Integrand size = 19, antiderivative size = 19 \[ \int e^{c+d x^2} x^4 \text {erfc}(a+b x) \, dx=\frac {3 b e^{-a^2+c-2 a b x-\left (b^2-d\right ) x^2}}{4 \left (b^2-d\right ) d^2 \sqrt {\pi }}-\frac {a^2 b^3 e^{-a^2+c-2 a b x-\left (b^2-d\right ) x^2}}{2 \left (b^2-d\right )^3 d \sqrt {\pi }}-\frac {b e^{-a^2+c-2 a b x-\left (b^2-d\right ) x^2}}{2 \left (b^2-d\right )^2 d \sqrt {\pi }}+\frac {a b^2 e^{-a^2+c-2 a b x-\left (b^2-d\right ) x^2} x}{2 \left (b^2-d\right )^2 d \sqrt {\pi }}-\frac {b e^{-a^2+c-2 a b x-\left (b^2-d\right ) x^2} x^2}{2 \left (b^2-d\right ) d \sqrt {\pi }}+\frac {3 a b^2 e^{c+\frac {a^2 d}{b^2-d}} \text {erf}\left (\frac {a b+\left (b^2-d\right ) x}{\sqrt {b^2-d}}\right )}{4 \left (b^2-d\right )^{3/2} d^2}-\frac {a^3 b^4 e^{c+\frac {a^2 d}{b^2-d}} \text {erf}\left (\frac {a b+\left (b^2-d\right ) x}{\sqrt {b^2-d}}\right )}{2 \left (b^2-d\right )^{7/2} d}-\frac {3 a b^2 e^{c+\frac {a^2 d}{b^2-d}} \text {erf}\left (\frac {a b+\left (b^2-d\right ) x}{\sqrt {b^2-d}}\right )}{4 \left (b^2-d\right )^{5/2} d}-\frac {3 e^{c+d x^2} x \text {erfc}(a+b x)}{4 d^2}+\frac {e^{c+d x^2} x^3 \text {erfc}(a+b x)}{2 d}+\frac {3 \text {Int}\left (e^{c+d x^2} \text {erfc}(a+b x),x\right )}{4 d^2} \]
3/4*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)^ (3/2)/d^2-1/2*a^3*b^4*exp(c+a^2*d/(b^2-d))*erf((a*b+(b^2-d)*x)/(b^2-d)^(1/ 2))/(b^2-d)^(7/2)/d-3/4*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)^(5/2)/d-3/4*exp(d*x^2+c)*x*erfc(b*x+a)/d^2+1/2*exp(d*x ^2+c)*x^3*erfc(b*x+a)/d+3/4*b*exp(-a^2+c-2*a*b*x-(b^2-d)*x^2)/(b^2-d)/d^2/ Pi^(1/2)-1/2*a^2*b^3*exp(-a^2+c-2*a*b*x-(b^2-d)*x^2)/(b^2-d)^3/d/Pi^(1/2)- 1/2*b*exp(-a^2+c-2*a*b*x-(b^2-d)*x^2)/(b^2-d)^2/d/Pi^(1/2)+1/2*a*b^2*exp(- a^2+c-2*a*b*x-(b^2-d)*x^2)*x/(b^2-d)^2/d/Pi^(1/2)-1/2*b*exp(-a^2+c-2*a*b*x -(b^2-d)*x^2)*x^2/(b^2-d)/d/Pi^(1/2)+3/4*Unintegrable(exp(d*x^2+c)*erfc(b* x+a),x)/d^2
Not integrable
Time = 0.79 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.11 \[ \int e^{c+d x^2} x^4 \text {erfc}(a+b x) \, dx=\int e^{c+d x^2} x^4 \text {erfc}(a+b x) \, dx \]
Not integrable
Time = 3.69 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {6940, 2671, 2670, 2664, 2634, 2671, 2664, 2634, 2670, 2664, 2634, 6940, 2670, 2664, 2634, 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.
