Integrand size = 17, antiderivative size = 66 \[ \int \frac {\coth ^5\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=-\frac {\coth ^2\left (a+b \log \left (c x^n\right )\right )}{2 b n}-\frac {\coth ^4\left (a+b \log \left (c x^n\right )\right )}{4 b n}+\frac {\log \left (\sinh \left (a+b \log \left (c x^n\right )\right )\right )}{b n} \] Output:
-1/2*coth(a+b*ln(c*x^n))^2/b/n-1/4*coth(a+b*ln(c*x^n))^4/b/n+ln(sinh(a+b*l n(c*x^n)))/b/n
Time = 0.11 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.80 \[ \int \frac {\coth ^5\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=-\frac {4 \text {csch}^2\left (a+b \log \left (c x^n\right )\right )+\text {csch}^4\left (a+b \log \left (c x^n\right )\right )-4 \log \left (\sinh \left (a+b \log \left (c x^n\right )\right )\right )}{4 b n} \] Input:
Integrate[Coth[a + b*Log[c*x^n]]^5/x,x]
Output:
-1/4*(4*Csch[a + b*Log[c*x^n]]^2 + Csch[a + b*Log[c*x^n]]^4 - 4*Log[Sinh[a + b*Log[c*x^n]]])/(b*n)
Result contains complex when optimal does not.
Time = 0.40 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.14, 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, 3954, 26, 3042, 26, 3954, 26, 3042, 26, 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.
| \(\displaystyle \int \frac {\coth ^5\left (a+b \log \left (c x^n\right )\right )}{x} \, dx\) |
| \(\Big \downarrow \) 3039 |
| \(\displaystyle \frac {\int \coth ^5\left (a+b \log \left (c x^n\right )\right )d\log \left (c x^n\right )}{n}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int -i \tan \left (i a+i b \log \left (c x^n\right )+\frac {\pi }{2}\right )^5d\log \left (c x^n\right )}{n}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {i \int \tan \left (\frac {1}{2} (2 i a+\pi )+i b \log \left (c x^n\right )\right )^5d\log \left (c x^n\right )}{n}\) |
| \(\Big \downarrow \) 3954 |
| \(\displaystyle -\frac {i \left (-\int -i \coth ^3\left (a+b \log \left (c x^n\right )\right )d\log \left (c x^n\right )-\frac {i \coth ^4\left (a+b \log \left (c x^n\right )\right )}{4 b}\right )}{n}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {i \left (i \int \coth ^3\left (a+b \log \left (c x^n\right )\right )d\log \left (c x^n\right )-\frac {i \coth ^4\left (a+b \log \left (c x^n\right )\right )}{4 b}\right )}{n}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {i \left (i \int i \tan \left (i a+i b \log \left (c x^n\right )+\frac {\pi }{2}\right )^3d\log \left (c x^n\right )-\frac {i \coth ^4\left (a+b \log \left (c x^n\right )\right )}{4 b}\right )}{n}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {i \left (-\int \tan \left (\frac {1}{2} (2 i a+\pi )+i b \log \left (c x^n\right )\right )^3d\log \left (c x^n\right )-\frac {i \coth ^4\left (a+b \log \left (c x^n\right )\right )}{4 b}\right )}{n}\) |
| \(\Big \downarrow \) 3954 |
| \(\displaystyle -\frac {i \left (\int i \coth \left (a+b \log \left (c x^n\right )\right )d\log \left (c x^n\right )-\frac {i \coth ^4\left (a+b \log \left (c x^n\right )\right )}{4 b}-\frac {i \coth ^2\left (a+b \log \left (c x^n\right )\right )}{2 b}\right )}{n}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {i \left (i \int \coth \left (a+b \log \left (c x^n\right )\right )d\log \left (c x^n\right )-\frac {i \coth ^4\left (a+b \log \left (c x^n\right )\right )}{4 b}-\frac {i \coth ^2\left (a+b \log \left (c x^n\right )\right )}{2 b}\right )}{n}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {i \left (i \int -i \tan \left (i a+i b \log \left (c x^n\right )+\frac {\pi }{2}\right )d\log \left (c x^n\right )-\frac {i \coth ^4\left (a+b \log \left (c x^n\right )\right )}{4 b}-\frac {i \coth ^2\left (a+b \log \left (c x^n\right )\right )}{2 b}\right )}{n}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {i \left (\int \tan \left (\frac {1}{2} (2 i a+\pi )+i b \log \left (c x^n\right )\right )d\log \left (c x^n\right )-\frac {i \coth ^4\left (a+b \log \left (c x^n\right )\right )}{4 b}-\frac {i \coth ^2\left (a+b \log \left (c x^n\right )\right )}{2 b}\right )}{n}\) |
