Integrand size = 17, antiderivative size = 43 \[ \int \frac {\csc ^4\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=-\frac {\cot \left (a+b \log \left (c x^n\right )\right )}{b n}-\frac {\cot ^3\left (a+b \log \left (c x^n\right )\right )}{3 b n} \] Output:
-cot(a+b*ln(c*x^n))/b/n-1/3*cot(a+b*ln(c*x^n))^3/b/n
Time = 0.08 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.30 \[ \int \frac {\csc ^4\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=-\frac {2 \cot \left (a+b \log \left (c x^n\right )\right )}{3 b n}-\frac {\cot \left (a+b \log \left (c x^n\right )\right ) \csc ^2\left (a+b \log \left (c x^n\right )\right )}{3 b n} \] Input:
Integrate[Csc[a + b*Log[c*x^n]]^4/x,x]
Output:
(-2*Cot[a + b*Log[c*x^n]])/(3*b*n) - (Cot[a + b*Log[c*x^n]]*Csc[a + b*Log[ c*x^n]]^2)/(3*b*n)
Time = 0.24 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.86, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {3039, 3042, 4254, 2009}
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.
\(\displaystyle \int \frac {\csc ^4\left (a+b \log \left (c x^n\right )\right )}{x} \, dx\) |
\(\Big \downarrow \) 3039 |
\(\displaystyle \frac {\int \csc ^4\left (a+b \log \left (c x^n\right )\right )d\log \left (c x^n\right )}{n}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\int \csc \left (a+b \log \left (c x^n\right )\right )^4d\log \left (c x^n\right )}{n}\) |
\(\Big \downarrow \) 4254 |
\(\displaystyle -\frac {\int \left (\cot ^2\left (a+b \log \left (c x^n\right )\right )+1\right )d\cot \left (a+b \log \left (c x^n\right )\right )}{b n}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {\frac {1}{3} \cot ^3\left (a+b \log \left (c x^n\right )\right )+\cot \left (a+b \log \left (c x^n\right )\right )}{b n}\) |
Input:
Int[Csc[a + b*Log[c*x^n]]^4/x,x]
Output:
-((Cot[a + b*Log[c*x^n]] + Cot[a + b*Log[c*x^n]]^3/3)/(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[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Simp[-d^(-1) Subst[Int[Exp andIntegrand[(1 + x^2)^(n/2 - 1), x], x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 0]
Time = 4.66 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.84
method | result | size |
derivativedivides | \(\frac {\left (-\frac {2}{3}-\frac {{\csc \left (a +b \ln \left (c \,x^{n}\right )\right )}^{2}}{3}\right ) \cot \left (a +b \ln \left (c \,x^{n}\right )\right )}{n b}\) | \(36\) |
default | \(\frac {\left (-\frac {2}{3}-\frac {{\csc \left (a +b \ln \left (c \,x^{n}\right )\right )}^{2}}{3}\right ) \cot \left (a +b \ln \left (c \,x^{n}\right )\right )}{n b}\) | \(36\) |
parallelrisch | \(\frac {-{\cot \left (\frac {a}{2}+b \ln \left (\sqrt {c \,x^{n}}\right )\right )}^{3}+{\tan \left (\frac {a}{2}+b \ln \left (\sqrt {c \,x^{n}}\right )\right )}^{3}+9 \tan \left (\frac {a}{2}+b \ln \left (\sqrt {c \,x^{n}}\right )\right )-9 \cot \left (\frac {a}{2}+b \ln \left (\sqrt {c \,x^{n}}\right )\right )}{24 b n}\) | \(80\) |
risch | \(\frac {4 i \left (3 \left (x^{n}\right )^{2 i b} c^{2 i b} {\mathrm e}^{-b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}} {\mathrm e}^{b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right ) \operatorname {csgn}\left (i c \right )} {\mathrm e}^{b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}} {\mathrm e}^{-b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i c \right )} {\mathrm e}^{2 i a}-1\right )}{3 b n {\left (\left (x^{n}\right )^{2 i b} c^{2 i b} {\mathrm e}^{-b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}} {\mathrm e}^{b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right ) \operatorname {csgn}\left (i c \right )} {\mathrm e}^{b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}} {\mathrm e}^{-b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i c \right )} {\mathrm e}^{2 i a}-1\right )}^{3}}\) | \(225\) |
Input:
int(csc(a+b*ln(c*x^n))^4/x,x,method=_RETURNVERBOSE)
Output:
1/n/b*(-2/3-1/3*csc(a+b*ln(c*x^n))^2)*cot(a+b*ln(c*x^n))
Time = 0.07 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.65 \[ \int \frac {\csc ^4\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=-\frac {2 \, \cos \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{3} - 3 \, \cos \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )}{3 \, {\left (b n \cos \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} - b n\right )} \sin \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )} \] Input:
integrate(csc(a+b*log(c*x^n))^4/x,x, algorithm="fricas")
Output:
-1/3*(2*cos(b*n*log(x) + b*log(c) + a)^3 - 3*cos(b*n*log(x) + b*log(c) + a ))/((b*n*cos(b*n*log(x) + b*log(c) + a)^2 - b*n)*sin(b*n*log(x) + b*log(c) + a))
\[ \int \frac {\csc ^4\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\int \frac {\csc ^{4}{\left (a + b \log {\left (c x^{n} \right )} \right )}}{x}\, dx \] Input:
integrate(csc(a+b*ln(c*x**n))**4/x,x)
Output:
Integral(csc(a + b*log(c*x**n))**4/x, x)
Leaf count of result is larger than twice the leaf count of optimal. 1332 vs. \(2 (41) = 82\).
