Integrand size = 19, antiderivative size = 19 \[ \int \frac {e^{c+d x^2} \text {erfc}(a+b x)}{x^4} \, dx=\frac {b e^{-a^2+c-2 a b x-\left (b^2-d\right ) x^2}}{3 \sqrt {\pi } x^2}-\frac {2 a b^2 e^{-a^2+c-2 a b x-\left (b^2-d\right ) x^2}}{3 \sqrt {\pi } x}-\frac {2}{3} a b^2 \sqrt {b^2-d} e^{c+\frac {a^2 d}{b^2-d}} \text {erf}\left (\frac {a b+\left (b^2-d\right ) x}{\sqrt {b^2-d}}\right )-\frac {e^{c+d x^2} \text {erfc}(a+b x)}{3 x^3}-\frac {2 d e^{c+d x^2} \text {erfc}(a+b x)}{3 x}-\frac {4 a^2 b^3 \text {Int}\left (\frac {e^{-a^2+c-2 a b x+\left (-b^2+d\right ) x^2}}{x},x\right )}{3 \sqrt {\pi }}+\frac {2 b \left (b^2-d\right ) \text {Int}\left (\frac {e^{-a^2+c-2 a b x+\left (-b^2+d\right ) x^2}}{x},x\right )}{3 \sqrt {\pi }}-\frac {4 b d \text {Int}\left (\frac {e^{-a^2+c-2 a b x+\left (-b^2+d\right ) x^2}}{x},x\right )}{3 \sqrt {\pi }}+\frac {4}{3} d^2 \text {Int}\left (e^{c+d x^2} \text {erfc}(a+b x),x\right ) \]
-1/3*exp(d*x^2+c)*erfc(b*x+a)/x^3-2/3*d*exp(d*x^2+c)*erfc(b*x+a)/x-2/3*a*b ^2*exp(c+a^2*d/(b^2-d))*erf((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)+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*b*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)*erfc(b*x+a),x)
Not integrable
Time = 0.83 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.11 \[ \int \frac {e^{c+d x^2} \text {erfc}(a+b x)}{x^4} \, dx=\int \frac {e^{c+d x^2} \text {erfc}(a+b x)}{x^4} \, dx \]
Not integrable
Time = 2.46 (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 = {6946, 2672, 2672, 2664, 2634, 2673, 6946, 2673, 6934}
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.
\(\displaystyle \int \frac {e^{c+d x^2} \text {erfc}(a+b x)}{x^4} \, dx\) |
\(\Big \downarrow \) 6946 |
\(\displaystyle -\frac {2 b \int \frac {e^{-a^2-2 b x a-\left (b^2-d\right ) x^2+c}}{x^3}dx}{3 \sqrt {\pi }}+\frac {2}{3} d \int \frac {e^{d x^2+c} \text {erfc}(a+b x)}{x^2}dx-\frac {e^{c+d x^2} \text {erfc}(a+b x)}{3 x^3}\) |
\(\Big \downarrow \) 2672 |
\(\displaystyle -\frac {2 b \left (-a b \int \frac {e^{-a^2-2 b x a-\left (b^2-d\right ) x^2+c}}{x^2}dx-\left (b^2-d\right ) \int \frac {e^{-a^2-2 b x a-\left (b^2-d\right ) x^2+c}}{x}dx-\frac {e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{2 x^2}\right )}{3 \sqrt {\pi }}+\frac {2}{3} d \int \frac {e^{d x^2+c} \text {erfc}(a+b x)}{x^2}dx-\frac {e^{c+d x^2} \text {erfc}(a+b x)}{3 x^3}\) |
\(\Big \downarrow \) 2672 |
\(\displaystyle -\frac {2 b \left (-\left (\left (b^2-d\right ) \int \frac {e^{-a^2-2 b x a-\left (b^2-d\right ) x^2+c}}{x}dx\right )-a b \left (-2 \left (b^2-d\right ) \int e^{-a^2-2 b x a-\left (b^2-d\right ) x^2+c}dx-2 a b \int \frac {e^{-a^2-2 b x a-\left (b^2-d\right ) x^2+c}}{x}dx-\frac {e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{x}\right )-\frac {e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{2 x^2}\right )}{3 \sqrt {\pi }}+\frac {2}{3} d \int \frac {e^{d x^2+c} \text {erfc}(a+b x)}{x^2}dx-\frac {e^{c+d x^2} \text {erfc}(a+b x)}{3 x^3}\) |
\(\Big \downarrow \) 2664 |
