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

Optimal. Leaf size=184 \[ \frac{(a+b x) (b c-a d) \text{Erfi}(a+b x)^2}{b^2}-\frac{2 e^{(a+b x)^2} (b c-a d) \text{Erfi}(a+b x)}{\sqrt{\pi } b^2}+\frac{\sqrt{\frac{2}{\pi }} (b c-a d) \text{Erfi}\left (\sqrt{2} (a+b x)\right )}{b^2}+\frac{d (a+b x)^2 \text{Erfi}(a+b x)^2}{2 b^2}+\frac{d \text{Erfi}(a+b x)^2}{4 b^2}-\frac{d e^{(a+b x)^2} (a+b x) \text{Erfi}(a+b x)}{\sqrt{\pi } b^2}+\frac{d e^{2 (a+b x)^2}}{2 \pi b^2} \]

[Out]

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

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Rubi [A]  time = 0.181994, antiderivative size = 184, 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 = {6369, 6354, 6384, 2204, 6366, 6387, 6375, 30, 2209} \[ \frac{(a+b x) (b c-a d) \text{Erfi}(a+b x)^2}{b^2}-\frac{2 e^{(a+b x)^2} (b c-a d) \text{Erfi}(a+b x)}{\sqrt{\pi } b^2}+\frac{\sqrt{\frac{2}{\pi }} (b c-a d) \text{Erfi}\left (\sqrt{2} (a+b x)\right )}{b^2}+\frac{d (a+b x)^2 \text{Erfi}(a+b x)^2}{2 b^2}+\frac{d \text{Erfi}(a+b x)^2}{4 b^2}-\frac{d e^{(a+b x)^2} (a+b x) \text{Erfi}(a+b x)}{\sqrt{\pi } b^2}+\frac{d e^{2 (a+b x)^2}}{2 \pi b^2} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x)*Erfi[a + b*x]^2,x]

[Out]

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

Rule 6369

Int[Erfi[(a_) + (b_.)*(x_)]^2*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Dist[1/b^(m + 1), Subst[Int[ExpandInteg
rand[Erfi[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 6354

Int[Erfi[(a_.) + (b_.)*(x_)]^2, x_Symbol] :> Simp[((a + b*x)*Erfi[a + b*x]^2)/b, x] - Dist[4/Sqrt[Pi], Int[(a
+ b*x)*E^(a + b*x)^2*Erfi[a + b*x], x], x] /; FreeQ[{a, b}, x]

Rule 6384

Int[E^((c_.) + (d_.)*(x_)^2)*Erfi[(a_.) + (b_.)*(x_)]*(x_), x_Symbol] :> Simp[(E^(c + d*x^2)*Erfi[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 2204

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[(F^a*Sqrt[Pi]*Erfi[(c + d*x)*Rt[b*Log[F], 2
]])/(2*d*Rt[b*Log[F], 2]), x] /; FreeQ[{F, a, b, c, d}, x] && PosQ[b]

Rule 6366

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

Rule 6387

Int[E^((c_.) + (d_.)*(x_)^2)*Erfi[(a_.) + (b_.)*(x_)]*(x_)^(m_), x_Symbol] :> Simp[(x^(m - 1)*E^(c + d*x^2)*Er
fi[a + b*x])/(2*d), x] + (-Dist[(m - 1)/(2*d), Int[x^(m - 2)*E^(c + d*x^2)*Erfi[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 6375

