\(\int \frac {(e x)^{-1+n}}{(a+b \text {csch}(c+d x^n))^2} \, dx\) [88]

Optimal result

Integrand size = 22, antiderivative size = 149 \[ \int \frac {(e x)^{-1+n}}{\left (a+b \text {csch}\left (c+d x^n\right )\right )^2} \, dx=\frac {(e x)^n}{a^2 e n}+\frac {2 b \left (2 a^2+b^2\right ) x^{-n} (e x)^n \text {arctanh}\left (\frac {a-b \tanh \left (\frac {1}{2} \left (c+d x^n\right )\right )}{\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d e n}-\frac {b^2 x^{-n} (e x)^n \coth \left (c+d x^n\right )}{a \left (a^2+b^2\right ) d e n \left (a+b \text {csch}\left (c+d x^n\right )\right )} \] Output:

(e*x)^n/a^2/e/n+2*b*(2*a^2+b^2)*(e*x)^n*arctanh((a-b*tanh(1/2*c+1/2*d*x^n) 
)/(a^2+b^2)^(1/2))/a^2/(a^2+b^2)^(3/2)/d/e/n/(x^n)-b^2*(e*x)^n*coth(c+d*x^ 
n)/a/(a^2+b^2)/d/e/n/(x^n)/(a+b*csch(c+d*x^n))
 

Mathematica [A] (verified)

Time = 0.86 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.12 \[ \int \frac {(e x)^{-1+n}}{\left (a+b \text {csch}\left (c+d x^n\right )\right )^2} \, dx=-\frac {x^{-n} (e x)^n \left (-a b^2 \sqrt {-a^2-b^2} \coth \left (c+d x^n\right )+\left (-\left (-a^2-b^2\right )^{3/2} \left (c+d x^n\right )-2 b \left (2 a^2+b^2\right ) \arctan \left (\frac {a-b \tanh \left (\frac {1}{2} \left (c+d x^n\right )\right )}{\sqrt {-a^2-b^2}}\right )\right ) \left (a+b \text {csch}\left (c+d x^n\right )\right )\right )}{a^2 \left (-a^2-b^2\right )^{3/2} d e n \left (a+b \text {csch}\left (c+d x^n\right )\right )} \] Input:

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

Output:

-(((e*x)^n*(-(a*b^2*Sqrt[-a^2 - b^2]*Coth[c + d*x^n]) + (-((-a^2 - b^2)^(3 
/2)*(c + d*x^n)) - 2*b*(2*a^2 + b^2)*ArcTan[(a - b*Tanh[(c + d*x^n)/2])/Sq 
rt[-a^2 - b^2]])*(a + b*Csch[c + d*x^n])))/(a^2*(-a^2 - b^2)^(3/2)*d*e*n*x 
^n*(a + b*Csch[c + d*x^n])))
 

Rubi [A] (warning: unable to verify)

Time = 0.89 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.98, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.682, Rules used = {5964, 5960, 3042, 4272, 25, 3042, 4407, 26, 3042, 26, 4318, 3042, 3139, 1083, 217}

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.

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

\(\Big \downarrow \) 5964

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

\(\Big \downarrow \) 5960

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4272

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4407

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 4318

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3139

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

\(\Big \downarrow \) 1083

\(\displaystyle \frac {x^{-n} (e x)^n \left (-\frac {b^2 \coth \left (c+d x^n\right )}{a d \left (a^2+b^2\right ) \left (a+b \text {csch}\left (c+d x^n\right )\right )}+\frac {\frac {\left (a^2+b^2\right ) x^n}{a}-\frac {4 i \left (2 a^2+b^2\right ) \int \frac {1}{-x^{2 n}-4 \left (\frac {a^2}{b^2}+1\right )}d\left (2 i \tanh \left (\frac {1}{2} \left (d x^n+c\right )\right )-\frac {2 i a}{b}\right )}{a d}}{a \left (a^2+b^2\right )}\right )}{e n}\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {x^{-n} (e x)^n \left (\frac {\frac {\left (a^2+b^2\right ) x^n}{a}-\frac {2 b \left (2 a^2+b^2\right ) \text {arctanh}\left (\frac {b \tanh \left (\frac {1}{2} \left (c+d x^n\right )\right )}{2 \sqrt {a^2+b^2}}\right )}{a d \sqrt {a^2+b^2}}}{a \left (a^2+b^2\right )}-\frac {b^2 \coth \left (c+d x^n\right )}{a d \left (a^2+b^2\right ) \left (a+b \text {csch}\left (c+d x^n\right )\right )}\right )}{e n}\)

