\(\int \frac {1}{x \coth ^{\frac {5}{2}}(a+b \log (c x^n))} \, dx\) [209]

Optimal result

Integrand size = 19, antiderivative size = 72 \[ \int \frac {1}{x \coth ^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx=\frac {\arctan \left (\sqrt {\coth \left (a+b \log \left (c x^n\right )\right )}\right )}{b n}+\frac {\text {arctanh}\left (\sqrt {\coth \left (a+b \log \left (c x^n\right )\right )}\right )}{b n}-\frac {2}{3 b n \coth ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )} \] 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-2/3/b/n/coth(a+b*ln(c*x^n))^(3/2)
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 0.17 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.40 \[ \int \frac {1}{x \coth ^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx=\frac {-2+3 \arctan \left (\sqrt [4]{\coth ^2\left (a+b \log \left (c x^n\right )\right )}\right ) \coth ^2\left (a+b \log \left (c x^n\right )\right )^{3/4}+3 \text {arctanh}\left (\sqrt [4]{\coth ^2\left (a+b \log \left (c x^n\right )\right )}\right ) \coth ^2\left (a+b \log \left (c x^n\right )\right )^{3/4}}{3 b n \coth ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )} \] Input:

Integrate[1/(x*Coth[a + b*Log[c*x^n]]^(5/2)),x]
 

Output:

(-2 + 3*ArcTan[(Coth[a + b*Log[c*x^n]]^2)^(1/4)]*(Coth[a + b*Log[c*x^n]]^2 
)^(3/4) + 3*ArcTanh[(Coth[a + b*Log[c*x^n]]^2)^(1/4)]*(Coth[a + b*Log[c*x^ 
n]]^2)^(3/4))/(3*b*n*Coth[a + b*Log[c*x^n]]^(3/2))
 

Rubi [A] (verified)

Time = 0.34 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.01, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.526, Rules used = {3039, 3042, 3955, 3042, 3957, 25, 266, 756, 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.

\(\displaystyle \int \frac {1}{x \coth ^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx\)

\(\Big \downarrow \) 3039

\(\displaystyle \frac {\int \frac {1}{\coth ^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )}d\log \left (c x^n\right )}{n}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\int \frac {1}{\left (-i \tan \left (i a+i b \log \left (c x^n\right )+\frac {\pi }{2}\right )\right )^{5/2}}d\log \left (c x^n\right )}{n}\)

\(\Big \downarrow \) 3955

\(\displaystyle \frac {\int \frac {1}{\sqrt {\coth \left (a+b \log \left (c x^n\right )\right )}}d\log \left (c x^n\right )-\frac {2}{3 b \coth ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}}{n}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {-\frac {2}{3 b \coth ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}+\int \frac {1}{\sqrt {-i \tan \left (i a+i b \log \left (c x^n\right )+\frac {\pi }{2}\right )}}d\log \left (c x^n\right )}{n}\)

\(\Big \downarrow \) 3957

\(\displaystyle \frac {-\frac {\int -\frac {1}{\sqrt {\coth \left (a+b \log \left (c x^n\right )\right )} \left (1-\coth ^2\left (a+b \log \left (c x^n\right )\right )\right )}d\coth \left (a+b \log \left (c x^n\right )\right )}{b}-\frac {2}{3 b \coth ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}}{n}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {\int \frac {1}{\sqrt {\coth \left (a+b \log \left (c x^n\right )\right )} \left (1-\coth ^2\left (a+b \log \left (c x^n\right )\right )\right )}d\coth \left (a+b \log \left (c x^n\right )\right )}{b}-\frac {2}{3 b \coth ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}}{n}\)

\(\Big \downarrow \) 266

\(\displaystyle \frac {\frac {2 \int \frac {1}{1-\coth ^2\left (a+b \log \left (c x^n\right )\right )}d\sqrt {\coth \left (a+b \log \left (c x^n\right )\right )}}{b}-\frac {2}{3 b \coth ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}}{n}\)

