\(\int x^3 \text {erfi}(b x)^2 \, dx\) [229]

Optimal result

Integrand size = 10, antiderivative size = 124 \[ \int x^3 \text {erfi}(b x)^2 \, dx=-\frac {e^{2 b^2 x^2}}{2 b^4 \pi }+\frac {e^{2 b^2 x^2} x^2}{4 b^2 \pi }+\frac {3 e^{b^2 x^2} x \text {erfi}(b x)}{4 b^3 \sqrt {\pi }}-\frac {e^{b^2 x^2} x^3 \text {erfi}(b x)}{2 b \sqrt {\pi }}-\frac {3 \text {erfi}(b x)^2}{16 b^4}+\frac {1}{4} x^4 \text {erfi}(b x)^2 \] Output:

-1/2*exp(2*b^2*x^2)/b^4/Pi+1/4*exp(2*b^2*x^2)*x^2/b^2/Pi+3/4*exp(b^2*x^2)* 
x*erfi(b*x)/b^3/Pi^(1/2)-1/2*exp(b^2*x^2)*x^3*erfi(b*x)/b/Pi^(1/2)-3/16*er 
fi(b*x)^2/b^4+1/4*x^4*erfi(b*x)^2
 

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.66 \[ \int x^3 \text {erfi}(b x)^2 \, dx=\frac {4 e^{2 b^2 x^2} \left (-2+b^2 x^2\right )-4 b e^{b^2 x^2} \sqrt {\pi } x \left (-3+2 b^2 x^2\right ) \text {erfi}(b x)+\pi \left (-3+4 b^4 x^4\right ) \text {erfi}(b x)^2}{16 b^4 \pi } \] Input:

Integrate[x^3*Erfi[b*x]^2,x]
 

Output:

(4*E^(2*b^2*x^2)*(-2 + b^2*x^2) - 4*b*E^(b^2*x^2)*Sqrt[Pi]*x*(-3 + 2*b^2*x 
^2)*Erfi[b*x] + Pi*(-3 + 4*b^4*x^4)*Erfi[b*x]^2)/(16*b^4*Pi)
 

Rubi [A] (verified)

Time = 0.80 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.31, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.800, Rules used = {6920, 6941, 2641, 2638, 6941, 2638, 6929, 15}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int x^3 \text {erfi}(b x)^2 \, dx\)

\(\Big \downarrow \) 6920

\(\displaystyle \frac {1}{4} x^4 \text {erfi}(b x)^2-\frac {b \int e^{b^2 x^2} x^4 \text {erfi}(b x)dx}{\sqrt {\pi }}\)

\(\Big \downarrow \) 6941

\(\displaystyle \frac {1}{4} x^4 \text {erfi}(b x)^2-\frac {b \left (-\frac {3 \int e^{b^2 x^2} x^2 \text {erfi}(b x)dx}{2 b^2}-\frac {\int e^{2 b^2 x^2} x^3dx}{\sqrt {\pi } b}+\frac {x^3 e^{b^2 x^2} \text {erfi}(b x)}{2 b^2}\right )}{\sqrt {\pi }}\)

\(\Big \downarrow \) 2641

\(\displaystyle \frac {1}{4} x^4 \text {erfi}(b x)^2-\frac {b \left (-\frac {3 \int e^{b^2 x^2} x^2 \text {erfi}(b x)dx}{2 b^2}-\frac {\frac {x^2 e^{2 b^2 x^2}}{4 b^2}-\frac {\int e^{2 b^2 x^2} xdx}{2 b^2}}{\sqrt {\pi } b}+\frac {x^3 e^{b^2 x^2} \text {erfi}(b x)}{2 b^2}\right )}{\sqrt {\pi }}\)

\(\Big \downarrow \) 2638

\(\displaystyle \frac {1}{4} x^4 \text {erfi}(b x)^2-\frac {b \left (-\frac {3 \int e^{b^2 x^2} x^2 \text {erfi}(b x)dx}{2 b^2}+\frac {x^3 e^{b^2 x^2} \text {erfi}(b x)}{2 b^2}-\frac {\frac {x^2 e^{2 b^2 x^2}}{4 b^2}-\frac {e^{2 b^2 x^2}}{8 b^4}}{\sqrt {\pi } b}\right )}{\sqrt {\pi }}\)

\(\Big \downarrow \) 6941

\(\displaystyle \frac {1}{4} x^4 \text {erfi}(b x)^2-\frac {b \left (-\frac {3 \left (-\frac {\int e^{b^2 x^2} \text {erfi}(b x)dx}{2 b^2}-\frac {\int e^{2 b^2 x^2} xdx}{\sqrt {\pi } b}+\frac {x e^{b^2 x^2} \text {erfi}(b x)}{2 b^2}\right )}{2 b^2}+\frac {x^3 e^{b^2 x^2} \text {erfi}(b x)}{2 b^2}-\frac {\frac {x^2 e^{2 b^2 x^2}}{4 b^2}-\frac {e^{2 b^2 x^2}}{8 b^4}}{\sqrt {\pi } b}\right )}{\sqrt {\pi }}\)

