Optimal. Leaf size=34 \[ \frac{\cos \left (a+b x^n\right )}{b^2 n}+\frac{x^n \sin \left (a+b x^n\right )}{b n} \]
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Rubi [A] time = 0.030392, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188, Rules used = {3380, 3296, 2638} \[ \frac{\cos \left (a+b x^n\right )}{b^2 n}+\frac{x^n \sin \left (a+b x^n\right )}{b n} \]
Antiderivative was successfully verified.
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Rule 3380
Rule 3296
Rule 2638
Rubi steps
\begin{align*} \int x^{-1+2 n} \cos \left (a+b x^n\right ) \, dx &=\frac{\operatorname{Subst}\left (\int x \cos (a+b x) \, dx,x,x^n\right )}{n}\\ &=\frac{x^n \sin \left (a+b x^n\right )}{b n}-\frac{\operatorname{Subst}\left (\int \sin (a+b x) \, dx,x,x^n\right )}{b n}\\ &=\frac{\cos \left (a+b x^n\right )}{b^2 n}+\frac{x^n \sin \left (a+b x^n\right )}{b n}\\ \end{align*}
Mathematica [A] time = 0.0628097, size = 29, normalized size = 0.85 \[ \frac{b x^n \sin \left (a+b x^n\right )+\cos \left (a+b x^n\right )}{b^2 n} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.013, size = 44, normalized size = 1.3 \begin{align*}{\frac{\cos \left ( a+b{x}^{n} \right ) + \left ( a+b{x}^{n} \right ) \sin \left ( a+b{x}^{n} \right ) -a\sin \left ( a+b{x}^{n} \right ) }{n{b}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.00682, size = 39, normalized size = 1.15 \begin{align*} \frac{b x^{n} \sin \left (b x^{n} + a\right ) + \cos \left (b x^{n} + a\right )}{b^{2} n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.84286, size = 66, normalized size = 1.94 \begin{align*} \frac{b x^{n} \sin \left (b x^{n} + a\right ) + \cos \left (b x^{n} + a\right )}{b^{2} n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{2 \, n - 1} \cos \left (b x^{n} + a\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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