Integrand size = 17, antiderivative size = 42 \[ \int \frac {\text {csch}^4\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {\coth \left (a+b \log \left (c x^n\right )\right )}{b n}-\frac {\coth ^3\left (a+b \log \left (c x^n\right )\right )}{3 b n} \] Output:
coth(a+b*ln(c*x^n))/b/n-1/3*coth(a+b*ln(c*x^n))^3/b/n
Time = 0.04 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.33 \[ \int \frac {\text {csch}^4\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {2 \coth \left (a+b \log \left (c x^n\right )\right )}{3 b n}-\frac {\coth \left (a+b \log \left (c x^n\right )\right ) \text {csch}^2\left (a+b \log \left (c x^n\right )\right )}{3 b n} \] Input:
Integrate[Csch[a + b*Log[c*x^n]]^4/x,x]
Output:
(2*Coth[a + b*Log[c*x^n]])/(3*b*n) - (Coth[a + b*Log[c*x^n]]*Csch[a + b*Lo g[c*x^n]]^2)/(3*b*n)
Result contains complex when optimal does not.
Time = 0.25 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.07, 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.
\(\displaystyle \int \frac {\text {csch}^4\left (a+b \log \left (c x^n\right )\right )}{x} \, dx\) |
\(\Big \downarrow \) 3039 |
\(\displaystyle \frac {\int \text {csch}^4\left (a+b \log \left (c x^n\right )\right )d\log \left (c x^n\right )}{n}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\int \csc \left (i a+i b \log \left (c x^n\right )\right )^4d\log \left (c x^n\right )}{n}\) |
\(\Big \downarrow \) 4254 |
\(\displaystyle \frac {i \int \left (1-\coth ^2\left (a+b \log \left (c x^n\right )\right )\right )d\left (-i \coth \left (a+b \log \left (c x^n\right )\right )\right )}{b n}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {i \left (\frac {1}{3} i \coth ^3\left (a+b \log \left (c x^n\right )\right )-i \coth \left (a+b \log \left (c x^n\right )\right )\right )}{b n}\) |
Input:
Int[Csch[a + b*Log[c*x^n]]^4/x,x]
Output:
(I*((-I)*Coth[a + b*Log[c*x^n]] + (I/3)*Coth[a + b*Log[c*x^n]]^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 = 6.62 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.86
method | result | size |
derivativedivides | \(\frac {\left (\frac {2}{3}-\frac {{\operatorname {csch}\left (a +b \ln \left (c \,x^{n}\right )\right )}^{2}}{3}\right ) \coth \left (a +b \ln \left (c \,x^{n}\right )\right )}{n b}\) | \(36\) |
default | \(\frac {\left (\frac {2}{3}-\frac {{\operatorname {csch}\left (a +b \ln \left (c \,x^{n}\right )\right )}^{2}}{3}\right ) \coth \left (a +b \ln \left (c \,x^{n}\right )\right )}{n b}\) | \(36\) |
parallelrisch | \(\frac {-{\tanh \left (\frac {a}{2}+b \ln \left (\sqrt {c \,x^{n}}\right )\right )}^{3}-{\coth \left (\frac {a}{2}+b \ln \left (\sqrt {c \,x^{n}}\right )\right )}^{3}+9 \tanh \left (\frac {a}{2}+b \ln \left (\sqrt {c \,x^{n}}\right )\right )+9 \coth \left (\frac {a}{2}+b \ln \left (\sqrt {c \,x^{n}}\right )\right )}{24 b n}\) | \(82\) |
risch | \(-\frac {4 \left (3 \left (x^{n}\right )^{2 b} c^{2 b} {\mathrm e}^{2 a} {\mathrm e}^{i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}} {\mathrm e}^{-i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right ) \operatorname {csgn}\left (i c \right )} {\mathrm e}^{-i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}} {\mathrm e}^{i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i c \right )}-1\right )}{3 b n {\left (\left (x^{n}\right )^{2 b} c^{2 b} {\mathrm e}^{2 a} {\mathrm e}^{i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}} {\mathrm e}^{-i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right ) \operatorname {csgn}\left (i c \right )} {\mathrm e}^{-i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}} {\mathrm e}^{i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i c \right )}-1\right )}^{3}}\) | \(222\) |
Input:
int(csch(a+b*ln(c*x^n))^4/x,x,method=_RETURNVERBOSE)
Output:
1/n/b*(2/3-1/3*csch(a+b*ln(c*x^n))^2)*coth(a+b*ln(c*x^n))
Leaf count of result is larger than twice the leaf count of optimal. 272 vs. \(2 (40) = 80\).
