Integrand size = 13, antiderivative size = 15 \[ \int \frac {f^{a+b x^2}}{x} \, dx=\frac {1}{2} f^a \operatorname {ExpIntegralEi}\left (b x^2 \log (f)\right ) \]
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Time = 0.02 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {2241} \[ \int \frac {f^{a+b x^2}}{x} \, dx=\frac {1}{2} f^a \operatorname {ExpIntegralEi}\left (b x^2 \log (f)\right ) \]
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Rule 2241
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} f^a \text {Ei}\left (b x^2 \log (f)\right ) \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00 \[ \int \frac {f^{a+b x^2}}{x} \, dx=\frac {1}{2} f^a \operatorname {ExpIntegralEi}\left (b x^2 \log (f)\right ) \]
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Time = 0.02 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.07
method | result | size |
risch | \(-\frac {f^{a} \operatorname {Ei}_{1}\left (-b \,x^{2} \ln \left (f \right )\right )}{2}\) | \(16\) |
meijerg | \(\frac {f^{a} \left (2 \ln \left (x \right )+\ln \left (-b \right )+\ln \left (\ln \left (f \right )\right )-\ln \left (-b \,x^{2} \ln \left (f \right )\right )-\operatorname {Ei}_{1}\left (-b \,x^{2} \ln \left (f \right )\right )\right )}{2}\) | \(41\) |
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Time = 0.30 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.87 \[ \int \frac {f^{a+b x^2}}{x} \, dx=\frac {1}{2} \, f^{a} {\rm Ei}\left (b x^{2} \log \left (f\right )\right ) \]
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\[ \int \frac {f^{a+b x^2}}{x} \, dx=\int \frac {f^{a + b x^{2}}}{x}\, dx \]
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none
Time = 0.22 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.87 \[ \int \frac {f^{a+b x^2}}{x} \, dx=\frac {1}{2} \, f^{a} {\rm Ei}\left (b x^{2} \log \left (f\right )\right ) \]
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\[ \int \frac {f^{a+b x^2}}{x} \, dx=\int { \frac {f^{b x^{2} + a}}{x} \,d x } \]
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Time = 0.13 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.87 \[ \int \frac {f^{a+b x^2}}{x} \, dx=\frac {f^a\,\mathrm {ei}\left (b\,x^2\,\ln \left (f\right )\right )}{2} \]
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