3.852 \(\int x^2 \text{csch}(x) \text{sech}(x) \sqrt{a \text{sech}^4(x)} \, dx\)
Optimal. Leaf size=204 \[ -x \cosh ^2(x) \text{PolyLog}\left (2,-e^{2 x}\right ) \sqrt{a \text{sech}^4(x)}+x \cosh ^2(x) \text{PolyLog}\left (2,e^{2 x}\right ) \sqrt{a \text{sech}^4(x)}+\frac{1}{2} \cosh ^2(x) \text{PolyLog}\left (3,-e^{2 x}\right ) \sqrt{a \text{sech}^4(x)}-\frac{1}{2} \cosh ^2(x) \text{PolyLog}\left (3,e^{2 x}\right ) \sqrt{a \text{sech}^4(x)}+\frac{1}{2} x^2 \cosh ^2(x) \sqrt{a \text{sech}^4(x)}-\frac{1}{2} x^2 \sinh ^2(x) \sqrt{a \text{sech}^4(x)}-2 x^2 \cosh ^2(x) \tanh ^{-1}\left (e^{2 x}\right ) \sqrt{a \text{sech}^4(x)}+\cosh ^2(x) \sqrt{a \text{sech}^4(x)} \log (\cosh (x))-x \sinh (x) \cosh (x) \sqrt{a \text{sech}^4(x)} \]
[Out]
(x^2*Cosh[x]^2*Sqrt[a*Sech[x]^4])/2 - 2*x^2*ArcTanh[E^(2*x)]*Cosh[x]^2*Sqrt[a*Sech[x]^4] + Cosh[x]^2*Log[Cosh[
x]]*Sqrt[a*Sech[x]^4] - x*Cosh[x]^2*PolyLog[2, -E^(2*x)]*Sqrt[a*Sech[x]^4] + x*Cosh[x]^2*PolyLog[2, E^(2*x)]*S
qrt[a*Sech[x]^4] + (Cosh[x]^2*PolyLog[3, -E^(2*x)]*Sqrt[a*Sech[x]^4])/2 - (Cosh[x]^2*PolyLog[3, E^(2*x)]*Sqrt[
a*Sech[x]^4])/2 - x*Cosh[x]*Sqrt[a*Sech[x]^4]*Sinh[x] - (x^2*Sqrt[a*Sech[x]^4]*Sinh[x]^2)/2
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Rubi [A] time = 0.638415, antiderivative size = 204, normalized size of antiderivative = 1.,
number of steps used = 16, number of rules used = 13, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.722, Rules
used = {6720, 2620, 14, 5462, 2551, 5461, 4182, 2531, 2282, 6589, 3720, 3475, 30}
\[ -x \cosh ^2(x) \text{PolyLog}\left (2,-e^{2 x}\right ) \sqrt{a \text{sech}^4(x)}+x \cosh ^2(x) \text{PolyLog}\left (2,e^{2 x}\right ) \sqrt{a \text{sech}^4(x)}+\frac{1}{2} \cosh ^2(x) \text{PolyLog}\left (3,-e^{2 x}\right ) \sqrt{a \text{sech}^4(x)}-\frac{1}{2} \cosh ^2(x) \text{PolyLog}\left (3,e^{2 x}\right ) \sqrt{a \text{sech}^4(x)}+\frac{1}{2} x^2 \cosh ^2(x) \sqrt{a \text{sech}^4(x)}-\frac{1}{2} x^2 \sinh ^2(x) \sqrt{a \text{sech}^4(x)}-2 x^2 \cosh ^2(x) \tanh ^{-1}\left (e^{2 x}\right ) \sqrt{a \text{sech}^4(x)}+\cosh ^2(x) \sqrt{a \text{sech}^4(x)} \log (\cosh (x))-x \sinh (x) \cosh (x) \sqrt{a \text{sech}^4(x)} \]
Antiderivative was successfully verified.
