Integrand size = 6, antiderivative size = 56 \[ \int \text {erfc}(b x)^2 \, dx=-\frac {\sqrt {\frac {2}{\pi }} \text {erf}\left (\sqrt {2} b x\right )}{b}-\frac {2 e^{-b^2 x^2} \text {erfc}(b x)}{b \sqrt {\pi }}+x \text {erfc}(b x)^2 \] Output:
-2^(1/2)/Pi^(1/2)*erf(2^(1/2)*b*x)/b-2*erfc(b*x)/b/exp(b^2*x^2)/Pi^(1/2)+x *erfc(b*x)^2
Time = 0.03 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.00 \[ \int \text {erfc}(b x)^2 \, dx=-\frac {\sqrt {\frac {2}{\pi }} \text {erf}\left (\sqrt {2} b x\right )}{b}-\frac {2 e^{-b^2 x^2} \text {erfc}(b x)}{b \sqrt {\pi }}+x \text {erfc}(b x)^2 \] Input:
Integrate[Erfc[b*x]^2,x]
Output:
-((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
Time = 0.30 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.07, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {6907, 27, 6937, 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.
\(\displaystyle \int \text {erfc}(b x)^2 \, dx\) |
\(\Big \downarrow \) 6907 |
\(\displaystyle \frac {4 \int b e^{-b^2 x^2} x \text {erfc}(b x)dx}{\sqrt {\pi }}+x \text {erfc}(b x)^2\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {4 b \int e^{-b^2 x^2} x \text {erfc}(b x)dx}{\sqrt {\pi }}+x \text {erfc}(b x)^2\) |
\(\Big \downarrow \) 6937 |
\(\displaystyle \frac {4 b \left (-\frac {\int e^{-2 b^2 x^2}dx}{\sqrt {\pi } b}-\frac {e^{-b^2 x^2} \text {erfc}(b x)}{2 b^2}\right )}{\sqrt {\pi }}+x \text {erfc}(b x)^2\) |
\(\Big \downarrow \) 2634 |
\(\displaystyle \frac {4 b \left (-\frac {\text {erf}\left (\sqrt {2} b x\right )}{2 \sqrt {2} b^2}-\frac {e^{-b^2 x^2} \text {erfc}(b x)}{2 b^2}\right )}{\sqrt {\pi }}+x \text {erfc}(b x)^2\) |
Input:
Int[Erfc[b*x]^2,x]
Output:
x*Erfc[b*x]^2 + (4*b*(-1/2*Erf[Sqrt[2]*b*x]/(Sqrt[2]*b^2) - Erfc[b*x]/(2*b ^2*E^(b^2*x^2))))/Sqrt[Pi]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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]
Int[Erfc[(a_.) + (b_.)*(x_)]^2, x_Symbol] :> Simp[(a + b*x)*(Erfc[a + b*x]^ 2/b), x] + Simp[4/Sqrt[Pi] Int[(a + b*x)*(Erfc[a + b*x]/E^(a + b*x)^2), x ], x] /; FreeQ[{a, b}, x]
Int[E^((c_.) + (d_.)*(x_)^2)*Erfc[(a_.) + (b_.)*(x_)]*(x_), x_Symbol] :> Si mp[E^(c + d*x^2)*(Erfc[a + b*x]/(2*d)), x] + Simp[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]
Time = 0.21 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.86
method | result | size |
derivativedivides | \(\frac {\operatorname {erf}\left (b x \right )^{2} b x +\frac {2 \,\operatorname {erf}\left (b x \right ) {\mathrm e}^{-b^{2} x^{2}}}{\sqrt {\pi }}-\frac {\sqrt {2}\, \operatorname {erf}\left (\sqrt {2}\, b x \right )}{\sqrt {\pi }}}{b}\) | \(48\) |
default | \(\frac {\operatorname {erf}\left (b x \right )^{2} b x +\frac {2 \,\operatorname {erf}\left (b x \right ) {\mathrm e}^{-b^{2} x^{2}}}{\sqrt {\pi }}-\frac {\sqrt {2}\, \operatorname {erf}\left (\sqrt {2}\, b x \right )}{\sqrt {\pi }}}{b}\) | \(48\) |
Input:
int(erfc(b*x)^2,x,method=_RETURNVERBOSE)
Output:
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)*e rf(2^(1/2)*b*x))
Time = 0.08 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.52 \[ \int \text {erfc}(b x)^2 \, dx=\frac {\pi b^{2} x \operatorname {erf}\left (b x\right )^{2} - 2 \, \pi b^{2} x \operatorname {erf}\left (b x\right ) + \pi b^{2} x - \sqrt {2} \sqrt {\pi } \sqrt {b^{2}} \operatorname {erf}\left (\sqrt {2} \sqrt {b^{2}} x\right ) + 2 \, \sqrt {\pi } {\left (b \operatorname {erf}\left (b x\right ) - b\right )} e^{\left (-b^{2} x^{2}\right )}}{\pi b^{2}} \] Input:
integrate(erfc(b*x)^2,x, algorithm="fricas")
Output:
(pi*b^2*x*erf(b*x)^2 - 2*pi*b^2*x*erf(b*x) + pi*b^2*x - sqrt(2)*sqrt(pi)*s qrt(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)
\[ \int \text {erfc}(b x)^2 \, dx=\int \operatorname {erfc}^{2}{\left (b x \right )}\, dx \] Input:
integrate(erfc(b*x)**2,x)
Output:
Integral(erfc(b*x)**2, x)
\[ \int \text {erfc}(b x)^2 \, dx=\int { \operatorname {erfc}\left (b x\right )^{2} \,d x } \] Input:
integrate(erfc(b*x)^2,x, algorithm="maxima")
Output:
integrate(erfc(b*x)^2, x)
\[ \int \text {erfc}(b x)^2 \, dx=\int { \operatorname {erfc}\left (b x\right )^{2} \,d x } \] Input:
integrate(erfc(b*x)^2,x, algorithm="giac")
Output:
integrate(erfc(b*x)^2, x)
Timed out. \[ \int \text {erfc}(b x)^2 \, dx=\int {\mathrm {erfc}\left (b\,x\right )}^2 \,d x \] Input:
int(erfc(b*x)^2,x)
Output:
int(erfc(b*x)^2, x)
\[ \int \text {erfc}(b x)^2 \, dx=\frac {-2 \sqrt {\pi }\, e^{b^{2} x^{2}} \mathrm {erf}\left (b x \right ) b x +\sqrt {\pi }\, e^{b^{2} x^{2}} \left (\int \mathrm {erf}\left (b x \right )^{2}d x \right ) b +\sqrt {\pi }\, e^{b^{2} x^{2}} b x -2}{\sqrt {\pi }\, e^{b^{2} x^{2}} b} \] Input:
int(erfc(b*x)^2,x)
Output:
( - 2*sqrt(pi)*e**(b**2*x**2)*erf(b*x)*b*x + sqrt(pi)*e**(b**2*x**2)*int(e rf(b*x)**2,x)*b + sqrt(pi)*e**(b**2*x**2)*b*x - 2)/(sqrt(pi)*e**(b**2*x**2 )*b)