3.49 \(\int \frac {e^{c-b^2 x^2}}{\text {erf}(b x)} \, dx\)

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

[Out]

1/2*exp(c)*ln(erf(b*x))*Pi^(1/2)/b

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Rubi [A]  time = 0.03, antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {6373, 29} \[ \frac {\sqrt {\pi } e^c \log (\text {Erf}(b x))}{2 b} \]

Antiderivative was successfully verified.

[In]

Int[E^(c - b^2*x^2)/Erf[b*x],x]

[Out]

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

Rule 29

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

Rule 6373

Int[E^((c_.) + (d_.)*(x_)^2)*Erf[(b_.)*(x_)]^(n_.), x_Symbol] :> Dist[(E^c*Sqrt[Pi])/(2*b), Subst[Int[x^n, x],
x, Erf[b*x]], x] /; FreeQ[{b, c, d, n}, x] && EqQ[d, -b^2]

Rubi steps

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

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Mathematica [A]  time = 0.01, size = 20, normalized size = 1.00 \[ \frac {\sqrt {\pi } e^c \log (\text {erf}(b x))}{2 b} \]

Antiderivative was successfully verified.

[In]

Integrate[E^(c - b^2*x^2)/Erf[b*x],x]

[Out]

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

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fricas [A]  time = 0.50, size = 15, normalized size = 0.75 \[ \frac {\sqrt {\pi } e^{c} \log \left (\operatorname {erf}\left (b x\right )\right )}{2 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F(-2)]  time = 180.00, size = 0, normalized size = 0.00 \[ \int \frac {{\mathrm e}^{-b^{2} x^{2}+c}}{\erf \left (b x \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.32, size = 15, normalized size = 0.75 \[ \frac {\sqrt {\pi } e^{c} \log \left (\operatorname {erf}\left (b x\right )\right )}{2 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.16, size = 15, normalized size = 0.75 \[ \frac {\sqrt {\pi }\,\ln \left (\mathrm {erf}\left (b\,x\right )\right )\,{\mathrm {e}}^c}{2\,b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(c - b^2*x^2)/erf(b*x),x)

[Out]

(pi^(1/2)*log(erf(b*x))*exp(c))/(2*b)

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sympy [A]  time = 0.41, size = 17, normalized size = 0.85 \[ \frac {\sqrt {\pi } e^{c} \log {\left (\operatorname {erf}{\left (b x \right )} \right )}}{2 b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(pi)*exp(c)*log(erf(b*x))/(2*b)

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