Integrand size = 18, antiderivative size = 165 \[ \int x^{-1-2 n} \cos ^3\left (a+b x^n\right ) \, dx=-\frac {3 x^{-2 n} \cos \left (a+b x^n\right )}{8 n}-\frac {x^{-2 n} \cos \left (3 \left (a+b x^n\right )\right )}{8 n}-\frac {3 b^2 \cos (a) \operatorname {CosIntegral}\left (b x^n\right )}{8 n}-\frac {9 b^2 \cos (3 a) \operatorname {CosIntegral}\left (3 b x^n\right )}{8 n}+\frac {3 b x^{-n} \sin \left (a+b x^n\right )}{8 n}+\frac {3 b x^{-n} \sin \left (3 \left (a+b x^n\right )\right )}{8 n}+\frac {3 b^2 \sin (a) \text {Si}\left (b x^n\right )}{8 n}+\frac {9 b^2 \sin (3 a) \text {Si}\left (3 b x^n\right )}{8 n} \]
-3/8*b^2*Ci(b*x^n)*cos(a)/n-9/8*b^2*Ci(3*b*x^n)*cos(3*a)/n-3/8*cos(a+b*x^n )/n/(x^(2*n))-1/8*cos(3*a+3*b*x^n)/n/(x^(2*n))+3/8*b^2*Si(b*x^n)*sin(a)/n+ 9/8*b^2*Si(3*b*x^n)*sin(3*a)/n+3/8*b*sin(a+b*x^n)/n/(x^n)+3/8*b*sin(3*a+3* b*x^n)/n/(x^n)
Time = 0.38 (sec) , antiderivative size = 141, normalized size of antiderivative = 0.85 \[ \int x^{-1-2 n} \cos ^3\left (a+b x^n\right ) \, dx=-\frac {x^{-2 n} \left (3 \cos \left (a+b x^n\right )+\cos \left (3 \left (a+b x^n\right )\right )+3 b^2 x^{2 n} \cos (a) \operatorname {CosIntegral}\left (b x^n\right )+9 b^2 x^{2 n} \cos (3 a) \operatorname {CosIntegral}\left (3 b x^n\right )-3 b x^n \sin \left (a+b x^n\right )-3 b x^n \sin \left (3 \left (a+b x^n\right )\right )-3 b^2 x^{2 n} \sin (a) \text {Si}\left (b x^n\right )-9 b^2 x^{2 n} \sin (3 a) \text {Si}\left (3 b x^n\right )\right )}{8 n} \]
-1/8*(3*Cos[a + b*x^n] + Cos[3*(a + b*x^n)] + 3*b^2*x^(2*n)*Cos[a]*CosInte gral[b*x^n] + 9*b^2*x^(2*n)*Cos[3*a]*CosIntegral[3*b*x^n] - 3*b*x^n*Sin[a + b*x^n] - 3*b*x^n*Sin[3*(a + b*x^n)] - 3*b^2*x^(2*n)*Sin[a]*SinIntegral[b *x^n] - 9*b^2*x^(2*n)*Sin[3*a]*SinIntegral[3*b*x^n])/(n*x^(2*n))
Time = 0.49 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {3907, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int x^{-2 n-1} \cos ^3\left (a+b x^n\right ) \, dx\) |
\(\Big \downarrow \) 3907 |
\(\displaystyle \int \left (\frac {3}{4} x^{-2 n-1} \cos \left (a+b x^n\right )+\frac {1}{4} x^{-2 n-1} \cos \left (3 a+3 b x^n\right )\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {3 b^2 \cos (a) \operatorname {CosIntegral}\left (b x^n\right )}{8 n}-\frac {9 b^2 \cos (3 a) \operatorname {CosIntegral}\left (3 b x^n\right )}{8 n}+\frac {3 b^2 \sin (a) \text {Si}\left (b x^n\right )}{8 n}+\frac {9 b^2 \sin (3 a) \text {Si}\left (3 b x^n\right )}{8 n}+\frac {3 b x^{-n} \sin \left (a+b x^n\right )}{8 n}+\frac {3 b x^{-n} \sin \left (3 \left (a+b x^n\right )\right )}{8 n}-\frac {3 x^{-2 n} \cos \left (a+b x^n\right )}{8 n}-\frac {x^{-2 n} \cos \left (3 \left (a+b x^n\right )\right )}{8 n}\) |
(-3*Cos[a + b*x^n])/(8*n*x^(2*n)) - Cos[3*(a + b*x^n)]/(8*n*x^(2*n)) - (3* b^2*Cos[a]*CosIntegral[b*x^n])/(8*n) - (9*b^2*Cos[3*a]*CosIntegral[3*b*x^n ])/(8*n) + (3*b*Sin[a + b*x^n])/(8*n*x^n) + (3*b*Sin[3*(a + b*x^n)])/(8*n* x^n) + (3*b^2*Sin[a]*SinIntegral[b*x^n])/(8*n) + (9*b^2*Sin[3*a]*SinIntegr al[3*b*x^n])/(8*n)
3.1.84.3.1 Defintions of rubi rules used
Int[((a_.) + Cos[(c_.) + (d_.)*(x_)^(n_)]*(b_.))^(p_)*((e_.)*(x_))^(m_.), x _Symbol] :> Int[ExpandTrigReduce[(e*x)^m, (a + b*Cos[c + d*x^n])^p, x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && IGtQ[p, 0]
Time = 10.52 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.87
