\(\int \frac {\coth ^5(a+b \log (c x^n))}{x} \, dx\) [193]

Optimal result

Integrand size = 17, antiderivative size = 66 \[ \int \frac {\coth ^5\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=-\frac {\coth ^2\left (a+b \log \left (c x^n\right )\right )}{2 b n}-\frac {\coth ^4\left (a+b \log \left (c x^n\right )\right )}{4 b n}+\frac {\log \left (\sinh \left (a+b \log \left (c x^n\right )\right )\right )}{b n} \]

[Out]

-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*ln(c*x^n)))/b/n

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {3554, 3556} \[ \int \frac {\coth ^5\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {\log \left (\sinh \left (a+b \log \left (c x^n\right )\right )\right )}{b n}-\frac {\coth ^4\left (a+b \log \left (c x^n\right )\right )}{4 b n}-\frac {\coth ^2\left (a+b \log \left (c x^n\right )\right )}{2 b n} \]

[In]

Int[Coth[a + b*Log[c*x^n]]^5/x,x]

[Out]

-1/2*Coth[a + b*Log[c*x^n]]^2/(b*n) - Coth[a + b*Log[c*x^n]]^4/(4*b*n) + Log[Sinh[a + b*Log[c*x^n]]]/(b*n)

Rule 3554

Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[b*((b*Tan[c + d*x])^(n - 1)/(d*(n - 1))), x] - Dis
t[b^2, Int[(b*Tan[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1]

Rule 3556

Int[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Log[RemoveContent[Cos[c + d*x], x]]/d, x] /; FreeQ[{c, d}, x]

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \coth ^5(a+b x) \, dx,x,\log \left (c x^n\right )\right )}{n} \\ & = -\frac {\coth ^4\left (a+b \log \left (c x^n\right )\right )}{4 b n}+\frac {\text {Subst}\left (\int \coth ^3(a+b x) \, dx,x,\log \left (c x^n\right )\right )}{n} \\ & = -\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 {\text {Subst}\left (\int \coth (a+b x) \, dx,x,\log \left (c x^n\right )\right )}{n} \\ & = -\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} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.30 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.02 \[ \int \frac {\coth ^5\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=-\frac {2 \coth ^2\left (a+b \log \left (c x^n\right )\right )+\coth ^4\left (a+b \log \left (c x^n\right )\right )-4 \log \left (\cosh \left (a+b \log \left (c x^n\right )\right )\right )-4 \log \left (\tanh \left (a+b \log \left (c x^n\right )\right )\right )}{4 b n} \]

[In]

Integrate[Coth[a + b*Log[c*x^n]]^5/x,x]

[Out]

-1/4*(2*Coth[a + b*Log[c*x^n]]^2 + Coth[a + b*Log[c*x^n]]^4 - 4*Log[Cosh[a + b*Log[c*x^n]]] - 4*Log[Tanh[a + b
*Log[c*x^n]]])/(b*n)

Maple [A] (verified)

Time = 1.68 (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}{b n}-\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}\)	\(658\)

[In]

int(coth(a+b*ln(c*x^n))^5/x,x,method=_RETURNVERBOSE)

[Out]

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))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1576 vs. \(2 (62) = 124\).

Time = 0.28 (sec) , antiderivative size = 1576, normalized size of antiderivative = 23.88 \[ \int \frac {\coth ^5\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\text {Too large to display} \]

[In]

integrate(coth(a+b*log(c*x^n))^5/x,x, algorithm="fricas")

[Out]

-(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*log(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*log(x) - 10*(b*n*log(x) - 1)*co
sh(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*l
og(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*lo
g(x) + b*log(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*log(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 - 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*log(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*n*log(x) + b*log(c) + a)^2 - 1)*
sinh(b*n*log(x) + b*log(c) + a)^2 - 4*cosh(b*n*log(x) + b*log(c) + a)^2 + 8*(cosh(b*n*log(x) + b*log(c) + a)^7
- 3*cosh(b*n*log(x) + b*log(c) + a)^5 + 3*cosh(b*n*log(x) + b*log(c) + a)^3 - cosh(b*n*log(x) + b*log(c) + a)
)*sinh(b*n*log(x) + b*log(c) + a) + 1)*log(2*sinh(b*n*log(x) + b*log(c) + a)/(cosh(b*n*log(x) + b*log(c) + a)
- sinh(b*n*log(x) + b*log(c) + a))) + 8*(b*n*cosh(b*n*log(x) + b*log(c) + a)^7*log(x) - 3*(b*n*log(x) - 1)*cos
h(b*n*log(x) + b*log(c) + a)^5 + (3*b*n*log(x) - 2)*cosh(b*n*log(x) + b*log(c) + a)^3 - (b*n*log(x) - 1)*cosh(
b*n*log(x) + b*log(c) + a))*sinh(b*n*log(x) + b*log(c) + a))/(b*n*cosh(b*n*log(x) + b*log(c) + a)^8 + 8*b*n*co
sh(b*n*log(x) + b*log(c) + a)*sinh(b*n*log(x) + b*log(c) + a)^7 + b*n*sinh(b*n*log(x) + b*log(c) + a)^8 - 4*b*
n*cosh(b*n*log(x) + b*log(c) + a)^6 + 4*(7*b*n*cosh(b*n*log(x) + b*log(c) + a)^2 - b*n)*sinh(b*n*log(x) + b*lo
g(c) + a)^6 + 6*b*n*cosh(b*n*log(x) + b*log(c) + a)^4 + 8*(7*b*n*cosh(b*n*log(x) + b*log(c) + a)^3 - 3*b*n*cos
h(b*n*log(x) + b*log(c) + a))*sinh(b*n*log(x) + b*log(c) + a)^5 + 2*(35*b*n*cosh(b*n*log(x) + b*log(c) + a)^4
- 30*b*n*cosh(b*n*log(x) + b*log(c) + a)^2 + 3*b*n)*sinh(b*n*log(x) + b*log(c) + a)^4 - 4*b*n*cosh(b*n*log(x)
+ b*log(c) + a)^2 + 8*(7*b*n*cosh(b*n*log(x) + b*log(c) + a)^5 - 10*b*n*cosh(b*n*log(x) + b*log(c) + a)^3 + 3*
b*n*cosh(b*n*log(x) + b*log(c) + a))*sinh(b*n*log(x) + b*log(c) + a)^3 + 4*(7*b*n*cosh(b*n*log(x) + b*log(c) +
a)^6 - 15*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 - b*n)*sinh(b*n*log
(x) + b*log(c) + a)^2 + b*n + 8*(b*n*cosh(b*n*log(x) + b*log(c) + a)^7 - 3*b*n*cosh(b*n*log(x) + b*log(c) + a)
^5 + 3*b*n*cosh(b*n*log(x) + b*log(c) + a)^3 - b*n*cosh(b*n*log(x) + b*log(c) + a))*sinh(b*n*log(x) + b*log(c)
+ a))

