Integrand size = 17, antiderivative size = 66 \[ \int \frac {\cot ^5\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {\cot ^2\left (a+b \log \left (c x^n\right )\right )}{2 b n}-\frac {\cot ^4\left (a+b \log \left (c x^n\right )\right )}{4 b n}+\frac {\log \left (\sin \left (a+b \log \left (c x^n\right )\right )\right )}{b n} \] Output:
1/2*cot(a+b*ln(c*x^n))^2/b/n-1/4*cot(a+b*ln(c*x^n))^4/b/n+ln(sin(a+b*ln(c* x^n)))/b/n
Time = 0.10 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.95 \[ \int \frac {\cot ^5\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {\csc ^2\left (a+b \log \left (c x^n\right )\right )}{b n}-\frac {\csc ^4\left (a+b \log \left (c x^n\right )\right )}{4 b n}+\frac {\log \left (\sin \left (a+b \log \left (c x^n\right )\right )\right )}{b n} \] Input:
Integrate[Cot[a + b*Log[c*x^n]]^5/x,x]
Output:
Csc[a + b*Log[c*x^n]]^2/(b*n) - Csc[a + b*Log[c*x^n]]^4/(4*b*n) + Log[Sin[ a + b*Log[c*x^n]]]/(b*n)
Time = 0.36 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.95, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.706, Rules used = {3039, 3042, 25, 3954, 25, 3042, 25, 3954, 25, 3042, 25, 3956}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {\cot ^5\left (a+b \log \left (c x^n\right )\right )}{x} \, dx\) |
| \(\Big \downarrow \) 3039 |
| \(\displaystyle \frac {\int \cot ^5\left (a+b \log \left (c x^n\right )\right )d\log \left (c x^n\right )}{n}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int -\tan \left (a+b \log \left (c x^n\right )+\frac {\pi }{2}\right )^5d\log \left (c x^n\right )}{n}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\int \tan \left (\frac {1}{2} (2 a+\pi )+b \log \left (c x^n\right )\right )^5d\log \left (c x^n\right )}{n}\) |
| \(\Big \downarrow \) 3954 |
| \(\displaystyle \frac {\int -\cot ^3\left (a+b \log \left (c x^n\right )\right )d\log \left (c x^n\right )-\frac {\cot ^4\left (a+b \log \left (c x^n\right )\right )}{4 b}}{n}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {-\int \cot ^3\left (a+b \log \left (c x^n\right )\right )d\log \left (c x^n\right )-\frac {\cot ^4\left (a+b \log \left (c x^n\right )\right )}{4 b}}{n}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\int -\tan \left (a+b \log \left (c x^n\right )+\frac {\pi }{2}\right )^3d\log \left (c x^n\right )-\frac {\cot ^4\left (a+b \log \left (c x^n\right )\right )}{4 b}}{n}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \tan \left (\frac {1}{2} (2 a+\pi )+b \log \left (c x^n\right )\right )^3d\log \left (c x^n\right )-\frac {\cot ^4\left (a+b \log \left (c x^n\right )\right )}{4 b}}{n}\) |
| \(\Big \downarrow \) 3954 |
| \(\displaystyle \frac {-\int -\cot \left (a+b \log \left (c x^n\right )\right )d\log \left (c x^n\right )-\frac {\cot ^4\left (a+b \log \left (c x^n\right )\right )}{4 b}+\frac {\cot ^2\left (a+b \log \left (c x^n\right )\right )}{2 b}}{n}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \cot \left (a+b \log \left (c x^n\right )\right )d\log \left (c x^n\right )-\frac {\cot ^4\left (a+b \log \left (c x^n\right )\right )}{4 b}+\frac {\cot ^2\left (a+b \log \left (c x^n\right )\right )}{2 b}}{n}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int -\tan \left (a+b \log \left (c x^n\right )+\frac {\pi }{2}\right )d\log \left (c x^n\right )-\frac {\cot ^4\left (a+b \log \left (c x^n\right )\right )}{4 b}+\frac {\cot ^2\left (a+b \log \left (c x^n\right )\right )}{2 b}}{n}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {-\int \tan \left (\frac {1}{2} (2 a+\pi )+b \log \left (c x^n\right )\right )d\log \left (c x^n\right )-\frac {\cot ^4\left (a+b \log \left (c x^n\right )\right )}{4 b}+\frac {\cot ^2\left (a+b \log \left (c x^n\right )\right )}{2 b}}{n}\) |
| \(\Big \downarrow \) 3956 |
| \(\displaystyle \frac {\frac {\log \left (-\sin \left (a+b \log \left (c x^n\right )\right )\right )}{b}-\frac {\cot ^4\left (a+b \log \left (c x^n\right )\right )}{4 b}+\frac {\cot ^2\left (a+b \log \left (c x^n\right )\right )}{2 b}}{n}\) |
Input:
Int[Cot[a + b*Log[c*x^n]]^5/x,x]
Output:
(Cot[a + b*Log[c*x^n]]^2/(2*b) - Cot[a + b*Log[c*x^n]]^4/(4*b) + Log[-Sin[ a + b*Log[c*x^n]]]/b)/n
Int[u_, x_Symbol] :> With[{lst = FunctionOfLog[Cancel[x*u], x]}, Simp[1/lst [[3]] Subst[Int[lst[[1]], x], x, Log[lst[[2]]]], x] /; !FalseQ[lst]] /; NonsumQ[u]
Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[b*((b*Tan[c + d *x])^(n - 1)/(d*(n - 1))), x] - Simp[b^2 Int[(b*Tan[c + d*x])^(n - 2), x] , x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1]
Int[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Log[RemoveContent[Cos[c + d *x], x]]/d, x] /; FreeQ[{c, d}, x]
Time = 1.38 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.86
| method | result | size |
| derivativedivides | \(\frac {-\frac {{\cot \left (a +b \ln \left (c \,x^{n}\right )\right )}^{4}}{4}+\frac {{\cot \left (a +b \ln \left (c \,x^{n}\right )\right )}^{2}}{2}-\frac {\ln \left ({\cot \left (a +b \ln \left (c \,x^{n}\right )\right )}^{2}+1\right )}{2}}{n b}\) | \(57\) |
| default | \(\frac {-\frac {{\cot \left (a +b \ln \left (c \,x^{n}\right )\right )}^{4}}{4}+\frac {{\cot \left (a +b \ln \left (c \,x^{n}\right )\right )}^{2}}{2}-\frac {\ln \left ({\cot \left (a +b \ln \left (c \,x^{n}\right )\right )}^{2}+1\right )}{2}}{n b}\) | \(57\) |
| parallelrisch | \(\frac {-{\cot \left (a +b \ln \left (c \,x^{n}\right )\right )}^{4}+4 \ln \left (\tan \left (a +b \ln \left (c \,x^{n}\right )\right )\right )-2 \ln \left ({\sec \left (a +b \ln \left (c \,x^{n}\right )\right )}^{2}\right )+2 {\cot \left (a +b \ln \left (c \,x^{n}\right )\right )}^{2}}{4 b n}\) | \(70\) |
| risch | \(i \ln \left (x \right )-\frac {2 i a}{b n}-\frac {2 i \ln \left (c \right )}{n}-\frac {2 i \ln \left (x^{n}\right )}{n}+\frac {\pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{n}-\frac {\pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right ) \operatorname {csgn}\left (i c \right )}{n}-\frac {\pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}}{n}+\frac {\pi \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i c \right )}{n}-\frac {4 \left (x^{n}\right )^{2 i b} c^{2 i b} \left (c^{4 i b} \left (x^{n}\right )^{4 i b} {\mathrm e}^{-3 b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}} {\mathrm e}^{3 b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right ) \operatorname {csgn}\left (i c \right )} {\mathrm e}^{3 b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}} {\mathrm e}^{-3 b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i c \right )} {\mathrm e}^{6 i a}-c^{2 i b} \left (x^{n}\right )^{2 i b} {\mathrm e}^{-2 b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}} {\mathrm e}^{2 b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right ) \operatorname {csgn}\left (i c \right )} {\mathrm e}^{2 b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}} {\mathrm e}^{-2 b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i c \right )} {\mathrm e}^{4 i a}+{\mathrm e}^{-b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}} {\mathrm e}^{b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right ) \operatorname {csgn}\left (i c \right )} {\mathrm e}^{b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}} {\mathrm e}^{-b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i c \right )} {\mathrm e}^{2 i a}\right )}{b n {\left (\left (x^{n}\right )^{2 i b} c^{2 i b} {\mathrm e}^{-b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}} {\mathrm e}^{b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right ) \operatorname {csgn}\left (i c \right )} {\mathrm e}^{b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}} {\mathrm e}^{-b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i c \right )} {\mathrm e}^{2 i a}-1\right )}^{4}}+\frac {\ln \left (\left (x^{n}\right )^{2 i b} c^{2 i b} {\mathrm e}^{-b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}} {\mathrm e}^{b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right ) \operatorname {csgn}\left (i c \right )} {\mathrm e}^{b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}} {\mathrm e}^{-b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i c \right )} {\mathrm e}^{2 i a}-1\right )}{b n}\) | \(667\) |
Input:
int(cot(a+b*ln(c*x^n))^5/x,x,method=_RETURNVERBOSE)
Output:
1/n/b*(-1/4*cot(a+b*ln(c*x^n))^4+1/2*cot(a+b*ln(c*x^n))^2-1/2*ln(cot(a+b*l n(c*x^n))^2+1))
Leaf count of result is larger than twice the leaf count of optimal. 129 vs. \(2 (62) = 124\).
