Integrand size = 17, antiderivative size = 89 \[ \int \frac {\text {csch}^5\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=-\frac {3 \text {arctanh}\left (\cosh \left (a+b \log \left (c x^n\right )\right )\right )}{8 b n}+\frac {3 \coth \left (a+b \log \left (c x^n\right )\right ) \text {csch}\left (a+b \log \left (c x^n\right )\right )}{8 b n}-\frac {\coth \left (a+b \log \left (c x^n\right )\right ) \text {csch}^3\left (a+b \log \left (c x^n\right )\right )}{4 b n} \] Output:
-3/8*arctanh(cosh(a+b*ln(c*x^n)))/b/n+3/8*coth(a+b*ln(c*x^n))*csch(a+b*ln( c*x^n))/b/n-1/4*coth(a+b*ln(c*x^n))*csch(a+b*ln(c*x^n))^3/b/n
Time = 0.05 (sec) , antiderivative size = 161, normalized size of antiderivative = 1.81 \[ \int \frac {\text {csch}^5\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {3 \text {csch}^2\left (\frac {1}{2} \left (a+b \log \left (c x^n\right )\right )\right )}{32 b n}-\frac {\text {csch}^4\left (\frac {1}{2} \left (a+b \log \left (c x^n\right )\right )\right )}{64 b n}-\frac {3 \log \left (\cosh \left (\frac {1}{2} \left (a+b \log \left (c x^n\right )\right )\right )\right )}{8 b n}+\frac {3 \log \left (\sinh \left (\frac {1}{2} \left (a+b \log \left (c x^n\right )\right )\right )\right )}{8 b n}+\frac {3 \text {sech}^2\left (\frac {1}{2} \left (a+b \log \left (c x^n\right )\right )\right )}{32 b n}+\frac {\text {sech}^4\left (\frac {1}{2} \left (a+b \log \left (c x^n\right )\right )\right )}{64 b n} \] Input:
Integrate[Csch[a + b*Log[c*x^n]]^5/x,x]
Output:
(3*Csch[(a + b*Log[c*x^n])/2]^2)/(32*b*n) - Csch[(a + b*Log[c*x^n])/2]^4/( 64*b*n) - (3*Log[Cosh[(a + b*Log[c*x^n])/2]])/(8*b*n) + (3*Log[Sinh[(a + b *Log[c*x^n])/2]])/(8*b*n) + (3*Sech[(a + b*Log[c*x^n])/2]^2)/(32*b*n) + Se ch[(a + b*Log[c*x^n])/2]^4/(64*b*n)
Result contains complex when optimal does not.
Time = 0.42 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.10, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.706, Rules used = {3039, 3042, 26, 4255, 26, 3042, 26, 4255, 26, 3042, 26, 4257}
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.
| \(\displaystyle \int \frac {\text {csch}^5\left (a+b \log \left (c x^n\right )\right )}{x} \, dx\) |
| \(\Big \downarrow \) 3039 |
| \(\displaystyle \frac {\int \text {csch}^5\left (a+b \log \left (c x^n\right )\right )d\log \left (c x^n\right )}{n}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int i \csc \left (i a+i b \log \left (c x^n\right )\right )^5d\log \left (c x^n\right )}{n}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {i \int \csc \left (i a+i b \log \left (c x^n\right )\right )^5d\log \left (c x^n\right )}{n}\) |
| \(\Big \downarrow \) 4255 |
