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

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

[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))

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Rubi [A]
time = 0.21, antiderivative size = 149, normalized size of antiderivative = 1.00, 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 = {5549, 5545, 3870, 4004, 3916, 2739, 632, 210} \begin {gather*} \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} \end {gather*}

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 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*} \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 ) \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 \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*}

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Mathematica [A]
time = 0.41, size = 167, normalized size = 1.12 \begin {gather*} -\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 ) \text {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 )} \end {gather*}

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] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 4.96, size = 490, normalized size = 3.29

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

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,method=_RETURNVERBOSE)

[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)/(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)-b^2*exp(2*c))^(1/2)*arctan(1/2*(2*a*exp(2*c+d*x^n)+2*exp(c)*b)/(-a^2*exp(2*c)-b^2*exp(2*c))^(1/2))

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1552 vs. \(2 (146) = 292\).
time = 0.39, size = 1552, normalized size = 10.42 \begin {gather*} \text {Too large to display} \end {gather*}

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) + (a^5 + 2*a^3*b^2 + a*b^4)*d*sinh(n - 1))*cosh(n*log(x)) + ((a^5 +
2*a^3*b^2 + a*b^4)*d*cosh(n - 1) + (a^5 + 2*a^3*b^2 + a*b^4)*d*sinh(n - 1))*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) + (a^5 + 2*a^3*b^2 + a*b^4)*d*sinh(
n - 1))*cosh(n*log(x)) + ((a^5 + 2*a^3*b^2 + a*b^4)*d*cosh(n - 1) + (a^5 + 2*a^3*b^2 + a*b^4)*d*sinh(n - 1))*s
inh(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) +
(a^4*b + 2*a^2*b^3 + b^5)*d*sinh(n - 1))*cosh(n*log(x)) + (a^2*b^3 + b^5)*cosh(n - 1) + ((a^4*b + 2*a^2*b^3 +
b^5)*d*cosh(n - 1) + (a^4*b + 2*a^2*b^3 + b^5)*d*sinh(n - 1))*sinh(n*log(x)) + (a^2*b^3 + b^5)*sinh(n - 1))*c
osh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c) - ((a^5 + 2*a^3*b^2 + a*b^4)*d*cosh(n - 1) + (a^5 + 2*a^3*b^2 + a
*b^4)*d*sinh(n - 1))*cosh(n*log(x)) - 2*(a^3*b^2 + a*b^4)*cosh(n - 1) + (sqrt(a^2 + b^2)*((2*a^3*b + a*b^3)*co
sh(n - 1) + (2*a^3*b + a*b^3)*sinh(n - 1))*cosh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c)^2 + sqrt(a^2 + b^2)*(
(2*a^3*b + a*b^3)*cosh(n - 1) + (2*a^3*b + a*b^3)*sinh(n - 1))*sinh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c)^2
+ 2*sqrt(a^2 + b^2)*((2*a^2*b^2 + b^4)*cosh(n - 1) + (2*a^2*b^2 + b^4)*sinh(n - 1))*cosh(d*cosh(n*log(x)) + d
*sinh(n*log(x)) + c) + 2*(sqrt(a^2 + b^2)*((2*a^3*b + a*b^3)*cosh(n - 1) + (2*a^3*b + a*b^3)*sinh(n - 1))*cosh
(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c) + sqrt(a^2 + b^2)*((2*a^2*b^2 + b^4)*cosh(n - 1) + (2*a^2*b^2 + b^4)
*sinh(n - 1)))*sinh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c) - sqrt(a^2 + b^2)*((2*a^3*b + a*b^3)*cosh(n - 1)
+ (2*a^3*b + a*b^3)*sinh(n - 1)))*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*cosh(n*log(x)) + d*sinh(n*log(x)) + c) + b)) + 2*((((a^5 + 2*a^3*b^2 + a*b^4)*d*cosh(n - 1) + (a^5 + 2
*a^3*b^2 + a*b^4)*d*sinh(n - 1))*cosh(n*log(x)) + ((a^5 + 2*a^3*b^2 + a*b^4)*d*cosh(n - 1) + (a^5 + 2*a^3*b^2
+ a*b^4)*d*sinh(n - 1))*sinh(n*log(x)))*cosh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c) + ((a^4*b + 2*a^2*b^3 +
b^5)*d*cosh(n - 1) + (a^4*b + 2*a^2*b^3 + b^5)*d*sinh(n - 1))*cosh(n*log(x)) + (a^2*b^3 + b^5)*cosh(n - 1) + (
(a^4*b + 2*a^2*b^3 + b^5)*d*cosh(n - 1) + (a^4*b + 2*a^2*b^3 + b^5)*d*sinh(n - 1))*sinh(n*log(x)) + (a^2*b^3 +
b^5)*sinh(n - 1))*sinh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c) - ((a^5 + 2*a^3*b^2 + a*b^4)*d*cosh(n - 1) +
(a^5 + 2*a^3*b^2 + a*b^4)*d*sinh(n - 1))*sinh(n*log(x)) - 2*(a^3*b^2 + a*b^4)*sinh(n - 1))/((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*cosh(n*log(x)) + d*sinh(n*log(x)) + c))

