3.76 \(\int (e x)^{-1+2 n} (a+b \text{csch}(c+d x^n))^2 \, dx\)
Optimal. Leaf size=198 \[ -\frac{2 a b x^{-2 n} (e x)^{2 n} \text{PolyLog}\left (2,-e^{c+d x^n}\right )}{d^2 e n}+\frac{2 a b x^{-2 n} (e x)^{2 n} \text{PolyLog}\left (2,e^{c+d x^n}\right )}{d^2 e n}+\frac{a^2 (e x)^{2 n}}{2 e n}-\frac{4 a b x^{-n} (e x)^{2 n} \tanh ^{-1}\left (e^{c+d x^n}\right )}{d e n}+\frac{b^2 x^{-2 n} (e x)^{2 n} \log \left (\sinh \left (c+d x^n\right )\right )}{d^2 e n}-\frac{b^2 x^{-n} (e x)^{2 n} \coth \left (c+d x^n\right )}{d e n} \]
[Out]
(a^2*(e*x)^(2*n))/(2*e*n) - (4*a*b*(e*x)^(2*n)*ArcTanh[E^(c + d*x^n)])/(d*e*n*x^n) - (b^2*(e*x)^(2*n)*Coth[c +
d*x^n])/(d*e*n*x^n) + (b^2*(e*x)^(2*n)*Log[Sinh[c + d*x^n]])/(d^2*e*n*x^(2*n)) - (2*a*b*(e*x)^(2*n)*PolyLog[2
, -E^(c + d*x^n)])/(d^2*e*n*x^(2*n)) + (2*a*b*(e*x)^(2*n)*PolyLog[2, E^(c + d*x^n)])/(d^2*e*n*x^(2*n))
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Rubi [A] time = 0.211037, antiderivative size = 198, normalized size of antiderivative = 1.,
number of steps used = 11, number of rules used = 8, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used
= {5441, 5437, 4190, 4182, 2279, 2391, 4184, 3475} \[ -\frac{2 a b x^{-2 n} (e x)^{2 n} \text{PolyLog}\left (2,-e^{c+d x^n}\right )}{d^2 e n}+\frac{2 a b x^{-2 n} (e x)^{2 n} \text{PolyLog}\left (2,e^{c+d x^n}\right )}{d^2 e n}+\frac{a^2 (e x)^{2 n}}{2 e n}-\frac{4 a b x^{-n} (e x)^{2 n} \tanh ^{-1}\left (e^{c+d x^n}\right )}{d e n}+\frac{b^2 x^{-2 n} (e x)^{2 n} \log \left (\sinh \left (c+d x^n\right )\right )}{d^2 e n}-\frac{b^2 x^{-n} (e x)^{2 n} \coth \left (c+d x^n\right )}{d e n} \]
Antiderivative was successfully verified.