\(\displaystyle \int x^4 e^{c+d x^2} \text {erfc}(a+b x) \, dx\) |
\(\Big \downarrow \) 6940 |
\(\displaystyle \frac {b \int e^{-a^2-2 b x a-\left (b^2-d\right ) x^2+c} x^3dx}{\sqrt {\pi } d}-\frac {3 \int e^{d x^2+c} x^2 \text {erfc}(a+b x)dx}{2 d}+\frac {x^3 e^{c+d x^2} \text {erfc}(a+b x)}{2 d}\) |
\(\Big \downarrow \) 2671 |
\(\displaystyle \frac {b \left (\frac {\int e^{-a^2-2 b x a-\left (b^2-d\right ) x^2+c} xdx}{b^2-d}-\frac {a b \int e^{-a^2-2 b x a-\left (b^2-d\right ) x^2+c} x^2dx}{b^2-d}-\frac {x^2 e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{2 \left (b^2-d\right )}\right )}{\sqrt {\pi } d}-\frac {3 \int e^{d x^2+c} x^2 \text {erfc}(a+b x)dx}{2 d}+\frac {x^3 e^{c+d x^2} \text {erfc}(a+b x)}{2 d}\) |
\(\Big \downarrow \) 2670 |
\(\displaystyle \frac {b \left (\frac {-\frac {a b \int e^{-a^2-2 b x a-\left (b^2-d\right ) x^2+c}dx}{b^2-d}-\frac {e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{2 \left (b^2-d\right )}}{b^2-d}-\frac {a b \int e^{-a^2-2 b x a-\left (b^2-d\right ) x^2+c} x^2dx}{b^2-d}-\frac {x^2 e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{2 \left (b^2-d\right )}\right )}{\sqrt {\pi } d}-\frac {3 \int e^{d x^2+c} x^2 \text {erfc}(a+b x)dx}{2 d}+\frac {x^3 e^{c+d x^2} \text {erfc}(a+b x)}{2 d}\) |
\(\Big \downarrow \) 2664 |
\(\displaystyle \frac {b \left (\frac {-\frac {a b e^{\frac {a^2 d+b^2 c-c d}{b^2-d}} \int e^{-\frac {\left (a b+\left (b^2-d\right ) x\right )^2}{b^2-d}}dx}{b^2-d}-\frac {e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{2 \left (b^2-d\right )}}{b^2-d}-\frac {a b \int e^{-a^2-2 b x a-\left (b^2-d\right ) x^2+c} x^2dx}{b^2-d}-\frac {x^2 e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{2 \left (b^2-d\right )}\right )}{\sqrt {\pi } d}-\frac {3 \int e^{d x^2+c} x^2 \text {erfc}(a+b x)dx}{2 d}+\frac {x^3 e^{c+d x^2} \text {erfc}(a+b x)}{2 d}\) |
\(\Big \downarrow \) 2634 |
\(\displaystyle \frac {b \left (-\frac {a b \int e^{-a^2-2 b x a-\left (b^2-d\right ) x^2+c} x^2dx}{b^2-d}+\frac {-\frac {\sqrt {\pi } a b e^{\frac {a^2 d+b^2 c-c d}{b^2-d}} \text {erf}\left (\frac {a b+x \left (b^2-d\right )}{\sqrt {b^2-d}}\right )}{2 \left (b^2-d\right )^{3/2}}-\frac {e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{2 \left (b^2-d\right )}}{b^2-d}-\frac {x^2 e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{2 \left (b^2-d\right )}\right )}{\sqrt {\pi } d}-\frac {3 \int e^{d x^2+c} x^2 \text {erfc}(a+b x)dx}{2 d}+\frac {x^3 e^{c+d x^2} \text {erfc}(a+b x)}{2 d}\) |
\(\Big \downarrow \) 2671 |
\(\displaystyle \frac {b \left (-\frac {a b \left (\frac {\int e^{-a^2-2 b x a-\left (b^2-d\right ) x^2+c}dx}{2 \left (b^2-d\right )}-\frac {a b \int e^{-a^2-2 b x a-\left (b^2-d\right ) x^2+c} xdx}{b^2-d}-\frac {x e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{2 \left (b^2-d\right )}\right )}{b^2-d}+\frac {-\frac {\sqrt {\pi } a b e^{\frac {a^2 d+b^2 c-c d}{b^2-d}} \text {erf}\left (\frac {a b+x \left (b^2-d\right )}{\sqrt {b^2-d}}\right )}{2 \left (b^2-d\right )^{3/2}}-\frac {e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{2 \left (b^2-d\right )}}{b^2-d}-\frac {x^2 e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{2 \left (b^2-d\right )}\right )}{\sqrt {\pi } d}-\frac {3 \int e^{d x^2+c} x^2 \text {erfc}(a+b x)dx}{2 d}+\frac {x^3 e^{c+d x^2} \text {erfc}(a+b x)}{2 d}\) |
