3.235 \(\int x^4 \text {erfi}(b x)^2 \, dx\)

Optimal. Leaf size=162 \[ \frac {43 \text {erfi}\left (\sqrt {2} b x\right )}{40 \sqrt {2 \pi } b^5}-\frac {2 x^4 e^{b^2 x^2} \text {erfi}(b x)}{5 \sqrt {\pi } b}+\frac {x^3 e^{2 b^2 x^2}}{5 \pi b^2}-\frac {4 e^{b^2 x^2} \text {erfi}(b x)}{5 \sqrt {\pi } b^5}-\frac {11 x e^{2 b^2 x^2}}{20 \pi b^4}+\frac {4 x^2 e^{b^2 x^2} \text {erfi}(b x)}{5 \sqrt {\pi } b^3}+\frac {1}{5} x^5 \text {erfi}(b x)^2 \]

[Out]

-11/20*exp(2*b^2*x^2)*x/b^4/Pi+1/5*exp(2*b^2*x^2)*x^3/b^2/Pi+1/5*x^5*erfi(b*x)^2-4/5*exp(b^2*x^2)*erfi(b*x)/b^
5/Pi^(1/2)+4/5*exp(b^2*x^2)*x^2*erfi(b*x)/b^3/Pi^(1/2)-2/5*exp(b^2*x^2)*x^4*erfi(b*x)/b/Pi^(1/2)+43/80*erfi(b*
x*2^(1/2))/b^5*2^(1/2)/Pi^(1/2)

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Rubi [A]  time = 0.23, antiderivative size = 162, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 5, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6366, 6387, 6384, 2204, 2212} \[ -\frac {2 x^4 e^{b^2 x^2} \text {Erfi}(b x)}{5 \sqrt {\pi } b}+\frac {4 x^2 e^{b^2 x^2} \text {Erfi}(b x)}{5 \sqrt {\pi } b^3}-\frac {4 e^{b^2 x^2} \text {Erfi}(b x)}{5 \sqrt {\pi } b^5}+\frac {43 \text {Erfi}\left (\sqrt {2} b x\right )}{40 \sqrt {2 \pi } b^5}+\frac {x^3 e^{2 b^2 x^2}}{5 \pi b^2}-\frac {11 x e^{2 b^2 x^2}}{20 \pi b^4}+\frac {1}{5} x^5 \text {Erfi}(b x)^2 \]

Antiderivative was successfully verified.

[In]

Int[x^4*Erfi[b*x]^2,x]

[Out]

(-11*E^(2*b^2*x^2)*x)/(20*b^4*Pi) + (E^(2*b^2*x^2)*x^3)/(5*b^2*Pi) - (4*E^(b^2*x^2)*Erfi[b*x])/(5*b^5*Sqrt[Pi]
) + (4*E^(b^2*x^2)*x^2*Erfi[b*x])/(5*b^3*Sqrt[Pi]) - (2*E^(b^2*x^2)*x^4*Erfi[b*x])/(5*b*Sqrt[Pi]) + (x^5*Erfi[
b*x]^2)/5 + (43*Erfi[Sqrt[2]*b*x])/(40*b^5*Sqrt[2*Pi])

Rule 2204

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 2212

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[((c + d*x)^(m
- n + 1)*F^(a + b*(c + d*x)^n))/(b*d*n*Log[F]), x] - Dist[(m - n + 1)/(b*n*Log[F]), Int[(c + d*x)^(m - n)*F^(
a + b*(c + d*x)^n), x], x] /; FreeQ[{F, a, b, c, d}, x] && IntegerQ[(2*(m + 1))/n] && LtQ[0, (m + 1)/n, 5] &&
IntegerQ[n] && (LtQ[0, n, m + 1] || LtQ[m, n, 0])

Rule 6366

Int[Erfi[(b_.)*(x_)]^2*(x_)^(m_.), x_Symbol] :> Simp[(x^(m + 1)*Erfi[b*x]^2)/(m + 1), x] - Dist[(4*b)/(Sqrt[Pi
]*(m + 1)), Int[x^(m + 1)*E^(b^2*x^2)*Erfi[b*x], x], x] /; FreeQ[b, x] && (IGtQ[m, 0] || ILtQ[(m + 1)/2, 0])

Rule 6384

Int[E^((c_.) + (d_.)*(x_)^2)*Erfi[(a_.) + (b_.)*(x_)]*(x_), x_Symbol] :> Simp[(E^(c + d*x^2)*Erfi[a + b*x])/(2
*d), x] - Dist[b/(d*Sqrt[Pi]), Int[E^(a^2 + c + 2*a*b*x + (b^2 + d)*x^2), x], x] /; FreeQ[{a, b, c, d}, x]