[In]
Int[(e*x)^(-1 + 2*n)*(a + b*Csch[c + d*x^n])^2,x]
[Out]
(a^2*(e*x)^(2*n))/(2*e*n) - (4*a*b*(e*x)^(2*n)*ArcTanh[E^(c + d*x^n)])/(d*e*n*x^n) - (b^2*(e*x)^(2*n)*Coth[c +
d*x^n])/(d*e*n*x^n) + (b^2*(e*x)^(2*n)*Log[Sinh[c + d*x^n]])/(d^2*e*n*x^(2*n)) - (2*a*b*(e*x)^(2*n)*PolyLog[2
, -E^(c + d*x^n)])/(d^2*e*n*x^(2*n)) + (2*a*b*(e*x)^(2*n)*PolyLog[2, E^(c + d*x^n)])/(d^2*e*n*x^(2*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 4190
Int[(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(n_.)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[
(c + d*x)^m, (a + b*Csc[e + f*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && IGtQ[m, 0] && IGtQ[n, 0]
Rule 4182
Int[csc[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[(-2*(c + d*x)^m*Ar
cTanh[E^(-(I*e) + f*fz*x)])/(f*fz*I), x] + (-Dist[(d*m)/(f*fz*I), Int[(c + d*x)^(m - 1)*Log[1 - E^(-(I*e) + f*
fz*x)], x], x] + Dist[(d*m)/(f*fz*I), Int[(c + d*x)^(m - 1)*Log[1 + E^(-(I*e) + f*fz*x)], x], x]) /; FreeQ[{c,
d, e, f, fz}, x] && IGtQ[m, 0]
Rule 2279
Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Dist[1/(d*e*n*Log[F]), Subst[Int
[Log[a + b*x]/x, x], x, (F^(e*(c + d*x)))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]
Rule 2391
Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> -Simp[PolyLog[2, -(c*e*x^n)]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]
Rule 4184
Int[csc[(e_.) + (f_.)*(x_)]^2*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> -Simp[((c + d*x)^m*Cot[e + f*x])/f, x]
+ Dist[(d*m)/f, Int[(c + d*x)^(m - 1)*Cot[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]
Rule 3475
Int[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[Log[RemoveContent[Cos[c + d*x], x]]/d, x] /; FreeQ[{c, d}, x]
Rubi steps
\begin{align*} \int (e x)^{-1+2 n} \left (a+b \text{csch}\left (c+d x^n\right )\right )^2 \, dx &=\frac{\left (x^{-2 n} (e x)^{2 n}\right ) \int x^{-1+2 n} \left (a+b \text{csch}\left (c+d x^n\right )\right )^2 \, dx}{e}\\ &=\frac{\left (x^{-2 n} (e x)^{2 n}\right ) \operatorname{Subst}\left (\int x (a+b \text{csch}(c+d x))^2 \, dx,x,x^n\right )}{e n}\\ &=\frac{\left (x^{-2 n} (e x)^{2 n}\right ) \operatorname{Subst}\left (\int \left (a^2 x+2 a b x \text{csch}(c+d x)+b^2 x \text{csch}^2(c+d x)\right ) \, dx,x,x^n\right )}{e n}\\ &=\frac{a^2 (e x)^{2 n}}{2 e n}+\frac{\left (2 a b x^{-2 n} (e x)^{2 n}\right ) \operatorname{Subst}\left (\int x \text{csch}(c+d x) \, dx,x,x^n\right )}{e n}+\frac{\left (b^2 x^{-2 n} (e x)^{2 n}\right ) \operatorname{Subst}\left (\int x \text{csch}^2(c+d x) \, dx,x,x^n\right )}{e n}\\ &=\frac{a^2 (e x)^{2 n}}{2 e n}-\frac{4 a b x^{-n} (e x)^{2 