Integrand size = 18, antiderivative size = 113 \[ \int x^{-1-n} \cos ^3\left (a+b x^n\right ) \, dx=-\frac {3 x^{-n} \cos \left (a+b x^n\right )}{4 n}-\frac {x^{-n} \cos \left (3 \left (a+b x^n\right )\right )}{4 n}-\frac {3 b \operatorname {CosIntegral}\left (b x^n\right ) \sin (a)}{4 n}-\frac {3 b \operatorname {CosIntegral}\left (3 b x^n\right ) \sin (3 a)}{4 n}-\frac {3 b \cos (a) \text {Si}\left (b x^n\right )}{4 n}-\frac {3 b \cos (3 a) \text {Si}\left (3 b x^n\right )}{4 n} \]
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Time = 0.30 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3507, 3461, 3378, 3384, 3380, 3383} \[ \int x^{-1-n} \cos ^3\left (a+b x^n\right ) \, dx=-\frac {3 b \sin (a) \operatorname {CosIntegral}\left (b x^n\right )}{4 n}-\frac {3 b \sin (3 a) \operatorname {CosIntegral}\left (3 b x^n\right )}{4 n}-\frac {3 b \cos (a) \text {Si}\left (b x^n\right )}{4 n}-\frac {3 b \cos (3 a) \text {Si}\left (3 b x^n\right )}{4 n}-\frac {3 x^{-n} \cos \left (a+b x^n\right )}{4 n}-\frac {x^{-n} \cos \left (3 \left (a+b x^n\right )\right )}{4 n} \]
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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 {3}{4} x^{-1-n} \cos \left (a+b x^n\right )+\frac {1}{4} x^{-1-n} \cos \left (3 a+3 b x^n\right )\right ) \, dx \\ & = \frac {1}{4} \int x^{-1-n} \cos \left (3 a+3 b x^n\right ) \, dx+\frac {3}{4} \int x^{-1-n} \cos \left (a+b x^n\right ) \, dx \\ & = \frac {\text {Subst}\left (\int \frac {\cos (3 a+3 b x)}{x^2} \, dx,x,x^n\right )}{4 n}+\frac {3 \text {Subst}\left (\int \frac {\cos (a+b x)}{x^2} \, dx,x,x^n\right )}{4 n} \\ & = -\frac {3 x^{-n} \cos \left (a+b x^n\right )}{4 n}-\frac {x^{-n} \cos \left (3 \left (a+b x^n\right )\right )}{4 n}-\frac {(3 b) \text {Subst}\left (\int \frac {\sin (a+b x)}{x} \, dx,x,x^n\right )}{4 n}-\frac {(3 b) \text {Subst}\left (\int \frac {\sin (3 a+3 b x)}{x} \, dx,x,x^n\right )}{4 n} \\ & = -\frac {3 x^{-n} \cos \left (a+b x^n\right )}{4 n}-\frac {x^{-n} \cos \left (3 \left (a+b x^n\right )\right )}{4 n}-\frac {(3 b \cos (a)) \text {Subst}\left (\int \frac {\sin (b x)}{x} \, dx,x,x^n\right )}{4 n}-\frac {(3 b \cos (3 a)) \text {Subst}\left (\int \frac {\sin (3 b x)}{x} \, dx,x,x^n\right )}{4 n}-\frac {(3 b \sin (a)) \text {Subst}\left (\int \frac {\cos (b x)}{x} \, dx,x,x^n\right )}{4 n}-\frac {(3 b \sin (3 a)) \text {Subst}\left (\int \frac {\cos (3 b x)}{x} \, dx,x,x^n\right )}{4 n} \\ & = -\frac {3 x^{-n} \cos \left (a+b x^n\right )}{4 n}-\frac {x^{-n} \cos \left (3 \left (a+b x^n\right )\right )}{4 n}-\frac {3 b \operatorname {CosIntegral}\left (b x^n\right ) \sin (a)}{4 n}-\frac {3 b \operatorname {CosIntegral}\left (3 b x^n\right ) \sin (3 a)}{4 n}-\frac {3 b \cos (a) \text {Si}\left (b x^n\right )}{4 n}-\frac {3 b \cos (3 a) \text {Si}\left (3 b x^n\right )}{4 n} \\ \end{align*}
