Optimal. Leaf size=155 \[ -\frac {b e^{c-\left (b^2-d\right ) x^2}}{3 \sqrt {\pi } x^2}-\frac {e^{c+d x^2} \text {Erf}(b x)}{3 x^3}-\frac {2 d e^{c+d x^2} \text {Erf}(b x)}{3 x}-\frac {b \left (b^2-d\right ) e^c \text {Ei}\left (-\left (\left (b^2-d\right ) x^2\right )\right )}{3 \sqrt {\pi }}+\frac {2 b d e^c \text {Ei}\left (-\left (\left (b^2-d\right ) x^2\right )\right )}{3 \sqrt {\pi }}+\frac {4}{3} d^2 \text {Int}\left (e^{c+d x^2} \text {Erf}(b x),x\right ) \]
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Rubi [A]
time = 0.19, 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 = {}
\begin {gather*} \int \frac {e^{c+d x^2} \text {Erf}(b x)}{x^4} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {e^{c+d x^2} \text {erf}(b x)}{x^4} \, dx &=-\frac {e^{c+d x^2} \text {erf}(b x)}{3 x^3}+\frac {1}{3} (2 d) \int \frac {e^{c+d x^2} \text {erf}(b x)}{x^2} \, dx+\frac {(2 b) \int \frac {e^{c-\left (b^2-d\right ) x^2}}{x^3} \, dx}{3 \sqrt {\pi }}\\ &=-\frac {b e^{c-\left (b^2-d\right ) x^2}}{3 \sqrt {\pi } x^2}-\frac {e^{c+d x^2} \text {erf}(b x)}{3 x^3}-\frac {2 d e^{c+d x^2} \text {erf}(b x)}{3 x}+\frac {1}{3} \left (4 d^2\right ) \int e^{c+d x^2} \text {erf}(b x) \, dx-\frac {\left (2 b \left (b^2-d\right )\right ) \int \frac {e^{c+\left (-b^2+d\right ) x^2}}{x} \, dx}{3 \sqrt {\pi }}+\frac {(4 b d) \int \frac {e^{c-\left (b^2-d\right ) x^2}}{x} \, dx}{3 \sqrt {\pi }}\\ &=-\frac {b e^{c-\left (b^2-d\right ) x^2}}{3 \sqrt {\pi } x^2}-\frac {e^{c+d x^2} \text {erf}(b x)}{3 x^3}-\frac {2 d e^{c+d x^2} \text {erf}(b x)}{3 x}-\frac {b \left (b^2-d\right ) e^c \text {Ei}\left (-\left (b^2-d\right ) x^2\right )}{3 \sqrt {\pi }}+\frac {2 b d e^c \text {Ei}\left (-\left (b^2-d\right ) x^2\right )}{3 \sqrt {\pi }}+\frac {1}{3} \left (4 d^2\right ) \int e^{c+d x^2} \text {erf}(b x) \, dx\\ \end {align*}
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Mathematica [A]
time = 0.22, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {e^{c+d x^2} \text {Erf}(b x)}{x^4} \, dx \end {gather*}
Verification is not applicable to the result.
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Maple [A]
time = 0.18, size = 0, normalized size = 0.00 \[\int \frac {{\mathrm e}^{d \,x^{2}+c} \erf \left (b x \right )}{x^{4}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
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.
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Fricas [A]
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.
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Sympy [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} e^{c} \int \frac {e^{d x^{2}} \operatorname {erf}{\left (b x \right )}}{x^{4}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
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.
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Mupad [A]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\mathrm {e}}^{d\,x^2+c}\,\mathrm {erf}\left (b\,x\right )}{x^4} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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