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

Optimal. Leaf size=115 \[ \frac {4 b^5 e^c x^2 \, _2F_2\left (1,1;\frac {3}{2},2;b^2 x^2\right )}{3 \sqrt {\pi }}+\frac {4 b^3 e^c \log (x)}{3 \sqrt {\pi }}-\frac {2 b^2 e^{b^2 x^2+c} \text {erf}(b x)}{3 x}-\frac {e^{b^2 x^2+c} \text {erf}(b x)}{3 x^3}-\frac {b e^c}{3 \sqrt {\pi } x^2} \]

[Out]

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

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Rubi [A]  time = 0.11, antiderivative size = 115, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {6391, 6376, 12, 29, 30} \[ \frac {4 b^5 e^c x^2 \, _2F_2\left (1,1;\frac {3}{2},2;b^2 x^2\right )}{3 \sqrt {\pi }}-\frac {2 b^2 e^{b^2 x^2+c} \text {Erf}(b x)}{3 x}-\frac {e^{b^2 x^2+c} \text {Erf}(b x)}{3 x^3}+\frac {4 b^3 e^c \log (x)}{3 \sqrt {\pi }}-\frac {b e^c}{3 \sqrt {\pi } x^2} \]

Antiderivative was successfully verified.

[In]

Int[(E^(c + b^2*x^2)*Erf[b*x])/x^4,x]

[Out]

-(b*E^c)/(3*Sqrt[Pi]*x^2) - (E^(c + b^2*x^2)*Erf[b*x])/(3*x^3) - (2*b^2*E^(c + b^2*x^2)*Erf[b*x])/(3*x) + (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 6376

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

Rule 6391

Int[E^((c_.) + (d_.)*(x_)^2)*Erf[(a_.) + (b_.)*(x_)]*(x_)^(m_), x_Symbol] :> Simp[(x^(m + 1)*E^(c + d*x^2)*Erf
[a + b*x])/(m + 1), x] + (-Dist[(2*d)/(m + 1), Int[x^(m + 2)*E^(c + d*x^2)*Erf[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 {erf}(b x)}{x^4} \, dx &=-\frac {e^{c+b^2 x^2} \text {erf}(b x)}{3 x^3}+\frac {1}{3} \left (2 b^2\right ) \int \frac {e^{c+b^2 x^2} \text {erf}(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 {erf}(b x)}{3 x^3}-\frac {2 b^2 e^{c+b^2 x^2} \text {erf}(b x)}{3 x}+\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\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 {erf}(b x)}{3 x^3}-\frac {2 b^2 e^{c+b^2 x^2} \text {erf}(b x)}{3 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 {\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 {erf}(b x)}{3 x^3}-\frac {2 b^2 e^{c+b^2 x^2} \text {erf}(b x)}{3 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 = 0.38, size = 100, normalized size = 0.87 \[ -\frac {e^c \left (4 b^5 x^5 \, _2F_2\left (1,1;\frac {3}{2},2;-b^2 x^2\right )-2 \pi b^3 x^3 \text {erf}(b x) \text {erfi}(b x)-4 b^3 x^3 \log (x)+\sqrt {\pi } e^{b^2 x^2} \left (2 b^2 x^2+1\right ) \text {erf}(b x)+b x\right )}{3 \sqrt {\pi } x^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F]  time = 0.23, size = 0, normalized size = 0.00 \[ \int \frac {{\mathrm e}^{b^{2} x^{2}+c} \erf \left (b x \right )}{x^{4}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(erf(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 \[ \int \frac {{\mathrm {e}}^{b^2\,x^2+c}\,\mathrm {erf}\left (b\,x\right )}{x^4} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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