Integrand size = 17, antiderivative size = 38 \[ \int f^{a+b x^n} x^{-1-n} \, dx=-\frac {f^{a+b x^n} x^{-n}}{n}+\frac {b f^a \operatorname {ExpIntegralEi}\left (b x^n \log (f)\right ) \log (f)}{n} \]
[Out]
Time = 0.03 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {2246, 2241} \[ \int f^{a+b x^n} x^{-1-n} \, dx=\frac {b f^a \log (f) \operatorname {ExpIntegralEi}\left (b x^n \log (f)\right )}{n}-\frac {x^{-n} f^{a+b x^n}}{n} \]
[In]
[Out]
Rule 2241
Rule 2246
Rubi steps \begin{align*} \text {integral}& = -\frac {f^{a+b x^n} x^{-n}}{n}+(b \log (f)) \int \frac {f^{a+b x^n}}{x} \, dx \\ & = -\frac {f^{a+b x^n} x^{-n}}{n}+\frac {b f^a \text {Ei}\left (b x^n \log (f)\right ) \log (f)}{n} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.53 \[ \int f^{a+b x^n} x^{-1-n} \, dx=\frac {b f^a \Gamma \left (-1,-b x^n \log (f)\right ) \log (f)}{n} \]
[In]
[Out]
Time = 0.12 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.13
method | result | size |
risch | \(-\frac {f^{b \,x^{n}} f^{a} x^{-n}}{n}-\frac {\ln \left (f \right ) b \,f^{a} \operatorname {Ei}_{1}\left (-b \,x^{n} \ln \left (f \right )\right )}{n}\) | \(43\) |
meijerg | \(-\frac {f^{a} \ln \left (f \right ) b \left (-x^{\left (\frac {-1-n}{n}+\frac {1}{n}\right ) n} \left (-b \right )^{\frac {-1-n}{n}+\frac {1}{n}} \ln \left (f \right )^{\frac {-1-n}{n}+\frac {1}{n}}+\frac {\left (-1\right )^{-\frac {-1-n}{n}-\frac {1}{n}} \left (-\Psi \left (1-\frac {-1-n}{n}-\frac {1}{n}\right )+n \ln \left (x \right )+\ln \left (-b \right )+\ln \left (\ln \left (f \right )\right )\right )}{\Gamma \left (1-\frac {-1-n}{n}-\frac {1}{n}\right )}+\frac {\left (-1\right )^{-\frac {-1-n}{n}-\frac {1}{n}} x^{-n} \left (2+2 b \,x^{n} \ln \left (f \right )\right )}{\Gamma \left (2-\frac {-1-n}{n}-\frac {1}{n}\right ) b \ln \left (f \right )}-\frac {2 \left (-1\right )^{-\frac {-1-n}{n}-\frac {1}{n}} x^{-n} {\mathrm e}^{b \,x^{n} \ln \left (f \right )}}{\Gamma \left (2-\frac {-1-n}{n}-\frac {1}{n}\right ) b \ln \left (f \right )}+\frac {2 \left (-1\right )^{-\frac {-1-n}{n}-\frac {1}{n}} \left (-\gamma -\ln \left (-b \,x^{n} \ln \left (f \right )\right )-\operatorname {Ei}_{1}\left (-b \,x^{n} \ln \left (f \right )\right )\right )}{\Gamma \left (2-\frac {-1-n}{n}-\frac {1}{n}\right )}\right )}{n}\) | \(324\) |
[In]
[Out]
none
Time = 0.28 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.13 \[ \int f^{a+b x^n} x^{-1-n} \, dx=\frac {b f^{a} x^{n} {\rm Ei}\left (b x^{n} \log \left (f\right )\right ) \log \left (f\right ) - e^{\left (b x^{n} \log \left (f\right ) + a \log \left (f\right )\right )}}{n x^{n}} \]
[In]
[Out]
\[ \int f^{a+b x^n} x^{-1-n} \, dx=\int f^{a + b x^{n}} x^{- n - 1}\, dx \]
[In]
[Out]
none
Time = 0.23 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.53 \[ \int f^{a+b x^n} x^{-1-n} \, dx=\frac {b f^{a} \Gamma \left (-1, -b x^{n} \log \left (f\right )\right ) \log \left (f\right )}{n} \]
[In]
[Out]
\[ \int f^{a+b x^n} x^{-1-n} \, dx=\int { f^{b x^{n} + a} x^{-n - 1} \,d x } \]
[In]
[Out]
Timed out. \[ \int f^{a+b x^n} x^{-1-n} \, dx=\int \frac {f^{a+b\,x^n}}{x^{n+1}} \,d x \]
[In]
[Out]