Int[E^((c_.) + (d_.)*(x_)^2)*Erfi[(b_.)*(x_)]^(n_.), x_Symbol] :> Dist[(E^c*Sqrt[Pi])/(2*b), Subst[Int[x^n, x]
, x, Erfi[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{erfi}(a+b x)^2 \, dx &=\frac{\operatorname{Subst}\left (\int \left (b c \left (1-\frac{a d}{b c}\right ) \text{erfi}(x)^2+d x \text{erfi}(x)^2\right ) \, dx,x,a+b x\right )}{b^2}\\ &=\frac{d \operatorname{Subst}\left (\int x \text{erfi}(x)^2 \, dx,x,a+b x\right )}{b^2}+\frac{(b c-a d) \operatorname{Subst}\left (\int \text{erfi}(x)^2 \, dx,x,a+b x\right )}{b^2}\\ &=\frac{(b c-a d) (a+b x) \text{erfi}(a+b x)^2}{b^2}+\frac{d (a+b x)^2 \text{erfi}(a+b x)^2}{2 b^2}-\frac{(2 d) \operatorname{Subst}\left (\int e^{x^2} x^2 \text{erfi}(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{erfi}(x) \, dx,x,a+b x\right )}{b^2 \sqrt{\pi }}\\ &=-\frac{2 (b c-a d) e^{(a+b x)^2} \text{erfi}(a+b x)}{b^2 \sqrt{\pi }}-\frac{d e^{(a+b x)^2} (a+b x) \text{erfi}(a+b x)}{b^2 \sqrt{\pi }}+\frac{(b c-a d) (a+b x) \text{erfi}(a+b x)^2}{b^2}+\frac{d (a+b x)^2 \text{erfi}(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{erfi}(x) \, dx,x,a+b x\right )}{b^2 \sqrt{\pi }}\\ &=\frac{d e^{2 (a+b x)^2}}{2 b^2 \pi }-\frac{2 (b c-a d) e^{(a+b x)^2} \text{erfi}(a+b x)}{b^2 \sqrt{\pi }}-\frac{d e^{(a+b x)^2} (a+b x) \text{erfi}(a+b x)}{b^2 \sqrt{\pi }}+\frac{(b c-a d) (a+b x) \text{erfi}(a+b x)^2}{b^2}+\frac{d (a+b x)^2 \text{erfi}(a+b x)^2}{2 b^2}+\frac{(b c-a d) \sqrt{\frac{2}{\pi }} \text{erfi}\left (\sqrt{2} (a+b x)\right )}{b^2}+\frac{d \operatorname{Subst}(\int x \, dx,x,\text{erfi}(a+b x))}{2 b^2}\\ &=\frac{d e^{2 (a+b x)^2}}{2 b^2 \pi }-\frac{2 (b c-a d) e^{(a+b x)^2} \text{erfi}(a+b x)}{b^2 \sqrt{\pi }}-\frac{d e^{(a+b x)^2} (a+b x) \text{erfi}(a+b x)}{b^2 \sqrt{\pi }}+\frac{d \text{erfi}(a+b x)^2}{4 b^2}+\frac{(b c-a d) (a+b x) \text{erfi}(a+b x)^2}{b^2}+\frac{d (a+b x)^2 \text{erfi}(a+b x)^2}{2 b^2}+\frac{(b c-a d) \sqrt{\frac{2}{\pi }} \text{erfi}\left (\sqrt{2} (a+b x)\right )}{b^2}\\ \end{align*}

Mathematica [A]  time = 0.143423, size = 128, normalized size = 0.7 \[ \frac{\pi \text{Erfi}(a+b x)^2 \left (-2 a^2 d+4 a b c+4 b^2 c x+2 b^2 d x^2+d\right )-4 \sqrt{\pi } e^{(a+b x)^2} \text{Erfi}(a+b x) (-a d+2 b c+b d x)+4 \sqrt{2 \pi } (b c-a d) \text{Erfi}\left (\sqrt{2} (a+b x)\right )+2 d e^{2 (a+b x)^2}}{4 \pi b^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.046, size = 0, normalized size = 0. \begin{align*} \int \left ( dx+c \right ) \left ({\it erfi} \left ( bx+a \right ) \right ) ^{2}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int((d*x+c)*erfi(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{erfi}\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)*erfi(b*x+a)^2,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 2.42785, size = 406, normalized size = 2.21 \begin{align*} \frac{4 \, \sqrt{2} \sqrt{\pi } \sqrt{b^{2}}{\left (b c - a d\right )} \operatorname{erfi}\left (\frac{\sqrt{2} \sqrt{b^{2}}{\left (b x + a\right )}}{b}\right ) - 4 \, \sqrt{\pi }{\left (b^{2} d x + 2 \, b^{2} c - a b d\right )} \operatorname{erfi}\left (b x + a\right ) e^{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\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{erfi}\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 \, \pi b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(4*sqrt(2)*sqrt(pi)*sqrt(b^2)*(b*c - a*d)*erfi(sqrt(2)*sqrt(b^2)*(b*x + a)/b) - 4*sqrt(pi)*(b^2*d*x + 2*b^
2*c - a*b*d)*erfi(b*x + a)*e^(b^2*x^2 + 2*a*b*x + a^2) + (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))*erfi(b*x + a)^2 + 2*b*d*e^(2*b^2*x^2 + 4*a*b*x + 2*a^2))/(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{erfi}^{2}{\left (a + b x \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((c + d*x)*erfi(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{erfi}\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)*erfi(b*x+a)^2,x, algorithm="giac")

[Out]

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