\(\int \text {erfc}(a+b x)^2 \, dx\) [140]

Optimal result

Integrand size = 8, antiderivative size = 71 \[ \int \text {erfc}(a+b x)^2 \, dx=-\frac {\sqrt {\frac {2}{\pi }} \text {erf}\left (\sqrt {2} (a+b x)\right )}{b}-\frac {2 e^{-(a+b x)^2} \text {erfc}(a+b x)}{b \sqrt {\pi }}+\frac {(a+b x) \text {erfc}(a+b x)^2}{b} \]

[Out]

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

Rubi [A] (verified)

Time = 0.17 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {6488, 6518, 2236} \[ \int \text {erfc}(a+b x)^2 \, dx=-\frac {\sqrt {\frac {2}{\pi }} \text {erf}\left (\sqrt {2} (a+b x)\right )}{b}+\frac {(a+b x) \text {erfc}(a+b x)^2}{b}-\frac {2 e^{-(a+b x)^2} \text {erfc}(a+b x)}{\sqrt {\pi } b} \]

[In]

Int[Erfc[a + b*x]^2,x]

[Out]

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

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

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.93 \[ \int \text {erfc}(a+b x)^2 \, dx=\frac {-\sqrt {\frac {2}{\pi }} \text {erf}\left (\sqrt {2} (a+b x)\right )+\text {erfc}(a+b x) \left (-\frac {2 e^{-(a+b x)^2}}{\sqrt {\pi }}+(a+b x) \text {erfc}(a+b x)\right )}{b} \]

[In]

Integrate[Erfc[a + b*x]^2,x]

[Out]

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

Maple [A] (verified)

Time = 0.38 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.83

method	result	size
derivativedivides	\(\frac {\operatorname {erf}\left (b x +a \right )^{2} \left (b x +a \right )+\frac {2 \,\operatorname {erf}\left (b x +a \right ) {\mathrm e}^{-\left (b x +a \right )^{2}}}{\sqrt {\pi }}-\frac {\sqrt {2}\, \operatorname {erf}\left (\left (b x +a \right ) \sqrt {2}\right )}{\sqrt {\pi }}}{b}\)	\(59\)
default	\(\frac {\operatorname {erf}\left (b x +a \right )^{2} \left (b x +a \right )+\frac {2 \,\operatorname {erf}\left (b x +a \right ) {\mathrm e}^{-\left (b x +a \right )^{2}}}{\sqrt {\pi }}-\frac {\sqrt {2}\, \operatorname {erf}\left (\left (b x +a \right ) \sqrt {2}\right )}{\sqrt {\pi }}}{b}\)	\(59\)

[In]

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

[Out]

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((b*x+a)*2^(1/2)))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 141 vs. \(2 (63) = 126\).

Time = 0.26 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.99 \[ \int \text {erfc}(a+b x)^2 \, dx=-\frac {2 \, \pi b^{2} x \operatorname {erf}\left (b x + a\right ) - \pi b^{2} x + 2 \, \pi a \sqrt {b^{2}} \operatorname {erf}\left (\frac {\sqrt {b^{2}} {\left (b x + a\right )}}{b}\right ) - {\left (\pi b^{2} x + \pi a b\right )} \operatorname {erf}\left (b x + a\right )^{2} + \sqrt {2} \sqrt {\pi } \sqrt {b^{2}} \operatorname {erf}\left (\frac {\sqrt {2} \sqrt {b^{2}} {\left (b x + a\right )}}{b}\right ) - 2 \, \sqrt {\pi } {\left (b \operatorname {erf}\left (b x + a\right ) - b\right )} e^{\left (-b^{2} x^{2} - 2 \, a b x - a^{2}\right )}}{\pi b^{2}} \]

[In]

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

[Out]

-(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)

Sympy [F]

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

[In]

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

[Out]

Integral(erfc(a + b*x)**2, x)

Maxima [F]

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

[In]

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

[Out]

integrate(erfc(b*x + a)^2, x)

Giac [F]

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

[In]

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

[Out]

integrate(erfc(b*x + a)^2, x)

Mupad [F(-1)]

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

[In]

int(erfc(a + b*x)^2,x)

[Out]

int(erfc(a + b*x)^2, x)