Integrand size = 19, antiderivative size = 19 \[ \int \frac {e^{c+d x^2} \text {erfi}(a+b x)}{x^4} \, dx=-\frac {b e^{a^2+c+2 a b x+\left (b^2+d\right ) x^2}}{3 \sqrt {\pi } x^2}-\frac {2 a b^2 e^{a^2+c+2 a b x+\left (b^2+d\right ) x^2}}{3 \sqrt {\pi } x}-\frac {e^{c+d x^2} \text {erfi}(a+b x)}{3 x^3}-\frac {2 d e^{c+d x^2} \text {erfi}(a+b x)}{3 x}+\frac {2}{3} a b^2 \sqrt {b^2+d} e^{c+\frac {a^2 d}{b^2+d}} \text {erfi}\left (\frac {a b+\left (b^2+d\right ) x}{\sqrt {b^2+d}}\right )+\frac {4 a^2 b^3 \text {Int}\left (\frac {e^{a^2+c+2 a b x+\left (b^2+d\right ) x^2}}{x},x\right )}{3 \sqrt {\pi }}+\frac {4 b d \text {Int}\left (\frac {e^{a^2+c+2 a b x+\left (b^2+d\right ) x^2}}{x},x\right )}{3 \sqrt {\pi }}+\frac {2 b \left (b^2+d\right ) \text {Int}\left (\frac {e^{a^2+c+2 a b x+\left (b^2+d\right ) x^2}}{x},x\right )}{3 \sqrt {\pi }}+\frac {4}{3} d^2 \text {Int}\left (e^{c+d x^2} \text {erfi}(a+b x),x\right ) \]
-1/3*exp(d*x^2+c)*erfi(b*x+a)/x^3-2/3*d*exp(d*x^2+c)*erfi(b*x+a)/x+2/3*a*b ^2*exp(c+a^2*d/(b^2+d))*erfi((a*b+(b^2+d)*x)/(b^2+d)^(1/2))*(b^2+d)^(1/2)- 1/3*b*exp(a^2+c+2*a*b*x+(b^2+d)*x^2)/x^2/Pi^(1/2)-2/3*a*b^2*exp(a^2+c+2*a* b*x+(b^2+d)*x^2)/x/Pi^(1/2)+4/3*a^2*b^3*Unintegrable(exp(a^2+c+2*a*b*x+(b^ 2+d)*x^2)/x,x)/Pi^(1/2)+4/3*b*d*Unintegrable(exp(a^2+c+2*a*b*x+(b^2+d)*x^2 )/x,x)/Pi^(1/2)+2/3*b*(b^2+d)*Unintegrable(exp(a^2+c+2*a*b*x+(b^2+d)*x^2)/ x,x)/Pi^(1/2)+4/3*d^2*Unintegrable(exp(d*x^2+c)*erfi(b*x+a),x)
Not integrable
Time = 0.29 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.11 \[ \int \frac {e^{c+d x^2} \text {erfi}(a+b x)}{x^4} \, dx=\int \frac {e^{c+d x^2} \text {erfi}(a+b x)}{x^4} \, dx \]
Not integrable
Time = 2.47 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {6947, 2672, 2672, 2664, 2633, 2673, 6947, 2673, 6935}
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.