Time = 0.08 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.95 \[ \int \frac {\cot ^5\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {{\left (\cos \left (2 \, b n \log \left (x\right ) + 2 \, b \log \left (c\right ) + 2 \, a\right )^{2} - 2 \, \cos \left (2 \, b n \log \left (x\right ) + 2 \, b \log \left (c\right ) + 2 \, a\right ) + 1\right )} \log \left (-\frac {1}{2} \, \cos \left (2 \, b n \log \left (x\right ) + 2 \, b \log \left (c\right ) + 2 \, a\right ) + \frac {1}{2}\right ) - 4 \, \cos \left (2 \, b n \log \left (x\right ) + 2 \, b \log \left (c\right ) + 2 \, a\right ) + 2}{2 \, {\left (b n \cos \left (2 \, b n \log \left (x\right ) + 2 \, b \log \left (c\right ) + 2 \, a\right )^{2} - 2 \, b n \cos \left (2 \, b n \log \left (x\right ) + 2 \, b \log \left (c\right ) + 2 \, a\right ) + b n\right )}} \] Input:
integrate(cot(a+b*log(c*x^n))^5/x,x, algorithm="fricas")
Output:
1/2*((cos(2*b*n*log(x) + 2*b*log(c) + 2*a)^2 - 2*cos(2*b*n*log(x) + 2*b*lo g(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*log( c) + 2*a)^2 - 2*b*n*cos(2*b*n*log(x) + 2*b*log(c) + 2*a) + b*n)
Leaf count of result is larger than twice the leaf count of optimal. 117 vs. \(2 (53) = 106\).
Time = 20.70 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.77 \[ \int \frac {\cot ^5\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\begin {cases} \tilde {\infty } \log {\left (x \right )} & \text {for}\: a = 0 \wedge b = 0 \wedge n = 0 \\\log {\left (x \right )} \cot ^{5}{\left (a \right )} & \text {for}\: b = 0 \\\log {\left (x \right )} \cot ^{5}{\left (a + b \log {\left (c \right )} \right )} & \text {for}\: n = 0 \\\tilde {\infty } \log {\left (x \right )} & \text {for}\: a = - b \log {\left (c x^{n} \right )} \\- \frac {\log {\left (\tan ^{2}{\left (a + b \log {\left (c x^{n} \right )} \right )} + 1 \right )}}{2 b n} + \frac {\log {\left (\tan {\left (a + b \log {\left (c x^{n} \right )} \right )} \right )}}{b n} + \frac {1}{2 b n \tan ^{2}{\left (a + b \log {\left (c x^{n} \right )} \right )}} - \frac {1}{4 b n \tan ^{4}{\left (a + b \log {\left (c x^{n} \right )} \right )}} & \text {otherwise} \end {cases} \] Input:
integrate(cot(a+b*ln(c*x**n))**5/x,x)
Output:
Piecewise((zoo*log(x), Eq(a, 0) & Eq(b, 0) & Eq(n, 0)), (log(x)*cot(a)**5, Eq(b, 0)), (log(x)*cot(a + b*log(c))**5, Eq(n, 0)), (zoo*log(x), Eq(a, -b *log(c*x**n))), (-log(tan(a + b*log(c*x**n))**2 + 1)/(2*b*n) + log(tan(a + b*log(c*x**n)))/(b*n) + 1/(2*b*n*tan(a + b*log(c*x**n))**2) - 1/(4*b*n*ta n(a + b*log(c*x**n))**4), True))
Leaf count of result is larger than twice the leaf count of optimal. 5998 vs. \(2 (62) = 124\).
Time = 0.25 (sec) , antiderivative size = 5998, normalized size of antiderivative = 90.88 \[ \int \frac {\cot ^5\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\text {Too large to display} \] Input:
integrate(cot(a+b*log(c*x^n))^5/x,x, algorithm="maxima")
Output:
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*log(c))^2)*cos(4*b*log(x^n) + 4*a)^2 + 3 2*(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*l og(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*l og(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)))*sin(4*b*lo g(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)) - c os(6*b*log(c))*sin(4*b*log(c)))*sin(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)))*sin(2*b*log(x^n) + 2 *a) + cos(6*b*log(c)))*cos(6*b*log(x^n) + 6*a) - 8*(10*(cos(4*b*log(c))...
