3.225 \(\int \frac{\cot ^5(a+b \log (c x^n))}{x} \, dx\)

Optimal. Leaf size=66 \[ \frac{\log \left (\sin \left (a+b \log \left (c x^n\right )\right )\right )}{b n}-\frac{\cot ^4\left (a+b \log \left (c x^n\right )\right )}{4 b n}+\frac{\cot ^2\left (a+b \log \left (c x^n\right )\right )}{2 b n} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0455594, antiderivative size = 66, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {3473, 3475} \[ \frac{\log \left (\sin \left (a+b \log \left (c x^n\right )\right )\right )}{b n}-\frac{\cot ^4\left (a+b \log \left (c x^n\right )\right )}{4 b n}+\frac{\cot ^2\left (a+b \log \left (c x^n\right )\right )}{2 b n} \]

Antiderivative was successfully verified.

[In]

Int[Cot[a + b*Log[c*x^n]]^5/x,x]

[Out]

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

Rule 3473

Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*(b*Tan[c + d*x])^(n - 1))/(d*(n - 1)), x] - Dis
t[b^2, Int[(b*Tan[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1]

Rule 3475

Int[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[Log[RemoveContent[Cos[c + d*x], x]]/d, x] /; FreeQ[{c, d}, x]

Rubi steps

\begin{align*} \int \frac{\cot ^5\left (a+b \log \left (c x^n\right )\right )}{x} \, dx &=\frac{\operatorname{Subst}\left (\int \cot ^5(a+b x) \, dx,x,\log \left (c x^n\right )\right )}{n}\\ &=-\frac{\cot ^4\left (a+b \log \left (c x^n\right )\right )}{4 b n}-\frac{\operatorname{Subst}\left (\int \cot ^3(a+b x) \, dx,x,\log \left (c x^n\right )\right )}{n}\\ &=\frac{\cot ^2\left (a+b \log \left (c x^n\right )\right )}{2 b n}-\frac{\cot ^4\left (a+b \log \left (c x^n\right )\right )}{4 b n}+\frac{\operatorname{Subst}\left (\int \cot (a+b x) \, dx,x,\log \left (c x^n\right )\right )}{n}\\ &=\frac{\cot ^2\left (a+b \log \left (c x^n\right )\right )}{2 b n}-\frac{\cot ^4\left (a+b \log \left (c x^n\right )\right )}{4 b n}+\frac{\log \left (\sin \left (a+b \log \left (c x^n\right )\right )\right )}{b n}\\ \end{align*}

Mathematica [A]  time = 0.230863, size = 69, normalized size = 1.05 \[ \frac{4 \log \left (\tan \left (a+b \log \left (c x^n\right )\right )\right )+4 \log \left (\cos \left (a+b \log \left (c x^n\right )\right )\right )-\cot ^4\left (a+b \log \left (c x^n\right )\right )+2 \cot ^2\left (a+b \log \left (c x^n\right )\right )}{4 b n} \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[a + b*Log[c*x^n]]^5/x,x]

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.025, size = 68, normalized size = 1. \begin{align*} -{\frac{ \left ( \cot \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) \right ) ^{4}}{4\,bn}}+{\frac{ \left ( \cot \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) \right ) ^{2}}{2\,bn}}-{\frac{\ln \left ( \left ( \cot \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) \right ) ^{2}+1 \right ) }{2\,bn}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(a+b*ln(c*x^n))^5/x,x)

[Out]

-1/4*cot(a+b*ln(c*x^n))^4/b/n+1/2*cot(a+b*ln(c*x^n))^2/b/n-1/2/n/b*ln(cot(a+b*ln(c*x^n))^2+1)

________________________________________________________________________________________

Maxima [B]  time = 1.88761, size = 8097, normalized size = 122.68 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(a+b*log(c*x^n))^5/x,x, algorithm="maxima")

[Out]

