\(\int (e x)^{-1+2 n} (a+b \text {csch}(c+d x^n)) \, dx\) [80]

Optimal result

Integrand size = 22, antiderivative size = 124 \[ \int (e x)^{-1+2 n} \left (a+b \text {csch}\left (c+d x^n\right )\right ) \, dx=\frac {a (e x)^{2 n}}{2 e n}-\frac {2 b x^{-n} (e x)^{2 n} \text {arctanh}\left (e^{c+d x^n}\right )}{d e n}-\frac {b x^{-2 n} (e x)^{2 n} \operatorname {PolyLog}\left (2,-e^{c+d x^n}\right )}{d^2 e n}+\frac {b x^{-2 n} (e x)^{2 n} \operatorname {PolyLog}\left (2,e^{c+d x^n}\right )}{d^2 e n} \] Output:

1/2*a*(e*x)^(2*n)/e/n-2*b*(e*x)^(2*n)*arctanh(exp(c+d*x^n))/d/e/n/(x^n)-b* 
(e*x)^(2*n)*polylog(2,-exp(c+d*x^n))/d^2/e/n/(x^(2*n))+b*(e*x)^(2*n)*polyl 
og(2,exp(c+d*x^n))/d^2/e/n/(x^(2*n))
 

Mathematica [A] (verified)

Time = 0.50 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.41 \[ \int (e x)^{-1+2 n} \left (a+b \text {csch}\left (c+d x^n\right )\right ) \, dx=\frac {x^{-2 n} (e x)^{2 n} \left (a d^2 x^{2 n}+2 b c \log \left (1-e^{-c-d x^n}\right )+2 b d x^n \log \left (1-e^{-c-d x^n}\right )-2 b c \log \left (1+e^{-c-d x^n}\right )-2 b d x^n \log \left (1+e^{-c-d x^n}\right )-2 b c \log \left (\tanh \left (\frac {1}{2} \left (c+d x^n\right )\right )\right )+2 b \operatorname {PolyLog}\left (2,-e^{-c-d x^n}\right )-2 b \operatorname {PolyLog}\left (2,e^{-c-d x^n}\right )\right )}{2 d^2 e n} \] Input:

Integrate[(e*x)^(-1 + 2*n)*(a + b*Csch[c + d*x^n]),x]
 

Output:

((e*x)^(2*n)*(a*d^2*x^(2*n) + 2*b*c*Log[1 - E^(-c - d*x^n)] + 2*b*d*x^n*Lo 
g[1 - E^(-c - d*x^n)] - 2*b*c*Log[1 + E^(-c - d*x^n)] - 2*b*d*x^n*Log[1 + 
E^(-c - d*x^n)] - 2*b*c*Log[Tanh[(c + d*x^n)/2]] + 2*b*PolyLog[2, -E^(-c - 
 d*x^n)] - 2*b*PolyLog[2, E^(-c - d*x^n)]))/(2*d^2*e*n*x^(2*n))
 

Rubi [A] (verified)

Time = 0.33 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {2010, 2009}

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.

\(\displaystyle \int (e x)^{2 n-1} \left (a+b \text {csch}\left (c+d x^n\right )\right ) \, dx\)

\(\Big \downarrow \) 2010

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {a (e x)^{2 n}}{2 e n}-\frac {2 b x^{-n} (e x)^{2 n} \text {arctanh}\left (e^{c+d x^n}\right )}{d e n}-\frac {b x^{-2 n} (e x)^{2 n} \operatorname {PolyLog}\left (2,-e^{d x^n+c}\right )}{d^2 e n}+\frac {b x^{-2 n} (e x)^{2 n} \operatorname {PolyLog}\left (2,e^{d x^n+c}\right )}{d^2 e n}\)

Input:

Int[(e*x)^(-1 + 2*n)*(a + b*Csch[c + d*x^n]),x]
 

Output:

(a*(e*x)^(2*n))/(2*e*n) - (2*b*(e*x)^(2*n)*ArcTanh[E^(c + d*x^n)])/(d*e*n* 
x^n) - (b*(e*x)^(2*n)*PolyLog[2, -E^(c + d*x^n)])/(d^2*e*n*x^(2*n)) + (b*( 
e*x)^(2*n)*PolyLog[2, E^(c + d*x^n)])/(d^2*e*n*x^(2*n))
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2010

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x] 
, x] /; FreeQ[{c, m}, x] && SumQ[u] &&  !LinearQ[u, x] &&  !MatchQ[u, (a_) 
+ (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]
 

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.59 (sec) , antiderivative size = 326, normalized size of antiderivative = 2.63

method	result	size
risch	\(\frac {a x \,{\mathrm e}^{\frac {\left (-1+2 n \right ) \left (-i \pi \,\operatorname {csgn}\left (i e \right ) \operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i e x \right )+i \pi \,\operatorname {csgn}\left (i e \right ) \operatorname {csgn}\left (i e x \right )^{2}+i \pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i e x \right )^{2}-i \pi \operatorname {csgn}\left (i e x \right )^{3}+2 \ln \left (x \right )+2 \ln \left (e \right )\right )}{2}}}{2 n}+\frac {2 b \,{\mathrm e}^{-i \pi n \,\operatorname {csgn}\left (i e \right ) \operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i e x \right )} {\mathrm e}^{i \pi n \,\operatorname {csgn}\left (i e \right ) \operatorname {csgn}\left (i e x \right )^{2}} {\mathrm e}^{i \pi n \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i e x \right )^{2}} {\mathrm e}^{-i \pi n \operatorname {csgn}\left (i e x \right )^{3}} {\mathrm e}^{\frac {i \pi \,\operatorname {csgn}\left (i e \right ) \operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i e x \right )}{2}} {\mathrm e}^{-\frac {i \pi \,\operatorname {csgn}\left (i e \right ) \operatorname {csgn}\left (i e x \right )^{2}}{2}} {\mathrm e}^{-\frac {i \pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i e x \right )^{2}}{2}} {\mathrm e}^{\frac {i \pi \operatorname {csgn}\left (i e x \right )^{3}}{2}} e^{2 n} {\mathrm e}^{c} \left (\frac {\left (\ln \left (1-{\mathrm e}^{c +d \,x^{n}}\right )-\ln \left ({\mathrm e}^{c +d \,x^{n}}+1\right )\right ) d \,x^{n} {\mathrm e}^{-c}}{2}+\frac {\left (\operatorname {dilog}\left (1-{\mathrm e}^{c +d \,x^{n}}\right )-\operatorname {dilog}\left ({\mathrm e}^{c +d \,x^{n}}+1\right )\right ) {\mathrm e}^{-c}}{2}\right )}{e n \,d^{2}}\)	\(326\)

Input:

int((e*x)^(-1+2*n)*(a+b*csch(c+d*x^n)),x,method=_RETURNVERBOSE)
 

Output:

1/2*a/n*x*exp(1/2*(-1+2*n)*(-I*Pi*csgn(I*e)*csgn(I*x)*csgn(I*e*x)+I*Pi*csg 
n(I*e)*csgn(I*e*x)^2+I*Pi*csgn(I*x)*csgn(I*e*x)^2-I*Pi*csgn(I*e*x)^3+2*ln( 
x)+2*ln(e)))+2*b*exp(-I*Pi*n*csgn(I*e)*csgn(I*x)*csgn(I*e*x))*exp(I*Pi*n*c 
sgn(I*e)*csgn(I*e*x)^2)*exp(I*Pi*n*csgn(I*x)*csgn(I*e*x)^2)*exp(-I*Pi*n*cs 
gn(I*e*x)^3)*exp(1/2*I*Pi*csgn(I*e)*csgn(I*x)*csgn(I*e*x))*exp(-1/2*I*Pi*c 
sgn(I*e)*csgn(I*e*x)^2)*exp(-1/2*I*Pi*csgn(I*x)*csgn(I*e*x)^2)*exp(1/2*I*P 
i*csgn(I*e*x)^3)*(e^n)^2/e*exp(c)/n/d^2*(1/2*(ln(1-exp(c+d*x^n))-ln(exp(c+ 
d*x^n)+1))*d*x^n*exp(-c)+1/2*(dilog(1-exp(c+d*x^n))-dilog(exp(c+d*x^n)+1)) 
*exp(-c))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 555 vs. \(2 (121) = 242\).

