Integrand size = 8, antiderivative size = 71 \[ \int \text {erfc}(a+b x)^2 \, dx=-\frac {\sqrt {\frac {2}{\pi }} \text {erf}\left (\sqrt {2} (a+b x)\right )}{b}-\frac {2 e^{-(a+b x)^2} \text {erfc}(a+b x)}{b \sqrt {\pi }}+\frac {(a+b x) \text {erfc}(a+b x)^2}{b} \] Output:
-2^(1/2)/Pi^(1/2)*erf(2^(1/2)*(b*x+a))/b-2*erfc(b*x+a)/b/exp((b*x+a)^2)/Pi ^(1/2)+(b*x+a)*erfc(b*x+a)^2/b
Time = 0.08 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.93 \[ \int \text {erfc}(a+b x)^2 \, dx=\frac {-\sqrt {\frac {2}{\pi }} \text {erf}\left (\sqrt {2} (a+b x)\right )+\text {erfc}(a+b x) \left (-\frac {2 e^{-(a+b x)^2}}{\sqrt {\pi }}+(a+b x) \text {erfc}(a+b x)\right )}{b} \] Input:
Integrate[Erfc[a + b*x]^2,x]
Output:
(-(Sqrt[2/Pi]*Erf[Sqrt[2]*(a + b*x)]) + Erfc[a + b*x]*(-2/(E^(a + b*x)^2*S qrt[Pi]) + (a + b*x)*Erfc[a + b*x]))/b
Time = 0.46 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6907, 7281, 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.
\(\displaystyle \int \text {erfc}(a+b x)^2 \, dx\) |
\(\Big \downarrow \) 6907 |
\(\displaystyle \frac {4 \int e^{-(a+b x)^2} (a+b x) \text {erfc}(a+b x)dx}{\sqrt {\pi }}+\frac {(a+b x) \text {erfc}(a+b x)^2}{b}\) |
\(\Big \downarrow \) 7281 |
\(\displaystyle \frac {4 \int e^{-(a+b x)^2} (a+b x) \text {erfc}(a+b x)d(a+b x)}{\sqrt {\pi } b}+\frac {(a+b x) \text {erfc}(a+b x)^2}{b}\) |
\(\Big \downarrow \) 6937 |
\(\displaystyle \frac {4 \left (-\frac {\int e^{-2 (a+b x)^2}d(a+b x)}{\sqrt {\pi }}-\frac {1}{2} e^{-(a+b x)^2} \text {erfc}(a+b x)\right )}{\sqrt {\pi } b}+\frac {(a+b x) \text {erfc}(a+b x)^2}{b}\) |
\(\Big \downarrow \) 2634 |
\(\displaystyle \frac {4 \left (-\frac {\text {erf}\left (\sqrt {2} (a+b x)\right )}{2 \sqrt {2}}-\frac {1}{2} e^{-(a+b x)^2} \text {erfc}(a+b x)\right )}{\sqrt {\pi } b}+\frac {(a+b x) \text {erfc}(a+b x)^2}{b}\) |
Input:
Int[Erfc[a + b*x]^2,x]
Output:
((a + b*x)*Erfc[a + b*x]^2)/b + (4*(-1/2*Erf[Sqrt[2]*(a + b*x)]/Sqrt[2] - Erfc[a + b*x]/(2*E^(a + b*x)^2)))/(b*Sqrt[Pi])
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]
Int[u_, x_Symbol] :> With[{lst = FunctionOfLinear[u, x]}, Simp[1/lst[[3]] Subst[Int[lst[[1]], x], x, lst[[2]] + lst[[3]]*x], x] /; !FalseQ[lst]]
Time = 0.26 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.83
method | result | size |
derivativedivides | \(\frac {\operatorname {erf}\left (b x +a \right )^{2} \left (b x +a \right )+\frac {2 \,\operatorname {erf}\left (b x +a \right ) {\mathrm e}^{-\left (b x +a \right )^{2}}}{\sqrt {\pi }}-\frac {\sqrt {2}\, \operatorname {erf}\left (\sqrt {2}\, \left (b x +a \right )\right )}{\sqrt {\pi }}}{b}\) | \(59\) |
default | \(\frac {\operatorname {erf}\left (b x +a \right )^{2} \left (b x +a \right )+\frac {2 \,\operatorname {erf}\left (b x +a \right ) {\mathrm e}^{-\left (b x +a \right )^{2}}}{\sqrt {\pi }}-\frac {\sqrt {2}\, \operatorname {erf}\left (\sqrt {2}\, \left (b x +a \right )\right )}{\sqrt {\pi }}}{b}\) | \(59\) |
Input:
int(erfc(b*x+a)^2,x,method=_RETURNVERBOSE)
Output:
1/b*(erf(b*x+a)^2*(b*x+a)+2*erf(b*x+a)/Pi^(1/2)*exp(-(b*x+a)^2)-1/Pi^(1/2) *2^(1/2)*erf(2^(1/2)*(b*x+a)))
Leaf count of result is larger than twice the leaf count of optimal. 141 vs. \(2 (63) = 126\).