1/2*(32*(cos(6*b*log(c))^2 + sin(6*b*log(c))^2)*cos(6*b*log(x^n) + 6*a)^2 + 48*(cos(4*b*log(c))^2 + sin(4*b*lo
g(c))^2)*cos(4*b*log(x^n) + 4*a)^2 + 32*(cos(2*b*log(c))^2 + sin(2*b*log(c))^2)*cos(2*b*log(x^n) + 2*a)^2 + 32
*(cos(6*b*log(c))^2 + sin(6*b*log(c))^2)*sin(6*b*log(x^n) + 6*a)^2 + 48*(cos(4*b*log(c))^2 + sin(4*b*log(c))^2
)*sin(4*b*log(x^n) + 4*a)^2 + 32*(cos(2*b*log(c))^2 + sin(2*b*log(c))^2)*sin(2*b*log(x^n) + 2*a)^2 - 8*((cos(8
*b*log(c))*cos(6*b*log(c)) + sin(8*b*log(c))*sin(6*b*log(c)))*cos(6*b*log(x^n) + 6*a) - (cos(8*b*log(c))*cos(4
*b*log(c)) + sin(8*b*log(c))*sin(4*b*log(c)))*cos(4*b*log(x^n) + 4*a) + (cos(8*b*log(c))*cos(2*b*log(c)) + sin
(8*b*log(c))*sin(2*b*log(c)))*cos(2*b*log(x^n) + 2*a) + (cos(6*b*log(c))*sin(8*b*log(c)) - cos(8*b*log(c))*sin
(6*b*log(c)))*sin(6*b*log(x^n) + 6*a) - (cos(4*b*log(c))*sin(8*b*log(c)) - cos(8*b*log(c))*sin(4*b*log(c)))*si
n(4*b*log(x^n) + 4*a) + (cos(2*b*log(c))*sin(8*b*log(c)) - cos(8*b*log(c))*sin(2*b*log(c)))*sin(2*b*log(x^n) +
2*a))*cos(8*b*log(x^n) + 8*a) - 8*(10*(cos(6*b*log(c))*cos(4*b*log(c)) + sin(6*b*log(c))*sin(4*b*log(c)))*cos
(4*b*log(x^n) + 4*a) - 8*(cos(6*b*log(c))*cos(2*b*log(c)) + sin(6*b*log(c))*sin(2*b*log(c)))*cos(2*b*log(x^n)
+ 2*a) + 10*(cos(4*b*log(c))*sin(6*b*log(c)) - cos(6*b*log(c))*sin(4*b*log(c)))*sin(4*b*log(x^n) + 4*a) - 8*(c
os(2*b*log(c))*sin(6*b*log(c)) - cos(6*b*log(c))*sin(2*b*log(c)))*sin(2*b*log(x^n) + 2*a) + cos(6*b*log(c)))*c
os(6*b*log(x^n) + 6*a) - 8*(10*(cos(4*b*log(c))*cos(2*b*log(c)) + sin(4*b*log(c))*sin(2*b*log(c)))*cos(2*b*log
(x^n) + 2*a) + 10*(cos(2*b*log(c))*sin(4*b*log(c)) - cos(4*b*log(c))*sin(2*b*log(c)))*sin(2*b*log(x^n) + 2*a)
- cos(4*b*log(c)))*cos(4*b*log(x^n) + 4*a) - 8*cos(2*b*log(c))*cos(2*b*log(x^n) + 2*a) + ((cos(8*b*log(c))^2 +
sin(8*b*log(c))^2)*cos(8*b*log(x^n) + 8*a)^2 + 16*(cos(6*b*log(c))^2 + sin(6*b*log(c))^2)*cos(6*b*log(x^n) +
6*a)^2 + 36*(cos(4*b*log(c))^2 + sin(4*b*log(c))^2)*cos(4*b*log(x^n) + 4*a)^2 + 16*(cos(2*b*log(c))^2 + sin(2*
b*log(c))^2)*cos(2*b*log(x^n) + 2*a)^2 + (cos(8*b*log(c))^2 + sin(8*b*log(c))^2)*sin(8*b*log(x^n) + 8*a)^2 + 1
6*(cos(6*b*log(c))^2 + sin(6*b*log(c))^2)*sin(6*b*log(x^n) + 6*a)^2 + 36*(cos(4*b*log(c))^2 + sin(4*b*log(c))^
