3.95 \(\int \frac{\cos ^2(a+b \log (c x^n))}{x^2} \, dx\)

Optimal. Leaf size=95 \[ -\frac{\cos ^2\left (a+b \log \left (c x^n\right )\right )}{x \left (4 b^2 n^2+1\right )}+\frac{2 b n \sin \left (a+b \log \left (c x^n\right )\right ) \cos \left (a+b \log \left (c x^n\right )\right )}{x \left (4 b^2 n^2+1\right )}-\frac{2 b^2 n^2}{x \left (4 b^2 n^2+1\right )} \]

[Out]

(-2*b^2*n^2)/((1 + 4*b^2*n^2)*x) - Cos[a + b*Log[c*x^n]]^2/((1 + 4*b^2*n^2)*x) + (2*b*n*Cos[a + b*Log[c*x^n]]*
Sin[a + b*Log[c*x^n]])/((1 + 4*b^2*n^2)*x)

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Rubi [A]  time = 0.026947, antiderivative size = 95, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {4488, 30} \[ -\frac{\cos ^2\left (a+b \log \left (c x^n\right )\right )}{x \left (4 b^2 n^2+1\right )}+\frac{2 b n \sin \left (a+b \log \left (c x^n\right )\right ) \cos \left (a+b \log \left (c x^n\right )\right )}{x \left (4 b^2 n^2+1\right )}-\frac{2 b^2 n^2}{x \left (4 b^2 n^2+1\right )} \]

Antiderivative was successfully verified.

[In]

Int[Cos[a + b*Log[c*x^n]]^2/x^2,x]

[Out]

(-2*b^2*n^2)/((1 + 4*b^2*n^2)*x) - Cos[a + b*Log[c*x^n]]^2/((1 + 4*b^2*n^2)*x) + (2*b*n*Cos[a + b*Log[c*x^n]]*
Sin[a + b*Log[c*x^n]])/((1 + 4*b^2*n^2)*x)

Rule 4488

Int[Cos[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(d_.)]^(p_)*((e_.)*(x_))^(m_.), x_Symbol] :> Simp[((m + 1)*(e*x)
^(m + 1)*Cos[d*(a + b*Log[c*x^n])]^p)/(b^2*d^2*e*n^2*p^2 + e*(m + 1)^2), x] + (Dist[(b^2*d^2*n^2*p*(p - 1))/(b
^2*d^2*n^2*p^2 + (m + 1)^2), Int[(e*x)^m*Cos[d*(a + b*Log[c*x^n])]^(p - 2), x], x] + Simp[(b*d*n*p*(e*x)^(m +
1)*Sin[d*(a + b*Log[c*x^n])]*Cos[d*(a + b*Log[c*x^n])]^(p - 1))/(b^2*d^2*e*n^2*p^2 + e*(m + 1)^2), x]) /; Free
Q[{a, b, c, d, e, m, n}, x] && IGtQ[p, 1] && NeQ[b^2*d^2*n^2*p^2 + (m + 1)^2, 0]

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rubi steps

\begin{align*} \int \frac{\cos ^2\left (a+b \log \left (c x^n\right )\right )}{x^2} \, dx &=-\frac{\cos ^2\left (a+b \log \left (c x^n\right )\right )}{\left (1+4 b^2 n^2\right ) x}+\frac{2 b n \cos \left (a+b \log \left (c x^n\right )\right ) \sin \left (a+b \log \left (c x^n\right )\right )}{\left (1+4 b^2 n^2\right ) x}+\frac{\left (2 b^2 n^2\right ) \int \frac{1}{x^2} \, dx}{1+4 b^2 n^2}\\ &=-\frac{2 b^2 n^2}{\left (1+4 b^2 n^2\right ) x}-\frac{\cos ^2\left (a+b \log \left (c x^n\right )\right )}{\left (1+4 b^2 n^2\right ) x}+\frac{2 b n \cos \left (a+b \log \left (c x^n\right )\right ) \sin \left (a+b \log \left (c x^n\right )\right )}{\left (1+4 b^2 n^2\right ) x}\\ \end{align*}

Mathematica [A]  time = 0.117084, size = 57, normalized size = 0.6 \[ -\frac{-2 b n \sin \left (2 \left (a+b \log \left (c x^n\right )\right )\right )+\cos \left (2 \left (a+b \log \left (c x^n\right )\right )\right )+4 b^2 n^2+1}{2 \left (4 b^2 n^2 x+x\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[a + b*Log[c*x^n]]^2/x^2,x]

[Out]

