Integrand size = 8, antiderivative size = 93 \[ \int x^5 \text {erfi}(b x) \, dx=-\frac {5 e^{b^2 x^2} x}{8 b^5 \sqrt {\pi }}+\frac {5 e^{b^2 x^2} x^3}{12 b^3 \sqrt {\pi }}-\frac {e^{b^2 x^2} x^5}{6 b \sqrt {\pi }}+\frac {5 \text {erfi}(b x)}{16 b^6}+\frac {1}{6} x^6 \text {erfi}(b x) \] Output:
-5/8*exp(b^2*x^2)*x/b^5/Pi^(1/2)+5/12*exp(b^2*x^2)*x^3/b^3/Pi^(1/2)-1/6*ex p(b^2*x^2)*x^5/b/Pi^(1/2)+5/16*erfi(b*x)/b^6+1/6*x^6*erfi(b*x)
Time = 0.02 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.69 \[ \int x^5 \text {erfi}(b x) \, dx=\frac {-2 b e^{b^2 x^2} x \left (15-10 b^2 x^2+4 b^4 x^4\right )+\sqrt {\pi } \left (15+8 b^6 x^6\right ) \text {erfi}(b x)}{48 b^6 \sqrt {\pi }} \] Input:
Integrate[x^5*Erfi[b*x],x]
Output:
(-2*b*E^(b^2*x^2)*x*(15 - 10*b^2*x^2 + 4*b^4*x^4) + Sqrt[Pi]*(15 + 8*b^6*x ^6)*Erfi[b*x])/(48*b^6*Sqrt[Pi])
Time = 0.38 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.18, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.625, Rules used = {6917, 2641, 2641, 2641, 2633}
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.
\(\displaystyle \int x^5 \text {erfi}(b x) \, dx\) |
\(\Big \downarrow \) 6917 |
\(\displaystyle \frac {1}{6} x^6 \text {erfi}(b x)-\frac {b \int e^{b^2 x^2} x^6dx}{3 \sqrt {\pi }}\) |
\(\Big \downarrow \) 2641 |
\(\displaystyle \frac {1}{6} x^6 \text {erfi}(b x)-\frac {b \left (\frac {x^5 e^{b^2 x^2}}{2 b^2}-\frac {5 \int e^{b^2 x^2} x^4dx}{2 b^2}\right )}{3 \sqrt {\pi }}\) |
\(\Big \downarrow \) 2641 |
\(\displaystyle \frac {1}{6} x^6 \text {erfi}(b x)-\frac {b \left (\frac {x^5 e^{b^2 x^2}}{2 b^2}-\frac {5 \left (\frac {x^3 e^{b^2 x^2}}{2 b^2}-\frac {3 \int e^{b^2 x^2} x^2dx}{2 b^2}\right )}{2 b^2}\right )}{3 \sqrt {\pi }}\) |
\(\Big \downarrow \) 2641 |
\(\displaystyle \frac {1}{6} x^6 \text {erfi}(b x)-\frac {b \left (\frac {x^5 e^{b^2 x^2}}{2 b^2}-\frac {5 \left (\frac {x^3 e^{b^2 x^2}}{2 b^2}-\frac {3 \left (\frac {x e^{b^2 x^2}}{2 b^2}-\frac {\int e^{b^2 x^2}dx}{2 b^2}\right )}{2 b^2}\right )}{2 b^2}\right )}{3 \sqrt {\pi }}\) |
\(\Big \downarrow \) 2633 |
\(\displaystyle \frac {1}{6} x^6 \text {erfi}(b x)-\frac {b \left (\frac {x^5 e^{b^2 x^2}}{2 b^2}-\frac {5 \left (\frac {x^3 e^{b^2 x^2}}{2 b^2}-\frac {3 \left (\frac {x e^{b^2 x^2}}{2 b^2}-\frac {\sqrt {\pi } \text {erfi}(b x)}{4 b^3}\right )}{2 b^2}\right )}{2 b^2}\right )}{3 \sqrt {\pi }}\) |
Input:
Int[x^5*Erfi[b*x],x]
Output:
(x^6*Erfi[b*x])/6 - (b*((E^(b^2*x^2)*x^5)/(2*b^2) - (5*((E^(b^2*x^2)*x^3)/ (2*b^2) - (3*((E^(b^2*x^2)*x)/(2*b^2) - (Sqrt[Pi]*Erfi[b*x])/(4*b^3)))/(2* b^2)))/(2*b^2)))/(3*Sqrt[Pi])
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]
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])
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] - Simp[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]
Result contains complex when optimal does not.
