Optimal. Leaf size=95 \[ -\frac {b^2 \cos (2 a) \text {Ci}\left (2 b x^n\right )}{n}+\frac {b^2 \sin (2 a) \text {Si}\left (2 b x^n\right )}{n}+\frac {b x^{-n} \sin \left (2 \left (a+b x^n\right )\right )}{2 n}-\frac {x^{-2 n} \cos \left (2 \left (a+b x^n\right )\right )}{4 n}-\frac {x^{-2 n}}{4 n} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.15, antiderivative size = 95, normalized size of antiderivative = 1.00, number of steps used = 8, 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} \[ -\frac {b^2 \cos (2 a) \text {CosIntegral}\left (2 b x^n\right )}{n}+\frac {b^2 \sin (2 a) \text {Si}\left (2 b x^n\right )}{n}+\frac {b x^{-n} \sin \left (2 \left (a+b x^n\right )\right )}{2 n}-\frac {x^{-2 n} \cos \left (2 \left (a+b x^n\right )\right )}{4 n}-\frac {x^{-2 n}}{4 n} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3297
Rule 3299
Rule 3302
Rule 3303
Rule 3380
Rule 3426
Rubi steps
\begin {align*} \int x^{-1-2 n} \cos ^2\left (a+b x^n\right ) \, dx &=\int \left (\frac {1}{2} x^{-1-2 n}+\frac {1}{2} x^{-1-2 n} \cos \left (2 a+2 b x^n\right )\right ) \, dx\\ &=-\frac {x^{-2 n}}{4 n}+\frac {1}{2} \int x^{-1-2 n} \cos \left (2 a+2 b x^n\right ) \, dx\\ &=-\frac {x^{-2 n}}{4 n}+\frac {\operatorname {Subst}\left (\int \frac {\cos (2 a+2 b x)}{x^3} \, dx,x,x^n\right )}{2 n}\\ &=-\frac {x^{-2 n}}{4 n}-\frac {x^{-2 n} \cos \left (2 \left (a+b x^n\right )\right )}{4 n}-\frac {b \operatorname {Subst}\left (\int \frac {\sin (2 a+2 b x)}{x^2} \, dx,x,x^n\right )}{2 n}\\ &=-\frac {x^{-2 n}}{4 n}-\frac {x^{-2 n} \cos \left (2 \left (a+b x^n\right )\right )}{4 n}+\frac {b x^{-n} \sin \left (2 \left (a+b x^n\right )\right )}{2 n}-\frac {b^2 \operatorname {Subst}\left (\int \frac {\cos (2 a+2 b x)}{x} \, dx,x,x^n\right )}{n}\\ &=-\frac {x^{-2 n}}{4 n}-\frac {x^{-2 n} \cos \left (2 \left (a+b x^n\right )\right )}{4 n}+\frac {b x^{-n} \sin \left (2 \left (a+b x^n\right )\right )}{2 n}-\frac {\left (b^2 \cos (2 a)\right ) \operatorname {Subst}\left (\int \frac {\cos (2 b x)}{x} \, dx,x,x^n\right )}{n}+\frac {\left (b^2 \sin (2 a)\right ) \operatorname {Subst}\left (\int \frac {\sin (2 b x)}{x} \, dx,x,x^n\right )}{n}\\ &=-\frac {x^{-2 n}}{4 n}-\frac {x^{-2 n} \cos \left (2 \left (a+b x^n\right )\right )}{4 n}-\frac {b^2 \cos (2 a) \text {Ci}\left (2 b x^n\right )}{n}+\frac {b x^{-n} \sin \left (2 \left (a+b x^n\right )\right )}{2 n}+\frac {b^2 \sin (2 a) \text {Si}\left (2 b x^n\right )}{n}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.22, size = 82, normalized size = 0.86 \[ -\frac {x^{-2 n} \left (4 b^2 \cos (2 a) x^{2 n} \text {Ci}\left (2 b x^n\right )-4 b^2 \sin (2 a) x^{2 n} \text {Si}\left (2 b x^n\right )-2 b x^n \sin \left (2 \left (a+b x^n\right )\right )+\cos \left (2 \left (a+b x^n\right )\right )+1\right )}{4 n} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.70, size = 106, normalized size = 1.12 \[ -\frac {b^{2} x^{2 \, n} \cos \left (2 \, a\right ) \operatorname {Ci}\left (2 \, b x^{n}\right ) + b^{2} x^{2 \, n} \cos \left (2 \, a\right ) \operatorname {Ci}\left (-2 \, b x^{n}\right ) - 2 \, b^{2} x^{2 \, n} \sin \left (2 \, a\right ) \operatorname {Si}\left (2 \, b x^{n}\right ) - 2 \, b x^{n} \cos \left (b x^{n} + a\right ) \sin \left (b x^{n} + a\right ) + \cos \left (b x^{n} + a\right )^{2}}{2 \, n x^{2 \, n}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{-2 \, n - 1} \cos \left (b x^{n} + a\right )^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.06, size = 89, normalized size = 0.94 \[ -\frac {x^{-2 n}}{4 n}+\frac {2 b^{2} \left (-\frac {x^{-2 n} \cos \left (2 a +2 b \,x^{n}\right )}{8 b^{2}}+\frac {\sin \left (2 a +2 b \,x^{n}\right ) x^{-n}}{4 b}+\frac {\Si \left (2 b \,x^{n}\right ) \sin \left (2 a \right )}{2}-\frac {\Ci \left (2 b \,x^{n}\right ) \cos \left (2 a \right )}{2}\right )}{n} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {2 \, n x^{2 \, n} \int \frac {\cos \left (2 \, b x^{n} + 2 \, a\right )}{x x^{2 \, n}}\,{d x} - 1}{4 \, n x^{2 \, n}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\cos \left (a+b\,x^n\right )}^2}{x^{2\,n+1}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________