Time = 0.13 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.99 \[ \int \text {erfc}(a+b x)^2 \, dx=-\frac {2 \, \pi b^{2} x \operatorname {erf}\left (b x + a\right ) - \pi b^{2} x + 2 \, \pi a \sqrt {b^{2}} \operatorname {erf}\left (\frac {\sqrt {b^{2}} {\left (b x + a\right )}}{b}\right ) - {\left (\pi b^{2} x + \pi a b\right )} \operatorname {erf}\left (b x + a\right )^{2} + \sqrt {2} \sqrt {\pi } \sqrt {b^{2}} \operatorname {erf}\left (\frac {\sqrt {2} \sqrt {b^{2}} {\left (b x + a\right )}}{b}\right ) - 2 \, \sqrt {\pi } {\left (b \operatorname {erf}\left (b x + a\right ) - b\right )} e^{\left (-b^{2} x^{2} - 2 \, a b x - a^{2}\right )}}{\pi b^{2}} \] Input:
integrate(erfc(b*x+a)^2,x, algorithm="fricas")
Output:
-(2*pi*b^2*x*erf(b*x + a) - pi*b^2*x + 2*pi*a*sqrt(b^2)*erf(sqrt(b^2)*(b*x + a)/b) - (pi*b^2*x + pi*a*b)*erf(b*x + a)^2 + sqrt(2)*sqrt(pi)*sqrt(b^2) *erf(sqrt(2)*sqrt(b^2)*(b*x + a)/b) - 2*sqrt(pi)*(b*erf(b*x + a) - b)*e^(- b^2*x^2 - 2*a*b*x - a^2))/(pi*b^2)
\[ \int \text {erfc}(a+b x)^2 \, dx=\int \operatorname {erfc}^{2}{\left (a + b x \right )}\, dx \] Input:
integrate(erfc(b*x+a)**2,x)
Output:
Integral(erfc(a + b*x)**2, x)
\[ \int \text {erfc}(a+b x)^2 \, dx=\int { \operatorname {erfc}\left (b x + a\right )^{2} \,d x } \] Input:
integrate(erfc(b*x+a)^2,x, algorithm="maxima")
Output:
integrate(erfc(b*x + a)^2, x)
\[ \int \text {erfc}(a+b x)^2 \, dx=\int { \operatorname {erfc}\left (b x + a\right )^{2} \,d x } \] Input:
integrate(erfc(b*x+a)^2,x, algorithm="giac")
Output:
integrate(erfc(b*x + a)^2, x)
Timed out. \[ \int \text {erfc}(a+b x)^2 \, dx=\int {\mathrm {erfc}\left (a+b\,x\right )}^2 \,d x \] Input:
int(erfc(a + b*x)^2,x)
Output:
int(erfc(a + b*x)^2, x)
\[ \int \text {erfc}(a+b x)^2 \, dx=\frac {-2 \sqrt {\pi }\, e^{b^{2} x^{2}+2 a b x +a^{2}} \mathrm {erf}\left (b x +a \right ) a -2 \sqrt {\pi }\, e^{b^{2} x^{2}+2 a b x +a^{2}} \mathrm {erf}\left (b x +a \right ) b x +\sqrt {\pi }\, e^{b^{2} x^{2}+2 a b x +a^{2}} \left (\int \mathrm {erf}\left (b x +a \right )^{2}d x \right ) b +\sqrt {\pi }\, e^{b^{2} x^{2}+2 a b x +a^{2}} b x -2}{\sqrt {\pi }\, e^{b^{2} x^{2}+2 a b x +a^{2}} b} \] Input:
int(erfc(b*x+a)^2,x)
Output:
( - 2*sqrt(pi)*e**(a**2 + 2*a*b*x + b**2*x**2)*erf(a + b*x)*a - 2*sqrt(pi) *e**(a**2 + 2*a*b*x + b**2*x**2)*erf(a + b*x)*b*x + sqrt(pi)*e**(a**2 + 2* a*b*x + b**2*x**2)*int(erf(a + b*x)**2,x)*b + sqrt(pi)*e**(a**2 + 2*a*b*x + b**2*x**2)*b*x - 2)/(sqrt(pi)*e**(a**2 + 2*a*b*x + b**2*x**2)*b)