Optimal. Leaf size=88 \[ -\frac{2 b^3 e^c x \text{HypergeometricPFQ}\left (\left \{\frac{1}{2},1\right \},\left \{\frac{3}{2},\frac{3}{2}\right \},b^2 x^2\right )}{\sqrt{\pi }}-\frac{e^{b^2 x^2+c} \text{Erfc}(b x)}{2 x^2}+\frac{1}{2} b^2 e^c \text{ExpIntegralEi}\left (b^2 x^2\right )+\frac{b e^c}{\sqrt{\pi } x} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.165603, antiderivative size = 88, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.316, Rules used = {6392, 6389, 2210, 6388, 12, 30} \[ -\frac{2 b^3 e^c x \, _2F_2\left (\frac{1}{2},1;\frac{3}{2},\frac{3}{2};b^2 x^2\right )}{\sqrt{\pi }}-\frac{e^{b^2 x^2+c} \text{Erfc}(b x)}{2 x^2}+\frac{1}{2} b^2 e^c \text{Ei}\left (b^2 x^2\right )+\frac{b e^c}{\sqrt{\pi } x} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 6392
Rule 6389
Rule 2210
Rule 6388
Rule 12
Rule 30
Rubi steps
\begin{align*} \int \frac{e^{c+b^2 x^2} \text{erfc}(b x)}{x^3} \, dx &=-\frac{e^{c+b^2 x^2} \text{erfc}(b x)}{2 x^2}+b^2 \int \frac{e^{c+b^2 x^2} \text{erfc}(b x)}{x} \, dx-\frac{b \int \frac{e^c}{x^2} \, dx}{\sqrt{\pi }}\\ &=-\frac{e^{c+b^2 x^2} \text{erfc}(b x)}{2 x^2}+b^2 \int \frac{e^{c+b^2 x^2}}{x} \, dx-b^2 \int \frac{e^{c+b^2 x^2} \text{erf}(b x)}{x} \, dx-\frac{\left (b e^c\right ) \int \frac{1}{x^2} \, dx}{\sqrt{\pi }}\\ &=\frac{b e^c}{\sqrt{\pi } x}-\frac{e^{c+b^2 x^2} \text{erfc}(b x)}{2 x^2}+\frac{1}{2} b^2 e^c \text{Ei}\left (b^2 x^2\right )-\frac{2 b^3 e^c x \, _2F_2\left (\frac{1}{2},1;\frac{3}{2},\frac{3}{2};b^2 x^2\right )}{\sqrt{\pi }}\\ \end{align*}
Mathematica [A] time = 0.207471, size = 65, normalized size = 0.74 \[ -\frac{e^c \left (-\frac{4 b x \text{HypergeometricPFQ}\left (\left \{-\frac{1}{2},1\right \},\left \{\frac{1}{2},\frac{3}{2}\right \},b^2 x^2\right )}{\sqrt{\pi }}-b^2 x^2 \text{ExpIntegralEi}\left (b^2 x^2\right )+e^{b^2 x^2}\right )}{2 x^2} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.277, size = 0, normalized size = 0. \begin{align*} \int{\frac{{{\rm e}^{{b}^{2}{x}^{2}+c}}{\it erfc} \left ( bx \right ) }{{x}^{3}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{erfc}\left (b x\right ) e^{\left (b^{2} x^{2} + c\right )}}{x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{{\left (\operatorname{erf}\left (b x\right ) - 1\right )} e^{\left (b^{2} x^{2} + c\right )}}{x^{3}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{erfc}\left (b x\right ) e^{\left (b^{2} x^{2} + c\right )}}{x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]