\(\displaystyle -\frac {2 b \left (-\left (\left (b^2-d\right ) \int \frac {e^{-a^2-2 b x a-\left (b^2-d\right ) x^2+c}}{x}dx\right )-a b \left (-2 a b \int \frac {e^{-a^2-2 b x a-\left (b^2-d\right ) x^2+c}}{x}dx-2 \left (b^2-d\right ) e^{\frac {a^2 d+b^2 c-c d}{b^2-d}} \int e^{-\frac {\left (a b+\left (b^2-d\right ) x\right )^2}{b^2-d}}dx-\frac {e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{x}\right )-\frac {e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{2 x^2}\right )}{3 \sqrt {\pi }}+\frac {2}{3} d \int \frac {e^{d x^2+c} \text {erfc}(a+b x)}{x^2}dx-\frac {e^{c+d x^2} \text {erfc}(a+b x)}{3 x^3}\) |
\(\Big \downarrow \) 2634 |
\(\displaystyle -\frac {2 b \left (-a b \left (-2 a b \int \frac {e^{-a^2-2 b x a-\left (b^2-d\right ) x^2+c}}{x}dx+\sqrt {\pi } \left (-\sqrt {b^2-d}\right ) e^{\frac {a^2 d+b^2 c-c d}{b^2-d}} \text {erf}\left (\frac {a b+x \left (b^2-d\right )}{\sqrt {b^2-d}}\right )-\frac {e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{x}\right )-\left (\left (b^2-d\right ) \int \frac {e^{-a^2-2 b x a-\left (b^2-d\right ) x^2+c}}{x}dx\right )-\frac {e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{2 x^2}\right )}{3 \sqrt {\pi }}+\frac {2}{3} d \int \frac {e^{d x^2+c} \text {erfc}(a+b x)}{x^2}dx-\frac {e^{c+d x^2} \text {erfc}(a+b x)}{3 x^3}\) |
\(\Big \downarrow \) 2673 |
\(\displaystyle -\frac {2 b \left (-a b \left (-2 a b \int \frac {e^{-a^2-2 b x a+\left (d-b^2\right ) x^2+c}}{x}dx+\sqrt {\pi } \left (-\sqrt {b^2-d}\right ) e^{\frac {a^2 d+b^2 c-c d}{b^2-d}} \text {erf}\left (\frac {a b+x \left (b^2-d\right )}{\sqrt {b^2-d}}\right )-\frac {e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{x}\right )-\left (\left (b^2-d\right ) \int \frac {e^{-a^2-2 b x a+\left (d-b^2\right ) x^2+c}}{x}dx\right )-\frac {e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{2 x^2}\right )}{3 \sqrt {\pi }}+\frac {2}{3} d \int \frac {e^{d x^2+c} \text {erfc}(a+b x)}{x^2}dx-\frac {e^{c+d x^2} \text {erfc}(a+b x)}{3 x^3}\) |
\(\Big \downarrow \) 6946 |
\(\displaystyle -\frac {2 b \left (-a b \left (-2 a b \int \frac {e^{-a^2-2 b x a+\left (d-b^2\right ) x^2+c}}{x}dx+\sqrt {\pi } \left (-\sqrt {b^2-d}\right ) e^{\frac {a^2 d+b^2 c-c d}{b^2-d}} \text {erf}\left (\frac {a b+x \left (b^2-d\right )}{\sqrt {b^2-d}}\right )-\frac {e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{x}\right )-\left (\left (b^2-d\right ) \int \frac {e^{-a^2-2 b x a+\left (d-b^2\right ) x^2+c}}{x}dx\right )-\frac {e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{2 x^2}\right )}{3 \sqrt {\pi }}+\frac {2}{3} d \left (-\frac {2 b \int \frac {e^{-a^2-2 b x a-\left (b^2-d\right ) x^2+c}}{x}dx}{\sqrt {\pi }}+2 d \int e^{d x^2+c} \text {erfc}(a+b x)dx-\frac {e^{c+d x^2} \text {erfc}(a+b x)}{x}\right )-\frac {e^{c+d x^2} \text {erfc}(a+b x)}{3 x^3}\) |
\(\Big \downarrow \) 2673 |
\(\displaystyle -\frac {2 b \left (-a b \left (-2 a b \int \frac {e^{-a^2-2 b x a+\left (d-b^2\right ) x^2+c}}{x}dx+\sqrt {\pi } \left (-\sqrt {b^2-d}\right ) e^{\frac {a^2 d+b^2 c-c d}{b^2-d}} \text {erf}\left (\frac {a b+x \left (b^2-d\right )}{\sqrt {b^2-d}}\right )-\frac {e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{x}\right )-\left (\left (b^2-d\right ) \int \frac {e^{-a^2-2 b x a+\left (d-b^2\right ) x^2+c}}{x}dx\right )-\frac {e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{2 x^2}\right )}{3 \sqrt {\pi }}+\frac {2}{3} d \left (-\frac {2 b \int \frac {e^{-a^2-2 b x a+\left (d-b^2\right ) x^2+c}}{x}dx}{\sqrt {\pi }}+2 d \int e^{d x^2+c} \text {erfc}(a+b x)dx-\frac {e^{c+d x^2} \text {erfc}(a+b x)}{x}\right )-\frac {e^{c+d x^2} \text {erfc}(a+b x)}{3 x^3}\) |