2)*sin(4*b*log(x^n) + 4*a)^2 + 16*(cos(2*b*log(c))^2 + sin(2*b*log(c))^2)*sin(2*b*log(x^n) + 2*a)^2 - 2*(4*(co
s(8*b*log(c))*cos(6*b*log(c)) + sin(8*b*log(c))*sin(6*b*log(c)))*cos(6*b*log(x^n) + 6*a) - 6*(cos(8*b*log(c))*
cos(4*b*log(c)) + sin(8*b*log(c))*sin(4*b*log(c)))*cos(4*b*log(x^n) + 4*a) + 4*(cos(8*b*log(c))*cos(2*b*log(c)
) + sin(8*b*log(c))*sin(2*b*log(c)))*cos(2*b*log(x^n) + 2*a) + 4*(cos(6*b*log(c))*sin(8*b*log(c)) - cos(8*b*lo
g(c))*sin(6*b*log(c)))*sin(6*b*log(x^n) + 6*a) - 6*(cos(4*b*log(c))*sin(8*b*log(c)) - cos(8*b*log(c))*sin(4*b*
log(c)))*sin(4*b*log(x^n) + 4*a) + 4*(cos(2*b*log(c))*sin(8*b*log(c)) - cos(8*b*log(c))*sin(2*b*log(c)))*sin(2
*b*log(x^n) + 2*a) - cos(8*b*log(c)))*cos(8*b*log(x^n) + 8*a) - 8*(6*(cos(6*b*log(c))*cos(4*b*log(c)) + sin(6*
b*log(c))*sin(4*b*log(c)))*cos(4*b*log(x^n) + 4*a) - 4*(cos(6*b*log(c))*cos(2*b*log(c)) + sin(6*b*log(c))*sin(
2*b*log(c)))*cos(2*b*log(x^n) + 2*a) + 6*(cos(4*b*log(c))*sin(6*b*log(c)) - cos(6*b*log(c))*sin(4*b*log(c)))*s
in(4*b*log(x^n) + 4*a) - 4*(cos(2*b*log(c))*sin(6*b*log(c)) - cos(6*b*log(c))*sin(2*b*log(c)))*sin(2*b*log(x^n
) + 2*a) + cos(6*b*log(c)))*cos(6*b*log(x^n) + 6*a) - 12*(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(x^n) + 2*a) + 4*(cos(2*b*log(c))*sin(4*b*log(c)) - cos(4*b*log(c))*sin(2*b*log(c
)))*sin(2*b*log(x^n) + 2*a) - cos(4*b*log(c)))*cos(4*b*log(x^n) + 4*a) - 8*cos(2*b*log(c))*cos(2*b*log(x^n) +
2*a) + 2*(4*(cos(6*b*log(c))*sin(8*b*log(c)) - cos(8*b*log(c))*sin(6*b*log(c)))*cos(6*b*log(x^n) + 6*a) - 6*(c
os(4*b*log(c))*sin(8*b*log(c)) - cos(8*b*log(c))*sin(4*b*log(c)))*cos(4*b*log(x^n) + 4*a) + 4*(cos(2*b*log(c))
*sin(8*b*log(c)) - cos(8*b*log(c))*sin(2*b*log(c)))*cos(2*b*log(x^n) + 2*a) - 4*(cos(8*b*log(c))*cos(6*b*log(c
)) + sin(8*b*log(c))*sin(6*b*log(c)))*sin(6*b*log(x^n) + 6*a) + 6*(cos(8*b*log(c))*cos(4*b*log(c)) + sin(8*b*l
og(c))*sin(4*b*log(c)))*sin(4*b*log(x^n) + 4*a) - 4*(cos(8*b*log(c))*cos(2*b*log(c)) + sin(8*b*log(c))*sin(2*b
*log(c)))*sin(2*b*log(x^n) + 2*a) - sin(8*b*log(c)))*sin(8*b*log(x^n) + 8*a) + 8*(6*(cos(4*b*log(c))*sin(6*b*l
og(c)) - cos(6*b*log(c))*sin(4*b*log(c)))*cos(4*b*log(x^n) + 4*a) - 4*(cos(2*b*log(c))*sin(6*b*log(c)) - cos(6
*b*log(c))*sin(2*b*log(c)))*cos(2*b*log(x^n) + 2*a) - 6*(cos(6*b*log(c))*cos(4*b*log(c)) + sin(6*b*log(c))*sin