n} \tanh ^{-1}\left (e^{c+d x^n}\right )}{d e n}-\frac{b^2 x^{-n} (e x)^{2 n} \coth \left (c+d x^n\right )}{d e n}-\frac{\left (2 a b x^{-2 n} (e x)^{2 n}\right ) \operatorname{Subst}\left (\int \log \left (1-e^{c+d x}\right ) \, dx,x,x^n\right )}{d e n}+\frac{\left (2 a b x^{-2 n} (e x)^{2 n}\right ) \operatorname{Subst}\left (\int \log \left (1+e^{c+d x}\right ) \, dx,x,x^n\right )}{d e n}+\frac{\left (b^2 x^{-2 n} (e x)^{2 n}\right ) \operatorname{Subst}\left (\int \coth (c+d x) \, dx,x,x^n\right )}{d e n}\\ &=\frac{a^2 (e x)^{2 n}}{2 e n}-\frac{4 a b x^{-n} (e x)^{2 n} \tanh ^{-1}\left (e^{c+d x^n}\right )}{d e n}-\frac{b^2 x^{-n} (e x)^{2 n} \coth \left (c+d x^n\right )}{d e n}+\frac{b^2 x^{-2 n} (e x)^{2 n} \log \left (\sinh \left (c+d x^n\right )\right )}{d^2 e n}-\frac{\left (2 a b x^{-2 n} (e x)^{2 n}\right ) \operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{c+d x^n}\right )}{d^2 e n}+\frac{\left (2 a b x^{-2 n} (e x)^{2 n}\right ) \operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,e^{c+d x^n}\right )}{d^2 e n}\\ &=\frac{a^2 (e x)^{2 n}}{2 e n}-\frac{4 a b x^{-n} (e x)^{2 n} \tanh ^{-1}\left (e^{c+d x^n}\right )}{d e n}-\frac{b^2 x^{-n} (e x)^{2 n} \coth \left (c+d x^n\right )}{d e n}+\frac{b^2 x^{-2 n} (e x)^{2 n} \log \left (\sinh \left (c+d x^n\right )\right )}{d^2 e n}-\frac{2 a b x^{-2 n} (e x)^{2 n} \text{Li}_2\left (-e^{c+d x^n}\right )}{d^2 e n}+\frac{2 a b x^{-2 n} (e x)^{2 n} \text{Li}_2\left (e^{c+d x^n}\right )}{d^2 e n}\\ \end{align*}
Mathematica [A] time = 5.71998, size = 278, normalized size = 1.4 \[ \frac{x^{-2 n} (e x)^{2 n} \left (8 a b \left (\frac{\text{sech}(c) \left (\text{PolyLog}\left (2,-e^{-\tanh ^{-1}(\tanh (c))-d x^n}\right )-\text{PolyLog}\left (2,e^{-\tanh ^{-1}(\tanh (c))-d x^n}\right )+\left (\tanh ^{-1}(\tanh (c))+d x^n\right ) \left (\log \left (1-e^{-\tanh ^{-1}(\tanh (c))-d x^n}\right )-\log \left (e^{-\tanh ^{-1}(\tanh (c))-d x^n}+1\right )\right )\right )}{\sqrt{\text{sech}^2(c)}}+2 \tanh ^{-1}(\tanh (c)) \tanh ^{-1}\left (\sinh (c) \tanh \left (\frac{d x^n}{2}\right )+\cosh (c)\right )\right )+2 d x^n \left (a^2 d x^n-2 b^2 \coth (c)\right )+4 b^2 d \coth (c) x^n+2 b^2 d \text{csch}\left (\frac{c}{2}\right ) x^n \sinh \left (\frac{d x^n}{2}\right ) \text{csch}\left (\frac{1}{2} \left (c+d x^n\right )\right )-2 b^2 d \text{sech}\left (\frac{c}{2}\right ) x^n \sinh \left (\frac{d x^n}{2}\right ) \text{sech}\left (\frac{1}{2} \left (c+d x^n\right )\right )-4 b^2 \left (d \coth (c) x^n-\log \left (\sinh \left (c+d x^n\right )\right )\right )\right )}{4 d^2 e n} \]
Warning: Unable to verify antiderivative.
