Integrand size = 16, antiderivative size = 78 \[ \int x^{-1-2 n} \cos \left (a+b x^n\right ) \, dx=-\frac {x^{-2 n} \cos \left (a+b x^n\right )}{2 n}-\frac {b^2 \cos (a) \operatorname {CosIntegral}\left (b x^n\right )}{2 n}+\frac {b x^{-n} \sin \left (a+b x^n\right )}{2 n}+\frac {b^2 \sin (a) \text {Si}\left (b x^n\right )}{2 n} \]
-1/2*b^2*Ci(b*x^n)*cos(a)/n-1/2*cos(a+b*x^n)/n/(x^(2*n))+1/2*b^2*Si(b*x^n) *sin(a)/n+1/2*b*sin(a+b*x^n)/n/(x^n)
Time = 0.14 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.90 \[ \int x^{-1-2 n} \cos \left (a+b x^n\right ) \, dx=-\frac {x^{-2 n} \left (\cos \left (a+b x^n\right )+b^2 x^{2 n} \cos (a) \operatorname {CosIntegral}\left (b x^n\right )-b x^n \sin \left (a+b x^n\right )-b^2 x^{2 n} \sin (a) \text {Si}\left (b x^n\right )\right )}{2 n} \]
-1/2*(Cos[a + b*x^n] + b^2*x^(2*n)*Cos[a]*CosIntegral[b*x^n] - b*x^n*Sin[a + b*x^n] - b^2*x^(2*n)*Sin[a]*SinIntegral[b*x^n])/(n*x^(2*n))
Time = 0.54 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.83, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.688, Rules used = {3861, 3042, 3778, 25, 3042, 3778, 3042, 3784, 3042, 3780, 3783}
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.
| \(\displaystyle \int x^{-2 n-1} \cos \left (a+b x^n\right ) \, dx\) |
| \(\Big \downarrow \) 3861 |
| \(\displaystyle \frac {\int x^{-3 n} \cos \left (b x^n+a\right )dx^n}{n}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int x^{-3 n} \sin \left (b x^n+a+\frac {\pi }{2}\right )dx^n}{n}\) |
| \(\Big \downarrow \) 3778 |
| \(\displaystyle \frac {\frac {1}{2} b \int -x^{-2 n} \sin \left (b x^n+a\right )dx^n-\frac {1}{2} x^{-2 n} \cos \left (a+b x^n\right )}{n}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {-\frac {1}{2} b \int x^{-2 n} \sin \left (b x^n+a\right )dx^n-\frac {1}{2} x^{-2 n} \cos \left (a+b x^n\right )}{n}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {1}{2} b \int x^{-2 n} \sin \left (b x^n+a\right )dx^n-\frac {1}{2} x^{-2 n} \cos \left (a+b x^n\right )}{n}\) |
| \(\Big \downarrow \) 3778 |
| \(\displaystyle \frac {-\frac {1}{2} b \left (b \int x^{-n} \cos \left (b x^n+a\right )dx^n-x^{-n} \sin \left (a+b x^n\right )\right )-\frac {1}{2} x^{-2 n} \cos \left (a+b x^n\right )}{n}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {1}{2} b \left (b \int x^{-n} \sin \left (b x^n+a+\frac {\pi }{2}\right )dx^n-x^{-n} \sin \left (a+b x^n\right )\right )-\frac {1}{2} x^{-2 n} \cos \left (a+b x^n\right )}{n}\) |
| \(\Big \downarrow \) 3784 |
| \(\displaystyle \frac {-\frac {1}{2} b \left (b \left (\cos (a) \int x^{-n} \cos \left (b x^n\right )dx^n-\sin (a) \int x^{-n} \sin \left (b x^n\right )dx^n\right )-x^{-n} \sin \left (a+b x^n\right )\right )-\frac {1}{2} x^{-2 n} \cos \left (a+b x^n\right )}{n}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {1}{2} b \left (b \left (\cos (a) \int x^{-n} \sin \left (b x^n+\frac {\pi }{2}\right )dx^n-\sin (a) \int x^{-n} \sin \left (b x^n\right )dx^n\right )-x^{-n} \sin \left (a+b x^n\right )\right )-\frac {1}{2} x^{-2 n} \cos \left (a+b x^n\right )}{n}\) |
| \(\Big \downarrow \) 3780 |