(4*b*log(c)))*sin(4*b*log(x^n) + 4*a) + 4*(cos(6*b*log(c))*cos(2*b*log(c)) + sin(6*b*log(c))*sin(2*b*log(c)))*
sin(2*b*log(x^n) + 2*a) + sin(6*b*log(c)))*sin(6*b*log(x^n) + 6*a) + 12*(4*(cos(2*b*log(c))*sin(4*b*log(c)) -
cos(4*b*log(c))*sin(2*b*log(c)))*cos(2*b*log(x^n) + 2*a) - 4*(cos(4*b*log(c))*cos(2*b*log(c)) + sin(4*b*log(c)
)*sin(2*b*log(c)))*sin(2*b*log(x^n) + 2*a) - sin(4*b*log(c)))*sin(4*b*log(x^n) + 4*a) + 8*sin(2*b*log(c))*sin(
2*b*log(x^n) + 2*a) + 1)*log((cos(a)^2 + sin(a)^2)*cos(b*log(c))^2 + (cos(a)^2 + sin(a)^2)*sin(b*log(c))^2 + 2
*(cos(b*log(c))*cos(a) - sin(b*log(c))*sin(a))*cos(b*log(x^n)) + cos(b*log(x^n))^2 - 2*(cos(a)*sin(b*log(c)) +
cos(b*log(c))*sin(a))*sin(b*log(x^n)) + sin(b*log(x^n))^2) + ((cos(8*b*log(c))^2 + sin(8*b*log(c))^2)*cos(8*b
*log(x^n) + 8*a)^2 + 16*(cos(6*b*log(c))^2 + sin(6*b*log(c))^2)*cos(6*b*log(x^n) + 6*a)^2 + 36*(cos(4*b*log(c)
)^2 + sin(4*b*log(c))^2)*cos(4*b*log(x^n) + 4*a)^2 + 16*(cos(2*b*log(c))^2 + sin(2*b*log(c))^2)*cos(2*b*log(x^
n) + 2*a)^2 + (cos(8*b*log(c))^2 + sin(8*b*log(c))^2)*sin(8*b*log(x^n) + 8*a)^2 + 16*(cos(6*b*log(c))^2 + sin(
6*b*log(c))^2)*sin(6*b*log(x^n) + 6*a)^2 + 36*(cos(4*b*log(c))^2 + sin(4*b*log(c))^2)*sin(4*b*log(x^n) + 4*a)^
2 + 16*(cos(2*b*log(c))^2 + sin(2*b*log(c))^2)*sin(2*b*log(x^n) + 2*a)^2 - 2*(4*(cos(8*b*log(c))*cos(6*b*log(c
)) + sin(8*b*log(c))*sin(6*b*log(c)))*cos(6*b*log(x^n) + 6*a) - 6*(cos(8*b*log(c))*cos(4*b*log(c)) + sin(8*b*l
og(c))*sin(4*b*log(c)))*cos(4*b*log(x^n) + 4*a) + 4*(cos(8*b*log(c))*cos(2*b*log(c)) + sin(8*b*log(c))*sin(2*b
*log(c)))*cos(2*b*log(x^n) + 2*a) + 4*(cos(6*b*log(c))*sin(8*b*log(c)) - cos(8*b*log(c))*sin(6*b*log(c)))*sin(
6*b*log(x^n) + 6*a) - 6*(cos(4*b*log(c))*sin(8*b*log(c)) - cos(8*b*log(c))*sin(4*b*log(c)))*sin(4*b*log(x^n) +
4*a) + 4*(cos(2*b*log(c))*sin(8*b*log(c)) - cos(8*b*log(c))*sin(2*b*log(c)))*sin(2*b*log(x^n) + 2*a) - cos(8*
b*log(c)))*cos(8*b*log(x^n) + 8*a) - 8*(6*(cos(6*b*log(c))*cos(4*b*log(c)) + sin(6*b*log(c))*sin(4*b*log(c)))*
cos(4*b*log(x^n) + 4*a) - 4*(cos(6*b*log(c))*cos(2*b*log(c)) + sin(6*b*log(c))*sin(2*b*log(c)))*cos(2*b*log(x^
n) + 2*a) + 6*(cos(4*b*log(c))*sin(6*b*log(c)) - cos(6*b*log(c))*sin(4*b*log(c)))*sin(4*b*log(x^n) + 4*a) - 4*