\(\Big \downarrow \) 2638

\(\displaystyle \frac {1}{4} x^4 \text {erfi}(b x)^2-\frac {b \left (-\frac {3 \left (-\frac {\int e^{b^2 x^2} \text {erfi}(b x)dx}{2 b^2}+\frac {x e^{b^2 x^2} \text {erfi}(b x)}{2 b^2}-\frac {e^{2 b^2 x^2}}{4 \sqrt {\pi } b^3}\right )}{2 b^2}+\frac {x^3 e^{b^2 x^2} \text {erfi}(b x)}{2 b^2}-\frac {\frac {x^2 e^{2 b^2 x^2}}{4 b^2}-\frac {e^{2 b^2 x^2}}{8 b^4}}{\sqrt {\pi } b}\right )}{\sqrt {\pi }}\)

\(\Big \downarrow \) 6929

\(\displaystyle \frac {1}{4} x^4 \text {erfi}(b x)^2-\frac {b \left (-\frac {3 \left (-\frac {\sqrt {\pi } \int \text {erfi}(b x)d\text {erfi}(b x)}{4 b^3}+\frac {x e^{b^2 x^2} \text {erfi}(b x)}{2 b^2}-\frac {e^{2 b^2 x^2}}{4 \sqrt {\pi } b^3}\right )}{2 b^2}+\frac {x^3 e^{b^2 x^2} \text {erfi}(b x)}{2 b^2}-\frac {\frac {x^2 e^{2 b^2 x^2}}{4 b^2}-\frac {e^{2 b^2 x^2}}{8 b^4}}{\sqrt {\pi } b}\right )}{\sqrt {\pi }}\)

\(\Big \downarrow \) 15

\(\displaystyle \frac {1}{4} x^4 \text {erfi}(b x)^2-\frac {b \left (\frac {x^3 e^{b^2 x^2} \text {erfi}(b x)}{2 b^2}-\frac {\frac {x^2 e^{2 b^2 x^2}}{4 b^2}-\frac {e^{2 b^2 x^2}}{8 b^4}}{\sqrt {\pi } b}-\frac {3 \left (-\frac {\sqrt {\pi } \text {erfi}(b x)^2}{8 b^3}+\frac {x e^{b^2 x^2} \text {erfi}(b x)}{2 b^2}-\frac {e^{2 b^2 x^2}}{4 \sqrt {\pi } b^3}\right )}{2 b^2}\right )}{\sqrt {\pi }}\)

Input:

Int[x^3*Erfi[b*x]^2,x]
 

Output:

(x^4*Erfi[b*x]^2)/4 - (b*(-((-1/8*E^(2*b^2*x^2)/b^4 + (E^(2*b^2*x^2)*x^2)/ 
(4*b^2))/(b*Sqrt[Pi])) + (E^(b^2*x^2)*x^3*Erfi[b*x])/(2*b^2) - (3*(-1/4*E^ 
(2*b^2*x^2)/(b^3*Sqrt[Pi]) + (E^(b^2*x^2)*x*Erfi[b*x])/(2*b^2) - (Sqrt[Pi] 
*Erfi[b*x]^2)/(8*b^3)))/(2*b^2)))/Sqrt[Pi]
 

Defintions of rubi rules used

rule 15

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

rule 2638

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 2641

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*L 
og[F])), x] - Simp[(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 6920

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

rule 6929

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

rule 6941

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

Maple [A] (verified)

Time = 0.15 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.90

method	result	size
parallelrisch	\(\frac {4 \operatorname {erfi}\left (b x \right )^{2} x^{4} \pi ^{\frac {3}{2}} b^{4}-8 \,\operatorname {erfi}\left (b x \right ) {\mathrm e}^{b^{2} x^{2}} x^{3} b^{3} \pi +4 \,{\mathrm e}^{2 b^{2} x^{2}} x^{2} b^{2} \sqrt {\pi }+12 \,\operatorname {erfi}\left (b x \right ) x \,{\mathrm e}^{b^{2} x^{2}} b \pi -3 \operatorname {erfi}\left (b x \right )^{2} \pi ^{\frac {3}{2}}-8 \,{\mathrm e}^{2 b^{2} x^{2}} \sqrt {\pi }}{16 \pi ^{\frac {3}{2}} b^{4}}\)	\(112\)

Input:

int(x^3*erfi(b*x)^2,x,method=_RETURNVERBOSE)
 

Output:

1/16*(4*erfi(b*x)^2*x^4*Pi^(3/2)*b^4-8*erfi(b*x)*exp(b^2*x^2)*x^3*b^3*Pi+4 
*exp(b^2*x^2)^2*x^2*b^2*Pi^(1/2)+12*erfi(b*x)*x*exp(b^2*x^2)*b*Pi-3*erfi(b 
*x)^2*Pi^(3/2)-8*exp(b^2*x^2)^2*Pi^(1/2))/Pi^(3/2)/b^4
 

