3.101 \(\int \cos ^4(a+b \log (c x^n)) \, dx\)

Optimal. Leaf size=191 \[ \frac{12 b^2 n^2 x \cos ^2\left (a+b \log \left (c x^n\right )\right )}{64 b^4 n^4+20 b^2 n^2+1}+\frac{x \cos ^4\left (a+b \log \left (c x^n\right )\right )}{16 b^2 n^2+1}+\frac{4 b n x \sin \left (a+b \log \left (c x^n\right )\right ) \cos ^3\left (a+b \log \left (c x^n\right )\right )}{16 b^2 n^2+1}+\frac{24 b^3 n^3 x \sin \left (a+b \log \left (c x^n\right )\right ) \cos \left (a+b \log \left (c x^n\right )\right )}{64 b^4 n^4+20 b^2 n^2+1}+\frac{24 b^4 n^4 x}{64 b^4 n^4+20 b^2 n^2+1} \]

[Out]

(24*b^4*n^4*x)/(1 + 20*b^2*n^2 + 64*b^4*n^4) + (12*b^2*n^2*x*Cos[a + b*Log[c*x^n]]^2)/(1 + 20*b^2*n^2 + 64*b^4
*n^4) + (x*Cos[a + b*Log[c*x^n]]^4)/(1 + 16*b^2*n^2) + (24*b^3*n^3*x*Cos[a + b*Log[c*x^n]]*Sin[a + b*Log[c*x^n
]])/(1 + 20*b^2*n^2 + 64*b^4*n^4) + (4*b*n*x*Cos[a + b*Log[c*x^n]]^3*Sin[a + b*Log[c*x^n]])/(1 + 16*b^2*n^2)

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Rubi [A]  time = 0.0449368, antiderivative size = 191, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {4478, 8} \[ \frac{12 b^2 n^2 x \cos ^2\left (a+b \log \left (c x^n\right )\right )}{64 b^4 n^4+20 b^2 n^2+1}+\frac{x \cos ^4\left (a+b \log \left (c x^n\right )\right )}{16 b^2 n^2+1}+\frac{4 b n x \sin \left (a+b \log \left (c x^n\right )\right ) \cos ^3\left (a+b \log \left (c x^n\right )\right )}{16 b^2 n^2+1}+\frac{24 b^3 n^3 x \sin \left (a+b \log \left (c x^n\right )\right ) \cos \left (a+b \log \left (c x^n\right )\right )}{64 b^4 n^4+20 b^2 n^2+1}+\frac{24 b^4 n^4 x}{64 b^4 n^4+20 b^2 n^2+1} \]

Antiderivative was successfully verified.

[In]

Int[Cos[a + b*Log[c*x^n]]^4,x]

[Out]

(24*b^4*n^4*x)/(1 + 20*b^2*n^2 + 64*b^4*n^4) + (12*b^2*n^2*x*Cos[a + b*Log[c*x^n]]^2)/(1 + 20*b^2*n^2 + 64*b^4
*n^4) + (x*Cos[a + b*Log[c*x^n]]^4)/(1 + 16*b^2*n^2) + (24*b^3*n^3*x*Cos[a + b*Log[c*x^n]]*Sin[a + b*Log[c*x^n
]])/(1 + 20*b^2*n^2 + 64*b^4*n^4) + (4*b*n*x*Cos[a + b*Log[c*x^n]]^3*Sin[a + b*Log[c*x^n]])/(1 + 16*b^2*n^2)

Rule 4478

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

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rubi steps

\begin{align*} \int \cos ^4\left (a+b \log \left (c x^n\right )\right ) \, dx &=\frac{x \cos ^4\left (a+b \log \left (c x^n\right )\right )}{1+16 b^2 n^2}+\frac{4 b n x \cos ^3\left (a+b \log \left (c x^n\right )\right ) \sin \left (a+b \log \left (c x^n\right )\right )}{1+16 b^2 n^2}+\frac{\left (12 b^2 n^2\right ) \int \cos ^2\left (a+b \log \left (c x^n\right )\right ) \, dx}{1+16 b^2 n^2}\\ &=\frac{12 b^2 n^2 x \cos ^2\left (a+b \log \left (c x^n\right )\right )}{1+20 b^2 n^2+64 b^4 n^4}+\frac{x \cos ^4\left (a+b \log \left (c x^n\right )\right )}{1+16 b^2 n^2}+\frac{24 b^3 n^3 x \cos \left (a+b \log \left (c x^n\right )\right ) \sin \left (a+b \log \left (c x^n\right )\right )}{1+20 b^2 n^2+64 b^4 n^4}+\frac{4 b n x \cos ^3\left (a+b \log \left (c x^n\right )\right ) \sin \left (a+b \log \left (c x^n\right )\right )}{1+16 b^2 n^2}+\frac{\left (24 b^4 n^4\right ) \int 1 \, dx}{1+20 b^2 n^2+64 b^4 n^4}\\ &=\frac{24 b^4 n^4 x}{1+20 b^2 n^2+64 b^4 n^4}+\frac{12 b^2 n^2 x \cos ^2\left (a+b \log \left (c x^n\right )\right )}{1+20 b^2 n^2+64 b^4 n^4}+\frac{x \cos ^4\left (a+b \log \left (c x^n\right )\right )}{1+16 b^2 n^2}+\frac{24 b^3 n^3 x \cos \left (a+b \log \left (c x^n\right )\right ) \sin \left (a+b \log \left (c x^n\right )\right )}{1+20 b^2 n^2+64 b^4 n^4}+\frac{4 b n x \cos ^3\left (a+b \log \left (c x^n\right )\right ) \sin \left (a+b \log \left (c x^n\right )\right )}{1+16 b^2 n^2}\\ \end{align*}

