\(\int (c+d x) \text {erfi}(a+b x)^2 \, dx\) [242]

Optimal result

Integrand size = 14, antiderivative size = 184 \[ \int (c+d x) \text {erfi}(a+b x)^2 \, dx=\frac {d e^{2 (a+b x)^2}}{2 b^2 \pi }-\frac {2 (b c-a d) e^{(a+b x)^2} \text {erfi}(a+b x)}{b^2 \sqrt {\pi }}-\frac {d e^{(a+b x)^2} (a+b x) \text {erfi}(a+b x)}{b^2 \sqrt {\pi }}+\frac {d \text {erfi}(a+b x)^2}{4 b^2}+\frac {(b c-a d) (a+b x) \text {erfi}(a+b x)^2}{b^2}+\frac {d (a+b x)^2 \text {erfi}(a+b x)^2}{2 b^2}+\frac {(b c-a d) \sqrt {\frac {2}{\pi }} \text {erfi}\left (\sqrt {2} (a+b x)\right )}{b^2} \]

[Out]

1/2*d*exp(2*(b*x+a)^2)/b^2/Pi+1/4*d*erfi(b*x+a)^2/b^2+(-a*d+b*c)*(b*x+a)*erfi(b*x+a)^2/b^2+1/2*d*(b*x+a)^2*erf
i(b*x+a)^2/b^2+(-a*d+b*c)*erfi((b*x+a)*2^(1/2))*2^(1/2)/Pi^(1/2)/b^2-2*(-a*d+b*c)*exp((b*x+a)^2)*erfi(b*x+a)/b
^2/Pi^(1/2)-d*exp((b*x+a)^2)*(b*x+a)*erfi(b*x+a)/b^2/Pi^(1/2)

Rubi [A] (verified)

Time = 0.13 (sec) , antiderivative size = 184, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.643, Rules used = {6504, 6489, 6519, 2235, 6501, 6522, 6510, 30, 2240} \[ \int (c+d x) \text {erfi}(a+b x)^2 \, dx=\frac {(a+b x) (b c-a d) \text {erfi}(a+b x)^2}{b^2}-\frac {2 e^{(a+b x)^2} (b c-a d) \text {erfi}(a+b x)}{\sqrt {\pi } b^2}+\frac {\sqrt {\frac {2}{\pi }} (b c-a d) \text {erfi}\left (\sqrt {2} (a+b x)\right )}{b^2}+\frac {d (a+b x)^2 \text {erfi}(a+b x)^2}{2 b^2}+\frac {d \text {erfi}(a+b x)^2}{4 b^2}-\frac {d e^{(a+b x)^2} (a+b x) \text {erfi}(a+b x)}{\sqrt {\pi } b^2}+\frac {d e^{2 (a+b x)^2}}{2 \pi b^2} \]

[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 30

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

Rule 2235

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 2240

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 6489

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 6501

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 6504

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 6510

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 6519

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 6522

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]

Rubi steps \begin{align*} \text {integral}& = \frac {\text {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 \text {Subst}\left (\int x \text {erfi}(x)^2 \, dx,x,a+b x\right )}{b^2}+\frac {(b c-a d) \text {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) \text {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)) \text {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) \text {Subst}\left (\int e^{2 x^2} x \, dx,x,a+b x\right )}{b^2 \pi }+\frac {(4 (b c-a d)) \text {Subst}\left (\int e^{2 x^2} \, dx,x,a+b x\right )}{b^2 \pi }+\frac {d \text {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 \text {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] (verified)

Time = 0.16 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.70 \[ \int (c+d x) \text {erfi}(a+b x)^2 \, dx=\frac {2 d e^{2 (a+b x)^2}-4 e^{(a+b x)^2} \sqrt {\pi } (2 b c-a d+b d x) \text {erfi}(a+b x)+\pi \left (4 a b c+d-2 a^2 d+4 b^2 c x+2 b^2 d x^2\right ) \text {erfi}(a+b x)^2+4 (b c-a d) \sqrt {2 \pi } \text {erfi}\left (\sqrt {2} (a+b x)\right )}{4 b^2 \pi } \]

[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)

Maple [F]

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

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 172, normalized size of antiderivative = 0.93 \[ \int (c+d x) \text {erfi}(a+b x)^2 \, dx=-\frac {4 \, \sqrt {2} \sqrt {\pi } \sqrt {-b^{2}} {\left (b c - a d\right )} \operatorname {erf}\left (\frac {\sqrt {2} \sqrt {-b^{2}} {\left (b x + a\right )}}{b}\right ) + 4 \, \sqrt {\pi } {\left (b^{2} d x + 2 \, b^{2} c - a b d\right )} \operatorname {erfi}\left (b x + a\right ) e^{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} - {\left (2 \, \pi b^{3} d x^{2} + 4 \, \pi b^{3} c x + \pi {\left (4 \, a b^{2} c - {\left (2 \, a^{2} - 1\right )} b d\right )}\right )} \operatorname {erfi}\left (b x + a\right )^{2} - 2 \, b d e^{\left (2 \, b^{2} x^{2} + 4 \, a b x + 2 \, a^{2}\right )}}{4 \, \pi b^{3}} \]

[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)*erf(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)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

int(erfi(a + b*x)^2*(c + d*x),x)

[Out]

int(erfi(a + b*x)^2*(c + d*x), x)