(cos(2*b*log(c))*sin(6*b*log(c)) - cos(6*b*log(c))*sin(2*b*log(c)))*sin(2*b*log(x^n) + 2*a) + cos(6*b*log(c)))
*cos(6*b*log(x^n) + 6*a) - 12*(4*(cos(4*b*log(c))*cos(2*b*log(c)) + sin(4*b*log(c))*sin(2*b*log(c)))*cos(2*b*l
og(x^n) + 2*a) + 4*(cos(2*b*log(c))*sin(4*b*log(c)) - cos(4*b*log(c))*sin(2*b*log(c)))*sin(2*b*log(x^n) + 2*a)
- cos(4*b*log(c)))*cos(4*b*log(x^n) + 4*a) - 8*cos(2*b*log(c))*cos(2*b*log(x^n) + 2*a) + 2*(4*(cos(6*b*log(c)
)*sin(8*b*log(c)) - cos(8*b*log(c))*sin(6*b*log(c)))*cos(6*b*log(x^n) + 6*a) - 6*(cos(4*b*log(c))*sin(8*b*log(
c)) - cos(8*b*log(c))*sin(4*b*log(c)))*cos(4*b*log(x^n) + 4*a) + 4*(cos(2*b*log(c))*sin(8*b*log(c)) - cos(8*b*
log(c))*sin(2*b*log(c)))*cos(2*b*log(x^n) + 2*a) - 4*(cos(8*b*log(c))*cos(6*b*log(c)) + sin(8*b*log(c))*sin(6*
b*log(c)))*sin(6*b*log(x^n) + 6*a) + 6*(cos(8*b*log(c))*cos(4*b*log(c)) + sin(8*b*log(c))*sin(4*b*log(c)))*sin
(4*b*log(x^n) + 4*a) - 4*(cos(8*b*log(c))*cos(2*b*log(c)) + sin(8*b*log(c))*sin(2*b*log(c)))*sin(2*b*log(x^n)
+ 2*a) - sin(8*b*log(c)))*sin(8*b*log(x^n) + 8*a) + 8*(6*(cos(4*b*log(c))*sin(6*b*log(c)) - cos(6*b*log(c))*si
n(4*b*log(c)))*cos(4*b*log(x^n) + 4*a) - 4*(cos(2*b*log(c))*sin(6*b*log(c)) - cos(6*b*log(c))*sin(2*b*log(c)))
*cos(2*b*log(x^n) + 2*a) - 6*(cos(6*b*log(c))*cos(4*b*log(c)) + sin(6*b*log(c))*sin(4*b*log(c)))*sin(4*b*log(x
^n) + 4*a) + 4*(cos(6*b*log(c))*cos(2*b*log(c)) + sin(6*b*log(c))*sin(2*b*log(c)))*sin(2*b*log(x^n) + 2*a) + s
in(6*b*log(c)))*sin(6*b*log(x^n) + 6*a) + 12*(4*(cos(2*b*log(c))*sin(4*b*log(c)) - cos(4*b*log(c))*sin(2*b*log
(c)))*cos(2*b*log(x^n) + 2*a) - 4*(cos(4*b*log(c))*cos(2*b*log(c)) + sin(4*b*log(c))*sin(2*b*log(c)))*sin(2*b*
log(x^n) + 2*a) - sin(4*b*log(c)))*sin(4*b*log(x^n) + 4*a) + 8*sin(2*b*log(c))*sin(2*b*log(x^n) + 2*a) + 1)*lo
g((cos(a)^2 + sin(a)^2)*cos(b*log(c))^2 + (cos(a)^2 + sin(a)^2)*sin(b*log(c))^2 - 2*(cos(b*log(c))*cos(a) - si
n(b*log(c))*sin(a))*cos(b*log(x^n)) + cos(b*log(x^n))^2 + 2*(cos(a)*sin(b*log(c)) + cos(b*log(c))*sin(a))*sin(
b*log(x^n)) + sin(b*log(x^n))^2) + 8*((cos(6*b*log(c))*sin(8*b*log(c)) - cos(8*b*log(c))*sin(6*b*log(c)))*cos(
6*b*log(x^n) + 6*a) - (cos(4*b*log(c))*sin(8*b*log(c)) - cos(8*b*log(c))*sin(4*b*log(c)))*cos(4*b*log(x^n) + 4
*a) + (cos(2*b*log(c))*sin(8*b*log(c)) - cos(8*b*log(c))*sin(2*b*log(c)))*cos(2*b*log(x^n) + 2*a) - (cos(8*b*l