[In]
Integrate[(e*x)^(-1 + 2*n)*(a + b*Csch[c + d*x^n])^2,x]
[Out]
((e*x)^(2*n)*(4*b^2*d*x^n*Coth[c] + 2*d*x^n*(a^2*d*x^n - 2*b^2*Coth[c]) - 4*b^2*(d*x^n*Coth[c] - Log[Sinh[c +
d*x^n]]) + 8*a*b*(2*ArcTanh[Tanh[c]]*ArcTanh[Cosh[c] + Sinh[c]*Tanh[(d*x^n)/2]] + (((d*x^n + ArcTanh[Tanh[c]])
*(Log[1 - E^(-(d*x^n) - ArcTanh[Tanh[c]])] - Log[1 + E^(-(d*x^n) - ArcTanh[Tanh[c]])]) + PolyLog[2, -E^(-(d*x^
n) - ArcTanh[Tanh[c]])] - PolyLog[2, E^(-(d*x^n) - ArcTanh[Tanh[c]])])*Sech[c])/Sqrt[Sech[c]^2]) + 2*b^2*d*x^n
*Csch[c/2]*Csch[(c + d*x^n)/2]*Sinh[(d*x^n)/2] - 2*b^2*d*x^n*Sech[c/2]*Sech[(c + d*x^n)/2]*Sinh[(d*x^n)/2]))/(
4*d^2*e*n*x^(2*n))
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Maple [F] time = 0.308, size = 0, normalized size = 0. \begin{align*} \int \left ( ex \right ) ^{-1+2\,n} \left ( a+b{\rm csch} \left (c+d{x}^{n}\right ) \right ) ^{2}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((e*x)^(-1+2*n)*(a+b*csch(c+d*x^n))^2,x)
[Out]
int((e*x)^(-1+2*n)*(a+b*csch(c+d*x^n))^2,x)
________________________________________________________________________________________
Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((e*x)^(-1+2*n)*(a+b*csch(c+d*x^n))^2,x, algorithm="maxima")
[Out]
Exception raised: ValueError
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Fricas [B] time = 2.64767, size = 8325, normalized size = 42.05 \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+2*n)*(a+b*csch(c+d*x^n))^2,x, algorithm="fricas")
[Out]
-1/2*(a^2*d^2*cosh((2*n - 1)*log(e))*cosh(n*log(x))^2 - 4*b^2*c*cosh((2*n - 1)*log(e)) - (a^2*d^2*cosh((2*n -
1)*log(e))*cosh(n*log(x))^2 - 4*b^2*d*cosh((2*n - 1)*log(e))*cosh(n*log(x)) - 4*b^2*c*cosh((2*n - 1)*log(e)) +
(a^2*d^2*cosh((2*n - 1)*log(e)) + a^2*d^2*sinh((2*n - 1)*log(e)))*sinh(n*log(x))^2 + (a^2*d^2*cosh(n*log(x))^
2 - 4*b^2*d*cosh(n*log(x)) - 4*b^2*c)*sinh((2*n - 1)*log(e)) + 2*(a^2*d^2*cosh((2*n - 1)*log(e))*cosh(n*log(x)
) - 2*b^2*d*cosh((2*n - 1)*log(e)) + (a^2*d^2*cosh(n*log(x)) - 2*b^2*d)*sinh((2*n - 1)*log(e)))*sinh(n*log(x))
)*cosh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c)^2 - 2*(a^2*d^2*cosh((2*n - 1)*log(e))*cosh(n*log(x))^2 - 4*b^2
*d*cosh((2*n - 1)*log(e))*cosh(n*log(x)) - 4*b^2*c*cosh((2*n - 1)*log(e)) + (a^2*d^2*cosh((2*n - 1)*log(e)) +
a^2*d^2*sinh((2*n - 1)*log(e)))*sinh(n*log(x))^2 + (a^2*d^2*cosh(n*log(x))^2 - 4*b^2*d*cosh(n*log(x)) - 4*b^2*
c)*sinh((2*n - 1)*log(e)) + 2*(a^2*d^2*cosh((2*n - 1)*log(e))*cosh(n*log(x)) - 2*b^2*d*cosh((2*n - 1)*log(e))
+ (a^2*d^2*cosh(n*log(x)) - 2*b^2*d)*sinh((2*n - 1)*log(e)))*sinh(n*log(x)))*cosh(d*cosh(n*log(x)) + d*sinh(n*
log(x)) + c)*sinh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c) - (a^2*d^2*cosh((2*n - 1)*log(e))*cosh(n*log(x))^2