[In]
Int[x^2*Csch[x]*Sech[x]*Sqrt[a*Sech[x]^4],x]
[Out]
(x^2*Cosh[x]^2*Sqrt[a*Sech[x]^4])/2 - 2*x^2*ArcTanh[E^(2*x)]*Cosh[x]^2*Sqrt[a*Sech[x]^4] + Cosh[x]^2*Log[Cosh[
x]]*Sqrt[a*Sech[x]^4] - x*Cosh[x]^2*PolyLog[2, -E^(2*x)]*Sqrt[a*Sech[x]^4] + x*Cosh[x]^2*PolyLog[2, E^(2*x)]*S
qrt[a*Sech[x]^4] + (Cosh[x]^2*PolyLog[3, -E^(2*x)]*Sqrt[a*Sech[x]^4])/2 - (Cosh[x]^2*PolyLog[3, E^(2*x)]*Sqrt[
a*Sech[x]^4])/2 - x*Cosh[x]*Sqrt[a*Sech[x]^4]*Sinh[x] - (x^2*Sqrt[a*Sech[x]^4]*Sinh[x]^2)/2
Rule 6720
Int[(u_.)*((a_.)*(v_)^(m_.))^(p_), x_Symbol] :> Dist[(a^IntPart[p]*(a*v^m)^FracPart[p])/v^(m*FracPart[p]), Int
[u*v^(m*p), x], x] /; FreeQ[{a, m, p}, x] && !IntegerQ[p] && !FreeQ[v, x] && !(EqQ[a, 1] && EqQ[m, 1]) &&
!(EqQ[v, x] && EqQ[m, 1])
Rule 2620
Int[csc[(e_.) + (f_.)*(x_)]^(m_.)*sec[(e_.) + (f_.)*(x_)]^(n_.), x_Symbol] :> Dist[1/f, Subst[Int[(1 + x^2)^((
m + n)/2 - 1)/x^m, x], x, Tan[e + f*x]], x] /; FreeQ[{e, f}, x] && IntegersQ[m, n, (m + n)/2]
Rule 14
Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&& !LinearQ[u, x] && !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]
Rule 5462
Int[Csch[(a_.) + (b_.)*(x_)]^(n_.)*((c_.) + (d_.)*(x_))^(m_.)*Sech[(a_.) + (b_.)*(x_)]^(p_.), x_Symbol] :> Wit
h[{u = IntHide[Csch[a + b*x]^n*Sech[a + b*x]^p, x]}, Dist[(c + d*x)^m, u, x] - Dist[d*m, Int[(c + d*x)^(m - 1)
*u, x], x]] /; FreeQ[{a, b, c, d}, x] && IntegersQ[n, p] && GtQ[m, 0] && NeQ[n, p]
Rule 2551
Int[Log[u_]*((a_.) + (b_.)*(x_))^(m_.), x_Symbol] :> Simp[((a + b*x)^(m + 1)*Log[u])/(b*(m + 1)), x] - Dist[1/
(b*(m + 1)), Int[SimplifyIntegrand[((a + b*x)^(m + 1)*D[u, x])/u, x], x], x] /; FreeQ[{a, b, m}, x] && Inverse
FunctionFreeQ[u, x] && NeQ[m, -1]
Rule 5461
Int[Csch[(a_.) + (b_.)*(x_)]^(n_.)*((c_.) + (d_.)*(x_))^(m_.)*Sech[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Dis
t[2^n, Int[(c + d*x)^m*Csch[2*a + 2*b*x]^n, x], x] /; FreeQ[{a, b, c, d}, x] && RationalQ[m] && IntegerQ[n]
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 2531
Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.)*(x_))^(m_.), x_Symbol] :> -Simp[((
f + g*x)^m*PolyLog[2, -(e*(F^(c*(a + b*x)))^n)])/(b*c*n*Log[F]), x] + Dist[(g*m)/(b*c*n*Log[F]), Int[(f + g*x)
^(m - 1)*PolyLog[2, -(e*(F^(c*(a + b*x)))^n)], x], x] /; FreeQ[{F, a, b, c, e, f, g, n}, x] && GtQ[m, 0]
Rule 2282
Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] && !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] && !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[x]]]
Rule 6589
Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[PolyLog[n + 1, c*(a
+ b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[b*d, a*e]
Rule 3720
Int[((c_.) + (d_.)*(x_))^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*(c + d*x)^m*(b*Tan[e
