\(\int (c+d x) \text {erfc}(a+b x)^2 \, dx\) [139]

Optimal result

Integrand size = 14, antiderivative size = 189 \[ \int (c+d x) \text {erfc}(a+b x)^2 \, dx=\frac {d e^{-2 (a+b x)^2}}{2 b^2 \pi }-\frac {(b c-a d) \sqrt {\frac {2}{\pi }} \text {erf}\left (\sqrt {2} (a+b x)\right )}{b^2}-\frac {2 (b c-a d) e^{-(a+b x)^2} \text {erfc}(a+b x)}{b^2 \sqrt {\pi }}-\frac {d e^{-(a+b x)^2} (a+b x) \text {erfc}(a+b x)}{b^2 \sqrt {\pi }}-\frac {d \text {erfc}(a+b x)^2}{4 b^2}+\frac {(b c-a d) (a+b x) \text {erfc}(a+b x)^2}{b^2}+\frac {d (a+b x)^2 \text {erfc}(a+b x)^2}{2 b^2} \] Output:

1/2*d/b^2/exp(2*(b*x+a)^2)/Pi-(-a*d+b*c)*2^(1/2)/Pi^(1/2)*erf(2^(1/2)*(b*x 
+a))/b^2-2*(-a*d+b*c)*erfc(b*x+a)/b^2/exp((b*x+a)^2)/Pi^(1/2)-d*(b*x+a)*er 
fc(b*x+a)/b^2/exp((b*x+a)^2)/Pi^(1/2)-1/4*d*erfc(b*x+a)^2/b^2+(-a*d+b*c)*( 
b*x+a)*erfc(b*x+a)^2/b^2+1/2*d*(b*x+a)^2*erfc(b*x+a)^2/b^2
 

Mathematica [A] (verified)

Time = 2.21 (sec) , antiderivative size = 301, normalized size of antiderivative = 1.59 \[ \int (c+d x) \text {erfc}(a+b x)^2 \, dx=\frac {4 b (c+d x) \left (-\sqrt {\frac {2}{\pi }} \text {erf}\left (\sqrt {2} (a+b x)\right )+\text {erfc}(a+b x) \left (-\frac {2 e^{-(a+b x)^2}}{\sqrt {\pi }}+(a+b x) \text {erfc}(a+b x)\right )\right )+\frac {d \left (2 e^{-2 (a+b x)^2}+4 e^{-(a+b x)^2} \sqrt {\pi } (a+b x)-2 \pi (a+b x)^2-2 \pi \text {erf}(a+b x)-4 e^{-(a+b x)^2} \sqrt {\pi } (a+b x) \text {erf}(a+b x)+4 \pi (a+b x)^2 \text {erf}(a+b x)+\pi \text {erf}(a+b x)^2-2 \pi (a+b x)^2 \text {erf}(a+b x)^2+4 a \sqrt {2 \pi } \text {erf}\left (\sqrt {2} (a+b x)\right )+4 b \sqrt {2 \pi } x \text {erf}\left (\sqrt {2} (a+b x)\right )+2 \pi (2+\text {erfc}(-a-b x) \text {erfc}(a+b x))-4 \sqrt {\pi } (a+b x) \operatorname {ExpIntegralE}\left (\frac {1}{2},(a+b x)^2\right )\right )}{\pi }}{4 b^2} \] Input:

Integrate[(c + d*x)*Erfc[a + b*x]^2,x]
 

Output:

(4*b*(c + d*x)*(-(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])) + (d*(2/E^(2*(a + b*x) 
^2) + (4*Sqrt[Pi]*(a + b*x))/E^(a + b*x)^2 - 2*Pi*(a + b*x)^2 - 2*Pi*Erf[a 
 + b*x] - (4*Sqrt[Pi]*(a + b*x)*Erf[a + b*x])/E^(a + b*x)^2 + 4*Pi*(a + b* 
x)^2*Erf[a + b*x] + Pi*Erf[a + b*x]^2 - 2*Pi*(a + b*x)^2*Erf[a + b*x]^2 + 
4*a*Sqrt[2*Pi]*Erf[Sqrt[2]*(a + b*x)] + 4*b*Sqrt[2*Pi]*x*Erf[Sqrt[2]*(a + 
b*x)] + 2*Pi*(2 + Erfc[-a - b*x]*Erfc[a + b*x]) - 4*Sqrt[Pi]*(a + b*x)*Exp 
IntegralE[1/2, (a + b*x)^2]))/Pi)/(4*b^2)
 

Rubi [A] (verified)

Time = 0.39 (sec) , antiderivative size = 172, normalized size of antiderivative = 0.91, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {6922, 2009}

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.