| \(\Big \downarrow \) 3956 |
| \(\displaystyle -\frac {i \left (\frac {i \log \left (-i \sinh \left (a+b \log \left (c x^n\right )\right )\right )}{b}-\frac {i \coth ^4\left (a+b \log \left (c x^n\right )\right )}{4 b}-\frac {i \coth ^2\left (a+b \log \left (c x^n\right )\right )}{2 b}\right )}{n}\) |
Input:
Int[Coth[a + b*Log[c*x^n]]^5/x,x]
Output:
((-I)*(((-1/2*I)*Coth[a + b*Log[c*x^n]]^2)/b - ((I/4)*Coth[a + b*Log[c*x^n ]]^4)/b + (I*Log[(-I)*Sinh[a + b*Log[c*x^n]]])/b))/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[((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 = 1.83 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.08
| method | result | size |
| derivativedivides | \(\frac {-\frac {{\coth \left (a +b \ln \left (c \,x^{n}\right )\right )}^{4}}{4}-\frac {{\coth \left (a +b \ln \left (c \,x^{n}\right )\right )}^{2}}{2}-\frac {\ln \left (\coth \left (a +b \ln \left (c \,x^{n}\right )\right )-1\right )}{2}-\frac {\ln \left (\coth \left (a +b \ln \left (c \,x^{n}\right )\right )+1\right )}{2}}{n b}\) | \(71\) |
| default | \(\frac {-\frac {{\coth \left (a +b \ln \left (c \,x^{n}\right )\right )}^{4}}{4}-\frac {{\coth \left (a +b \ln \left (c \,x^{n}\right )\right )}^{2}}{2}-\frac {\ln \left (\coth \left (a +b \ln \left (c \,x^{n}\right )\right )-1\right )}{2}-\frac {\ln \left (\coth \left (a +b \ln \left (c \,x^{n}\right )\right )+1\right )}{2}}{n b}\) | \(71\) |
| parallelrisch | \(\frac {-{\coth \left (a +b \ln \left (c \,x^{n}\right )\right )}^{4}-4 \ln \left (x \right ) b n +4 \ln \left (\tanh \left (a +b \ln \left (c \,x^{n}\right )\right )\right )-4 \ln \left (1-\tanh \left (a +b \ln \left (c \,x^{n}\right )\right )\right )-2 {\coth \left (a +b \ln \left (c \,x^{n}\right )\right )}^{2}}{4 b n}\) | \(78\) |
| risch | \(\ln \left (x \right )-\frac {2 a}{n b}-\frac {2 \ln \left (c \right )}{n}-\frac {2 \ln \left (x^{n}\right )}{n}-\frac {i \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{n}+\frac {i \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right ) \operatorname {csgn}\left (i c \right )}{n}+\frac {i \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}}{n}-\frac {i \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i c \right )}{n}-\frac {4 \left (x^{n}\right )^{2 b} c^{2 b} \left ({\mathrm e}^{3 i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}} {\mathrm e}^{-3 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}^{-3 i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}} {\mathrm e}^{3 i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i c \right )} {\mathrm e}^{6 a} \left (x^{n}\right )^{4 b} c^{4 b}-{\mathrm e}^{2 i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}} {\mathrm e}^{-2 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}^{-2 i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}} {\mathrm e}^{2 i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{2} \operatorname {csgn}\left (i c \right )} {\mathrm e}^{4 a} \left (x^{n}\right )^{2 b} c^{2 b}+{\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 )} {\mathrm e}^{2 a}\right )}{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 )}^{4}}+\frac {\ln \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 )}{b n}\) | \(658\) |
Input:
int(coth(a+b*ln(c*x^n))^5/x,x,method=_RETURNVERBOSE)
Output:
1/n/b*(-1/4*coth(a+b*ln(c*x^n))^4-1/2*coth(a+b*ln(c*x^n))^2-1/2*ln(coth(a+ b*ln(c*x^n))-1)-1/2*ln(coth(a+b*ln(c*x^n))+1))
Leaf count of result is larger than twice the leaf count of optimal. 1576 vs. \(2 (62) = 124\).
