Integrand size = 19, antiderivative size = 19 \[ \int \frac {e^{c+d x^2} \text {erfc}(a+b x)}{x^4} \, dx=\frac {b e^{-a^2+c-2 a b x-\left (b^2-d\right ) x^2}}{3 \sqrt {\pi } x^2}-\frac {2 a b^2 e^{-a^2+c-2 a b x-\left (b^2-d\right ) x^2}}{3 \sqrt {\pi } x}-\frac {2}{3} a b^2 \sqrt {b^2-d} e^{c+\frac {a^2 d}{b^2-d}} \text {erf}\left (\frac {a b+\left (b^2-d\right ) x}{\sqrt {b^2-d}}\right )-\frac {e^{c+d x^2} \text {erfc}(a+b x)}{3 x^3}-\frac {2 d e^{c+d x^2} \text {erfc}(a+b x)}{3 x}-\frac {4 a^2 b^3 \text {Int}\left (\frac {e^{-a^2+c-2 a b x+\left (-b^2+d\right ) x^2}}{x},x\right )}{3 \sqrt {\pi }}+\frac {2 b \left (b^2-d\right ) \text {Int}\left (\frac {e^{-a^2+c-2 a b x+\left (-b^2+d\right ) x^2}}{x},x\right )}{3 \sqrt {\pi }}-\frac {4 b d \text {Int}\left (\frac {e^{-a^2+c-2 a b x+\left (-b^2+d\right ) x^2}}{x},x\right )}{3 \sqrt {\pi }}+\frac {4}{3} d^2 \text {Int}\left (e^{c+d x^2} \text {erfc}(a+b x),x\right ) \]
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Not integrable
Time = 0.62 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 0, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {e^{c+d x^2} \text {erfc}(a+b x)}{x^4} \, dx=\int \frac {e^{c+d x^2} \text {erfc}(a+b x)}{x^4} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = -\frac {e^{c+d x^2} \text {erfc}(a+b x)}{3 x^3}+\frac {1}{3} (2 d) \int \frac {e^{c+d x^2} \text {erfc}(a+b x)}{x^2} \, dx-\frac {(2 b) \int \frac {e^{-a^2+c-2 a b x+\left (-b^2+d\right ) x^2}}{x^3} \, dx}{3 \sqrt {\pi }} \\ & = \frac {b e^{-a^2+c-2 a b x-\left (b^2-d\right ) x^2}}{3 \sqrt {\pi } x^2}-\frac {e^{c+d x^2} \text {erfc}(a+b x)}{3 x^3}-\frac {2 d e^{c+d x^2} \text {erfc}(a+b x)}{3 x}+\frac {1}{3} \left (4 d^2\right ) \int e^{c+d x^2} \text {erfc}(a+b x) \, dx+\frac {\left (2 a b^2\right ) \int \frac {e^{-a^2+c-2 a b x+\left (-b^2+d\right ) x^2}}{x^2} \, dx}{3 \sqrt {\pi }}+\frac {\left (2 b \left (b^2-d\right )\right ) \int \frac {e^{-a^2+c-2 a b x+\left (-b^2+d\right ) x^2}}{x} \, dx}{3 \sqrt {\pi }}-\frac {(4 b d) \int \frac {e^{-a^2+c-2 a b x+\left (-b^2+d\right ) x^2}}{x} \, dx}{3 \sqrt {\pi }} \\ & = \frac {b e^{-a^2+c-2 a b x-\left (b^2-d\right ) x^2}}{3 \sqrt {\pi } x^2}-\frac {2 a b^2 e^{-a^2+c-2 a b x-\left (b^2-d\right ) x^2}}{3 \sqrt {\pi } x}-\frac {e^{c+d x^2} \text {erfc}(a+b x)}{3 x^3}-\frac {2 d e^{c+d x^2} \text {erfc}(a+b x)}{3 x}+\frac {1}{3} \left (4 d^2\right ) \int e^{c+d x^2} \text {erfc}(a+b x) \, dx-\frac {\left (4 a^2 b^3\right ) \int \frac {e^{-a^2+c-2 a b x+\left (-b^2+d\right ) x^2}}{x} \, dx}{3 \sqrt {\pi }}+\frac {\left (2 b \left (b^2-d\right )\right ) \int \frac {e^{-a^2+c-2 a b x+\left (-b^2+d\right ) x^2}}{x} \, dx}{3 \sqrt {\pi }}-\frac {\left (4 a b^2 \left (b^2-d\right )\right ) \int e^{-a^2+c-2 a b x+\left (-b^2+d\right ) x^2} \, dx}{3 \sqrt {\pi }}-\frac {(4 b d) \int \frac {e^{-a^2+c-2 a b x+\left (-b^2+d\right ) x^2}}{x} \, dx}{3 \sqrt {\pi }} \\ & = \frac {b e^{-a^2+c-2 a b x-\left (b^2-d\right ) x^2}}{3 \sqrt {\pi } x^2}-\frac {2 a b^2 e^{-a^2+c-2 a b x-\left (b^2-d\right ) x^2}}{3 \sqrt {\pi } x}-\frac {e^{c+d x^2} \text {erfc}(a+b x)}{3 x^3}-\frac {2 d e^{c+d x^2} \text {erfc}(a+b x)}{3 x}+\frac {1}{3} \left (4 d^2\right ) \int e^{c+d x^2} \text {erfc}(a+b x) \, dx-\frac {\left (4 a^2 b^3\right ) \int \frac {e^{-a^2+c-2 a b x+\left (-b^2+d\right ) x^2}}{x} \, dx}{3 \sqrt {\pi }}+\frac {\left (2 b \left (b^2-d\right )\right ) \int \frac {e^{-a^2+c-2 a b x+\left (-b^2+d\right ) x^2}}{x} \, dx}{3 \sqrt {\pi }}-\frac {(4 b d) \int \frac {e^{-a^2+c-2 a b x+\left (-b^2+d\right ) x^2}}{x} \, dx}{3 \sqrt {\pi }}-\frac {\left (4 a b^2 \left (b^2-d\right ) e^{\frac {b^2 c+a^2 d-c d}{b^2-d}}\right ) \int \exp \left (\frac {\left (-2 a b+2 \left (-b^2+d\right ) x\right )^2}{4 \left (-b^2+d\right )}\right ) \, dx}{3 \sqrt {\pi }} \\ & = \frac {b e^{-a^2+c-2 a b x-\left (b^2-d\right ) x^2}}{3 \sqrt {\pi } x^2}-\frac {2 a b^2 e^{-a^2+c-2 a b x-\left (b^2-d\right ) x^2}}{3 \sqrt {\pi } x}-\frac {2}{3} a b^2 \sqrt {b^2-d} e^{\frac {b^2 c+a^2 d-c d}{b^2-d}} \text {erf}\left (\frac {a b+\left (b^2-d\right ) x}{\sqrt {b^2-d}}\right )-\frac {e^{c+d x^2} \text {erfc}(a+b x)}{3 x^3}-\frac {2 d e^{c+d x^2} \text {erfc}(a+b x)}{3 x}+\frac {1}{3} \left (4 d^2\right ) \int e^{c+d x^2} \text {erfc}(a+b x) \, dx-\frac {\left (4 a^2 b^3\right ) \int \frac {e^{-a^2+c-2 a b x+\left (-b^2+d\right ) x^2}}{x} \, dx}{3 \sqrt {\pi }}+\frac {\left (2 b \left (b^2-d\right )\right ) \int \frac {e^{-a^2+c-2 a b x+\left (-b^2+d\right ) x^2}}{x} \, dx}{3 \sqrt {\pi }}-\frac {(4 b d) \int \frac {e^{-a^2+c-2 a b x+\left (-b^2+d\right ) x^2}}{x} \, dx}{3 \sqrt {\pi }} \\ \end{align*}
Not integrable
Time = 0.83 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.11 \[ \int \frac {e^{c+d x^2} \text {erfc}(a+b x)}{x^4} \, dx=\int \frac {e^{c+d x^2} \text {erfc}(a+b x)}{x^4} \, dx \]
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Not integrable
Time = 0.20 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.95
\[\int \frac {{\mathrm e}^{d \,x^{2}+c} \operatorname {erfc}\left (b x +a \right )}{x^{4}}d x\]
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Not integrable
Time = 0.25 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.21 \[ \int \frac {e^{c+d x^2} \text {erfc}(a+b x)}{x^4} \, dx=\int { \frac {\operatorname {erfc}\left (b x + a\right ) e^{\left (d x^{2} + c\right )}}{x^{4}} \,d x } \]
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Not integrable
Time = 87.83 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.05 \[ \int \frac {e^{c+d x^2} \text {erfc}(a+b x)}{x^4} \, dx=e^{c} \int \frac {e^{d x^{2}} \operatorname {erfc}{\left (a + b x \right )}}{x^{4}}\, dx \]
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Not integrable
Time = 0.25 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.05 \[ \int \frac {e^{c+d x^2} \text {erfc}(a+b x)}{x^4} \, dx=\int { \frac {\operatorname {erfc}\left (b x + a\right ) e^{\left (d x^{2} + c\right )}}{x^{4}} \,d x } \]
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Not integrable
Time = 0.27 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.05 \[ \int \frac {e^{c+d x^2} \text {erfc}(a+b x)}{x^4} \, dx=\int { \frac {\operatorname {erfc}\left (b x + a\right ) e^{\left (d x^{2} + c\right )}}{x^{4}} \,d x } \]
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Not integrable
Time = 4.81 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.05 \[ \int \frac {e^{c+d x^2} \text {erfc}(a+b x)}{x^4} \, dx=\int \frac {\mathrm {erfc}\left (a+b\,x\right )\,{\mathrm {e}}^{d\,x^2+c}}{x^4} \,d x \]
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