Integrand size = 8, antiderivative size = 78 \[ \int \frac {\text {erfi}(b x)}{x^6} \, dx=-\frac {b e^{b^2 x^2}}{10 \sqrt {\pi } x^4}-\frac {b^3 e^{b^2 x^2}}{10 \sqrt {\pi } x^2}-\frac {\text {erfi}(b x)}{5 x^5}+\frac {b^5 \operatorname {ExpIntegralEi}\left (b^2 x^2\right )}{10 \sqrt {\pi }} \] Output:
-1/10*b*exp(b^2*x^2)/Pi^(1/2)/x^4-1/10*b^3*exp(b^2*x^2)/Pi^(1/2)/x^2-1/5*e rfi(b*x)/x^5+1/10*b^5*Ei(b^2*x^2)/Pi^(1/2)
Time = 0.02 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.78 \[ \int \frac {\text {erfi}(b x)}{x^6} \, dx=\frac {-b e^{b^2 x^2} x \left (1+b^2 x^2\right )-2 \sqrt {\pi } \text {erfi}(b x)+b^5 x^5 \operatorname {ExpIntegralEi}\left (b^2 x^2\right )}{10 \sqrt {\pi } x^5} \] Input:
Integrate[Erfi[b*x]/x^6,x]
Output:
(-(b*E^(b^2*x^2)*x*(1 + b^2*x^2)) - 2*Sqrt[Pi]*Erfi[b*x] + b^5*x^5*ExpInte gralEi[b^2*x^2])/(10*Sqrt[Pi]*x^5)
Time = 0.31 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.00, 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, 2639}
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.
\(\displaystyle \int \frac {\text {erfi}(b x)}{x^6} \, dx\) |
\(\Big \downarrow \) 6917 |
\(\displaystyle \frac {2 b \int \frac {e^{b^2 x^2}}{x^5}dx}{5 \sqrt {\pi }}-\frac {\text {erfi}(b x)}{5 x^5}\) |
\(\Big \downarrow \) 2643 |
\(\displaystyle \frac {2 b \left (\frac {1}{2} b^2 \int \frac {e^{b^2 x^2}}{x^3}dx-\frac {e^{b^2 x^2}}{4 x^4}\right )}{5 \sqrt {\pi }}-\frac {\text {erfi}(b x)}{5 x^5}\) |
\(\Big \downarrow \) 2643 |
\(\displaystyle \frac {2 b \left (\frac {1}{2} b^2 \left (b^2 \int \frac {e^{b^2 x^2}}{x}dx-\frac {e^{b^2 x^2}}{2 x^2}\right )-\frac {e^{b^2 x^2}}{4 x^4}\right )}{5 \sqrt {\pi }}-\frac {\text {erfi}(b x)}{5 x^5}\) |
\(\Big \downarrow \) 2639 |
\(\displaystyle \frac {2 b \left (\frac {1}{2} b^2 \left (\frac {1}{2} b^2 \operatorname {ExpIntegralEi}\left (b^2 x^2\right )-\frac {e^{b^2 x^2}}{2 x^2}\right )-\frac {e^{b^2 x^2}}{4 x^4}\right )}{5 \sqrt {\pi }}-\frac {\text {erfi}(b x)}{5 x^5}\) |
Input:
Int[Erfi[b*x]/x^6,x]
Output:
-1/5*Erfi[b*x]/x^5 + (2*b*(-1/4*E^(b^2*x^2)/x^4 + (b^2*(-1/2*E^(b^2*x^2)/x ^2 + (b^2*ExpIntegralEi[b^2*x^2])/2))/2))/(5*Sqrt[Pi])
Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_ Symbol] :> Simp[F^a*(ExpIntegralEi[b*(c + d*x)^n*Log[F]]/(f*n)), x] /; Free Q[{F, a, b, c, d, e, f, n}, x] && EqQ[d*e - c*f, 0]
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 = 0.64 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.83
method | result | size |
parts | \(-\frac {\operatorname {erfi}\left (b x \right )}{5 x^{5}}+\frac {2 b \left (-\frac {{\mathrm e}^{b^{2} x^{2}}}{4 x^{4}}+\frac {b^{2} \left (-\frac {{\mathrm e}^{b^{2} x^{2}}}{2 x^{2}}-\frac {b^{2} \operatorname {expIntegral}_{1}\left (-b^{2} x^{2}\right )}{2}\right )}{2}\right )}{5 \sqrt {\pi }}\) | \(65\) |
derivativedivides | \(b^{5} \left (-\frac {\operatorname {erfi}\left (b x \right )}{5 b^{5} x^{5}}+\frac {-\frac {{\mathrm e}^{b^{2} x^{2}}}{10 b^{4} x^{4}}-\frac {{\mathrm e}^{b^{2} x^{2}}}{10 x^{2} b^{2}}-\frac {\operatorname {expIntegral}_{1}\left (-b^{2} x^{2}\right )}{10}}{\sqrt {\pi }}\right )\) | \(68\) |
default | \(b^{5} \left (-\frac {\operatorname {erfi}\left (b x \right )}{5 b^{5} x^{5}}+\frac {-\frac {{\mathrm e}^{b^{2} x^{2}}}{10 b^{4} x^{4}}-\frac {{\mathrm e}^{b^{2} x^{2}}}{10 x^{2} b^{2}}-\frac {\operatorname {expIntegral}_{1}\left (-b^{2} x^{2}\right )}{10}}{\sqrt {\pi }}\right )\) | \(68\) |