Time = 0.25 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.84 \[ \int x^{-1-n} \cos ^3\left (a+b x^n\right ) \, dx=-\frac {x^{-n} \left (3 \cos \left (a+b x^n\right )+\cos \left (3 \left (a+b x^n\right )\right )+3 b x^n \operatorname {CosIntegral}\left (b x^n\right ) \sin (a)+3 b x^n \operatorname {CosIntegral}\left (3 b x^n\right ) \sin (3 a)+3 b x^n \cos (a) \text {Si}\left (b x^n\right )+3 b x^n \cos (3 a) \text {Si}\left (3 b x^n\right )\right )}{4 n} \]
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Time = 10.36 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.89
method | result | size |
default | \(\frac {3 b \left (-\frac {\cos \left (a +b \,x^{n}\right ) x^{-n}}{b}-\operatorname {Si}\left (b \,x^{n}\right ) \cos \left (a \right )-\operatorname {Ci}\left (b \,x^{n}\right ) \sin \left (a \right )\right )}{4 n}+\frac {3 b \left (-\frac {\cos \left (3 a +3 b \,x^{n}\right ) x^{-n}}{3 b}-\operatorname {Si}\left (3 b \,x^{n}\right ) \cos \left (3 a \right )-\operatorname {Ci}\left (3 b \,x^{n}\right ) \sin \left (3 a \right )\right )}{4 n}\) | \(101\) |
risch | \(-\frac {\left (-3 b \,{\mathrm e}^{-3 i a} \pi \,\operatorname {csgn}\left (b \,x^{n}\right ) x^{n}-3 b \,{\mathrm e}^{-i a} \pi \,\operatorname {csgn}\left (b \,x^{n}\right ) x^{n}-3 i b \,{\mathrm e}^{-3 i a} \operatorname {Ei}_{1}\left (-3 i b \,x^{n}\right ) x^{n}-3 i b \,{\mathrm e}^{-i a} \operatorname {Ei}_{1}\left (-i b \,x^{n}\right ) x^{n}+3 i b \,{\mathrm e}^{i a} \operatorname {Ei}_{1}\left (-i b \,x^{n}\right ) x^{n}+3 i b \,{\mathrm e}^{3 i a} \operatorname {Ei}_{1}\left (-3 i b \,x^{n}\right ) x^{n}+6 b \,{\mathrm e}^{-3 i a} \operatorname {Si}\left (3 b \,x^{n}\right ) x^{n}+6 b \,{\mathrm e}^{-i a} \operatorname {Si}\left (b \,x^{n}\right ) x^{n}+6 \cos \left (a +b \,x^{n}\right )+2 \cos \left (3 a +3 b \,x^{n}\right )\right ) x^{-n}}{8 n}\) | \(190\) |
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Time = 0.26 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.75 \[ \int x^{-1-n} \cos ^3\left (a+b x^n\right ) \, dx=-\frac {3 \, b x^{n} \operatorname {Ci}\left (3 \, b x^{n}\right ) \sin \left (3 \, a\right ) + 3 \, b x^{n} \operatorname {Ci}\left (b x^{n}\right ) \sin \left (a\right ) + 3 \, b x^{n} \cos \left (3 \, a\right ) \operatorname {Si}\left (3 \, b x^{n}\right ) + 3 \, b x^{n} \cos \left (a\right ) \operatorname {Si}\left (b x^{n}\right ) + 4 \, \cos \left (b x^{n} + a\right )^{3}}{4 \, n x^{n}} \]
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\[ \int x^{-1-n} \cos ^3\left (a+b x^n\right ) \, dx=\int x^{- n - 1} \cos ^{3}{\left (a + b x^{n} \right )}\, dx \]
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\[ \int x^{-1-n} \cos ^3\left (a+b x^n\right ) \, dx=\int { x^{-n - 1} \cos \left (b x^{n} + a\right )^{3} \,d x } \]
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\[ \int x^{-1-n} \cos ^3\left (a+b x^n\right ) \, dx=\int { x^{-n - 1} \cos \left (b x^{n} + a\right )^{3} \,d x } \]
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Timed out. \[ \int x^{-1-n} \cos ^3\left (a+b x^n\right ) \, dx=\int \frac {{\cos \left (a+b\,x^n\right )}^3}{x^{n+1}} \,d x \]
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