Input:

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

Output:

((e*x)^n*((((a^2 + b^2)*x^n)/a - (2*b*(2*a^2 + b^2)*ArcTanh[(b*Tanh[(c + d 
*x^n)/2])/(2*Sqrt[a^2 + b^2])])/(a*Sqrt[a^2 + b^2]*d))/(a*(a^2 + b^2)) - ( 
b^2*Coth[c + d*x^n])/(a*(a^2 + b^2)*d*(a + b*Csch[c + d*x^n]))))/(e*n*x^n)
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 26

Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a])   I 
nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
 

rule 217

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( 
-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & 
& (LtQ[a, 0] || LtQ[b, 0])
 

rule 1083

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Simp[-2   Subst[I 
nt[1/Simp[b^2 - 4*a*c - x^2, x], x], x, b + 2*c*x], x] /; FreeQ[{a, b, c}, 
x]
 

rule 3042

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

rule 3139

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{e = Fre 
eFactors[Tan[(c + d*x)/2], x]}, Simp[2*(e/d)   Subst[Int[1/(a + 2*b*e*x + a 
*e^2*x^2), x], x, Tan[(c + d*x)/2]/e], x]] /; FreeQ[{a, b, c, d}, x] && NeQ 
[a^2 - b^2, 0]
 

rule 4272

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_))^(n_), x_Symbol] :> Simp[b^2*Cot[ 
c + d*x]*((a + b*Csc[c + d*x])^(n + 1)/(a*d*(n + 1)*(a^2 - b^2))), x] + Sim 
p[1/(a*(n + 1)*(a^2 - b^2))   Int[(a + b*Csc[c + d*x])^(n + 1)*Simp[(a^2 - 
b^2)*(n + 1) - a*b*(n + 1)*Csc[c + d*x] + b^2*(n + 2)*Csc[c + d*x]^2, x], x 
], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && LtQ[n, -1] && Integ 
erQ[2*n]
 

rule 4318

Int[csc[(e_.) + (f_.)*(x_)]/(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbo 
l] :> Simp[1/b   Int[1/(1 + (a/b)*Sin[e + f*x]), x], x] /; FreeQ[{a, b, e, 
f}, x] && NeQ[a^2 - b^2, 0]
 

rule 4407

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.) + (c_))/(csc[(e_.) + (f_.)*(x_)]*(b_.) + 
 (a_)), x_Symbol] :> Simp[c*(x/a), x] - Simp[(b*c - a*d)/a   Int[Csc[e + f* 
x]/(a + b*Csc[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c 
- a*d, 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 [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 5.82 (sec) , antiderivative size = 490, normalized size of antiderivative = 3.29

method	result	size
risch	\(\frac {x \,{\mathrm e}^{\frac {\left (-1+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}}}{a^{2} n}-\frac {2 b^{2} {\mathrm e}^{\frac {\left (-1+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}} x \left (-b \,{\mathrm e}^{c +d \,x^{n}}+a \right ) x^{-n}}{a^{2} \left (a^{2}+b^{2}\right ) d n \left (a \,{\mathrm e}^{2 c +2 d \,x^{n}}+2 b \,{\mathrm e}^{c +d \,x^{n}}-a \right )}-\frac {2 b \left (2 a^{2}+b^{2}\right ) {\mathrm e}^{-\frac {i \pi n \,\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 n \,\operatorname {csgn}\left (i e \right ) \operatorname {csgn}\left (i e x \right )^{2}}{2}} {\mathrm e}^{\frac {i \pi n \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i e x \right )^{2}}{2}} {\mathrm e}^{-\frac {i \pi n \operatorname {csgn}\left (i e x \right )^{3}}{2}} {\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^{n} {\mathrm e}^{c} \arctan \left (\frac {2 a \,{\mathrm e}^{2 c +d \,x^{n}}+2 \,{\mathrm e}^{c} b}{2 \sqrt {-a^{2} {\mathrm e}^{2 c}-{\mathrm e}^{2 c} b^{2}}}\right )}{a^{2} \left (a^{2}+b^{2}\right ) n e d \sqrt {-a^{2} {\mathrm e}^{2 c}-{\mathrm e}^{2 c} b^{2}}}\)	\(490\)

Input:

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

Output:

1/a^2/n*x*exp(1/2*(-1+n)*(-I*Pi*csgn(I*e)*csgn(I*x)*csgn(I*e*x)+I*Pi*csgn( 
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^2*exp(1/2*(-1+n)*(-I*Pi*csgn(I*e)*csgn(I*x)*csgn(I*e*x)+I*P 
i*csgn(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)))*x*(-b*exp(c+d*x^n)+a)/a^2/(a^2+b^2)/d/n/(x^n)/(a*exp(2*c 
+2*d*x^n)+2*b*exp(c+d*x^n)-a)-2*b/a^2*(2*a^2+b^2)/(a^2+b^2)/n*exp(-1/2*I*P 
i*n*csgn(I*e)*csgn(I*x)*csgn(I*e*x))*exp(1/2*I*Pi*n*csgn(I*e)*csgn(I*e*x)^ 
2)*exp(1/2*I*Pi*n*csgn(I*x)*csgn(I*e*x)^2)*exp(-1/2*I*Pi*n*csgn(I*e*x)^3)* 
exp(1/2*I*Pi*csgn(I*e)*csgn(I*x)*csgn(I*e*x))*exp(-1/2*I*Pi*csgn(I*e)*csgn 
(I*e*x)^2)*exp(-1/2*I*Pi*csgn(I*x)*csgn(I*e*x)^2)*exp(1/2*I*Pi*csgn(I*e*x) 
^3)*e^n/e*exp(c)/d/(-a^2*exp(2*c)-exp(2*c)*b^2)^(1/2)*arctan(1/2*(2*a*exp( 
2*c+d*x^n)+2*exp(c)*b)/(-a^2*exp(2*c)-exp(2*c)*b^2)^(1/2))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1729 vs. \(2 (146) = 292\).

Time = 0.12 (sec) , antiderivative size = 1729, normalized size of antiderivative = 11.60 \[ \int \frac {(e x)^{-1+n}}{\left (a+b \text {csch}\left (c+d x^n\right )\right )^2} \, dx=\text {Too large to display} \] Input:

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

Output:

-((a^5 + 2*a^3*b^2 + a*b^4)*d*cosh((n - 1)*log(e))*cosh(n*log(x)) - ((a^5 
+ 2*a^3*b^2 + a*b^4)*d*cosh((n - 1)*log(e))*cosh(n*log(x)) + (a^5 + 2*a^3* 
b^2 + a*b^4)*d*cosh(n*log(x))*sinh((n - 1)*log(e)) + ((a^5 + 2*a^3*b^2 + a 
*b^4)*d*cosh((n - 1)*log(e)) + (a^5 + 2*a^3*b^2 + a*b^4)*d*sinh((n - 1)*lo 
g(e)))*sinh(n*log(x)))*cosh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c)^2 - ( 
(a^5 + 2*a^3*b^2 + a*b^4)*d*cosh((n - 1)*log(e))*cosh(n*log(x)) + (a^5 + 2 
*a^3*b^2 + a*b^4)*d*cosh(n*log(x))*sinh((n - 1)*log(e)) + ((a^5 + 2*a^3*b^ 
2 + a*b^4)*d*cosh((n - 1)*log(e)) + (a^5 + 2*a^3*b^2 + a*b^4)*d*sinh((n - 
1)*log(e)))*sinh(n*log(x)))*sinh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c)^ 
2 - 2*((a^4*b + 2*a^2*b^3 + b^5)*d*cosh((n - 1)*log(e))*cosh(n*log(x)) + ( 
a^2*b^3 + b^5)*cosh((n - 1)*log(e)) + (a^2*b^3 + b^5 + (a^4*b + 2*a^2*b^3 
+ b^5)*d*cosh(n*log(x)))*sinh((n - 1)*log(e)) + ((a^4*b + 2*a^2*b^3 + b^5) 
*d*cosh((n - 1)*log(e)) + (a^4*b + 2*a^2*b^3 + b^5)*d*sinh((n - 1)*log(e)) 
)*sinh(n*log(x)))*cosh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c) + 2*(a^3*b 
^2 + a*b^4)*cosh((n - 1)*log(e)) - (((2*a^3*b + a*b^3)*sqrt(a^2 + b^2)*cos 
h((n - 1)*log(e)) + (2*a^3*b + a*b^3)*sqrt(a^2 + b^2)*sinh((n - 1)*log(e)) 
)*cosh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c)^2 + ((2*a^3*b + a*b^3)*sqr 
t(a^2 + b^2)*cosh((n - 1)*log(e)) + (2*a^3*b + a*b^3)*sqrt(a^2 + b^2)*sinh 
((n - 1)*log(e)))*sinh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c)^2 - (2*a^3 
*b + a*b^3)*sqrt(a^2 + b^2)*cosh((n - 1)*log(e)) - (2*a^3*b + a*b^3)*sq...
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [B] (verification not implemented)

