\(\int e^{-b^2 x^2} x^4 \text {erfi}(b x) \, dx\) [277]

Optimal result

Integrand size = 18, antiderivative size = 109 \[ \int e^{-b^2 x^2} x^4 \text {erfi}(b x) \, dx=\frac {3 x^2}{4 b^3 \sqrt {\pi }}+\frac {x^4}{4 b \sqrt {\pi }}-\frac {3 e^{-b^2 x^2} x \text {erfi}(b x)}{4 b^4}-\frac {e^{-b^2 x^2} x^3 \text {erfi}(b x)}{2 b^2}+\frac {3 x^2 \, _2F_2\left (1,1;\frac {3}{2},2;-b^2 x^2\right )}{4 b^3 \sqrt {\pi }} \]

[Out]

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

Rubi [A] (verified)

Time = 0.08 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {6522, 6513, 30} \[ \int e^{-b^2 x^2} x^4 \text {erfi}(b x) \, dx=\frac {3 x^2 \, _2F_2\left (1,1;\frac {3}{2},2;-b^2 x^2\right )}{4 \sqrt {\pi } b^3}+\frac {3 x^2}{4 \sqrt {\pi } b^3}-\frac {x^3 e^{-b^2 x^2} \text {erfi}(b x)}{2 b^2}-\frac {3 x e^{-b^2 x^2} \text {erfi}(b x)}{4 b^4}+\frac {x^4}{4 \sqrt {\pi } b} \]

[In]

Int[(x^4*Erfi[b*x])/E^(b^2*x^2),x]

[Out]

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

Rule 30

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

Rule 6513

Int[E^((c_.) + (d_.)*(x_)^2)*Erfi[(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 6522

Int[E^((c_.) + (d_.)*(x_)^2)*Erfi[(a_.) + (b_.)*(x_)]*(x_)^(m_), x_Symbol] :> Simp[x^(m - 1)*E^(c + d*x^2)*(Er
fi[a + b*x]/(2*d)), x] + (-Dist[(m - 1)/(2*d), Int[x^(m - 2)*E^(c + d*x^2)*Erfi[a + b*x], x], x] - Dist[b/(d*S
qrt[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] && IGtQ[m, 1]

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

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.39 \[ \int e^{-b^2 x^2} x^4 \text {erfi}(b x) \, dx=\frac {x^2 \left (1+b^2 x^2-\, _2F_2\left (1,1;-\frac {1}{2},2;-b^2 x^2\right )\right )}{4 b^3 \sqrt {\pi }} \]

[In]

Integrate[(x^4*Erfi[b*x])/E^(b^2*x^2),x]

[Out]

(x^2*(1 + b^2*x^2 - HypergeometricPFQ[{1, 1}, {-1/2, 2}, -(b^2*x^2)]))/(4*b^3*Sqrt[Pi])

Maple [F]

\[\int x^{4} \operatorname {erfi}\left (b x \right ) {\mathrm e}^{-b^{2} x^{2}}d x\]

[In]

int(x^4*erfi(b*x)/exp(b^2*x^2),x)

[Out]

int(x^4*erfi(b*x)/exp(b^2*x^2),x)

Fricas [F]

\[ \int e^{-b^2 x^2} x^4 \text {erfi}(b x) \, dx=\int { x^{4} \operatorname {erfi}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )} \,d x } \]

[In]

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

[Out]

integral(x^4*erfi(b*x)*e^(-b^2*x^2), x)

Sympy [F(-1)]

Timed out. \[ \int e^{-b^2 x^2} x^4 \text {erfi}(b x) \, dx=\text {Timed out} \]

[In]

integrate(x**4*erfi(b*x)/exp(b**2*x**2),x)

[Out]

Timed out

Maxima [F]

\[ \int e^{-b^2 x^2} x^4 \text {erfi}(b x) \, dx=\int { x^{4} \operatorname {erfi}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )} \,d x } \]

[In]

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

[Out]

integrate(x^4*erfi(b*x)*e^(-b^2*x^2), x)

Giac [F]

\[ \int e^{-b^2 x^2} x^4 \text {erfi}(b x) \, dx=\int { x^{4} \operatorname {erfi}\left (b x\right ) e^{\left (-b^{2} x^{2}\right )} \,d x } \]

[In]

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

[Out]

integrate(x^4*erfi(b*x)*e^(-b^2*x^2), x)

Mupad [F(-1)]

Timed out. \[ \int e^{-b^2 x^2} x^4 \text {erfi}(b x) \, dx=\int x^4\,{\mathrm {e}}^{-b^2\,x^2}\,\mathrm {erfi}\left (b\,x\right ) \,d x \]

[In]

int(x^4*exp(-b^2*x^2)*erfi(b*x),x)

[Out]

int(x^4*exp(-b^2*x^2)*erfi(b*x), x)