Integrand size = 11, antiderivative size = 35 \[ \int f^{a+\frac {b}{x^2}} x \, dx=\frac {1}{2} f^{a+\frac {b}{x^2}} x^2-\frac {1}{2} b f^a \operatorname {ExpIntegralEi}\left (\frac {b \log (f)}{x^2}\right ) \log (f) \]
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Time = 0.02 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {2245, 2241} \[ \int f^{a+\frac {b}{x^2}} x \, dx=\frac {1}{2} x^2 f^{a+\frac {b}{x^2}}-\frac {1}{2} b f^a \log (f) \operatorname {ExpIntegralEi}\left (\frac {b \log (f)}{x^2}\right ) \]
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Rule 2241
Rule 2245
Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} f^{a+\frac {b}{x^2}} x^2+(b \log (f)) \int \frac {f^{a+\frac {b}{x^2}}}{x} \, dx \\ & = \frac {1}{2} f^{a+\frac {b}{x^2}} x^2-\frac {1}{2} b f^a \text {Ei}\left (\frac {b \log (f)}{x^2}\right ) \log (f) \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.91 \[ \int f^{a+\frac {b}{x^2}} x \, dx=\frac {1}{2} f^a \left (f^{\frac {b}{x^2}} x^2-b \operatorname {ExpIntegralEi}\left (\frac {b \log (f)}{x^2}\right ) \log (f)\right ) \]
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Time = 0.04 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.00
method | result | size |
risch | \(\frac {f^{a} x^{2} f^{\frac {b}{x^{2}}}}{2}+\frac {f^{a} \ln \left (f \right ) b \,\operatorname {Ei}_{1}\left (-\frac {b \ln \left (f \right )}{x^{2}}\right )}{2}\) | \(35\) |
meijerg | \(\frac {f^{a} b \ln \left (f \right ) \left (\frac {x^{2}}{b \ln \left (f \right )}+1+2 \ln \left (x \right )-\ln \left (-b \right )-\ln \left (\ln \left (f \right )\right )-\frac {x^{2} \left (2+\frac {2 b \ln \left (f \right )}{x^{2}}\right )}{2 b \ln \left (f \right )}+\frac {x^{2} {\mathrm e}^{\frac {b \ln \left (f \right )}{x^{2}}}}{b \ln \left (f \right )}+\ln \left (-\frac {b \ln \left (f \right )}{x^{2}}\right )+\operatorname {Ei}_{1}\left (-\frac {b \ln \left (f \right )}{x^{2}}\right )\right )}{2}\) | \(97\) |
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Time = 0.27 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.00 \[ \int f^{a+\frac {b}{x^2}} x \, dx=-\frac {1}{2} \, b f^{a} {\rm Ei}\left (\frac {b \log \left (f\right )}{x^{2}}\right ) \log \left (f\right ) + \frac {1}{2} \, f^{\frac {a x^{2} + b}{x^{2}}} x^{2} \]
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\[ \int f^{a+\frac {b}{x^2}} x \, dx=\int f^{a + \frac {b}{x^{2}}} x\, dx \]
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Time = 0.22 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.51 \[ \int f^{a+\frac {b}{x^2}} x \, dx=-\frac {1}{2} \, b f^{a} \Gamma \left (-1, -\frac {b \log \left (f\right )}{x^{2}}\right ) \log \left (f\right ) \]
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\[ \int f^{a+\frac {b}{x^2}} x \, dx=\int { f^{a + \frac {b}{x^{2}}} x \,d x } \]
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Time = 0.27 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.94 \[ \int f^{a+\frac {b}{x^2}} x \, dx=\frac {f^a\,f^{\frac {b}{x^2}}\,x^2}{2}+\frac {b\,f^a\,\ln \left (f\right )\,\mathrm {expint}\left (-\frac {b\,\ln \left (f\right )}{x^2}\right )}{2} \]
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