\(\displaystyle \int \frac {e^{c+d x^2} \text {erfi}(a+b x)}{x^4} \, dx\) |
\(\Big \downarrow \) 6947 |
\(\displaystyle \frac {2 b \int \frac {e^{a^2+2 b x a+\left (b^2+d\right ) x^2+c}}{x^3}dx}{3 \sqrt {\pi }}+\frac {2}{3} d \int \frac {e^{d x^2+c} \text {erfi}(a+b x)}{x^2}dx-\frac {e^{c+d x^2} \text {erfi}(a+b x)}{3 x^3}\) |
\(\Big \downarrow \) 2672 |
\(\displaystyle \frac {2 b \left (a b \int \frac {e^{a^2+2 b x a+\left (b^2+d\right ) x^2+c}}{x^2}dx+\left (b^2+d\right ) \int \frac {e^{a^2+2 b x a+\left (b^2+d\right ) x^2+c}}{x}dx-\frac {e^{a^2+2 a b x+x^2 \left (b^2+d\right )+c}}{2 x^2}\right )}{3 \sqrt {\pi }}+\frac {2}{3} d \int \frac {e^{d x^2+c} \text {erfi}(a+b x)}{x^2}dx-\frac {e^{c+d x^2} \text {erfi}(a+b x)}{3 x^3}\) |
\(\Big \downarrow \) 2672 |
\(\displaystyle \frac {2 b \left (\left (b^2+d\right ) \int \frac {e^{a^2+2 b x a+\left (b^2+d\right ) x^2+c}}{x}dx+a b \left (2 \left (b^2+d\right ) \int e^{a^2+2 b x a+\left (b^2+d\right ) x^2+c}dx+2 a b \int \frac {e^{a^2+2 b x a+\left (b^2+d\right ) x^2+c}}{x}dx-\frac {e^{a^2+2 a b x+x^2 \left (b^2+d\right )+c}}{x}\right )-\frac {e^{a^2+2 a b x+x^2 \left (b^2+d\right )+c}}{2 x^2}\right )}{3 \sqrt {\pi }}+\frac {2}{3} d \int \frac {e^{d x^2+c} \text {erfi}(a+b x)}{x^2}dx-\frac {e^{c+d x^2} \text {erfi}(a+b x)}{3 x^3}\) |
\(\Big \downarrow \) 2664 |
\(\displaystyle \frac {2 b \left (\left (b^2+d\right ) \int \frac {e^{a^2+2 b x a+\left (b^2+d\right ) x^2+c}}{x}dx+a b \left (2 a b \int \frac {e^{a^2+2 b x a+\left (b^2+d\right ) x^2+c}}{x}dx+2 \left (b^2+d\right ) e^{\frac {a^2 d}{b^2+d}+c} \int e^{\frac {\left (a b+\left (b^2+d\right ) x\right )^2}{b^2+d}}dx-\frac {e^{a^2+2 a b x+x^2 \left (b^2+d\right )+c}}{x}\right )-\frac {e^{a^2+2 a b x+x^2 \left (b^2+d\right )+c}}{2 x^2}\right )}{3 \sqrt {\pi }}+\frac {2}{3} d \int \frac {e^{d x^2+c} \text {erfi}(a+b x)}{x^2}dx-\frac {e^{c+d x^2} \text {erfi}(a+b x)}{3 x^3}\) |
\(\Big \downarrow \) 2633 |
\(\displaystyle \frac {2 b \left (a b \left (2 a b \int \frac {e^{a^2+2 b x a+\left (b^2+d\right ) x^2+c}}{x}dx+\sqrt {\pi } \sqrt {b^2+d} e^{\frac {a^2 d}{b^2+d}+c} \text {erfi}\left (\frac {a b+x \left (b^2+d\right )}{\sqrt {b^2+d}}\right )-\frac {e^{a^2+2 a b x+x^2 \left (b^2+d\right )+c}}{x}\right )+\left (b^2+d\right ) \int \frac {e^{a^2+2 b x a+\left (b^2+d\right ) x^2+c}}{x}dx-\frac {e^{a^2+2 a b x+x^2 \left (b^2+d\right )+c}}{2 x^2}\right )}{3 \sqrt {\pi }}+\frac {2}{3} d \int \frac {e^{d x^2+c} \text {erfi}(a+b x)}{x^2}dx-\frac {e^{c+d x^2} \text {erfi}(a+b x)}{3 x^3}\) |
\(\Big \downarrow \) 2673 |
