Integrand size = 18, antiderivative size = 69 \[ \int x^{-1-n} \cos ^2\left (a+b x^n\right ) \, dx=-\frac {x^{-n}}{2 n}-\frac {x^{-n} \cos \left (2 \left (a+b x^n\right )\right )}{2 n}-\frac {b \operatorname {CosIntegral}\left (2 b x^n\right ) \sin (2 a)}{n}-\frac {b \cos (2 a) \text {Si}\left (2 b x^n\right )}{n} \]
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Time = 0.19 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3507, 3461, 3378, 3384, 3380, 3383} \[ \int x^{-1-n} \cos ^2\left (a+b x^n\right ) \, dx=-\frac {b \sin (2 a) \operatorname {CosIntegral}\left (2 b x^n\right )}{n}-\frac {b \cos (2 a) \text {Si}\left (2 b x^n\right )}{n}-\frac {x^{-n} \cos \left (2 \left (a+b x^n\right )\right )}{2 n}-\frac {x^{-n}}{2 n} \]
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Rule 3378
Rule 3380
Rule 3383
Rule 3384
Rule 3461
Rule 3507
Rubi steps \begin{align*} \text {integral}& = \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 {\text {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 \text {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)) \text {Subst}\left (\int \frac {\sin (2 b x)}{x} \, dx,x,x^n\right )}{n}-\frac {(b \sin (2 a)) \text {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 \operatorname {CosIntegral}\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*}
Time = 0.20 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.77 \[ \int x^{-1-n} \cos ^2\left (a+b x^n\right ) \, dx=-\frac {x^{-n} \left (\cos ^2\left (a+b x^n\right )+b x^n \operatorname {CosIntegral}\left (2 b x^n\right ) \sin (2 a)+b x^n \cos (2 a) \text {Si}\left (2 b x^n\right )\right )}{n} \]
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Time = 2.57 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.94
method | result | size |
default | \(-\frac {x^{-n}}{2 n}+\frac {b \left (-\frac {\cos \left (2 a +2 b \,x^{n}\right ) x^{-n}}{2 b}-\operatorname {Si}\left (2 b \,x^{n}\right ) \cos \left (2 a \right )-\operatorname {Ci}\left (2 b \,x^{n}\right ) \sin \left (2 a \right )\right )}{n}\) | \(65\) |
risch | \(\frac {\left (b \,{\mathrm e}^{-2 i a} \pi \,\operatorname {csgn}\left (b \,x^{n}\right ) x^{n}+i b \,{\mathrm e}^{-2 i a} \operatorname {Ei}_{1}\left (-2 i b \,x^{n}\right ) x^{n}-i b \,{\mathrm e}^{2 i a} \operatorname {Ei}_{1}\left (-2 i b \,x^{n}\right ) x^{n}-2 b \,{\mathrm e}^{-2 i a} \operatorname {Si}\left (2 b \,x^{n}\right ) x^{n}-\cos \left (2 a +2 b \,x^{n}\right )-1\right ) x^{-n}}{2 n}\) | \(103\) |
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Time = 0.26 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.77 \[ \int x^{-1-n} \cos ^2\left (a+b x^n\right ) \, dx=-\frac {b x^{n} \operatorname {Ci}\left (2 \, b x^{n}\right ) \sin \left (2 \, a\right ) + b x^{n} \cos \left (2 \, a\right ) \operatorname {Si}\left (2 \, b x^{n}\right ) + \cos \left (b x^{n} + a\right )^{2}}{n x^{n}} \]
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\[ \int x^{-1-n} \cos ^2\left (a+b x^n\right ) \, dx=\int x^{- n - 1} \cos ^{2}{\left (a + b x^{n} \right )}\, dx \]
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\[ \int x^{-1-n} \cos ^2\left (a+b x^n\right ) \, dx=\int { x^{-n - 1} \cos \left (b x^{n} + a\right )^{2} \,d x } \]
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\[ \int x^{-1-n} \cos ^2\left (a+b x^n\right ) \, dx=\int { x^{-n - 1} \cos \left (b x^{n} + a\right )^{2} \,d x } \]
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Timed out. \[ \int x^{-1-n} \cos ^2\left (a+b x^n\right ) \, dx=\int \frac {{\cos \left (a+b\,x^n\right )}^2}{x^{n+1}} \,d x \]
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