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

Optimal. Leaf size=68 \[ \frac{1}{8} x e^{-2 a \sqrt{-\frac{1}{n^2}} n} \left (c x^n\right )^{\frac{1}{n}}+\frac{1}{4} x e^{2 a \sqrt{-\frac{1}{n^2}} n} \log (x) \left (c x^n\right )^{-1/n}+\frac{x}{2} \]

[Out]

x/2 + (x*(c*x^n)^n^(-1))/(8*E^(2*a*Sqrt[-n^(-2)]*n)) + (E^(2*a*Sqrt[-n^(-2)]*n)*x*Log[x])/(4*(c*x^n)^n^(-1))

________________________________________________________________________________________

Rubi [A]  time = 0.0557935, antiderivative size = 68, 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} \[ \frac{1}{8} x e^{-2 a \sqrt{-\frac{1}{n^2}} n} \left (c x^n\right )^{\frac{1}{n}}+\frac{1}{4} x e^{2 a \sqrt{-\frac{1}{n^2}} n} \log (x) \left (c x^n\right )^{-1/n}+\frac{x}{2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

x/2 + (x*(c*x^n)^n^(-1))/(8*E^(2*a*Sqrt[-n^(-2)]*n)) + (E^(2*a*Sqrt[-n^(-2)]*n)*x*Log[x])/(4*(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 ^2\left (a+\frac{1}{2} \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 ^2\left (a+\frac{1}{2} \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^{2 a \sqrt{-\frac{1}{n^2}} n}}{x}+2 x^{-1+\frac{1}{n}}+e^{-2 a \sqrt{-\frac{1}{n^2}} n} x^{-1+\frac{2}{n}}\right ) \, dx,x,c x^n\right )}{4 n}\\ &=\frac{x}{2}+\frac{1}{8} e^{-2 a \sqrt{-\frac{1}{n^2}} n} x \left (c x^n\right )^{\frac{1}{n}}+\frac{1}{4} e^{2 a \sqrt{-\frac{1}{n^2}} n} x \left (c x^n\right )^{-1/n} \log (x)\\ \end{align*}

Mathematica [F]  time = 0.100558, size = 0, normalized size = 0. \[ \int \cos ^2\left (a+\frac{1}{2} \sqrt{-\frac{1}{n^2}} \log \left (c x^n\right )\right ) \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [F]  time = 0.079, size = 0, normalized size = 0. \begin{align*} \int \left ( \cos \left ( a+{\frac{\ln \left ( c{x}^{n} \right ) }{2}\sqrt{-{n}^{-2}}} \right ) \right ) ^{2}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.29058, size = 55, normalized size = 0.81 \begin{align*} \frac{c^{\frac{2}{n}} x^{2} \cos \left (2 \, a\right ) + 4 \, c^{\left (\frac{1}{n}\right )} x + 2 \, \cos \left (2 \, a\right ) \log \left (x\right )}{8 \, c^{\left (\frac{1}{n}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*(c^(2/n)*x^2*cos(2*a) + 4*c^(1/n)*x + 2*cos(2*a)*log(x))/c^(1/n)

________________________________________________________________________________________

Fricas [C]  time = 0.471724, size = 143, normalized size = 2.1 \begin{align*} \frac{1}{8} \,{\left (x^{2} + 4 \, x e^{\left (\frac{2 i \, a n - \log \left (c\right )}{n}\right )} + 2 \, e^{\left (\frac{2 \,{\left (2 i \, a n - \log \left (c\right )\right )}}{n}\right )} \log \left (x\right )\right )} e^{\left (-\frac{2 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+1/2*log(c*x^n)*(-1/n^2)^(1/2))^2,x, algorithm="fricas")

[Out]

1/8*(x^2 + 4*x*e^((2*I*a*n - log(c))/n) + 2*e^(2*(2*I*a*n - log(c))/n)*log(x))*e^(-(2*I*a*n - log(c))/n)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [A]  time = 1.61904, size = 1, normalized size = 0.01 \begin{align*} +\infty \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

+Infinity