method | result | size |
default | \(\frac {3 b^{2} \left (-\frac {\cos \left (a +b \,x^{n}\right ) x^{-2 n}}{2 b^{2}}+\frac {\sin \left (a +b \,x^{n}\right ) x^{-n}}{2 b}+\frac {\operatorname {Si}\left (b \,x^{n}\right ) \sin \left (a \right )}{2}-\frac {\operatorname {Ci}\left (b \,x^{n}\right ) \cos \left (a \right )}{2}\right )}{4 n}+\frac {9 b^{2} \left (-\frac {\cos \left (3 a +3 b \,x^{n}\right ) x^{-2 n}}{18 b^{2}}+\frac {\sin \left (3 a +3 b \,x^{n}\right ) x^{-n}}{6 b}+\frac {\operatorname {Si}\left (3 b \,x^{n}\right ) \sin \left (3 a \right )}{2}-\frac {\operatorname {Ci}\left (3 b \,x^{n}\right ) \cos \left (3 a \right )}{2}\right )}{4 n}\) | \(144\) |
risch | \(-\frac {\left (9 i b^{2} {\mathrm e}^{-3 i a} \pi \,\operatorname {csgn}\left (b \,x^{n}\right ) x^{2 n}+3 i b^{2} {\mathrm e}^{-i a} \pi \,\operatorname {csgn}\left (b \,x^{n}\right ) x^{2 n}-18 i b^{2} {\mathrm e}^{-3 i a} \operatorname {Si}\left (3 b \,x^{n}\right ) x^{2 n}-6 i b^{2} {\mathrm e}^{-i a} \operatorname {Si}\left (b \,x^{n}\right ) x^{2 n}-9 b^{2} {\mathrm e}^{-3 i a} \operatorname {Ei}_{1}\left (-3 i b \,x^{n}\right ) x^{2 n}-3 b^{2} {\mathrm e}^{-i a} \operatorname {Ei}_{1}\left (-i b \,x^{n}\right ) x^{2 n}-3 b^{2} {\mathrm e}^{i a} \operatorname {Ei}_{1}\left (-i b \,x^{n}\right ) x^{2 n}-9 b^{2} {\mathrm e}^{3 i a} \operatorname {Ei}_{1}\left (-3 i b \,x^{n}\right ) x^{2 n}-6 \sin \left (a +b \,x^{n}\right ) x^{n} b -6 \sin \left (3 a +3 b \,x^{n}\right ) b \,x^{n}+6 \cos \left (a +b \,x^{n}\right )+2 \cos \left (3 a +3 b \,x^{n}\right )\right ) x^{-2 n}}{16 n}\) | \(253\) |
3/4/n*b^2*(-1/2*cos(a+b*x^n)/b^2/(x^n)^2+1/2*sin(a+b*x^n)/b/(x^n)+1/2*Si(b *x^n)*sin(a)-1/2*Ci(b*x^n)*cos(a))+9/4/n*b^2*(-1/18*cos(3*a+3*b*x^n)/(x^n) ^2/b^2+1/6*sin(3*a+3*b*x^n)/(x^n)/b+1/2*Si(3*b*x^n)*sin(3*a)-1/2*Ci(3*b*x^ n)*cos(3*a))
Time = 0.26 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.77 \[ \int x^{-1-2 n} \cos ^3\left (a+b x^n\right ) \, dx=-\frac {9 \, b^{2} x^{2 \, n} \cos \left (3 \, a\right ) \operatorname {Ci}\left (3 \, b x^{n}\right ) + 3 \, b^{2} x^{2 \, n} \cos \left (a\right ) \operatorname {Ci}\left (b x^{n}\right ) - 12 \, b x^{n} \cos \left (b x^{n} + a\right )^{2} \sin \left (b x^{n} + a\right ) - 9 \, b^{2} x^{2 \, n} \sin \left (3 \, a\right ) \operatorname {Si}\left (3 \, b x^{n}\right ) - 3 \, b^{2} x^{2 \, n} \sin \left (a\right ) \operatorname {Si}\left (b x^{n}\right ) + 4 \, \cos \left (b x^{n} + a\right )^{3}}{8 \, n x^{2 \, n}} \]
-1/8*(9*b^2*x^(2*n)*cos(3*a)*cos_integral(3*b*x^n) + 3*b^2*x^(2*n)*cos(a)* cos_integral(b*x^n) - 12*b*x^n*cos(b*x^n + a)^2*sin(b*x^n + a) - 9*b^2*x^( 2*n)*sin(3*a)*sin_integral(3*b*x^n) - 3*b^2*x^(2*n)*sin(a)*sin_integral(b* x^n) + 4*cos(b*x^n + a)^3)/(n*x^(2*n))
\[ \int x^{-1-2 n} \cos ^3\left (a+b x^n\right ) \, dx=\int x^{- 2 n - 1} \cos ^{3}{\left (a + b x^{n} \right )}\, dx \]
\[ \int x^{-1-2 n} \cos ^3\left (a+b x^n\right ) \, dx=\int { x^{-2 \, n - 1} \cos \left (b x^{n} + a\right )^{3} \,d x } \]
\[ \int x^{-1-2 n} \cos ^3\left (a+b x^n\right ) \, dx=\int { x^{-2 \, n - 1} \cos \left (b x^{n} + a\right )^{3} \,d x } \]
Timed out. \[ \int x^{-1-2 n} \cos ^3\left (a+b x^n\right ) \, dx=\int \frac {{\cos \left (a+b\,x^n\right )}^3}{x^{2\,n+1}} \,d x \]