\(\int \frac {\coth (d (a+b \log (c x^n)))}{x^2} \, dx\) [187]

Optimal result

Integrand size = 17, antiderivative size = 58 \[ \int \frac {\coth \left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^2} \, dx=-\frac {1}{x}+\frac {2 \operatorname {Hypergeometric2F1}\left (1,-\frac {1}{2 b d n},1-\frac {1}{2 b d n},e^{2 a d} \left (c x^n\right )^{2 b d}\right )}{x} \] Output:

-1/x+2*hypergeom([1, -1/2/b/d/n],[1-1/2/b/d/n],exp(2*a*d)*(c*x^n)^(2*b*d)) 
/x
 

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(197\) vs. \(2(58)=116\).

Time = 2.56 (sec) , antiderivative size = 197, normalized size of antiderivative = 3.40 \[ \int \frac {\coth \left (d \left (a+b \log \left (c x^n\right )\right )\right )}{x^2} \, dx=\frac {\coth \left (d \left (a+b \log \left (c x^n\right )\right )\right )-\coth \left (d \left (a-b n \log (x)+b \log \left (c x^n\right )\right )\right )-\frac {e^{2 d \left (a+b \log \left (c x^n\right )\right )} \operatorname {Hypergeometric2F1}\left (1,1-\frac {1}{2 b d n},2-\frac {1}{2 b d n},e^{2 d \left (a+b \log \left (c x^n\right )\right )}\right )}{-1+2 b d n}+\operatorname {Hypergeometric2F1}\left (1,-\frac {1}{2 b d n},1-\frac {1}{2 b d n},e^{2 d \left (a+b \log \left (c x^n\right )\right )}\right )+\text {csch}\left (d \left (a+b \log \left (c x^n\right )\right )\right ) \text {csch}\left (d \left (a-b n \log (x)+b \log \left (c x^n\right )\right )\right ) \sinh (b d n \log (x))}{x} \] Input:

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

Output:

(Coth[d*(a + b*Log[c*x^n])] - Coth[d*(a - b*n*Log[x] + b*Log[c*x^n])] - (E 
^(2*d*(a + b*Log[c*x^n]))*Hypergeometric2F1[1, 1 - 1/(2*b*d*n), 2 - 1/(2*b 
*d*n), E^(2*d*(a + b*Log[c*x^n]))])/(-1 + 2*b*d*n) + Hypergeometric2F1[1, 
-1/2*1/(b*d*n), 1 - 1/(2*b*d*n), E^(2*d*(a + b*Log[c*x^n]))] + Csch[d*(a + 
 b*Log[c*x^n])]*Csch[d*(a - b*n*Log[x] + b*Log[c*x^n])]*Sinh[b*d*n*Log[x]] 
)/x
 

Rubi [A] (verified)

Time = 0.34 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.59, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {6074, 6072, 959, 888}

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.

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

\(\Big \downarrow \) 6074

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

\(\Big \downarrow \) 6072

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

\(\Big \downarrow \) 959

\(\displaystyle \frac {\left (c x^n\right )^{\frac {1}{n}} \left (-2 \int \frac {\left (c x^n\right )^{-1-\frac {1}{n}}}{1-e^{2 a d} \left (c x^n\right )^{2 b d}}d\left (c x^n\right )-n \left (c x^n\right )^{-1/n}\right )}{n x}\)

\(\Big \downarrow \) 888

\(\displaystyle \frac {\left (c x^n\right )^{\frac {1}{n}} \left (2 n \left (c x^n\right )^{-1/n} \operatorname {Hypergeometric2F1}\left (1,-\frac {1}{2 b d n},1-\frac {1}{2 b d n},e^{2 a d} \left (c x^n\right )^{2 b d}\right )-n \left (c x^n\right )^{-1/n}\right )}{n x}\)

Input:

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

Output:

((c*x^n)^n^(-1)*(-(n/(c*x^n)^n^(-1)) + (2*n*Hypergeometric2F1[1, -1/2*1/(b 
*d*n), 1 - 1/(2*b*d*n), E^(2*a*d)*(c*x^n)^(2*b*d)])/(c*x^n)^n^(-1)))/(n*x)
 

Defintions of rubi rules used

rule 888

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p 
*((c*x)^(m + 1)/(c*(m + 1)))*Hypergeometric2F1[-p, (m + 1)/n, (m + 1)/n + 1 
, (-b)*(x^n/a)], x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] && (ILt 
Q[p, 0] || GtQ[a, 0])
 

rule 959

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n 
_)), x_Symbol] :> Simp[d*(e*x)^(m + 1)*((a + b*x^n)^(p + 1)/(b*e*(m + n*(p 
+ 1) + 1))), x] - Simp[(a*d*(m + 1) - b*c*(m + n*(p + 1) + 1))/(b*(m + n*(p 
 + 1) + 1))   Int[(e*x)^m*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d, e, m, 
 n, p}, x] && NeQ[b*c - a*d, 0] && NeQ[m + n*(p + 1) + 1, 0]
 

rule 6072

Int[Coth[((a_.) + Log[x_]*(b_.))*(d_.)]^(p_.)*((e_.)*(x_))^(m_.), x_Symbol] 
 :> Int[(e*x)^m*((-1 - E^(2*a*d)*x^(2*b*d))^p/(1 - E^(2*a*d)*x^(2*b*d))^p), 
 x] /; FreeQ[{a, b, d, e, m, p}, x]
 

rule 6074

Int[Coth[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(d_.)]^(p_.)*((e_.)*(x_))^(m 
_.), x_Symbol] :> Simp[(e*x)^(m + 1)/(e*n*(c*x^n)^((m + 1)/n))   Subst[Int[ 
x^((m + 1)/n - 1)*Coth[d*(a + b*Log[x])]^p, x], x, c*x^n], x] /; FreeQ[{a, 
b, c, d, e, m, n, p}, x] && (NeQ[c, 1] || NeQ[n, 1])
 

Maple [F]

\[\int \frac {\coth \left (d \left (a +b \ln \left (c \,x^{n}\right )\right )\right )}{x^{2}}d x\]

Input:

int(coth(d*(a+b*ln(c*x^n)))/x^2,x)
 

Output:

int(coth(d*(a+b*ln(c*x^n)))/x^2,x)
 

Fricas [F]

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

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

Output:

integral(coth(b*d*log(c*x^n) + a*d)/x^2, x)
 

Sympy [F]

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

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

Output:

Integral(coth(a*d + b*d*log(c*x**n))/x**2, x)
 

Maxima [F]

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

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

Output:

-1/x - integrate(1/(c^(b*d)*x^2*e^(b*d*log(x^n) + a*d) + x^2), x) + integr 
ate(1/(c^(b*d)*x^2*e^(b*d*log(x^n) + a*d) - x^2), x)
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

(2*e**(2*a*d)*c**(2*b*d)*int(x**(2*b*d*n)/(x**(2*b*d*n)*e**(2*a*d)*c**(2*b 
*d)*x**2 - x**2),x)*x + 1)/x