3.255 \(\int \frac{e^{c+b^2 x^2}}{\text{Erfi}(b x)} \, dx\)

Optimal. Leaf size=20 \[ \frac{\sqrt{\pi } e^c \log (\text{Erfi}(b x))}{2 b} \]

[Out]

(E^c*Sqrt[Pi]*Log[Erfi[b*x]])/(2*b)

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Rubi [A]  time = 0.0287121, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {6375, 29} \[ \frac{\sqrt{\pi } e^c \log (\text{Erfi}(b x))}{2 b} \]

Antiderivative was successfully verified.

[In]

Int[E^(c + b^2*x^2)/Erfi[b*x],x]

[Out]

(E^c*Sqrt[Pi]*Log[Erfi[b*x]])/(2*b)

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 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rubi steps

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

Mathematica [A]  time = 0.0105771, size = 20, normalized size = 1. \[ \frac{\sqrt{\pi } e^c \log (\text{Erfi}(b x))}{2 b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(E^c*Sqrt[Pi]*Log[Erfi[b*x]])/(2*b)

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Maple [F]  time = 0.061, size = 0, normalized size = 0. \begin{align*} \int{\frac{{{\rm e}^{{b}^{2}{x}^{2}+c}}}{{\it erfi} \left ( bx \right ) }}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 2.94376, size = 47, normalized size = 2.35 \begin{align*} \frac{\sqrt{\pi } e^{c} \log \left (\operatorname{erfi}\left (b x\right )\right )}{2 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*sqrt(pi)*e^c*log(erfi(b*x))/b

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Sympy [A]  time = 0.422322, size = 24, normalized size = 1.2 \begin{align*} \begin{cases} \frac{\sqrt{\pi } e^{c} \log{\left (\operatorname{erfi}{\left (b x \right )} \right )}}{2 b} & \text{for}\: b \neq 0 \\\tilde{\infty } x e^{c} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((sqrt(pi)*exp(c)*log(erfi(b*x))/(2*b), Ne(b, 0)), (zoo*x*exp(c), True))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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