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (e x\right )^{n - 1}}{\left (a + b \operatorname {csch}{\left (c + d x^{n} \right )}\right )^{2}}\, dx \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

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Mupad [B]
time = 19.09, size = 1449, normalized size = 9.72 \begin {gather*} \frac {\left (2\,\mathrm {atan}\left (\left (\frac {a^5\,\sqrt {-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}}}{2}+\frac {a^3\,b^2\,\sqrt {-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}}}{2}\right )\,\left ({\mathrm {e}}^{d\,x^n}\,{\mathrm {e}}^c\,\left (\frac {2\,{\left (e\,x\right )}^{1-n}\,\left (a^4\,b\,d\,n\,x^n\,\sqrt {b^6\,x^2\,{\left (e\,x\right )}^{2\,n-2}+4\,a^2\,b^4\,x^2\,{\left (e\,x\right )}^{2\,n-2}+4\,a^4\,b^2\,x^2\,{\left (e\,x\right )}^{2\,n-2}}+a^2\,b^3\,d\,n\,x^n\,\sqrt {b^6\,x^2\,{\left (e\,x\right )}^{2\,n-2}+4\,a^2\,b^4\,x^2\,{\left (e\,x\right )}^{2\,n-2}+4\,a^4\,b^2\,x^2\,{\left (e\,x\right )}^{2\,n-2}}\right )}{a^2\,x\,\left (a^4+a^2\,b^2\right )\,\left (2\,a^2+b^2\right )\,\sqrt {-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}}\,\sqrt {-a^4\,d^2\,n^2\,x^{2\,n}\,{\left (a^2+b^2\right )}^3}}+\frac {2\,\left (b^3\,x\,{\left (e\,x\right )}^{n-1}\,\sqrt {-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}}+2\,a^2\,b\,x\,{\left (e\,x\right )}^{n-1}\,\sqrt {-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}}\right )}{a^4\,d\,n\,x^n\,\left (a^4+a^2\,b^2\right )\,\left (a^2+b^2\right )\,\sqrt {b^2\,x^2\,{\left (e\,x\right )}^{2\,n-2}\,{\left (2\,a^2+b^2\right )}^2}\,\sqrt {-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}}}\right )-\frac {2\,{\left (e\,x\right )}^{1-n}\,\left (a^5\,d\,n\,x^n\,\sqrt {b^6\,x^2\,{\left (e\,x\right )}^{2\,n-2}+4\,a^2\,b^4\,x^2\,{\left (e\,x\right )}^{2\,n-2}+4\,a^4\,b^2\,x^2\,{\left (e\,x\right )}^{2\,n-2}}+a^3\,b^2\,d\,n\,x^n\,\sqrt {b^6\,x^2\,{\left (e\,x\right )}^{2\,n-2}+4\,a^2\,b^4\,x^2\,{\left (e\,x\right )}^{2\,n-2}+4\,a^4\,b^2\,x^2\,{\left (e\,x\right )}^{2\,n-2}}\right )}{a^2\,x\,\left (a^4+a^2\,b^2\right )\,\left (2\,a^2+b^2\right )\,\sqrt {-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}}\,\sqrt {-a^4\,d^2\,n^2\,x^{2\,n}\,{\left (a^2+b^2\right )}^3}}\right )\right )-2\,\mathrm {atan}\left (\frac {x\,{\left (e\,x\right )}^{n-1}\,\left (2\,a^2+b^2\right )\,\sqrt {-a^4\,d^2\,n^2\,x^{2\,n}\,{\left (a^2+b^2\right )}^3}}{a^2\,d\,n\,x^n\,\left (a^2+b^2\right )\,\sqrt {b^2\,x^2\,{\left (e\,x\right )}^{2\,n-2}\,{\left (2\,a^2+b^2\right )}^2}}\right )\right )\,\sqrt {b^6\,x^2\,{\left (e\,x\right )}^{2\,n-2}+4\,a^2\,b^4\,x^2\,{\left (e\,x\right )}^{2\,n-2}+4\,a^4\,b^2\,x^2\,{\left (e\,x\right )}^{2\,n-2}}}{\sqrt {-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}}}-\frac {\frac {2\,b^2\,x^2\,{\left (e\,x\right )}^{n-1}}{d\,n\,x^n\,\left (x\,a^3+x\,a\,b^2\right )}-\frac {2\,b^3\,x^2\,{\mathrm {e}}^{c+d\,x^n}\,{\left (e\,x\right )}^{n-1}}{a\,d\,n\,x^n\,\left (x\,a^3+x\,a\,b^2\right )}}{a\,{\mathrm {e}}^{2\,c+2\,d\,x^n}-a+2\,b\,{\mathrm {e}}^{c+d\,x^n}}+\frac {x\,{\left (e\,x\right )}^{n-1}}{a^2\,n} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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