+ f*x])^(n - 1))/(f*(n - 1)), x] + (-Dist[(b*d*m)/(f*(n - 1)), Int[(c + d*x)^(m - 1)*(b*Tan[e + f*x])^(n - 1)
, x], x] - Dist[b^2, Int[(c + d*x)^m*(b*Tan[e + f*x])^(n - 2), x], x]) /; FreeQ[{b, c, d, e, f}, x] && GtQ[n,
1] && 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]
Rule 30
Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]
Rubi steps
\begin{align*} \int x^2 \text{csch}(x) \text{sech}(x) \sqrt{a \text{sech}^4(x)} \, dx &=\left (\cosh ^2(x) \sqrt{a \text{sech}^4(x)}\right ) \int x^2 \text{csch}(x) \text{sech}^3(x) \, dx\\ &=x^2 \cosh ^2(x) \log (\tanh (x)) \sqrt{a \text{sech}^4(x)}-\frac{1}{2} x^2 \sqrt{a \text{sech}^4(x)} \sinh ^2(x)-\left (2 \cosh ^2(x) \sqrt{a \text{sech}^4(x)}\right ) \int x \left (\log (\tanh (x))-\frac{\tanh ^2(x)}{2}\right ) \, dx\\ &=x^2 \cosh ^2(x) \log (\tanh (x)) \sqrt{a \text{sech}^4(x)}-\frac{1}{2} x^2 \sqrt{a \text{sech}^4(x)} \sinh ^2(x)-\left (2 \cosh ^2(x) \sqrt{a \text{sech}^4(x)}\right ) \int \left (x \log (\tanh (x))-\frac{1}{2} x \tanh ^2(x)\right ) \, dx\\ &=x^2 \cosh ^2(x) \log (\tanh (x)) \sqrt{a \text{sech}^4(x)}-\frac{1}{2} x^2 \sqrt{a \text{sech}^4(x)} \sinh ^2(x)+\left (\cosh ^2(x) \sqrt{a \text{sech}^4(x)}\right ) \int x \tanh ^2(x) \, dx-\left (2 \cosh ^2(x) \sqrt{a \text{sech}^4(x)}\right ) \int x \log (\tanh (x)) \, dx\\ &=-x \cosh (x) \sqrt{a \text{sech}^4(x)} \sinh (x)-\frac{1}{2} x^2 \sqrt{a \text{sech}^4(x)} \sinh ^2(x)+\left (\cosh ^2(x) \sqrt{a \text{sech}^4(x)}\right ) \int x \, dx+\left (\cosh ^2(x) \sqrt{a \text{sech}^4(x)}\right ) \int x^2 \text{csch}(x) \text{sech}(x) \, dx+\left (\cosh ^2(x) \sqrt{a \text{sech}^4(x)}\right ) \int \tanh (x) \, dx\\ &=\frac{1}{2} x^2 \cosh ^2(x) \sqrt{a \text{sech}^4(x)}+\cosh ^2(x) \log (\cosh (x)) \sqrt{a \text{sech}^4(x)}-x \cosh (x) \sqrt{a \text{sech}^4(x)} \sinh (x)-\frac{1}{2} x^2 \sqrt{a \text{sech}^4(x)} \sinh ^2(x)+\left (2 \cosh ^2(x) \sqrt{a \text{sech}^4(x)}\right ) \int x^2 \text{csch}(2 x) \, dx\\ &=\frac{1}{2} x^2 \cosh ^2(x) \sqrt{a \text{sech}^4(x)}-2 x^2 \tanh ^{-1}\left (e^{2 x}\right ) \cosh ^2(x) \sqrt{a \text{sech}^4(x)}+\cosh ^2(x) \log (\cosh (x)) \sqrt{a \text{sech}^4(x)}-x \cosh (x) \sqrt{a \text{sech}^4(x)} \sinh (x)-\frac{1}{2} x^2 \sqrt{a \text{sech}^4(x)} \sinh ^2(x)-\left (2 \cosh ^2(x) \sqrt{a \text{sech}^4(x)}\right ) \int x \log \left (1-e^{2 x}\right ) \, dx+\left (2 \cosh ^2(x) \sqrt{a \text{sech}^4(x)}\right ) \int x \log \left (1+e^{2 x}\right ) \, dx\\ &=\frac{1}{2} x^2 \cosh ^2(x) \sqrt{a \text{sech}^4(x)}-2 x^2 \tanh ^{-1}\left (e^{2 x}\right ) \cosh ^2(x) \sqrt{a \text{sech}^4(x)}+\cosh ^2(x) \log (\cosh (x)) \sqrt{a \text{sech}^4(x)}-x \cosh ^2(x) \text{Li}_2\left (-e^{2 x}\right ) \sqrt{a \text{sech}^4(x)}+x \cosh ^2(x) \text{Li}_2\left (e^{2 x}\right ) \sqrt{a \text{sech}^4(x)}-x \cosh (x) \sqrt{a \text{sech}^4(x)} \sinh (x)-\frac{1}{2} x^2 \sqrt{a \text{sech}^4(x)} \sinh ^2(x)+\left (\cosh ^2(x) \sqrt{a \text{sech}^4(x)}\right ) \int \text{Li}_2\left (-e^{2 x}\right ) \, dx-\left (\cosh ^2(x) \sqrt{a \text{sech}^4(x)}\right ) \int \text{Li}_2\left (e^{2 x}\right ) \, dx\\ &=\frac{1}{2} x^2 \cosh ^2(x) \sqrt{a \text{sech}^4(x)}-2 x^2 \tanh ^{-1}\left (e^{2 x}\right ) \cosh ^2(x) \sqrt{a \text{sech}^4(x)}+\cosh ^2(x) \log (\cosh (x)) \sqrt{a \text{sech}^4(x)}-x \cosh ^2(x) \text{Li}_2\left (-e^{2 x}\right ) \sqrt{a \text{sech}^4(x)}+x \cosh ^2(x) \text{Li}_2\left (e^{2 x}\right ) \sqrt{a \text{sech}^4(x)}-x \cosh (x) \sqrt{a \text{sech}^4(x)} \sinh (x)-\frac{1}{2} x^2 \sqrt{a \text{sech}^4(x)} \sinh ^2(x)+\frac{1}{2} \left (\cosh ^2(x) \sqrt{a \text{sech}^4(x)}\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(-x)}{x} \, dx,x,e^{2 x}\right )-\frac{1}{2} \left (\cosh ^2(x) \sqrt{a \text{sech}^4(x)}\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2(x)}{x} \, dx,x,e^{2 x}\right )\\ &=\frac{1}{2} x^2 \cosh ^2(x) \sqrt{a \text{sech}^4(x)}-2 x^2 \tanh ^{-1}\left (e^{2 x}\right ) \cosh ^2(x) \sqrt{a \text{sech}^4(x)}+\cosh ^2(x) \log (\cosh (x)) \sqrt{a \text{sech}^4(x)}-x \cosh ^2(x) \text{Li}_2\left (-e^{2 x}\right ) \sqrt{a \text{sech}^4(x)}+x \cosh ^2(x) \text{Li}_2\left (e^{2 x}\right ) \sqrt{a \text{sech}^4(x)}+\frac{1}{2} \cosh ^2(x) \text{Li}_3\left (-e^{2 x}\right ) \sqrt{a \text{sech}^4(x)}-\frac{1}{2} \cosh ^2(x) \text{Li}_3\left (e^{2 x}\right ) \sqrt{a \text{sech}^4(x)}-x \cosh (x) \sqrt{a \text{sech}^4(x)} \sinh (x)-\frac{1}{2} x^2 \sqrt{a \text{sech}^4(x)} \sinh ^2(x)\\ \end{align*}
Mathematica [C] time = 0.619928, size = 120, normalized size = 0.59 \[ \frac{1}{24} \cosh ^2(x) \sqrt{a \text{sech}^4(x)} \left (24 x \text{PolyLog}\left (2,-e^{-2 x}\right )+24 x \text{PolyLog}\left (2,e^{2 x}\right )+12 \text{PolyLog}\left (3,-e^{-2 x}\right )-12 \text{PolyLog}\left (3,e^{2 x}\right )-16 x^3-24 x^2 \log \left (e^{-2 x}+1\right )+24 x^2 \log \left (1-e^{2 x}\right )+12 x^2 \text{sech}^2(x)-24 x \tanh (x)+24 \log (\cosh (x))+i \pi ^3\right ) \]
Antiderivative was successfully verified.
[In]
Integrate[x^2*Csch[x]*Sech[x]*Sqrt[a*Sech[x]^4],x]
[Out]
(Cosh[x]^2*Sqrt[a*Sech[x]^4]*(I*Pi^3 - 16*x^3 - 24*x^2*Log[1 + E^(-2*x)] + 24*x^2*Log[1 - E^(2*x)] + 24*Log[Co
sh[x]] + 24*x*PolyLog[2, -E^(-2*x)] + 24*x*PolyLog[2, E^(2*x)] + 12*PolyLog[3, -E^(-2*x)] - 12*PolyLog[3, E^(2
*x)] + 12*x^2*Sech[x]^2 - 24*x*Tanh[x]))/24
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Maple [B] time = 0.083, size = 441, normalized size = 2.2 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(x^2*csch(x)*sech(x)*(a*sech(x)^4)^(1/2),x)
