Optimal. Leaf size=69 \[ -\frac {b \sin (2 a) \text {Ci}\left (2 b x^n\right )}{n}-\frac {b \cos (2 a) \text {Si}\left (2 b x^n\right )}{n}-\frac {x^{-n} \cos \left (2 \left (a+b x^n\right )\right )}{2 n}-\frac {x^{-n}}{2 n} \]
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Rubi [A] time = 0.12, antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3426, 3380, 3297, 3303, 3299, 3302} \[ -\frac {b \sin (2 a) \text {CosIntegral}\left (2 b x^n\right )}{n}-\frac {b \cos (2 a) \text {Si}\left (2 b x^n\right )}{n}-\frac {x^{-n} \cos \left (2 \left (a+b x^n\right )\right )}{2 n}-\frac {x^{-n}}{2 n} \]
Antiderivative was successfully verified.
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Rule 3297
Rule 3299
Rule 3302
Rule 3303
Rule 3380
Rule 3426
Rubi steps
\begin {align*} \int x^{-1-n} \cos ^2\left (a+b x^n\right ) \, dx &=\int \left (\frac {x^{-1-n}}{2}+\frac {1}{2} x^{-1-n} \cos \left (2 a+2 b x^n\right )\right ) \, dx\\ &=-\frac {x^{-n}}{2 n}+\frac {1}{2} \int x^{-1-n} \cos \left (2 a+2 b x^n\right ) \, dx\\ &=-\frac {x^{-n}}{2 n}+\frac {\operatorname {Subst}\left (\int \frac {\cos (2 a+2 b x)}{x^2} \, dx,x,x^n\right )}{2 n}\\ &=-\frac {x^{-n}}{2 n}-\frac {x^{-n} \cos \left (2 \left (a+b x^n\right )\right )}{2 n}-\frac {b \operatorname {Subst}\left (\int \frac {\sin (2 a+2 b x)}{x} \, dx,x,x^n\right )}{n}\\ &=-\frac {x^{-n}}{2 n}-\frac {x^{-n} \cos \left (2 \left (a+b x^n\right )\right )}{2 n}-\frac {(b \cos (2 a)) \operatorname {Subst}\left (\int \frac {\sin (2 b x)}{x} \, dx,x,x^n\right )}{n}-\frac {(b \sin (2 a)) \operatorname {Subst}\left (\int \frac {\cos (2 b x)}{x} \, dx,x,x^n\right )}{n}\\ &=-\frac {x^{-n}}{2 n}-\frac {x^{-n} \cos \left (2 \left (a+b x^n\right )\right )}{2 n}-\frac {b \text {Ci}\left (2 b x^n\right ) \sin (2 a)}{n}-\frac {b \cos (2 a) \text {Si}\left (2 b x^n\right )}{n}\\ \end {align*}
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Mathematica [A] time = 0.19, size = 53, normalized size = 0.77 \[ -\frac {x^{-n} \left (b \sin (2 a) x^n \text {Ci}\left (2 b x^n\right )+b \cos (2 a) x^n \text {Si}\left (2 b x^n\right )+\cos ^2\left (a+b x^n\right )\right )}{n} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.96, size = 72, normalized size = 1.04 \[ -\frac {b x^{n} \operatorname {Ci}\left (2 \, b x^{n}\right ) \sin \left (2 \, a\right ) + b x^{n} \operatorname {Ci}\left (-2 \, b x^{n}\right ) \sin \left (2 \, a\right ) + 2 \, b x^{n} \cos \left (2 \, a\right ) \operatorname {Si}\left (2 \, b x^{n}\right ) + 2 \, \cos \left (b x^{n} + a\right )^{2}}{2 \, n x^{n}} \]
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 \[ \int x^{-n - 1} \cos \left (b x^{n} + a\right )^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 65, normalized size = 0.94 \[ -\frac {x^{-n}}{2 n}+\frac {b \left (-\frac {\cos \left (2 a +2 b \,x^{n}\right ) x^{-n}}{2 b}-\Si \left (2 b \,x^{n}\right ) \cos \left (2 a \right )-\Ci \left (2 b \,x^{n}\right ) \sin \left (2 a \right )\right )}{n} \]
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 \[ \frac {n x^{n} \int \frac {\cos \left (2 \, b x^{n} + 2 \, a\right )}{x x^{n}}\,{d x} - 1}{2 \, n x^{n}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\cos \left (a+b\,x^n\right )}^2}{x^{n+1}} \,d x \]
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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