Mathematica [A]  time = 0.434809, size = 167, normalized size = 0.87 \[ \frac{x \left (128 b^3 n^3 \sin \left (2 \left (a+b \log \left (c x^n\right )\right )\right )+16 b^3 n^3 \sin \left (4 \left (a+b \log \left (c x^n\right )\right )\right )+\left (64 b^2 n^2+4\right ) \cos \left (2 \left (a+b \log \left (c x^n\right )\right )\right )+\left (4 b^2 n^2+1\right ) \cos \left (4 \left (a+b \log \left (c x^n\right )\right )\right )+8 b n \sin \left (2 \left (a+b \log \left (c x^n\right )\right )\right )+4 b n \sin \left (4 \left (a+b \log \left (c x^n\right )\right )\right )+192 b^4 n^4+60 b^2 n^2+3\right )}{8 \left (64 b^4 n^4+20 b^2 n^2+1\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[a + b*Log[c*x^n]]^4,x]

[Out]

(x*(3 + 60*b^2*n^2 + 192*b^4*n^4 + (4 + 64*b^2*n^2)*Cos[2*(a + b*Log[c*x^n])] + (1 + 4*b^2*n^2)*Cos[4*(a + b*L
og[c*x^n])] + 8*b*n*Sin[2*(a + b*Log[c*x^n])] + 128*b^3*n^3*Sin[2*(a + b*Log[c*x^n])] + 4*b*n*Sin[4*(a + b*Log
[c*x^n])] + 16*b^3*n^3*Sin[4*(a + b*Log[c*x^n])]))/(8*(1 + 20*b^2*n^2 + 64*b^4*n^4))

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Maple [F]  time = 0.09, size = 0, normalized size = 0. \begin{align*} \int \left ( \cos \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) \right ) ^{4}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(a+b*ln(c*x^n))^4,x)

[Out]

int(cos(a+b*ln(c*x^n))^4,x)

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Maxima [B]  time = 1.27288, size = 1455, normalized size = 7.62 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(a+b*log(c*x^n))^4,x, algorithm="maxima")

[Out]