Rule 6387

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*} \int x^4 \text {erfi}(b x)^2 \, dx &=\frac {1}{5} x^5 \text {erfi}(b x)^2-\frac {(4 b) \int e^{b^2 x^2} x^5 \text {erfi}(b x) \, dx}{5 \sqrt {\pi }}\\ &=-\frac {2 e^{b^2 x^2} x^4 \text {erfi}(b x)}{5 b \sqrt {\pi }}+\frac {1}{5} x^5 \text {erfi}(b x)^2+\frac {4 \int e^{2 b^2 x^2} x^4 \, dx}{5 \pi }+\frac {8 \int e^{b^2 x^2} x^3 \text {erfi}(b x) \, dx}{5 b \sqrt {\pi }}\\ &=\frac {e^{2 b^2 x^2} x^3}{5 b^2 \pi }+\frac {4 e^{b^2 x^2} x^2 \text {erfi}(b x)}{5 b^3 \sqrt {\pi }}-\frac {2 e^{b^2 x^2} x^4 \text {erfi}(b x)}{5 b \sqrt {\pi }}+\frac {1}{5} x^5 \text {erfi}(b x)^2-\frac {3 \int e^{2 b^2 x^2} x^2 \, dx}{5 b^2 \pi }-\frac {8 \int e^{2 b^2 x^2} x^2 \, dx}{5 b^2 \pi }-\frac {8 \int e^{b^2 x^2} x \text {erfi}(b x) \, dx}{5 b^3 \sqrt {\pi }}\\ &=-\frac {11 e^{2 b^2 x^2} x}{20 b^4 \pi }+\frac {e^{2 b^2 x^2} x^3}{5 b^2 \pi }-\frac {4 e^{b^2 x^2} \text {erfi}(b x)}{5 b^5 \sqrt {\pi }}+\frac {4 e^{b^2 x^2} x^2 \text {erfi}(b x)}{5 b^3 \sqrt {\pi }}-\frac {2 e^{b^2 x^2} x^4 \text {erfi}(b x)}{5 b \sqrt {\pi }}+\frac {1}{5} x^5 \text {erfi}(b x)^2+\frac {3 \int e^{2 b^2 x^2} \, dx}{20 b^4 \pi }+\frac {2 \int e^{2 b^2 x^2} \, dx}{5 b^4 \pi }+\frac {8 \int e^{2 b^2 x^2} \, dx}{5 b^4 \pi }\\ &=-\frac {11 e^{2 b^2 x^2} x}{20 b^4 \pi }+\frac {e^{2 b^2 x^2} x^3}{5 b^2 \pi }-\frac {4 e^{b^2 x^2} \text {erfi}(b x)}{5 b^5 \sqrt {\pi }}+\frac {4 e^{b^2 x^2} x^2 \text {erfi}(b x)}{5 b^3 \sqrt {\pi }}-\frac {2 e^{b^2 x^2} x^4 \text {erfi}(b x)}{5 b \sqrt {\pi }}+\frac {1}{5} x^5 \text {erfi}(b x)^2+\frac {2 \sqrt {\frac {2}{\pi }} \text {erfi}\left (\sqrt {2} b x\right )}{5 b^5}+\frac {11 \text {erfi}\left (\sqrt {2} b x\right )}{40 b^5 \sqrt {2 \pi }}\\ \end {align*}

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Mathematica [A]  time = 0.05, size = 105, normalized size = 0.65 \[ \frac {16 \pi b^5 x^5 \text {erfi}(b x)^2+4 b x e^{2 b^2 x^2} \left (4 b^2 x^2-11\right )-32 \sqrt {\pi } e^{b^2 x^2} \left (b^4 x^4-2 b^2 x^2+2\right ) \text {erfi}(b x)+43 \sqrt {2 \pi } \text {erfi}\left (\sqrt {2} b x\right )}{80 \pi b^5} \]

Antiderivative was successfully verified.

[In]

Integrate[x^4*Erfi[b*x]^2,x]

[Out]

(4*b*E^(2*b^2*x^2)*x*(-11 + 4*b^2*x^2) - 32*E^(b^2*x^2)*Sqrt[Pi]*(2 - 2*b^2*x^2 + b^4*x^4)*Erfi[b*x] + 16*b^5*
Pi*x^5*Erfi[b*x]^2 + 43*Sqrt[2*Pi]*Erfi[Sqrt[2]*b*x])/(80*b^5*Pi)

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fricas [A]  time = 0.61, size = 110, normalized size = 0.68 \[ \frac {16 \, \pi b^{6} x^{5} \operatorname {erfi}\left (b x\right )^{2} - 32 \, \sqrt {\pi } {\left (b^{5} x^{4} - 2 \, b^{3} x^{2} + 2 \, b\right )} \operatorname {erfi}\left (b x\right ) e^{\left (b^{2} x^{2}\right )} + 43 \, \sqrt {2} \sqrt {\pi } \sqrt {b^{2}} \operatorname {erfi}\left (\sqrt {2} \sqrt {b^{2}} x\right ) + 4 \, {\left (4 \, b^{4} x^{3} - 11 \, b^{2} x\right )} e^{\left (2 \, b^{2} x^{2}\right )}}{80 \, \pi b^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/80*(16*pi*b^6*x^5*erfi(b*x)^2 - 32*sqrt(pi)*(b^5*x^4 - 2*b^3*x^2 + 2*b)*erfi(b*x)*e^(b^2*x^2) + 43*sqrt(2)*s
qrt(pi)*sqrt(b^2)*erfi(sqrt(2)*sqrt(b^2)*x) + 4*(4*b^4*x^3 - 11*b^2*x)*e^(2*b^2*x^2))/(pi*b^6)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x^4*erfi(b*x)^2, x)

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maple [F]  time = 0.01, size = 0, normalized size = 0.00 \[ \int x^{4} \erfi \left (b x \right )^{2}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x^4*erfi(b*x)^2, x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int x^4\,{\mathrm {erfi}\left (b\,x\right )}^2 \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(x^4*erfi(b*x)^2, x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{4} \operatorname {erfi}^{2}{\left (b x \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x**4*erfi(b*x)**2, x)

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