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]

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

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Rubi [A]  time = 0.19, antiderivative size = 189, normalized size of antiderivative = 1.00, 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 30

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

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 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]

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 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 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 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 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 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
]

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*}

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Mathematica [A]  time = 1.36, size = 301, normalized size = 1.59 \[ \frac {4 b (c+d x) \left (\text {erfc}(a+b x) \left ((a+b x) \text {erfc}(a+b x)-\frac {2 e^{-(a+b x)^2}}{\sqrt {\pi }}\right )-\sqrt {\frac {2}{\pi }} \text {erf}\left (\sqrt {2} (a+b x)\right )\right )+\frac {d \left (-2 \pi (a+b x)^2 \text {erf}(a+b x)^2+4 \pi (a+b x)^2 \text {erf}(a+b x)-4 \sqrt {\pi } e^{-(a+b x)^2} (a+b x) \text {erf}(a+b x)+\pi \text {erf}(a+b x)^2-2 \pi \text {erf}(a+b x)+4 \sqrt {2 \pi } b x \text {erf}\left (\sqrt {2} (a+b x)\right )+4 \sqrt {2 \pi } a \text {erf}\left (\sqrt {2} (a+b x)\right )+2 \pi (\text {erfc}(-a-b x) \text {erfc}(a+b x)+2)-4 \sqrt {\pi } (a+b x) E_{\frac {1}{2}}\left ((a+b x)^2\right )-2 \pi (a+b x)^2+4 \sqrt {\pi } e^{-(a+b x)^2} (a+b x)+2 e^{-2 (a+b x)^2}\right )}{\pi }}{4 b^2} \]

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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fricas [A]  time = 0.42, size = 273, normalized size = 1.44 \[ \frac {2 \, \pi b^{3} d x^{2} + 4 \, \pi b^{3} c x - 4 \, \sqrt {2} \sqrt {\pi } \sqrt {b^{2}} {\left (b c - a d\right )} \operatorname {erf}\left (\frac {\sqrt {2} \sqrt {b^{2}} {\left (b x + a\right )}}{b}\right ) - 2 \, \pi {\left (4 \, a b c - {\left (2 \, a^{2} + 1\right )} d\right )} \sqrt {b^{2}} \operatorname {erf}\left (\frac {\sqrt {b^{2}} {\left (b x + a\right )}}{b}\right ) + {\left (2 \, \pi b^{3} d x^{2} + 4 \, \pi b^{3} c x + \pi {\left (4 \, a b^{2} c - {\left (2 \, a^{2} + 1\right )} b d\right )}\right )} \operatorname {erf}\left (b x + a\right )^{2} + 2 \, b d e^{\left (-2 \, b^{2} x^{2} - 4 \, a b x - 2 \, a^{2}\right )} - 4 \, \sqrt {\pi } {\left (b^{2} d x + 2 \, b^{2} c - a b d - {\left (b^{2} d x + 2 \, b^{2} c - a b d\right )} \operatorname {erf}\left (b x + a\right )\right )} e^{\left (-b^{2} x^{2} - 2 \, a b x - a^{2}\right )} - 4 \, {\left (\pi b^{3} d x^{2} + 2 \, \pi b^{3} c x\right )} \operatorname {erf}\left (b x + a\right )}{4 \, \pi b^{3}} \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (d x + c\right )} \operatorname {erfc}\left (b x + a\right )^{2}\,{d x} \]

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)

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maple [F]  time = 0.02, size = 0, normalized size = 0.00 \[ \int \left (d x +c \right ) \mathrm {erfc}\left (b x +a \right )^{2}\, dx \]

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 = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (d x + c\right )} \operatorname {erfc}\left (b x + a\right )^{2}\,{d x} \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int {\mathrm {erfc}\left (a+b\,x\right )}^2\,\left (c+d\,x\right ) \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(erfc(a + b*x)^2*(c + d*x),x)

[Out]

int(erfc(a + b*x)^2*(c + d*x), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (c + d x\right ) \operatorname {erfc}^{2}{\left (a + b x \right )}\, dx \]

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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