3.197 \(\int \frac{e^{c+d x^2} \text{Erfc}(a+b x)}{x^4} \, dx\)

Optimal. Leaf size=351 \[ -\frac{4 a^2 b^3 \text{Unintegrable}\left (\frac{e^{-a^2-2 a b x+x^2 \left (d-b^2\right )+c}}{x},x\right )}{3 \sqrt{\pi }}+\frac{2 b \left (b^2-d\right ) \text{Unintegrable}\left (\frac{e^{-a^2-2 a b x+x^2 \left (d-b^2\right )+c}}{x},x\right )}{3 \sqrt{\pi }}-\frac{4 b d \text{Unintegrable}\left (\frac{e^{-a^2-2 a b x+x^2 \left (d-b^2\right )+c}}{x},x\right )}{3 \sqrt{\pi }}+\frac{4}{3} d^2 \text{Unintegrable}\left (e^{c+d x^2} \text{Erfc}(a+b x),x\right )-\frac{2}{3} a b^2 \sqrt{b^2-d} e^{\frac{a^2 d}{b^2-d}+c} \text{Erf}\left (\frac{a b+x \left (b^2-d\right )}{\sqrt{b^2-d}}\right )-\frac{2 a b^2 e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{3 \sqrt{\pi } x}+\frac{b e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{3 \sqrt{\pi } x^2}-\frac{2 d e^{c+d x^2} \text{Erfc}(a+b x)}{3 x}-\frac{e^{c+d x^2} \text{Erfc}(a+b x)}{3 x^3} \]

[Out]

(b*E^(-a^2 + c - 2*a*b*x - (b^2 - d)*x^2))/(3*Sqrt[Pi]*x^2) - (2*a*b^2*E^(-a^2 + c - 2*a*b*x - (b^2 - d)*x^2))
/(3*Sqrt[Pi]*x) - (2*a*b^2*Sqrt[b^2 - d]*E^(c + (a^2*d)/(b^2 - d))*Erf[(a*b + (b^2 - d)*x)/Sqrt[b^2 - d]])/3 -
(E^(c + d*x^2)*Erfc[a + b*x])/(3*x^3) - (2*d*E^(c + d*x^2)*Erfc[a + b*x])/(3*x) - (4*a^2*b^3*Unintegrable[E^(
-a^2 + c - 2*a*b*x + (-b^2 + d)*x^2)/x, x])/(3*Sqrt[Pi]) + (2*b*(b^2 - d)*Unintegrable[E^(-a^2 + c - 2*a*b*x +
(-b^2 + d)*x^2)/x, x])/(3*Sqrt[Pi]) - (4*b*d*Unintegrable[E^(-a^2 + c - 2*a*b*x + (-b^2 + d)*x^2)/x, x])/(3*S
qrt[Pi]) + (4*d^2*Unintegrable[E^(c + d*x^2)*Erfc[a + b*x], x])/3

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Rubi [A]  time = 0.868551, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{e^{c+d x^2} \text{Erfc}(a+b x)}{x^4} \, dx \]

Verification is Not applicable to the result.

[In]

Int[(E^(c + d*x^2)*Erfc[a + b*x])/x^4,x]

[Out]

(b*E^(-a^2 + c - 2*a*b*x - (b^2 - d)*x^2))/(3*Sqrt[Pi]*x^2) - (2*a*b^2*E^(-a^2 + c - 2*a*b*x - (b^2 - d)*x^2))
/(3*Sqrt[Pi]*x) - (2*a*b^2*Sqrt[b^2 - d]*E^(c + (a^2*d)/(b^2 - d))*Erf[(a*b + (b^2 - d)*x)/Sqrt[b^2 - d]])/3 -
(E^(c + d*x^2)*Erfc[a + b*x])/(3*x^3) - (2*d*E^(c + d*x^2)*Erfc[a + b*x])/(3*x) - (4*a^2*b^3*Defer[Int][E^(-a
^2 + c - 2*a*b*x + (-b^2 + d)*x^2)/x, x])/(3*Sqrt[Pi]) + (2*b*(b^2 - d)*Defer[Int][E^(-a^2 + c - 2*a*b*x + (-b
^2 + d)*x^2)/x, x])/(3*Sqrt[Pi]) - (4*b*d*Defer[Int][E^(-a^2 + c - 2*a*b*x + (-b^2 + d)*x^2)/x, x])/(3*Sqrt[Pi
]) + (4*d^2*Defer[Int][E^(c + d*x^2)*Erfc[a + b*x], x])/3

