Optimal. Leaf size=88 \[ \frac {b e^c}{\sqrt {\pi } x}-\frac {e^{c+b^2 x^2} \text {Erfc}(b x)}{2 x^2}+\frac {1}{2} b^2 e^c \text {Ei}\left (b^2 x^2\right )-\frac {2 b^3 e^c x \, _2F_2\left (\frac {1}{2},1;\frac {3}{2},\frac {3}{2};b^2 x^2\right )}{\sqrt {\pi }} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.10, antiderivative size = 88, normalized size of antiderivative = 1.00, 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 = {6527, 6524,
2241, 6523, 12, 30} \begin {gather*} -\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} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 30
Rule 2241
Rule 6523
Rule 6524
Rule 6527
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 = 0.15, size = 65, normalized size = 0.74 \begin {gather*} -\frac {e^c \left (e^{b^2 x^2}-b^2 x^2 \text {Ei}\left (b^2 x^2\right )-\frac {4 b x \, _2F_2\left (-\frac {1}{2},1;\frac {1}{2},\frac {3}{2};b^2 x^2\right )}{\sqrt {\pi }}\right )}{2 x^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 0.21, size = 0, normalized size = 0.00 \[\int \frac {{\mathrm e}^{b^{2} x^{2}+c} \mathrm {erfc}\left (b x \right )}{x^{3}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A]
time = 33.79, size = 61, normalized size = 0.69 \begin {gather*} \frac {b^{2} e^{c} \operatorname {Ei}{\left (b^{2} x^{2} \right )}}{2} + \frac {2 b e^{c} {{}_{2}F_{2}\left (\begin {matrix} - \frac {1}{2}, 1 \\ \frac {1}{2}, \frac {3}{2} \end {matrix}\middle | {b^{2} x^{2}} \right )}}{\sqrt {\pi } x} - \frac {e^{c} e^{b^{2} x^{2}}}{2 x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\mathrm {e}}^{b^2\,x^2+c}\,\mathrm {erfc}\left (b\,x\right )}{x^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________