3.2.1 \(\int \cos ^4(a+b \log (c x^n)) \, dx\) [101]

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

[Out]

24*b^4*n^4*x/(64*b^4*n^4+20*b^2*n^2+1)+12*b^2*n^2*x*cos(a+b*ln(c*x^n))^2/(64*b^4*n^4+20*b^2*n^2+1)+x*cos(a+b*l
n(c*x^n))^4/(16*b^2*n^2+1)+24*b^3*n^3*x*cos(a+b*ln(c*x^n))*sin(a+b*ln(c*x^n))/(64*b^4*n^4+20*b^2*n^2+1)+4*b*n*
x*cos(a+b*ln(c*x^n))^3*sin(a+b*ln(c*x^n))/(16*b^2*n^2+1)

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Rubi [A]
time = 0.03, antiderivative size = 191, normalized size of antiderivative = 1.00, 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 = {4566, 8} \begin {gather*} \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 {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 {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} \end {gather*}

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 8

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

Rule 4566

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]

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*}

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Mathematica [A]
time = 0.46, size = 167, normalized size = 0.87 \begin {gather*} \frac {x \left (3+60 b^2 n^2+192 b^4 n^4+\left (4+64 b^2 n^2\right ) \cos \left (2 \left (a+b \log \left (c x^n\right )\right )\right )+\left (1+4 b^2 n^2\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 )+128 b^3 n^3 \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 )+16 b^3 n^3 \sin \left (4 \left (a+b \log \left (c x^n\right )\right )\right )\right )}{8 \left (1+20 b^2 n^2+64 b^4 n^4\right )} \end {gather*}

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 = 0.07, size = 0, normalized size = 0.00 \[\int \cos ^{4}\left (a +b \ln \left (c \,x^{n}\right )\right )\, dx\]

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] Leaf count of result is larger than twice the leaf count of optimal. 1078 vs. \(2 (191) = 382\).
time = 0.32, size = 1078, normalized size = 5.64 \begin {gather*} \text {Too large to display} \end {gather*}

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 = 2.58, size = 144, normalized size = 0.75 \begin {gather*} \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 {gather*}

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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \begin {cases} \int \cos ^{4}{\left (a - \frac {i \log {\left (c x^{n} \right )}}{2 n} \right )}\, dx & \text {for}\: b = - \frac {i}{2 n} \\\int \cos ^{4}{\left (a - \frac {i \log {\left (c x^{n} \right )}}{4 n} \right )}\, dx & \text {for}\: b = - \frac {i}{4 n} \\\int \cos ^{4}{\left (a + \frac {i \log {\left (c x^{n} \right )}}{4 n} \right )}\, dx & \text {for}\: b = \frac {i}{4 n} \\\int \cos ^{4}{\left (a + \frac {i \log {\left (c x^{n} \right )}}{2 n} \right )}\, dx & \text {for}\: b = \frac {i}{2 n} \\\frac {24 b^{4} n^{4} x \sin ^{4}{\left (a + b \log {\left (c x^{n} \right )} \right )}}{64 b^{4} n^{4} + 20 b^{2} n^{2} + 1} + \frac {48 b^{4} n^{4} x \sin ^{2}{\left (a + b \log {\left (c x^{n} \right )} \right )} \cos ^{2}{\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 \cos ^{4}{\left (a + b \log {\left (c x^{n} \right )} \right )}}{64 b^{4} n^{4} + 20 b^{2} n^{2} + 1} + \frac {24 b^{3} n^{3} x \sin ^{3}{\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 {40 b^{3} n^{3} x \sin {\left (a + b \log {\left (c x^{n} \right )} \right )} \cos ^{3}{\left (a + b \log {\left (c x^{n} \right )} \right )}}{64 b^{4} n^{4} + 20 b^{2} n^{2} + 1} + \frac {12 b^{2} n^{2} x \sin ^{2}{\left (a + b \log {\left (c x^{n} \right )} \right )} \cos ^{2}{\left (a + b \log {\left (c x^{n} \right )} \right )}}{64 b^{4} n^{4} + 20 b^{2} n^{2} + 1} + \frac {16 b^{2} n^{2} x \cos ^{4}{\left (a + b \log {\left (c x^{n} \right )} \right )}}{64 b^{4} n^{4} + 20 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 )}}{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 )}}{64 b^{4} n^{4} + 20 b^{2} n^{2} + 1} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((Integral(cos(a - I*log(c*x**n)/(2*n))**4, x), Eq(b, -I/(2*n))), (Integral(cos(a - I*log(c*x**n)/(4*
n))**4, x), Eq(b, -I/(4*n))), (Integral(cos(a + I*log(c*x**n)/(4*n))**4, x), Eq(b, I/(4*n))), (Integral(cos(a
+ I*log(c*x**n)/(2*n))**4, x), Eq(b, I/(2*n))), (24*b**4*n**4*x*sin(a + b*log(c*x**n))**4/(64*b**4*n**4 + 20*b
**2*n**2 + 1) + 48*b**4*n**4*x*sin(a + b*log(c*x**n))**2*cos(a + b*log(c*x**n))**2/(64*b**4*n**4 + 20*b**2*n**
2 + 1) + 24*b**4*n**4*x*cos(a + b*log(c*x**n))**4/(64*b**4*n**4 + 20*b**2*n**2 + 1) + 24*b**3*n**3*x*sin(a + b
*log(c*x**n))**3*cos(a + b*log(c*x**n))/(64*b**4*n**4 + 20*b**2*n**2 + 1) + 40*b**3*n**3*x*sin(a + b*log(c*x**
n))*cos(a + b*log(c*x**n))**3/(64*b**4*n**4 + 20*b**2*n**2 + 1) + 12*b**2*n**2*x*sin(a + b*log(c*x**n))**2*cos
(a + b*log(c*x**n))**2/(64*b**4*n**4 + 20*b**2*n**2 + 1) + 16*b**2*n**2*x*cos(a + b*log(c*x**n))**4/(64*b**4*n
**4 + 20*b**2*n**2 + 1) + 4*b*n*x*sin(a + b*log(c*x**n))*cos(a + b*log(c*x**n))**3/(64*b**4*n**4 + 20*b**2*n**
2 + 1) + x*cos(a + b*log(c*x**n))**4/(64*b**4*n**4 + 20*b**2*n**2 + 1), True))

