3.2.77 \(\int \frac {e^{c+b^2 x^2} \text {Erfc}(b x)}{x^4} \, dx\) [177]

Optimal. Leaf size=134 \[ \frac {b e^c}{3 \sqrt {\pi } x^2}-\frac {e^{c+b^2 x^2} \text {Erfc}(b x)}{3 x^3}-\frac {2 b^2 e^{c+b^2 x^2} \text {Erfc}(b x)}{3 x}+\frac {2}{3} b^3 e^c \sqrt {\pi } \text {Erfi}(b x)-\frac {4 b^5 e^c x^2 \, _2F_2\left (1,1;\frac {3}{2},2;b^2 x^2\right )}{3 \sqrt {\pi }}-\frac {4 b^3 e^c \log (x)}{3 \sqrt {\pi }} \]

[Out]

-1/3*exp(b^2*x^2+c)*erfc(b*x)/x^3-2/3*b^2*exp(b^2*x^2+c)*erfc(b*x)/x+1/3*b*exp(c)/x^2/Pi^(1/2)-4/3*b^5*exp(c)*
x^2*hypergeom([1, 1],[3/2, 2],b^2*x^2)/Pi^(1/2)-4/3*b^3*exp(c)*ln(x)/Pi^(1/2)+2/3*b^3*exp(c)*erfi(b*x)*Pi^(1/2
)

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Rubi [A]
time = 0.09, antiderivative size = 134, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.368, Rules used = {6527, 6512, 2235, 6511, 12, 29, 30} \begin {gather*} -\frac {4 b^5 e^c x^2 \, _2F_2\left (1,1;\frac {3}{2},2;b^2 x^2\right )}{3 \sqrt {\pi }}+\frac {2}{3} \sqrt {\pi } b^3 e^c \text {Erfi}(b x)-\frac {4 b^3 e^c \log (x)}{3 \sqrt {\pi }}-\frac {2 b^2 e^{b^2 x^2+c} \text {Erfc}(b x)}{3 x}-\frac {e^{b^2 x^2+c} \text {Erfc}(b x)}{3 x^3}+\frac {b e^c}{3 \sqrt {\pi } x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(b*E^c)/(3*Sqrt[Pi]*x^2) - (E^(c + b^2*x^2)*Erfc[b*x])/(3*x^3) - (2*b^2*E^(c + b^2*x^2)*Erfc[b*x])/(3*x) + (2*
b^3*E^c*Sqrt[Pi]*Erfi[b*x])/3 - (4*b^5*E^c*x^2*HypergeometricPFQ[{1, 1}, {3/2, 2}, b^2*x^2])/(3*Sqrt[Pi]) - (4
*b^3*E^c*Log[x])/(3*Sqrt[Pi])

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 2235

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[F^a*Sqrt[Pi]*(Erfi[(c + d*x)*Rt[b*Log[F], 2
]]/(2*d*Rt[b*Log[F], 2])), x] /; FreeQ[{F, a, b, c, d}, x] && PosQ[b]

Rule 6511

Int[E^((c_.) + (d_.)*(x_)^2)*Erf[(b_.)*(x_)], x_Symbol] :> Simp[b*E^c*(x^2/Sqrt[Pi])*HypergeometricPFQ[{1, 1},
{3/2, 2}, b^2*x^2], x] /; FreeQ[{b, c, d}, x] && EqQ[d, b^2]

Rule 6512

Int[E^((c_.) + (d_.)*(x_)^2)*Erfc[(b_.)*(x_)], x_Symbol] :> Int[E^(c + d*x^2), x] - Int[E^(c + d*x^2)*Erf[b*x]
, x] /; FreeQ[{b, c, d}, x] && EqQ[d, b^2]

Rule 6527

Int[E^((c_.) + (d_.)*(x_)^2)*Erfc[(a_.) + (b_.)*(x_)]*(x_)^(m_), x_Symbol] :> Simp[x^(m + 1)*E^(c + d*x^2)*(Er
fc[a + b*x]/(m + 1)), x] + (-Dist[2*(d/(m + 1)), Int[x^(m + 2)*E^(c + d*x^2)*Erfc[a + b*x], x], x] + Dist[2*(b
/((m + 1)*Sqrt[Pi])), Int[x^(m + 1)*E^(-a^2 + c - 2*a*b*x - (b^2 - d)*x^2), x], x]) /; FreeQ[{a, b, c, d}, x]
&& ILtQ[m, -1]

