\(\int \frac {\sqrt {\coth (a+b \log (c x^n))}}{x} \, dx\) [206]

Optimal result

Integrand size = 19, antiderivative size = 48 \[ \int \frac {\sqrt {\coth \left (a+b \log \left (c x^n\right )\right )}}{x} \, dx=-\frac {\arctan \left (\sqrt {\coth \left (a+b \log \left (c x^n\right )\right )}\right )}{b n}+\frac {\text {arctanh}\left (\sqrt {\coth \left (a+b \log \left (c x^n\right )\right )}\right )}{b n} \] Output:

-arctan(coth(a+b*ln(c*x^n))^(1/2))/b/n+arctanh(coth(a+b*ln(c*x^n))^(1/2))/ 
b/n
 

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.90 \[ \int \frac {\sqrt {\coth \left (a+b \log \left (c x^n\right )\right )}}{x} \, dx=-\frac {\arctan \left (\sqrt {\coth \left (a+b \log \left (c x^n\right )\right )}\right )-\text {arctanh}\left (\sqrt {\coth \left (a+b \log \left (c x^n\right )\right )}\right )}{b n} \] Input:

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

Output:

-((ArcTan[Sqrt[Coth[a + b*Log[c*x^n]]]] - ArcTanh[Sqrt[Coth[a + b*Log[c*x^ 
n]]]])/(b*n))
 

Rubi [A] (verified)

Time = 0.26 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.02, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.421, Rules used = {3039, 3042, 3957, 25, 266, 827, 216, 219}

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.

\(\displaystyle \int \frac {\sqrt {\coth \left (a+b \log \left (c x^n\right )\right )}}{x} \, dx\)

\(\Big \downarrow \) 3039

\(\displaystyle \frac {\int \sqrt {\coth \left (a+b \log \left (c x^n\right )\right )}d\log \left (c x^n\right )}{n}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\int \sqrt {-i \tan \left (i a+i b \log \left (c x^n\right )+\frac {\pi }{2}\right )}d\log \left (c x^n\right )}{n}\)

\(\Big \downarrow \) 3957

\(\displaystyle -\frac {\int -\frac {\sqrt {\coth \left (a+b \log \left (c x^n\right )\right )}}{1-\coth ^2\left (a+b \log \left (c x^n\right )\right )}d\coth \left (a+b \log \left (c x^n\right )\right )}{b n}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\int \frac {\sqrt {\coth \left (a+b \log \left (c x^n\right )\right )}}{1-\coth ^2\left (a+b \log \left (c x^n\right )\right )}d\coth \left (a+b \log \left (c x^n\right )\right )}{b n}\)

\(\Big \downarrow \) 266

\(\displaystyle \frac {2 \int \frac {\coth \left (a+b \log \left (c x^n\right )\right )}{1-\coth ^2\left (a+b \log \left (c x^n\right )\right )}d\sqrt {\coth \left (a+b \log \left (c x^n\right )\right )}}{b n}\)

\(\Big \downarrow \) 827

\(\displaystyle \frac {2 \left (\frac {1}{2} \int \frac {1}{1-\coth \left (a+b \log \left (c x^n\right )\right )}d\sqrt {\coth \left (a+b \log \left (c x^n\right )\right )}-\frac {1}{2} \int \frac {1}{\coth \left (a+b \log \left (c x^n\right )\right )+1}d\sqrt {\coth \left (a+b \log \left (c x^n\right )\right )}\right )}{b n}\)

\(\Big \downarrow \) 216

\(\displaystyle \frac {2 \left (\frac {1}{2} \int \frac {1}{1-\coth \left (a+b \log \left (c x^n\right )\right )}d\sqrt {\coth \left (a+b \log \left (c x^n\right )\right )}-\frac {1}{2} \arctan \left (\sqrt {\coth \left (a+b \log \left (c x^n\right )\right )}\right )\right )}{b n}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {2 \left (\frac {1}{2} \text {arctanh}\left (\sqrt {\coth \left (a+b \log \left (c x^n\right )\right )}\right )-\frac {1}{2} \arctan \left (\sqrt {\coth \left (a+b \log \left (c x^n\right )\right )}\right )\right )}{b n}\)

Input:

Int[Sqrt[Coth[a + b*Log[c*x^n]]]/x,x]
 

Output:

(2*(-1/2*ArcTan[Sqrt[Coth[a + b*Log[c*x^n]]]] + ArcTanh[Sqrt[Coth[a + b*Lo 
g[c*x^n]]]]/2))/(b*n)
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 216

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A 
rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a 
, 0] || GtQ[b, 0])
 

rule 219

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* 
ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

rule 266

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{k = De 
nominator[m]}, Simp[k/c   Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(2*k)/c^2)) 
^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && FractionQ[m] && I 
ntBinomialQ[a, b, c, 2, m, p, x]
 

rule 827

Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 
 2]], s = Denominator[Rt[-a/b, 2]]}, Simp[s/(2*b)   Int[1/(r + s*x^2), x], 
x] - Simp[s/(2*b)   Int[1/(r - s*x^2), x], x]] /; FreeQ[{a, b}, x] &&  !GtQ 
[a/b, 0]
 

rule 3039

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]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3957

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

Maple [A] (verified)

