\(\int \frac {(e x)^{-1+2 n}}{(a+b \text {csch}(c+d x^n))^2} \, dx\) [89]

Optimal result

Integrand size = 24, antiderivative size = 681 \[ \int \frac {(e x)^{-1+2 n}}{\left (a+b \text {csch}\left (c+d x^n\right )\right )^2} \, dx=\frac {(e x)^{2 n}}{2 a^2 e n}+\frac {b^3 x^{-n} (e x)^{2 n} \log \left (1+\frac {a e^{c+d x^n}}{b-\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d e n}-\frac {2 b x^{-n} (e x)^{2 n} \log \left (1+\frac {a e^{c+d x^n}}{b-\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d e n}-\frac {b^3 x^{-n} (e x)^{2 n} \log \left (1+\frac {a e^{c+d x^n}}{b+\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d e n}+\frac {2 b x^{-n} (e x)^{2 n} \log \left (1+\frac {a e^{c+d x^n}}{b+\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d e n}+\frac {b^2 x^{-2 n} (e x)^{2 n} \log \left (b+a \sinh \left (c+d x^n\right )\right )}{a^2 \left (a^2+b^2\right ) d^2 e n}+\frac {b^3 x^{-2 n} (e x)^{2 n} \operatorname {PolyLog}\left (2,-\frac {a e^{c+d x^n}}{b-\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d^2 e n}-\frac {2 b x^{-2 n} (e x)^{2 n} \operatorname {PolyLog}\left (2,-\frac {a e^{c+d x^n}}{b-\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d^2 e n}-\frac {b^3 x^{-2 n} (e x)^{2 n} \operatorname {PolyLog}\left (2,-\frac {a e^{c+d x^n}}{b+\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d^2 e n}+\frac {2 b x^{-2 n} (e x)^{2 n} \operatorname {PolyLog}\left (2,-\frac {a e^{c+d x^n}}{b+\sqrt {a^2+b^2}}\right )}{a^2 \sqrt {a^2+b^2} d^2 e n}-\frac {b^2 x^{-n} (e x)^{2 n} \cosh \left (c+d x^n\right )}{a \left (a^2+b^2\right ) d e n \left (b+a \sinh \left (c+d x^n\right )\right )} \] Output:

1/2*(e*x)^(2*n)/a^2/e/n+b^3*(e*x)^(2*n)*ln(1+a*exp(c+d*x^n)/(b-(a^2+b^2)^( 
1/2)))/a^2/(a^2+b^2)^(3/2)/d/e/n/(x^n)-2*b*(e*x)^(2*n)*ln(1+a*exp(c+d*x^n) 
/(b-(a^2+b^2)^(1/2)))/a^2/(a^2+b^2)^(1/2)/d/e/n/(x^n)-b^3*(e*x)^(2*n)*ln(1 
+a*exp(c+d*x^n)/(b+(a^2+b^2)^(1/2)))/a^2/(a^2+b^2)^(3/2)/d/e/n/(x^n)+2*b*( 
e*x)^(2*n)*ln(1+a*exp(c+d*x^n)/(b+(a^2+b^2)^(1/2)))/a^2/(a^2+b^2)^(1/2)/d/ 
e/n/(x^n)+b^2*(e*x)^(2*n)*ln(b+a*sinh(c+d*x^n))/a^2/(a^2+b^2)/d^2/e/n/(x^( 
2*n))+b^3*(e*x)^(2*n)*polylog(2,-a*exp(c+d*x^n)/(b-(a^2+b^2)^(1/2)))/a^2/( 
a^2+b^2)^(3/2)/d^2/e/n/(x^(2*n))-2*b*(e*x)^(2*n)*polylog(2,-a*exp(c+d*x^n) 
/(b-(a^2+b^2)^(1/2)))/a^2/(a^2+b^2)^(1/2)/d^2/e/n/(x^(2*n))-b^3*(e*x)^(2*n 
)*polylog(2,-a*exp(c+d*x^n)/(b+(a^2+b^2)^(1/2)))/a^2/(a^2+b^2)^(3/2)/d^2/e 
/n/(x^(2*n))+2*b*(e*x)^(2*n)*polylog(2,-a*exp(c+d*x^n)/(b+(a^2+b^2)^(1/2)) 
)/a^2/(a^2+b^2)^(1/2)/d^2/e/n/(x^(2*n))-b^2*(e*x)^(2*n)*cosh(c+d*x^n)/a/(a 
^2+b^2)/d/e/n/(x^n)/(b+a*sinh(c+d*x^n))
 