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.64 \[ \int x^3 \text {erfi}(b x)^2 \, dx=-\frac {4 \, \sqrt {\pi } {\left (2 \, b^{3} x^{3} - 3 \, b x\right )} \operatorname {erfi}\left (b x\right ) e^{\left (b^{2} x^{2}\right )} + {\left (3 \, \pi - 4 \, \pi b^{4} x^{4}\right )} \operatorname {erfi}\left (b x\right )^{2} - 4 \, {\left (b^{2} x^{2} - 2\right )} e^{\left (2 \, b^{2} x^{2}\right )}}{16 \, \pi b^{4}} \] Input:

integrate(x^3*erfi(b*x)^2,x, algorithm="fricas")
                                                                                    
                                                                                    
 

Output:

-1/16*(4*sqrt(pi)*(2*b^3*x^3 - 3*b*x)*erfi(b*x)*e^(b^2*x^2) + (3*pi - 4*pi 
*b^4*x^4)*erfi(b*x)^2 - 4*(b^2*x^2 - 2)*e^(2*b^2*x^2))/(pi*b^4)
 

Sympy [A] (verification not implemented)

Time = 0.36 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.94 \[ \int x^3 \text {erfi}(b x)^2 \, dx=\begin {cases} \frac {x^{4} \operatorname {erfi}^{2}{\left (b x \right )}}{4} - \frac {x^{3} e^{b^{2} x^{2}} \operatorname {erfi}{\left (b x \right )}}{2 \sqrt {\pi } b} + \frac {x^{2} e^{2 b^{2} x^{2}}}{4 \pi b^{2}} + \frac {3 x e^{b^{2} x^{2}} \operatorname {erfi}{\left (b x \right )}}{4 \sqrt {\pi } b^{3}} - \frac {e^{2 b^{2} x^{2}}}{2 \pi b^{4}} - \frac {3 \operatorname {erfi}^{2}{\left (b x \right )}}{16 b^{4}} & \text {for}\: b \neq 0 \\0 & \text {otherwise} \end {cases} \] Input:

integrate(x**3*erfi(b*x)**2,x)
 

Output:

Piecewise((x**4*erfi(b*x)**2/4 - x**3*exp(b**2*x**2)*erfi(b*x)/(2*sqrt(pi) 
*b) + x**2*exp(2*b**2*x**2)/(4*pi*b**2) + 3*x*exp(b**2*x**2)*erfi(b*x)/(4* 
sqrt(pi)*b**3) - exp(2*b**2*x**2)/(2*pi*b**4) - 3*erfi(b*x)**2/(16*b**4), 
Ne(b, 0)), (0, True))
 

Maxima [F]

\[ \int x^3 \text {erfi}(b x)^2 \, dx=\int { x^{3} \operatorname {erfi}\left (b x\right )^{2} \,d x } \] Input:

integrate(x^3*erfi(b*x)^2,x, algorithm="maxima")
 

Output:

integrate(x^3*erfi(b*x)^2, x)
 

Giac [F]

\[ \int x^3 \text {erfi}(b x)^2 \, dx=\int { x^{3} \operatorname {erfi}\left (b x\right )^{2} \,d x } \] Input:

integrate(x^3*erfi(b*x)^2,x, algorithm="giac")
 

Output:

integrate(x^3*erfi(b*x)^2, x)
 

Mupad [B] (verification not implemented)

Time = 3.97 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.81 \[ \int x^3 \text {erfi}(b x)^2 \, dx=\frac {x^4\,{\mathrm {erfi}\left (b\,x\right )}^2}{4}-\frac {\frac {{\mathrm {e}}^{2\,b^2\,x^2}}{2}+\frac {3\,\pi \,{\mathrm {erfi}\left (b\,x\right )}^2}{16}-\frac {b^2\,x^2\,{\mathrm {e}}^{2\,b^2\,x^2}}{4}+\frac {b^3\,x^3\,\sqrt {\pi }\,{\mathrm {e}}^{b^2\,x^2}\,\mathrm {erfi}\left (b\,x\right )}{2}-\frac {3\,b\,x\,\sqrt {\pi }\,{\mathrm {e}}^{b^2\,x^2}\,\mathrm {erfi}\left (b\,x\right )}{4}}{b^4\,\pi } \] Input:

int(x^3*erfi(b*x)^2,x)
 

Output:

(x^4*erfi(b*x)^2)/4 - (exp(2*b^2*x^2)/2 + (3*pi*erfi(b*x)^2)/16 - (b^2*x^2 
*exp(2*b^2*x^2))/4 + (b^3*x^3*pi^(1/2)*exp(b^2*x^2)*erfi(b*x))/2 - (3*b*x* 
pi^(1/2)*exp(b^2*x^2)*erfi(b*x))/4)/(b^4*pi)
 

Reduce [F]

\[ \int x^3 \text {erfi}(b x)^2 \, dx=-\left (\int \mathrm {erf}\left (b i x \right )^{2} x^{3}d x \right ) \] Input:

int(x^3*erfi(b*x)^2,x)
 

Output:

 - int(erf(b*i*x)**2*x**3,x)