| \(\displaystyle \frac {i \left (\frac {3}{4} \int i \text {csch}^3\left (a+b \log \left (c x^n\right )\right )d\log \left (c x^n\right )+\frac {i \coth \left (a+b \log \left (c x^n\right )\right ) \text {csch}^3\left (a+b \log \left (c x^n\right )\right )}{4 b}\right )}{n}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {i \left (\frac {3}{4} i \int \text {csch}^3\left (a+b \log \left (c x^n\right )\right )d\log \left (c x^n\right )+\frac {i \coth \left (a+b \log \left (c x^n\right )\right ) \text {csch}^3\left (a+b \log \left (c x^n\right )\right )}{4 b}\right )}{n}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {i \left (\frac {3}{4} i \int -i \csc \left (i a+i b \log \left (c x^n\right )\right )^3d\log \left (c x^n\right )+\frac {i \coth \left (a+b \log \left (c x^n\right )\right ) \text {csch}^3\left (a+b \log \left (c x^n\right )\right )}{4 b}\right )}{n}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {i \left (\frac {3}{4} \int \csc \left (i a+i b \log \left (c x^n\right )\right )^3d\log \left (c x^n\right )+\frac {i \coth \left (a+b \log \left (c x^n\right )\right ) \text {csch}^3\left (a+b \log \left (c x^n\right )\right )}{4 b}\right )}{n}\) |
| \(\Big \downarrow \) 4255 |
| \(\displaystyle \frac {i \left (\frac {3}{4} \left (\frac {1}{2} \int -i \text {csch}\left (a+b \log \left (c x^n\right )\right )d\log \left (c x^n\right )-\frac {i \coth \left (a+b \log \left (c x^n\right )\right ) \text {csch}\left (a+b \log \left (c x^n\right )\right )}{2 b}\right )+\frac {i \coth \left (a+b \log \left (c x^n\right )\right ) \text {csch}^3\left (a+b \log \left (c x^n\right )\right )}{4 b}\right )}{n}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {i \left (\frac {3}{4} \left (-\frac {1}{2} i \int \text {csch}\left (a+b \log \left (c x^n\right )\right )d\log \left (c x^n\right )-\frac {i \coth \left (a+b \log \left (c x^n\right )\right ) \text {csch}\left (a+b \log \left (c x^n\right )\right )}{2 b}\right )+\frac {i \coth \left (a+b \log \left (c x^n\right )\right ) \text {csch}^3\left (a+b \log \left (c x^n\right )\right )}{4 b}\right )}{n}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {i \left (\frac {3}{4} \left (-\frac {1}{2} i \int i \csc \left (i a+i b \log \left (c x^n\right )\right )d\log \left (c x^n\right )-\frac {i \coth \left (a+b \log \left (c x^n\right )\right ) \text {csch}\left (a+b \log \left (c x^n\right )\right )}{2 b}\right )+\frac {i \coth \left (a+b \log \left (c x^n\right )\right ) \text {csch}^3\left (a+b \log \left (c x^n\right )\right )}{4 b}\right )}{n}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {i \left (\frac {3}{4} \left (\frac {1}{2} \int \csc \left (i a+i b \log \left (c x^n\right )\right )d\log \left (c x^n\right )-\frac {i \coth \left (a+b \log \left (c x^n\right )\right ) \text {csch}\left (a+b \log \left (c x^n\right )\right )}{2 b}\right )+\frac {i \coth \left (a+b \log \left (c x^n\right )\right ) \text {csch}^3\left (a+b \log \left (c x^n\right )\right )}{4 b}\right )}{n}\) |
| \(\Big \downarrow \) 4257 |
| \(\displaystyle \frac {i \left (\frac {3}{4} \left (\frac {i \text {arctanh}\left (\cosh \left (a+b \log \left (c x^n\right )\right )\right )}{2 b}-\frac {i \coth \left (a+b \log \left (c x^n\right )\right ) \text {csch}\left (a+b \log \left (c x^n\right )\right )}{2 b}\right )+\frac {i \coth \left (a+b \log \left (c x^n\right )\right ) \text {csch}^3\left (a+b \log \left (c x^n\right )\right )}{4 b}\right )}{n}\) |
Input:
Int[Csch[a + b*Log[c*x^n]]^5/x,x]
Output:
(I*(((I/4)*Coth[a + b*Log[c*x^n]]*Csch[a + b*Log[c*x^n]]^3)/b + (3*(((I/2) *ArcTanh[Cosh[a + b*Log[c*x^n]]])/b - ((I/2)*Coth[a + b*Log[c*x^n]]*Csch[a + b*Log[c*x^n]])/b))/4))/n
Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a]) I nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
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_)]*(b_.))^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d* x]*((b*Csc[c + d*x])^(n - 1)/(d*(n - 1))), x] + Simp[b^2*((n - 2)/(n - 1)) Int[(b*Csc[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && IntegerQ[2*n]
Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]
Time = 22.18 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.72
| method | result | size |
| derivativedivides | \(\frac {\left (-\frac {{\operatorname {csch}\left (a +b \ln \left (c \,x^{n}\right )\right )}^{3}}{4}+\frac {3 \,\operatorname {csch}\left (a +b \ln \left (c \,x^{n}\right )\right )}{8}\right ) \coth \left (a +b \ln \left (c \,x^{n}\right )\right )-\frac {3 \,\operatorname {arctanh}\left ({\mathrm e}^{a +b \ln \left (c \,x^{n}\right )}\right )}{4}}{n b}\) | \(64\) |
| default | \(\frac {\left (-\frac {{\operatorname {csch}\left (a +b \ln \left (c \,x^{n}\right )\right )}^{3}}{4}+\frac {3 \,\operatorname {csch}\left (a +b \ln \left (c \,x^{n}\right )\right )}{8}\right ) \coth \left (a +b \ln \left (c \,x^{n}\right )\right )-\frac {3 \,\operatorname {arctanh}\left ({\mathrm e}^{a +b \ln \left (c \,x^{n}\right )}\right )}{4}}{n b}\) | \(64\) |
| parallelrisch | \(\frac {-{\coth \left (\frac {a}{2}+b \ln \left (\sqrt {c \,x^{n}}\right )\right )}^{4}+{\tanh \left (\frac {a}{2}+b \ln \left (\sqrt {c \,x^{n}}\right )\right )}^{4}-8 {\tanh \left (\frac {a}{2}+b \ln \left (\sqrt {c \,x^{n}}\right )\right )}^{2}+24 \ln \left (\tanh \left (\frac {a}{2}+b \ln \left (\sqrt {c \,x^{n}}\right )\right )\right )+8 {\coth \left (\frac {a}{2}+b \ln \left (\sqrt {c \,x^{n}}\right )\right )}^{2}}{64 b n}\) | \(102\) |
| risch | \(\text {Expression too large to display}\) | \(744\) |
Input:
int(csch(a+b*ln(c*x^n))^5/x,x,method=_RETURNVERBOSE)
Output:
1/n/b*((-1/4*csch(a+b*ln(c*x^n))^3+3/8*csch(a+b*ln(c*x^n)))*coth(a+b*ln(c* x^n))-3/4*arctanh(exp(a+b*ln(c*x^n))))
Leaf count of result is larger than twice the leaf count of optimal. 1806 vs. \(2 (83) = 166\).
Time = 0.10 (sec) , antiderivative size = 1806, normalized size of antiderivative = 20.29 \[ \int \frac {\text {csch}^5\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\text {Too large to display} \] Input:
integrate(csch(a+b*log(c*x^n))^5/x,x, algorithm="fricas")
Output:
1/8*(6*cosh(b*n*log(x) + b*log(c) + a)^7 + 42*cosh(b*n*log(x) + b*log(c) + a)*sinh(b*n*log(x) + b*log(c) + a)^6 + 6*sinh(b*n*log(x) + b*log(c) + a)^ 7 + 2*(63*cosh(b*n*log(x) + b*log(c) + a)^2 - 11)*sinh(b*n*log(x) + b*log( c) + a)^5 - 22*cosh(b*n*log(x) + b*log(c) + a)^5 + 10*(21*cosh(b*n*log(x) + b*log(c) + a)^3 - 11*cosh(b*n*log(x) + b*log(c) + a))*sinh(b*n*log(x) + b*log(c) + a)^4 + 2*(105*cosh(b*n*log(x) + b*log(c) + a)^4 - 110*cosh(b*n* log(x) + b*log(c) + a)^2 - 11)*sinh(b*n*log(x) + b*log(c) + a)^3 - 22*cosh (b*n*log(x) + b*log(c) + a)^3 + 2*(63*cosh(b*n*log(x) + b*log(c) + a)^5 - 110*cosh(b*n*log(x) + b*log(c) + a)^3 - 33*cosh(b*n*log(x) + b*log(c) + a) )*sinh(b*n*log(x) + b*log(c) + a)^2 - 3*(cosh(b*n*log(x) + b*log(c) + a)^8 + 8*cosh(b*n*log(x) + b*log(c) + a)*sinh(b*n*log(x) + b*log(c) + a)^7 + s inh(b*n*log(x) + b*log(c) + a)^8 + 4*(7*cosh(b*n*log(x) + b*log(c) + a)^2 - 1)*sinh(b*n*log(x) + b*log(c) + a)^6 - 4*cosh(b*n*log(x) + b*log(c) + a) ^6 + 8*(7*cosh(b*n*log(x) + b*log(c) + a)^3 - 3*cosh(b*n*log(x) + b*log(c) + a))*sinh(b*n*log(x) + b*log(c) + a)^5 + 2*(35*cosh(b*n*log(x) + b*log(c ) + a)^4 - 30*cosh(b*n*log(x) + b*log(c) + a)^2 + 3)*sinh(b*n*log(x) + b*l og(c) + a)^4 + 6*cosh(b*n*log(x) + b*log(c) + a)^4 + 8*(7*cosh(b*n*log(x) + b*log(c) + a)^5 - 10*cosh(b*n*log(x) + b*log(c) + a)^3 + 3*cosh(b*n*log( x) + b*log(c) + a))*sinh(b*n*log(x) + b*log(c) + a)^3 + 4*(7*cosh(b*n*log( x) + b*log(c) + a)^6 - 15*cosh(b*n*log(x) + b*log(c) + a)^4 + 9*cosh(b*...
\[ \int \frac {\text {csch}^5\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\int \frac {\operatorname {csch}^{5}{\left (a + b \log {\left (c x^{n} \right )} \right )}}{x}\, dx \] Input:
integrate(csch(a+b*ln(c*x**n))**5/x,x)
Output:
Integral(csch(a + b*log(c*x**n))**5/x, x)
Leaf count of result is larger than twice the leaf count of optimal. 232 vs. \(2 (83) = 166\).
Time = 0.06 (sec) , antiderivative size = 232, normalized size of antiderivative = 2.61 \[ \int \frac {\text {csch}^5\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {3 \, c^{7 \, b} e^{\left (7 \, b \log \left (x^{n}\right ) + 7 \, a\right )} - 11 \, c^{5 \, b} e^{\left (5 \, b \log \left (x^{n}\right ) + 5 \, a\right )} - 11 \, c^{3 \, b} e^{\left (3 \, b \log \left (x^{n}\right ) + 3 \, a\right )} + 3 \, c^{b} e^{\left (b \log \left (x^{n}\right ) + a\right )}}{4 \, {\left (b c^{8 \, b} n e^{\left (8 \, b \log \left (x^{n}\right ) + 8 \, a\right )} - 4 \, b c^{6 \, b} n e^{\left (6 \, b \log \left (x^{n}\right ) + 6 \, a\right )} + 6 \, b c^{4 \, b} n e^{\left (4 \, b \log \left (x^{n}\right ) + 4 \, a\right )} - 4 \, b c^{2 \, b} n e^{\left (2 \, b \log \left (x^{n}\right ) + 2 \, a\right )} + b n\right )}} - \frac {3 \, \log \left (\frac {{\left (c^{b} e^{\left (b \log \left (x^{n}\right ) + a\right )} + 1\right )} e^{\left (-a\right )}}{c^{b}}\right )}{8 \, b n} + \frac {3 \, \log \left (\frac {{\left (c^{b} e^{\left (b \log \left (x^{n}\right ) + a\right )} - 1\right )} e^{\left (-a\right )}}{c^{b}}\right )}{8 \, b n} \] Input:
integrate(csch(a+b*log(c*x^n))^5/x,x, algorithm="maxima")
Output:
1/4*(3*c^(7*b)*e^(7*b*log(x^n) + 7*a) - 11*c^(5*b)*e^(5*b*log(x^n) + 5*a) - 11*c^(3*b)*e^(3*b*log(x^n) + 3*a) + 3*c^b*e^(b*log(x^n) + a))/(b*c^(8*b) *n*e^(8*b*log(x^n) + 8*a) - 4*b*c^(6*b)*n*e^(6*b*log(x^n) + 6*a) + 6*b*c^( 4*b)*n*e^(4*b*log(x^n) + 4*a) - 4*b*c^(2*b)*n*e^(2*b*log(x^n) + 2*a) + b*n ) - 3/8*log((c^b*e^(b*log(x^n) + a) + 1)*e^(-a)/c^b)/(b*n) + 3/8*log((c^b* e^(b*log(x^n) + a) - 1)*e^(-a)/c^b)/(b*n)
Leaf count of result is larger than twice the leaf count of optimal. 276 vs. \(2 (83) = 166\).