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 19.27 (sec) , antiderivative size = 3219, normalized size of antiderivative = 4.73 \[ \int \frac {(e x)^{-1+2 n}}{\left (a+b \text {csch}\left (c+d x^n\right )\right )^2} \, dx=\text {Result too large to show} \] Input:

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

Output:

(b^2*x^(1 - n)*(e*x)^(-1 + 2*n)*Csch[c/2]*Csch[c + d*x^n]^2*Sech[c/2]*(b*C 
osh[c] + a*Sinh[d*x^n])*(b + a*Sinh[c + d*x^n]))/(2*a^2*(a^2 + b^2)*d*n*(a 
 + b*Csch[c + d*x^n])^2) + (b^2*x^(1 - n)*(e*x)^(-1 + 2*n)*Coth[c]*Csch[c 
+ d*x^n]^2*(b + a*Sinh[c + d*x^n])^2)/(a^2*(a^2 + b^2)*d*n*(a + b*Csch[c + 
 d*x^n])^2) + (2*b^3*x^(1 - 2*n)*(e*x)^(-1 + 2*n)*ArcTanh[(b + a*E^(c + d* 
x^n))/Sqrt[a^2 + b^2]]*Coth[c]*Csch[c + d*x^n]^2*(b + a*Sinh[c + d*x^n])^2 
)/(a^2*(a^2 + b^2)^(3/2)*d^2*n*(a + b*Csch[c + d*x^n])^2) - (b^2*E^c*x^(1 
- 2*n)*(e*x)^(-1 + 2*n)*Csch[c]*Csch[c + d*x^n]^2*((d*x^n)/a + (b*(1 + E^( 
2*c))*ArcTanh[(b*E^c + a*E^(2*c + d*x^n))/(Sqrt[a^2 + b^2]*E^c)])/(a*Sqrt[ 
a^2 + b^2]*E^(2*c)) - ((1 - E^(-2*c))*Log[a - 2*b*E^(c + d*x^n) - a*E^(2*c 
 + 2*d*x^n)])/(2*a))*(b + a*Sinh[c + d*x^n])^2)/(a*(a^2 + b^2)*d^2*n*(a + 
b*Csch[c + d*x^n])^2) - (2*b*x^(1 - 2*n)*(e*x)^(-1 + 2*n)*Csch[c + d*x^n]^ 
2*(((-I)*Pi*ArcTanh[(-a + b*Tanh[(c + d*x^n)/2])/Sqrt[a^2 + b^2]])/Sqrt[a^ 
2 + b^2] - (2*((-I)*c + Pi/2 - I*d*x^n)*ArcTanh[(((-I)*a + b)*Cot[((-I)*c 
+ Pi/2 - I*d*x^n)/2])/Sqrt[-a^2 - b^2]] - 2*((-I)*c + ArcCos[((-I)*b)/a])* 
ArcTanh[(((-I)*a - b)*Tan[((-I)*c + Pi/2 - I*d*x^n)/2])/Sqrt[-a^2 - b^2]] 
+ (ArcCos[((-I)*b)/a] - (2*I)*(ArcTanh[(((-I)*a + b)*Cot[((-I)*c + Pi/2 - 
I*d*x^n)/2])/Sqrt[-a^2 - b^2]] - ArcTanh[(((-I)*a - b)*Tan[((-I)*c + Pi/2 
- I*d*x^n)/2])/Sqrt[-a^2 - b^2]]))*Log[Sqrt[-a^2 - b^2]/(Sqrt[2]*Sqrt[(-I) 
*a]*E^((I/2)*((-I)*c + Pi/2 - I*d*x^n))*Sqrt[b + a*Sinh[c + d*x^n]])] +...
 

Rubi [A] (verified)

Time = 1.56 (sec) , antiderivative size = 527, normalized size of antiderivative = 0.77, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {5964, 5960, 3042, 4679, 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.