\(\Big \downarrow \) 756

\(\displaystyle \frac {\frac {2 \left (\frac {1}{2} \int \frac {1}{1-\coth \left (a+b \log \left (c x^n\right )\right )}d\sqrt {\coth \left (a+b \log \left (c x^n\right )\right )}+\frac {1}{2} \int \frac {1}{\coth \left (a+b \log \left (c x^n\right )\right )+1}d\sqrt {\coth \left (a+b \log \left (c x^n\right )\right )}\right )}{b}-\frac {2}{3 b \coth ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}}{n}\)

\(\Big \downarrow \) 216

\(\displaystyle \frac {\frac {2 \left (\frac {1}{2} \int \frac {1}{1-\coth \left (a+b \log \left (c x^n\right )\right )}d\sqrt {\coth \left (a+b \log \left (c x^n\right )\right )}+\frac {1}{2} \arctan \left (\sqrt {\coth \left (a+b \log \left (c x^n\right )\right )}\right )\right )}{b}-\frac {2}{3 b \coth ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}}{n}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\frac {2 \left (\frac {1}{2} \arctan \left (\sqrt {\coth \left (a+b \log \left (c x^n\right )\right )}\right )+\frac {1}{2} \text {arctanh}\left (\sqrt {\coth \left (a+b \log \left (c x^n\right )\right )}\right )\right )}{b}-\frac {2}{3 b \coth ^{\frac {3}{2}}\left (a+b \log \left (c x^n\right )\right )}}{n}\)

Input:

Int[1/(x*Coth[a + b*Log[c*x^n]]^(5/2)),x]
 

Output:

((2*(ArcTan[Sqrt[Coth[a + b*Log[c*x^n]]]]/2 + ArcTanh[Sqrt[Coth[a + b*Log[ 
c*x^n]]]]/2))/b - 2/(3*b*Coth[a + b*Log[c*x^n]]^(3/2)))/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 756

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2 
]], s = Denominator[Rt[-a/b, 2]]}, Simp[r/(2*a)   Int[1/(r - s*x^2), x], x] 
 + Simp[r/(2*a)   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 3955

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

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 = 0.08 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.03

method	result	size
derivativedivides	\(\frac {-\frac {2}{3 {\coth \left (a +b \ln \left (c \,x^{n}\right )\right )}^{\frac {3}{2}}}+\arctan \left (\sqrt {\coth \left (a +b \ln \left (c \,x^{n}\right )\right )}\right )+\frac {\ln \left (\sqrt {\coth \left (a +b \ln \left (c \,x^{n}\right )\right )}+1\right )}{2}-\frac {\ln \left (\sqrt {\coth \left (a +b \ln \left (c \,x^{n}\right )\right )}-1\right )}{2}}{n b}\)	\(74\)
default	\(\frac {-\frac {2}{3 {\coth \left (a +b \ln \left (c \,x^{n}\right )\right )}^{\frac {3}{2}}}+\arctan \left (\sqrt {\coth \left (a +b \ln \left (c \,x^{n}\right )\right )}\right )+\frac {\ln \left (\sqrt {\coth \left (a +b \ln \left (c \,x^{n}\right )\right )}+1\right )}{2}-\frac {\ln \left (\sqrt {\coth \left (a +b \ln \left (c \,x^{n}\right )\right )}-1\right )}{2}}{n b}\)	\(74\)

Input:

int(1/x/coth(a+b*ln(c*x^n))^(5/2),x,method=_RETURNVERBOSE)
 

Output:

1/n/b*(-2/3/coth(a+b*ln(c*x^n))^(3/2)+arctan(coth(a+b*ln(c*x^n))^(1/2))+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))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1104 vs. \(2 (64) = 128\).

Time = 0.10 (sec) , antiderivative size = 1104, normalized size of antiderivative = 15.33 \[ \int \frac {1}{x \coth ^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx=\text {Too large to display} \] Input:

integrate(1/x/coth(a+b*log(c*x^n))^(5/2),x, algorithm="fricas")
 

Output:

-1/6*(4*cosh(b*n*log(x) + b*log(c) + a)^4 + 16*cosh(b*n*log(x) + b*log(c) 
+ a)*sinh(b*n*log(x) + b*log(c) + a)^3 + 4*sinh(b*n*log(x) + b*log(c) + a) 
^4 + 8*(3*cosh(b*n*log(x) + b*log(c) + a)^2 + 1)*sinh(b*n*log(x) + b*log(c 
) + a)^2 + 6*(cosh(b*n*log(x) + b*log(c) + a)^4 + 4*cosh(b*n*log(x) + b*lo 
g(c) + a)*sinh(b*n*log(x) + b*log(c) + a)^3 + sinh(b*n*log(x) + b*log(c) + 
 a)^4 + 2*(3*cosh(b*n*log(x) + b*log(c) + a)^2 + 1)*sinh(b*n*log(x) + b*lo 
g(c) + a)^2 + 2*cosh(b*n*log(x) + b*log(c) + a)^2 + 4*(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)*arctan(-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*lo 
g(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))) + 8*cosh(b*n*log(x) + b*log(c) + a)^2 + 3*(cosh(b*n*log(x) + b*log(c 
) + a)^4 + 4*cosh(b*n*log(x) + b*log(c) + a)*sinh(b*n*log(x) + b*log(c) + 
a)^3 + sinh(b*n*log(x) + b*log(c) + a)^4 + 2*(3*cosh(b*n*log(x) + b*log(c) 
 + a)^2 + 1)*sinh(b*n*log(x) + b*log(c) + a)^2 + 2*cosh(b*n*log(x) + b*log 
(c) + a)^2 + 4*(cosh(b*n*log(x) + b*log(c) + a)^3 + cosh(b*n*log(x) + b*lo 
g(c) + a))*sinh(b*n*log(x) + b*log(c) + a) + 1)*log(-cosh(b*n*log(x) + b*l 
og(c) + a)^2 - 2*cosh(b*n*log(x) + b*log(c) + a)*sinh(b*n*log(x) + b*lo...
 

Sympy [F(-1)]

Timed out. \[ \int \frac {1}{x \coth ^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx=\text {Timed out} \] Input:

integrate(1/x/coth(a+b*ln(c*x**n))**(5/2),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {1}{x \coth ^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx=\int { \frac {1}{x \coth \left (b \log \left (c x^{n}\right ) + a\right )^{\frac {5}{2}}} \,d x } \] Input:

integrate(1/x/coth(a+b*log(c*x^n))^(5/2),x, algorithm="maxima")
 

Output:

integrate(1/(x*coth(b*log(c*x^n) + a)^(5/2)), x)
 

Giac [F(-1)]

Timed out. \[ \int \frac {1}{x \coth ^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx=\text {Timed out} \] Input:

integrate(1/x/coth(a+b*log(c*x^n))^(5/2),x, algorithm="giac")
 

Output:

Timed out
 

Mupad [B] (verification not implemented)

Time = 3.79 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.89 \[ \int \frac {1}{x \coth ^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx=\frac {\mathrm {atan}\left (\sqrt {\mathrm {coth}\left (a+b\,\ln \left (c\,x^n\right )\right )}\right )}{b\,n}+\frac {\mathrm {atanh}\left (\sqrt {\mathrm {coth}\left (a+b\,\ln \left (c\,x^n\right )\right )}\right )}{b\,n}-\frac {2}{3\,b\,n\,{\mathrm {coth}\left (a+b\,\ln \left (c\,x^n\right )\right )}^{3/2}} \] Input:

int(1/(x*coth(a + b*log(c*x^n))^(5/2)),x)
 

Output:

atan(coth(a + b*log(c*x^n))^(1/2))/(b*n) + atanh(coth(a + b*log(c*x^n))^(1 
/2))/(b*n) - 2/(3*b*n*coth(a + b*log(c*x^n))^(3/2))
 

Reduce [F]

\[ \int \frac {1}{x \coth ^{\frac {5}{2}}\left (a+b \log \left (c x^n\right )\right )} \, dx=\int \frac {\sqrt {\coth \left (\mathrm {log}\left (x^{n} c \right ) b +a \right )}}{{\coth \left (\mathrm {log}\left (x^{n} c \right ) b +a \right )}^{3} x}d x \] Input:

int(1/x/coth(a+b*log(c*x^n))^(5/2),x)
 

Output:

int(sqrt(coth(log(x**n*c)*b + a))/(coth(log(x**n*c)*b + a)**3*x),x)