Sympy [F(-2)]

Exception generated. \[ \int \frac {\coth ^5\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate(coth(a+b*ln(c*x**n))**5/x,x)

[Out]

Exception raised: TypeError >> Invalid NaN comparison

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 855 vs. \(2 (62) = 124\).

Time = 0.32 (sec) , antiderivative size = 855, normalized size of antiderivative = 12.95 \[ \int \frac {\coth ^5\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\text {Too large to display} \]

[In]

integrate(coth(a+b*log(c*x^n))^5/x,x, algorithm="maxima")

[Out]

-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*lo
g(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*log(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) + 1)*e^(-a)/c^b)/(b*n) + log((c^b*e^(b*log(x^n) + a) - 1)*e^(-
a)/c^b)/(b*n) - log(x)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 161 vs. \(2 (62) = 124\).

Time = 0.40 (sec) , antiderivative size = 161, normalized size of antiderivative = 2.44 \[ \int \frac {\coth ^5\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {\log \left (\sqrt {-2 \, x^{2 \, b n} {\left | c \right |}^{2 \, b} \cos \left (\pi b \mathrm {sgn}\left (c\right ) - \pi b\right ) e^{\left (2 \, a\right )} + x^{4 \, b n} {\left | c \right |}^{4 \, b} e^{\left (4 \, a\right )} + 1}\right )}{b n} - \frac {25 \, c^{8 \, b} x^{8 \, b n} e^{\left (8 \, a\right )} - 52 \, c^{6 \, b} x^{6 \, b n} e^{\left (6 \, a\right )} + 102 \, c^{4 \, b} x^{4 \, b n} e^{\left (4 \, a\right )} - 52 \, c^{2 \, b} x^{2 \, b n} e^{\left (2 \, a\right )} + 25}{12 \, {\left (c^{2 \, b} x^{2 \, b n} e^{\left (2 \, a\right )} - 1\right )}^{4} b n} - \log \left (x\right ) \]

[In]

integrate(coth(a+b*log(c*x^n))^5/x,x, algorithm="giac")

[Out]

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) - log(x)

Mupad [B] (verification not implemented)

Time = 1.88 (sec) , antiderivative size = 229, normalized size of antiderivative = 3.47 \[ \int \frac {\coth ^5\left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {8}{b\,n-3\,b\,n\,{\mathrm {e}}^{2\,a}\,{\left (c\,x^n\right )}^{2\,b}+3\,b\,n\,{\mathrm {e}}^{4\,a}\,{\left (c\,x^n\right )}^{4\,b}-b\,n\,{\mathrm {e}}^{6\,a}\,{\left (c\,x^n\right )}^{6\,b}}-\ln \left (x\right )+\frac {4}{b\,n-b\,n\,{\mathrm {e}}^{2\,a}\,{\left (c\,x^n\right )}^{2\,b}}-\frac {4}{b\,n-4\,b\,n\,{\mathrm {e}}^{2\,a}\,{\left (c\,x^n\right )}^{2\,b}+6\,b\,n\,{\mathrm {e}}^{4\,a}\,{\left (c\,x^n\right )}^{4\,b}-4\,b\,n\,{\mathrm {e}}^{6\,a}\,{\left (c\,x^n\right )}^{6\,b}+b\,n\,{\mathrm {e}}^{8\,a}\,{\left (c\,x^n\right )}^{8\,b}}-\frac {8}{b\,n-2\,b\,n\,{\mathrm {e}}^{2\,a}\,{\left (c\,x^n\right )}^{2\,b}+b\,n\,{\mathrm {e}}^{4\,a}\,{\left (c\,x^n\right )}^{4\,b}}+\frac {\ln \left ({\mathrm {e}}^{2\,a}\,{\left (c\,x^n\right )}^{2\,b}-1\right )}{b\,n} \]

[In]

int(coth(a + b*log(c*x^n))^5/x,x)

[Out]

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)