Time = 18.80 (sec) , antiderivative size = 1449, normalized size of antiderivative = 9.72 \[ \int \frac {(e x)^{-1+n}}{\left (a+b \text {csch}\left (c+d x^n\right )\right )^2} \, dx=\text {Too large to display} \] Input:

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

Output:

((2*atan(((a^5*(- a^10*d^2*n^2*x^(2*n) - a^4*b^6*d^2*n^2*x^(2*n) - 3*a^6*b 
^4*d^2*n^2*x^(2*n) - 3*a^8*b^2*d^2*n^2*x^(2*n))^(1/2))/2 + (a^3*b^2*(- a^1 
0*d^2*n^2*x^(2*n) - a^4*b^6*d^2*n^2*x^(2*n) - 3*a^6*b^4*d^2*n^2*x^(2*n) - 
3*a^8*b^2*d^2*n^2*x^(2*n))^(1/2))/2)*(exp(d*x^n)*exp(c)*((2*(e*x)^(1 - n)* 
(a^4*b*d*n*x^n*(b^6*x^2*(e*x)^(2*n - 2) + 4*a^2*b^4*x^2*(e*x)^(2*n - 2) + 
4*a^4*b^2*x^2*(e*x)^(2*n - 2))^(1/2) + a^2*b^3*d*n*x^n*(b^6*x^2*(e*x)^(2*n 
 - 2) + 4*a^2*b^4*x^2*(e*x)^(2*n - 2) + 4*a^4*b^2*x^2*(e*x)^(2*n - 2))^(1/ 
2)))/(a^2*x*(a^4 + a^2*b^2)*(2*a^2 + b^2)*(- a^10*d^2*n^2*x^(2*n) - a^4*b^ 
6*d^2*n^2*x^(2*n) - 3*a^6*b^4*d^2*n^2*x^(2*n) - 3*a^8*b^2*d^2*n^2*x^(2*n)) 
^(1/2)*(-a^4*d^2*n^2*x^(2*n)*(a^2 + b^2)^3)^(1/2)) + (2*(b^3*x*(e*x)^(n - 
1)*(- a^10*d^2*n^2*x^(2*n) - a^4*b^6*d^2*n^2*x^(2*n) - 3*a^6*b^4*d^2*n^2*x 
^(2*n) - 3*a^8*b^2*d^2*n^2*x^(2*n))^(1/2) + 2*a^2*b*x*(e*x)^(n - 1)*(- a^1 
0*d^2*n^2*x^(2*n) - a^4*b^6*d^2*n^2*x^(2*n) - 3*a^6*b^4*d^2*n^2*x^(2*n) - 
3*a^8*b^2*d^2*n^2*x^(2*n))^(1/2)))/(a^4*d*n*x^n*(a^4 + a^2*b^2)*(a^2 + b^2 
)*(b^2*x^2*(e*x)^(2*n - 2)*(2*a^2 + b^2)^2)^(1/2)*(- a^10*d^2*n^2*x^(2*n) 
- a^4*b^6*d^2*n^2*x^(2*n) - 3*a^6*b^4*d^2*n^2*x^(2*n) - 3*a^8*b^2*d^2*n^2* 
x^(2*n))^(1/2))) - (2*(e*x)^(1 - n)*(a^5*d*n*x^n*(b^6*x^2*(e*x)^(2*n - 2) 
+ 4*a^2*b^4*x^2*(e*x)^(2*n - 2) + 4*a^4*b^2*x^2*(e*x)^(2*n - 2))^(1/2) + a 
^3*b^2*d*n*x^n*(b^6*x^2*(e*x)^(2*n - 2) + 4*a^2*b^4*x^2*(e*x)^(2*n - 2) + 
4*a^4*b^2*x^2*(e*x)^(2*n - 2))^(1/2)))/(a^2*x*(a^4 + a^2*b^2)*(2*a^2 + ...
 