[Out]
2*(a*exp(4*x)/(exp(2*x)+1)^4)^(1/2)*exp(-2*x)*x*(x*exp(2*x)+exp(2*x)+1)-2*(a*exp(4*x)/(exp(2*x)+1)^4)^(1/2)*ex
p(-2*x)*(exp(2*x)+1)^2*ln(exp(x))+(a*exp(4*x)/(exp(2*x)+1)^4)^(1/2)*exp(-2*x)*(exp(2*x)+1)^2*ln(exp(2*x)+1)+(a
*exp(4*x)/(exp(2*x)+1)^4)^(1/2)*exp(-2*x)*(exp(2*x)+1)^2*x^2*ln(exp(x)+1)+2*(a*exp(4*x)/(exp(2*x)+1)^4)^(1/2)*
exp(-2*x)*(exp(2*x)+1)^2*x*polylog(2,-exp(x))-2*(a*exp(4*x)/(exp(2*x)+1)^4)^(1/2)*exp(-2*x)*(exp(2*x)+1)^2*pol
ylog(3,-exp(x))+(a*exp(4*x)/(exp(2*x)+1)^4)^(1/2)*exp(-2*x)*(exp(2*x)+1)^2*x^2*ln(1-exp(x))+2*(a*exp(4*x)/(exp
(2*x)+1)^4)^(1/2)*exp(-2*x)*(exp(2*x)+1)^2*x*polylog(2,exp(x))-2*(a*exp(4*x)/(exp(2*x)+1)^4)^(1/2)*exp(-2*x)*(
exp(2*x)+1)^2*polylog(3,exp(x))-(a*exp(4*x)/(exp(2*x)+1)^4)^(1/2)*exp(-2*x)*(exp(2*x)+1)^2*x^2*ln(exp(2*x)+1)-
(a*exp(4*x)/(exp(2*x)+1)^4)^(1/2)*exp(-2*x)*(exp(2*x)+1)^2*x*polylog(2,-exp(2*x))+1/2*(a*exp(4*x)/(exp(2*x)+1)
^4)^(1/2)*exp(-2*x)*(exp(2*x)+1)^2*polylog(3,-exp(2*x))
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Maxima [A] time = 1.8728, size = 208, normalized size = 1.02 \begin{align*} -\frac{1}{2} \,{\left (2 \, x^{2} \log \left (e^{\left (2 \, x\right )} + 1\right ) + 2 \, x{\rm Li}_2\left (-e^{\left (2 \, x\right )}\right ) -{\rm Li}_{3}(-e^{\left (2 \, x\right )})\right )} \sqrt{a} +{\left (x^{2} \log \left (e^{x} + 1\right ) + 2 \, x{\rm Li}_2\left (-e^{x}\right ) - 2 \,{\rm Li}_{3}(-e^{x})\right )} \sqrt{a} +{\left (x^{2} \log \left (-e^{x} + 1\right ) + 2 \, x{\rm Li}_2\left (e^{x}\right ) - 2 \,{\rm Li}_{3}(e^{x})\right )} \sqrt{a} - 2 \, \sqrt{a} x + \sqrt{a} \log \left (e^{\left (2 \, x\right )} + 1\right ) + \frac{2 \,{\left ({\left (\sqrt{a} x^{2} + \sqrt{a} x\right )} e^{\left (2 \, x\right )} + \sqrt{a} x\right )}}{e^{\left (4 \, x\right )} + 2 \, e^{\left (2 \, x\right )} + 1} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^2*csch(x)*sech(x)*(a*sech(x)^4)^(1/2),x, algorithm="maxima")
[Out]
-1/2*(2*x^2*log(e^(2*x) + 1) + 2*x*dilog(-e^(2*x)) - polylog(3, -e^(2*x)))*sqrt(a) + (x^2*log(e^x + 1) + 2*x*d
ilog(-e^x) - 2*polylog(3, -e^x))*sqrt(a) + (x^2*log(-e^x + 1) + 2*x*dilog(e^x) - 2*polylog(3, e^x))*sqrt(a) -
2*sqrt(a)*x + sqrt(a)*log(e^(2*x) + 1) + 2*((sqrt(a)*x^2 + sqrt(a)*x)*e^(2*x) + sqrt(a)*x)/(e^(4*x) + 2*e^(2*x
) + 1)
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Fricas [C] time = 3.07512, size = 10301, normalized size = 50.5 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^2*csch(x)*sech(x)*(a*sech(x)^4)^(1/2),x, algorithm="fricas")
[Out]
-(2*((e^(4*x) + 2*e^(2*x) + 1)*sinh(x)^4 + cosh(x)^4 + 4*(cosh(x)*e^(4*x) + 2*cosh(x)*e^(2*x) + cosh(x))*sinh(
x)^3 + 2*(3*cosh(x)^2 + (3*cosh(x)^2 + 1)*e^(4*x) + 2*(3*cosh(x)^2 + 1)*e^(2*x) + 1)*sinh(x)^2 + 2*cosh(x)^2 +