Time = 0.23 (sec) , antiderivative size = 276, normalized size of antiderivative = 3.10 \[ \int \frac {\text {csch}^5\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=-\frac {1}{8} \, c^{5 \, b} {\left (\frac {3 \, c^{b} e^{\left (-5 \, a\right )} \log \left (\sqrt {2 \, {\left | c \right |}^{b} {\left | x \right |}^{b n} \cos \left (\frac {1}{2} \, \pi b n \mathrm {sgn}\left (x\right ) - \frac {1}{2} \, \pi b n + \frac {1}{2} \, \pi b \mathrm {sgn}\left (c\right ) - \frac {1}{2} \, \pi b\right ) e^{a} + {\left | c \right |}^{2 \, b} {\left | x \right |}^{2 \, b n} e^{\left (2 \, a\right )} + 1}\right )}{b c^{6 \, b} n} - \frac {3 \, c^{b} e^{\left (-5 \, a\right )} \log \left (\sqrt {-2 \, {\left | c \right |}^{b} {\left | x \right |}^{b n} \cos \left (\frac {1}{2} \, \pi b n \mathrm {sgn}\left (x\right ) - \frac {1}{2} \, \pi b n + \frac {1}{2} \, \pi b \mathrm {sgn}\left (c\right ) - \frac {1}{2} \, \pi b\right ) e^{a} + {\left | c \right |}^{2 \, b} {\left | x \right |}^{2 \, b n} e^{\left (2 \, a\right )} + 1}\right )}{b c^{6 \, b} n} - \frac {2 \, {\left (3 \, c^{6 \, b} x^{7 \, b n} e^{\left (6 \, a\right )} - 11 \, c^{4 \, b} x^{5 \, b n} e^{\left (4 \, a\right )} - 11 \, c^{2 \, b} x^{3 \, b n} e^{\left (2 \, a\right )} + 3 \, x^{b n}\right )} e^{\left (-4 \, a\right )}}{{\left (c^{2 \, b} x^{2 \, b n} e^{\left (2 \, a\right )} - 1\right )}^{4} b c^{4 \, b} n}\right )} e^{\left (5 \, a\right )} \] Input:
integrate(csch(a+b*log(c*x^n))^5/x,x, algorithm="giac")
Output:
-1/8*c^(5*b)*(3*c^b*e^(-5*a)*log(sqrt(2*abs(c)^b*abs(x)^(b*n)*cos(1/2*pi*b *n*sgn(x) - 1/2*pi*b*n + 1/2*pi*b*sgn(c) - 1/2*pi*b)*e^a + abs(c)^(2*b)*ab s(x)^(2*b*n)*e^(2*a) + 1))/(b*c^(6*b)*n) - 3*c^b*e^(-5*a)*log(sqrt(-2*abs( c)^b*abs(x)^(b*n)*cos(1/2*pi*b*n*sgn(x) - 1/2*pi*b*n + 1/2*pi*b*sgn(c) - 1 /2*pi*b)*e^a + abs(c)^(2*b)*abs(x)^(2*b*n)*e^(2*a) + 1))/(b*c^(6*b)*n) - 2 *(3*c^(6*b)*x^(7*b*n)*e^(6*a) - 11*c^(4*b)*x^(5*b*n)*e^(4*a) - 11*c^(2*b)* x^(3*b*n)*e^(2*a) + 3*x^(b*n))*e^(-4*a)/((c^(2*b)*x^(2*b*n)*e^(2*a) - 1)^4 *b*c^(4*b)*n))*e^(5*a)