meijerg | \(\frac {b^{5} \left (-\frac {1}{b^{4} x^{4}}-\frac {2}{3 b^{2} x^{2}}-\frac {19}{50}+\frac {2 \ln \left (x \right )}{5}+\frac {2 \ln \left (i b \right )}{5}+\frac {399 b^{4} x^{4}+700 b^{2} x^{2}+1050}{1050 b^{4} x^{4}}-\frac {\left (21 b^{2} x^{2}+21\right ) {\mathrm e}^{b^{2} x^{2}}}{105 b^{4} x^{4}}-\frac {2 \sqrt {\pi }\, \operatorname {erfi}\left (b x \right )}{5 b^{5} x^{5}}-\frac {\ln \left (-b^{2} x^{2}\right )}{5}-\frac {\operatorname {expIntegral}_{1}\left (-b^{2} x^{2}\right )}{5}\right )}{2 \sqrt {\pi }}\) | \(128\) |
Input:
int(erfi(b*x)/x^6,x,method=_RETURNVERBOSE)
Output:
-1/5*erfi(b*x)/x^5+2/5/Pi^(1/2)*b*(-1/4/x^4*exp(b^2*x^2)+1/2*b^2*(-1/2/x^2 *exp(b^2*x^2)-1/2*b^2*Ei(1,-b^2*x^2)))
Time = 0.09 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.74 \[ \int \frac {\text {erfi}(b x)}{x^6} \, dx=-\frac {2 \, \pi \operatorname {erfi}\left (b x\right ) - \sqrt {\pi } {\left (b^{5} x^{5} {\rm Ei}\left (b^{2} x^{2}\right ) - {\left (b^{3} x^{3} + b x\right )} e^{\left (b^{2} x^{2}\right )}\right )}}{10 \, \pi x^{5}} \] Input:
integrate(erfi(b*x)/x^6,x, algorithm="fricas")
Output:
-1/10*(2*pi*erfi(b*x) - sqrt(pi)*(b^5*x^5*Ei(b^2*x^2) - (b^3*x^3 + b*x)*e^ (b^2*x^2)))/(pi*x^5)
Result contains complex when optimal does not.
Time = 2.01 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.09 \[ \int \frac {\text {erfi}(b x)}{x^6} \, dx=- \frac {b^{5} \operatorname {E}_{1}\left (b^{2} x^{2} e^{i \pi }\right )}{10 \sqrt {\pi }} - \frac {b^{3} e^{b^{2} x^{2}}}{10 \sqrt {\pi } x^{2}} - \frac {b e^{b^{2} x^{2}}}{10 \sqrt {\pi } x^{4}} - \frac {i \operatorname {erfc}{\left (i b x \right )}}{5 x^{5}} + \frac {i}{5 x^{5}} \] Input:
integrate(erfi(b*x)/x**6,x)
Output:
-b**5*expint(1, b**2*x**2*exp_polar(I*pi))/(10*sqrt(pi)) - b**3*exp(b**2*x **2)/(10*sqrt(pi)*x**2) - b*exp(b**2*x**2)/(10*sqrt(pi)*x**4) - I*erfc(I*b *x)/(5*x**5) + I/(5*x**5)
Time = 0.07 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.36 \[ \int \frac {\text {erfi}(b x)}{x^6} \, dx=-\frac {b^{5} \Gamma \left (-2, -b^{2} x^{2}\right )}{5 \, \sqrt {\pi }} - \frac {\operatorname {erfi}\left (b x\right )}{5 \, x^{5}} \] Input:
integrate(erfi(b*x)/x^6,x, algorithm="maxima")
Output:
-1/5*b^5*gamma(-2, -b^2*x^2)/sqrt(pi) - 1/5*erfi(b*x)/x^5
\[ \int \frac {\text {erfi}(b x)}{x^6} \, dx=\int { \frac {\operatorname {erfi}\left (b x\right )}{x^{6}} \,d x } \] Input:
integrate(erfi(b*x)/x^6,x, algorithm="giac")
Output:
integrate(erfi(b*x)/x^6, x)
Time = 3.97 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.79 \[ \int \frac {\text {erfi}(b x)}{x^6} \, dx=\frac {b^5\,\mathrm {ei}\left (b^2\,x^2\right )}{10\,\sqrt {\pi }}-\frac {\mathrm {erfi}\left (b\,x\right )}{5\,x^5}-\frac {\frac {b\,{\mathrm {e}}^{b^2\,x^2}}{2}+\frac {b^3\,x^2\,{\mathrm {e}}^{b^2\,x^2}}{2}}{5\,x^4\,\sqrt {\pi }} \] Input:
int(erfi(b*x)/x^6,x)
Output:
(b^5*ei(b^2*x^2))/(10*pi^(1/2)) - erfi(b*x)/(5*x^5) - ((b*exp(b^2*x^2))/2 + (b^3*x^2*exp(b^2*x^2))/2)/(5*x^4*pi^(1/2))
\[ \int \frac {\text {erfi}(b x)}{x^6} \, dx=\frac {-\mathrm {erf}\left (b i x \right ) b^{4} i \pi \,x^{4}+2 \,\mathrm {erf}\left (b i x \right ) i \pi -\sqrt {\pi }\, e^{b^{2} x^{2}} b^{3} x^{3}-\sqrt {\pi }\, e^{b^{2} x^{2}} b x -\left (\int \frac {\mathrm {erf}\left (b i x \right )}{x^{2}}d x \right ) b^{4} i \pi \,x^{5}}{10 \pi \,x^{5}} \] Input:
int(erfi(b*x)/x^6,x)
Output:
( - erf(b*i*x)*b**4*i*pi*x**4 + 2*erf(b*i*x)*i*pi - sqrt(pi)*e**(b**2*x**2 )*b**3*x**3 - sqrt(pi)*e**(b**2*x**2)*b*x - int(erf(b*i*x)/x**2,x)*b**4*i* pi*x**5)/(10*pi*x**5)