3.1.94 \(\int \frac {e^{c+d x^2} \text {Erf}(a+b x)}{x^4} \, dx\) [94]

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

[Out]

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

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Rubi [A]
time = 0.61, 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 {Erf}(a+b x)}{x^4} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

-1/3*(b*E^(-a^2 + c - 2*a*b*x - (b^2 - d)*x^2))/(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) - (E^(c + d*x^2)*Erf[a + b*x])/(3*x^3) - (2*d*E^(c + d*x^2)*Erf[a + b*x])/(3*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 + (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[P
i]) + (4*d^2*Defer[Int][E^(c + d*x^2)*Erf[a + b*x], x])/3

Rubi steps

\begin {align*} \int \frac {e^{c+d x^2} \text {erf}(a+b x)}{x^4} \, dx &=-\frac {e^{c+d x^2} \text {erf}(a+b x)}{3 x^3}+\frac {1}{3} (2 d) \int \frac {e^{c+d x^2} \text {erf}(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 {erf}(a+b x)}{3 x^3}-\frac {2 d e^{c+d x^2} \text {erf}(a+b x)}{3 x}+\frac {1}{3} \left (4 d^2\right ) \int e^{c+d x^2} \text {erf}(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 {erf}(a+b x)}{3 x^3}-\frac {2 d e^{c+d x^2} \text {erf}(a+b x)}{3 x}+\frac {1}{3} \left (4 d^2\right ) \int e^{c+d x^2} \text {erf}(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 {erf}(a+b x)}{3 x^3}-\frac {2 d e^{c+d x^2} \text {erf}(a+b x)}{3 x}+\frac {1}{3} \left (4 d^2\right ) \int e^{c+d x^2} \text {erf}(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 {e^{c+d x^2} \text {erf}(a+b x)}{3 x^3}-\frac {2 d e^{c+d x^2} \text {erf}(a+b x)}{3 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 {1}{3} \left (4 d^2\right ) \int e^{c+d x^2} \text {erf}(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 = 0.38, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {e^{c+d x^2} \text {Erf}(a+b x)}{x^4} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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Maple [A]
time = 0.19, size = 0, normalized size = 0.00 \[\int \frac {{\mathrm e}^{d \,x^{2}+c} \erf \left (b x +a \right )}{x^{4}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

[Out]

integral(erf(b*x + a)*e^(d*x^2 + c)/x^4, 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 {erf}{\left (a + b x \right )}}{x^{4}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

integrate(erf(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 \begin {gather*} \int \frac {\mathrm {erf}\left (a+b\,x\right )\,{\mathrm {e}}^{d\,x^2+c}}{x^4} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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