- 4*b^2*d*cosh((2*n - 1)*log(e))*cosh(n*log(x)) - 4*b^2*c*cosh((2*n - 1)*log(e)) + (a^2*d^2*cosh((2*n - 1)*log
(e)) + a^2*d^2*sinh((2*n - 1)*log(e)))*sinh(n*log(x))^2 + (a^2*d^2*cosh(n*log(x))^2 - 4*b^2*d*cosh(n*log(x)) -
4*b^2*c)*sinh((2*n - 1)*log(e)) + 2*(a^2*d^2*cosh((2*n - 1)*log(e))*cosh(n*log(x)) - 2*b^2*d*cosh((2*n - 1)*l
og(e)) + (a^2*d^2*cosh(n*log(x)) - 2*b^2*d)*sinh((2*n - 1)*log(e)))*sinh(n*log(x)))*sinh(d*cosh(n*log(x)) + d*
sinh(n*log(x)) + c)^2 + (a^2*d^2*cosh((2*n - 1)*log(e)) + a^2*d^2*sinh((2*n - 1)*log(e)))*sinh(n*log(x))^2 - 4
*((a*b*cosh((2*n - 1)*log(e)) + a*b*sinh((2*n - 1)*log(e)))*cosh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c)^2 -
a*b*cosh((2*n - 1)*log(e)) + 2*(a*b*cosh((2*n - 1)*log(e)) + a*b*sinh((2*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) + (a*b*cosh((2*n - 1)*log(e)) + a*b*sin
h((2*n - 1)*log(e)))*sinh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c)^2 - a*b*sinh((2*n - 1)*log(e)))*dilog(cosh(
d*cosh(n*log(x)) + d*sinh(n*log(x)) + c) + sinh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c)) + 4*((a*b*cosh((2*n
- 1)*log(e)) + a*b*sinh((2*n - 1)*log(e)))*cosh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c)^2 - a*b*cosh((2*n - 1
)*log(e)) + 2*(a*b*cosh((2*n - 1)*log(e)) + a*b*sinh((2*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) + (a*b*cosh((2*n - 1)*log(e)) + a*b*sinh((2*n - 1)*log(e
)))*sinh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c)^2 - a*b*sinh((2*n - 1)*log(e)))*dilog(-cosh(d*cosh(n*log(x))
+ d*sinh(n*log(x)) + c) - sinh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c)) - 2*(2*a*b*d*cosh((2*n - 1)*log(e))*
cosh(n*log(x)) - (2*a*b*d*cosh((2*n - 1)*log(e))*cosh(n*log(x)) - b^2*cosh((2*n - 1)*log(e)) + (2*a*b*d*cosh(n
*log(x)) - b^2)*sinh((2*n - 1)*log(e)) + 2*(a*b*d*cosh((2*n - 1)*log(e)) + a*b*d*sinh((2*n - 1)*log(e)))*sinh(
n*log(x)))*cosh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c)^2 - b^2*cosh((2*n - 1)*log(e)) - 2*(2*a*b*d*cosh((2*n
- 1)*log(e))*cosh(n*log(x)) - b^2*cosh((2*n - 1)*log(e)) + (2*a*b*d*cosh(n*log(x)) - b^2)*sinh((2*n - 1)*log(
e)) + 2*(a*b*d*cosh((2*n - 1)*log(e)) + a*b*d*sinh((2*n - 1)*log(e)))*sinh(n*log(x)))*cosh(d*cosh(n*log(x)) +
d*sinh(n*log(x)) + c)*sinh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c) - (2*a*b*d*cosh((2*n - 1)*log(e))*cosh(n*l
og(x)) - b^2*cosh((2*n - 1)*log(e)) + (2*a*b*d*cosh(n*log(x)) - b^2)*sinh((2*n - 1)*log(e)) + 2*(a*b*d*cosh((2
*n - 1)*log(e)) + a*b*d*sinh((2*n - 1)*log(e)))*sinh(n*log(x)))*sinh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c)^
2 + (2*a*b*d*cosh(n*log(x)) - b^2)*sinh((2*n - 1)*log(e)) + 2*(a*b*d*cosh((2*n - 1)*log(e)) + a*b*d*sinh((2*n
- 1)*log(e)))*sinh(n*log(x)))*log(cosh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c) + sinh(d*cosh(n*log(x)) + d*si
