3.63 \(\int \frac {e^{c+d x^2} \text {erf}(b x)}{x^4} \, dx\)

Optimal. Leaf size=155 \[ \frac {4}{3} d^2 \text {Int}\left (\text {erf}(b x) e^{c+d x^2},x\right )+\frac {2 b e^c d \text {Ei}\left (-\left (\left (b^2-d\right ) x^2\right )\right )}{3 \sqrt {\pi }}-\frac {b e^c \left (b^2-d\right ) \text {Ei}\left (-\left (\left (b^2-d\right ) x^2\right )\right )}{3 \sqrt {\pi }}-\frac {b e^{c-x^2 \left (b^2-d\right )}}{3 \sqrt {\pi } x^2}-\frac {2 d \text {erf}(b x) e^{c+d x^2}}{3 x}-\frac {\text {erf}(b x) e^{c+d x^2}}{3 x^3} \]

[Out]

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

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Rubi [A]  time = 0.29, 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 = {} \[ \int \frac {e^{c+d x^2} \text {Erf}(b x)}{x^4} \, dx \]

Verification is Not applicable to the result.

[In]

Int[(E^(c + d*x^2)*Erf[b*x])/x^4,x]

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 0.34, size = 0, normalized size = 0.00 \[ \int \frac {e^{c+d x^2} \text {erf}(b x)}{x^4} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.51, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\operatorname {erf}\left (b x\right ) e^{\left (d x^{2} + c\right )}}{x^{4}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.26, size = 0, normalized size = 0.00 \[ \int \frac {{\mathrm e}^{d \,x^{2}+c} \erf \left (b x \right )}{x^{4}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [A]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\mathrm {e}}^{d\,x^2+c}\,\mathrm {erf}\left (b\,x\right )}{x^4} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(c + d*x^2)*erf(b*x))/x^4,x)

[Out]

int((exp(c + d*x^2)*erf(b*x))/x^4, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

exp(c)*Integral(exp(d*x**2)*erf(b*x)/x**4, x)

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