| \(\displaystyle \frac {-\frac {1}{2} b \left (b \left (\cos (a) \int x^{-n} \sin \left (b x^n+\frac {\pi }{2}\right )dx^n-\sin (a) \text {Si}\left (b x^n\right )\right )-x^{-n} \sin \left (a+b x^n\right )\right )-\frac {1}{2} x^{-2 n} \cos \left (a+b x^n\right )}{n}\) |
| \(\Big \downarrow \) 3783 |
| \(\displaystyle \frac {-\frac {1}{2} b \left (b \left (\cos (a) \operatorname {CosIntegral}\left (b x^n\right )-\sin (a) \text {Si}\left (b x^n\right )\right )-x^{-n} \sin \left (a+b x^n\right )\right )-\frac {1}{2} x^{-2 n} \cos \left (a+b x^n\right )}{n}\) |
(-1/2*Cos[a + b*x^n]/x^(2*n) - (b*(-(Sin[a + b*x^n]/x^n) + b*(Cos[a]*CosIn tegral[b*x^n] - Sin[a]*SinIntegral[b*x^n])))/2)/n
3.1.82.3.1 Defintions of rubi rules used
Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[(c + d*x)^(m + 1)*(Sin[e + f*x]/(d*(m + 1))), x] - Simp[f/(d*(m + 1)) Int[( c + d*x)^(m + 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && LtQ[m, - 1]
Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[SinInte gral[e + f*x]/d, x] /; FreeQ[{c, d, e, f}, x] && EqQ[d*e - c*f, 0]
Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[CosInte gral[e - Pi/2 + f*x]/d, x] /; FreeQ[{c, d, e, f}, x] && EqQ[d*(e - Pi/2) - c*f, 0]
Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[Cos[(d* e - c*f)/d] Int[Sin[c*(f/d) + f*x]/(c + d*x), x], x] + Simp[Sin[(d*e - c* f)/d] Int[Cos[c*(f/d) + f*x]/(c + d*x), x], x] /; FreeQ[{c, d, e, f}, x] && NeQ[d*e - c*f, 0]
Int[((a_.) + Cos[(c_.) + (d_.)*(x_)^(n_)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbol ] :> Simp[1/n Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*Cos[c + d*x])^ p, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p}, x] && IntegerQ[Simplify[ (m + 1)/n]] && (EqQ[p, 1] || EqQ[m, n - 1] || (IntegerQ[p] && GtQ[Simplify[ (m + 1)/n], 0]))
Time = 1.54 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.83
| method | result | size |
| default | \(\frac {b^{2} \left (-\frac {\cos \left (a +b \,x^{n}\right ) x^{-2 n}}{2 b^{2}}+\frac {\sin \left (a +b \,x^{n}\right ) x^{-n}}{2 b}+\frac {\operatorname {Si}\left (b \,x^{n}\right ) \sin \left (a \right )}{2}-\frac {\operatorname {Ci}\left (b \,x^{n}\right ) \cos \left (a \right )}{2}\right )}{n}\) | \(65\) |
| risch | \(-\frac {\left (i b^{2} {\mathrm e}^{-i a} \pi \,\operatorname {csgn}\left (b \,x^{n}\right ) x^{2 n}-2 i b^{2} {\mathrm e}^{-i a} \operatorname {Si}\left (b \,x^{n}\right ) x^{2 n}-b^{2} {\mathrm e}^{-i a} \operatorname {Ei}_{1}\left (-i b \,x^{n}\right ) x^{2 n}-b^{2} {\mathrm e}^{i a} \operatorname {Ei}_{1}\left (-i b \,x^{n}\right ) x^{2 n}-2 \sin \left (a +b \,x^{n}\right ) x^{n} b +2 \cos \left (a +b \,x^{n}\right )\right ) x^{-2 n}}{4 n}\) | \(129\) |
