(1/2)))

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.52 \[ \int \text {erfc}(b x)^2 \, dx=\frac {\pi b^{2} x \operatorname {erf}\left (b x\right )^{2} - 2 \, \pi b^{2} x \operatorname {erf}\left (b x\right ) + \pi b^{2} x - \sqrt {2} \sqrt {\pi } \sqrt {b^{2}} \operatorname {erf}\left (\sqrt {2} \sqrt {b^{2}} x\right ) + 2 \, \sqrt {\pi } {\left (b \operatorname {erf}\left (b x\right ) - b\right )} e^{\left (-b^{2} x^{2}\right )}}{\pi b^{2}} \]

[In]

integrate(erfc(b*x)^2,x, algorithm="fricas")

[Out]

(pi*b^2*x*erf(b*x)^2 - 2*pi*b^2*x*erf(b*x) + pi*b^2*x - sqrt(2)*sqrt(pi)*sqrt(b^2)*erf(sqrt(2)*sqrt(b^2)*x) +
2*sqrt(pi)*(b*erf(b*x) - b)*e^(-b^2*x^2))/(pi*b^2)

Sympy [F]

\[ \int \text {erfc}(b x)^2 \, dx=\int \operatorname {erfc}^{2}{\left (b x \right )}\, dx \]

[In]

integrate(erfc(b*x)**2,x)

[Out]

Integral(erfc(b*x)**2, x)

Maxima [F]

\[ \int \text {erfc}(b x)^2 \, dx=\int { \operatorname {erfc}\left (b x\right )^{2} \,d x } \]

[In]

integrate(erfc(b*x)^2,x, algorithm="maxima")

[Out]

integrate(erfc(b*x)^2, x)

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.30 \[ \int \text {erfc}(b x)^2 \, dx=x \operatorname {erf}\left (b x\right )^{2} - 2 \, x \operatorname {erf}\left (b x\right ) + \frac {b {\left (\frac {2 \, \operatorname {erf}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )}}{b^{2}} + \frac {\sqrt {2} \operatorname {erf}\left (-\sqrt {2} b x\right )}{b^{2}}\right )}}{\sqrt {\pi }} + x - \frac {2 \, e^{\left (-b^{2} x^{2}\right )}}{\sqrt {\pi } b} \]

[In]

integrate(erfc(b*x)^2,x, algorithm="giac")

[Out]

x*erf(b*x)^2 - 2*x*erf(b*x) + b*(2*erf(b*x)*e^(-b^2*x^2)/b^2 + sqrt(2)*erf(-sqrt(2)*b*x)/b^2)/sqrt(pi) + x - 2
*e^(-b^2*x^2)/(sqrt(pi)*b)

Mupad [F(-1)]

Timed out. \[ \int \text {erfc}(b x)^2 \, dx=\int {\mathrm {erfc}\left (b\,x\right )}^2 \,d x \]

[In]

int(erfc(b*x)^2,x)

[Out]

int(erfc(b*x)^2, x)