Integrand size = 8, antiderivative size = 69 \[ \int \frac {\text {erfi}(b x)}{x^5} \, dx=-\frac {b e^{b^2 x^2}}{6 \sqrt {\pi } x^3}-\frac {b^3 e^{b^2 x^2}}{3 \sqrt {\pi } x}+\frac {1}{3} b^4 \text {erfi}(b x)-\frac {\text {erfi}(b x)}{4 x^4} \] Output:
-1/6*b*exp(b^2*x^2)/Pi^(1/2)/x^3-1/3*b^3*exp(b^2*x^2)/Pi^(1/2)/x+1/3*b^4*e rfi(b*x)-1/4*erfi(b*x)/x^4
Time = 0.02 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.74 \[ \int \frac {\text {erfi}(b x)}{x^5} \, dx=\frac {-\frac {2 b e^{b^2 x^2} x \left (1+2 b^2 x^2\right )}{\sqrt {\pi }}+\left (-3+4 b^4 x^4\right ) \text {erfi}(b x)}{12 x^4} \] Input:
Integrate[Erfi[b*x]/x^5,x]
Output:
((-2*b*E^(b^2*x^2)*x*(1 + 2*b^2*x^2))/Sqrt[Pi] + (-3 + 4*b^4*x^4)*Erfi[b*x ])/(12*x^4)
Time = 0.30 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.04, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6917, 2643, 2643, 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.
\(\displaystyle \int \frac {\text {erfi}(b x)}{x^5} \, dx\) |
\(\Big \downarrow \) 6917 |
\(\displaystyle \frac {b \int \frac {e^{b^2 x^2}}{x^4}dx}{2 \sqrt {\pi }}-\frac {\text {erfi}(b x)}{4 x^4}\) |
\(\Big \downarrow \) 2643 |
\(\displaystyle \frac {b \left (\frac {2}{3} b^2 \int \frac {e^{b^2 x^2}}{x^2}dx-\frac {e^{b^2 x^2}}{3 x^3}\right )}{2 \sqrt {\pi }}-\frac {\text {erfi}(b x)}{4 x^4}\) |
\(\Big \downarrow \) 2643 |
\(\displaystyle \frac {b \left (\frac {2}{3} b^2 \left (2 b^2 \int e^{b^2 x^2}dx-\frac {e^{b^2 x^2}}{x}\right )-\frac {e^{b^2 x^2}}{3 x^3}\right )}{2 \sqrt {\pi }}-\frac {\text {erfi}(b x)}{4 x^4}\) |
\(\Big \downarrow \) 2633 |
\(\displaystyle \frac {b \left (\frac {2}{3} b^2 \left (\sqrt {\pi } b \text {erfi}(b x)-\frac {e^{b^2 x^2}}{x}\right )-\frac {e^{b^2 x^2}}{3 x^3}\right )}{2 \sqrt {\pi }}-\frac {\text {erfi}(b x)}{4 x^4}\) |
Input:
Int[Erfi[b*x]/x^5,x]
Output:
-1/4*Erfi[b*x]/x^4 + (b*(-1/3*E^(b^2*x^2)/x^3 + (2*b^2*(-(E^(b^2*x^2)/x) + b*Sqrt[Pi]*Erfi[b*x]))/3))/(2*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 + 1)*(F^(a + b*(c + d*x)^n)/(d*(m + 1))) , x] - Simp[b*n*(Log[F]/(m + 1)) 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[ -4, (m + 1)/n, 5] && IntegerQ[n] && ((GtQ[n, 0] && LtQ[m, -1]) || (GtQ[-n, 0] && LeQ[-n, m + 1]))
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]
Time = 0.24 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.90
method | result | size |
parallelrisch | \(\frac {4 \,\operatorname {erfi}\left (b x \right ) x^{4} \sqrt {\pi }\, b^{4}-4 b^{3} x^{3} {\mathrm e}^{b^{2} x^{2}}-2 \,{\mathrm e}^{b^{2} x^{2}} b x -3 \,\operatorname {erfi}\left (b x \right ) \sqrt {\pi }}{12 \sqrt {\pi }\, x^{4}}\) | \(62\) |
parts | \(-\frac {\operatorname {erfi}\left (b x \right )}{4 x^{4}}+\frac {b \left (-\frac {{\mathrm e}^{b^{2} x^{2}}}{3 x^{3}}+\frac {2 b^{2} \left (-\frac {{\mathrm e}^{b^{2} x^{2}}}{x}-i b \sqrt {\pi }\, \operatorname {erf}\left (i b x \right )\right )}{3}\right )}{2 \sqrt {\pi }}\) | \(63\) |
meijerg | \(-\frac {i b^{4} \left (-\frac {4 i \left (\frac {b^{2} x^{2}}{2}+\frac {1}{4}\right ) {\mathrm e}^{b^{2} x^{2}}}{3 x^{3} b^{3}}-\frac {i \left (-4 b^{4} x^{4}+3\right ) \operatorname {erfi}\left (b x \right ) \sqrt {\pi }}{6 x^{4} b^{4}}\right )}{2 \sqrt {\pi }}\) | \(64\) |
derivativedivides | \(b^{4} \left (-\frac {\operatorname {erfi}\left (b x \right )}{4 b^{4} x^{4}}+\frac {-\frac {{\mathrm e}^{b^{2} x^{2}}}{3 b^{3} x^{3}}-\frac {2 \,{\mathrm e}^{b^{2} x^{2}}}{3 b x}+\frac {2 \,\operatorname {erfi}\left (b x \right ) \sqrt {\pi }}{3}}{2 \sqrt {\pi }}\right )\) | \(65\) |