\(\Big \downarrow \) 2664 |
\(\displaystyle \frac {b \left (-\frac {a b \left (-\frac {a b \int e^{-a^2-2 b x a-\left (b^2-d\right ) x^2+c} xdx}{b^2-d}+\frac {e^{\frac {a^2 d+b^2 c-c d}{b^2-d}} \int e^{-\frac {\left (a b+\left (b^2-d\right ) x\right )^2}{b^2-d}}dx}{2 \left (b^2-d\right )}-\frac {x e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{2 \left (b^2-d\right )}\right )}{b^2-d}+\frac {-\frac {\sqrt {\pi } a b e^{\frac {a^2 d+b^2 c-c d}{b^2-d}} \text {erf}\left (\frac {a b+x \left (b^2-d\right )}{\sqrt {b^2-d}}\right )}{2 \left (b^2-d\right )^{3/2}}-\frac {e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{2 \left (b^2-d\right )}}{b^2-d}-\frac {x^2 e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{2 \left (b^2-d\right )}\right )}{\sqrt {\pi } d}-\frac {3 \int e^{d x^2+c} x^2 \text {erfc}(a+b x)dx}{2 d}+\frac {x^3 e^{c+d x^2} \text {erfc}(a+b x)}{2 d}\) |
\(\Big \downarrow \) 2634 |
\(\displaystyle \frac {b \left (-\frac {a b \left (-\frac {a b \int e^{-a^2-2 b x a-\left (b^2-d\right ) x^2+c} xdx}{b^2-d}+\frac {\sqrt {\pi } e^{\frac {a^2 d+b^2 c-c d}{b^2-d}} \text {erf}\left (\frac {a b+x \left (b^2-d\right )}{\sqrt {b^2-d}}\right )}{4 \left (b^2-d\right )^{3/2}}-\frac {x e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{2 \left (b^2-d\right )}\right )}{b^2-d}+\frac {-\frac {\sqrt {\pi } a b e^{\frac {a^2 d+b^2 c-c d}{b^2-d}} \text {erf}\left (\frac {a b+x \left (b^2-d\right )}{\sqrt {b^2-d}}\right )}{2 \left (b^2-d\right )^{3/2}}-\frac {e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{2 \left (b^2-d\right )}}{b^2-d}-\frac {x^2 e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{2 \left (b^2-d\right )}\right )}{\sqrt {\pi } d}-\frac {3 \int e^{d x^2+c} x^2 \text {erfc}(a+b x)dx}{2 d}+\frac {x^3 e^{c+d x^2} \text {erfc}(a+b x)}{2 d}\) |
\(\Big \downarrow \) 2670 |
\(\displaystyle \frac {b \left (-\frac {a b \left (-\frac {a b \left (-\frac {a b \int e^{-a^2-2 b x a-\left (b^2-d\right ) x^2+c}dx}{b^2-d}-\frac {e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{2 \left (b^2-d\right )}\right )}{b^2-d}+\frac {\sqrt {\pi } e^{\frac {a^2 d+b^2 c-c d}{b^2-d}} \text {erf}\left (\frac {a b+x \left (b^2-d\right )}{\sqrt {b^2-d}}\right )}{4 \left (b^2-d\right )^{3/2}}-\frac {x e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{2 \left (b^2-d\right )}\right )}{b^2-d}+\frac {-\frac {\sqrt {\pi } a b e^{\frac {a^2 d+b^2 c-c d}{b^2-d}} \text {erf}\left (\frac {a b+x \left (b^2-d\right )}{\sqrt {b^2-d}}\right )}{2 \left (b^2-d\right )^{3/2}}-\frac {e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{2 \left (b^2-d\right )}}{b^2-d}-\frac {x^2 e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{2 \left (b^2-d\right )}\right )}{\sqrt {\pi } d}-\frac {3 \int e^{d x^2+c} x^2 \text {erfc}(a+b x)dx}{2 d}+\frac {x^3 e^{c+d x^2} \text {erfc}(a+b x)}{2 d}\) |
