3.75 \(\int \cos ^3(a+b x^n) \, dx\)

Optimal. Leaf size=179 \[ -\frac{3 e^{i a} x \left (-i b x^n\right )^{-1/n} \text{Gamma}\left (\frac{1}{n},-i b x^n\right )}{8 n}-\frac{e^{3 i a} 3^{-1/n} x \left (-i b x^n\right )^{-1/n} \text{Gamma}\left (\frac{1}{n},-3 i b x^n\right )}{8 n}-\frac{3 e^{-i a} x \left (i b x^n\right )^{-1/n} \text{Gamma}\left (\frac{1}{n},i b x^n\right )}{8 n}-\frac{e^{-3 i a} 3^{-1/n} x \left (i b x^n\right )^{-1/n} \text{Gamma}\left (\frac{1}{n},3 i b x^n\right )}{8 n} \]

[Out]

(-3*E^(I*a)*x*Gamma[n^(-1), (-I)*b*x^n])/(8*n*((-I)*b*x^n)^n^(-1)) - (3*x*Gamma[n^(-1), I*b*x^n])/(8*E^(I*a)*n
*(I*b*x^n)^n^(-1)) - (E^((3*I)*a)*x*Gamma[n^(-1), (-3*I)*b*x^n])/(8*3^n^(-1)*n*((-I)*b*x^n)^n^(-1)) - (x*Gamma
[n^(-1), (3*I)*b*x^n])/(8*3^n^(-1)*E^((3*I)*a)*n*(I*b*x^n)^n^(-1))

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Rubi [A]  time = 0.0789607, antiderivative size = 179, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {3368, 3366, 2208} \[ -\frac{3 e^{i a} x \left (-i b x^n\right )^{-1/n} \text{Gamma}\left (\frac{1}{n},-i b x^n\right )}{8 n}-\frac{e^{3 i a} 3^{-1/n} x \left (-i b x^n\right )^{-1/n} \text{Gamma}\left (\frac{1}{n},-3 i b x^n\right )}{8 n}-\frac{3 e^{-i a} x \left (i b x^n\right )^{-1/n} \text{Gamma}\left (\frac{1}{n},i b x^n\right )}{8 n}-\frac{e^{-3 i a} 3^{-1/n} x \left (i b x^n\right )^{-1/n} \text{Gamma}\left (\frac{1}{n},3 i b x^n\right )}{8 n} \]

Antiderivative was successfully verified.

[In]

Int[Cos[a + b*x^n]^3,x]

[Out]

(-3*E^(I*a)*x*Gamma[n^(-1), (-I)*b*x^n])/(8*n*((-I)*b*x^n)^n^(-1)) - (3*x*Gamma[n^(-1), I*b*x^n])/(8*E^(I*a)*n
*(I*b*x^n)^n^(-1)) - (E^((3*I)*a)*x*Gamma[n^(-1), (-3*I)*b*x^n])/(8*3^n^(-1)*n*((-I)*b*x^n)^n^(-1)) - (x*Gamma
[n^(-1), (3*I)*b*x^n])/(8*3^n^(-1)*E^((3*I)*a)*n*(I*b*x^n)^n^(-1))

Rule 3368

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]

Rule 3366

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 2208

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]

Rubi steps

\begin{align*} \int \cos ^3\left (a+b x^n\right ) \, dx &=\int \left (\frac{3}{4} \cos \left (a+b x^n\right )+\frac{1}{4} \cos \left (3 a+3 b x^n\right )\right ) \, dx\\ &=\frac{1}{4} \int \cos \left (3 a+3 b x^n\right ) \, dx+\frac{3}{4} \int \cos \left (a+b x^n\right ) \, dx\\ &=\frac{1}{8} \int e^{-3 i a-3 i b x^n} \, dx+\frac{1}{8} \int e^{3 i a+3 i b x^n} \, dx+\frac{3}{8} \int e^{-i a-i b x^n} \, dx+\frac{3}{8} \int e^{i a+i b x^n} \, dx\\ &=-\frac{3 e^{i a} x \left (-i b x^n\right )^{-1/n} \Gamma \left (\frac{1}{n},-i b x^n\right )}{8 n}-\frac{3 e^{-i a} x \left (i b x^n\right )^{-1/n} \Gamma \left (\frac{1}{n},i b x^n\right )}{8 n}-\frac{3^{-1/n} e^{3 i a} x \left (-i b x^n\right )^{-1/n} \Gamma \left (\frac{1}{n},-3 i b x^n\right )}{8 n}-\frac{3^{-1/n} e^{-3 i a} x \left (i b x^n\right )^{-1/n} \Gamma \left (\frac{1}{n},3 i b x^n\right )}{8 n}\\ \end{align*}

Mathematica [A]  time = 0.239687, size = 173, normalized size = 0.97 \[ -\frac{e^{-3 i a} 3^{-1/n} x \left (b^2 x^{2 n}\right )^{-1/n} \left (e^{2 i a} 3^{\frac{1}{n}+1} \left (-i b x^n\right )^{\frac{1}{n}} \text{Gamma}\left (\frac{1}{n},i b x^n\right )+e^{4 i a} 3^{\frac{1}{n}+1} \left (i b x^n\right )^{\frac{1}{n}} \text{Gamma}\left (\frac{1}{n},-i b x^n\right )+e^{6 i a} \left (i b x^n\right )^{\frac{1}{n}} \text{Gamma}\left (\frac{1}{n},-3 i b x^n\right )+\left (-i b x^n\right )^{\frac{1}{n}} \text{Gamma}\left (\frac{1}{n},3 i b x^n\right )\right )}{8 n} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[a + b*x^n]^3,x]

[Out]

-(x*(3^(1 + n^(-1))*E^((4*I)*a)*(I*b*x^n)^n^(-1)*Gamma[n^(-1), (-I)*b*x^n] + 3^(1 + n^(-1))*E^((2*I)*a)*((-I)*
b*x^n)^n^(-1)*Gamma[n^(-1), I*b*x^n] + E^((6*I)*a)*(I*b*x^n)^n^(-1)*Gamma[n^(-1), (-3*I)*b*x^n] + ((-I)*b*x^n)
^n^(-1)*Gamma[n^(-1), (3*I)*b*x^n]))/(8*3^n^(-1)*E^((3*I)*a)*n*(b^2*x^(2*n))^n^(-1))

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Maple [F]  time = 180., size = 0, normalized size = 0. \begin{align*} \int \left ( \cos \left ( a+b{x}^{n} \right ) \right ) ^{3}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(a+b*x^n)^3,x)

[Out]

int(cos(a+b*x^n)^3,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \cos \left (b x^{n} + a\right )^{3}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(a+b*x^n)^3,x, algorithm="maxima")

[Out]

integrate(cos(b*x^n + a)^3, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\cos \left (b x^{n} + a\right )^{3}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(a+b*x^n)^3,x, algorithm="fricas")

[Out]

integral(cos(b*x^n + a)^3, x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \cos ^{3}{\left (a + b x^{n} \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(a+b*x**n)**3,x)

[Out]

Integral(cos(a + b*x**n)**3, x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \cos \left (b x^{n} + a\right )^{3}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(a+b*x^n)^3,x, algorithm="giac")

[Out]

integrate(cos(b*x^n + a)^3, x)