Optimal. Leaf size=41 \[ -\frac{x^{m+1} \left (\log (f) \left (-x^n\right )\right )^{-\frac{m+1}{n}} \text{Gamma}\left (\frac{m+1}{n},\log (f) \left (-x^n\right )\right )}{n} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.015585, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {2218} \[ -\frac{x^{m+1} \left (\log (f) \left (-x^n\right )\right )^{-\frac{m+1}{n}} \text{Gamma}\left (\frac{m+1}{n},\log (f) \left (-x^n\right )\right )}{n} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2218
Rubi steps
\begin{align*} \int f^{x^n} x^m \, dx &=-\frac{x^{1+m} \Gamma \left (\frac{1+m}{n},-x^n \log (f)\right ) \left (-x^n \log (f)\right )^{-\frac{1+m}{n}}}{n}\\ \end{align*}
Mathematica [A] time = 0.0084263, size = 41, normalized size = 1. \[ -\frac{x^{m+1} \left (\log (f) \left (-x^n\right )\right )^{-\frac{m+1}{n}} \text{Gamma}\left (\frac{m+1}{n},\log (f) \left (-x^n\right )\right )}{n} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.181, size = 0, normalized size = 0. \begin{align*} \int{f}^{{x}^{n}}{x}^{m}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.11877, size = 57, normalized size = 1.39 \begin{align*} -\frac{x^{m + 1} \Gamma \left (\frac{m + 1}{n}, -x^{n} \log \left (f\right )\right )}{\left (-x^{n} \log \left (f\right )\right )^{\frac{m + 1}{n}} n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (f^{\left (x^{n}\right )} x^{m}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int f^{x^{n}} x^{m}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int f^{\left (x^{n}\right )} x^{m}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]