\(\Big \downarrow \) 2664 |
\(\displaystyle \frac {b \left (-\frac {a b \left (-\frac {a b \left (-\frac {a b e^{\frac {a^2 d+b^2 c-c d}{b^2-d}} \int e^{-\frac {\left (a b+\left (b^2-d\right ) x\right )^2}{b^2-d}}dx}{b^2-d}-\frac {e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{2 \left (b^2-d\right )}\right )}{b^2-d}+\frac {\sqrt {\pi } e^{\frac {a^2 d+b^2 c-c d}{b^2-d}} \text {erf}\left (\frac {a b+x \left (b^2-d\right )}{\sqrt {b^2-d}}\right )}{4 \left (b^2-d\right )^{3/2}}-\frac {x e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{2 \left (b^2-d\right )}\right )}{b^2-d}+\frac {-\frac {\sqrt {\pi } a b e^{\frac {a^2 d+b^2 c-c d}{b^2-d}} \text {erf}\left (\frac {a b+x \left (b^2-d\right )}{\sqrt {b^2-d}}\right )}{2 \left (b^2-d\right )^{3/2}}-\frac {e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{2 \left (b^2-d\right )}}{b^2-d}-\frac {x^2 e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{2 \left (b^2-d\right )}\right )}{\sqrt {\pi } d}-\frac {3 \int e^{d x^2+c} x^2 \text {erfc}(a+b x)dx}{2 d}+\frac {x^3 e^{c+d x^2} \text {erfc}(a+b x)}{2 d}\) |
\(\Big \downarrow \) 2634 |
\(\displaystyle -\frac {3 \int e^{d x^2+c} x^2 \text {erfc}(a+b x)dx}{2 d}+\frac {b \left (\frac {-\frac {\sqrt {\pi } a b e^{\frac {a^2 d+b^2 c-c d}{b^2-d}} \text {erf}\left (\frac {a b+x \left (b^2-d\right )}{\sqrt {b^2-d}}\right )}{2 \left (b^2-d\right )^{3/2}}-\frac {e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{2 \left (b^2-d\right )}}{b^2-d}-\frac {a b \left (-\frac {a b \left (-\frac {\sqrt {\pi } a b e^{\frac {a^2 d+b^2 c-c d}{b^2-d}} \text {erf}\left (\frac {a b+x \left (b^2-d\right )}{\sqrt {b^2-d}}\right )}{2 \left (b^2-d\right )^{3/2}}-\frac {e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{2 \left (b^2-d\right )}\right )}{b^2-d}+\frac {\sqrt {\pi } e^{\frac {a^2 d+b^2 c-c d}{b^2-d}} \text {erf}\left (\frac {a b+x \left (b^2-d\right )}{\sqrt {b^2-d}}\right )}{4 \left (b^2-d\right )^{3/2}}-\frac {x e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{2 \left (b^2-d\right )}\right )}{b^2-d}-\frac {x^2 e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{2 \left (b^2-d\right )}\right )}{\sqrt {\pi } d}+\frac {x^3 e^{c+d x^2} \text {erfc}(a+b x)}{2 d}\) |
\(\Big \downarrow \) 6940 |