\(\displaystyle \frac {2 b \left (a b \left (2 a b \int \frac {e^{a^2+2 b x a+\left (b^2+d\right ) x^2+c}}{x}dx+\sqrt {\pi } \sqrt {b^2+d} e^{\frac {a^2 d}{b^2+d}+c} \text {erfi}\left (\frac {a b+x \left (b^2+d\right )}{\sqrt {b^2+d}}\right )-\frac {e^{a^2+2 a b x+x^2 \left (b^2+d\right )+c}}{x}\right )+\left (b^2+d\right ) \int \frac {e^{a^2+2 b x a+\left (b^2+d\right ) x^2+c}}{x}dx-\frac {e^{a^2+2 a b x+x^2 \left (b^2+d\right )+c}}{2 x^2}\right )}{3 \sqrt {\pi }}+\frac {2}{3} d \int \frac {e^{d x^2+c} \text {erfi}(a+b x)}{x^2}dx-\frac {e^{c+d x^2} \text {erfi}(a+b x)}{3 x^3}\) |
\(\Big \downarrow \) 6947 |
\(\displaystyle \frac {2 b \left (a b \left (2 a b \int \frac {e^{a^2+2 b x a+\left (b^2+d\right ) x^2+c}}{x}dx+\sqrt {\pi } \sqrt {b^2+d} e^{\frac {a^2 d}{b^2+d}+c} \text {erfi}\left (\frac {a b+x \left (b^2+d\right )}{\sqrt {b^2+d}}\right )-\frac {e^{a^2+2 a b x+x^2 \left (b^2+d\right )+c}}{x}\right )+\left (b^2+d\right ) \int \frac {e^{a^2+2 b x a+\left (b^2+d\right ) x^2+c}}{x}dx-\frac {e^{a^2+2 a b x+x^2 \left (b^2+d\right )+c}}{2 x^2}\right )}{3 \sqrt {\pi }}+\frac {2}{3} d \left (\frac {2 b \int \frac {e^{a^2+2 b x a+\left (b^2+d\right ) x^2+c}}{x}dx}{\sqrt {\pi }}+2 d \int e^{d x^2+c} \text {erfi}(a+b x)dx-\frac {e^{c+d x^2} \text {erfi}(a+b x)}{x}\right )-\frac {e^{c+d x^2} \text {erfi}(a+b x)}{3 x^3}\) |
\(\Big \downarrow \) 2673 |
\(\displaystyle \frac {2 b \left (a b \left (2 a b \int \frac {e^{a^2+2 b x a+\left (b^2+d\right ) x^2+c}}{x}dx+\sqrt {\pi } \sqrt {b^2+d} e^{\frac {a^2 d}{b^2+d}+c} \text {erfi}\left (\frac {a b+x \left (b^2+d\right )}{\sqrt {b^2+d}}\right )-\frac {e^{a^2+2 a b x+x^2 \left (b^2+d\right )+c}}{x}\right )+\left (b^2+d\right ) \int \frac {e^{a^2+2 b x a+\left (b^2+d\right ) x^2+c}}{x}dx-\frac {e^{a^2+2 a b x+x^2 \left (b^2+d\right )+c}}{2 x^2}\right )}{3 \sqrt {\pi }}+\frac {2}{3} d \left (\frac {2 b \int \frac {e^{a^2+2 b x a+\left (b^2+d\right ) x^2+c}}{x}dx}{\sqrt {\pi }}+2 d \int e^{d x^2+c} \text {erfi}(a+b x)dx-\frac {e^{c+d x^2} \text {erfi}(a+b x)}{x}\right )-\frac {e^{c+d x^2} \text {erfi}(a+b x)}{3 x^3}\) |
\(\Big \downarrow \) 6935 |
\(\displaystyle \frac {2 b \left (a b \left (2 a b \int \frac {e^{a^2+2 b x a+\left (b^2+d\right ) x^2+c}}{x}dx+\sqrt {\pi } \sqrt {b^2+d} e^{\frac {a^2 d}{b^2+d}+c} \text {erfi}\left (\frac {a b+x \left (b^2+d\right )}{\sqrt {b^2+d}}\right )-\frac {e^{a^2+2 a b x+x^2 \left (b^2+d\right )+c}}{x}\right )+\left (b^2+d\right ) \int \frac {e^{a^2+2 b x a+\left (b^2+d\right ) x^2+c}}{x}dx-\frac {e^{a^2+2 a b x+x^2 \left (b^2+d\right )+c}}{2 x^2}\right )}{3 \sqrt {\pi }}+\frac {2}{3} d \left (\frac {2 b \int \frac {e^{a^2+2 b x a+\left (b^2+d\right ) x^2+c}}{x}dx}{\sqrt {\pi }}+2 d \int e^{d x^2+c} \text {erfi}(a+b x)dx-\frac {e^{c+d x^2} \text {erfi}(a+b x)}{x}\right )-\frac {e^{c+d x^2} \text {erfi}(a+b x)}{3 x^3}\) |
