3.1.75 \(\int \cos ^3(a+b x^n) \, dx\) [75]

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

[Out]

-3/8*exp(I*a)*x*GAMMA(1/n,-I*b*x^n)/n/((-I*b*x^n)^(1/n))-3/8*x*GAMMA(1/n,I*b*x^n)/exp(I*a)/n/((I*b*x^n)^(1/n))
-1/8*exp(3*I*a)*x*GAMMA(1/n,-3*I*b*x^n)/(3^(1/n))/n/((-I*b*x^n)^(1/n))-1/8*x*GAMMA(1/n,3*I*b*x^n)/(3^(1/n))/ex
p(3*I*a)/n/((I*b*x^n)^(1/n))

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Rubi [A]
time = 0.06, antiderivative size = 179, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {3449, 3447, 2239} \begin {gather*} -\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} \end {gather*}

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 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*} \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*}

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Mathematica [A]
time = 0.27, size = 173, normalized size = 0.97 \begin {gather*} -\frac {3^{-1/n} e^{-3 i a} x \left (b^2 x^{2 n}\right )^{-1/n} \left (3^{1+\frac {1}{n}} e^{4 i a} \left (i b x^n\right )^{\frac {1}{n}} \text {Gamma}\left (\frac {1}{n},-i b x^n\right )+3^{1+\frac {1}{n}} e^{2 i a} \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} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Cos[a + b*x^n]^3,x]

[Out]

-1/8*(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]))/(3^n^(-1)*E^((3*I)*a)*n*(b^2*x^(2*n))^n^(-1))

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Maple [F]
time = 0.08, size = 0, normalized size = 0.00 \[\int \cos ^{3}\left (a +b \,x^{n}\right )\, dx\]

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \cos ^{3}{\left (a + b x^{n} \right )}\, dx \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int {\cos \left (a+b\,x^n\right )}^3 \,d x \end {gather*}

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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