\(\int x^4 \text {erfi}(b x) \, dx\) [215]

Optimal result

Integrand size = 8, antiderivative size = 81 \[ \int x^4 \text {erfi}(b x) \, dx=-\frac {2 e^{b^2 x^2}}{5 b^5 \sqrt {\pi }}+\frac {2 e^{b^2 x^2} x^2}{5 b^3 \sqrt {\pi }}-\frac {e^{b^2 x^2} x^4}{5 b \sqrt {\pi }}+\frac {1}{5} x^5 \text {erfi}(b x) \]

[Out]

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

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {6498, 2243, 2240} \[ \int x^4 \text {erfi}(b x) \, dx=-\frac {x^4 e^{b^2 x^2}}{5 \sqrt {\pi } b}-\frac {2 e^{b^2 x^2}}{5 \sqrt {\pi } b^5}+\frac {2 x^2 e^{b^2 x^2}}{5 \sqrt {\pi } b^3}+\frac {1}{5} x^5 \text {erfi}(b x) \]

[In]

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

[Out]

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

Rule 2240

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Simp[(e + f*x)^n*(
F^(a + b*(c + d*x)^n)/(b*f*n*(c + d*x)^n*Log[F])), x] /; FreeQ[{F, a, b, c, d, e, f, n}, x] && EqQ[m, n - 1] &
& EqQ[d*e - c*f, 0]

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 6498

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

Rubi steps \begin{align*} \text {integral}& = \frac {1}{5} x^5 \text {erfi}(b x)-\frac {(2 b) \int e^{b^2 x^2} x^5 \, dx}{5 \sqrt {\pi }} \\ & = -\frac {e^{b^2 x^2} x^4}{5 b \sqrt {\pi }}+\frac {1}{5} x^5 \text {erfi}(b x)+\frac {4 \int e^{b^2 x^2} x^3 \, dx}{5 b \sqrt {\pi }} \\ & = \frac {2 e^{b^2 x^2} x^2}{5 b^3 \sqrt {\pi }}-\frac {e^{b^2 x^2} x^4}{5 b \sqrt {\pi }}+\frac {1}{5} x^5 \text {erfi}(b x)-\frac {4 \int e^{b^2 x^2} x \, dx}{5 b^3 \sqrt {\pi }} \\ & = -\frac {2 e^{b^2 x^2}}{5 b^5 \sqrt {\pi }}+\frac {2 e^{b^2 x^2} x^2}{5 b^3 \sqrt {\pi }}-\frac {e^{b^2 x^2} x^4}{5 b \sqrt {\pi }}+\frac {1}{5} x^5 \text {erfi}(b x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.60 \[ \int x^4 \text {erfi}(b x) \, dx=\frac {1}{5} \left (-\frac {e^{b^2 x^2} \left (2-2 b^2 x^2+b^4 x^4\right )}{b^5 \sqrt {\pi }}+x^5 \text {erfi}(b x)\right ) \]

[In]

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

[Out]

(-((E^(b^2*x^2)*(2 - 2*b^2*x^2 + b^4*x^4))/(b^5*Sqrt[Pi])) + x^5*Erfi[b*x])/5

Maple [A] (verified)

Time = 0.21 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.67

method	result	size
meijerg	\(-\frac {-\frac {4}{5}+\frac {2 \left (3 b^{4} x^{4}-6 b^{2} x^{2}+6\right ) {\mathrm e}^{b^{2} x^{2}}}{15}-\frac {2 b^{5} x^{5} \sqrt {\pi }\, \operatorname {erfi}\left (b x \right )}{5}}{2 b^{5} \sqrt {\pi }}\)	\(54\)
derivativedivides	\(\frac {\frac {b^{5} x^{5} \operatorname {erfi}\left (b x \right )}{5}-\frac {2 \left (\frac {{\mathrm e}^{b^{2} x^{2}} b^{4} x^{4}}{2}-b^{2} x^{2} {\mathrm e}^{b^{2} x^{2}}+{\mathrm e}^{b^{2} x^{2}}\right )}{5 \sqrt {\pi }}}{b^{5}}\)	\(64\)
default	\(\frac {\frac {b^{5} x^{5} \operatorname {erfi}\left (b x \right )}{5}-\frac {2 \left (\frac {{\mathrm e}^{b^{2} x^{2}} b^{4} x^{4}}{2}-b^{2} x^{2} {\mathrm e}^{b^{2} x^{2}}+{\mathrm e}^{b^{2} x^{2}}\right )}{5 \sqrt {\pi }}}{b^{5}}\)	\(64\)
parallelrisch	\(\frac {b^{5} x^{5} \sqrt {\pi }\, \operatorname {erfi}\left (b x \right )-{\mathrm e}^{b^{2} x^{2}} b^{4} x^{4}+2 b^{2} x^{2} {\mathrm e}^{b^{2} x^{2}}-2 \,{\mathrm e}^{b^{2} x^{2}}}{5 b^{5} \sqrt {\pi }}\)	\(66\)
parts	\(\frac {x^{5} \operatorname {erfi}\left (b x \right )}{5}-\frac {2 b \left (\frac {x^{4} {\mathrm e}^{b^{2} x^{2}}}{2 b^{2}}-\frac {2 \left (\frac {x^{2} {\mathrm e}^{b^{2} x^{2}}}{2 b^{2}}-\frac {{\mathrm e}^{b^{2} x^{2}}}{2 b^{4}}\right )}{b^{2}}\right )}{5 \sqrt {\pi }}\)	\(69\)

