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 = 0.03, antiderivative size = 34, normalized size of antiderivative = 1.00, 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 2638
Rule 3296
Rule 3380
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*}
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Mathematica [A] time = 0.07, 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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fricas [A] time = 0.77, size = 29, normalized size = 0.85 \[ \frac {b x^{n} \sin \left (b x^{n} + a\right ) + \cos \left (b x^{n} + a\right )}{b^{2} n} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{2 \, n - 1} \cos \left (b x^{n} + a\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 44, normalized size = 1.29 \[ \frac {\cos \left (a +b \,x^{n}\right )+\left (a +b \,x^{n}\right ) \sin \left (a +b \,x^{n}\right )-a \sin \left (a +b \,x^{n}\right )}{n \,b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.41, size = 29, normalized size = 0.85 \[ \frac {b x^{n} \sin \left (b x^{n} + a\right ) + \cos \left (b x^{n} + a\right )}{b^{2} n} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.03 \[ \int x^{2\,n-1}\,\cos \left (a+b\,x^n\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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