3.192 \(\int \frac {e^{c+d x^2} \text {erfc}(a+b x)}{x^3} \, dx\)

Optimal. Leaf size=183 \[ \frac {2 a b^2 \text {Int}\left (\frac {e^{-a^2-2 a b x+x^2 \left (d-b^2\right )+c}}{x},x\right )}{\sqrt {\pi }}+d \text {Int}\left (\frac {e^{c+d x^2} \text {erfc}(a+b x)}{x},x\right )+b \sqrt {b^2-d} e^{\frac {a^2 d}{b^2-d}+c} \text {erf}\left (\frac {a b+x \left (b^2-d\right )}{\sqrt {b^2-d}}\right )+\frac {b e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{\sqrt {\pi } x}-\frac {e^{c+d x^2} \text {erfc}(a+b x)}{2 x^2} \]

[Out]

-1/2*exp(d*x^2+c)*erfc(b*x+a)/x^2+b*exp(c+a^2*d/(b^2-d))*erf((a*b+(b^2-d)*x)/(b^2-d)^(1/2))*(b^2-d)^(1/2)+b*ex
p(-a^2+c-2*a*b*x-(b^2-d)*x^2)/x/Pi^(1/2)+2*a*b^2*Unintegrable(exp(-a^2+c-2*a*b*x+(-b^2+d)*x^2)/x,x)/Pi^(1/2)+d
*Unintegrable(exp(d*x^2+c)*erfc(b*x+a)/x,x)

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Rubi [A]  time = 0.41, 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 = {} \[ \int \frac {e^{c+d x^2} \text {Erfc}(a+b x)}{x^3} \, dx \]

Verification is Not applicable to the result.

[In]

Int[(E^(c + d*x^2)*Erfc[a + b*x])/x^3,x]

[Out]

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

Rubi steps

\begin {align*} \int \frac {e^{c+d x^2} \text {erfc}(a+b x)}{x^3} \, dx &=-\frac {e^{c+d x^2} \text {erfc}(a+b x)}{2 x^2}+d \int \frac {e^{c+d x^2} \text {erfc}(a+b x)}{x} \, dx-\frac {b \int \frac {e^{-a^2+c-2 a b x+\left (-b^2+d\right ) x^2}}{x^2} \, dx}{\sqrt {\pi }}\\ &=\frac {b e^{-a^2+c-2 a b x-\left (b^2-d\right ) x^2}}{\sqrt {\pi } x}-\frac {e^{c+d x^2} \text {erfc}(a+b x)}{2 x^2}+d \int \frac {e^{c+d x^2} \text {erfc}(a+b x)}{x} \, dx+\frac {\left (2 a b^2\right ) \int \frac {e^{-a^2+c-2 a b x+\left (-b^2+d\right ) x^2}}{x} \, dx}{\sqrt {\pi }}+\frac {\left (2 b \left (b^2-d\right )\right ) \int e^{-a^2+c-2 a b x+\left (-b^2+d\right ) x^2} \, dx}{\sqrt {\pi }}\\ &=\frac {b e^{-a^2+c-2 a b x-\left (b^2-d\right ) x^2}}{\sqrt {\pi } x}-\frac {e^{c+d x^2} \text {erfc}(a+b x)}{2 x^2}+d \int \frac {e^{c+d x^2} \text {erfc}(a+b x)}{x} \, dx+\frac {\left (2 a b^2\right ) \int \frac {e^{-a^2+c-2 a b x+\left (-b^2+d\right ) x^2}}{x} \, dx}{\sqrt {\pi }}+\frac {\left (2 b \left (b^2-d\right ) e^{\frac {b^2 c+a^2 d-c d}{b^2-d}}\right ) \int \exp \left (\frac {\left (-2 a b+2 \left (-b^2+d\right ) x\right )^2}{4 \left (-b^2+d\right )}\right ) \, dx}{\sqrt {\pi }}\\ &=\frac {b e^{-a^2+c-2 a b x-\left (b^2-d\right ) x^2}}{\sqrt {\pi } x}+b \sqrt {b^2-d} e^{\frac {b^2 c+a^2 d-c d}{b^2-d}} \text {erf}\left (\frac {a b+\left (b^2-d\right ) x}{\sqrt {b^2-d}}\right )-\frac {e^{c+d x^2} \text {erfc}(a+b x)}{2 x^2}+d \int \frac {e^{c+d x^2} \text {erfc}(a+b x)}{x} \, dx+\frac {\left (2 a b^2\right ) \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 = 1.03, size = 0, normalized size = 0.00 \[ \int \frac {e^{c+d x^2} \text {erfc}(a+b x)}{x^3} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[(E^(c + d*x^2)*Erfc[a + b*x])/x^3,x]

[Out]

Integrate[(E^(c + d*x^2)*Erfc[a + b*x])/x^3, x]

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fricas [A]  time = 0.50, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {{\left (\operatorname {erf}\left (b x + a\right ) - 1\right )} e^{\left (d x^{2} + c\right )}}{x^{3}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(d*x^2+c)*erfc(b*x+a)/x^3,x, algorithm="fricas")

[Out]

integral(-(erf(b*x + a) - 1)*e^(d*x^2 + c)/x^3, x)

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giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {erfc}\left (b x + a\right ) e^{\left (d x^{2} + c\right )}}{x^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(d*x^2+c)*erfc(b*x+a)/x^3,x, algorithm="giac")

[Out]

integrate(erfc(b*x + a)*e^(d*x^2 + c)/x^3, x)

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maple [A]  time = 0.39, size = 0, normalized size = 0.00 \[ \int \frac {{\mathrm e}^{d \,x^{2}+c} \mathrm {erfc}\left (b x +a \right )}{x^{3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(d*x^2+c)*erfc(b*x+a)/x^3,x)

[Out]

int(exp(d*x^2+c)*erfc(b*x+a)/x^3,x)

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maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {erfc}\left (b x + a\right ) e^{\left (d x^{2} + c\right )}}{x^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(d*x^2+c)*erfc(b*x+a)/x^3,x, algorithm="maxima")

[Out]

integrate(erfc(b*x + a)*e^(d*x^2 + c)/x^3, x)

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mupad [A]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\mathrm {erfc}\left (a+b\,x\right )\,{\mathrm {e}}^{d\,x^2+c}}{x^3} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((erfc(a + b*x)*exp(c + d*x^2))/x^3,x)

[Out]

int((erfc(a + b*x)*exp(c + d*x^2))/x^3, x)

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sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ e^{c} \int \frac {e^{d x^{2}} \operatorname {erfc}{\left (a + b x \right )}}{x^{3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(d*x**2+c)*erfc(b*x+a)/x**3,x)

[Out]

exp(c)*Integral(exp(d*x**2)*erfc(a + b*x)/x**3, x)

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