\(\int \cos (a+b x^n) \, dx\) [73]

Optimal result

Integrand size = 8, antiderivative size = 83 \[ \int \cos \left (a+b x^n\right ) \, dx=-\frac {e^{i a} x \left (-i b x^n\right )^{-1/n} \Gamma \left (\frac {1}{n},-i b x^n\right )}{2 n}-\frac {e^{-i a} x \left (i b x^n\right )^{-1/n} \Gamma \left (\frac {1}{n},i b x^n\right )}{2 n} \]

[Out]

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

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3447, 2239} \[ \int \cos \left (a+b x^n\right ) \, dx=-\frac {e^{i a} x \left (-i b x^n\right )^{-1/n} \Gamma \left (\frac {1}{n},-i b x^n\right )}{2 n}-\frac {e^{-i a} x \left (i b x^n\right )^{-1/n} \Gamma \left (\frac {1}{n},i b x^n\right )}{2 n} \]

[In]

Int[Cos[a + b*x^n],x]

[Out]

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

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

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.11 \[ \int \cos \left (a+b x^n\right ) \, dx=-\frac {x \left (b^2 x^{2 n}\right )^{-1/n} \left (\left (-i b x^n\right )^{\frac {1}{n}} \Gamma \left (\frac {1}{n},i b x^n\right ) (\cos (a)-i \sin (a))+\left (i b x^n\right )^{\frac {1}{n}} \Gamma \left (\frac {1}{n},-i b x^n\right ) (\cos (a)+i \sin (a))\right )}{2 n} \]

[In]

Integrate[Cos[a + b*x^n],x]

[Out]

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

Maple [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 4.

Time = 0.52 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.90

method	result	size
meijerg	\(x {}_{1}^{}{\moversetsp {}{\mundersetsp {}{F_{2}^{}}}}\left (\frac {1}{2 n};\frac {1}{2},1+\frac {1}{2 n};-\frac {x^{2 n} b^{2}}{4}\right ) \cos \left (a \right )-\frac {b \,x^{1+n} {}_{1}^{}{\moversetsp {}{\mundersetsp {}{F_{2}^{}}}}\left (\frac {1}{2}+\frac {1}{2 n};\frac {3}{2},\frac {3}{2}+\frac {1}{2 n};-\frac {x^{2 n} b^{2}}{4}\right ) \sin \left (a \right )}{1+n}\)	\(75\)

[In]

int(cos(a+b*x^n),x,method=_RETURNVERBOSE)

[Out]

x*hypergeom([1/2/n],[1/2,1+1/2/n],-1/4*x^(2*n)*b^2)*cos(a)-b/(1+n)*x^(1+n)*hypergeom([1/2+1/2/n],[3/2,3/2+1/2/
n],-1/4*x^(2*n)*b^2)*sin(a)

Fricas [F]

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

[In]

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

[Out]

integral(cos(b*x^n + a), x)

Sympy [F]

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

[In]

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

[Out]

Integral(cos(a + b*x**n), x)

Maxima [F]

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

[In]

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

[Out]

integrate(cos(b*x^n + a), x)

Giac [F]

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

[In]

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

[Out]

integrate(cos(b*x^n + a), x)

Mupad [F(-1)]

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

[In]

int(cos(a + b*x^n),x)

[Out]

int(cos(a + b*x^n), x)