Integrand size = 17, antiderivative size = 17 \[ \int \frac {e^{c+d x^2} \text {erfi}(b x)}{x^4} \, dx=-\frac {b e^{c+\left (b^2+d\right ) x^2}}{3 \sqrt {\pi } x^2}-\frac {e^{c+d x^2} \text {erfi}(b x)}{3 x^3}-\frac {2 d e^{c+d x^2} \text {erfi}(b x)}{3 x}+\frac {2 b d e^c \operatorname {ExpIntegralEi}\left (\left (b^2+d\right ) x^2\right )}{3 \sqrt {\pi }}+\frac {b \left (b^2+d\right ) e^c \operatorname {ExpIntegralEi}\left (\left (b^2+d\right ) x^2\right )}{3 \sqrt {\pi }}+\frac {4}{3} d^2 \text {Int}\left (e^{c+d x^2} \text {erfi}(b x),x\right ) \]
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Not integrable
Time = 0.24 (sec) , antiderivative size = 17, 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 = {} \[ \int \frac {e^{c+d x^2} \text {erfi}(b x)}{x^4} \, dx=\int \frac {e^{c+d x^2} \text {erfi}(b x)}{x^4} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = -\frac {e^{c+d x^2} \text {erfi}(b x)}{3 x^3}+\frac {1}{3} (2 d) \int \frac {e^{c+d x^2} \text {erfi}(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 {erfi}(b x)}{3 x^3}-\frac {2 d e^{c+d x^2} \text {erfi}(b x)}{3 x}+\frac {1}{3} \left (4 d^2\right ) \int e^{c+d x^2} \text {erfi}(b x) \, dx+\frac {(4 b d) \int \frac {e^{c+\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^{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 {erfi}(b x)}{3 x^3}-\frac {2 d e^{c+d x^2} \text {erfi}(b x)}{3 x}+\frac {2 b d e^c \operatorname {ExpIntegralEi}\left (\left (b^2+d\right ) x^2\right )}{3 \sqrt {\pi }}+\frac {b \left (b^2+d\right ) e^c \operatorname {ExpIntegralEi}\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 {erfi}(b x) \, dx \\ \end{align*}
Not integrable
Time = 0.22 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.12 \[ \int \frac {e^{c+d x^2} \text {erfi}(b x)}{x^4} \, dx=\int \frac {e^{c+d x^2} \text {erfi}(b x)}{x^4} \, dx \]
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Not integrable
Time = 0.18 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.94
\[\int \frac {{\mathrm e}^{d \,x^{2}+c} \operatorname {erfi}\left (b x \right )}{x^{4}}d x\]
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Not integrable
Time = 0.24 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.06 \[ \int \frac {e^{c+d x^2} \text {erfi}(b x)}{x^4} \, dx=\int { \frac {\operatorname {erfi}\left (b x\right ) e^{\left (d x^{2} + c\right )}}{x^{4}} \,d x } \]
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Not integrable
Time = 13.67 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.12 \[ \int \frac {e^{c+d x^2} \text {erfi}(b x)}{x^4} \, dx=e^{c} \int \frac {e^{d x^{2}} \operatorname {erfi}{\left (b x \right )}}{x^{4}}\, dx \]
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Not integrable
Time = 0.25 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.06 \[ \int \frac {e^{c+d x^2} \text {erfi}(b x)}{x^4} \, dx=\int { \frac {\operatorname {erfi}\left (b x\right ) e^{\left (d x^{2} + c\right )}}{x^{4}} \,d x } \]
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Not integrable
Time = 0.26 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.06 \[ \int \frac {e^{c+d x^2} \text {erfi}(b x)}{x^4} \, dx=\int { \frac {\operatorname {erfi}\left (b x\right ) e^{\left (d x^{2} + c\right )}}{x^{4}} \,d x } \]
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Not integrable
Time = 5.82 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.06 \[ \int \frac {e^{c+d x^2} \text {erfi}(b x)}{x^4} \, dx=\int \frac {{\mathrm {e}}^{d\,x^2+c}\,\mathrm {erfi}\left (b\,x\right )}{x^4} \,d x \]
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