Time = 2.54 (sec) , antiderivative size = 318, normalized size of antiderivative = 3.57 \[ \int \frac {\text {csch}^5\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {2\,{\mathrm {e}}^{-a}}{{\left (c\,x^n\right )}^b\,\left (b\,n-\frac {3\,b\,n\,{\mathrm {e}}^{-2\,a}}{{\left (c\,x^n\right )}^{2\,b}}+\frac {3\,b\,n\,{\mathrm {e}}^{-4\,a}}{{\left (c\,x^n\right )}^{4\,b}}-\frac {b\,n\,{\mathrm {e}}^{-6\,a}}{{\left (c\,x^n\right )}^{6\,b}}\right )}-\frac {3\,\mathrm {atan}\left (\frac {{\mathrm {e}}^{-a}\,\sqrt {-b^2\,n^2}}{b\,n\,{\left (c\,x^n\right )}^b}\right )}{4\,\sqrt {-b^2\,n^2}}-\frac {3\,{\mathrm {e}}^{-a}}{4\,{\left (c\,x^n\right )}^b\,\left (b\,n-\frac {b\,n\,{\mathrm {e}}^{-2\,a}}{{\left (c\,x^n\right )}^{2\,b}}\right )}-\frac {4\,{\mathrm {e}}^{-3\,a}}{{\left (c\,x^n\right )}^{3\,b}\,\left (b\,n-\frac {4\,b\,n\,{\mathrm {e}}^{-2\,a}}{{\left (c\,x^n\right )}^{2\,b}}+\frac {6\,b\,n\,{\mathrm {e}}^{-4\,a}}{{\left (c\,x^n\right )}^{4\,b}}-\frac {4\,b\,n\,{\mathrm {e}}^{-6\,a}}{{\left (c\,x^n\right )}^{6\,b}}+\frac {b\,n\,{\mathrm {e}}^{-8\,a}}{{\left (c\,x^n\right )}^{8\,b}}\right )}-\frac {{\mathrm {e}}^{-a}}{2\,{\left (c\,x^n\right )}^b\,\left (b\,n-\frac {2\,b\,n\,{\mathrm {e}}^{-2\,a}}{{\left (c\,x^n\right )}^{2\,b}}+\frac {b\,n\,{\mathrm {e}}^{-4\,a}}{{\left (c\,x^n\right )}^{4\,b}}\right )} \] Input:
int(1/(x*sinh(a + b*log(c*x^n))^5),x)
Output:
(2*exp(-a))/((c*x^n)^b*(b*n - (3*b*n*exp(-2*a))/(c*x^n)^(2*b) + (3*b*n*exp (-4*a))/(c*x^n)^(4*b) - (b*n*exp(-6*a))/(c*x^n)^(6*b))) - (3*atan((exp(-a) *(-b^2*n^2)^(1/2))/(b*n*(c*x^n)^b)))/(4*(-b^2*n^2)^(1/2)) - (3*exp(-a))/(4 *(c*x^n)^b*(b*n - (b*n*exp(-2*a))/(c*x^n)^(2*b))) - (4*exp(-3*a))/((c*x^n) ^(3*b)*(b*n - (4*b*n*exp(-2*a))/(c*x^n)^(2*b) + (6*b*n*exp(-4*a))/(c*x^n)^ (4*b) - (4*b*n*exp(-6*a))/(c*x^n)^(6*b) + (b*n*exp(-8*a))/(c*x^n)^(8*b))) - exp(-a)/(2*(c*x^n)^b*(b*n - (2*b*n*exp(-2*a))/(c*x^n)^(2*b) + (b*n*exp(- 4*a))/(c*x^n)^(4*b)))