default | \(b^{4} \left (-\frac {\operatorname {erfi}\left (b x \right )}{4 b^{4} x^{4}}+\frac {-\frac {{\mathrm e}^{b^{2} x^{2}}}{3 b^{3} x^{3}}-\frac {2 \,{\mathrm e}^{b^{2} x^{2}}}{3 b x}+\frac {2 \,\operatorname {erfi}\left (b x \right ) \sqrt {\pi }}{3}}{2 \sqrt {\pi }}\right )\) | \(65\) |
Input:
int(erfi(b*x)/x^5,x,method=_RETURNVERBOSE)
Output:
1/12*(4*erfi(b*x)*x^4*Pi^(1/2)*b^4-4*b^3*x^3*exp(b^2*x^2)-2*exp(b^2*x^2)*b *x-3*erfi(b*x)*Pi^(1/2))/Pi^(1/2)/x^4
Time = 0.09 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.75 \[ \int \frac {\text {erfi}(b x)}{x^5} \, dx=-\frac {2 \, \sqrt {\pi } {\left (2 \, b^{3} x^{3} + b x\right )} e^{\left (b^{2} x^{2}\right )} + {\left (3 \, \pi - 4 \, \pi b^{4} x^{4}\right )} \operatorname {erfi}\left (b x\right )}{12 \, \pi x^{4}} \] Input:
integrate(erfi(b*x)/x^5,x, algorithm="fricas")
Output:
-1/12*(2*sqrt(pi)*(2*b^3*x^3 + b*x)*e^(b^2*x^2) + (3*pi - 4*pi*b^4*x^4)*er fi(b*x))/(pi*x^4)
Time = 0.33 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.87 \[ \int \frac {\text {erfi}(b x)}{x^5} \, dx=\frac {b^{4} \operatorname {erfi}{\left (b x \right )}}{3} - \frac {b^{3} e^{b^{2} x^{2}}}{3 \sqrt {\pi } x} - \frac {b e^{b^{2} x^{2}}}{6 \sqrt {\pi } x^{3}} - \frac {\operatorname {erfi}{\left (b x \right )}}{4 x^{4}} \] Input:
integrate(erfi(b*x)/x**5,x)
Output:
b**4*erfi(b*x)/3 - b**3*exp(b**2*x**2)/(3*sqrt(pi)*x) - b*exp(b**2*x**2)/( 6*sqrt(pi)*x**3) - erfi(b*x)/(4*x**4)
Time = 0.07 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.57 \[ \int \frac {\text {erfi}(b x)}{x^5} \, dx=-\frac {\left (-b^{2} x^{2}\right )^{\frac {3}{2}} b \Gamma \left (-\frac {3}{2}, -b^{2} x^{2}\right )}{4 \, \sqrt {\pi } x^{3}} - \frac {\operatorname {erfi}\left (b x\right )}{4 \, x^{4}} \] Input:
integrate(erfi(b*x)/x^5,x, algorithm="maxima")
Output:
-1/4*(-b^2*x^2)^(3/2)*b*gamma(-3/2, -b^2*x^2)/(sqrt(pi)*x^3) - 1/4*erfi(b* x)/x^4
\[ \int \frac {\text {erfi}(b x)}{x^5} \, dx=\int { \frac {\operatorname {erfi}\left (b x\right )}{x^{5}} \,d x } \] Input:
integrate(erfi(b*x)/x^5,x, algorithm="giac")
Output:
integrate(erfi(b*x)/x^5, x)
Time = 0.08 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.29 \[ \int \frac {\text {erfi}(b x)}{x^5} \, dx=\frac {b\,{\left (-b^2\,x^2\right )}^{3/2}}{3\,x^3}-\frac {\mathrm {erfi}\left (b\,x\right )}{4\,x^4}-\frac {b^3\,{\mathrm {e}}^{b^2\,x^2}}{3\,x\,\sqrt {\pi }}-\frac {b\,{\mathrm {e}}^{b^2\,x^2}}{6\,x^3\,\sqrt {\pi }}-\frac {b\,\mathrm {erfc}\left (\sqrt {-b^2\,x^2}\right )\,{\left (-b^2\,x^2\right )}^{3/2}}{3\,x^3} \] Input:
int(erfi(b*x)/x^5,x)
Output:
(b*(-b^2*x^2)^(3/2))/(3*x^3) - erfi(b*x)/(4*x^4) - (b^3*exp(b^2*x^2))/(3*x *pi^(1/2)) - (b*exp(b^2*x^2))/(6*x^3*pi^(1/2)) - (b*erfc((-b^2*x^2)^(1/2)) *(-b^2*x^2)^(3/2))/(3*x^3)
Time = 0.16 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.97 \[ \int \frac {\text {erfi}(b x)}{x^5} \, dx=\frac {-4 \,\mathrm {erf}\left (b i x \right ) b^{4} i \pi \,x^{4}+3 \,\mathrm {erf}\left (b i x \right ) i \pi -4 \sqrt {\pi }\, e^{b^{2} x^{2}} b^{3} x^{3}-2 \sqrt {\pi }\, e^{b^{2} x^{2}} b x}{12 \pi \,x^{4}} \] Input:
int(erfi(b*x)/x^5,x)
Output:
( - 4*erf(b*i*x)*b**4*i*pi*x**4 + 3*erf(b*i*x)*i*pi - 4*sqrt(pi)*e**(b**2* x**2)*b**3*x**3 - 2*sqrt(pi)*e**(b**2*x**2)*b*x)/(12*pi*x**4)