Time = 0.12 (sec) , antiderivative size = 1576, normalized size of antiderivative = 23.88 \[ \int \frac {\coth ^5\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\text {Too large to display} \] Input:
integrate(coth(a+b*log(c*x^n))^5/x,x, algorithm="fricas")
Output:
-(b*n*cosh(b*n*log(x) + b*log(c) + a)^8*log(x) + 8*b*n*cosh(b*n*log(x) + b *log(c) + a)*log(x)*sinh(b*n*log(x) + b*log(c) + a)^7 + b*n*log(x)*sinh(b* n*log(x) + b*log(c) + a)^8 - 4*(b*n*log(x) - 1)*cosh(b*n*log(x) + b*log(c) + a)^6 + 4*(7*b*n*cosh(b*n*log(x) + b*log(c) + a)^2*log(x) - b*n*log(x) + 1)*sinh(b*n*log(x) + b*log(c) + a)^6 + 8*(7*b*n*cosh(b*n*log(x) + b*log(c ) + a)^3*log(x) - 3*(b*n*log(x) - 1)*cosh(b*n*log(x) + b*log(c) + a))*sinh (b*n*log(x) + b*log(c) + a)^5 + 2*(3*b*n*log(x) - 2)*cosh(b*n*log(x) + b*l og(c) + a)^4 + 2*(35*b*n*cosh(b*n*log(x) + b*log(c) + a)^4*log(x) - 30*(b* n*log(x) - 1)*cosh(b*n*log(x) + b*log(c) + a)^2 + 3*b*n*log(x) - 2)*sinh(b *n*log(x) + b*log(c) + a)^4 + 8*(7*b*n*cosh(b*n*log(x) + b*log(c) + a)^5*l og(x) - 10*(b*n*log(x) - 1)*cosh(b*n*log(x) + b*log(c) + a)^3 + (3*b*n*log (x) - 2)*cosh(b*n*log(x) + b*log(c) + a))*sinh(b*n*log(x) + b*log(c) + a)^ 3 - 4*(b*n*log(x) - 1)*cosh(b*n*log(x) + b*log(c) + a)^2 + b*n*log(x) + 4* (7*b*n*cosh(b*n*log(x) + b*log(c) + a)^6*log(x) - 15*(b*n*log(x) - 1)*cosh (b*n*log(x) + b*log(c) + a)^4 + 3*(3*b*n*log(x) - 2)*cosh(b*n*log(x) + b*l og(c) + a)^2 - b*n*log(x) + 1)*sinh(b*n*log(x) + b*log(c) + a)^2 - (cosh(b *n*log(x) + b*log(c) + a)^8 + 8*cosh(b*n*log(x) + b*log(c) + a)*sinh(b*n*l og(x) + b*log(c) + a)^7 + sinh(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 ...
Exception generated. \[ \int \frac {\coth ^5\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\text {Exception raised: TypeError} \] Input:
integrate(coth(a+b*ln(c*x**n))**5/x,x)
Output:
Exception raised: TypeError >> Invalid NaN comparison
Leaf count of result is larger than twice the leaf count of optimal. 855 vs. \(2 (62) = 124\).
Time = 0.17 (sec) , antiderivative size = 855, normalized size of antiderivative = 12.95 \[ \int \frac {\coth ^5\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\text {Too large to display} \] Input:
integrate(coth(a+b*log(c*x^n))^5/x,x, algorithm="maxima")
Output:
-1/24*(48*c^(6*b)*e^(6*b*log(x^n) + 6*a) - 108*c^(4*b)*e^(4*b*log(x^n) + 4 *a) + 88*c^(2*b)*e^(2*b*log(x^n) + 2*a) - 25)/(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) + 1/24*(12*c^(6* b)*e^(6*b*log(x^n) + 6*a) - 42*c^(4*b)*e^(4*b*log(x^n) + 4*a) + 52*c^(2*b) *e^(2*b*log(x^n) + 2*a) - 25)/(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) - 5/8*(4*c^(6*b)*e^(6*b*log(x^n) + 6*a) - 6*c^(4*b)*e^(4*b*log(x^n) + 4*a) + 4*c^(2*b)*e^(2*b*log(x^n) + 2 *a) - 1)/(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*lo g(x^n) + 2*a) + b*n) - 5/12*(6*c^(4*b)*e^(4*b*log(x^n) + 4*a) - 4*c^(2*b)* e^(2*b*log(x^n) + 2*a) + 1)/(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) - 5/12*(4*c^(2*b)*e^(2*b*log(x^n) + 2*a) - 1)/(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) - 5/8/(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) + log((c^b*e^(b*log(x^n) + a)...
