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

Optimal. Leaf size=189 \[ -\frac{\sqrt{\frac{2}{\pi }} (b c-a d) \text{Erf}\left (\sqrt{2} (a+b x)\right )}{b^2}+\frac{(a+b x) (b c-a d) \text{Erfc}(a+b x)^2}{b^2}-\frac{2 e^{-(a+b x)^2} (b c-a d) \text{Erfc}(a+b x)}{\sqrt{\pi } b^2}+\frac{d (a+b x)^2 \text{Erfc}(a+b x)^2}{2 b^2}-\frac{d \text{Erfc}(a+b x)^2}{4 b^2}-\frac{d e^{-(a+b x)^2} (a+b x) \text{Erfc}(a+b x)}{\sqrt{\pi } b^2}+\frac{d e^{-2 (a+b x)^2}}{2 \pi b^2} \]

[Out]

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

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Rubi [A]  time = 0.189064, antiderivative size = 189, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 9, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.643, Rules used = {6368, 6353, 6383, 2205, 6365, 6386, 6374, 30, 2209} \[ -\frac{\sqrt{\frac{2}{\pi }} (b c-a d) \text{Erf}\left (\sqrt{2} (a+b x)\right )}{b^2}+\frac{(a+b x) (b c-a d) \text{Erfc}(a+b x)^2}{b^2}-\frac{2 e^{-(a+b x)^2} (b c-a d) \text{Erfc}(a+b x)}{\sqrt{\pi } b^2}+\frac{d (a+b x)^2 \text{Erfc}(a+b x)^2}{2 b^2}-\frac{d \text{Erfc}(a+b x)^2}{4 b^2}-\frac{d e^{-(a+b x)^2} (a+b x) \text{Erfc}(a+b x)}{\sqrt{\pi } b^2}+\frac{d e^{-2 (a+b x)^2}}{2 \pi b^2} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x)*Erfc[a + b*x]^2,x]

[Out]

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

Rule 6368

Int[Erfc[(a_) + (b_.)*(x_)]^2*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Dist[1/b^(m + 1), Subst[Int[ExpandInteg
rand[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]

Rule 6353

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 6383

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]

Rule 2205

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 6365

Int[Erfc[(b_.)*(x_)]^2*(x_)^(m_.), x_Symbol] :> Simp[(x^(m + 1)*Erfc[b*x]^2)/(m + 1), x] + Dist[(4*b)/(Sqrt[Pi
]*(m + 1)), Int[(x^(m + 1)*Erfc[b*x])/E^(b^2*x^2), x], x] /; FreeQ[b, x] && (IGtQ[m, 0] || ILtQ[(m + 1)/2, 0])

Rule 6386

Int[E^((c_.) + (d_.)*(x_)^2)*Erfc[(a_.) + (b_.)*(x_)]*(x_)^(m_), x_Symbol] :> Simp[(x^(m - 1)*E^(c + d*x^2)*Er
fc[a + b*x])/(2*d), x] + (-Dist[(m - 1)/(2*d), Int[x^(m - 2)*E^(c + d*x^2)*Erfc[a + b*x], x], x] + Dist[b/(d*S
qrt[Pi]), Int[x^(m - 1)*E^(-a^2 + c - 2*a*b*x - (b^2 - d)*x^2), x], x]) /; FreeQ[{a, b, c, d}, x] && IGtQ[m, 1
]

Rule 6374

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

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 2209

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Simp[((e + f*x)^n*
F^(a + b*(c + d*x)^n))/(b*f*n*(c + d*x)^n*Log[F]), x] /; FreeQ[{F, a, b, c, d, e, f, n}, x] && EqQ[m, n - 1] &
& EqQ[d*e - c*f, 0]

Rubi steps

\begin{align*} \int (c+d x) \text{erfc}(a+b x)^2 \, dx &=\frac{\operatorname{Subst}\left (\int \left (b c \left (1-\frac{a d}{b c}\right ) \text{erfc}(x)^2+d x \text{erfc}(x)^2\right ) \, dx,x,a+b x\right )}{b^2}\\ &=\frac{d \operatorname{Subst}\left (\int x \text{erfc}(x)^2 \, dx,x,a+b x\right )}{b^2}+\frac{(b c-a d) \operatorname{Subst}\left (\int \text{erfc}(x)^2 \, dx,x,a+b x\right )}{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}+\frac{(2 d) \operatorname{Subst}\left (\int e^{-x^2} x^2 \text{erfc}(x) \, dx,x,a+b x\right )}{b^2 \sqrt{\pi }}+\frac{(4 (b c-a d)) \operatorname{Subst}\left (\int e^{-x^2} x \text{erfc}(x) \, dx,x,a+b x\right )}{b^2 \sqrt{\pi }}\\ &=-\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{(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}-\frac{(2 d) \operatorname{Subst}\left (\int e^{-2 x^2} x \, dx,x,a+b x\right )}{b^2 \pi }-\frac{(4 (b c-a d)) \operatorname{Subst}\left (\int e^{-2 x^2} \, dx,x,a+b x\right )}{b^2 \pi }+\frac{d \operatorname{Subst}\left (\int e^{-x^2} \text{erfc}(x) \, dx,x,a+b x\right )}{b^2 \sqrt{\pi }}\\ &=\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{(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}-\frac{d \operatorname{Subst}(\int x \, dx,x,\text{erfc}(a+b x))}{2 b^2}\\ &=\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}\\ \end{align*}

Mathematica [A]  time = 2.04342, size = 301, normalized size = 1.59 \[ \frac{\frac{d \left (-4 \sqrt{\pi } (a+b x) \text{ExpIntegralE}\left (\frac{1}{2},(a+b x)^2\right )-2 \pi (a+b x)^2 \text{Erf}(a+b x)^2+4 \pi (a+b x)^2 \text{Erf}(a+b x)-4 \sqrt{\pi } e^{-(a+b x)^2} (a+b x) \text{Erf}(a+b x)+\pi \text{Erf}(a+b x)^2-2 \pi \text{Erf}(a+b x)+4 \sqrt{2 \pi } b x \text{Erf}\left (\sqrt{2} (a+b x)\right )+4 \sqrt{2 \pi } a \text{Erf}\left (\sqrt{2} (a+b x)\right )+2 \pi (\text{Erfc}(-a-b x) \text{Erfc}(a+b x)+2)-2 \pi (a+b x)^2+4 \sqrt{\pi } e^{-(a+b x)^2} (a+b x)+2 e^{-2 (a+b x)^2}\right )}{\pi }+4 b (c+d x) \left (\text{Erfc}(a+b x) \left ((a+b x) \text{Erfc}(a+b x)-\frac{2 e^{-(a+b x)^2}}{\sqrt{\pi }}\right )-\sqrt{\frac{2}{\pi }} \text{Erf}\left (\sqrt{2} (a+b x)\right )\right )}{4 b^2} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(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)*ExpIntegralE[1/2, (a + b*x)^2]))/Pi
)/(4*b^2)

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Maple [F]  time = 0.068, size = 0, normalized size = 0. \begin{align*} \int \left ( dx+c \right ) \left ({\it erfc} \left ( bx+a \right ) \right ) ^{2}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (d x + c\right )} \operatorname{erfc}\left (b x + a\right )^{2}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 2.17829, size = 647, normalized size = 3.42 \begin{align*} \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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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)*sqrt(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*pi*b^3*c*x)*erf(b*x + a))/(pi*b^3)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (c + d x\right ) \operatorname{erfc}^{2}{\left (a + b x \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (d x + c\right )} \operatorname{erfc}\left (b x + a\right )^{2}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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