Integrand size = 22, antiderivative size = 149 \[ \int \frac {(e x)^{-1+n}}{\left (a+b \text {csch}\left (c+d x^n\right )\right )^2} \, dx=\frac {(e x)^n}{a^2 e n}+\frac {2 b \left (2 a^2+b^2\right ) x^{-n} (e x)^n \text {arctanh}\left (\frac {a-b \tanh \left (\frac {1}{2} \left (c+d x^n\right )\right )}{\sqrt {a^2+b^2}}\right )}{a^2 \left (a^2+b^2\right )^{3/2} d e n}-\frac {b^2 x^{-n} (e x)^n \coth \left (c+d x^n\right )}{a \left (a^2+b^2\right ) d e n \left (a+b \text {csch}\left (c+d x^n\right )\right )} \]
(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))
Time = 0.92 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.12 \[ \int \frac {(e x)^{-1+n}}{\left (a+b \text {csch}\left (c+d x^n\right )\right )^2} \, dx=-\frac {x^{-n} (e x)^n \left (-a b^2 \sqrt {-a^2-b^2} \coth \left (c+d x^n\right )+\left (-\left (-a^2-b^2\right )^{3/2} \left (c+d x^n\right )-2 b \left (2 a^2+b^2\right ) \arctan \left (\frac {a-b \tanh \left (\frac {1}{2} \left (c+d x^n\right )\right )}{\sqrt {-a^2-b^2}}\right )\right ) \left (a+b \text {csch}\left (c+d x^n\right )\right )\right )}{a^2 \left (-a^2-b^2\right )^{3/2} d e n \left (a+b \text {csch}\left (c+d x^n\right )\right )} \]
-(((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])))
Time = 0.88 (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 |
| \(\displaystyle \frac {x^{-n} (e x)^n \left (\frac {\frac {\left (a^2+b^2\right ) x^n}{a}-\frac {2 b \left (2 a^2+b^2\right ) \text {arctanh}\left (\frac {b \tanh \left (\frac {1}{2} \left (c+d x^n\right )\right )}{2 \sqrt {a^2+b^2}}\right )}{a d \sqrt {a^2+b^2}}}{a \left (a^2+b^2\right )}-\frac {b^2 \coth \left (c+d x^n\right )}{a d \left (a^2+b^2\right ) \left (a+b \text {csch}\left (c+d x^n\right )\right )}\right )}{e n}\) |
((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)
3.1.81.3.1 Defintions of rubi rules used
Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a]) I nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
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])
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]
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]
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]
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]
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]
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]
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]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 5.43 (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 \operatorname {csgn}\left (i e \right ) \operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i e x \right ) \pi +i \operatorname {csgn}\left (i e \right ) \operatorname {csgn}\left (i e x \right )^{2} \pi +i \operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i e x \right )^{2} \pi -i \operatorname {csgn}\left (i e x \right )^{3} \pi +2 \ln \left (e \right )+2 \ln \left (x \right )\right )}{2}}}{a^{2} n}-\frac {2 \,{\mathrm e}^{\frac {\left (-1+n \right ) \left (-i \operatorname {csgn}\left (i e \right ) \operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i e x \right ) \pi +i \operatorname {csgn}\left (i e \right ) \operatorname {csgn}\left (i e x \right )^{2} \pi +i \operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i e x \right )^{2} \pi -i \operatorname {csgn}\left (i e x \right )^{3} \pi +2 \ln \left (e \right )+2 \ln \left (x \right )\right )}{2}} b^{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 \operatorname {csgn}\left (i e \right ) \operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i e x \right ) \pi }{2}} {\mathrm e}^{-\frac {i \operatorname {csgn}\left (i e \right ) \operatorname {csgn}\left (i e x \right )^{2} \pi }{2}} {\mathrm e}^{-\frac {i \operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i e x \right )^{2} \pi }{2}} {\mathrm e}^{\frac {i \operatorname {csgn}\left (i e x \right )^{3} \pi }{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}}}\) | \(490\) |
1/a^2/n*x*exp(1/2*(-1+n)*(-I*csgn(I*e)*csgn(I*x)*csgn(I*e*x)*Pi+I*csgn(I*e )*csgn(I*e*x)^2*Pi+I*csgn(I*x)*csgn(I*e*x)^2*Pi-I*csgn(I*e*x)^3*Pi+2*ln(e) +2*ln(x)))-2*exp(1/2*(-1+n)*(-I*csgn(I*e)*csgn(I*x)*csgn(I*e*x)*Pi+I*csgn( I*e)*csgn(I*e*x)^2*Pi+I*csgn(I*x)*csgn(I*e*x)^2*Pi-I*csgn(I*e*x)^3*Pi+2*ln (e)+2*ln(x)))*b^2*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))
Leaf count of result is larger than twice the leaf count of optimal. 1729 vs. \(2 (146) = 292\).
Time = 0.29 (sec) , antiderivative size = 1729, normalized size of antiderivative = 11.60 \[ \int \frac {(e x)^{-1+n}}{\left (a+b \text {csch}\left (c+d x^n\right )\right )^2} \, dx=\text {Too large to display} \]
-((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...
\[ \int \frac {(e x)^{-1+n}}{\left (a+b \text {csch}\left (c+d x^n\right )\right )^2} \, dx=\int \frac {\left (e x\right )^{n - 1}}{\left (a + b \operatorname {csch}{\left (c + d x^{n} \right )}\right )^{2}}\, dx \]
\[ \int \frac {(e x)^{-1+n}}{\left (a+b \text {csch}\left (c+d x^n\right )\right )^2} \, dx=\int { \frac {\left (e x\right )^{n - 1}}{{\left (b \operatorname {csch}\left (d x^{n} + c\right ) + a\right )}^{2}} \,d x } \]
-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))
\[ \int \frac {(e x)^{-1+n}}{\left (a+b \text {csch}\left (c+d x^n\right )\right )^2} \, dx=\int { \frac {\left (e x\right )^{n - 1}}{{\left (b \operatorname {csch}\left (d x^{n} + c\right ) + a\right )}^{2}} \,d x } \]
Time = 17.96 (sec) , antiderivative size = 1449, normalized size of antiderivative = 9.72 \[ \int \frac {(e x)^{-1+n}}{\left (a+b \text {csch}\left (c+d x^n\right )\right )^2} \, dx=\text {Too large to display} \]
((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 + ...