og(c))*cos(6*b*log(c)) + sin(8*b*log(c))*sin(6*b*log(c)))*sin(6*b*log(x^n) + 6*a) + (cos(8*b*log(c))*cos(4*b*l
og(c)) + sin(8*b*log(c))*sin(4*b*log(c)))*sin(4*b*log(x^n) + 4*a) - (cos(8*b*log(c))*cos(2*b*log(c)) + sin(8*b
*log(c))*sin(2*b*log(c)))*sin(2*b*log(x^n) + 2*a))*sin(8*b*log(x^n) + 8*a) + 8*(10*(cos(4*b*log(c))*sin(6*b*lo
g(c)) - cos(6*b*log(c))*sin(4*b*log(c)))*cos(4*b*log(x^n) + 4*a) - 8*(cos(2*b*log(c))*sin(6*b*log(c)) - cos(6*
b*log(c))*sin(2*b*log(c)))*cos(2*b*log(x^n) + 2*a) - 10*(cos(6*b*log(c))*cos(4*b*log(c)) + sin(6*b*log(c))*sin
(4*b*log(c)))*sin(4*b*log(x^n) + 4*a) + 8*(cos(6*b*log(c))*cos(2*b*log(c)) + sin(6*b*log(c))*sin(2*b*log(c)))*
sin(2*b*log(x^n) + 2*a) + sin(6*b*log(c)))*sin(6*b*log(x^n) + 6*a) + 8*(10*(cos(2*b*log(c))*sin(4*b*log(c)) -
cos(4*b*log(c))*sin(2*b*log(c)))*cos(2*b*log(x^n) + 2*a) - 10*(cos(4*b*log(c))*cos(2*b*log(c)) + sin(4*b*log(c
))*sin(2*b*log(c)))*sin(2*b*log(x^n) + 2*a) - sin(4*b*log(c)))*sin(4*b*log(x^n) + 4*a) + 8*sin(2*b*log(c))*sin
(2*b*log(x^n) + 2*a))/((b*cos(8*b*log(c))^2 + b*sin(8*b*log(c))^2)*n*cos(8*b*log(x^n) + 8*a)^2 + 16*(b*cos(6*b
*log(c))^2 + b*sin(6*b*log(c))^2)*n*cos(6*b*log(x^n) + 6*a)^2 + 36*(b*cos(4*b*log(c))^2 + b*sin(4*b*log(c))^2)
*n*cos(4*b*log(x^n) + 4*a)^2 - 8*b*n*cos(2*b*log(c))*cos(2*b*log(x^n) + 2*a) + 16*(b*cos(2*b*log(c))^2 + b*sin
(2*b*log(c))^2)*n*cos(2*b*log(x^n) + 2*a)^2 + (b*cos(8*b*log(c))^2 + b*sin(8*b*log(c))^2)*n*sin(8*b*log(x^n) +
8*a)^2 + 16*(b*cos(6*b*log(c))^2 + b*sin(6*b*log(c))^2)*n*sin(6*b*log(x^n) + 6*a)^2 + 36*(b*cos(4*b*log(c))^2
+ b*sin(4*b*log(c))^2)*n*sin(4*b*log(x^n) + 4*a)^2 + 8*b*n*sin(2*b*log(c))*sin(2*b*log(x^n) + 2*a) + 16*(b*co
s(2*b*log(c))^2 + b*sin(2*b*log(c))^2)*n*sin(2*b*log(x^n) + 2*a)^2 + b*n + 2*(b*n*cos(8*b*log(c)) - 4*(b*cos(8
*b*log(c))*cos(6*b*log(c)) + b*sin(8*b*log(c))*sin(6*b*log(c)))*n*cos(6*b*log(x^n) + 6*a) + 6*(b*cos(8*b*log(c
))*cos(4*b*log(c)) + b*sin(8*b*log(c))*sin(4*b*log(c)))*n*cos(4*b*log(x^n) + 4*a) - 4*(b*cos(8*b*log(c))*cos(2
*b*log(c)) + b*sin(8*b*log(c))*sin(2*b*log(c)))*n*cos(2*b*log(x^n) + 2*a) - 4*(b*cos(6*b*log(c))*sin(8*b*log(c
)) - b*cos(8*b*log(c))*sin(6*b*log(c)))*n*sin(6*b*log(x^n) + 6*a) + 6*(b*cos(4*b*log(c))*sin(8*b*log(c)) - b*c
