\(\int \frac {(e x)^{-1+n}}{(a+b \text {csch}(c+d x^n))^2} \, dx\) [81]

Optimal result

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

[Out]

(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))

Rubi [A] (verified)

Time = 0.21 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {5549, 5545, 3870, 4004, 3916, 2739, 632, 210} \[ \int \frac {(e x)^{-1+n}}{\left (a+b \text {csch}\left (c+d x^n\right )\right )^2} \, dx=\frac {2 b \left (2 a^2+b^2\right ) x^{-n} (e x)^n \text {arctanh}\left (\frac {a-b \tanh \left (\frac {1}{2} \left (c+d x^n\right )\right )}{\sqrt {a^2+b^2}}\right )}{a^2 d e n \left (a^2+b^2\right )^{3/2}}-\frac {b^2 x^{-n} (e x)^n \coth \left (c+d x^n\right )}{a d e n \left (a^2+b^2\right ) \left (a+b \text {csch}\left (c+d x^n\right )\right )}+\frac {(e x)^n}{a^2 e n} \]

[In]

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

[Out]

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

Rule 210

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 632

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 2739

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{e = FreeFactors[Tan[(c + d*x)/2], x]}, Dis
t[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 3870

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] + Dist[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] && IntegerQ[2*n]

Rule 3916

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

Rule 4004

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.) + (c_))/(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Simp[c*(x/a),
x] - Dist[(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 5545

Int[((a_.) + Csch[(c_.) + (d_.)*(x_)^(n_)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simpli
fy[(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[Simplif
y[(m + 1)/n], 0] && IntegerQ[p]

Rule 5549

Int[((a_.) + Csch[(c_.) + (d_.)*(x_)^(n_)]*(b_.))^(p_.)*((e_)*(x_))^(m_.), x_Symbol] :> Dist[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]

Rubi steps \begin{align*} \text {integral}& = \frac {\left (x^{-n} (e x)^n\right ) \int \frac {x^{-1+n}}{\left (a+b \text {csch}\left (c+d x^n\right )\right )^2} \, dx}{e} \\ & = \frac {\left (x^{-n} (e x)^n\right ) \text {Subst}\left (\int \frac {1}{(a+b \text {csch}(c+d x))^2} \, dx,x,x^n\right )}{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 )}-\frac {\left (x^{-n} (e x)^n\right ) \text {Subst}\left (\int \frac {-a^2-b^2+a b \text {csch}(c+d x)}{a+b \text {csch}(c+d x)} \, dx,x,x^n\right )}{a \left (a^2+b^2\right ) e n} \\ & = \frac {(e x)^n}{a^2 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 )}-\frac {\left (i \left (-i a^2 b+i b \left (-a^2-b^2\right )\right ) x^{-n} (e x)^n\right ) \text {Subst}\left (\int \frac {\text {csch}(c+d x)}{a+b \text {csch}(c+d x)} \, dx,x,x^n\right )}{a^2 \left (a^2+b^2\right ) e n} \\ & = \frac {(e x)^n}{a^2 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 )}-\frac {\left (i \left (-i a^2 b+i b \left (-a^2-b^2\right )\right ) x^{-n} (e x)^n\right ) \text {Subst}\left (\int \frac {1}{1+\frac {a \sinh (c+d x)}{b}} \, dx,x,x^n\right )}{a^2 b \left (a^2+b^2\right ) e n} \\ & = \frac {(e x)^n}{a^2 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 )}-\frac {\left (2 \left (-i a^2 b+i b \left (-a^2-b^2\right )\right ) x^{-n} (e x)^n\right ) \text {Subst}\left (\int \frac {1}{1-\frac {2 i a x}{b}+x^2} \, dx,x,i \tanh \left (\frac {1}{2} \left (c+d x^n\right )\right )\right )}{a^2 b \left (a^2+b^2\right ) d e n} \\ & = \frac {(e x)^n}{a^2 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 )}+\frac {\left (4 \left (-i a^2 b+i b \left (-a^2-b^2\right )\right ) x^{-n} (e x)^n\right ) \text {Subst}\left (\int \frac {1}{-4 \left (1+\frac {a^2}{b^2}\right )-x^2} \, dx,x,-\frac {2 i a}{b}+2 i \tanh \left (\frac {1}{2} \left (c+d x^n\right )\right )\right )}{a^2 b \left (a^2+b^2\right ) d e n} \\ & = \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 {b \left (\frac {a}{b}-\tanh \left (\frac {1}{2} \left (c+d x^n\right )\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 )} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.92 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.12 \[ \int \frac {(e x)^{-1+n}}{\left (a+b \text {csch}\left (c+d x^n\right )\right )^2} \, dx=-\frac {x^{-n} (e x)^n \left (-a b^2 \sqrt {-a^2-b^2} \coth \left (c+d x^n\right )+\left (-\left (-a^2-b^2\right )^{3/2} \left (c+d x^n\right )-2 b \left (2 a^2+b^2\right ) \arctan \left (\frac {a-b \tanh \left (\frac {1}{2} \left (c+d x^n\right )\right )}{\sqrt {-a^2-b^2}}\right )\right ) \left (a+b \text {csch}\left (c+d x^n\right )\right )\right )}{a^2 \left (-a^2-b^2\right )^{3/2} d e n \left (a+b \text {csch}\left (c+d x^n\right )\right )} \]

