Integrand size = 9, antiderivative size = 37 \[ \int f^{a+b x^2} \, dx=\frac {f^a \sqrt {\pi } \text {erfi}\left (\sqrt {b} x \sqrt {\log (f)}\right )}{2 \sqrt {b} \sqrt {\log (f)}} \]
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Time = 0.00 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {2235} \[ \int f^{a+b x^2} \, dx=\frac {\sqrt {\pi } f^a \text {erfi}\left (\sqrt {b} x \sqrt {\log (f)}\right )}{2 \sqrt {b} \sqrt {\log (f)}} \]
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Rule 2235
Rubi steps \begin{align*} \text {integral}& = \frac {f^a \sqrt {\pi } \text {erfi}\left (\sqrt {b} x \sqrt {\log (f)}\right )}{2 \sqrt {b} \sqrt {\log (f)}} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.00 \[ \int f^{a+b x^2} \, dx=\frac {f^a \sqrt {\pi } \text {erfi}\left (\sqrt {b} x \sqrt {\log (f)}\right )}{2 \sqrt {b} \sqrt {\log (f)}} \]
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Time = 0.02 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.70
method | result | size |
meijerg | \(\frac {f^{a} \operatorname {erfi}\left (x \sqrt {b}\, \sqrt {\ln \left (f \right )}\right ) \sqrt {\pi }}{2 \sqrt {b}\, \sqrt {\ln \left (f \right )}}\) | \(26\) |
risch | \(\frac {f^{a} \sqrt {\pi }\, \operatorname {erf}\left (\sqrt {-b \ln \left (f \right )}\, x \right )}{2 \sqrt {-b \ln \left (f \right )}}\) | \(26\) |
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none
Time = 0.28 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.86 \[ \int f^{a+b x^2} \, dx=-\frac {\sqrt {\pi } \sqrt {-b \log \left (f\right )} f^{a} \operatorname {erf}\left (\sqrt {-b \log \left (f\right )} x\right )}{2 \, b \log \left (f\right )} \]
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\[ \int f^{a+b x^2} \, dx=\int f^{a + b x^{2}}\, dx \]
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none
Time = 0.18 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.68 \[ \int f^{a+b x^2} \, dx=\frac {\sqrt {\pi } f^{a} \operatorname {erf}\left (\sqrt {-b \log \left (f\right )} x\right )}{2 \, \sqrt {-b \log \left (f\right )}} \]
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none
Time = 0.32 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.70 \[ \int f^{a+b x^2} \, dx=-\frac {\sqrt {\pi } f^{a} \operatorname {erf}\left (-\sqrt {-b \log \left (f\right )} x\right )}{2 \, \sqrt {-b \log \left (f\right )}} \]
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Time = 0.12 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.70 \[ \int f^{a+b x^2} \, dx=\frac {f^a\,\sqrt {\pi }\,\mathrm {erfi}\left (\frac {b\,x\,\ln \left (f\right )}{\sqrt {b\,\ln \left (f\right )}}\right )}{2\,\sqrt {b\,\ln \left (f\right )}} \]
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