Time = 0.07 (sec) , antiderivative size = 1332, normalized size of antiderivative = 30.98 \[ \int \frac {\csc ^4\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\text {Too large to display} \] Input:
integrate(csc(a+b*log(c*x^n))^4/x,x, algorithm="maxima")
Output:
4/3*((3*(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) - 3*(cos(6*b*log(c))*cos(2*b*log(c)) + sin(6*b*l og(c))*sin(2*b*log(c)))*sin(2*b*log(x^n) + 2*a) - sin(6*b*log(c)))*cos(6*b *log(x^n) + 6*a) - 3*(3*(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) - 3*(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)))*cos(4*b*log(x^n) + 4*a) + (3*(cos(6*b*log(c))*cos(2*b*log(c)) + s in(6*b*log(c))*sin(2*b*log(c)))*cos(2*b*log(x^n) + 2*a) + 3*(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)))*sin(6*b*log(x^n) + 6*a) - 3*(3*(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) + 3*(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)))*sin(4*b*log(x^n) + 4*a))/((b*cos(6* b*log(c))^2 + b*sin(6*b*log(c))^2)*n*cos(6*b*log(x^n) + 6*a)^2 + 9*(b*cos( 4*b*log(c))^2 + b*sin(4*b*log(c))^2)*n*cos(4*b*log(x^n) + 4*a)^2 - 6*b*n*c os(2*b*log(c))*cos(2*b*log(x^n) + 2*a) + 9*(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(6*b*log(c))^2 + b*sin(6* b*log(c))^2)*n*sin(6*b*log(x^n) + 6*a)^2 + 9*(b*cos(4*b*log(c))^2 + b*sin( 4*b*log(c))^2)*n*sin(4*b*log(x^n) + 4*a)^2 + 6*b*n*sin(2*b*log(c))*sin(2*b *log(x^n) + 2*a) + 9*(b*cos(2*b*log(c))^2 + b*sin(2*b*log(c))^2)*n*sin(...
\[ \int \frac {\csc ^4\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\int { \frac {\csc \left (b \log \left (c x^{n}\right ) + a\right )^{4}}{x} \,d x } \] Input:
integrate(csc(a+b*log(c*x^n))^4/x,x, algorithm="giac")
Output:
integrate(csc(b*log(c*x^n) + a)^4/x, x)
Time = 34.46 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.14 \[ \int \frac {\csc ^4\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {4\,\left ({\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\left (c\,x^n\right )}^{b\,2{}\mathrm {i}}\,3{}\mathrm {i}-\mathrm {i}\right )}{3\,b\,n\,{\left ({\mathrm {e}}^{a\,2{}\mathrm {i}}\,{\left (c\,x^n\right )}^{b\,2{}\mathrm {i}}-1\right )}^3} \] Input:
int(1/(x*sin(a + b*log(c*x^n))^4),x)
Output:
(4*(exp(a*2i)*(c*x^n)^(b*2i)*3i - 1i))/(3*b*n*(exp(a*2i)*(c*x^n)^(b*2i) - 1)^3)
Time = 0.16 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.14 \[ \int \frac {\csc ^4\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {\cos \left (\mathrm {log}\left (x^{n} c \right ) b +a \right ) \left (-2 {\sin \left (\mathrm {log}\left (x^{n} c \right ) b +a \right )}^{2}-1\right )}{3 {\sin \left (\mathrm {log}\left (x^{n} c \right ) b +a \right )}^{3} b n} \] Input:
int(csc(a+b*log(c*x^n))^4/x,x)
Output:
(cos(log(x**n*c)*b + a)*( - 2*sin(log(x**n*c)*b + a)**2 - 1))/(3*sin(log(x **n*c)*b + a)**3*b*n)