Optimal. Leaf size=69 \[ -\frac{b \sin (2 a) \text{CosIntegral}\left (2 b x^n\right )}{n}-\frac{b \cos (2 a) \text{Si}\left (2 b x^n\right )}{n}-\frac{x^{-n} \cos \left (2 \left (a+b x^n\right )\right )}{2 n}-\frac{x^{-n}}{2 n} \]
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Rubi [A] time = 0.120562, antiderivative size = 69, normalized size of antiderivative = 1., 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 3426
Rule 3380
Rule 3297
Rule 3303
Rule 3299
Rule 3302
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*}
Mathematica [A] time = 0.186908, size = 53, normalized size = 0.77 \[ -\frac{x^{-n} \left (b \sin (2 a) x^n \text{CosIntegral}\left (2 b x^n\right )+b \cos (2 a) x^n \text{Si}\left (2 b x^n\right )+\cos ^2\left (a+b x^n\right )\right )}{n} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.044, size = 65, normalized size = 0.9 \begin{align*} -{\frac{1}{2\,n{x}^{n}}}+{\frac{b}{n} \left ( -{\frac{\cos \left ( 2\,a+2\,b{x}^{n} \right ) }{2\,b{x}^{n}}}-{\it Si} \left ( 2\,b{x}^{n} \right ) \cos \left ( 2\,a \right ) -{\it Ci} \left ( 2\,b{x}^{n} \right ) \sin \left ( 2\,a \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.73607, size = 209, normalized size = 3.03 \begin{align*} -\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}} \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^{-n - 1} \cos \left (b x^{n} + a\right )^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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