\(\displaystyle \int (c+d x) \text {erfc}(a+b x)^2 \, dx\)

\(\Big \downarrow \) 6922

\(\displaystyle \frac {\int \left ((b c-a d) \text {erfc}(a+b x)^2+d (a+b x) \text {erfc}(a+b x)^2\right )d(a+b x)}{b^2}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {-\sqrt {\frac {2}{\pi }} (b c-a d) \text {erf}\left (\sqrt {2} (a+b x)\right )+(a+b x) (b c-a d) \text {erfc}(a+b x)^2-\frac {2 e^{-(a+b x)^2} (b c-a d) \text {erfc}(a+b x)}{\sqrt {\pi }}+\frac {1}{2} d (a+b x)^2 \text {erfc}(a+b x)^2-\frac {1}{4} d \text {erfc}(a+b x)^2-\frac {d e^{-(a+b x)^2} (a+b x) \text {erfc}(a+b x)}{\sqrt {\pi }}+\frac {d e^{-2 (a+b x)^2}}{2 \pi }}{b^2}\)

Input:

Int[(c + d*x)*Erfc[a + b*x]^2,x]
 

Output:

(d/(2*E^(2*(a + b*x)^2)*Pi) - (b*c - a*d)*Sqrt[2/Pi]*Erf[Sqrt[2]*(a + b*x) 
] - (2*(b*c - a*d)*Erfc[a + b*x])/(E^(a + b*x)^2*Sqrt[Pi]) - (d*(a + b*x)* 
Erfc[a + b*x])/(E^(a + b*x)^2*Sqrt[Pi]) - (d*Erfc[a + b*x]^2)/4 + (b*c - a 
*d)*(a + b*x)*Erfc[a + b*x]^2 + (d*(a + b*x)^2*Erfc[a + b*x]^2)/2)/b^2
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 6922

Int[Erfc[(a_) + (b_.)*(x_)]^2*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp 
[1/b^(m + 1)   Subst[Int[ExpandIntegrand[Erfc[x]^2, (b*c - a*d + d*x)^m, x] 
, x], x, a + b*x], x] /; FreeQ[{a, b, c, d}, x] && IGtQ[m, 0]
 

Maple [F]

\[\int \left (d x +c \right ) \operatorname {erfc}\left (b x +a \right )^{2}d x\]

Input:

int((d*x+c)*erfc(b*x+a)^2,x)
 

Output:

int((d*x+c)*erfc(b*x+a)^2,x)
 

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 273, normalized size of antiderivative = 1.44 \[ \int (c+d x) \text {erfc}(a+b x)^2 \, dx=\frac {2 \, \pi b^{3} d x^{2} + 4 \, \pi b^{3} c x - 4 \, \sqrt {2} \sqrt {\pi } \sqrt {b^{2}} {\left (b c - a d\right )} \operatorname {erf}\left (\frac {\sqrt {2} \sqrt {b^{2}} {\left (b x + a\right )}}{b}\right ) - 2 \, \pi {\left (4 \, a b c - {\left (2 \, a^{2} + 1\right )} d\right )} \sqrt {b^{2}} \operatorname {erf}\left (\frac {\sqrt {b^{2}} {\left (b x + a\right )}}{b}\right ) + {\left (2 \, \pi b^{3} d x^{2} + 4 \, \pi b^{3} c x + \pi {\left (4 \, a b^{2} c - {\left (2 \, a^{2} + 1\right )} b d\right )}\right )} \operatorname {erf}\left (b x + a\right )^{2} + 2 \, b d e^{\left (-2 \, b^{2} x^{2} - 4 \, a b x - 2 \, a^{2}\right )} - 4 \, \sqrt {\pi } {\left (b^{2} d x + 2 \, b^{2} c - a b d - {\left (b^{2} d x + 2 \, b^{2} c - a b d\right )} \operatorname {erf}\left (b x + a\right )\right )} e^{\left (-b^{2} x^{2} - 2 \, a b x - a^{2}\right )} - 4 \, {\left (\pi b^{3} d x^{2} + 2 \, \pi b^{3} c x\right )} \operatorname {erf}\left (b x + a\right )}{4 \, \pi b^{3}} \] Input:

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

Output:

1/4*(2*pi*b^3*d*x^2 + 4*pi*b^3*c*x - 4*sqrt(2)*sqrt(pi)*sqrt(b^2)*(b*c - a 
*d)*erf(sqrt(2)*sqrt(b^2)*(b*x + a)/b) - 2*pi*(4*a*b*c - (2*a^2 + 1)*d)*sq 
rt(b^2)*erf(sqrt(b^2)*(b*x + a)/b) + (2*pi*b^3*d*x^2 + 4*pi*b^3*c*x + pi*( 
4*a*b^2*c - (2*a^2 + 1)*b*d))*erf(b*x + a)^2 + 2*b*d*e^(-2*b^2*x^2 - 4*a*b 
*x - 2*a^2) - 4*sqrt(pi)*(b^2*d*x + 2*b^2*c - a*b*d - (b^2*d*x + 2*b^2*c - 
 a*b*d)*erf(b*x + a))*e^(-b^2*x^2 - 2*a*b*x - a^2) - 4*(pi*b^3*d*x^2 + 2*p 
i*b^3*c*x)*erf(b*x + a))/(pi*b^3)
 

Sympy [F]

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

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

Output:

Integral((c + d*x)*erfc(a + b*x)**2, x)
 

Maxima [F]

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

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

Output:

integrate((d*x + c)*erfc(b*x + a)^2, x)
 

Giac [F]

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

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

Output:

integrate((d*x + c)*erfc(b*x + a)^2, x)
 

Mupad [F(-1)]

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

int(erfc(a + b*x)^2*(c + d*x),x)
 

Output:

int(erfc(a + b*x)^2*(c + d*x), x)
 

Reduce [F]

\[ \int (c+d x) \text {erfc}(a+b x)^2 \, dx=\frac {-4 \sqrt {\pi }\, e^{b^{2} x^{2}+2 a b x +a^{2}} \mathrm {erf}\left (b x +a \right ) a c -4 \sqrt {\pi }\, e^{b^{2} x^{2}+2 a b x +a^{2}} \mathrm {erf}\left (b x +a \right ) b c x +2 \sqrt {\pi }\, e^{b^{2} x^{2}+2 a b x +a^{2}} \left (\int \mathrm {erf}\left (b x +a \right )^{2}d x \right ) b c -4 \sqrt {\pi }\, e^{b^{2} x^{2}+2 a b x +a^{2}} \left (\int \mathrm {erf}\left (b x +a \right ) x d x \right ) b d +2 \sqrt {\pi }\, e^{b^{2} x^{2}+2 a b x +a^{2}} \left (\int \mathrm {erf}\left (b x +a \right )^{2} x d x \right ) b d +2 \sqrt {\pi }\, e^{b^{2} x^{2}+2 a b x +a^{2}} b c x +\sqrt {\pi }\, e^{b^{2} x^{2}+2 a b x +a^{2}} b d \,x^{2}-4 c}{2 \sqrt {\pi }\, e^{b^{2} x^{2}+2 a b x +a^{2}} b} \] Input:

int((d*x+c)*erfc(b*x+a)^2,x)
 

Output:

( - 4*sqrt(pi)*e**(a**2 + 2*a*b*x + b**2*x**2)*erf(a + b*x)*a*c - 4*sqrt(p 
i)*e**(a**2 + 2*a*b*x + b**2*x**2)*erf(a + b*x)*b*c*x + 2*sqrt(pi)*e**(a** 
2 + 2*a*b*x + b**2*x**2)*int(erf(a + b*x)**2,x)*b*c - 4*sqrt(pi)*e**(a**2 
+ 2*a*b*x + b**2*x**2)*int(erf(a + b*x)*x,x)*b*d + 2*sqrt(pi)*e**(a**2 + 2 
*a*b*x + b**2*x**2)*int(erf(a + b*x)**2*x,x)*b*d + 2*sqrt(pi)*e**(a**2 + 2 
*a*b*x + b**2*x**2)*b*c*x + sqrt(pi)*e**(a**2 + 2*a*b*x + b**2*x**2)*b*d*x 
**2 - 4*c)/(2*sqrt(pi)*e**(a**2 + 2*a*b*x + b**2*x**2)*b)