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 16422 vs. \(2 (191) = 382\).
time = 0.80, size = 16422, normalized size = 85.98 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/8*x - 1/16*(256*b^3*n^3*x*e^(pi*b*n*sgn(x) - pi*b*n + pi*b*sgn(c) - pi*b)*tan(2*b*n*log(abs(x)) + 2*b*log(ab
s(c)))^2*tan(b*n*log(abs(x)) + b*log(abs(c)))^2*tan(2*a)^2*tan(a) + 256*b^3*n^3*x*e^(-pi*b*n*sgn(x) + pi*b*n -
pi*b*sgn(c) + pi*b)*tan(2*b*n*log(abs(x)) + 2*b*log(abs(c)))^2*tan(b*n*log(abs(x)) + b*log(abs(c)))^2*tan(2*a
)^2*tan(a) + 32*b^3*n^3*x*e^(2*pi*b*n*sgn(x) - 2*pi*b*n + 2*pi*b*sgn(c) - 2*pi*b)*tan(2*b*n*log(abs(x)) + 2*b*
log(abs(c)))^2*tan(b*n*log(abs(x)) + b*log(abs(c)))^2*tan(2*a)*tan(a)^2 + 32*b^3*n^3*x*e^(-2*pi*b*n*sgn(x) + 2
*pi*b*n - 2*pi*b*sgn(c) + 2*pi*b)*tan(2*b*n*log(abs(x)) + 2*b*log(abs(c)))^2*tan(b*n*log(abs(x)) + b*log(abs(c
)))^2*tan(2*a)*tan(a)^2 + 256*b^3*n^3*x*e^(pi*b*n*sgn(x) - pi*b*n + pi*b*sgn(c) - pi*b)*tan(2*b*n*log(abs(x))
+ 2*b*log(abs(c)))^2*tan(b*n*log(abs(x)) + b*log(abs(c)))*tan(2*a)^2*tan(a)^2 + 256*b^3*n^3*x*e^(-pi*b*n*sgn(x
) + pi*b*n - pi*b*sgn(c) + pi*b)*tan(2*b*n*log(abs(x)) + 2*b*log(abs(c)))^2*tan(b*n*log(abs(x)) + b*log(abs(c)
))*tan(2*a)^2*tan(a)^2 + 32*b^3*n^3*x*e^(2*pi*b*n*sgn(x) - 2*pi*b*n + 2*pi*b*sgn(c) - 2*pi*b)*tan(2*b*n*log(ab
s(x)) + 2*b*log(abs(c)))*tan(b*n*log(abs(x)) + b*log(abs(c)))^2*tan(2*a)^2*tan(a)^2 + 32*b^3*n^3*x*e^(-2*pi*b*
n*sgn(x) + 2*pi*b*n - 2*pi*b*sgn(c) + 2*pi*b)*tan(2*b*n*log(abs(x)) + 2*b*log(abs(c)))*tan(b*n*log(abs(x)) + b
*log(abs(c)))^2*tan(2*a)^2*tan(a)^2 - 4*b^2*n^2*x*e^(2*pi*b*n*sgn(x) - 2*pi*b*n + 2*pi*b*sgn(c) - 2*pi*b)*tan(
2*b*n*log(abs(x)) + 2*b*log(abs(c)))^2*tan(b*n*log(abs(x)) + b*log(abs(c)))^2*tan(2*a)^2*tan(a)^2 - 64*b^2*n^2
*x*e^(pi*b*n*sgn(x) - pi*b*n + pi*b*sgn(c) - pi*b)*tan(2*b*n*log(abs(x)) + 2*b*log(abs(c)))^2*tan(b*n*log(abs(
x)) + b*log(abs(c)))^2*tan(2*a)^2*tan(a)^2 - 64*b^2*n^2*x*e^(-pi*b*n*sgn(x) + pi*b*n - pi*b*sgn(c) + pi*b)*tan
(2*b*n*log(abs(x)) + 2*b*log(abs(c)))^2*tan(b*n*log(abs(x)) + b*log(abs(c)))^2*tan(2*a)^2*tan(a)^2 - 4*b^2*n^2
*x*e^(-2*pi*b*n*sgn(x) + 2*pi*b*n - 2*pi*b*sgn(c) + 2*pi*b)*tan(2*b*n*log(abs(x)) + 2*b*log(abs(c)))^2*tan(b*n
*log(abs(x)) + b*log(abs(c)))^2*tan(2*a)^2*tan(a)^2 + 32*b^3*n^3*x*e^(2*pi*b*n*sgn(x) - 2*pi*b*n + 2*pi*b*sgn(
c) - 2*pi*b)*tan(2*b*n*log(abs(x)) + 2*b*log(abs(c)))^2*tan(b*n*log(abs(x)) + b*log(abs(c)))^2*tan(2*a) + 32*b
^3*n^3*x*e^(-2*pi*b*n*sgn(x) + 2*pi*b*n - 2*pi*b*sgn(c) + 2*pi*b)*tan(2*b*n*log(abs(x)) + 2*b*log(abs(c)))^2*t
an(b*n*log(abs(x)) + b*log(abs(c)))^2*tan(2*a) - 256*b^3*n^3*x*e^(pi*b*n*sgn(x) - pi*b*n + pi*b*sgn(c) - pi*b)