3.4.2.3.1 Defintions of rubi rules used
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_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[F^(a - b^2/ (4*c)) Int[F^((b + 2*c*x)^2/(4*c)), x], x] /; FreeQ[{F, a, b, c}, x]
Int[(F_)^((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)*((d_.) + (e_.)*(x_))^(m_), x_S ymbol] :> Simp[(d + e*x)^(m + 1)*(F^(a + b*x + c*x^2)/(e*(m + 1))), x] + (- Simp[2*c*(Log[F]/(e^2*(m + 1))) Int[(d + e*x)^(m + 2)*F^(a + b*x + c*x^2) , x], x] - Simp[(b*e - 2*c*d)*(Log[F]/(e^2*(m + 1))) Int[(d + e*x)^(m + 1 )*F^(a + b*x + c*x^2), x], x]) /; FreeQ[{F, a, b, c, d, e}, x] && NeQ[b*e - 2*c*d, 0] && LtQ[m, -1]
Int[(F_)^((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)*((d_.) + (e_.)*(x_))^(m_.), x_ Symbol] :> Unintegrable[F^(a + b*x + c*x^2)*(d + e*x)^m, x] /; FreeQ[{F, a, b, c, d, e, m}, x]
Int[E^((c_.) + (d_.)*(x_)^2)*Erfi[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> U nintegrable[E^(c + d*x^2)*Erfi[a + b*x]^n, x] /; FreeQ[{a, b, c, d, n}, x]
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]/(m + 1)), x] + (-Simp[2*(d/( m + 1)) Int[x^(m + 2)*E^(c + d*x^2)*Erfi[a + b*x], x], x] - Simp[2*(b/((m + 1)*Sqrt[Pi])) Int[x^(m + 1)*E^(a^2 + c + 2*a*b*x + (b^2 + d)*x^2), x], x]) /; FreeQ[{a, b, c, d}, x] && ILtQ[m, -1]
Not integrable
Time = 0.19 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.95
\[\int \frac {{\mathrm e}^{d \,x^{2}+c} \operatorname {erfi}\left (b x +a \right )}{x^{4}}d x\]
Not integrable
Time = 0.26 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.05 \[ \int \frac {e^{c+d x^2} \text {erfi}(a+b x)}{x^4} \, dx=\int { \frac {\operatorname {erfi}\left (b x + a\right ) e^{\left (d x^{2} + c\right )}}{x^{4}} \,d x } \]
Not integrable
Time = 65.54 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.05 \[ \int \frac {e^{c+d x^2} \text {erfi}(a+b x)}{x^4} \, dx=e^{c} \int \frac {e^{d x^{2}} \operatorname {erfi}{\left (a + b x \right )}}{x^{4}}\, dx \]
Not integrable
Time = 0.28 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.05 \[ \int \frac {e^{c+d x^2} \text {erfi}(a+b x)}{x^4} \, dx=\int { \frac {\operatorname {erfi}\left (b x + a\right ) e^{\left (d x^{2} + c\right )}}{x^{4}} \,d x } \]
Not integrable
Time = 0.25 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.05 \[ \int \frac {e^{c+d x^2} \text {erfi}(a+b x)}{x^4} \, dx=\int { \frac {\operatorname {erfi}\left (b x + a\right ) e^{\left (d x^{2} + c\right )}}{x^{4}} \,d x } \]
Not integrable
Time = 5.86 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.05 \[ \int \frac {e^{c+d x^2} \text {erfi}(a+b x)}{x^4} \, dx=\int \frac {\mathrm {erfi}\left (a+b\,x\right )\,{\mathrm {e}}^{d\,x^2+c}}{x^4} \,d x \]