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

Optimal. Leaf size=355 \[ \frac {2 b \left (b^2-d\right ) \text {Int}\left (\frac {e^{-a^2-2 a b x+x^2 \left (d-b^2\right )+c}}{x},x\right )}{3 \sqrt {\pi }}-\frac {4 b d \text {Int}\left (\frac {e^{-a^2-2 a b x+x^2 \left (d-b^2\right )+c}}{x},x\right )}{3 \sqrt {\pi }}-\frac {4 a^2 b^3 \text {Int}\left (\frac {e^{-a^2-2 a b x+x^2 \left (d-b^2\right )+c}}{x},x\right )}{3 \sqrt {\pi }}+\frac {4}{3} d^2 \text {Int}\left (e^{c+d x^2} \text {erfc}(a+b x),x\right )-\frac {2}{3} a b^2 \sqrt {b^2-d} e^{\frac {a^2 d}{b^2-d}+c} \text {erf}\left (\frac {a b+x \left (b^2-d\right )}{\sqrt {b^2-d}}\right )-\frac {2 a b^2 e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{3 \sqrt {\pi } x}+\frac {b e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{3 \sqrt {\pi } x^2}-\frac {2 d e^{c+d x^2} \text {erfc}(a+b x)}{3 x}-\frac {e^{c+d x^2} \text {erfc}(a+b x)}{3 x^3} \]

[Out]

-1/3*exp(d*x^2+c)*erfc(b*x+a)/x^3-2/3*d*exp(d*x^2+c)*erfc(b*x+a)/x-2/3*a*b^2*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)+1/3*b*exp(-a^2+c-2*a*b*x-(b^2-d)*x^2)/x^2/Pi^(1/2)-2/3*a*b^2*exp(-a^2+c-2
*a*b*x-(b^2-d)*x^2)/x/Pi^(1/2)-4/3*a^2*b^3*Unintegrable(exp(-a^2+c-2*a*b*x+(-b^2+d)*x^2)/x,x)/Pi^(1/2)+2/3*b*(
b^2-d)*Unintegrable(exp(-a^2+c-2*a*b*x+(-b^2+d)*x^2)/x,x)/Pi^(1/2)-4/3*b*d*Unintegrable(exp(-a^2+c-2*a*b*x+(-b
^2+d)*x^2)/x,x)/Pi^(1/2)+4/3*d^2*Unintegrable(exp(d*x^2+c)*erfc(b*x+a),x)

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Rubi [A]  time = 0.87, 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 = {} \[ \int \frac {e^{c+d x^2} \text {Erfc}(a+b x)}{x^4} \, dx \]

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*}

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Mathematica [A]  time = 1.30, size = 0, normalized size = 0.00 \[ \int \frac {e^{c+d x^2} \text {erfc}(a+b x)}{x^4} \, dx \]

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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fricas [A]  time = 0.41, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {{\left (\operatorname {erf}\left (b x + a\right ) - 1\right )} e^{\left (d x^{2} + c\right )}}{x^{4}}, x\right ) \]

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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giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {erfc}\left (b x + a\right ) e^{\left (d x^{2} + c\right )}}{x^{4}}\,{d x} \]

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)

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maple [A]  time = 0.34, size = 0, normalized size = 0.00 \[ \int \frac {{\mathrm e}^{d \,x^{2}+c} \mathrm {erfc}\left (b x +a \right )}{x^{4}}\, dx \]

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 = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {erfc}\left (b x + a\right ) e^{\left (d x^{2} + c\right )}}{x^{4}}\,{d x} \]

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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mupad [A]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\mathrm {erfc}\left (a+b\,x\right )\,{\mathrm {e}}^{d\,x^2+c}}{x^4} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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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