3.109 \(\int \cos ^3(a+\frac{1}{3} \sqrt{-\frac{1}{n^2}} \log (c x^n)) \, dx\)

Optimal. Leaf size=128 \[ \frac{9}{16} x e^{a \sqrt{-\frac{1}{n^2}} n} \left (c x^n\right )^{\left .-\frac{1}{3}\right /n}+\frac{9}{32} x e^{-a \sqrt{-\frac{1}{n^2}} n} \left (c x^n\right )^{\left .\frac{1}{3}\right /n}+\frac{1}{16} x e^{-3 a \sqrt{-\frac{1}{n^2}} n} \left (c x^n\right )^{\frac{1}{n}}+\frac{1}{8} x e^{3 a \sqrt{-\frac{1}{n^2}} n} \log (x) \left (c x^n\right )^{-1/n} \]

[Out]

(9*E^(a*Sqrt[-n^(-2)]*n)*x)/(16*(c*x^n)^(1/(3*n))) + (9*x*(c*x^n)^(1/(3*n)))/(32*E^(a*Sqrt[-n^(-2)]*n)) + (x*(
c*x^n)^n^(-1))/(16*E^(3*a*Sqrt[-n^(-2)]*n)) + (E^(3*a*Sqrt[-n^(-2)]*n)*x*Log[x])/(8*(c*x^n)^n^(-1))

________________________________________________________________________________________

Rubi [A]  time = 0.0958812, antiderivative size = 128, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {4484, 4490} \[ \frac{9}{16} x e^{a \sqrt{-\frac{1}{n^2}} n} \left (c x^n\right )^{\left .-\frac{1}{3}\right /n}+\frac{9}{32} x e^{-a \sqrt{-\frac{1}{n^2}} n} \left (c x^n\right )^{\left .\frac{1}{3}\right /n}+\frac{1}{16} x e^{-3 a \sqrt{-\frac{1}{n^2}} n} \left (c x^n\right )^{\frac{1}{n}}+\frac{1}{8} x e^{3 a \sqrt{-\frac{1}{n^2}} n} \log (x) \left (c x^n\right )^{-1/n} \]

Antiderivative was successfully verified.

[In]

Int[Cos[a + (Sqrt[-n^(-2)]*Log[c*x^n])/3]^3,x]

[Out]

(9*E^(a*Sqrt[-n^(-2)]*n)*x)/(16*(c*x^n)^(1/(3*n))) + (9*x*(c*x^n)^(1/(3*n)))/(32*E^(a*Sqrt[-n^(-2)]*n)) + (x*(
c*x^n)^n^(-1))/(16*E^(3*a*Sqrt[-n^(-2)]*n)) + (E^(3*a*Sqrt[-n^(-2)]*n)*x*Log[x])/(8*(c*x^n)^n^(-1))

Rule 4484

Int[Cos[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(d_.)]^(p_.), x_Symbol] :> Dist[x/(n*(c*x^n)^(1/n)), Subst[Int[x
^(1/n - 1)*Cos[d*(a + b*Log[x])]^p, x], x, c*x^n], x] /; FreeQ[{a, b, c, d, n, p}, x] && (NeQ[c, 1] || NeQ[n,
1])

Rule 4490

Int[Cos[((a_.) + Log[x_]*(b_.))*(d_.)]^(p_.)*((e_.)*(x_))^(m_.), x_Symbol] :> Dist[1/2^p, Int[ExpandIntegrand[
(e*x)^m*(E^((a*b*d^2*p)/(m + 1))/x^((m + 1)/p) + x^((m + 1)/p)/E^((a*b*d^2*p)/(m + 1)))^p, x], x], x] /; FreeQ
[{a, b, d, e, m}, x] && IGtQ[p, 0] && EqQ[b^2*d^2*p^2 + (m + 1)^2, 0]

Rubi steps

\begin{align*} \int \cos ^3\left (a+\frac{1}{3} \sqrt{-\frac{1}{n^2}} \log \left (c x^n\right )\right ) \, dx &=\frac{\left (x \left (c x^n\right )^{-1/n}\right ) \operatorname{Subst}\left (\int x^{-1+\frac{1}{n}} \cos ^3\left (a+\frac{1}{3} \sqrt{-\frac{1}{n^2}} \log (x)\right ) \, dx,x,c x^n\right )}{n}\\ &=\frac{\left (x \left (c x^n\right )^{-1/n}\right ) \operatorname{Subst}\left (\int \left (\frac{e^{3 a \sqrt{-\frac{1}{n^2}} n}}{x}+3 e^{a \sqrt{-\frac{1}{n^2}} n} x^{-1+\frac{2}{3 n}}+3 e^{-a \sqrt{-\frac{1}{n^2}} n} x^{-1+\frac{4}{3 n}}+e^{-3 a \sqrt{-\frac{1}{n^2}} n} x^{-1+\frac{2}{n}}\right ) \, dx,x,c x^n\right )}{8 n}\\ &=\frac{9}{16} e^{a \sqrt{-\frac{1}{n^2}} n} x \left (c x^n\right )^{\left .-\frac{1}{3}\right /n}+\frac{9}{32} e^{-a \sqrt{-\frac{1}{n^2}} n} x \left (c x^n\right )^{\left .\frac{1}{3}\right /n}+\frac{1}{16} e^{-3 a \sqrt{-\frac{1}{n^2}} n} x \left (c x^n\right )^{\frac{1}{n}}+\frac{1}{8} e^{3 a \sqrt{-\frac{1}{n^2}} n} x \left (c x^n\right )^{-1/n} \log (x)\\ \end{align*}