(cosh(x)^4 + 2*cosh(x)^2 + 1)*e^(4*x) + 2*(cosh(x)^4 + 2*cosh(x)^2 + 1)*e^(2*x) + 4*(cosh(x)^3 + (cosh(x)^3 +
cosh(x))*e^(4*x) + 2*(cosh(x)^3 + cosh(x))*e^(2*x) + cosh(x))*sinh(x) + 1)*sqrt(a/(e^(8*x) + 4*e^(6*x) + 6*e^
(4*x) + 4*e^(2*x) + 1))*e^(2*x)*polylog(3, cosh(x) + sinh(x)) - 2*((e^(4*x) + 2*e^(2*x) + 1)*sinh(x)^4 + cosh(
x)^4 + 4*(cosh(x)*e^(4*x) + 2*cosh(x)*e^(2*x) + cosh(x))*sinh(x)^3 + 2*(3*cosh(x)^2 + (3*cosh(x)^2 + 1)*e^(4*x
) + 2*(3*cosh(x)^2 + 1)*e^(2*x) + 1)*sinh(x)^2 + 2*cosh(x)^2 + (cosh(x)^4 + 2*cosh(x)^2 + 1)*e^(4*x) + 2*(cosh
(x)^4 + 2*cosh(x)^2 + 1)*e^(2*x) + 4*(cosh(x)^3 + (cosh(x)^3 + cosh(x))*e^(4*x) + 2*(cosh(x)^3 + cosh(x))*e^(2
*x) + cosh(x))*sinh(x) + 1)*sqrt(a/(e^(8*x) + 4*e^(6*x) + 6*e^(4*x) + 4*e^(2*x) + 1))*e^(2*x)*polylog(3, I*cos
h(x) + I*sinh(x)) - 2*((e^(4*x) + 2*e^(2*x) + 1)*sinh(x)^4 + cosh(x)^4 + 4*(cosh(x)*e^(4*x) + 2*cosh(x)*e^(2*x
) + cosh(x))*sinh(x)^3 + 2*(3*cosh(x)^2 + (3*cosh(x)^2 + 1)*e^(4*x) + 2*(3*cosh(x)^2 + 1)*e^(2*x) + 1)*sinh(x)
^2 + 2*cosh(x)^2 + (cosh(x)^4 + 2*cosh(x)^2 + 1)*e^(4*x) + 2*(cosh(x)^4 + 2*cosh(x)^2 + 1)*e^(2*x) + 4*(cosh(x
)^3 + (cosh(x)^3 + cosh(x))*e^(4*x) + 2*(cosh(x)^3 + cosh(x))*e^(2*x) + cosh(x))*sinh(x) + 1)*sqrt(a/(e^(8*x)
+ 4*e^(6*x) + 6*e^(4*x) + 4*e^(2*x) + 1))*e^(2*x)*polylog(3, -I*cosh(x) - I*sinh(x)) + 2*((e^(4*x) + 2*e^(2*x)
+ 1)*sinh(x)^4 + cosh(x)^4 + 4*(cosh(x)*e^(4*x) + 2*cosh(x)*e^(2*x) + cosh(x))*sinh(x)^3 + 2*(3*cosh(x)^2 + (
3*cosh(x)^2 + 1)*e^(4*x) + 2*(3*cosh(x)^2 + 1)*e^(2*x) + 1)*sinh(x)^2 + 2*cosh(x)^2 + (cosh(x)^4 + 2*cosh(x)^2
+ 1)*e^(4*x) + 2*(cosh(x)^4 + 2*cosh(x)^2 + 1)*e^(2*x) + 4*(cosh(x)^3 + (cosh(x)^3 + cosh(x))*e^(4*x) + 2*(co
sh(x)^3 + cosh(x))*e^(2*x) + cosh(x))*sinh(x) + 1)*sqrt(a/(e^(8*x) + 4*e^(6*x) + 6*e^(4*x) + 4*e^(2*x) + 1))*e
^(2*x)*polylog(3, -cosh(x) - sinh(x)) + (2*x*cosh(x)^4 + 2*(x*e^(4*x) + 2*x*e^(2*x) + x)*sinh(x)^4 + 8*(x*cosh
(x)*e^(4*x) + 2*x*cosh(x)*e^(2*x) + x*cosh(x))*sinh(x)^3 - 2*(x^2 - x)*cosh(x)^2 + 2*(6*x*cosh(x)^2 - x^2 + (6
*x*cosh(x)^2 - x^2 + x)*e^(4*x) + 2*(6*x*cosh(x)^2 - x^2 + x)*e^(2*x) + x)*sinh(x)^2 - 2*(x*cosh(x)^4 + (x*e^(
4*x) + 2*x*e^(2*x) + x)*sinh(x)^4 + 4*(x*cosh(x)*e^(4*x) + 2*x*cosh(x)*e^(2*x) + x*cosh(x))*sinh(x)^3 + 2*x*co
sh(x)^2 + 2*(3*x*cosh(x)^2 + (3*x*cosh(x)^2 + x)*e^(4*x) + 2*(3*x*cosh(x)^2 + x)*e^(2*x) + x)*sinh(x)^2 + (x*c
osh(x)^4 + 2*x*cosh(x)^2 + x)*e^(4*x) + 2*(x*cosh(x)^4 + 2*x*cosh(x)^2 + x)*e^(2*x) + 4*(x*cosh(x)^3 + x*cosh(
x) + (x*cosh(x)^3 + x*cosh(x))*e^(4*x) + 2*(x*cosh(x)^3 + x*cosh(x))*e^(2*x))*sinh(x) + x)*dilog(cosh(x) + sin
h(x)) + 2*(x*cosh(x)^4 + (x*e^(4*x) + 2*x*e^(2*x) + x)*sinh(x)^4 + 4*(x*cosh(x)*e^(4*x) + 2*x*cosh(x)*e^(2*x)
