Optimal. Leaf size=84 \[ -\frac {\text {Int}\left (\text {erfc}(b x) e^{c+d x^2},x\right )}{2 d}-\frac {b e^{c-x^2 \left (b^2-d\right )}}{2 \sqrt {\pi } d \left (b^2-d\right )}+\frac {x \text {erfc}(b x) e^{c+d x^2}}{2 d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.09, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int e^{c+d x^2} x^2 \text {Erfc}(b x) \, dx \]
Verification is Not applicable to the result.
[In]
[Out]
Rubi steps
\begin {align*} \int e^{c+d x^2} x^2 \text {erfc}(b x) \, dx &=\frac {e^{c+d x^2} x \text {erfc}(b x)}{2 d}-\frac {\int e^{c+d x^2} \text {erfc}(b x) \, dx}{2 d}+\frac {b \int e^{c-\left (b^2-d\right ) x^2} x \, dx}{d \sqrt {\pi }}\\ &=-\frac {b e^{c-\left (b^2-d\right ) x^2}}{2 \left (b^2-d\right ) d \sqrt {\pi }}+\frac {e^{c+d x^2} x \text {erfc}(b x)}{2 d}-\frac {\int e^{c+d x^2} \text {erfc}(b x) \, dx}{2 d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.83, size = 0, normalized size = 0.00 \[ \int e^{c+d x^2} x^2 \text {erfc}(b x) \, dx \]
Verification is Not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.57, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-{\left (x^{2} \operatorname {erf}\left (b x\right ) - x^{2}\right )} e^{\left (d x^{2} + c\right )}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{2} \operatorname {erfc}\left (b x\right ) e^{\left (d x^{2} + c\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.19, size = 0, normalized size = 0.00 \[ \int {\mathrm e}^{d \,x^{2}+c} x^{2} \mathrm {erfc}\left (b x \right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{2} \operatorname {erfc}\left (b x\right ) e^{\left (d x^{2} + c\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [A] time = 0.00, size = -1, normalized size = -0.01 \[ \int x^2\,{\mathrm {e}}^{d\,x^2+c}\,\mathrm {erfc}\left (b\,x\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.00, size = 0, normalized size = 0.00 \[ e^{c} \int x^{2} e^{d x^{2}} \operatorname {erfc}{\left (b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________