Leaf count of result is larger than twice the leaf count of optimal. 171 vs. \(2 (62) = 124\).
Time = 0.19 (sec) , antiderivative size = 171, normalized size of antiderivative = 2.59 \[ \int \frac {\coth ^5\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=-\frac {\log \left (x^{b n}\right )}{b n} + \frac {\log \left (\sqrt {-2 \, x^{2 \, b n} {\left | c \right |}^{2 \, b} \cos \left (\pi b \mathrm {sgn}\left (c\right ) - \pi b\right ) e^{\left (2 \, a\right )} + x^{4 \, b n} {\left | c \right |}^{4 \, b} e^{\left (4 \, a\right )} + 1}\right )}{b n} - \frac {25 \, c^{8 \, b} x^{8 \, b n} e^{\left (8 \, a\right )} - 52 \, c^{6 \, b} x^{6 \, b n} e^{\left (6 \, a\right )} + 102 \, c^{4 \, b} x^{4 \, b n} e^{\left (4 \, a\right )} - 52 \, c^{2 \, b} x^{2 \, b n} e^{\left (2 \, a\right )} + 25}{12 \, {\left (c^{2 \, b} x^{2 \, b n} e^{\left (2 \, a\right )} - 1\right )}^{4} b n} \] Input:
integrate(coth(a+b*log(c*x^n))^5/x,x, algorithm="giac")
Output:
-log(x^(b*n))/(b*n) + log(sqrt(-2*x^(2*b*n)*abs(c)^(2*b)*cos(pi*b*sgn(c) - pi*b)*e^(2*a) + x^(4*b*n)*abs(c)^(4*b)*e^(4*a) + 1))/(b*n) - 1/12*(25*c^( 8*b)*x^(8*b*n)*e^(8*a) - 52*c^(6*b)*x^(6*b*n)*e^(6*a) + 102*c^(4*b)*x^(4*b *n)*e^(4*a) - 52*c^(2*b)*x^(2*b*n)*e^(2*a) + 25)/((c^(2*b)*x^(2*b*n)*e^(2* a) - 1)^4*b*n)
Time = 2.47 (sec) , antiderivative size = 229, normalized size of antiderivative = 3.47 \[ \int \frac {\coth ^5\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {8}{b\,n-3\,b\,n\,{\mathrm {e}}^{2\,a}\,{\left (c\,x^n\right )}^{2\,b}+3\,b\,n\,{\mathrm {e}}^{4\,a}\,{\left (c\,x^n\right )}^{4\,b}-b\,n\,{\mathrm {e}}^{6\,a}\,{\left (c\,x^n\right )}^{6\,b}}-\ln \left (x\right )+\frac {4}{b\,n-b\,n\,{\mathrm {e}}^{2\,a}\,{\left (c\,x^n\right )}^{2\,b}}-\frac {4}{b\,n-4\,b\,n\,{\mathrm {e}}^{2\,a}\,{\left (c\,x^n\right )}^{2\,b}+6\,b\,n\,{\mathrm {e}}^{4\,a}\,{\left (c\,x^n\right )}^{4\,b}-4\,b\,n\,{\mathrm {e}}^{6\,a}\,{\left (c\,x^n\right )}^{6\,b}+b\,n\,{\mathrm {e}}^{8\,a}\,{\left (c\,x^n\right )}^{8\,b}}-\frac {8}{b\,n-2\,b\,n\,{\mathrm {e}}^{2\,a}\,{\left (c\,x^n\right )}^{2\,b}+b\,n\,{\mathrm {e}}^{4\,a}\,{\left (c\,x^n\right )}^{4\,b}}+\frac {\ln \left ({\mathrm {e}}^{2\,a}\,{\left (c\,x^n\right )}^{2\,b}-1\right )}{b\,n} \] Input:
int(coth(a + b*log(c*x^n))^5/x,x)
Output:
8/(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)) - log(x) + 4/(b*n - b*n*exp(2*a)*(c*x^n)^(2*b)) - 4/(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)) - 8/(b*n - 2*b*n* exp(2*a)*(c*x^n)^(2*b) + b*n*exp(4*a)*(c*x^n)^(4*b)) + log(exp(2*a)*(c*x^n )^(2*b) - 1)/(b*n)