*tan(2*b*n*log(abs(x)) + 2*b*log(abs(c)))^2*tan(b*n*log(abs(x)) + b*log(abs(c)))*tan(2*a)^2 - 256*b^3*n^3*x*e^
(-pi*b*n*sgn(x) + pi*b*n - pi*b*sgn(c) + pi*b)*tan(2*b*n*log(abs(x)) + 2*b*log(abs(c)))^2*tan(b*n*log(abs(x))
+ b*log(abs(c)))*tan(2*a)^2 + 32*b^3*n^3*x*e^(2*pi*b*n*sgn(x) - 2*pi*b*n + 2*pi*b*sgn(c) - 2*pi*b)*tan(2*b*n*l
og(abs(x)) + 2*b*log(abs(c)))*tan(b*n*log(abs(x)) + b*log(abs(c)))^2*tan(2*a)^2 + 32*b^3*n^3*x*e^(-2*pi*b*n*sg
n(x) + 2*pi*b*n - 2*pi*b*sgn(c) + 2*pi*b)*tan(2*b*n*log(abs(x)) + 2*b*log(abs(c)))*tan(b*n*log(abs(x)) + b*log
(abs(c)))^2*tan(2*a)^2 + 256*b^3*n^3*x*e^(pi*b*n*sgn(x) - pi*b*n + pi*b*sgn(c) - pi*b)*tan(2*b*n*log(abs(x)) +
2*b*log(abs(c)))^2*tan(b*n*log(abs(x)) + b*log(abs(c)))^2*tan(a) + 256*b^3*n^3*x*e^(-pi*b*n*sgn(x) + pi*b*n -
pi*b*sgn(c) + pi*b)*tan(2*b*n*log(abs(x)) + 2*b*log(abs(c)))^2*tan(b*n*log(abs(x)) + b*log(abs(c)))^2*tan(a)
- 256*b^3*n^3*x*e^(pi*b*n*sgn(x) - pi*b*n + pi*b*sgn(c) - pi*b)*tan(2*b*n*log(abs(x)) + 2*b*log(abs(c)))^2*tan
(2*a)^2*tan(a) - 256*b^3*n^3*x*e^(-pi*b*n*sgn(x) + pi*b*n - pi*b*sgn(c) + pi*b)*tan(2*b*n*log(abs(x)) + 2*b*lo
g(abs(c)))^2*tan(2*a)^2*tan(a) + 256*b^3*n^3*x*e^(pi*b*n*sgn(x) - pi*b*n + pi*b*sgn(c) - pi*b)*tan(b*n*log(abs
(x)) + b*log(abs(c)))^2*tan(2*a)^2*tan(a) + 256*b^3*n^3*x*e^(-pi*b*n*sgn(x) + pi*b*n - pi*b*sgn(c) + pi*b)*tan
(b*n*log(abs(x)) + b*log(abs(c)))^2*tan(2*a)^2*tan(a) + 256*b^3*n^3*x*e^(pi*b*n*sgn(x) - pi*b*n + pi*b*sgn(c)
- pi*b)*tan(2*b*n*log(abs(x)) + 2*b*log(abs(c)))^2*tan(b*n*log(abs(x)) + b*log(abs(c)))*tan(a)^2 + 256*b^3*n^3
*x*e^(-pi*b*n*sgn(x) + pi*b*n - pi*b*sgn(c) + pi*b)*tan(2*b*n*log(abs(x)) + 2*b*log(abs(c)))^2*tan(b*n*log(abs
(x)) + b*log(abs(c)))*tan(a)^2 - 32*b^3*n^3*x*e^(2*pi*b*n*sgn(x) - 2*pi*b*n + 2*pi*b*sgn(c) - 2*pi*b)*tan(2*b*
n*log(abs(x)) + 2*b*log(abs(c)))*tan(b*n*log(abs(x)) + b*log(abs(c)))^2*tan(a)^2 - 32*b^3*n^3*x*e^(-2*pi*b*n*s
gn(x) + 2*pi*b*n - 2*pi*b*sgn(c) + 2*pi*b)*tan(2*b*n*log(abs(x)) + 2*b*log(abs(c)))*tan(b*n*log(abs(x)) + b*lo
g(abs(c)))^2*tan(a)^2 + 32*b^3*n^3*x*e^(2*pi*b*n*sgn(x) - 2*pi*b*n + 2*pi*b*sgn(c) - 2*pi*b)*tan(2*b*n*log(abs
(x)) + 2*b*log(abs(c)))^2*tan(2*a)*tan(a)^2 + 32*b^3*n^3*x*e^(-2*pi*b*n*sgn(x) + 2*pi*b*n - 2*pi*b*sgn(c) + 2*
pi*b)*tan(2*b*n*log(abs(x)) + 2*b*log(abs(c)))^2*tan(2*a)*tan(a)^2 - 32*b^3*n^3*x*e^(2*pi*b*n*sgn(x) - 2*pi*b*
n + 2*pi*b*sgn(c) - 2*pi*b)*tan(b*n*log(abs(x)) + b*log(abs(c)))^2*tan(2*a)*tan(a)^2 - 32*b^3*n^3*x*e^(-2*pi*b
*n*sgn(x) + 2*pi*b*n - 2*pi*b*sgn(c) + 2*pi*b)*tan(b*n*log(abs(x)) + b*log(abs(c)))^2*tan(2*a)*tan(a)^2 + 32*b
^3*n^3*x*e^(2*pi*b*n*sgn(x) - 2*pi*b*n + 2*pi*b...

