3.125 \(\int x^m \cos ^2(a+b \log (c x^n)) \, dx\)

Optimal. Leaf size=120 \[ \frac{(m+1) x^{m+1} \cos ^2\left (a+b \log \left (c x^n\right )\right )}{4 b^2 n^2+(m+1)^2}+\frac{2 b n x^{m+1} \sin \left (a+b \log \left (c x^n\right )\right ) \cos \left (a+b \log \left (c x^n\right )\right )}{4 b^2 n^2+(m+1)^2}+\frac{2 b^2 n^2 x^{m+1}}{(m+1) \left (4 b^2 n^2+(m+1)^2\right )} \]

[Out]

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

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Rubi [A]  time = 0.0318313, antiderivative size = 120, 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} \[ \frac{(m+1) x^{m+1} \cos ^2\left (a+b \log \left (c x^n\right )\right )}{4 b^2 n^2+(m+1)^2}+\frac{2 b n x^{m+1} \sin \left (a+b \log \left (c x^n\right )\right ) \cos \left (a+b \log \left (c x^n\right )\right )}{4 b^2 n^2+(m+1)^2}+\frac{2 b^2 n^2 x^{m+1}}{(m+1) \left (4 b^2 n^2+(m+1)^2\right )} \]

Antiderivative was successfully verified.

[In]

Int[x^m*Cos[a + b*Log[c*x^n]]^2,x]

[Out]

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

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

Mathematica [C]  time = 0.319739, size = 91, normalized size = 0.76 \[ \frac{x^{m+1} \left (2 b (m+1) n \sin \left (2 \left (a+b \log \left (c x^n\right )\right )\right )+(m+1)^2 \cos \left (2 \left (a+b \log \left (c x^n\right )\right )\right )+4 b^2 n^2+m^2+2 m+1\right )}{2 (m+1) (-2 i b n+m+1) (2 i b n+m+1)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.07, size = 0, normalized size = 0. \begin{align*} \int{x}^{m} \left ( \cos \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) \right ) ^{2}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 1.25935, size = 872, normalized size = 7.27 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(((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)))*m^2 + 2*(cos(4*b*l
og(c))*cos(2*b*log(c)) + sin(4*b*log(c))*sin(2*b*log(c)) + cos(2*b*log(c)))*m + 2*(b*cos(2*b*log(c))*sin(4*b*l
og(c)) - b*cos(4*b*log(c))*sin(2*b*log(c)) + (b*cos(2*b*log(c))*sin(4*b*log(c)) - b*cos(4*b*log(c))*sin(2*b*lo
g(c)) + b*sin(2*b*log(c)))*m + 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)))*x*x^m*cos(2*b*log(x^n) + 2*a) - ((cos(2*b*log(c))*sin(4*b*log(c)) - cos(4*b*log(c
))*sin(2*b*log(c)) + sin(2*b*log(c)))*m^2 + 2*(cos(2*b*log(c))*sin(4*b*log(c)) - cos(4*b*log(c))*sin(2*b*log(c
)) + sin(2*b*log(c)))*m - 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(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)))*m + b*cos(2*b*log(c)))*n +
cos(2*b*log(c))*sin(4*b*log(c)) - cos(4*b*log(c))*sin(2*b*log(c)) + sin(2*b*log(c)))*x*x^m*sin(2*b*log(x^n) +
2*a) + 2*((cos(2*b*log(c))^2 + sin(2*b*log(c))^2)*m^2 + 4*(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 + sin(2*b*log(c))^2)*m + cos(2*b*log(c))^2 + sin(2*b*log(c))^2)*x*x^m)/((cos(2*b*log(c)
)^2 + sin(2*b*log(c))^2)*m^3 + 3*(cos(2*b*log(c))^2 + sin(2*b*log(c))^2)*m^2 + 4*(b^2*cos(2*b*log(c))^2 + b^2*
sin(2*b*log(c))^2 + (b^2*cos(2*b*log(c))^2 + b^2*sin(2*b*log(c))^2)*m)*n^2 + 3*(cos(2*b*log(c))^2 + sin(2*b*lo
g(c))^2)*m + cos(2*b*log(c))^2 + sin(2*b*log(c))^2)

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Fricas [A]  time = 0.501269, size = 275, normalized size = 2.29 \begin{align*} \frac{2 \,{\left (b m + b\right )} n x x^{m} \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 ) +{\left (2 \, b^{2} n^{2} x +{\left (m^{2} + 2 \, m + 1\right )} x \cos \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2}\right )} x^{m}}{m^{3} + 4 \,{\left (b^{2} m + b^{2}\right )} n^{2} + 3 \, m^{2} + 3 \, m + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2*(b*m + b)*n*x*x^m*cos(b*n*log(x) + b*log(c) + a)*sin(b*n*log(x) + b*log(c) + a) + (2*b^2*n^2*x + (m^2 + 2*m
+ 1)*x*cos(b*n*log(x) + b*log(c) + a)^2)*x^m)/(m^3 + 4*(b^2*m + b^2)*n^2 + 3*m^2 + 3*m + 1)

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

[Out]

Exception raised: TypeError

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Giac [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out