Optimal. Leaf size=134 \[ \frac {b e^c}{3 \sqrt {\pi } x^2}-\frac {e^{c+b^2 x^2} \text {Erfc}(b x)}{3 x^3}-\frac {2 b^2 e^{c+b^2 x^2} \text {Erfc}(b x)}{3 x}+\frac {2}{3} b^3 e^c \sqrt {\pi } \text {Erfi}(b x)-\frac {4 b^5 e^c x^2 \, _2F_2\left (1,1;\frac {3}{2},2;b^2 x^2\right )}{3 \sqrt {\pi }}-\frac {4 b^3 e^c \log (x)}{3 \sqrt {\pi }} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.09, antiderivative size = 134, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 7, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.368, Rules used = {6527, 6512,
2235, 6511, 12, 29, 30} \begin {gather*} -\frac {4 b^5 e^c x^2 \, _2F_2\left (1,1;\frac {3}{2},2;b^2 x^2\right )}{3 \sqrt {\pi }}+\frac {2}{3} \sqrt {\pi } b^3 e^c \text {Erfi}(b x)-\frac {4 b^3 e^c \log (x)}{3 \sqrt {\pi }}-\frac {2 b^2 e^{b^2 x^2+c} \text {Erfc}(b x)}{3 x}-\frac {e^{b^2 x^2+c} \text {Erfc}(b x)}{3 x^3}+\frac {b e^c}{3 \sqrt {\pi } x^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 29
Rule 30
Rule 2235
Rule 6511
Rule 6512
Rule 6527
Rubi steps
\begin {align*} \int \frac {e^{c+b^2 x^2} \text {erfc}(b x)}{x^4} \, dx &=-\frac {e^{c+b^2 x^2} \text {erfc}(b x)}{3 x^3}+\frac {1}{3} \left (2 b^2\right ) \int \frac {e^{c+b^2 x^2} \text {erfc}(b x)}{x^2} \, dx-\frac {(2 b) \int \frac {e^c}{x^3} \, dx}{3 \sqrt {\pi }}\\ &=-\frac {e^{c+b^2 x^2} \text {erfc}(b x)}{3 x^3}-\frac {2 b^2 e^{c+b^2 x^2} \text {erfc}(b x)}{3 x}+\frac {1}{3} \left (4 b^4\right ) \int e^{c+b^2 x^2} \text {erfc}(b x) \, dx-\frac {\left (4 b^3\right ) \int \frac {e^c}{x} \, dx}{3 \sqrt {\pi }}-\frac {\left (2 b e^c\right ) \int \frac {1}{x^3} \, dx}{3 \sqrt {\pi }}\\ &=\frac {b e^c}{3 \sqrt {\pi } x^2}-\frac {e^{c+b^2 x^2} \text {erfc}(b x)}{3 x^3}-\frac {2 b^2 e^{c+b^2 x^2} \text {erfc}(b x)}{3 x}+\frac {1}{3} \left (4 b^4\right ) \int e^{c+b^2 x^2} \, dx-\frac {1}{3} \left (4 b^4\right ) \int e^{c+b^2 x^2} \text {erf}(b x) \, dx-\frac {\left (4 b^3 e^c\right ) \int \frac {1}{x} \, dx}{3 \sqrt {\pi }}\\ &=\frac {b e^c}{3 \sqrt {\pi } x^2}-\frac {e^{c+b^2 x^2} \text {erfc}(b x)}{3 x^3}-\frac {2 b^2 e^{c+b^2 x^2} \text {erfc}(b x)}{3 x}+\frac {2}{3} b^3 e^c \sqrt {\pi } \text {erfi}(b x)-\frac {4 b^5 e^c x^2 \, _2F_2\left (1,1;\frac {3}{2},2;b^2 x^2\right )}{3 \sqrt {\pi }}-\frac {4 b^3 e^c \log (x)}{3 \sqrt {\pi }}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.31, size = 151, normalized size = 1.13 \begin {gather*} \frac {e^c \left (-e^{b^2 x^2} \sqrt {\pi }+b x-2 b^2 e^{b^2 x^2} \sqrt {\pi } x^2+e^{b^2 x^2} \sqrt {\pi } \left (1+2 b^2 x^2\right ) \text {Erf}(b x)+2 b^3 \pi x^3 \text {Erfi}(b x)-2 b^3 \pi x^3 \text {Erf}(b x) \text {Erfi}(b x)+4 b^5 x^5 \, _2F_2\left (1,1;\frac {3}{2},2;-b^2 x^2\right )-4 b^3 x^3 \log (x)\right )}{3 \sqrt {\pi } x^3} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 0.14, size = 0, normalized size = 0.00 \[\int \frac {{\mathrm e}^{b^{2} x^{2}+c} \mathrm {erfc}\left (b x \right )}{x^{4}}\, dx\]
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 = 90.79, size = 22, normalized size = 0.16 \begin {gather*} - \frac {b^{3} {G_{3, 2}^{1, 3}\left (\begin {matrix} 2, \frac {5}{2}, 1 & \\2 & 0 \end {matrix} \middle | {\frac {1}{b^{2} x^{2}}} \right )} e^{c}}{2 \pi } \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^4} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________