\(\displaystyle -\frac {3 \left (\frac {b \int e^{-a^2-2 b x a-\left (b^2-d\right ) x^2+c} xdx}{\sqrt {\pi } d}-\frac {\int e^{d x^2+c} \text {erfc}(a+b x)dx}{2 d}+\frac {x e^{c+d x^2} \text {erfc}(a+b x)}{2 d}\right )}{2 d}+\frac {b \left (\frac {-\frac {\sqrt {\pi } a b e^{\frac {a^2 d+b^2 c-c d}{b^2-d}} \text {erf}\left (\frac {a b+x \left (b^2-d\right )}{\sqrt {b^2-d}}\right )}{2 \left (b^2-d\right )^{3/2}}-\frac {e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{2 \left (b^2-d\right )}}{b^2-d}-\frac {a b \left (-\frac {a b \left (-\frac {\sqrt {\pi } a b e^{\frac {a^2 d+b^2 c-c d}{b^2-d}} \text {erf}\left (\frac {a b+x \left (b^2-d\right )}{\sqrt {b^2-d}}\right )}{2 \left (b^2-d\right )^{3/2}}-\frac {e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{2 \left (b^2-d\right )}\right )}{b^2-d}+\frac {\sqrt {\pi } e^{\frac {a^2 d+b^2 c-c d}{b^2-d}} \text {erf}\left (\frac {a b+x \left (b^2-d\right )}{\sqrt {b^2-d}}\right )}{4 \left (b^2-d\right )^{3/2}}-\frac {x e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{2 \left (b^2-d\right )}\right )}{b^2-d}-\frac {x^2 e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{2 \left (b^2-d\right )}\right )}{\sqrt {\pi } d}+\frac {x^3 e^{c+d x^2} \text {erfc}(a+b x)}{2 d}\) |
\(\Big \downarrow \) 2670 |
\(\displaystyle -\frac {3 \left (\frac {b \left (-\frac {a b \int e^{-a^2-2 b x a-\left (b^2-d\right ) x^2+c}dx}{b^2-d}-\frac {e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{2 \left (b^2-d\right )}\right )}{\sqrt {\pi } d}-\frac {\int e^{d x^2+c} \text {erfc}(a+b x)dx}{2 d}+\frac {x e^{c+d x^2} \text {erfc}(a+b x)}{2 d}\right )}{2 d}+\frac {b \left (\frac {-\frac {\sqrt {\pi } a b e^{\frac {a^2 d+b^2 c-c d}{b^2-d}} \text {erf}\left (\frac {a b+x \left (b^2-d\right )}{\sqrt {b^2-d}}\right )}{2 \left (b^2-d\right )^{3/2}}-\frac {e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{2 \left (b^2-d\right )}}{b^2-d}-\frac {a b \left (-\frac {a b \left (-\frac {\sqrt {\pi } a b e^{\frac {a^2 d+b^2 c-c d}{b^2-d}} \text {erf}\left (\frac {a b+x \left (b^2-d\right )}{\sqrt {b^2-d}}\right )}{2 \left (b^2-d\right )^{3/2}}-\frac {e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{2 \left (b^2-d\right )}\right )}{b^2-d}+\frac {\sqrt {\pi } e^{\frac {a^2 d+b^2 c-c d}{b^2-d}} \text {erf}\left (\frac {a b+x \left (b^2-d\right )}{\sqrt {b^2-d}}\right )}{4 \left (b^2-d\right )^{3/2}}-\frac {x e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{2 \left (b^2-d\right )}\right )}{b^2-d}-\frac {x^2 e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{2 \left (b^2-d\right )}\right )}{\sqrt {\pi } d}+\frac {x^3 e^{c+d x^2} \text {erfc}(a+b x)}{2 d}\) |
\(\Big \downarrow \) 2664 |
\(\displaystyle -\frac {3 \left (\frac {b \left (-\frac {a b e^{\frac {a^2 d+b^2 c-c d}{b^2-d}} \int e^{-\frac {\left (a b+\left (b^2-d\right ) x\right )^2}{b^2-d}}dx}{b^2-d}-\frac {e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{2 \left (b^2-d\right )}\right )}{\sqrt {\pi } d}-\frac {\int e^{d x^2+c} \text {erfc}(a+b x)dx}{2 d}+\frac {x e^{c+d x^2} \text {erfc}(a+b x)}{2 d}\right )}{2 d}+\frac {b \left (\frac {-\frac {\sqrt {\pi } a b e^{\frac {a^2 d+b^2 c-c d}{b^2-d}} \text {erf}\left (\frac {a b+x \left (b^2-d\right )}{\sqrt {b^2-d}}\right )}{2 \left (b^2-d\right )^{3/2}}-\frac {e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{2 \left (b^2-d\right )}}{b^2-d}-\frac {a b \left (-\frac {a b \left (-\frac {\sqrt {\pi } a b e^{\frac {a^2 d+b^2 c-c d}{b^2-d}} \text {erf}\left (\frac {a b+x \left (b^2-d\right )}{\sqrt {b^2-d}}\right )}{2 \left (b^2-d\right )^{3/2}}-\frac {e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{2 \left (b^2-d\right )}\right )}{b^2-d}+\frac {\sqrt {\pi } e^{\frac {a^2 d+b^2 c-c d}{b^2-d}} \text {erf}\left (\frac {a b+x \left (b^2-d\right )}{\sqrt {b^2-d}}\right )}{4 \left (b^2-d\right )^{3/2}}-\frac {x e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{2 \left (b^2-d\right )}\right )}{b^2-d}-\frac {x^2 e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{2 \left (b^2-d\right )}\right )}{\sqrt {\pi } d}+\frac {x^3 e^{c+d x^2} \text {erfc}(a+b x)}{2 d}\) |
\(\Big \downarrow \) 2634 |
\(\displaystyle -\frac {3 \left (-\frac {\int e^{d x^2+c} \text {erfc}(a+b x)dx}{2 d}+\frac {b \left (-\frac {\sqrt {\pi } a b e^{\frac {a^2 d+b^2 c-c d}{b^2-d}} \text {erf}\left (\frac {a b+x \left (b^2-d\right )}{\sqrt {b^2-d}}\right )}{2 \left (b^2-d\right )^{3/2}}-\frac {e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{2 \left (b^2-d\right )}\right )}{\sqrt {\pi } d}+\frac {x e^{c+d x^2} \text {erfc}(a+b x)}{2 d}\right )}{2 d}+\frac {b \left (\frac {-\frac {\sqrt {\pi } a b e^{\frac {a^2 d+b^2 c-c d}{b^2-d}} \text {erf}\left (\frac {a b+x \left (b^2-d\right )}{\sqrt {b^2-d}}\right )}{2 \left (b^2-d\right )^{3/2}}-\frac {e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{2 \left (b^2-d\right )}}{b^2-d}-\frac {a b \left (-\frac {a b \left (-\frac {\sqrt {\pi } a b e^{\frac {a^2 d+b^2 c-c d}{b^2-d}} \text {erf}\left (\frac {a b+x \left (b^2-d\right )}{\sqrt {b^2-d}}\right )}{2 \left (b^2-d\right )^{3/2}}-\frac {e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{2 \left (b^2-d\right )}\right )}{b^2-d}+\frac {\sqrt {\pi } e^{\frac {a^2 d+b^2 c-c d}{b^2-d}} \text {erf}\left (\frac {a b+x \left (b^2-d\right )}{\sqrt {b^2-d}}\right )}{4 \left (b^2-d\right )^{3/2}}-\frac {x e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{2 \left (b^2-d\right )}\right )}{b^2-d}-\frac {x^2 e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{2 \left (b^2-d\right )}\right )}{\sqrt {\pi } d}+\frac {x^3 e^{c+d x^2} \text {erfc}(a+b x)}{2 d}\) |
\(\Big \downarrow \) 6934 |
\(\displaystyle -\frac {3 \left (-\frac {\int e^{d x^2+c} \text {erfc}(a+b x)dx}{2 d}+\frac {b \left (-\frac {\sqrt {\pi } a b e^{\frac {a^2 d+b^2 c-c d}{b^2-d}} \text {erf}\left (\frac {a b+x \left (b^2-d\right )}{\sqrt {b^2-d}}\right )}{2 \left (b^2-d\right )^{3/2}}-\frac {e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{2 \left (b^2-d\right )}\right )}{\sqrt {\pi } d}+\frac {x e^{c+d x^2} \text {erfc}(a+b x)}{2 d}\right )}{2 d}+\frac {b \left (\frac {-\frac {\sqrt {\pi } a b e^{\frac {a^2 d+b^2 c-c d}{b^2-d}} \text {erf}\left (\frac {a b+x \left (b^2-d\right )}{\sqrt {b^2-d}}\right )}{2 \left (b^2-d\right )^{3/2}}-\frac {e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{2 \left (b^2-d\right )}}{b^2-d}-\frac {a b \left (-\frac {a b \left (-\frac {\sqrt {\pi } a b e^{\frac {a^2 d+b^2 c-c d}{b^2-d}} \text {erf}\left (\frac {a b+x \left (b^2-d\right )}{\sqrt {b^2-d}}\right )}{2 \left (b^2-d\right )^{3/2}}-\frac {e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{2 \left (b^2-d\right )}\right )}{b^2-d}+\frac {\sqrt {\pi } e^{\frac {a^2 d+b^2 c-c d}{b^2-d}} \text {erf}\left (\frac {a b+x \left (b^2-d\right )}{\sqrt {b^2-d}}\right )}{4 \left (b^2-d\right )^{3/2}}-\frac {x e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{2 \left (b^2-d\right )}\right )}{b^2-d}-\frac {x^2 e^{-a^2-2 a b x-x^2 \left (b^2-d\right )+c}}{2 \left (b^2-d\right )}\right )}{\sqrt {\pi } d}+\frac {x^3 e^{c+d x^2} \text {erfc}(a+b x)}{2 d}\) |
3.2.93.3.1 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_)), x_Symbol ] :> Simp[e*(F^(a + b*x + c*x^2)/(2*c*Log[F])), x] - Simp[(b*e - 2*c*d)/(2* c) Int[F^(a + b*x + c*x^2), x], x] /; FreeQ[{F, a, b, c, d, e}, x] && NeQ [b*e - 2*c*d, 0]
Int[(F_)^((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)*((d_.) + (e_.)*(x_))^(m_), x_S ymbol] :> Simp[e*(d + e*x)^(m - 1)*(F^(a + b*x + c*x^2)/(2*c*Log[F])), x] + (-Simp[(b*e - 2*c*d)/(2*c) Int[(d + e*x)^(m - 1)*F^(a + b*x + c*x^2), x] , x] - Simp[(m - 1)*(e^2/(2*c*Log[F])) Int[(d + e*x)^(m - 2)*F^(a + b*x + c*x^2), x], x]) /; FreeQ[{F, a, b, c, d, e}, x] && NeQ[b*e - 2*c*d, 0] && GtQ[m, 1]
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]/(2*d)), x] + (-Simp[(m - 1)/ (2*d) Int[x^(m - 2)*E^(c + d*x^2)*Erfc[a + b*x], x], x] + Simp[b/(d*Sqrt[ Pi]) Int[x^(m - 1)*E^(-a^2 + c - 2*a*b*x - (b^2 - d)*x^2), x], x]) /; Fre eQ[{a, b, c, d}, x] && IGtQ[m, 1]
Not integrable
Time = 0.08 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.95
\[\int {\mathrm e}^{d \,x^{2}+c} x^{4} \operatorname {erfc}\left (b x +a \right )d x\]
Not integrable
Time = 0.25 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.47 \[ \int e^{c+d x^2} x^4 \text {erfc}(a+b x) \, dx=\int { x^{4} \operatorname {erfc}\left (b x + a\right ) e^{\left (d x^{2} + c\right )} \,d x } \]
Timed out. \[ \int e^{c+d x^2} x^4 \text {erfc}(a+b x) \, dx=\text {Timed out} \]
Not integrable
Time = 0.25 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.05 \[ \int e^{c+d x^2} x^4 \text {erfc}(a+b x) \, dx=\int { x^{4} \operatorname {erfc}\left (b x + a\right ) e^{\left (d x^{2} + c\right )} \,d x } \]
Not integrable
Time = 0.28 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.05 \[ \int e^{c+d x^2} x^4 \text {erfc}(a+b x) \, dx=\int { x^{4} \operatorname {erfc}\left (b x + a\right ) e^{\left (d x^{2} + c\right )} \,d x } \]
Not integrable
Time = 5.16 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.05 \[ \int e^{c+d x^2} x^4 \text {erfc}(a+b x) \, dx=\int x^4\,\mathrm {erfc}\left (a+b\,x\right )\,{\mathrm {e}}^{d\,x^2+c} \,d x \]