[In]

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

[Out]

-(((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])/Sqrt[-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])))

Maple [C] (warning: unable to verify)

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}}}\)	\(490\)

[In]

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

[Out]

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*csg
n(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*Pi*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 = 0.29 (sec) , antiderivative size = 1729, normalized size of antiderivative = 11.60 \[ \int \frac {(e x)^{-1+n}}{\left (a+b \text {csch}\left (c+d x^n\right )\right )^2} \, dx=\text {Too large to display} \]

[In]

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

[Out]

-((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)*log(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)*lo
g(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*cos
h(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)*cosh((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)*sqrt(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)*sqrt(a^2 + b^2)*sinh((n - 1)*log(e)) + 2*((2*a^2*b^2 + b^4)*sqrt(
a^2 + b^2)*cosh((n - 1)*log(e)) + (2*a^2*b^2 + b^4)*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^2*b^2 + b^4)*sqrt(a^2 + b^2)*cosh((n - 1)*log(e)) + (2*a^2*b^2 + b^4)*sqrt
(a^2 + b^2)*sinh((n - 1)*log(e)) + ((2*a^3*b + a*b^3)*sqrt(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)))*cosh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c))*sinh(d*cosh(n*log(x)) +
d*sinh(n*log(x)) + c))*log((a*b + (a^2 + b^2 + sqrt(a^2 + b^2)*b)*cosh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c
) - (b^2 + sqrt(a^2 + b^2)*b)*sinh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c) + sqrt(a^2 + b^2)*a)/(a*sinh(d*cos
h(n*log(x)) + d*sinh(n*log(x)) + c) + b)) - 2*((a^4*b + 2*a^2*b^3 + b^5)*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)*log(e)))*sinh(n*log(x)))*cosh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c) + (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)))*sinh(d*
cosh(n*log(x)) + d*sinh(n*log(x)) + c) + (2*a^3*b^2 + 2*a*b^4 + (a^5 + 2*a^3*b^2 + a*b^4)*d*cosh(n*log(x)))*si
nh((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)))/((a^7 + 2*a^5*b^2 + a^3*b^4)*d*n*cosh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c)^2
+ (a^7 + 2*a^5*b^2 + a^3*b^4)*d*n*sinh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c)^2 + 2*(a^6*b + 2*a^4*b^3 + a^2
*b^5)*d*n*cosh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c) - (a^7 + 2*a^5*b^2 + a^3*b^4)*d*n + 2*((a^7 + 2*a^5*b^
2 + a^3*b^4)*d*n*cosh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c) + (a^6*b + 2*a^4*b^3 + a^2*b^5)*d*n)*sinh(d*cos
h(n*log(x)) + d*sinh(n*log(x)) + c))

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

-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]

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

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 17.96 (sec) , antiderivative size = 1449, normalized size of antiderivative = 9.72 \[ \int \frac {(e x)^{-1+n}}{\left (a+b \text {csch}\left (c+d x^n\right )\right )^2} \, dx=\text {Too large to display} \]

[In]

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

[Out]

((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^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)*(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^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*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 + 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*atan(
(x*(e*x)^(n - 1)*(2*a^2 + b^2)*(-a^4*d^2*n^2*x^(2*n)*(a^2 + b^2)^3)^(1/2))/(a^2*d*n*x^n*(a^2 + b^2)*(b^2*x^2*(
e*x)^(2*n - 2)*(2*a^2 + b^2)^2)^(1/2))))*(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^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*b^2*x^2*(e*x)^(n - 1))/(d*n*x^n*(a^3*x + a*b^2*x)) - (2*b^3*x^2*exp(c + d
*x^n)*(e*x)^(n - 1))/(a*d*n*x^n*(a^3*x + a*b^2*x)))/(a*exp(2*c + 2*d*x^n) - a + 2*b*exp(c + d*x^n)) + (x*(e*x)
^(n - 1))/(a^2*n)