\(\int e^{c+b^2 x^2} \text {erfi}(b x)^2 \, dx\) [253]

Optimal result

Integrand size = 18, antiderivative size = 21 \[ \int e^{c+b^2 x^2} \text {erfi}(b x)^2 \, dx=\frac {e^c \sqrt {\pi } \text {erfi}(b x)^3}{6 b} \]

[Out]

1/6*exp(c)*erfi(b*x)^3*Pi^(1/2)/b

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {6510, 30} \[ \int e^{c+b^2 x^2} \text {erfi}(b x)^2 \, dx=\frac {\sqrt {\pi } e^c \text {erfi}(b x)^3}{6 b} \]

[In]

Int[E^(c + b^2*x^2)*Erfi[b*x]^2,x]

[Out]

(E^c*Sqrt[Pi]*Erfi[b*x]^3)/(6*b)

Rule 30

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

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]

Rubi steps \begin{align*} \text {integral}& = \frac {\left (e^c \sqrt {\pi }\right ) \text {Subst}\left (\int x^2 \, dx,x,\text {erfi}(b x)\right )}{2 b} \\ & = \frac {e^c \sqrt {\pi } \text {erfi}(b x)^3}{6 b} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00 \[ \int e^{c+b^2 x^2} \text {erfi}(b x)^2 \, dx=\frac {e^c \sqrt {\pi } \text {erfi}(b x)^3}{6 b} \]

[In]

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

[Out]

(E^c*Sqrt[Pi]*Erfi[b*x]^3)/(6*b)

Maple [F]

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

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.76 \[ \int e^{c+b^2 x^2} \text {erfi}(b x)^2 \, dx=\frac {\sqrt {\pi } \operatorname {erfi}\left (b x\right )^{3} e^{c}}{6 \, b} \]

[In]

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

[Out]

1/6*sqrt(pi)*erfi(b*x)^3*e^c/b

Sympy [A] (verification not implemented)

Time = 0.36 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.90 \[ \int e^{c+b^2 x^2} \text {erfi}(b x)^2 \, dx=\begin {cases} \frac {\sqrt {\pi } e^{c} \operatorname {erfi}^{3}{\left (b x \right )}}{6 b} & \text {for}\: b \neq 0 \\0 & \text {otherwise} \end {cases} \]

[In]

integrate(exp(b**2*x**2+c)*erfi(b*x)**2,x)

[Out]

Piecewise((sqrt(pi)*exp(c)*erfi(b*x)**3/(6*b), Ne(b, 0)), (0, True))

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.76 \[ \int e^{c+b^2 x^2} \text {erfi}(b x)^2 \, dx=\frac {\sqrt {\pi }\,{\mathrm {e}}^c\,{\mathrm {erfi}\left (b\,x\right )}^3}{6\,b} \]

[In]

int(exp(c + b^2*x^2)*erfi(b*x)^2,x)

[Out]

(pi^(1/2)*exp(c)*erfi(b*x)^3)/(6*b)