Time = 0.09 (sec) , antiderivative size = 272, normalized size of antiderivative = 6.48 \[ \int \frac {\text {csch}^4\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=-\frac {8 \, {\left (\cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) + 2 \, \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )\right )}}{3 \, {\left (b n \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{5} + 5 \, b n \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{4} + b n \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{5} - 3 \, b n \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{3} + {\left (10 \, b n \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} - 3 \, b n\right )} \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{3} + 2 \, b n \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) + {\left (10 \, b n \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{3} - 9 \, b n \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )\right )} \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} + {\left (5 \, b n \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{4} - 9 \, b n \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} + 4 \, b n\right )} \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )\right )}} \] Input:
integrate(csch(a+b*log(c*x^n))^4/x,x, algorithm="fricas")
Output:
-8/3*(cosh(b*n*log(x) + b*log(c) + a) + 2*sinh(b*n*log(x) + b*log(c) + a)) /(b*n*cosh(b*n*log(x) + b*log(c) + a)^5 + 5*b*n*cosh(b*n*log(x) + b*log(c) + a)*sinh(b*n*log(x) + b*log(c) + a)^4 + b*n*sinh(b*n*log(x) + b*log(c) + a)^5 - 3*b*n*cosh(b*n*log(x) + b*log(c) + a)^3 + (10*b*n*cosh(b*n*log(x) + b*log(c) + a)^2 - 3*b*n)*sinh(b*n*log(x) + b*log(c) + a)^3 + 2*b*n*cosh( b*n*log(x) + b*log(c) + a) + (10*b*n*cosh(b*n*log(x) + b*log(c) + a)^3 - 9 *b*n*cosh(b*n*log(x) + b*log(c) + a))*sinh(b*n*log(x) + b*log(c) + a)^2 + (5*b*n*cosh(b*n*log(x) + b*log(c) + a)^4 - 9*b*n*cosh(b*n*log(x) + b*log(c ) + a)^2 + 4*b*n)*sinh(b*n*log(x) + b*log(c) + a))
\[ \int \frac {\text {csch}^4\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\int \frac {\operatorname {csch}^{4}{\left (a + b \log {\left (c x^{n} \right )} \right )}}{x}\, dx \] Input:
integrate(csch(a+b*ln(c*x**n))**4/x,x)
Output:
Integral(csch(a + b*log(c*x**n))**4/x, x)
Leaf count of result is larger than twice the leaf count of optimal. 92 vs. \(2 (40) = 80\).
Time = 0.05 (sec) , antiderivative size = 92, normalized size of antiderivative = 2.19 \[ \int \frac {\text {csch}^4\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=-\frac {4 \, {\left (3 \, c^{2 \, b} e^{\left (2 \, b \log \left (x^{n}\right ) + 2 \, a\right )} - 1\right )}}{3 \, {\left (b c^{6 \, b} n e^{\left (6 \, b \log \left (x^{n}\right ) + 6 \, a\right )} - 3 \, b c^{4 \, b} n e^{\left (4 \, b \log \left (x^{n}\right ) + 4 \, a\right )} + 3 \, b c^{2 \, b} n e^{\left (2 \, b \log \left (x^{n}\right ) + 2 \, a\right )} - b n\right )}} \] Input:
integrate(csch(a+b*log(c*x^n))^4/x,x, algorithm="maxima")
Output:
-4/3*(3*c^(2*b)*e^(2*b*log(x^n) + 2*a) - 1)/(b*c^(6*b)*n*e^(6*b*log(x^n) + 6*a) - 3*b*c^(4*b)*n*e^(4*b*log(x^n) + 4*a) + 3*b*c^(2*b)*n*e^(2*b*log(x^ n) + 2*a) - b*n)
Time = 0.13 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.12 \[ \int \frac {\text {csch}^4\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=-\frac {4 \, {\left (3 \, c^{2 \, b} x^{2 \, b n} e^{\left (2 \, a\right )} - 1\right )}}{3 \, {\left (c^{2 \, b} x^{2 \, b n} e^{\left (2 \, a\right )} - 1\right )}^{3} b n} \] Input:
integrate(csch(a+b*log(c*x^n))^4/x,x, algorithm="giac")
Output:
-4/3*(3*c^(2*b)*x^(2*b*n)*e^(2*a) - 1)/((c^(2*b)*x^(2*b*n)*e^(2*a) - 1)^3* b*n)
Time = 2.56 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.31 \[ \int \frac {\text {csch}^4\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {4\,{\mathrm {e}}^{4\,a}\,{\left (c\,x^n\right )}^{4\,b}\,\left ({\mathrm {e}}^{2\,a}\,{\left (c\,x^n\right )}^{2\,b}-3\right )}{3\,b\,n\,{\left ({\mathrm {e}}^{2\,a}\,{\left (c\,x^n\right )}^{2\,b}-1\right )}^3} \] Input:
int(1/(x*sinh(a + b*log(c*x^n))^4),x)
Output:
(4*exp(4*a)*(c*x^n)^(4*b)*(exp(2*a)*(c*x^n)^(2*b) - 3))/(3*b*n*(exp(2*a)*( c*x^n)^(2*b) - 1)^3)
Time = 0.21 (sec) , antiderivative size = 85, normalized size of antiderivative = 2.02 \[ \int \frac {\text {csch}^4\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {-4 x^{2 b n} e^{2 a} c^{2 b}+\frac {4}{3}}{b n \left (x^{6 b n} e^{6 a} c^{6 b}-3 x^{4 b n} e^{4 a} c^{4 b}+3 x^{2 b n} e^{2 a} c^{2 b}-1\right )} \] Input:
int(csch(a+b*log(c*x^n))^4/x,x)
Output:
(4*( - 3*x**(2*b*n)*e**(2*a)*c**(2*b) + 1))/(3*b*n*(x**(6*b*n)*e**(6*a)*c* *(6*b) - 3*x**(4*b*n)*e**(4*a)*c**(4*b) + 3*x**(2*b*n)*e**(2*a)*c**(2*b) - 1))