+ x*cosh(x))*sinh(x)^3 + 2*x*cosh(x)^2 + 2*(3*x*cosh(x)^2 + (3*x*cosh(x)^2 + x)*e^(4*x) + 2*(3*x*cosh(x)^2 + x
)*e^(2*x) + x)*sinh(x)^2 + (x*cosh(x)^4 + 2*x*cosh(x)^2 + x)*e^(4*x) + 2*(x*cosh(x)^4 + 2*x*cosh(x)^2 + x)*e^(
2*x) + 4*(x*cosh(x)^3 + x*cosh(x) + (x*cosh(x)^3 + x*cosh(x))*e^(4*x) + 2*(x*cosh(x)^3 + x*cosh(x))*e^(2*x))*s
inh(x) + x)*dilog(I*cosh(x) + I*sinh(x)) + 2*(x*cosh(x)^4 + (x*e^(4*x) + 2*x*e^(2*x) + x)*sinh(x)^4 + 4*(x*cos
h(x)*e^(4*x) + 2*x*cosh(x)*e^(2*x) + x*cosh(x))*sinh(x)^3 + 2*x*cosh(x)^2 + 2*(3*x*cosh(x)^2 + (3*x*cosh(x)^2
+ x)*e^(4*x) + 2*(3*x*cosh(x)^2 + x)*e^(2*x) + x)*sinh(x)^2 + (x*cosh(x)^4 + 2*x*cosh(x)^2 + x)*e^(4*x) + 2*(x
*cosh(x)^4 + 2*x*cosh(x)^2 + x)*e^(2*x) + 4*(x*cosh(x)^3 + x*cosh(x) + (x*cosh(x)^3 + x*cosh(x))*e^(4*x) + 2*(
x*cosh(x)^3 + x*cosh(x))*e^(2*x))*sinh(x) + x)*dilog(-I*cosh(x) - I*sinh(x)) - 2*(x*cosh(x)^4 + (x*e^(4*x) + 2
*x*e^(2*x) + x)*sinh(x)^4 + 4*(x*cosh(x)*e^(4*x) + 2*x*cosh(x)*e^(2*x) + x*cosh(x))*sinh(x)^3 + 2*x*cosh(x)^2
+ 2*(3*x*cosh(x)^2 + (3*x*cosh(x)^2 + x)*e^(4*x) + 2*(3*x*cosh(x)^2 + x)*e^(2*x) + x)*sinh(x)^2 + (x*cosh(x)^4
+ 2*x*cosh(x)^2 + x)*e^(4*x) + 2*(x*cosh(x)^4 + 2*x*cosh(x)^2 + x)*e^(2*x) + 4*(x*cosh(x)^3 + x*cosh(x) + (x*
cosh(x)^3 + x*cosh(x))*e^(4*x) + 2*(x*cosh(x)^3 + x*cosh(x))*e^(2*x))*sinh(x) + x)*dilog(-cosh(x) - sinh(x)) +
2*(x*cosh(x)^4 - (x^2 - x)*cosh(x)^2)*e^(4*x) + 4*(x*cosh(x)^4 - (x^2 - x)*cosh(x)^2)*e^(2*x) - (x^2*cosh(x)^
4 + (x^2*e^(4*x) + 2*x^2*e^(2*x) + x^2)*sinh(x)^4 + 2*x^2*cosh(x)^2 + 4*(x^2*cosh(x)*e^(4*x) + 2*x^2*cosh(x)*e
^(2*x) + x^2*cosh(x))*sinh(x)^3 + 2*(3*x^2*cosh(x)^2 + x^2 + (3*x^2*cosh(x)^2 + x^2)*e^(4*x) + 2*(3*x^2*cosh(x
)^2 + x^2)*e^(2*x))*sinh(x)^2 + x^2 + (x^2*cosh(x)^4 + 2*x^2*cosh(x)^2 + x^2)*e^(4*x) + 2*(x^2*cosh(x)^4 + 2*x
^2*cosh(x)^2 + x^2)*e^(2*x) + 4*(x^2*cosh(x)^3 + x^2*cosh(x) + (x^2*cosh(x)^3 + x^2*cosh(x))*e^(4*x) + 2*(x^2*
cosh(x)^3 + x^2*cosh(x))*e^(2*x))*sinh(x))*log(cosh(x) + sinh(x) + 1) - ((e^(4*x) + 2*e^(2*x) + 1)*sinh(x)^4 +
cosh(x)^4 + 4*(cosh(x)*e^(4*x) + 2*cosh(x)*e^(2*x) + cosh(x))*sinh(x)^3 + 2*(3*cosh(x)^2 + (3*cosh(x)^2 + 1)*
e^(4*x) + 2*(3*cosh(x)^2 + 1)*e^(2*x) + 1)*sinh(x)^2 + 2*cosh(x)^2 + (cosh(x)^4 + 2*cosh(x)^2 + 1)*e^(4*x) + 2
*(cosh(x)^4 + 2*cosh(x)^2 + 1)*e^(2*x) + 4*(cosh(x)^3 + (cosh(x)^3 + cosh(x))*e^(4*x) + 2*(cosh(x)^3 + cosh(x)
)*e^(2*x) + cosh(x))*sinh(x) + 1)*log(cosh(x) + sinh(x) + I) - ((e^(4*x) + 2*e^(2*x) + 1)*sinh(x)^4 + cosh(x)^
4 + 4*(cosh(x)*e^(4*x) + 2*cosh(x)*e^(2*x) + cosh(x))*sinh(x)^3 + 2*(3*cosh(x)^2 + (3*cosh(x)^2 + 1)*e^(4*x) +
2*(3*cosh(x)^2 + 1)*e^(2*x) + 1)*sinh(x)^2 + 2*cosh(x)^2 + (cosh(x)^4 + 2*cosh(x)^2 + 1)*e^(4*x) + 2*(cosh(x)
^4 + 2*cosh(x)^2 + 1)*e^(2*x) + 4*(cosh(x)^3 + (cosh(x)^3 + cosh(x))*e^(4*x) + 2*(cosh(x)^3 + cosh(x))*e^(2*x)
