Optimal. Leaf size=47 \[ -\frac {b \sin (a) \text {Ci}\left (b x^n\right )}{n}-\frac {b \cos (a) \text {Si}\left (b x^n\right )}{n}-\frac {x^{-n} \cos \left (a+b x^n\right )}{n} \]
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Rubi [A] time = 0.09, antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {3380, 3297, 3303, 3299, 3302} \[ -\frac {b \sin (a) \text {CosIntegral}\left (b x^n\right )}{n}-\frac {b \cos (a) \text {Si}\left (b x^n\right )}{n}-\frac {x^{-n} \cos \left (a+b x^n\right )}{n} \]
Antiderivative was successfully verified.
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Rule 3297
Rule 3299
Rule 3302
Rule 3303
Rule 3380
Rubi steps
\begin {align*} \int x^{-1-n} \cos \left (a+b x^n\right ) \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\cos (a+b x)}{x^2} \, dx,x,x^n\right )}{n}\\ &=-\frac {x^{-n} \cos \left (a+b x^n\right )}{n}-\frac {b \operatorname {Subst}\left (\int \frac {\sin (a+b x)}{x} \, dx,x,x^n\right )}{n}\\ &=-\frac {x^{-n} \cos \left (a+b x^n\right )}{n}-\frac {(b \cos (a)) \operatorname {Subst}\left (\int \frac {\sin (b x)}{x} \, dx,x,x^n\right )}{n}-\frac {(b \sin (a)) \operatorname {Subst}\left (\int \frac {\cos (b x)}{x} \, dx,x,x^n\right )}{n}\\ &=-\frac {x^{-n} \cos \left (a+b x^n\right )}{n}-\frac {b \text {Ci}\left (b x^n\right ) \sin (a)}{n}-\frac {b \cos (a) \text {Si}\left (b x^n\right )}{n}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 45, normalized size = 0.96 \[ -\frac {x^{-n} \left (b \sin (a) x^n \text {Ci}\left (b x^n\right )+b \cos (a) x^n \text {Si}\left (b x^n\right )+\cos \left (a+b x^n\right )\right )}{n} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.84, size = 62, normalized size = 1.32 \[ -\frac {b x^{n} \operatorname {Ci}\left (b x^{n}\right ) \sin \relax (a) + b x^{n} \operatorname {Ci}\left (-b x^{n}\right ) \sin \relax (a) + 2 \, b x^{n} \cos \relax (a) \operatorname {Si}\left (b x^{n}\right ) + 2 \, \cos \left (b x^{n} + a\right )}{2 \, n x^{n}} \]
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 \[ \int x^{-n - 1} \cos \left (b x^{n} + a\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 45, normalized size = 0.96 \[ \frac {b \left (-\frac {\cos \left (a +b \,x^{n}\right ) x^{-n}}{b}-\Si \left (b \,x^{n}\right ) \cos \relax (a )-\Ci \left (b \,x^{n}\right ) \sin \relax (a )\right )}{n} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{-n - 1} \cos \left (b x^{n} + a\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {\cos \left (a+b\,x^n\right )}{x^{n+1}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{- n - 1} \cos {\left (a + b x^{n} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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