Timed out. \[ \int \frac {\cot ^5\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\text {Timed out} \] Input:
integrate(cot(a+b*log(c*x^n))^5/x,x, algorithm="giac")
Output:
Timed out
Time = 24.46 (sec) , antiderivative size = 246, normalized size of antiderivative = 3.73 \[ \int \frac {\cot ^5\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=-\ln \left (x\right )\,1{}\mathrm {i}-\frac {8}{b\,n\,\left (1+{\mathrm {e}}^{a\,4{}\mathrm {i}}\,{\left (c\,x^n\right )}^{b\,4{}\mathrm {i}}-2\,{\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\left (c\,x^n\right )}^{b\,2{}\mathrm {i}}\right )}-\frac {4}{b\,n\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\left (c\,x^n\right )}^{b\,2{}\mathrm {i}}-1\right )}-\frac {4}{b\,n\,\left (1+6\,{\mathrm {e}}^{a\,4{}\mathrm {i}}\,{\left (c\,x^n\right )}^{b\,4{}\mathrm {i}}-4\,{\mathrm {e}}^{a\,6{}\mathrm {i}}\,{\left (c\,x^n\right )}^{b\,6{}\mathrm {i}}+{\mathrm {e}}^{a\,8{}\mathrm {i}}\,{\left (c\,x^n\right )}^{b\,8{}\mathrm {i}}-4\,{\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\left (c\,x^n\right )}^{b\,2{}\mathrm {i}}\right )}+\frac {\ln \left ({\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\left (c\,x^n\right )}^{b\,2{}\mathrm {i}}-1\right )}{b\,n}-\frac {8}{b\,n\,\left (3\,{\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\left (c\,x^n\right )}^{b\,2{}\mathrm {i}}-3\,{\mathrm {e}}^{a\,4{}\mathrm {i}}\,{\left (c\,x^n\right )}^{b\,4{}\mathrm {i}}+{\mathrm {e}}^{a\,6{}\mathrm {i}}\,{\left (c\,x^n\right )}^{b\,6{}\mathrm {i}}-1\right )} \] Input:
int(cot(a + b*log(c*x^n))^5/x,x)
Output:
log(exp(a*2i)*(c*x^n)^(b*2i) - 1)/(b*n) - 8/(b*n*(exp(a*4i)*(c*x^n)^(b*4i) - 2*exp(a*2i)*(c*x^n)^(b*2i) + 1)) - 4/(b*n*(exp(a*2i)*(c*x^n)^(b*2i) - 1 )) - 4/(b*n*(6*exp(a*4i)*(c*x^n)^(b*4i) - 4*exp(a*2i)*(c*x^n)^(b*2i) - 4*e xp(a*6i)*(c*x^n)^(b*6i) + exp(a*8i)*(c*x^n)^(b*8i) + 1)) - log(x)*1i - 8/( b*n*(3*exp(a*2i)*(c*x^n)^(b*2i) - 3*exp(a*4i)*(c*x^n)^(b*4i) + exp(a*6i)*( c*x^n)^(b*6i) - 1))
Time = 0.21 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.77 \[ \int \frac {\cot ^5\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {-32 \,\mathrm {log}\left ({\tan \left (\frac {\mathrm {log}\left (x^{n} c \right ) b}{2}+\frac {a}{2}\right )}^{2}+1\right ) {\sin \left (\mathrm {log}\left (x^{n} c \right ) b +a \right )}^{4}+32 \,\mathrm {log}\left (\tan \left (\frac {\mathrm {log}\left (x^{n} c \right ) b}{2}+\frac {a}{2}\right )\right ) {\sin \left (\mathrm {log}\left (x^{n} c \right ) b +a \right )}^{4}-13 {\sin \left (\mathrm {log}\left (x^{n} c \right ) b +a \right )}^{4}+32 {\sin \left (\mathrm {log}\left (x^{n} c \right ) b +a \right )}^{2}-8}{32 {\sin \left (\mathrm {log}\left (x^{n} c \right ) b +a \right )}^{4} b n} \] Input:
int(cot(a+b*log(c*x^n))^5/x,x)
Output:
( - 32*log(tan((log(x**n*c)*b + a)/2)**2 + 1)*sin(log(x**n*c)*b + a)**4 + 32*log(tan((log(x**n*c)*b + a)/2))*sin(log(x**n*c)*b + a)**4 - 13*sin(log( x**n*c)*b + a)**4 + 32*sin(log(x**n*c)*b + a)**2 - 8)/(32*sin(log(x**n*c)* b + a)**4*b*n)