Time = 0.10 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.27

method	result	size
derivativedivides	\(\frac {-\frac {\ln \left (\sqrt {\coth \left (a +b \ln \left (c \,x^{n}\right )\right )}-1\right )}{2}+\frac {\ln \left (\sqrt {\coth \left (a +b \ln \left (c \,x^{n}\right )\right )}+1\right )}{2}-\arctan \left (\sqrt {\coth \left (a +b \ln \left (c \,x^{n}\right )\right )}\right )}{n b}\)	\(61\)
default	\(\frac {-\frac {\ln \left (\sqrt {\coth \left (a +b \ln \left (c \,x^{n}\right )\right )}-1\right )}{2}+\frac {\ln \left (\sqrt {\coth \left (a +b \ln \left (c \,x^{n}\right )\right )}+1\right )}{2}-\arctan \left (\sqrt {\coth \left (a +b \ln \left (c \,x^{n}\right )\right )}\right )}{n b}\)	\(61\)

Input:

int(coth(a+b*ln(c*x^n))^(1/2)/x,x,method=_RETURNVERBOSE)
 

Output:

1/n/b*(-1/2*ln(coth(a+b*ln(c*x^n))^(1/2)-1)+1/2*ln(coth(a+b*ln(c*x^n))^(1/ 
2)+1)-arctan(coth(a+b*ln(c*x^n))^(1/2)))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 305 vs. \(2 (44) = 88\).

Time = 0.10 (sec) , antiderivative size = 305, normalized size of antiderivative = 6.35 \[ \int \frac {\sqrt {\coth \left (a+b \log \left (c x^n\right )\right )}}{x} \, dx=\frac {2 \, \arctan \left (-\cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} - 2 \, \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) - \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} + {\left (\cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} + 2 \, \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) + \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} - 1\right )} \sqrt {\frac {\cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )}{\sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )}}\right ) - \log \left (-\cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} - 2 \, \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) - \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} + {\left (\cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} + 2 \, \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) + \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} - 1\right )} \sqrt {\frac {\cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )}{\sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )}}\right )}{2 \, b n} \] Input:

integrate(coth(a+b*log(c*x^n))^(1/2)/x,x, algorithm="fricas")
 

Output:

1/2*(2*arctan(-cosh(b*n*log(x) + b*log(c) + a)^2 - 2*cosh(b*n*log(x) + b*l 
og(c) + a)*sinh(b*n*log(x) + b*log(c) + a) - sinh(b*n*log(x) + b*log(c) + 
a)^2 + (cosh(b*n*log(x) + b*log(c) + a)^2 + 2*cosh(b*n*log(x) + b*log(c) + 
 a)*sinh(b*n*log(x) + b*log(c) + a) + sinh(b*n*log(x) + b*log(c) + a)^2 - 
1)*sqrt(cosh(b*n*log(x) + b*log(c) + a)/sinh(b*n*log(x) + b*log(c) + a))) 
- log(-cosh(b*n*log(x) + b*log(c) + a)^2 - 2*cosh(b*n*log(x) + b*log(c) + 
a)*sinh(b*n*log(x) + b*log(c) + a) - sinh(b*n*log(x) + b*log(c) + a)^2 + ( 
cosh(b*n*log(x) + b*log(c) + a)^2 + 2*cosh(b*n*log(x) + b*log(c) + a)*sinh 
(b*n*log(x) + b*log(c) + a) + sinh(b*n*log(x) + b*log(c) + a)^2 - 1)*sqrt( 
cosh(b*n*log(x) + b*log(c) + a)/sinh(b*n*log(x) + b*log(c) + a))))/(b*n)
 

Sympy [F]

\[ \int \frac {\sqrt {\coth \left (a+b \log \left (c x^n\right )\right )}}{x} \, dx=\int \frac {\sqrt {\coth {\left (a + b \log {\left (c x^{n} \right )} \right )}}}{x}\, dx \] Input:

integrate(coth(a+b*ln(c*x**n))**(1/2)/x,x)
 

Output:

Integral(sqrt(coth(a + b*log(c*x**n)))/x, x)
 

Maxima [F]

\[ \int \frac {\sqrt {\coth \left (a+b \log \left (c x^n\right )\right )}}{x} \, dx=\int { \frac {\sqrt {\coth \left (b \log \left (c x^{n}\right ) + a\right )}}{x} \,d x } \] Input:

integrate(coth(a+b*log(c*x^n))^(1/2)/x,x, algorithm="maxima")
 

Output:

integrate(sqrt(coth(b*log(c*x^n) + a))/x, x)
 

Giac [F(-1)]

Timed out. \[ \int \frac {\sqrt {\coth \left (a+b \log \left (c x^n\right )\right )}}{x} \, dx=\text {Timed out} \] Input:

integrate(coth(a+b*log(c*x^n))^(1/2)/x,x, algorithm="giac")
 

Output:

Timed out
 

Mupad [B] (verification not implemented)

Time = 2.61 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.81 \[ \int \frac {\sqrt {\coth \left (a+b \log \left (c x^n\right )\right )}}{x} \, dx=-\frac {\mathrm {atan}\left (\sqrt {\mathrm {coth}\left (a+b\,\ln \left (c\,x^n\right )\right )}\right )-\mathrm {atanh}\left (\sqrt {\mathrm {coth}\left (a+b\,\ln \left (c\,x^n\right )\right )}\right )}{b\,n} \] Input:

int(coth(a + b*log(c*x^n))^(1/2)/x,x)
 

Output:

-(atan(coth(a + b*log(c*x^n))^(1/2)) - atanh(coth(a + b*log(c*x^n))^(1/2)) 
)/(b*n)
 

Reduce [F]

\[ \int \frac {\sqrt {\coth \left (a+b \log \left (c x^n\right )\right )}}{x} \, dx=\int \frac {\sqrt {\coth \left (\mathrm {log}\left (x^{n} c \right ) b +a \right )}}{x}d x \] Input:

int(coth(a+b*log(c*x^n))^(1/2)/x,x)
 

Output:

int(sqrt(coth(log(x**n*c)*b + a))/x,x)