| meijerg | \(\frac {b^{2} \sqrt {\pi }\, \left (-\frac {x^{2 \left (\frac {-1-2 n}{2 n}+\frac {1}{2 n}\right ) n} 2^{-\frac {-1-2 n}{n}-\frac {1}{n}}}{\sqrt {\pi }\, b^{2}}+\frac {\left (-1\right )^{-\frac {-1-2 n}{2 n}-\frac {1}{2 n}} \left (-\Psi \left (1-\frac {-1-2 n}{2 n}-\frac {1}{2 n}\right )-\Psi \left (\frac {1}{2}-\frac {-1-2 n}{2 n}-\frac {1}{2 n}\right )+2 n \ln \left (x \right )-2 \ln \left (2\right )+\ln \left (b^{2}\right )\right ) \sqrt {2}\, 2^{-\frac {-1-2 n}{n}-\frac {1}{n}-\frac {1}{2}}}{2 \sqrt {\pi }\, \Gamma \left (-\frac {-1-2 n}{n}-\frac {1}{n}\right )}-\frac {\left (-1\right )^{-\frac {-1-2 n}{2 n}-\frac {1}{2 n}} 2^{1-\frac {-1-2 n}{n}-\frac {1}{n}} x^{-2 n} \left (-\frac {9 x^{2 n} b^{2}}{2}+3\right )}{\sqrt {\pi }\, b^{2} \Gamma \left (2-\frac {-1-2 n}{n}-\frac {1}{n}\right )}-\frac {3 \left (-1\right )^{-\frac {-1-2 n}{2 n}-\frac {1}{2 n}} 2^{1-\frac {-1-2 n}{n}-\frac {1}{n}} \gamma }{\sqrt {\pi }\, \Gamma \left (2-\frac {-1-2 n}{n}-\frac {1}{n}\right )}-\frac {3 \left (-1\right )^{-\frac {-1-2 n}{2 n}-\frac {1}{2 n}} 2^{1-\frac {-1-2 n}{n}-\frac {1}{n}} \ln \left (2\right )}{\sqrt {\pi }\, \Gamma \left (2-\frac {-1-2 n}{n}-\frac {1}{n}\right )}-\frac {3 \left (-1\right )^{-\frac {-1-2 n}{2 n}-\frac {1}{2 n}} 2^{1-\frac {-1-2 n}{n}-\frac {1}{n}} \ln \left (\frac {b \,x^{n}}{2}\right )}{\sqrt {\pi }\, \Gamma \left (2-\frac {-1-2 n}{n}-\frac {1}{n}\right )}+\frac {3 \,2^{1-\frac {-1-2 n}{n}-\frac {1}{n}} \left (-1\right )^{-\frac {-1-2 n}{2 n}-\frac {1}{2 n}} x^{-2 n} \cos \left (b \,x^{n}\right )}{\sqrt {\pi }\, b^{2} \Gamma \left (2-\frac {-1-2 n}{n}-\frac {1}{n}\right )}-\frac {3 \left (-1\right )^{-\frac {-1-2 n}{2 n}-\frac {1}{2 n}} 2^{1-\frac {-1-2 n}{n}-\frac {1}{n}} x^{-n} \sin \left (b \,x^{n}\right )}{\sqrt {\pi }\, b \Gamma \left (2-\frac {-1-2 n}{n}-\frac {1}{n}\right )}+\frac {3 \,2^{1-\frac {-1-2 n}{n}-\frac {1}{n}} \left (-1\right )^{-\frac {-1-2 n}{2 n}-\frac {1}{2 n}} \operatorname {Ci}\left (b \,x^{n}\right )}{\sqrt {\pi }\, \Gamma \left (2-\frac {-1-2 n}{n}-\frac {1}{n}\right )}\right ) \cos \left (a \right )}{8 n}-\frac {b^{2} \sqrt {\pi }\, \left (-\frac {4 x^{-n} \cos \left (b \,x^{n}\right )}{\sqrt {\pi }\, b}-\frac {4 x^{-2 n} \sin \left (b \,x^{n}\right )}{\sqrt {\pi }\, b^{2}}-\frac {4 \,\operatorname {Si}\left (b \,x^{n}\right )}{\sqrt {\pi }}\right ) \sin \left (a \right )}{8 n}\) | \(761\) |
1/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))
Time = 0.29 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.90 \[ \int x^{-1-2 n} \cos \left (a+b x^n\right ) \, dx=-\frac {b^{2} x^{2 \, n} \cos \left (a\right ) \operatorname {Ci}\left (b x^{n}\right ) - b^{2} x^{2 \, n} \sin \left (a\right ) \operatorname {Si}\left (b x^{n}\right ) - b x^{n} \sin \left (b x^{n} + a\right ) + \cos \left (b x^{n} + a\right )}{2 \, n x^{2 \, n}} \]
-1/2*(b^2*x^(2*n)*cos(a)*cos_integral(b*x^n) - b^2*x^(2*n)*sin(a)*sin_inte gral(b*x^n) - b*x^n*sin(b*x^n + a) + cos(b*x^n + a))/(n*x^(2*n))
\[ \int x^{-1-2 n} \cos \left (a+b x^n\right ) \, dx=\int x^{- 2 n - 1} \cos {\left (a + b x^{n} \right )}\, dx \]
\[ \int x^{-1-2 n} \cos \left (a+b x^n\right ) \, dx=\int { x^{-2 \, n - 1} \cos \left (b x^{n} + a\right ) \,d x } \]
\[ \int x^{-1-2 n} \cos \left (a+b x^n\right ) \, dx=\int { x^{-2 \, n - 1} \cos \left (b x^{n} + a\right ) \,d x } \]
Timed out. \[ \int x^{-1-2 n} \cos \left (a+b x^n\right ) \, dx=\int \frac {\cos \left (a+b\,x^n\right )}{x^{2\,n+1}} \,d x \]