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Mupad [B]
time = 2.82, size = 116, normalized size = 0.61 \begin {gather*} \frac {3\,x}{8}+\frac {x\,{\mathrm {e}}^{-a\,2{}\mathrm {i}}\,\frac {1}{{\left (c\,x^n\right )}^{b\,2{}\mathrm {i}}}\,1{}\mathrm {i}}{8\,b\,n+4{}\mathrm {i}}+\frac {x\,{\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\left (c\,x^n\right )}^{b\,2{}\mathrm {i}}}{4+b\,n\,8{}\mathrm {i}}+\frac {x\,{\mathrm {e}}^{-a\,4{}\mathrm {i}}\,\frac {1}{{\left (c\,x^n\right )}^{b\,4{}\mathrm {i}}}\,1{}\mathrm {i}}{64\,b\,n+16{}\mathrm {i}}+\frac {x\,{\mathrm {e}}^{a\,4{}\mathrm {i}}\,{\left (c\,x^n\right )}^{b\,4{}\mathrm {i}}}{16+b\,n\,64{}\mathrm {i}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(a + b*log(c*x^n))^4,x)

[Out]

(3*x)/8 + (x*exp(-a*2i)/(c*x^n)^(b*2i)*1i)/(8*b*n + 4i) + (x*exp(a*2i)*(c*x^n)^(b*2i))/(b*n*8i + 4) + (x*exp(-
a*4i)/(c*x^n)^(b*4i)*1i)/(64*b*n + 16i) + (x*exp(a*4i)*(c*x^n)^(b*4i))/(b*n*64i + 16)

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