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

\(\Big \downarrow \) 5964

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

\(\Big \downarrow \) 5960

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4679

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {x^{-2 n} (e x)^{2 n} \left (-\frac {2 b \operatorname {PolyLog}\left (2,-\frac {a e^{d x^n+c}}{b-\sqrt {a^2+b^2}}\right )}{a^2 d^2 \sqrt {a^2+b^2}}+\frac {2 b \operatorname {PolyLog}\left (2,-\frac {a e^{d x^n+c}}{b+\sqrt {a^2+b^2}}\right )}{a^2 d^2 \sqrt {a^2+b^2}}+\frac {b^2 \log \left (a \sinh \left (c+d x^n\right )+b\right )}{a^2 d^2 \left (a^2+b^2\right )}-\frac {2 b x^n \log \left (\frac {a e^{c+d x^n}}{b-\sqrt {a^2+b^2}}+1\right )}{a^2 d \sqrt {a^2+b^2}}+\frac {2 b x^n \log \left (\frac {a e^{c+d x^n}}{\sqrt {a^2+b^2}+b}+1\right )}{a^2 d \sqrt {a^2+b^2}}-\frac {b^2 x^n \cosh \left (c+d x^n\right )}{a d \left (a^2+b^2\right ) \left (a \sinh \left (c+d x^n\right )+b\right )}+\frac {b^3 \operatorname {PolyLog}\left (2,-\frac {a e^{d x^n+c}}{b-\sqrt {a^2+b^2}}\right )}{a^2 d^2 \left (a^2+b^2\right )^{3/2}}-\frac {b^3 \operatorname {PolyLog}\left (2,-\frac {a e^{d x^n+c}}{b+\sqrt {a^2+b^2}}\right )}{a^2 d^2 \left (a^2+b^2\right )^{3/2}}+\frac {b^3 x^n \log \left (\frac {a e^{c+d x^n}}{b-\sqrt {a^2+b^2}}+1\right )}{a^2 d \left (a^2+b^2\right )^{3/2}}-\frac {b^3 x^n \log \left (\frac {a e^{c+d x^n}}{\sqrt {a^2+b^2}+b}+1\right )}{a^2 d \left (a^2+b^2\right )^{3/2}}+\frac {x^{2 n}}{2 a^2}\right )}{e n}\)

Input:

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

Output:

((e*x)^(2*n)*(x^(2*n)/(2*a^2) + (b^3*x^n*Log[1 + (a*E^(c + d*x^n))/(b - Sq 
rt[a^2 + b^2])])/(a^2*(a^2 + b^2)^(3/2)*d) - (2*b*x^n*Log[1 + (a*E^(c + d* 
x^n))/(b - Sqrt[a^2 + b^2])])/(a^2*Sqrt[a^2 + b^2]*d) - (b^3*x^n*Log[1 + ( 
a*E^(c + d*x^n))/(b + Sqrt[a^2 + b^2])])/(a^2*(a^2 + b^2)^(3/2)*d) + (2*b* 
x^n*Log[1 + (a*E^(c + d*x^n))/(b + Sqrt[a^2 + b^2])])/(a^2*Sqrt[a^2 + b^2] 
*d) + (b^2*Log[b + a*Sinh[c + d*x^n]])/(a^2*(a^2 + b^2)*d^2) + (b^3*PolyLo 
g[2, -((a*E^(c + d*x^n))/(b - Sqrt[a^2 + b^2]))])/(a^2*(a^2 + b^2)^(3/2)*d 
^2) - (2*b*PolyLog[2, -((a*E^(c + d*x^n))/(b - Sqrt[a^2 + b^2]))])/(a^2*Sq 
rt[a^2 + b^2]*d^2) - (b^3*PolyLog[2, -((a*E^(c + d*x^n))/(b + Sqrt[a^2 + b 
^2]))])/(a^2*(a^2 + b^2)^(3/2)*d^2) + (2*b*PolyLog[2, -((a*E^(c + d*x^n))/ 
(b + Sqrt[a^2 + b^2]))])/(a^2*Sqrt[a^2 + b^2]*d^2) - (b^2*x^n*Cosh[c + d*x 
^n])/(a*(a^2 + b^2)*d*(b + a*Sinh[c + d*x^n]))))/(e*n*x^(2*n))
 

Defintions of rubi rules used

rule 2009

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

rule 3042

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

rule 4679

Int[(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(n_.)*((c_.) + (d_.)*(x_))^(m_.) 
, x_Symbol] :> Int[ExpandIntegrand[(c + d*x)^m, 1/(Sin[e + f*x]^n/(b + a*Si 
n[e + f*x])^n), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && ILtQ[n, 0] && IGt 
Q[m, 0]
 

rule 5960

Int[((a_.) + Csch[(c_.) + (d_.)*(x_)^(n_)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbo 
l] :> Simp[1/n   Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*Csch[c + d*x] 
)^p, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p}, x] && IGtQ[Simplify[(m 
 + 1)/n], 0] && IntegerQ[p]
 

rule 5964

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

Maple [F]

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

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 8453 vs. \(2 (645) = 1290\).