Rubi steps

\begin{align*} \int \frac{e^{c+d x^2} \text{erfc}(a+b x)}{x^4} \, dx &=-\frac{e^{c+d x^2} \text{erfc}(a+b x)}{3 x^3}+\frac{1}{3} (2 d) \int \frac{e^{c+d x^2} \text{erfc}(a+b x)}{x^2} \, dx-\frac{(2 b) \int \frac{e^{-a^2+c-2 a b x+\left (-b^2+d\right ) x^2}}{x^3} \, dx}{3 \sqrt{\pi }}\\ &=\frac{b e^{-a^2+c-2 a b x-\left (b^2-d\right ) x^2}}{3 \sqrt{\pi } x^2}-\frac{e^{c+d x^2} \text{erfc}(a+b x)}{3 x^3}-\frac{2 d e^{c+d x^2} \text{erfc}(a+b x)}{3 x}+\frac{1}{3} \left (4 d^2\right ) \int e^{c+d x^2} \text{erfc}(a+b 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^2} \, dx}{3 \sqrt{\pi }}+\frac{\left (2 b \left (b^2-d\right )\right ) \int \frac{e^{-a^2+c-2 a b x+\left (-b^2+d\right ) x^2}}{x} \, dx}{3 \sqrt{\pi }}-\frac{(4 b d) \int \frac{e^{-a^2+c-2 a b x+\left (-b^2+d\right ) x^2}}{x} \, dx}{3 \sqrt{\pi }}\\ &=\frac{b e^{-a^2+c-2 a b x-\left (b^2-d\right ) x^2}}{3 \sqrt{\pi } x^2}-\frac{2 a b^2 e^{-a^2+c-2 a b x-\left (b^2-d\right ) x^2}}{3 \sqrt{\pi } x}-\frac{e^{c+d x^2} \text{erfc}(a+b x)}{3 x^3}-\frac{2 d e^{c+d x^2} \text{erfc}(a+b x)}{3 x}+\frac{1}{3} \left (4 d^2\right ) \int e^{c+d x^2} \text{erfc}(a+b x) \, dx-\frac{\left (4 a^2 b^3\right ) \int \frac{e^{-a^2+c-2 a b x+\left (-b^2+d\right ) x^2}}{x} \, dx}{3 \sqrt{\pi }}+\frac{\left (2 b \left (b^2-d\right )\right ) \int \frac{e^{-a^2+c-2 a b x+\left (-b^2+d\right ) x^2}}{x} \, dx}{3 \sqrt{\pi }}-\frac{\left (4 a b^2 \left (b^2-d\right )\right ) \int e^{-a^2+c-2 a b x+\left (-b^2+d\right ) x^2} \, dx}{3 \sqrt{\pi }}-\frac{(4 b d) \int \frac{e^{-a^2+c-2 a b x+\left (-b^2+d\right ) x^2}}{x} \, dx}{3 \sqrt{\pi }}\\ &=\frac{b e^{-a^2+c-2 a b x-\left (b^2-d\right ) x^2}}{3 \sqrt{\pi } x^2}-\frac{2 a b^2 e^{-a^2+c-2 a b x-\left (b^2-d\right ) x^2}}{3 \sqrt{\pi } x}-\frac{e^{c+d x^2} \text{erfc}(a+b x)}{3 x^3}-\frac{2 d e^{c+d x^2} \text{erfc}(a+b x)}{3 x}+\frac{1}{3} \left (4 d^2\right ) \int e^{c+d x^2} \text{erfc}(a+b x) \, dx-\frac{\left (4 a^2 b^3\right ) \int \frac{e^{-a^2+c-2 a b x+\left (-b^2+d\right ) x^2}}{x} \, dx}{3 \sqrt{\pi }}+\frac{\left (2 b \left (b^2-d\right )\right ) \int \frac{e^{-a^2+c-2 a b x+\left (-b^2+d\right ) x^2}}{x} \, dx}{3 \sqrt{\pi }}-\frac{(4 b d) \int \frac{e^{-a^2+c-2 a b x+\left (-b^2+d\right ) x^2}}{x} \, dx}{3 \sqrt{\pi }}-\frac{\left (4 a b^2 \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}{3 \sqrt{\pi }}\\ &=\frac{b e^{-a^2+c-2 a b x-\left (b^2-d\right ) x^2}}{3 \sqrt{\pi } x^2}-\frac{2 a b^2 e^{-a^2+c-2 a b x-\left (b^2-d\right ) x^2}}{3 \sqrt{\pi } x}-\frac{2}{3} a b^2 \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)}{3 x^3}-\frac{2 d e^{c+d x^2} \text{erfc}(a+b x)}{3 x}+\frac{1}{3} \left (4 d^2\right ) \int e^{c+d x^2} \text{erfc}(a+b x) \, dx-\frac{\left (4 a^2 b^3\right ) \int \frac{e^{-a^2+c-2 a b x+\left (-b^2+d\right ) x^2}}{x} \, dx}{3 \sqrt{\pi }}+\frac{\left (2 b \left (b^2-d\right )\right ) \int \frac{e^{-a^2+c-2 a b x+\left (-b^2+d\right ) x^2}}{x} \, dx}{3 \sqrt{\pi }}-\frac{(4 b d) \int \frac{e^{-a^2+c-2 a b x+\left (-b^2+d\right ) x^2}}{x} \, dx}{3 \sqrt{\pi }}\\ \end{align*}