1/16*((16*(b^3*cos(4*b*log(c))*sin(8*b*log(c)) - b^3*cos(8*b*log(c))*sin(4*b*log(c)) + b^3*sin(4*b*log(c)))*n^
3 + 4*(b^2*cos(8*b*log(c))*cos(4*b*log(c)) + b^2*sin(8*b*log(c))*sin(4*b*log(c)) + b^2*cos(4*b*log(c)))*n^2 +
4*(b*cos(4*b*log(c))*sin(8*b*log(c)) - b*cos(8*b*log(c))*sin(4*b*log(c)) + b*sin(4*b*log(c)))*n + cos(8*b*log(
c))*cos(4*b*log(c)) + sin(8*b*log(c))*sin(4*b*log(c)) + cos(4*b*log(c)))*x*cos(4*b*log(x^n) + 4*a) + 4*(32*(b^
3*cos(4*b*log(c))*sin(6*b*log(c)) - b^3*cos(6*b*log(c))*sin(4*b*log(c)) + b^3*cos(2*b*log(c))*sin(4*b*log(c))
- b^3*cos(4*b*log(c))*sin(2*b*log(c)))*n^3 + 16*(b^2*cos(6*b*log(c))*cos(4*b*log(c)) + b^2*cos(4*b*log(c))*cos
(2*b*log(c)) + b^2*sin(6*b*log(c))*sin(4*b*log(c)) + b^2*sin(4*b*log(c))*sin(2*b*log(c)))*n^2 + 2*(b*cos(4*b*l
og(c))*sin(6*b*log(c)) - b*cos(6*b*log(c))*sin(4*b*log(c)) + b*cos(2*b*log(c))*sin(4*b*log(c)) - b*cos(4*b*log
(c))*sin(2*b*log(c)))*n + cos(6*b*log(c))*cos(4*b*log(c)) + cos(4*b*log(c))*cos(2*b*log(c)) + sin(6*b*log(c))*
sin(4*b*log(c)) + sin(4*b*log(c))*sin(2*b*log(c)))*x*cos(2*b*log(x^n) + 2*a) + (16*(b^3*cos(8*b*log(c))*cos(4*
b*log(c)) + b^3*sin(8*b*log(c))*sin(4*b*log(c)) + b^3*cos(4*b*log(c)))*n^3 - 4*(b^2*cos(4*b*log(c))*sin(8*b*lo
g(c)) - b^2*cos(8*b*log(c))*sin(4*b*log(c)) + b^2*sin(4*b*log(c)))*n^2 + 4*(b*cos(8*b*log(c))*cos(4*b*log(c))
+ b*sin(8*b*log(c))*sin(4*b*log(c)) + b*cos(4*b*log(c)))*n - cos(4*b*log(c))*sin(8*b*log(c)) + cos(8*b*log(c))
*sin(4*b*log(c)) - sin(4*b*log(c)))*x*sin(4*b*log(x^n) + 4*a) + 4*(32*(b^3*cos(6*b*log(c))*cos(4*b*log(c)) + b
^3*cos(4*b*log(c))*cos(2*b*log(c)) + b^3*sin(6*b*log(c))*sin(4*b*log(c)) + b^3*sin(4*b*log(c))*sin(2*b*log(c))
)*n^3 - 16*(b^2*cos(4*b*log(c))*sin(6*b*log(c)) - b^2*cos(6*b*log(c))*sin(4*b*log(c)) + b^2*cos(2*b*log(c))*si
n(4*b*log(c)) - b^2*cos(4*b*log(c))*sin(2*b*log(c)))*n^2 + 2*(b*cos(6*b*log(c))*cos(4*b*log(c)) + b*cos(4*b*lo
g(c))*cos(2*b*log(c)) + b*sin(6*b*log(c))*sin(4*b*log(c)) + b*sin(4*b*log(c))*sin(2*b*log(c)))*n - cos(4*b*log
(c))*sin(6*b*log(c)) + cos(6*b*log(c))*sin(4*b*log(c)) - cos(2*b*log(c))*sin(4*b*log(c)) + cos(4*b*log(c))*sin
(2*b*log(c)))*x*sin(2*b*log(x^n) + 2*a) + 6*(64*(b^4*cos(4*b*log(c))^2 + b^4*sin(4*b*log(c))^2)*n^4 + 20*(b^2*
cos(4*b*log(c))^2 + b^2*sin(4*b*log(c))^2)*n^2 + cos(4*b*log(c))^2 + sin(4*b*log(c))^2)*x)/(64*(b^4*cos(4*b*lo
g(c))^2 + b^4*sin(4*b*log(c))^2)*n^4 + 20*(b^2*cos(4*b*log(c))^2 + b^2*sin(4*b*log(c))^2)*n^2 + cos(4*b*log(c)
)^2 + sin(4*b*log(c))^2)

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Fricas [A]  time = 0.514719, size = 381, normalized size = 1.99 \begin{align*} \frac{24 \, b^{4} n^{4} x + 12 \, b^{2} n^{2} x \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 \cos \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{4} + 4 \,{\left (6 \, b^{3} n^{3} x \cos \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) +{\left (4 \, b^{3} n^{3} + b n\right )} x \cos \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{3}\right )} \sin \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )}{64 \, b^{4} n^{4} + 20 \, b^{2} n^{2} + 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(a+b*log(c*x^n))^4,x, algorithm="fricas")

[Out]

(24*b^4*n^4*x + 12*b^2*n^2*x*cos(b*n*log(x) + b*log(c) + a)^2 + (4*b^2*n^2 + 1)*x*cos(b*n*log(x) + b*log(c) +
a)^4 + 4*(6*b^3*n^3*x*cos(b*n*log(x) + b*log(c) + a) + (4*b^3*n^3 + b*n)*x*cos(b*n*log(x) + b*log(c) + a)^3)*s
in(b*n*log(x) + b*log(c) + a))/(64*b^4*n^4 + 20*b^2*n^2 + 1)

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Sympy [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(cos(a+b*ln(c*x**n))**4,x)

[Out]

Timed out

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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(cos(a+b*log(c*x^n))^4,x, algorithm="giac")

[Out]

Timed out