\(\int \cos ^2(a+b x^n) \, dx\) [74]

Optimal result

Integrand size = 10, antiderivative size = 102 \[ \int \cos ^2\left (a+b x^n\right ) \, dx=\frac {x}{2}-\frac {2^{-2-\frac {1}{n}} e^{2 i a} x \left (-i b x^n\right )^{-1/n} \Gamma \left (\frac {1}{n},-2 i b x^n\right )}{n}-\frac {2^{-2-\frac {1}{n}} e^{-2 i a} x \left (i b x^n\right )^{-1/n} \Gamma \left (\frac {1}{n},2 i b x^n\right )}{n} \]

[Out]

1/2*x-2^(-2-1/n)*exp(2*I*a)*x*GAMMA(1/n,-2*I*b*x^n)/n/((-I*b*x^n)^(1/n))-2^(-2-1/n)*x*GAMMA(1/n,2*I*b*x^n)/exp
(2*I*a)/n/((I*b*x^n)^(1/n))

Rubi [A] (verified)

Time = 0.10 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {3449, 3447, 2239} \[ \int \cos ^2\left (a+b x^n\right ) \, dx=-\frac {e^{2 i a} 2^{-\frac {1}{n}-2} x \left (-i b x^n\right )^{-1/n} \Gamma \left (\frac {1}{n},-2 i b x^n\right )}{n}-\frac {e^{-2 i a} 2^{-\frac {1}{n}-2} x \left (i b x^n\right )^{-1/n} \Gamma \left (\frac {1}{n},2 i b x^n\right )}{n}+\frac {x}{2} \]

[In]

Int[Cos[a + b*x^n]^2,x]

[Out]

x/2 - (2^(-2 - n^(-1))*E^((2*I)*a)*x*Gamma[n^(-1), (-2*I)*b*x^n])/(n*((-I)*b*x^n)^n^(-1)) - (2^(-2 - n^(-1))*x
*Gamma[n^(-1), (2*I)*b*x^n])/(E^((2*I)*a)*n*(I*b*x^n)^n^(-1))

Rule 2239

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_)), x_Symbol] :> Simp[(-F^a)*(c + d*x)*(Gamma[1/n, (-b)*(c + d
*x)^n*Log[F]]/(d*n*((-b)*(c + d*x)^n*Log[F])^(1/n))), x] /; FreeQ[{F, a, b, c, d, n}, x] &&  !IntegerQ[2/n]

Rule 3447

Int[Cos[(c_.) + (d_.)*((e_.) + (f_.)*(x_))^(n_)], x_Symbol] :> Dist[1/2, Int[E^((-c)*I - d*I*(e + f*x)^n), x],
x] + Dist[1/2, Int[E^(c*I + d*I*(e + f*x)^n), x], x] /; FreeQ[{c, d, e, f, n}, x]

Rule 3449

Int[((a_.) + Cos[(c_.) + (d_.)*((e_.) + (f_.)*(x_))^(n_)]*(b_.))^(p_), x_Symbol] :> Int[ExpandTrigReduce[(a +
b*Cos[c + d*(e + f*x)^n])^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && IGtQ[p, 1]

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {1}{2}+\frac {1}{2} \cos \left (2 a+2 b x^n\right )\right ) \, dx \\ & = \frac {x}{2}+\frac {1}{2} \int \cos \left (2 a+2 b x^n\right ) \, dx \\ & = \frac {x}{2}+\frac {1}{4} \int e^{-2 i a-2 i b x^n} \, dx+\frac {1}{4} \int e^{2 i a+2 i b x^n} \, dx \\ & = \frac {x}{2}-\frac {2^{-2-\frac {1}{n}} e^{2 i a} x \left (-i b x^n\right )^{-1/n} \Gamma \left (\frac {1}{n},-2 i b x^n\right )}{n}-\frac {2^{-2-\frac {1}{n}} e^{-2 i a} x \left (i b x^n\right )^{-1/n} \Gamma \left (\frac {1}{n},2 i b x^n\right )}{n} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.27 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.92 \[ \int \cos ^2\left (a+b x^n\right ) \, dx=-\frac {x \left (-2 n+2^{-1/n} e^{2 i a} \left (-i b x^n\right )^{-1/n} \Gamma \left (\frac {1}{n},-2 i b x^n\right )+2^{-1/n} e^{-2 i a} \left (i b x^n\right )^{-1/n} \Gamma \left (\frac {1}{n},2 i b x^n\right )\right )}{4 n} \]

[In]

Integrate[Cos[a + b*x^n]^2,x]

[Out]

-1/4*(x*(-2*n + (E^((2*I)*a)*Gamma[n^(-1), (-2*I)*b*x^n])/(2^n^(-1)*((-I)*b*x^n)^n^(-1)) + Gamma[n^(-1), (2*I)
*b*x^n]/(2^n^(-1)*E^((2*I)*a)*(I*b*x^n)^n^(-1))))/n

Maple [F]

\[\int \left (\cos ^{2}\left (a +b \,x^{n}\right )\right )d x\]

[In]

int(cos(a+b*x^n)^2,x)

[Out]

int(cos(a+b*x^n)^2,x)

Fricas [F]

\[ \int \cos ^2\left (a+b x^n\right ) \, dx=\int { \cos \left (b x^{n} + a\right )^{2} \,d x } \]

[In]

integrate(cos(a+b*x^n)^2,x, algorithm="fricas")

[Out]

integral(cos(b*x^n + a)^2, x)

Sympy [F]

\[ \int \cos ^2\left (a+b x^n\right ) \, dx=\int \cos ^{2}{\left (a + b x^{n} \right )}\, dx \]

[In]

integrate(cos(a+b*x**n)**2,x)

[Out]

Integral(cos(a + b*x**n)**2, x)

Maxima [F]

\[ \int \cos ^2\left (a+b x^n\right ) \, dx=\int { \cos \left (b x^{n} + a\right )^{2} \,d x } \]

[In]

integrate(cos(a+b*x^n)^2,x, algorithm="maxima")

[Out]

1/2*x + 1/2*integrate(cos(2*b*x^n + 2*a), x)

Giac [F]

\[ \int \cos ^2\left (a+b x^n\right ) \, dx=\int { \cos \left (b x^{n} + a\right )^{2} \,d x } \]

[In]

integrate(cos(a+b*x^n)^2,x, algorithm="giac")

[Out]

integrate(cos(b*x^n + a)^2, x)

Mupad [F(-1)]

Timed out. \[ \int \cos ^2\left (a+b x^n\right ) \, dx=\int {\cos \left (a+b\,x^n\right )}^2 \,d x \]

[In]

int(cos(a + b*x^n)^2,x)

[Out]

int(cos(a + b*x^n)^2, x)