Mathematica [A]  time = 1.13716, size = 0, normalized size = 0. \[ \int \frac{e^{c+d x^2} \text{Erfc}(a+b x)}{x^4} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[(E^(c + d*x^2)*Erfc[a + b*x])/x^4,x]

[Out]

Integrate[(E^(c + d*x^2)*Erfc[a + b*x])/x^4, x]

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Maple [A]  time = 0.362, size = 0, normalized size = 0. \begin{align*} \int{\frac{{{\rm e}^{d{x}^{2}+c}}{\it erfc} \left ( bx+a \right ) }{{x}^{4}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(d*x^2+c)*erfc(b*x+a)/x^4,x)

[Out]

int(exp(d*x^2+c)*erfc(b*x+a)/x^4,x)

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{erfc}\left (b x + a\right ) e^{\left (d x^{2} + c\right )}}{x^{4}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(d*x^2+c)*erfc(b*x+a)/x^4,x, algorithm="maxima")

[Out]

integrate(erfc(b*x + a)*e^(d*x^2 + c)/x^4, x)

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{{\left (\operatorname{erf}\left (b x + a\right ) - 1\right )} e^{\left (d x^{2} + c\right )}}{x^{4}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(d*x^2+c)*erfc(b*x+a)/x^4,x, algorithm="fricas")

[Out]

integral(-(erf(b*x + a) - 1)*e^(d*x^2 + c)/x^4, x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(d*x**2+c)*erfc(b*x+a)/x**4,x)

[Out]

Timed out

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{erfc}\left (b x + a\right ) e^{\left (d x^{2} + c\right )}}{x^{4}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(d*x^2+c)*erfc(b*x+a)/x^4,x, algorithm="giac")

[Out]

integrate(erfc(b*x + a)*e^(d*x^2 + c)/x^4, x)