Time = 0.11 (sec) , antiderivative size = 555, normalized size of antiderivative = 4.48 \[ \int (e x)^{-1+2 n} \left (a+b \text {csch}\left (c+d x^n\right )\right ) \, dx=\text {Too large to display} \] Input:

integrate((e*x)^(-1+2*n)*(a+b*csch(c+d*x^n)),x, algorithm="fricas")
 

Output:

1/2*(a*d^2*cosh((2*n - 1)*log(e))*cosh(n*log(x))^2 + a*d^2*cosh(n*log(x))^ 
2*sinh((2*n - 1)*log(e)) + (a*d^2*cosh((2*n - 1)*log(e)) + a*d^2*sinh((2*n 
 - 1)*log(e)))*sinh(n*log(x))^2 + 2*(b*cosh((2*n - 1)*log(e)) + b*sinh((2* 
n - 1)*log(e)))*dilog(cosh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c) + sinh 
(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c)) - 2*(b*cosh((2*n - 1)*log(e)) + 
 b*sinh((2*n - 1)*log(e)))*dilog(-cosh(d*cosh(n*log(x)) + d*sinh(n*log(x)) 
 + c) - sinh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c)) - 2*(b*d*cosh((2*n 
- 1)*log(e))*cosh(n*log(x)) + b*d*cosh(n*log(x))*sinh((2*n - 1)*log(e)) + 
(b*d*cosh((2*n - 1)*log(e)) + b*d*sinh((2*n - 1)*log(e)))*sinh(n*log(x)))* 
log(cosh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c) + sinh(d*cosh(n*log(x)) 
+ d*sinh(n*log(x)) + c) + 1) - 2*(b*c*cosh((2*n - 1)*log(e)) + b*c*sinh((2 
*n - 1)*log(e)))*log(cosh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c) + sinh( 
d*cosh(n*log(x)) + d*sinh(n*log(x)) + c) - 1) + 2*(b*d*cosh((2*n - 1)*log( 
e))*cosh(n*log(x)) + b*c*cosh((2*n - 1)*log(e)) + (b*d*cosh(n*log(x)) + b* 
c)*sinh((2*n - 1)*log(e)) + (b*d*cosh((2*n - 1)*log(e)) + b*d*sinh((2*n - 
1)*log(e)))*sinh(n*log(x)))*log(-cosh(d*cosh(n*log(x)) + d*sinh(n*log(x)) 
+ c) - sinh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c) + 1) + 2*(a*d^2*cosh( 
(2*n - 1)*log(e))*cosh(n*log(x)) + a*d^2*cosh(n*log(x))*sinh((2*n - 1)*log 
(e)))*sinh(n*log(x)))/(d^2*n)
 

Sympy [F]

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

integrate((e*x)**(-1+2*n)*(a+b*csch(c+d*x**n)),x)
                                                                                    
                                                                                    
 

Output:

Integral((e*x)**(2*n - 1)*(a + b*csch(c + d*x**n)), x)
 

Maxima [F]

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

integrate((e*x)^(-1+2*n)*(a+b*csch(c+d*x^n)),x, algorithm="maxima")
 

Output:

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

Giac [F]

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

integrate((e*x)^(-1+2*n)*(a+b*csch(c+d*x^n)),x, algorithm="giac")
 

Output:

integrate((b*csch(d*x^n + c) + a)*(e*x)^(2*n - 1), x)
 

Mupad [F(-1)]

Timed out. \[ \int (e x)^{-1+2 n} \left (a+b \text {csch}\left (c+d x^n\right )\right ) \, dx=\int \left (a+\frac {b}{\mathrm {sinh}\left (c+d\,x^n\right )}\right )\,{\left (e\,x\right )}^{2\,n-1} \,d x \] Input:

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

Output:

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

Reduce [F]

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

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

Output:

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