os(8*b*log(c))*sin(4*b*log(c)))*n*sin(4*b*log(x^n) + 4*a) - 4*(b*cos(2*b*log(c))*sin(8*b*log(c)) - b*cos(8*b*l
og(c))*sin(2*b*log(c)))*n*sin(2*b*log(x^n) + 2*a))*cos(8*b*log(x^n) + 8*a) - 8*(b*n*cos(6*b*log(c)) + 6*(b*cos
(6*b*log(c))*cos(4*b*log(c)) + b*sin(6*b*log(c))*sin(4*b*log(c)))*n*cos(4*b*log(x^n) + 4*a) - 4*(b*cos(6*b*log
(c))*cos(2*b*log(c)) + b*sin(6*b*log(c))*sin(2*b*log(c)))*n*cos(2*b*log(x^n) + 2*a) + 6*(b*cos(4*b*log(c))*sin
(6*b*log(c)) - b*cos(6*b*log(c))*sin(4*b*log(c)))*n*sin(4*b*log(x^n) + 4*a) - 4*(b*cos(2*b*log(c))*sin(6*b*log
(c)) - b*cos(6*b*log(c))*sin(2*b*log(c)))*n*sin(2*b*log(x^n) + 2*a))*cos(6*b*log(x^n) + 6*a) + 12*(b*n*cos(4*b
*log(c)) - 4*(b*cos(4*b*log(c))*cos(2*b*log(c)) + b*sin(4*b*log(c))*sin(2*b*log(c)))*n*cos(2*b*log(x^n) + 2*a)
- 4*(b*cos(2*b*log(c))*sin(4*b*log(c)) - b*cos(4*b*log(c))*sin(2*b*log(c)))*n*sin(2*b*log(x^n) + 2*a))*cos(4*
b*log(x^n) + 4*a) + 2*(4*(b*cos(6*b*log(c))*sin(8*b*log(c)) - b*cos(8*b*log(c))*sin(6*b*log(c)))*n*cos(6*b*log
(x^n) + 6*a) - 6*(b*cos(4*b*log(c))*sin(8*b*log(c)) - b*cos(8*b*log(c))*sin(4*b*log(c)))*n*cos(4*b*log(x^n) +
4*a) + 4*(b*cos(2*b*log(c))*sin(8*b*log(c)) - b*cos(8*b*log(c))*sin(2*b*log(c)))*n*cos(2*b*log(x^n) + 2*a) - b
*n*sin(8*b*log(c)) - 4*(b*cos(8*b*log(c))*cos(6*b*log(c)) + b*sin(8*b*log(c))*sin(6*b*log(c)))*n*sin(6*b*log(x
^n) + 6*a) + 6*(b*cos(8*b*log(c))*cos(4*b*log(c)) + b*sin(8*b*log(c))*sin(4*b*log(c)))*n*sin(4*b*log(x^n) + 4*
a) - 4*(b*cos(8*b*log(c))*cos(2*b*log(c)) + b*sin(8*b*log(c))*sin(2*b*log(c)))*n*sin(2*b*log(x^n) + 2*a))*sin(
8*b*log(x^n) + 8*a) + 8*(6*(b*cos(4*b*log(c))*sin(6*b*log(c)) - b*cos(6*b*log(c))*sin(4*b*log(c)))*n*cos(4*b*l
og(x^n) + 4*a) - 4*(b*cos(2*b*log(c))*sin(6*b*log(c)) - b*cos(6*b*log(c))*sin(2*b*log(c)))*n*cos(2*b*log(x^n)
+ 2*a) + b*n*sin(6*b*log(c)) - 6*(b*cos(6*b*log(c))*cos(4*b*log(c)) + b*sin(6*b*log(c))*sin(4*b*log(c)))*n*sin
(4*b*log(x^n) + 4*a) + 4*(b*cos(6*b*log(c))*cos(2*b*log(c)) + b*sin(6*b*log(c))*sin(2*b*log(c)))*n*sin(2*b*log
(x^n) + 2*a))*sin(6*b*log(x^n) + 6*a) + 12*(4*(b*cos(2*b*log(c))*sin(4*b*log(c)) - b*cos(4*b*log(c))*sin(2*b*l
og(c)))*n*cos(2*b*log(x^n) + 2*a) - b*n*sin(4*b*log(c)) - 4*(b*cos(4*b*log(c))*cos(2*b*log(c)) + b*sin(4*b*log
(c))*sin(2*b*log(c)))*n*sin(2*b*log(x^n) + 2*a))*sin(4*b*log(x^n) + 4*a))