\(\Big \downarrow \) 6934 |
\(\displaystyle -\frac {2 b \left (-a b \left (-2 a b \int \frac {e^{-a^2-2 b x a+\left (d-b^2\right ) x^2+c}}{x}dx+\sqrt {\pi } \left (-\sqrt {b^2-d}\right ) e^{\frac {a^2 d+b^2 c-c d}{b^2-d}} \text {erf}\left (\frac {a b+x \left (b^2-d\right )}{\sqrt {b^2-d}}\right )-\frac {e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{x}\right )-\left (\left (b^2-d\right ) \int \frac {e^{-a^2-2 b x a+\left (d-b^2\right ) x^2+c}}{x}dx\right )-\frac {e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{2 x^2}\right )}{3 \sqrt {\pi }}+\frac {2}{3} d \left (-\frac {2 b \int \frac {e^{-a^2-2 b x a+\left (d-b^2\right ) x^2+c}}{x}dx}{\sqrt {\pi }}+2 d \int e^{d x^2+c} \text {erfc}(a+b x)dx-\frac {e^{c+d x^2} \text {erfc}(a+b x)}{x}\right )-\frac {e^{c+d x^2} \text {erfc}(a+b x)}{3 x^3}\) |
3.2.97.3.1 Defintions of rubi rules used
Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[F^a*Sqrt [Pi]*(Erf[(c + d*x)*Rt[(-b)*Log[F], 2]]/(2*d*Rt[(-b)*Log[F], 2])), x] /; Fr eeQ[{F, a, b, c, d}, x] && NegQ[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)*Erfc[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> U nintegrable[E^(c + d*x^2)*Erfc[a + b*x]^n, x] /; FreeQ[{a, b, c, d, n}, x]
Int[E^((c_.) + (d_.)*(x_)^2)*Erfc[(a_.) + (b_.)*(x_)]*(x_)^(m_), x_Symbol] :> Simp[x^(m + 1)*E^(c + d*x^2)*(Erfc[a + b*x]/(m + 1)), x] + (-Simp[2*(d/( m + 1)) Int[x^(m + 2)*E^(c + d*x^2)*Erfc[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 = 0.20 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.95
\[\int \frac {{\mathrm e}^{d \,x^{2}+c} \operatorname {erfc}\left (b x +a \right )}{x^{4}}d x\]
Not integrable
Time = 0.25 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.21 \[ \int \frac {e^{c+d x^2} \text {erfc}(a+b x)}{x^4} \, dx=\int { \frac {\operatorname {erfc}\left (b x + a\right ) e^{\left (d x^{2} + c\right )}}{x^{4}} \,d x } \]
Not integrable
Time = 87.83 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.05 \[ \int \frac {e^{c+d x^2} \text {erfc}(a+b x)}{x^4} \, dx=e^{c} \int \frac {e^{d x^{2}} \operatorname {erfc}{\left (a + b x \right )}}{x^{4}}\, dx \]
Not integrable
Time = 0.25 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.05 \[ \int \frac {e^{c+d x^2} \text {erfc}(a+b x)}{x^4} \, dx=\int { \frac {\operatorname {erfc}\left (b x + a\right ) e^{\left (d x^{2} + c\right )}}{x^{4}} \,d x } \]
Not integrable
Time = 0.27 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.05 \[ \int \frac {e^{c+d x^2} \text {erfc}(a+b x)}{x^4} \, dx=\int { \frac {\operatorname {erfc}\left (b x + a\right ) e^{\left (d x^{2} + c\right )}}{x^{4}} \,d x } \]
Not integrable
Time = 4.81 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.05 \[ \int \frac {e^{c+d x^2} \text {erfc}(a+b x)}{x^4} \, dx=\int \frac {\mathrm {erfc}\left (a+b\,x\right )\,{\mathrm {e}}^{d\,x^2+c}}{x^4} \,d x \]