Time = 0.22 (sec) , antiderivative size = 8453, normalized size of antiderivative = 12.41 \[ \int \frac {(e x)^{-1+2 n}}{\left (a+b \text {csch}\left (c+d x^n\right )\right )^2} \, dx=\text {Too large to display} \] Input:

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

Output:

Too large to include
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

1/2*(4*a*b^2*e^(2*n)*x^n + (a^3*d*e^(2*n) + a*b^2*d*e^(2*n))*x^(2*n) - (a^ 
3*d*e^(2*n)*e^(2*c) + a*b^2*d*e^(2*n)*e^(2*c))*e^(2*d*x^n + 2*n*log(x)) - 
2*(2*b^3*e^(2*n)*e^(n*log(x) + c) + (a^2*b*d*e^(2*n)*e^c + b^3*d*e^(2*n)*e 
^c)*x^(2*n))*e^(d*x^n))/(a^5*d*e*n + a^3*b^2*d*e*n - (a^5*d*e*n*e^(2*c) + 
a^3*b^2*d*e*n*e^(2*c))*e^(2*d*x^n) - 2*(a^4*b*d*e*n*e^c + a^2*b^3*d*e*n*e^ 
c)*e^(d*x^n)) - integrate(-2*(a*b^2*e^(2*n)*x^n - (b^3*e^(2*n)*e^(n*log(x) 
 + c) + (2*a^2*b*d*e^(2*n)*e^c + b^3*d*e^(2*n)*e^c)*x^(2*n))*e^(d*x^n))/(( 
a^5*d*e*e^(2*c) + a^3*b^2*d*e*e^(2*c))*x*e^(2*d*x^n) + 2*(a^4*b*d*e*e^c + 
a^2*b^3*d*e*e^c)*x*e^(d*x^n) - (a^5*d*e + a^3*b^2*d*e)*x), x)
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

(e**(2*n)*( - 8*e**(2*x**n*d + 2*c)*sqrt(a**2 + b**2)*atan((e**(x**n*d + c 
)*a*i + b*i)/sqrt(a**2 + b**2))*a**3*b*i - 4*e**(2*x**n*d + 2*c)*sqrt(a**2 
 + b**2)*atan((e**(x**n*d + c)*a*i + b*i)/sqrt(a**2 + b**2))*a*b**3*i - 16 
*e**(x**n*d + c)*sqrt(a**2 + b**2)*atan((e**(x**n*d + c)*a*i + b*i)/sqrt(a 
**2 + b**2))*a**2*b**2*i - 8*e**(x**n*d + c)*sqrt(a**2 + b**2)*atan((e**(x 
**n*d + c)*a*i + b*i)/sqrt(a**2 + b**2))*b**4*i + 8*sqrt(a**2 + b**2)*atan 
((e**(x**n*d + c)*a*i + b*i)/sqrt(a**2 + b**2))*a**3*b*i + 4*sqrt(a**2 + b 
**2)*atan((e**(x**n*d + c)*a*i + b*i)/sqrt(a**2 + b**2))*a*b**3*i + 16*e** 
(2*x**n*d + 3*c)*int((x**(2*n)*e**(x**n*d))/(e**(4*x**n*d + 4*c)*a**2*x + 
4*e**(3*x**n*d + 3*c)*a*b*x - 2*e**(2*x**n*d + 2*c)*a**2*x + 4*e**(2*x**n* 
d + 2*c)*b**2*x - 4*e**(x**n*d + c)*a*b*x + a**2*x),x)*a**6*b*d**2*n + 24* 
e**(2*x**n*d + 3*c)*int((x**(2*n)*e**(x**n*d))/(e**(4*x**n*d + 4*c)*a**2*x 
 + 4*e**(3*x**n*d + 3*c)*a*b*x - 2*e**(2*x**n*d + 2*c)*a**2*x + 4*e**(2*x* 
*n*d + 2*c)*b**2*x - 4*e**(x**n*d + c)*a*b*x + a**2*x),x)*a**4*b**3*d**2*n 
 + 8*e**(2*x**n*d + 3*c)*int((x**(2*n)*e**(x**n*d))/(e**(4*x**n*d + 4*c)*a 
**2*x + 4*e**(3*x**n*d + 3*c)*a*b*x - 2*e**(2*x**n*d + 2*c)*a**2*x + 4*e** 
(2*x**n*d + 2*c)*b**2*x - 4*e**(x**n*d + c)*a*b*x + a**2*x),x)*a**2*b**5*d 
**2*n + x**(2*n)*e**(2*x**n*d + 2*c)*a**5*d**2 + x**(2*n)*e**(2*x**n*d + 2 
*c)*a**3*b**2*d**2 + 4*x**n*e**(2*x**n*d + 2*c)*a**3*b**2*d + 4*x**n*e**(2 
*x**n*d + 2*c)*a*b**4*d - 2*e**(2*x**n*d + 2*c)*log(e**(2*x**n*d + 2*c)...