Time = 0.30 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.67
method | result | size |
meijerg | \(\frac {i \left (\frac {i x b \left (28 b^{4} x^{4}-70 b^{2} x^{2}+105\right ) {\mathrm e}^{b^{2} x^{2}}}{84}-\frac {i \left (56 b^{6} x^{6}+105\right ) \operatorname {erfi}\left (b x \right ) \sqrt {\pi }}{168}\right )}{2 b^{6} \sqrt {\pi }}\) | \(62\) |
derivativedivides | \(\frac {\frac {b^{6} x^{6} \operatorname {erfi}\left (b x \right )}{6}-\frac {\frac {b^{5} x^{5} {\mathrm e}^{b^{2} x^{2}}}{2}-\frac {5 b^{3} x^{3} {\mathrm e}^{b^{2} x^{2}}}{4}+\frac {15 \,{\mathrm e}^{b^{2} x^{2}} b x}{8}-\frac {15 \,\operatorname {erfi}\left (b x \right ) \sqrt {\pi }}{16}}{3 \sqrt {\pi }}}{b^{6}}\) | \(77\) |
default | \(\frac {\frac {b^{6} x^{6} \operatorname {erfi}\left (b x \right )}{6}-\frac {\frac {b^{5} x^{5} {\mathrm e}^{b^{2} x^{2}}}{2}-\frac {5 b^{3} x^{3} {\mathrm e}^{b^{2} x^{2}}}{4}+\frac {15 \,{\mathrm e}^{b^{2} x^{2}} b x}{8}-\frac {15 \,\operatorname {erfi}\left (b x \right ) \sqrt {\pi }}{16}}{3 \sqrt {\pi }}}{b^{6}}\) | \(77\) |
parallelrisch | \(\frac {8 \,\operatorname {erfi}\left (b x \right ) x^{6} b^{6} \sqrt {\pi }-8 b^{5} x^{5} {\mathrm e}^{b^{2} x^{2}}+20 b^{3} x^{3} {\mathrm e}^{b^{2} x^{2}}-30 \,{\mathrm e}^{b^{2} x^{2}} b x +15 \,\operatorname {erfi}\left (b x \right ) \sqrt {\pi }}{48 b^{6} \sqrt {\pi }}\) | \(78\) |
parts | \(\frac {x^{6} \operatorname {erfi}\left (b x \right )}{6}-\frac {b \left (\frac {x^{5} {\mathrm e}^{b^{2} x^{2}}}{2 b^{2}}-\frac {5 \left (\frac {x^{3} {\mathrm e}^{b^{2} x^{2}}}{2 b^{2}}-\frac {3 \left (\frac {x \,{\mathrm e}^{b^{2} x^{2}}}{2 b^{2}}+\frac {i \sqrt {\pi }\, \operatorname {erf}\left (i b x \right )}{4 b^{3}}\right )}{2 b^{2}}\right )}{2 b^{2}}\right )}{3 \sqrt {\pi }}\) | \(91\) |
Input:
int(x^5*erfi(b*x),x,method=_RETURNVERBOSE)
Output:
1/2*I/b^6/Pi^(1/2)*(1/84*I*x*b*(28*b^4*x^4-70*b^2*x^2+105)*exp(b^2*x^2)-1/ 168*I*(56*b^6*x^6+105)*erfi(b*x)*Pi^(1/2))
Time = 0.10 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.67 \[ \int x^5 \text {erfi}(b x) \, dx=-\frac {2 \, \sqrt {\pi } {\left (4 \, b^{5} x^{5} - 10 \, b^{3} x^{3} + 15 \, b x\right )} e^{\left (b^{2} x^{2}\right )} - {\left (15 \, \pi + 8 \, \pi b^{6} x^{6}\right )} \operatorname {erfi}\left (b x\right )}{48 \, \pi b^{6}} \] Input:
integrate(x^5*erfi(b*x),x, algorithm="fricas")
Output:
-1/48*(2*sqrt(pi)*(4*b^5*x^5 - 10*b^3*x^3 + 15*b*x)*e^(b^2*x^2) - (15*pi + 8*pi*b^6*x^6)*erfi(b*x))/(pi*b^6)
Time = 0.48 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.95 \[ \int x^5 \text {erfi}(b x) \, dx=\begin {cases} \frac {x^{6} \operatorname {erfi}{\left (b x \right )}}{6} - \frac {x^{5} e^{b^{2} x^{2}}}{6 \sqrt {\pi } b} + \frac {5 x^{3} e^{b^{2} x^{2}}}{12 \sqrt {\pi } b^{3}} - \frac {5 x e^{b^{2} x^{2}}}{8 \sqrt {\pi } b^{5}} + \frac {5 \operatorname {erfi}{\left (b x \right )}}{16 b^{6}} & \text {for}\: b \neq 0 \\0 & \text {otherwise} \end {cases} \] Input:
integrate(x**5*erfi(b*x),x)
Output:
Piecewise((x**6*erfi(b*x)/6 - x**5*exp(b**2*x**2)/(6*sqrt(pi)*b) + 5*x**3* exp(b**2*x**2)/(12*sqrt(pi)*b**3) - 5*x*exp(b**2*x**2)/(8*sqrt(pi)*b**5) + 5*erfi(b*x)/(16*b**6), Ne(b, 0)), (0, True))
Result contains complex when optimal does not.
