Integrand size = 19, antiderivative size = 66 \[ \int f^{a+b x^n} x^{-1-\frac {n}{2}} \, dx=-\frac {2 f^{a+b x^n} x^{-n/2}}{n}+\frac {2 \sqrt {b} f^a \sqrt {\pi } \text {erfi}\left (\sqrt {b} x^{n/2} \sqrt {\log (f)}\right ) \sqrt {\log (f)}}{n} \]
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Time = 0.04 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {2246, 2242, 2235} \[ \int f^{a+b x^n} x^{-1-\frac {n}{2}} \, dx=\frac {2 \sqrt {\pi } \sqrt {b} f^a \sqrt {\log (f)} \text {erfi}\left (\sqrt {b} \sqrt {\log (f)} x^{n/2}\right )}{n}-\frac {2 x^{-n/2} f^{a+b x^n}}{n} \]
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Rule 2235
Rule 2242
Rule 2246
Rubi steps \begin{align*} \text {integral}& = -\frac {2 f^{a+b x^n} x^{-n/2}}{n}+(2 b \log (f)) \int f^{a+b x^n} x^{\frac {1}{2} (-2+n)} \, dx \\ & = -\frac {2 f^{a+b x^n} x^{-n/2}}{n}+\frac {(4 b \log (f)) \text {Subst}\left (\int f^{a+b x^2} \, dx,x,x^{1+\frac {1}{2} (-2+n)}\right )}{n} \\ & = -\frac {2 f^{a+b x^n} x^{-n/2}}{n}+\frac {2 \sqrt {b} f^a \sqrt {\pi } \text {erfi}\left (\sqrt {b} x^{n/2} \sqrt {\log (f)}\right ) \sqrt {\log (f)}}{n} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.59 \[ \int f^{a+b x^n} x^{-1-\frac {n}{2}} \, dx=-\frac {f^a x^{-n/2} \Gamma \left (-\frac {1}{2},-b x^n \log (f)\right ) \sqrt {-b x^n \log (f)}}{n} \]
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Time = 0.12 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.89
method | result | size |
risch | \(-\frac {2 f^{a} x^{-\frac {n}{2}} f^{b \,x^{n}}}{n}+\frac {2 f^{a} \ln \left (f \right ) b \sqrt {\pi }\, \operatorname {erf}\left (\sqrt {-b \ln \left (f \right )}\, x^{\frac {n}{2}}\right )}{n \sqrt {-b \ln \left (f \right )}}\) | \(59\) |
meijerg | \(\frac {f^{a} \sqrt {-b}\, \sqrt {\ln \left (f \right )}\, \left (-\frac {2 x^{-\frac {n}{2}} {\mathrm e}^{b \,x^{n} \ln \left (f \right )}}{\sqrt {-b}\, \sqrt {\ln \left (f \right )}}+\frac {2 \sqrt {b}\, \sqrt {\pi }\, \operatorname {erfi}\left (x^{\frac {n}{2}} \sqrt {b}\, \sqrt {\ln \left (f \right )}\right )}{\sqrt {-b}}\right )}{n}\) | \(69\) |
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Time = 0.31 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.26 \[ \int f^{a+b x^n} x^{-1-\frac {n}{2}} \, dx=-\frac {2 \, {\left (\sqrt {\pi } \sqrt {-b \log \left (f\right )} f^{a} \operatorname {erf}\left (\frac {\sqrt {-b \log \left (f\right )}}{x x^{-\frac {1}{2} \, n - 1}}\right ) + x x^{-\frac {1}{2} \, n - 1} e^{\left (\frac {a x^{2} x^{-n - 2} \log \left (f\right ) + b \log \left (f\right )}{x^{2} x^{-n - 2}}\right )}\right )}}{n} \]
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\[ \int f^{a+b x^n} x^{-1-\frac {n}{2}} \, dx=\int f^{a + b x^{n}} x^{- \frac {n}{2} - 1}\, dx \]
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Time = 0.23 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.53 \[ \int f^{a+b x^n} x^{-1-\frac {n}{2}} \, dx=-\frac {\sqrt {-b x^{n} \log \left (f\right )} f^{a} \Gamma \left (-\frac {1}{2}, -b x^{n} \log \left (f\right )\right )}{n x^{\frac {1}{2} \, n}} \]
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\[ \int f^{a+b x^n} x^{-1-\frac {n}{2}} \, dx=\int { f^{b x^{n} + a} x^{-\frac {1}{2} \, n - 1} \,d x } \]
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Timed out. \[ \int f^{a+b x^n} x^{-1-\frac {n}{2}} \, dx=\int \frac {f^{a+b\,x^n}}{x^{\frac {n}{2}+1}} \,d x \]
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