nh(n*log(x)) + c) + 1) + 2*(((2*a*b*c - b^2)*cosh((2*n - 1)*log(e)) + (2*a*b*c - b^2)*sinh((2*n - 1)*log(e)))*
cosh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c)^2 + 2*((2*a*b*c - b^2)*cosh((2*n - 1)*log(e)) + (2*a*b*c - b^2)*
sinh((2*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) + ((2*a*b*c - b^2)*cosh((2*n - 1)*log(e)) + (2*a*b*c - b^2)*sinh((2*n - 1)*log(e)))*sinh(d*cosh(n*log(x)
) + d*sinh(n*log(x)) + c)^2 - (2*a*b*c - b^2)*cosh((2*n - 1)*log(e)) - (2*a*b*c - b^2)*sinh((2*n - 1)*log(e)))
*log(cosh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c) + sinh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c) - 1) + 4*(a
*b*d*cosh((2*n - 1)*log(e))*cosh(n*log(x)) + a*b*c*cosh((2*n - 1)*log(e)) - (a*b*d*cosh((2*n - 1)*log(e))*cosh
(n*log(x)) + a*b*c*cosh((2*n - 1)*log(e)) + (a*b*d*cosh(n*log(x)) + a*b*c)*sinh((2*n - 1)*log(e)) + (a*b*d*cos
h((2*n - 1)*log(e)) + a*b*d*sinh((2*n - 1)*log(e)))*sinh(n*log(x)))*cosh(d*cosh(n*log(x)) + d*sinh(n*log(x)) +
c)^2 - 2*(a*b*d*cosh((2*n - 1)*log(e))*cosh(n*log(x)) + a*b*c*cosh((2*n - 1)*log(e)) + (a*b*d*cosh(n*log(x))
+ a*b*c)*sinh((2*n - 1)*log(e)) + (a*b*d*cosh((2*n - 1)*log(e)) + a*b*d*sinh((2*n - 1)*log(e)))*sinh(n*log(x))
)*cosh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c)*sinh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c) - (a*b*d*cosh((2
*n - 1)*log(e))*cosh(n*log(x)) + a*b*c*cosh((2*n - 1)*log(e)) + (a*b*d*cosh(n*log(x)) + a*b*c)*sinh((2*n - 1)*
log(e)) + (a*b*d*cosh((2*n - 1)*log(e)) + a*b*d*sinh((2*n - 1)*log(e)))*sinh(n*log(x)))*sinh(d*cosh(n*log(x))
+ d*sinh(n*log(x)) + c)^2 + (a*b*d*cosh(n*log(x)) + a*b*c)*sinh((2*n - 1)*log(e)) + (a*b*d*cosh((2*n - 1)*log(
e)) + a*b*d*sinh((2*n - 1)*log(e)))*sinh(n*log(x)))*log(-cosh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c) - sinh(
d*cosh(n*log(x)) + d*sinh(n*log(x)) + c) + 1) + (a^2*d^2*cosh(n*log(x))^2 - 4*b^2*c)*sinh((2*n - 1)*log(e)) +
2*(a^2*d^2*cosh((2*n - 1)*log(e))*cosh(n*log(x)) + a^2*d^2*cosh(n*log(x))*sinh((2*n - 1)*log(e)))*sinh(n*log(x
)))/(d^2*n*cosh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c)^2 + 2*d^2*n*cosh(d*cosh(n*log(x)) + d*sinh(n*log(x))
+ c)*sinh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c) + d^2*n*sinh(d*cosh(n*log(x)) + d*sinh(n*log(x)) + c)^2 - d
^2*n)
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (e x\right )^{2 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+2*n)*(a+b*csch(c+d*x**n))**2,x)
[Out]
Integral((e*x)**(2*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{\left (b \operatorname{csch}\left (d x^{n} + c\right ) + a\right )}^{2} \left (e x\right )^{2 \, n - 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((e*x)^(-1+2*n)*(a+b*csch(c+d*x^n))^2,x, algorithm="giac")
[Out]
integrate((b*csch(d*x^n + c) + a)^2*(e*x)^(2*n - 1), x)