Time = 0.03 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.68 \[ \int x^5 \text {erfi}(b x) \, dx=\frac {1}{6} \, x^{6} \operatorname {erfi}\left (b x\right ) - \frac {b {\left (\frac {2 \, {\left (4 \, b^{4} x^{5} - 10 \, b^{2} x^{3} + 15 \, x\right )} e^{\left (b^{2} x^{2}\right )}}{b^{6}} + \frac {15 i \, \sqrt {\pi } \operatorname {erf}\left (i \, b x\right )}{b^{7}}\right )}}{48 \, \sqrt {\pi }} \] Input:
integrate(x^5*erfi(b*x),x, algorithm="maxima")
Output:
1/6*x^6*erfi(b*x) - 1/48*b*(2*(4*b^4*x^5 - 10*b^2*x^3 + 15*x)*e^(b^2*x^2)/ b^6 + 15*I*sqrt(pi)*erf(I*b*x)/b^7)/sqrt(pi)
\[ \int x^5 \text {erfi}(b x) \, dx=\int { x^{5} \operatorname {erfi}\left (b x\right ) \,d x } \] Input:
integrate(x^5*erfi(b*x),x, algorithm="giac")
Output:
integrate(x^5*erfi(b*x), x)
Time = 0.12 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.16 \[ \int x^5 \text {erfi}(b x) \, dx=\frac {x^6\,\mathrm {erfi}\left (b\,x\right )}{6}-\frac {5\,b\,x^7}{16\,{\left (-b^2\,x^2\right )}^{7/2}}-\frac {x^5\,{\mathrm {e}}^{b^2\,x^2}}{6\,b\,\sqrt {\pi }}+\frac {5\,x^3\,{\mathrm {e}}^{b^2\,x^2}}{12\,b^3\,\sqrt {\pi }}-\frac {5\,x\,{\mathrm {e}}^{b^2\,x^2}}{8\,b^5\,\sqrt {\pi }}+\frac {5\,b\,x^7\,\mathrm {erfc}\left (\sqrt {-b^2\,x^2}\right )}{16\,{\left (-b^2\,x^2\right )}^{7/2}} \] Input:
int(x^5*erfi(b*x),x)
Output:
(x^6*erfi(b*x))/6 - (5*b*x^7)/(16*(-b^2*x^2)^(7/2)) - (x^5*exp(b^2*x^2))/( 6*b*pi^(1/2)) + (5*x^3*exp(b^2*x^2))/(12*b^3*pi^(1/2)) - (5*x*exp(b^2*x^2) )/(8*b^5*pi^(1/2)) + (5*b*x^7*erfc((-b^2*x^2)^(1/2)))/(16*(-b^2*x^2)^(7/2) )
Time = 0.16 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.92 \[ \int x^5 \text {erfi}(b x) \, dx=\frac {-8 \,\mathrm {erf}\left (b i x \right ) b^{6} i \pi \,x^{6}-15 \,\mathrm {erf}\left (b i x \right ) i \pi -8 \sqrt {\pi }\, e^{b^{2} x^{2}} b^{5} x^{5}+20 \sqrt {\pi }\, e^{b^{2} x^{2}} b^{3} x^{3}-30 \sqrt {\pi }\, e^{b^{2} x^{2}} b x}{48 b^{6} \pi } \] Input:
int(x^5*erfi(b*x),x)
Output:
( - 8*erf(b*i*x)*b**6*i*pi*x**6 - 15*erf(b*i*x)*i*pi - 8*sqrt(pi)*e**(b**2 *x**2)*b**5*x**5 + 20*sqrt(pi)*e**(b**2*x**2)*b**3*x**3 - 30*sqrt(pi)*e**( b**2*x**2)*b*x)/(48*b**6*pi)