Optimal. Leaf size=105 \[ -\frac{e^{i a} x^{m+1} \left (-i b x^n\right )^{-\frac{m+1}{n}} \text{Gamma}\left (\frac{m+1}{n},-i b x^n\right )}{2 n}-\frac{e^{-i a} x^{m+1} \left (i b x^n\right )^{-\frac{m+1}{n}} \text{Gamma}\left (\frac{m+1}{n},i b x^n\right )}{2 n} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0748536, antiderivative size = 105, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {3424, 2218} \[ -\frac{e^{i a} x^{m+1} \left (-i b x^n\right )^{-\frac{m+1}{n}} \text{Gamma}\left (\frac{m+1}{n},-i b x^n\right )}{2 n}-\frac{e^{-i a} x^{m+1} \left (i b x^n\right )^{-\frac{m+1}{n}} \text{Gamma}\left (\frac{m+1}{n},i b x^n\right )}{2 n} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3424
Rule 2218
Rubi steps
\begin{align*} \int x^m \cos \left (a+b x^n\right ) \, dx &=\frac{1}{2} \int e^{-i a-i b x^n} x^m \, dx+\frac{1}{2} \int e^{i a+i b x^n} x^m \, dx\\ &=-\frac{e^{i a} x^{1+m} \left (-i b x^n\right )^{-\frac{1+m}{n}} \Gamma \left (\frac{1+m}{n},-i b x^n\right )}{2 n}-\frac{e^{-i a} x^{1+m} \left (i b x^n\right )^{-\frac{1+m}{n}} \Gamma \left (\frac{1+m}{n},i b x^n\right )}{2 n}\\ \end{align*}
Mathematica [A] time = 0.200067, size = 115, normalized size = 1.1 \[ -\frac{x^{m+1} \left (b^2 x^{2 n}\right )^{-\frac{m+1}{n}} \left ((\cos (a)-i \sin (a)) \left (-i b x^n\right )^{\frac{m+1}{n}} \text{Gamma}\left (\frac{m+1}{n},i b x^n\right )+(\cos (a)+i \sin (a)) \left (i b x^n\right )^{\frac{m+1}{n}} \text{Gamma}\left (\frac{m+1}{n},-i b x^n\right )\right )}{2 n} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [C] time = 0.162, size = 111, normalized size = 1.1 \begin{align*}{\frac{{x}^{1+m}\cos \left ( a \right ) }{1+m}{\mbox{$_1$F$_2$}({\frac{m}{2\,n}}+{\frac{1}{2\,n}};\,{\frac{1}{2}},1+{\frac{m}{2\,n}}+{\frac{1}{2\,n}};\,-{\frac{{x}^{2\,n}{b}^{2}}{4}})}}-{\frac{b{x}^{m+n+1}\sin \left ( a \right ) }{m+n+1}{\mbox{$_1$F$_2$}({\frac{1}{2}}+{\frac{m}{2\,n}}+{\frac{1}{2\,n}};\,{\frac{3}{2}},{\frac{3}{2}}+{\frac{m}{2\,n}}+{\frac{1}{2\,n}};\,-{\frac{{x}^{2\,n}{b}^{2}}{4}})}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{m} \cos \left (b x^{n} + a\right )\,{d x} \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 (x^{m} \cos \left (b x^{n} + a\right ), 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 x^{m} \cos{\left (a + b x^{n} \right )}\, 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 x^{m} \cos \left (b x^{n} + a\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]