3.3.35 \(\int x^4 \text {Erfi}(b x)^2 \, dx\) [235]

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

[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)

________________________________________________________________________________________

Rubi [A]
time = 0.15, 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 = {6501, 6522, 6519, 2235, 2243} \begin {gather*} \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 \end {gather*}

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 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 2243

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 6501

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 6519

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 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*} \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*}

________________________________________________________________________________________

Mathematica [A]
time = 0.03, size = 105, normalized size = 0.65 \begin {gather*} \frac {4 b e^{2 b^2 x^2} x \left (-11+4 b^2 x^2\right )-32 e^{b^2 x^2} \sqrt {\pi } \left (2-2 b^2 x^2+b^4 x^4\right ) \text {Erfi}(b x)+16 b^5 \pi x^5 \text {Erfi}(b x)^2+43 \sqrt {2 \pi } \text {Erfi}\left (\sqrt {2} b x\right )}{80 b^5 \pi } \end {gather*}

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)

________________________________________________________________________________________

Maple [F]
time = 0.02, 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)

________________________________________________________________________________________

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

[Out]

integrate(x^4*erfi(b*x)^2, x)

________________________________________________________________________________________

Fricas [A]
time = 0.37, size = 110, normalized size = 0.68 \begin {gather*} \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}} \end {gather*}

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)

________________________________________________________________________________________

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

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)

________________________________________________________________________________________

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(x^4*erfi(b*x)^2,x, algorithm="giac")

[Out]

integrate(x^4*erfi(b*x)^2, x)

________________________________________________________________________________________

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

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)

________________________________________________________________________________________