Integrand size = 13, antiderivative size = 39 \[ \int \frac {f^{a+\frac {b}{x^2}}}{x^2} \, dx=-\frac {f^a \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {b} \sqrt {\log (f)}}{x}\right )}{2 \sqrt {b} \sqrt {\log (f)}} \]
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Time = 0.02 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2242, 2235} \[ \int \frac {f^{a+\frac {b}{x^2}}}{x^2} \, dx=-\frac {\sqrt {\pi } f^a \text {erfi}\left (\frac {\sqrt {b} \sqrt {\log (f)}}{x}\right )}{2 \sqrt {b} \sqrt {\log (f)}} \]
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Rule 2235
Rule 2242
Rubi steps \begin{align*} \text {integral}& = -\text {Subst}\left (\int f^{a+b x^2} \, dx,x,\frac {1}{x}\right ) \\ & = -\frac {f^a \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {b} \sqrt {\log (f)}}{x}\right )}{2 \sqrt {b} \sqrt {\log (f)}} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.00 \[ \int \frac {f^{a+\frac {b}{x^2}}}{x^2} \, dx=-\frac {f^a \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {b} \sqrt {\log (f)}}{x}\right )}{2 \sqrt {b} \sqrt {\log (f)}} \]
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Time = 0.04 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.72
method | result | size |
meijerg | \(-\frac {f^{a} \operatorname {erfi}\left (\frac {\sqrt {b}\, \sqrt {\ln \left (f \right )}}{x}\right ) \sqrt {\pi }}{2 \sqrt {b}\, \sqrt {\ln \left (f \right )}}\) | \(28\) |
risch | \(-\frac {f^{a} \sqrt {\pi }\, \operatorname {erf}\left (\frac {\sqrt {-b \ln \left (f \right )}}{x}\right )}{2 \sqrt {-b \ln \left (f \right )}}\) | \(28\) |
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Time = 0.30 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.87 \[ \int \frac {f^{a+\frac {b}{x^2}}}{x^2} \, dx=\frac {\sqrt {\pi } \sqrt {-b \log \left (f\right )} f^{a} \operatorname {erf}\left (\frac {\sqrt {-b \log \left (f\right )}}{x}\right )}{2 \, b \log \left (f\right )} \]
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\[ \int \frac {f^{a+\frac {b}{x^2}}}{x^2} \, dx=\int \frac {f^{a + \frac {b}{x^{2}}}}{x^{2}}\, dx \]
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Time = 0.22 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.87 \[ \int \frac {f^{a+\frac {b}{x^2}}}{x^2} \, dx=-\frac {\sqrt {\pi } f^{a} {\left (\operatorname {erf}\left (\sqrt {-\frac {b \log \left (f\right )}{x^{2}}}\right ) - 1\right )}}{2 \, x \sqrt {-\frac {b \log \left (f\right )}{x^{2}}}} \]
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\[ \int \frac {f^{a+\frac {b}{x^2}}}{x^2} \, dx=\int { \frac {f^{a + \frac {b}{x^{2}}}}{x^{2}} \,d x } \]
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Time = 0.19 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.72 \[ \int \frac {f^{a+\frac {b}{x^2}}}{x^2} \, dx=-\frac {f^a\,\sqrt {\pi }\,\mathrm {erfi}\left (\frac {b\,\ln \left (f\right )}{x\,\sqrt {b\,\ln \left (f\right )}}\right )}{2\,\sqrt {b\,\ln \left (f\right )}} \]
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