Optimal. Leaf size=78 \[ -\frac {b^2 \cos (a) \text {Ci}\left (b x^n\right )}{2 n}+\frac {b^2 \sin (a) \text {Si}\left (b x^n\right )}{2 n}+\frac {b x^{-n} \sin \left (a+b x^n\right )}{2 n}-\frac {x^{-2 n} \cos \left (a+b x^n\right )}{2 n} \]
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Rubi [A] time = 0.11, antiderivative size = 78, normalized size of antiderivative = 1.00, number of steps used = 6, 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} \[ -\frac {b^2 \cos (a) \text {CosIntegral}\left (b x^n\right )}{2 n}+\frac {b^2 \sin (a) \text {Si}\left (b x^n\right )}{2 n}+\frac {b x^{-n} \sin \left (a+b x^n\right )}{2 n}-\frac {x^{-2 n} \cos \left (a+b x^n\right )}{2 n} \]
Antiderivative was successfully verified.
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Rule 3297
Rule 3299
Rule 3302
Rule 3303
Rule 3380
Rubi steps
\begin {align*} \int x^{-1-2 n} \cos \left (a+b x^n\right ) \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\cos (a+b x)}{x^3} \, dx,x,x^n\right )}{n}\\ &=-\frac {x^{-2 n} \cos \left (a+b x^n\right )}{2 n}-\frac {b \operatorname {Subst}\left (\int \frac {\sin (a+b x)}{x^2} \, dx,x,x^n\right )}{2 n}\\ &=-\frac {x^{-2 n} \cos \left (a+b x^n\right )}{2 n}+\frac {b x^{-n} \sin \left (a+b x^n\right )}{2 n}-\frac {b^2 \operatorname {Subst}\left (\int \frac {\cos (a+b x)}{x} \, dx,x,x^n\right )}{2 n}\\ &=-\frac {x^{-2 n} \cos \left (a+b x^n\right )}{2 n}+\frac {b x^{-n} \sin \left (a+b x^n\right )}{2 n}-\frac {\left (b^2 \cos (a)\right ) \operatorname {Subst}\left (\int \frac {\cos (b x)}{x} \, dx,x,x^n\right )}{2 n}+\frac {\left (b^2 \sin (a)\right ) \operatorname {Subst}\left (\int \frac {\sin (b x)}{x} \, dx,x,x^n\right )}{2 n}\\ &=-\frac {x^{-2 n} \cos \left (a+b x^n\right )}{2 n}-\frac {b^2 \cos (a) \text {Ci}\left (b x^n\right )}{2 n}+\frac {b x^{-n} \sin \left (a+b x^n\right )}{2 n}+\frac {b^2 \sin (a) \text {Si}\left (b x^n\right )}{2 n}\\ \end {align*}
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Mathematica [A] time = 0.14, size = 70, normalized size = 0.90 \[ -\frac {x^{-2 n} \left (b^2 \cos (a) x^{2 n} \text {Ci}\left (b x^n\right )-b^2 \sin (a) x^{2 n} \text {Si}\left (b x^n\right )-b x^n \sin \left (a+b x^n\right )+\cos \left (a+b x^n\right )\right )}{2 n} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.92, size = 90, normalized size = 1.15 \[ -\frac {b^{2} x^{2 \, n} \cos \relax (a) \operatorname {Ci}\left (b x^{n}\right ) + b^{2} x^{2 \, n} \cos \relax (a) \operatorname {Ci}\left (-b x^{n}\right ) - 2 \, b^{2} x^{2 \, n} \sin \relax (a) \operatorname {Si}\left (b x^{n}\right ) - 2 \, b x^{n} \sin \left (b x^{n} + a\right ) + 2 \, \cos \left (b x^{n} + a\right )}{4 \, n x^{2 \, n}} \]
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 = 65, normalized size = 0.83 \[ \frac {b^{2} \left (-\frac {\cos \left (a +b \,x^{n}\right ) x^{-2 n}}{2 b^{2}}+\frac {\sin \left (a +b \,x^{n}\right ) x^{-n}}{2 b}+\frac {\Si \left (b \,x^{n}\right ) \sin \relax (a )}{2}-\frac {\Ci \left (b \,x^{n}\right ) \cos \relax (a )}{2}\right )}{n} \]
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 \[ \int x^{-2 \, n - 1} \cos \left (b x^{n} + a\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\cos \left (a+b\,x^n\right )}{x^{2\,n+1}} \,d x \]
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 \[ \int x^{- 2 n - 1} \cos {\left (a + b x^{n} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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