Time = 0.18 (sec) , antiderivative size = 556, normalized size of antiderivative = 8.42 \[ \int \frac {\coth ^5\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {x^{8 b n} e^{8 a} c^{8 b} \mathrm {log}\left (x^{b n} e^{a} c^{2 b}+c^{b}\right )+x^{8 b n} e^{8 a} c^{8 b} \mathrm {log}\left (x^{b n} e^{a} c^{2 b}-c^{b}\right )-x^{8 b n} e^{8 a} c^{8 b} \mathrm {log}\left (x \right ) b n -x^{8 b n} e^{8 a} c^{8 b}-4 x^{6 b n} e^{6 a} c^{6 b} \mathrm {log}\left (x^{b n} e^{a} c^{2 b}+c^{b}\right )-4 x^{6 b n} e^{6 a} c^{6 b} \mathrm {log}\left (x^{b n} e^{a} c^{2 b}-c^{b}\right )+4 x^{6 b n} e^{6 a} c^{6 b} \mathrm {log}\left (x \right ) b n +6 x^{4 b n} e^{4 a} c^{4 b} \mathrm {log}\left (x^{b n} e^{a} c^{2 b}+c^{b}\right )+6 x^{4 b n} e^{4 a} c^{4 b} \mathrm {log}\left (x^{b n} e^{a} c^{2 b}-c^{b}\right )-6 x^{4 b n} e^{4 a} c^{4 b} \mathrm {log}\left (x \right ) b n -2 x^{4 b n} e^{4 a} c^{4 b}-4 x^{2 b n} e^{2 a} c^{2 b} \mathrm {log}\left (x^{b n} e^{a} c^{2 b}+c^{b}\right )-4 x^{2 b n} e^{2 a} c^{2 b} \mathrm {log}\left (x^{b n} e^{a} c^{2 b}-c^{b}\right )+4 x^{2 b n} e^{2 a} c^{2 b} \mathrm {log}\left (x \right ) b n +\mathrm {log}\left (x^{b n} e^{a} c^{2 b}+c^{b}\right )+\mathrm {log}\left (x^{b n} e^{a} c^{2 b}-c^{b}\right )-\mathrm {log}\left (x \right ) b n -1}{b n \left (x^{8 b n} e^{8 a} c^{8 b}-4 x^{6 b n} e^{6 a} c^{6 b}+6 x^{4 b n} e^{4 a} c^{4 b}-4 x^{2 b n} e^{2 a} c^{2 b}+1\right )} \] Input:
int(coth(a+b*log(c*x^n))^5/x,x)
Output:
(x**(8*b*n)*e**(8*a)*c**(8*b)*log(x**(b*n)*e**a*c**(2*b) + c**b) + x**(8*b *n)*e**(8*a)*c**(8*b)*log(x**(b*n)*e**a*c**(2*b) - c**b) - x**(8*b*n)*e**( 8*a)*c**(8*b)*log(x)*b*n - x**(8*b*n)*e**(8*a)*c**(8*b) - 4*x**(6*b*n)*e** (6*a)*c**(6*b)*log(x**(b*n)*e**a*c**(2*b) + c**b) - 4*x**(6*b*n)*e**(6*a)* c**(6*b)*log(x**(b*n)*e**a*c**(2*b) - c**b) + 4*x**(6*b*n)*e**(6*a)*c**(6* b)*log(x)*b*n + 6*x**(4*b*n)*e**(4*a)*c**(4*b)*log(x**(b*n)*e**a*c**(2*b) + c**b) + 6*x**(4*b*n)*e**(4*a)*c**(4*b)*log(x**(b*n)*e**a*c**(2*b) - c**b ) - 6*x**(4*b*n)*e**(4*a)*c**(4*b)*log(x)*b*n - 2*x**(4*b*n)*e**(4*a)*c**( 4*b) - 4*x**(2*b*n)*e**(2*a)*c**(2*b)*log(x**(b*n)*e**a*c**(2*b) + c**b) - 4*x**(2*b*n)*e**(2*a)*c**(2*b)*log(x**(b*n)*e**a*c**(2*b) - c**b) + 4*x** (2*b*n)*e**(2*a)*c**(2*b)*log(x)*b*n + log(x**(b*n)*e**a*c**(2*b) + c**b) + log(x**(b*n)*e**a*c**(2*b) - c**b) - log(x)*b*n - 1)/(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))