\(\int \text {csch}(a+b \log (c x^n)) \, dx\) [156]

Optimal result

Integrand size = 11, antiderivative size = 70 \[ \int \text {csch}\left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {2 e^{-a} x \left (c x^n\right )^{-b} \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2} \left (1-\frac {1}{b n}\right ),\frac {1}{2} \left (3-\frac {1}{b n}\right ),e^{-2 a} \left (c x^n\right )^{-2 b}\right )}{1-b n} \] Output:

2*x*hypergeom([1, 1/2-1/2/b/n],[3/2-1/2/b/n],1/exp(2*a)/((c*x^n)^(2*b)))/e 
xp(a)/(-b*n+1)/((c*x^n)^b)
 

Mathematica [A] (verified)

Time = 0.75 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.89 \[ \int \text {csch}\left (a+b \log \left (c x^n\right )\right ) \, dx=-\frac {2 e^a x \left (c x^n\right )^b \operatorname {Hypergeometric2F1}\left (1,\frac {1}{2} \left (1+\frac {1}{b n}\right ),\frac {1}{2} \left (3+\frac {1}{b n}\right ),e^{2 \left (a+b \log \left (c x^n\right )\right )}\right )}{1+b n} \] Input:

Integrate[Csch[a + b*Log[c*x^n]],x]
 

Output:

(-2*E^a*x*(c*x^n)^b*Hypergeometric2F1[1, (1 + 1/(b*n))/2, (3 + 1/(b*n))/2, 
 E^(2*(a + b*Log[c*x^n]))])/(1 + b*n)
 

Rubi [A] (verified)

Time = 0.33 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.89, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {6080, 6082, 795, 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.

\(\displaystyle \int \text {csch}\left (a+b \log \left (c x^n\right )\right ) \, dx\)

\(\Big \downarrow \) 6080

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

\(\Big \downarrow \) 6082

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

\(\Big \downarrow \) 795

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

\(\Big \downarrow \) 888

\(\displaystyle -\frac {2 e^a x \left (c x^n\right )^b \operatorname {Hypergeometric2F1}\left (1,\frac {b+\frac {1}{n}}{2 b},\frac {1}{2} \left (3+\frac {1}{b n}\right ),e^{2 a} \left (c x^n\right )^{2 b}\right )}{b n+1}\)

Input:

Int[Csch[a + b*Log[c*x^n]],x]
 

Output:

(-2*E^a*x*(c*x^n)^b*Hypergeometric2F1[1, (b + n^(-1))/(2*b), (3 + 1/(b*n)) 
/2, E^(2*a)*(c*x^n)^(2*b)])/(1 + b*n)
 

Defintions of rubi rules used

rule 795

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[x^(m + n*p)* 
(b + a/x^n)^p, x] /; FreeQ[{a, b, m, n}, x] && IntegerQ[p] && NegQ[n]
 

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 6080

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

rule 6082

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

Maple [F]

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

Input:

int(csch(a+b*ln(c*x^n)),x)
 

Output:

int(csch(a+b*ln(c*x^n)),x)
 

Fricas [F]

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

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

Output:

integral(csch(b*log(c*x^n) + a), x)
 

Sympy [F]

\[ \int \text {csch}\left (a+b \log \left (c x^n\right )\right ) \, dx=\int \operatorname {csch}{\left (a + b \log {\left (c x^{n} \right )} \right )}\, dx \] Input:

integrate(csch(a+b*ln(c*x**n)),x)
 

Output:

Integral(csch(a + b*log(c*x**n)), x)
 

Maxima [F]

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

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

Output:

integrate(csch(b*log(c*x^n) + a), x)
 

Giac [F]

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

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

Output:

integrate(csch(b*log(c*x^n) + a), x)
 

Mupad [F(-1)]

Timed out. \[ \int \text {csch}\left (a+b \log \left (c x^n\right )\right ) \, dx=\int \frac {1}{\mathrm {sinh}\left (a+b\,\ln \left (c\,x^n\right )\right )} \,d x \] Input:

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

Output:

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

Reduce [F]

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

int(csch(a+b*log(c*x^n)),x)
 

Output:

int(csch(log(x**n*c)*b + a),x)