+ cosh(x))*sinh(x) + 1)*log(cosh(x) + sinh(x) - I) + (x^2*cosh(x)^4 + (x^2*e^(4*x) + 2*x^2*e^(2*x) + x^2)*sin
h(x)^4 + 2*x^2*cosh(x)^2 + 4*(x^2*cosh(x)*e^(4*x) + 2*x^2*cosh(x)*e^(2*x) + x^2*cosh(x))*sinh(x)^3 + 2*(3*x^2*
cosh(x)^2 + x^2 + (3*x^2*cosh(x)^2 + x^2)*e^(4*x) + 2*(3*x^2*cosh(x)^2 + x^2)*e^(2*x))*sinh(x)^2 + x^2 + (x^2*
cosh(x)^4 + 2*x^2*cosh(x)^2 + x^2)*e^(4*x) + 2*(x^2*cosh(x)^4 + 2*x^2*cosh(x)^2 + x^2)*e^(2*x) + 4*(x^2*cosh(x
)^3 + x^2*cosh(x) + (x^2*cosh(x)^3 + x^2*cosh(x))*e^(4*x) + 2*(x^2*cosh(x)^3 + x^2*cosh(x))*e^(2*x))*sinh(x))*
log(I*cosh(x) + I*sinh(x) + 1) + (x^2*cosh(x)^4 + (x^2*e^(4*x) + 2*x^2*e^(2*x) + x^2)*sinh(x)^4 + 2*x^2*cosh(x
)^2 + 4*(x^2*cosh(x)*e^(4*x) + 2*x^2*cosh(x)*e^(2*x) + x^2*cosh(x))*sinh(x)^3 + 2*(3*x^2*cosh(x)^2 + x^2 + (3*
x^2*cosh(x)^2 + x^2)*e^(4*x) + 2*(3*x^2*cosh(x)^2 + x^2)*e^(2*x))*sinh(x)^2 + x^2 + (x^2*cosh(x)^4 + 2*x^2*cos
h(x)^2 + x^2)*e^(4*x) + 2*(x^2*cosh(x)^4 + 2*x^2*cosh(x)^2 + x^2)*e^(2*x) + 4*(x^2*cosh(x)^3 + x^2*cosh(x) + (
x^2*cosh(x)^3 + x^2*cosh(x))*e^(4*x) + 2*(x^2*cosh(x)^3 + x^2*cosh(x))*e^(2*x))*sinh(x))*log(-I*cosh(x) - I*si
nh(x) + 1) - (x^2*cosh(x)^4 + (x^2*e^(4*x) + 2*x^2*e^(2*x) + x^2)*sinh(x)^4 + 2*x^2*cosh(x)^2 + 4*(x^2*cosh(x)
*e^(4*x) + 2*x^2*cosh(x)*e^(2*x) + x^2*cosh(x))*sinh(x)^3 + 2*(3*x^2*cosh(x)^2 + x^2 + (3*x^2*cosh(x)^2 + x^2)
*e^(4*x) + 2*(3*x^2*cosh(x)^2 + x^2)*e^(2*x))*sinh(x)^2 + x^2 + (x^2*cosh(x)^4 + 2*x^2*cosh(x)^2 + x^2)*e^(4*x
) + 2*(x^2*cosh(x)^4 + 2*x^2*cosh(x)^2 + x^2)*e^(2*x) + 4*(x^2*cosh(x)^3 + x^2*cosh(x) + (x^2*cosh(x)^3 + x^2*
cosh(x))*e^(4*x) + 2*(x^2*cosh(x)^3 + x^2*cosh(x))*e^(2*x))*sinh(x))*log(-cosh(x) - sinh(x) + 1) + 4*(2*x*cosh
(x)^3 - (x^2 - x)*cosh(x) + (2*x*cosh(x)^3 - (x^2 - x)*cosh(x))*e^(4*x) + 2*(2*x*cosh(x)^3 - (x^2 - x)*cosh(x)
)*e^(2*x))*sinh(x))*sqrt(a/(e^(8*x) + 4*e^(6*x) + 6*e^(4*x) + 4*e^(2*x) + 1))*e^(2*x))/(4*cosh(x)*e^(2*x)*sinh
(x)^3 + e^(2*x)*sinh(x)^4 + 2*(3*cosh(x)^2 + 1)*e^(2*x)*sinh(x)^2 + 4*(cosh(x)^3 + cosh(x))*e^(2*x)*sinh(x) +
(cosh(x)^4 + 2*cosh(x)^2 + 1)*e^(2*x))
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{2} \sqrt{a \operatorname{sech}^{4}{\left (x \right )}} \operatorname{csch}{\left (x \right )} \operatorname{sech}{\left (x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x**2*csch(x)*sech(x)*(a*sech(x)**4)**(1/2),x)
[Out]
Integral(x**2*sqrt(a*sech(x)**4)*csch(x)*sech(x), x)
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{a \operatorname{sech}\left (x\right )^{4}} x^{2} \operatorname{csch}\left (x\right ) \operatorname{sech}\left (x\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x^2*csch(x)*sech(x)*(a*sech(x)^4)^(1/2),x, algorithm="giac")
[Out]
integrate(sqrt(a*sech(x)^4)*x^2*csch(x)*sech(x), x)