Integrand size = 19, antiderivative size = 19 \[ \int \frac {e^{c+d x^2} \text {erfi}(a+b x)}{x^3} \, dx=-\frac {b e^{a^2+c+2 a b x+\left (b^2+d\right ) x^2}}{\sqrt {\pi } x}-\frac {e^{c+d x^2} \text {erfi}(a+b x)}{2 x^2}+b \sqrt {b^2+d} e^{c+\frac {a^2 d}{b^2+d}} \text {erfi}\left (\frac {a b+\left (b^2+d\right ) x}{\sqrt {b^2+d}}\right )+\frac {2 a b^2 \text {Int}\left (\frac {e^{a^2+c+2 a b x+\left (b^2+d\right ) x^2}}{x},x\right )}{\sqrt {\pi }}+d \text {Int}\left (\frac {e^{c+d x^2} \text {erfi}(a+b x)}{x},x\right ) \]
[Out]
Not integrable
Time = 0.28 (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 = {} \[ \int \frac {e^{c+d x^2} \text {erfi}(a+b x)}{x^3} \, dx=\int \frac {e^{c+d x^2} \text {erfi}(a+b x)}{x^3} \, dx \]
[In]
[Out]
Rubi steps \begin{align*} \text {integral}& = -\frac {e^{c+d x^2} \text {erfi}(a+b x)}{2 x^2}+d \int \frac {e^{c+d x^2} \text {erfi}(a+b x)}{x} \, dx+\frac {b \int \frac {e^{a^2+c+2 a b x+\left (b^2+d\right ) x^2}}{x^2} \, dx}{\sqrt {\pi }} \\ & = -\frac {b e^{a^2+c+2 a b x+\left (b^2+d\right ) x^2}}{\sqrt {\pi } x}-\frac {e^{c+d x^2} \text {erfi}(a+b x)}{2 x^2}+d \int \frac {e^{c+d x^2} \text {erfi}(a+b x)}{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} \, dx}{\sqrt {\pi }}+\frac {\left (2 b \left (b^2+d\right )\right ) \int e^{a^2+c+2 a b x+\left (b^2+d\right ) x^2} \, dx}{\sqrt {\pi }} \\ & = -\frac {b e^{a^2+c+2 a b x+\left (b^2+d\right ) x^2}}{\sqrt {\pi } x}-\frac {e^{c+d x^2} \text {erfi}(a+b x)}{2 x^2}+d \int \frac {e^{c+d x^2} \text {erfi}(a+b x)}{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} \, dx}{\sqrt {\pi }}+\frac {\left (2 b \left (b^2+d\right ) e^{c+\frac {a^2 d}{b^2+d}}\right ) \int e^{\frac {\left (2 a b+2 \left (b^2+d\right ) x\right )^2}{4 \left (b^2+d\right )}} \, dx}{\sqrt {\pi }} \\ & = -\frac {b e^{a^2+c+2 a b x+\left (b^2+d\right ) x^2}}{\sqrt {\pi } x}-\frac {e^{c+d x^2} \text {erfi}(a+b x)}{2 x^2}+b \sqrt {b^2+d} e^{c+\frac {a^2 d}{b^2+d}} \text {erfi}\left (\frac {a b+\left (b^2+d\right ) x}{\sqrt {b^2+d}}\right )+d \int \frac {e^{c+d x^2} \text {erfi}(a+b x)}{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} \, dx}{\sqrt {\pi }} \\ \end{align*}
Not integrable
Time = 0.20 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.11 \[ \int \frac {e^{c+d x^2} \text {erfi}(a+b x)}{x^3} \, dx=\int \frac {e^{c+d x^2} \text {erfi}(a+b x)}{x^3} \, dx \]
[In]
[Out]
Not integrable
Time = 0.18 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.95
\[\int \frac {{\mathrm e}^{d \,x^{2}+c} \operatorname {erfi}\left (b x +a \right )}{x^{3}}d x\]
[In]
[Out]
Not integrable
Time = 0.24 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.05 \[ \int \frac {e^{c+d x^2} \text {erfi}(a+b x)}{x^3} \, dx=\int { \frac {\operatorname {erfi}\left (b x + a\right ) e^{\left (d x^{2} + c\right )}}{x^{3}} \,d x } \]
[In]
[Out]
Not integrable
Time = 23.65 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.05 \[ \int \frac {e^{c+d x^2} \text {erfi}(a+b x)}{x^3} \, dx=e^{c} \int \frac {e^{d x^{2}} \operatorname {erfi}{\left (a + b x \right )}}{x^{3}}\, dx \]
[In]
[Out]
Not integrable
Time = 0.27 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.05 \[ \int \frac {e^{c+d x^2} \text {erfi}(a+b x)}{x^3} \, dx=\int { \frac {\operatorname {erfi}\left (b x + a\right ) e^{\left (d x^{2} + c\right )}}{x^{3}} \,d x } \]
[In]
[Out]
Not integrable
Time = 0.27 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.05 \[ \int \frac {e^{c+d x^2} \text {erfi}(a+b x)}{x^3} \, dx=\int { \frac {\operatorname {erfi}\left (b x + a\right ) e^{\left (d x^{2} + c\right )}}{x^{3}} \,d x } \]
[In]
[Out]
Not integrable
Time = 5.84 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.05 \[ \int \frac {e^{c+d x^2} \text {erfi}(a+b x)}{x^3} \, dx=\int \frac {\mathrm {erfi}\left (a+b\,x\right )\,{\mathrm {e}}^{d\,x^2+c}}{x^3} \,d x \]
[In]
[Out]