Time = 0.22 (sec) , antiderivative size = 499, normalized size of antiderivative = 5.61 \[ \int \frac {\text {csch}^5\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {-3 x^{8 b n} e^{8 a} c^{8 b} \mathrm {log}\left (x^{b n} e^{a} c^{2 b}+c^{b}\right )+3 x^{8 b n} e^{8 a} c^{8 b} \mathrm {log}\left (x^{b n} e^{a} c^{2 b}-c^{b}\right )+6 x^{7 b n} e^{7 a} c^{7 b}+12 x^{6 b n} e^{6 a} c^{6 b} \mathrm {log}\left (x^{b n} e^{a} c^{2 b}+c^{b}\right )-12 x^{6 b n} e^{6 a} c^{6 b} \mathrm {log}\left (x^{b n} e^{a} c^{2 b}-c^{b}\right )-22 x^{5 b n} e^{5 a} c^{5 b}-18 x^{4 b n} e^{4 a} c^{4 b} \mathrm {log}\left (x^{b n} e^{a} c^{2 b}+c^{b}\right )+18 x^{4 b n} e^{4 a} c^{4 b} \mathrm {log}\left (x^{b n} e^{a} c^{2 b}-c^{b}\right )-22 x^{3 b n} e^{3 a} c^{3 b}+12 x^{2 b n} e^{2 a} c^{2 b} \mathrm {log}\left (x^{b n} e^{a} c^{2 b}+c^{b}\right )-12 x^{2 b n} e^{2 a} c^{2 b} \mathrm {log}\left (x^{b n} e^{a} c^{2 b}-c^{b}\right )+6 x^{b n} e^{a} c^{b}-3 \,\mathrm {log}\left (x^{b n} e^{a} c^{2 b}+c^{b}\right )+3 \,\mathrm {log}\left (x^{b n} e^{a} c^{2 b}-c^{b}\right )}{8 b n \left (x^{8 b n} e^{8 a} c^{8 b}-4 x^{6 b n} e^{6 a} c^{6 b}+6 x^{4 b n} e^{4 a} c^{4 b}-4 x^{2 b n} e^{2 a} c^{2 b}+1\right )} \] Input:
int(csch(a+b*log(c*x^n))^5/x,x)
Output:
( - 3*x**(8*b*n)*e**(8*a)*c**(8*b)*log(x**(b*n)*e**a*c**(2*b) + c**b) + 3* x**(8*b*n)*e**(8*a)*c**(8*b)*log(x**(b*n)*e**a*c**(2*b) - c**b) + 6*x**(7* b*n)*e**(7*a)*c**(7*b) + 12*x**(6*b*n)*e**(6*a)*c**(6*b)*log(x**(b*n)*e**a *c**(2*b) + c**b) - 12*x**(6*b*n)*e**(6*a)*c**(6*b)*log(x**(b*n)*e**a*c**( 2*b) - c**b) - 22*x**(5*b*n)*e**(5*a)*c**(5*b) - 18*x**(4*b*n)*e**(4*a)*c* *(4*b)*log(x**(b*n)*e**a*c**(2*b) + c**b) + 18*x**(4*b*n)*e**(4*a)*c**(4*b )*log(x**(b*n)*e**a*c**(2*b) - c**b) - 22*x**(3*b*n)*e**(3*a)*c**(3*b) + 1 2*x**(2*b*n)*e**(2*a)*c**(2*b)*log(x**(b*n)*e**a*c**(2*b) + c**b) - 12*x** (2*b*n)*e**(2*a)*c**(2*b)*log(x**(b*n)*e**a*c**(2*b) - c**b) + 6*x**(b*n)* e**a*c**b - 3*log(x**(b*n)*e**a*c**(2*b) + c**b) + 3*log(x**(b*n)*e**a*c** (2*b) - c**b))/(8*b*n*(x**(8*b*n)*e**(8*a)*c**(8*b) - 4*x**(6*b*n)*e**(6*a )*c**(6*b) + 6*x**(4*b*n)*e**(4*a)*c**(4*b) - 4*x**(2*b*n)*e**(2*a)*c**(2* b) + 1))