Rubi steps

\begin {align*} \int \frac {e^{c+b^2 x^2} \text {erfc}(b x)}{x^4} \, dx &=-\frac {e^{c+b^2 x^2} \text {erfc}(b x)}{3 x^3}+\frac {1}{3} \left (2 b^2\right ) \int \frac {e^{c+b^2 x^2} \text {erfc}(b x)}{x^2} \, dx-\frac {(2 b) \int \frac {e^c}{x^3} \, dx}{3 \sqrt {\pi }}\\ &=-\frac {e^{c+b^2 x^2} \text {erfc}(b x)}{3 x^3}-\frac {2 b^2 e^{c+b^2 x^2} \text {erfc}(b x)}{3 x}+\frac {1}{3} \left (4 b^4\right ) \int e^{c+b^2 x^2} \text {erfc}(b x) \, dx-\frac {\left (4 b^3\right ) \int \frac {e^c}{x} \, dx}{3 \sqrt {\pi }}-\frac {\left (2 b e^c\right ) \int \frac {1}{x^3} \, dx}{3 \sqrt {\pi }}\\ &=\frac {b e^c}{3 \sqrt {\pi } x^2}-\frac {e^{c+b^2 x^2} \text {erfc}(b x)}{3 x^3}-\frac {2 b^2 e^{c+b^2 x^2} \text {erfc}(b x)}{3 x}+\frac {1}{3} \left (4 b^4\right ) \int e^{c+b^2 x^2} \, dx-\frac {1}{3} \left (4 b^4\right ) \int e^{c+b^2 x^2} \text {erf}(b x) \, dx-\frac {\left (4 b^3 e^c\right ) \int \frac {1}{x} \, dx}{3 \sqrt {\pi }}\\ &=\frac {b e^c}{3 \sqrt {\pi } x^2}-\frac {e^{c+b^2 x^2} \text {erfc}(b x)}{3 x^3}-\frac {2 b^2 e^{c+b^2 x^2} \text {erfc}(b x)}{3 x}+\frac {2}{3} b^3 e^c \sqrt {\pi } \text {erfi}(b x)-\frac {4 b^5 e^c x^2 \, _2F_2\left (1,1;\frac {3}{2},2;b^2 x^2\right )}{3 \sqrt {\pi }}-\frac {4 b^3 e^c \log (x)}{3 \sqrt {\pi }}\\ \end {align*}

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Mathematica [A]
time = 0.31, size = 151, normalized size = 1.13 \begin {gather*} \frac {e^c \left (-e^{b^2 x^2} \sqrt {\pi }+b x-2 b^2 e^{b^2 x^2} \sqrt {\pi } x^2+e^{b^2 x^2} \sqrt {\pi } \left (1+2 b^2 x^2\right ) \text {Erf}(b x)+2 b^3 \pi x^3 \text {Erfi}(b x)-2 b^3 \pi x^3 \text {Erf}(b x) \text {Erfi}(b x)+4 b^5 x^5 \, _2F_2\left (1,1;\frac {3}{2},2;-b^2 x^2\right )-4 b^3 x^3 \log (x)\right )}{3 \sqrt {\pi } x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(E^c*(-(E^(b^2*x^2)*Sqrt[Pi]) + b*x - 2*b^2*E^(b^2*x^2)*Sqrt[Pi]*x^2 + E^(b^2*x^2)*Sqrt[Pi]*(1 + 2*b^2*x^2)*Er
f[b*x] + 2*b^3*Pi*x^3*Erfi[b*x] - 2*b^3*Pi*x^3*Erf[b*x]*Erfi[b*x] + 4*b^5*x^5*HypergeometricPFQ[{1, 1}, {3/2,
2}, -(b^2*x^2)] - 4*b^3*x^3*Log[x]))/(3*Sqrt[Pi]*x^3)

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Maple [F]
time = 0.14, size = 0, normalized size = 0.00 \[\int \frac {{\mathrm e}^{b^{2} x^{2}+c} \mathrm {erfc}\left (b x \right )}{x^{4}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
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(b^2*x^2+c)*erfc(b*x)/x^4,x, algorithm="maxima")

[Out]

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

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Fricas [F]
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(b^2*x^2+c)*erfc(b*x)/x^4,x, algorithm="fricas")

[Out]

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

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Sympy [A]
time = 90.79, size = 22, normalized size = 0.16 \begin {gather*} - \frac {b^{3} {G_{3, 2}^{1, 3}\left (\begin {matrix} 2, \frac {5}{2}, 1 & \\2 & 0 \end {matrix} \middle | {\frac {1}{b^{2} x^{2}}} \right )} e^{c}}{2 \pi } \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-b**3*meijerg(((2, 5/2, 1), ()), ((2,), (0,)), 1/(b**2*x**2))*exp(c)/(2*pi)

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Giac [F]
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(b^2*x^2+c)*erfc(b*x)/x^4,x, algorithm="giac")

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\mathrm {e}}^{b^2\,x^2+c}\,\mathrm {erfc}\left (b\,x\right )}{x^4} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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