-(1 + 4*b^2*n^2 + Cos[2*(a + b*Log[c*x^n])] - 2*b*n*Sin[2*(a + b*Log[c*x^n])])/(2*(x + 4*b^2*n^2*x))

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Maple [F]  time = 0.057, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( \cos \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) \right ) ^{2}}{{x}^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 1.37307, size = 385, normalized size = 4.05 \begin{align*} -\frac{8 \,{\left (b^{2} \cos \left (2 \, b \log \left (c\right )\right )^{2} + b^{2} \sin \left (2 \, b \log \left (c\right )\right )^{2}\right )} n^{2} + 2 \, \cos \left (2 \, b \log \left (c\right )\right )^{2} -{\left (2 \,{\left (b \cos \left (2 \, b \log \left (c\right )\right ) \sin \left (4 \, b \log \left (c\right )\right ) - b \cos \left (4 \, b \log \left (c\right )\right ) \sin \left (2 \, b \log \left (c\right )\right ) + b \sin \left (2 \, b \log \left (c\right )\right )\right )} n - \cos \left (4 \, b \log \left (c\right )\right ) \cos \left (2 \, b \log \left (c\right )\right ) - \sin \left (4 \, b \log \left (c\right )\right ) \sin \left (2 \, b \log \left (c\right )\right ) - \cos \left (2 \, b \log \left (c\right )\right )\right )} \cos \left (2 \, b \log \left (x^{n}\right ) + 2 \, a\right ) + 2 \, \sin \left (2 \, b \log \left (c\right )\right )^{2} -{\left (2 \,{\left (b \cos \left (4 \, b \log \left (c\right )\right ) \cos \left (2 \, b \log \left (c\right )\right ) + b \sin \left (4 \, b \log \left (c\right )\right ) \sin \left (2 \, b \log \left (c\right )\right ) + b \cos \left (2 \, b \log \left (c\right )\right )\right )} n + \cos \left (2 \, b \log \left (c\right )\right ) \sin \left (4 \, b \log \left (c\right )\right ) - \cos \left (4 \, b \log \left (c\right )\right ) \sin \left (2 \, b \log \left (c\right )\right ) + \sin \left (2 \, b \log \left (c\right )\right )\right )} \sin \left (2 \, b \log \left (x^{n}\right ) + 2 \, a\right )}{4 \,{\left (4 \,{\left (b^{2} \cos \left (2 \, b \log \left (c\right )\right )^{2} + b^{2} \sin \left (2 \, b \log \left (c\right )\right )^{2}\right )} n^{2} + \cos \left (2 \, b \log \left (c\right )\right )^{2} + \sin \left (2 \, b \log \left (c\right )\right )^{2}\right )} x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*(8*(b^2*cos(2*b*log(c))^2 + b^2*sin(2*b*log(c))^2)*n^2 + 2*cos(2*b*log(c))^2 - (2*(b*cos(2*b*log(c))*sin(
4*b*log(c)) - b*cos(4*b*log(c))*sin(2*b*log(c)) + b*sin(2*b*log(c)))*n - cos(4*b*log(c))*cos(2*b*log(c)) - sin
(4*b*log(c))*sin(2*b*log(c)) - cos(2*b*log(c)))*cos(2*b*log(x^n) + 2*a) + 2*sin(2*b*log(c))^2 - (2*(b*cos(4*b*
log(c))*cos(2*b*log(c)) + b*sin(4*b*log(c))*sin(2*b*log(c)) + b*cos(2*b*log(c)))*n + cos(2*b*log(c))*sin(4*b*l
og(c)) - cos(4*b*log(c))*sin(2*b*log(c)) + sin(2*b*log(c)))*sin(2*b*log(x^n) + 2*a))/((4*(b^2*cos(2*b*log(c))^
2 + b^2*sin(2*b*log(c))^2)*n^2 + cos(2*b*log(c))^2 + sin(2*b*log(c))^2)*x)

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Fricas [A]  time = 0.504295, size = 188, normalized size = 1.98 \begin{align*} -\frac{2 \, b^{2} n^{2} - 2 \, b n \cos \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) \sin \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) + \cos \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2}}{{\left (4 \, b^{2} n^{2} + 1\right )} x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(2*b^2*n^2 - 2*b*n*cos(b*n*log(x) + b*log(c) + a)*sin(b*n*log(x) + b*log(c) + a) + cos(b*n*log(x) + b*log(c)
+ a)^2)/((4*b^2*n^2 + 1)*x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(cos(b*log(c*x^n) + a)^2/x^2, x)