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

Optimal. Leaf size=149 \[ \frac{2 b \left (2 a^2+b^2\right ) x^{-n} (e x)^n \tanh ^{-1}\left (\frac{a-b \tanh \left (\frac{1}{2} \left (c+d x^n\right )\right )}{\sqrt{a^2+b^2}}\right )}{a^2 d e n \left (a^2+b^2\right )^{3/2}}-\frac{b^2 x^{-n} (e x)^n \coth \left (c+d x^n\right )}{a d e n \left (a^2+b^2\right ) \left (a+b \text{csch}\left (c+d x^n\right )\right )}+\frac{(e x)^n}{a^2 e n} \]

[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]))

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Rubi [A]  time = 0.289806, antiderivative size = 149, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364, Rules used = {5441, 5437, 3785, 3919, 3831, 2660, 618, 204} \[ \frac{2 b \left (2 a^2+b^2\right ) x^{-n} (e x)^n \tanh ^{-1}\left (\frac{a-b \tanh \left (\frac{1}{2} \left (c+d x^n\right )\right )}{\sqrt{a^2+b^2}}\right )}{a^2 d e n \left (a^2+b^2\right )^{3/2}}-\frac{b^2 x^{-n} (e x)^n \coth \left (c+d x^n\right )}{a d e n \left (a^2+b^2\right ) \left (a+b \text{csch}\left (c+d x^n\right )\right )}+\frac{(e x)^n}{a^2 e n} \]

Antiderivative was successfully verified.

[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 5441

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]

Rule 5437

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 3785

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 3919

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 3831

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

Rule 2660

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 618

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 204

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

Rubi steps

\begin{align*} \int \frac{(e x)^{-1+n}}{\left (a+b \text{csch}\left (c+d x^n\right )\right )^2} \, dx &=\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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 \tanh ^{-1}\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]  time = 0.576248, size = 167, normalized size = 1.12 \[ -\frac{x^{-n} (e x)^n \left (\left (a+b \text{csch}\left (c+d x^n\right )\right ) \left (\left (-a^2-b^2\right )^{3/2} \left (-\left (c+d x^n\right )\right )-2 b \left (2 a^2+b^2\right ) \tan ^{-1}\left (\frac{a-b \tanh \left (\frac{1}{2} \left (c+d x^n\right )\right )}{\sqrt{-a^2-b^2}}\right )\right )-a b^2 \sqrt{-a^2-b^2} \coth \left (c+d x^n\right )\right )}{a^2 d e n \left (-a^2-b^2\right )^{3/2} \left (a+b \text{csch}\left (c+d x^n\right )\right )} \]

Antiderivative was successfully verified.

[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])))

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Maple [C]  time = 0.151, size = 490, normalized size = 3.3 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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)*cs
gn(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
*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)))*x*(-b*exp(c+d*x^
n)+a)/a^2/(a^2+b^2)/d/n/(x^n)/(2*b*exp(c+d*x^n)+exp(2*c+2*d*x^n)*a-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))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -2 \,{\left (2 \, a^{2} b e^{n} e^{c} + b^{3} e^{n} e^{c}\right )} \int \frac{e^{\left (d x^{n} + n \log \left (x\right )\right )}}{{\left (a^{5} e e^{\left (2 \, c\right )} + a^{3} b^{2} e e^{\left (2 \, c\right )}\right )} x e^{\left (2 \, d x^{n}\right )} + 2 \,{\left (a^{4} b e e^{c} + a^{2} b^{3} e e^{c}\right )} x e^{\left (d x^{n}\right )} -{\left (a^{5} e + a^{3} b^{2} e\right )} x}\,{d x} + \frac{2 \, a b^{2} e^{n} +{\left (a^{3} d e^{n} + a b^{2} d e^{n}\right )} x^{n} -{\left (a^{3} d e^{n} e^{\left (2 \, c\right )} + a b^{2} d e^{n} e^{\left (2 \, c\right )}\right )} e^{\left (2 \, d x^{n} + n \log \left (x\right )\right )} - 2 \,{\left (b^{3} e^{n} e^{c} +{\left (a^{2} b d e^{n} e^{c} + b^{3} d e^{n} e^{c}\right )} x^{n}\right )} e^{\left (d x^{n}\right )}}{a^{5} d e n + a^{3} b^{2} d e n -{\left (a^{5} d e n e^{\left (2 \, c\right )} + a^{3} b^{2} d e n e^{\left (2 \, c\right )}\right )} e^{\left (2 \, d x^{n}\right )} - 2 \,{\left (a^{4} b d e n e^{c} + a^{2} b^{3} d e n e^{c}\right )} e^{\left (d x^{n}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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Fricas [B]  time = 2.45616, size = 4803, normalized size = 32.23 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (e x\right )^{n - 1}}{\left (a + b \operatorname{csch}{\left (c + d x^{n} \right )}\right )^{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (e x\right )^{n - 1}}{{\left (b \operatorname{csch}\left (d x^{n} + c\right ) + a\right )}^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)