Optimal. Leaf size=41 \[ -\frac {x^{m+1} \left (\log (f) \left (-x^n\right )\right )^{-\frac {m+1}{n}} \Gamma \left (\frac {m+1}{n},-x^n \log (f)\right )}{n} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.02, antiderivative size = 41, normalized size of antiderivative = 1.00, 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 = 0.01, size = 41, normalized size = 1.00 \[ -\frac {x^{m+1} \left (\log (f) \left (-x^n\right )\right )^{-\frac {m+1}{n}} \Gamma \left (\frac {m+1}{n},-x^n \log (f)\right )}{n} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.40, size = 0, normalized size = 0.00 \[ {\rm integral}\left (f^{\left (x^{n}\right )} x^{m}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int f^{\left (x^{n}\right )} x^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.15, size = 0, normalized size = 0.00 \[ \int f^{x^{n}} x^{m}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.01, size = 42, normalized size = 1.02 \[ -\frac {x^{m + 1} \Gamma \left (\frac {m + 1}{n}, -x^{n} \log \relax (f)\right )}{\left (-x^{n} \log \relax (f)\right )^{\frac {m + 1}{n}} n} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 3.78, size = 71, normalized size = 1.73 \[ \frac {f^{x^n}\,x^{m+1}\,{\mathrm {e}}^{-\frac {x^n\,\ln \relax (f)}{2}}\,{\mathrm {M}}_{1-\frac {m+n+1}{2\,n},\frac {m+n+1}{2\,n}-\frac {1}{2}}\left (x^n\,\ln \relax (f)\right )}{{\left (x^n\,\ln \relax (f)\right )}^{\frac {m+n+1}{2\,n}}\,\left (m+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int f^{x^{n}} x^{m}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________