________________________________________________________________________________________

Fricas [B]  time = 0.500595, size = 387, normalized size = 5.86 \begin{align*} \frac{{\left (\cos \left (2 \, b n \log \left (x\right ) + 2 \, b \log \left (c\right ) + 2 \, a\right )^{2} - 2 \, \cos \left (2 \, b n \log \left (x\right ) + 2 \, b \log \left (c\right ) + 2 \, a\right ) + 1\right )} \log \left (-\frac{1}{2} \, \cos \left (2 \, b n \log \left (x\right ) + 2 \, b \log \left (c\right ) + 2 \, a\right ) + \frac{1}{2}\right ) - 4 \, \cos \left (2 \, b n \log \left (x\right ) + 2 \, b \log \left (c\right ) + 2 \, a\right ) + 2}{2 \,{\left (b n \cos \left (2 \, b n \log \left (x\right ) + 2 \, b \log \left (c\right ) + 2 \, a\right )^{2} - 2 \, b n \cos \left (2 \, b n \log \left (x\right ) + 2 \, b \log \left (c\right ) + 2 \, a\right ) + b n\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(a+b*log(c*x^n))^5/x,x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

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(cot(a+b*ln(c*x**n))**5/x,x)

[Out]

Timed out

________________________________________________________________________________________

Giac [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(cot(a+b*log(c*x^n))^5/x,x, algorithm="giac")

[Out]

Exception raised: TypeError