Mathematica [F]  time = 0.138508, size = 0, normalized size = 0. \[ \int \cos ^3\left (a+\frac{1}{3} \sqrt{-\frac{1}{n^2}} \log \left (c x^n\right )\right ) \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[Cos[a + (Sqrt[-n^(-2)]*Log[c*x^n])/3]^3,x]

[Out]

Integrate[Cos[a + (Sqrt[-n^(-2)]*Log[c*x^n])/3]^3, x]

________________________________________________________________________________________

Maple [F]  time = 0.083, size = 0, normalized size = 0. \begin{align*} \int \left ( \cos \left ( a+{\frac{\ln \left ( c{x}^{n} \right ) }{3}\sqrt{-{n}^{-2}}} \right ) \right ) ^{3}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(a+1/3*ln(c*x^n)*(-1/n^2)^(1/2))^3,x)

[Out]

int(cos(a+1/3*ln(c*x^n)*(-1/n^2)^(1/2))^3,x)

________________________________________________________________________________________

Maxima [A]  time = 1.14431, size = 143, normalized size = 1.12 \begin{align*} \frac{9 \, c^{\frac{5}{3 \, n}} x{\left (x^{n}\right )}^{\frac{2}{3 \, n}} \cos \left (a\right ) + 4 \, c^{\frac{1}{3 \, n}}{\left (x^{n}\right )}^{\frac{1}{3 \, n}} \cos \left (3 \, a\right ) \log \left (x\right ) + 18 \, c^{\left (\frac{1}{n}\right )} x \cos \left (a\right ) + 2 \, c^{\frac{7}{3 \, n}} \cos \left (3 \, a\right ) e^{\left (\frac{\log \left (x^{n}\right )}{3 \, n} + 2 \, \log \left (x\right )\right )}}{32 \, c^{\frac{4}{3 \, n}}{\left (x^{n}\right )}^{\frac{1}{3 \, n}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(a+1/3*log(c*x^n)*(-1/n^2)^(1/2))^3,x, algorithm="maxima")

[Out]

1/32*(9*c^(5/3/n)*x*(x^n)^(2/3/n)*cos(a) + 4*c^(1/3/n)*(x^n)^(1/3/n)*cos(3*a)*log(x) + 18*c^(1/n)*x*cos(a) + 2
*c^(7/3/n)*cos(3*a)*e^(1/3*log(x^n)/n + 2*log(x)))/(c^(4/3/n)*(x^n)^(1/3/n))

________________________________________________________________________________________

Fricas [C]  time = 0.478143, size = 227, normalized size = 1.77 \begin{align*} \frac{1}{32} \,{\left (9 \, x^{\frac{4}{3}} e^{\left (\frac{2 \,{\left (3 i \, a n - \log \left (c\right )\right )}}{3 \, n}\right )} + 2 \, x^{2} + 12 \, e^{\left (\frac{2 \,{\left (3 i \, a n - \log \left (c\right )\right )}}{n}\right )} \log \left (x^{\frac{1}{3}}\right ) + 18 \, x^{\frac{2}{3}} e^{\left (\frac{4 \,{\left (3 i \, a n - \log \left (c\right )\right )}}{3 \, n}\right )}\right )} e^{\left (-\frac{3 i \, a n - \log \left (c\right )}{n}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(a+1/3*log(c*x^n)*(-1/n^2)^(1/2))^3,x, algorithm="fricas")

[Out]

1/32*(9*x^(4/3)*e^(2/3*(3*I*a*n - log(c))/n) + 2*x^2 + 12*e^(2*(3*I*a*n - log(c))/n)*log(x^(1/3)) + 18*x^(2/3)
*e^(4/3*(3*I*a*n - log(c))/n))*e^(-(3*I*a*n - log(c))/n)

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \cos ^{3}{\left (a + \frac{\sqrt{- \frac{1}{n^{2}}} \log{\left (c x^{n} \right )}}{3} \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(a+1/3*ln(c*x**n)*(-1/n**2)**(1/2))**3,x)

[Out]

Integral(cos(a + sqrt(-1/n**2)*log(c*x**n)/3)**3, x)

________________________________________________________________________________________

Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(a+1/3*log(c*x^n)*(-1/n^2)^(1/2))^3,x, algorithm="giac")

[Out]

Exception raised: NotImplementedError