[In]

int(x^4*erfi(b*x),x,method=_RETURNVERBOSE)

[Out]

-1/2/b^5/Pi^(1/2)*(-4/5+2/15*(3*b^4*x^4-6*b^2*x^2+6)*exp(b^2*x^2)-2/5*b^5*x^5*Pi^(1/2)*erfi(b*x))

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.63 \[ \int x^4 \text {erfi}(b x) \, dx=\frac {\pi b^{5} x^{5} \operatorname {erfi}\left (b x\right ) - \sqrt {\pi } {\left (b^{4} x^{4} - 2 \, b^{2} x^{2} + 2\right )} e^{\left (b^{2} x^{2}\right )}}{5 \, \pi b^{5}} \]

[In]

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

[Out]

1/5*(pi*b^5*x^5*erfi(b*x) - sqrt(pi)*(b^4*x^4 - 2*b^2*x^2 + 2)*e^(b^2*x^2))/(pi*b^5)

Sympy [A] (verification not implemented)

Time = 0.30 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.93 \[ \int x^4 \text {erfi}(b x) \, dx=\begin {cases} \frac {x^{5} \operatorname {erfi}{\left (b x \right )}}{5} - \frac {x^{4} e^{b^{2} x^{2}}}{5 \sqrt {\pi } b} + \frac {2 x^{2} e^{b^{2} x^{2}}}{5 \sqrt {\pi } b^{3}} - \frac {2 e^{b^{2} x^{2}}}{5 \sqrt {\pi } b^{5}} & \text {for}\: b \neq 0 \\0 & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise((x**5*erfi(b*x)/5 - x**4*exp(b**2*x**2)/(5*sqrt(pi)*b) + 2*x**2*exp(b**2*x**2)/(5*sqrt(pi)*b**3) - 2
*exp(b**2*x**2)/(5*sqrt(pi)*b**5), Ne(b, 0)), (0, True))

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.53 \[ \int x^4 \text {erfi}(b x) \, dx=\frac {1}{5} \, x^{5} \operatorname {erfi}\left (b x\right ) - \frac {{\left (b^{4} x^{4} - 2 \, b^{2} x^{2} + 2\right )} e^{\left (b^{2} x^{2}\right )}}{5 \, \sqrt {\pi } b^{5}} \]

[In]

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

[Out]

1/5*x^5*erfi(b*x) - 1/5*(b^4*x^4 - 2*b^2*x^2 + 2)*e^(b^2*x^2)/(sqrt(pi)*b^5)

Giac [F]

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

[In]

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

[Out]

integrate(x^4*erfi(b*x), x)

Mupad [B] (verification not implemented)

Time = 4.83 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.53 \[ \int x^4 \text {erfi}(b x) \, dx=\frac {x^5\,\mathrm {erfi}\left (b\,x\right )}{5}-\frac {{\mathrm {e}}^{b^2\,x^2}\,\left (b^4\,x^4-2\,b^2\,x^2+2\right )}{5\,b^5\,\sqrt {\pi }} \]

[In]

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

[Out]

(x^5*erfi(b*x))/5 - (exp(b^2*x^2)*(b^4*x^4 - 2*b^2*x^2 + 2))/(5*b^5*pi^(1/2))