3.2.96 \(\int \frac {e^{c+d x^2} \text {Erfc}(a+b x)}{x^2} \, dx\) [196]

Optimal. Leaf size=83 \[ -\frac {e^{c+d x^2} \text {Erfc}(a+b x)}{x}-\frac {2 b \text {Int}\left (\frac {e^{-a^2+c-2 a b x+\left (-b^2+d\right ) x^2}}{x},x\right )}{\sqrt {\pi }}+2 d \text {Int}\left (e^{c+d x^2} \text {Erfc}(a+b x),x\right ) \]

[Out]

-exp(d*x^2+c)*erfc(b*x+a)/x-2*b*Unintegrable(exp(-a^2+c-2*a*b*x+(-b^2+d)*x^2)/x,x)/Pi^(1/2)+2*d*Unintegrable(e
xp(d*x^2+c)*erfc(b*x+a),x)

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Rubi [A]
time = 0.14, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {e^{c+d x^2} \text {Erfc}(a+b x)}{x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^(c + d*x^2)*Erfc[a + b*x])/x^2,x]

[Out]

-((E^(c + d*x^2)*Erfc[a + b*x])/x) - (2*b*Defer[Int][E^(-a^2 + c - 2*a*b*x + (-b^2 + d)*x^2)/x, x])/Sqrt[Pi] +
2*d*Defer[Int][E^(c + d*x^2)*Erfc[a + b*x], x]

Rubi steps

\begin {align*} \int \frac {e^{c+d x^2} \text {erfc}(a+b x)}{x^2} \, dx &=-\frac {e^{c+d x^2} \text {erfc}(a+b x)}{x}+(2 d) \int e^{c+d x^2} \text {erfc}(a+b x) \, dx-\frac {(2 b) \int \frac {e^{-a^2+c-2 a b x+\left (-b^2+d\right ) x^2}}{x} \, dx}{\sqrt {\pi }}\\ \end {align*}

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Mathematica [A]
time = 0.68, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {e^{c+d x^2} \text {Erfc}(a+b x)}{x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[(E^(c + d*x^2)*Erfc[a + b*x])/x^2,x]

[Out]

Integrate[(E^(c + d*x^2)*Erfc[a + b*x])/x^2, x]

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Maple [A]
time = 0.09, size = 0, normalized size = 0.00 \[\int \frac {{\mathrm e}^{d \,x^{2}+c} \mathrm {erfc}\left (b x +a \right )}{x^{2}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(d*x^2+c)*erfc(b*x+a)/x^2,x)

[Out]

int(exp(d*x^2+c)*erfc(b*x+a)/x^2,x)

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Maxima [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(erfc(b*x + a)*e^(d*x^2 + c)/x^2, x)

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Fricas [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(-(erf(b*x + a) - 1)*e^(d*x^2 + c)/x^2, x)

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Sympy [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} e^{c} \int \frac {e^{d x^{2}} \operatorname {erfc}{\left (a + b x \right )}}{x^{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(d*x**2+c)*erfc(b*x+a)/x**2,x)

[Out]

exp(c)*Integral(exp(d*x**2)*erfc(a + b*x)/x**2, x)

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Giac [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(erfc(b*x + a)*e^(d*x^2 + c)/x^2, x)

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Mupad [A]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\mathrm {erfc}\left (a+b\,x\right )\,{\mathrm {e}}^{d\,x^2+c}}{x^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((erfc(a + b*x)*exp(c + d*x^2))/x^2,x)

[Out]

int((erfc(a + b*x)*exp(c + d*x^2))/x^2, x)

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