Reduce [B] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 645, normalized size of antiderivative = 4.33 \[ \int \frac {(e x)^{-1+n}}{\left (a+b \text {csch}\left (c+d x^n\right )\right )^2} \, dx=\frac {e^{n} \left (-4 e^{2 x^{n} d +2 c} \sqrt {a^{2}+b^{2}}\, \mathit {atan} \left (\frac {e^{x^{n} d +c} a i +b i}{\sqrt {a^{2}+b^{2}}}\right ) a^{3} b i -2 e^{2 x^{n} d +2 c} \sqrt {a^{2}+b^{2}}\, \mathit {atan} \left (\frac {e^{x^{n} d +c} a i +b i}{\sqrt {a^{2}+b^{2}}}\right ) a \,b^{3} i -8 e^{x^{n} d +c} \sqrt {a^{2}+b^{2}}\, \mathit {atan} \left (\frac {e^{x^{n} d +c} a i +b i}{\sqrt {a^{2}+b^{2}}}\right ) a^{2} b^{2} i -4 e^{x^{n} d +c} \sqrt {a^{2}+b^{2}}\, \mathit {atan} \left (\frac {e^{x^{n} d +c} a i +b i}{\sqrt {a^{2}+b^{2}}}\right ) b^{4} i +4 \sqrt {a^{2}+b^{2}}\, \mathit {atan} \left (\frac {e^{x^{n} d +c} a i +b i}{\sqrt {a^{2}+b^{2}}}\right ) a^{3} b i +2 \sqrt {a^{2}+b^{2}}\, \mathit {atan} \left (\frac {e^{x^{n} d +c} a i +b i}{\sqrt {a^{2}+b^{2}}}\right ) a \,b^{3} i +x^{n} e^{2 x^{n} d +2 c} a^{5} d +2 x^{n} e^{2 x^{n} d +2 c} a^{3} b^{2} d +x^{n} e^{2 x^{n} d +2 c} a \,b^{4} d -e^{2 x^{n} d +2 c} a^{3} b^{2}-e^{2 x^{n} d +2 c} a \,b^{4}+2 x^{n} e^{x^{n} d +c} a^{4} b d +4 x^{n} e^{x^{n} d +c} a^{2} b^{3} d +2 x^{n} e^{x^{n} d +c} b^{5} d -x^{n} a^{5} d -2 x^{n} a^{3} b^{2} d -x^{n} a \,b^{4} d -a^{3} b^{2}-a \,b^{4}\right )}{a^{2} d e n \left (e^{2 x^{n} d +2 c} a^{5}+2 e^{2 x^{n} d +2 c} a^{3} b^{2}+e^{2 x^{n} d +2 c} a \,b^{4}+2 e^{x^{n} d +c} a^{4} b +4 e^{x^{n} d +c} a^{2} b^{3}+2 e^{x^{n} d +c} b^{5}-a^{5}-2 a^{3} b^{2}-a \,b^{4}\right )} \] Input:

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

Output:

(e**n*( - 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**3*b*i - 2*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 - 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))*a**2*b**2*i - 4*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 + 4*sqrt(a**2 + b**2)*atan((e** 
(x**n*d + c)*a*i + b*i)/sqrt(a**2 + b**2))*a**3*b*i + 2*sqrt(a**2 + b**2)* 
atan((e**(x**n*d + c)*a*i + b*i)/sqrt(a**2 + b**2))*a*b**3*i + x**n*e**(2* 
x**n*d + 2*c)*a**5*d + 2*x**n*e**(2*x**n*d + 2*c)*a**3*b**2*d + x**n*e**(2 
*x**n*d + 2*c)*a*b**4*d - e**(2*x**n*d + 2*c)*a**3*b**2 - e**(2*x**n*d + 2 
*c)*a*b**4 + 2*x**n*e**(x**n*d + c)*a**4*b*d + 4*x**n*e**(x**n*d + c)*a**2 
*b**3*d + 2*x**n*e**(x**n*d + c)*b**5*d - x**n*a**5*d - 2*x**n*a**3*b**2*d 
 - x**n*a*b**4*d - a**3*b**2 - a*b**4))/(a**2*d*e*n*(e**(2*x**n*d + 2*c)*a 
**5 + 2*e**(2*x**n*d + 2*c)*a**3*b**2 + e**(2*x**n*d + 2*c)*a*b**4 + 2*e** 
(x**n*d + c)*a**4*b + 4*e**(x**n*d + c)*a**2*b**3 + 2*e**(x**n*d + c)*b**5 
 - a**5 - 2*a**3*b**2 - a*b**4))