Integrand size = 8, antiderivative size = 40 \[ \int \frac {\text {erfi}(b x)}{x^3} \, dx=-\frac {b e^{b^2 x^2}}{\sqrt {\pi } x}+b^2 \text {erfi}(b x)-\frac {\text {erfi}(b x)}{2 x^2} \]
[Out]
Time = 0.02 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {6498, 2245, 2235} \[ \int \frac {\text {erfi}(b x)}{x^3} \, dx=b^2 \text {erfi}(b x)-\frac {b e^{b^2 x^2}}{\sqrt {\pi } x}-\frac {\text {erfi}(b x)}{2 x^2} \]
[In]
[Out]
Rule 2235
Rule 2245
Rule 6498
Rubi steps \begin{align*} \text {integral}& = -\frac {\text {erfi}(b x)}{2 x^2}+\frac {b \int \frac {e^{b^2 x^2}}{x^2} \, dx}{\sqrt {\pi }} \\ & = -\frac {b e^{b^2 x^2}}{\sqrt {\pi } x}-\frac {\text {erfi}(b x)}{2 x^2}+\frac {\left (2 b^3\right ) \int e^{b^2 x^2} \, dx}{\sqrt {\pi }} \\ & = -\frac {b e^{b^2 x^2}}{\sqrt {\pi } x}+b^2 \text {erfi}(b x)-\frac {\text {erfi}(b x)}{2 x^2} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.92 \[ \int \frac {\text {erfi}(b x)}{x^3} \, dx=-\frac {b e^{b^2 x^2}}{\sqrt {\pi } x}+\left (b^2-\frac {1}{2 x^2}\right ) \text {erfi}(b x) \]
[In]
[Out]
Result contains complex when optimal does not.
Time = 0.19 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.08
method | result | size |
parts | \(-\frac {\operatorname {erfi}\left (b x \right )}{2 x^{2}}+\frac {b \left (-\frac {{\mathrm e}^{b^{2} x^{2}}}{x}-i b \sqrt {\pi }\, \operatorname {erf}\left (i b x \right )\right )}{\sqrt {\pi }}\) | \(43\) |
parallelrisch | \(\frac {2 x^{2} \operatorname {erfi}\left (b x \right ) \sqrt {\pi }\, b^{2}-2 \,{\mathrm e}^{b^{2} x^{2}} b x -\operatorname {erfi}\left (b x \right ) \sqrt {\pi }}{2 \sqrt {\pi }\, x^{2}}\) | \(46\) |
derivativedivides | \(b^{2} \left (-\frac {\operatorname {erfi}\left (b x \right )}{2 b^{2} x^{2}}+\frac {-\frac {{\mathrm e}^{b^{2} x^{2}}}{b x}+\operatorname {erfi}\left (b x \right ) \sqrt {\pi }}{\sqrt {\pi }}\right )\) | \(47\) |
default | \(b^{2} \left (-\frac {\operatorname {erfi}\left (b x \right )}{2 b^{2} x^{2}}+\frac {-\frac {{\mathrm e}^{b^{2} x^{2}}}{b x}+\operatorname {erfi}\left (b x \right ) \sqrt {\pi }}{\sqrt {\pi }}\right )\) | \(47\) |
meijerg | \(\frac {i b^{2} \left (\frac {2 i {\mathrm e}^{b^{2} x^{2}}}{x b}+\frac {i \left (-2 b^{2} x^{2}+1\right ) \operatorname {erfi}\left (b x \right ) \sqrt {\pi }}{x^{2} b^{2}}\right )}{2 \sqrt {\pi }}\) | \(54\) |
[In]
[Out]
none
Time = 0.26 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.00 \[ \int \frac {\text {erfi}(b x)}{x^3} \, dx=-\frac {2 \, \sqrt {\pi } b x e^{\left (b^{2} x^{2}\right )} + {\left (\pi - 2 \, \pi b^{2} x^{2}\right )} \operatorname {erfi}\left (b x\right )}{2 \, \pi x^{2}} \]
[In]
[Out]
Time = 0.20 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.85 \[ \int \frac {\text {erfi}(b x)}{x^3} \, dx=b^{2} \operatorname {erfi}{\left (b x \right )} - \frac {b e^{b^{2} x^{2}}}{\sqrt {\pi } x} - \frac {\operatorname {erfi}{\left (b x \right )}}{2 x^{2}} \]
[In]
[Out]
none
Time = 0.25 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.98 \[ \int \frac {\text {erfi}(b x)}{x^3} \, dx=-\frac {\sqrt {-b^{2} x^{2}} b \Gamma \left (-\frac {1}{2}, -b^{2} x^{2}\right )}{2 \, \sqrt {\pi } x} - \frac {\operatorname {erfi}\left (b x\right )}{2 \, x^{2}} \]
[In]
[Out]
\[ \int \frac {\text {erfi}(b x)}{x^3} \, dx=\int { \frac {\operatorname {erfi}\left (b x\right )}{x^{3}} \,d x } \]
[In]
[Out]
Time = 4.87 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.72 \[ \int \frac {\text {erfi}(b x)}{x^3} \, dx=\frac {b\,\mathrm {erfc}\left (\sqrt {-b^2\,x^2}\right )\,\sqrt {-b^2\,x^2}}{x}-\frac {b\,\sqrt {-b^2\,x^2}}{x}-\frac {b\,{\mathrm {e}}^{b^2\,x^2}}{x\,\sqrt {\pi }}-\frac {\mathrm {erfi}\left (b\,x\right )}{2\,x^2} \]
[In]
[Out]