3.851 \(\int x \text{csch}(x) \text{sech}(x) \sqrt{a \text{sech}^4(x)} \, dx\)
Optimal. Leaf size=132 \[ -\frac{1}{2} \cosh ^2(x) \text{PolyLog}\left (2,-e^{2 x}\right ) \sqrt{a \text{sech}^4(x)}+\frac{1}{2} \cosh ^2(x) \text{PolyLog}\left (2,e^{2 x}\right ) \sqrt{a \text{sech}^4(x)}+\frac{1}{2} x \cosh ^2(x) \sqrt{a \text{sech}^4(x)}-\frac{1}{2} x \sinh ^2(x) \sqrt{a \text{sech}^4(x)}-2 x \cosh ^2(x) \tanh ^{-1}\left (e^{2 x}\right ) \sqrt{a \text{sech}^4(x)}-\frac{1}{2} \sinh (x) \cosh (x) \sqrt{a \text{sech}^4(x)} \]
[Out]
(x*Cosh[x]^2*Sqrt[a*Sech[x]^4])/2 - 2*x*ArcTanh[E^(2*x)]*Cosh[x]^2*Sqrt[a*Sech[x]^4] - (Cosh[x]^2*PolyLog[2, -
E^(2*x)]*Sqrt[a*Sech[x]^4])/2 + (Cosh[x]^2*PolyLog[2, E^(2*x)]*Sqrt[a*Sech[x]^4])/2 - (Cosh[x]*Sqrt[a*Sech[x]^
4]*Sinh[x])/2 - (x*Sqrt[a*Sech[x]^4]*Sinh[x]^2)/2
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Rubi [A] time = 0.407189, antiderivative size = 132, normalized size of antiderivative =
1., number of steps used = 12, number of rules used = 11, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) =
0.688, Rules used = {6720, 2620, 14, 5462, 2548, 5461, 4182, 2279, 2391, 3473, 8}
\[ -\frac{1}{2} \cosh ^2(x) \text{PolyLog}\left (2,-e^{2 x}\right ) \sqrt{a \text{sech}^4(x)}+\frac{1}{2} \cosh ^2(x) \text{PolyLog}\left (2,e^{2 x}\right ) \sqrt{a \text{sech}^4(x)}+\frac{1}{2} x \cosh ^2(x) \sqrt{a \text{sech}^4(x)}-\frac{1}{2} x \sinh ^2(x) \sqrt{a \text{sech}^4(x)}-2 x \cosh ^2(x) \tanh ^{-1}\left (e^{2 x}\right ) \sqrt{a \text{sech}^4(x)}-\frac{1}{2} \sinh (x) \cosh (x) \sqrt{a \text{sech}^4(x)} \]
Antiderivative was successfully verified.
[In]
Int[x*Csch[x]*Sech[x]*Sqrt[a*Sech[x]^4],x]
[Out]
(x*Cosh[x]^2*Sqrt[a*Sech[x]^4])/2 - 2*x*ArcTanh[E^(2*x)]*Cosh[x]^2*Sqrt[a*Sech[x]^4] - (Cosh[x]^2*PolyLog[2, -
E^(2*x)]*Sqrt[a*Sech[x]^4])/2 + (Cosh[x]^2*PolyLog[2, E^(2*x)]*Sqrt[a*Sech[x]^4])/2 - (Cosh[x]*Sqrt[a*Sech[x]^
4]*Sinh[x])/2 - (x*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 2548
Int[Log[u_], x_Symbol] :> Simp[x*Log[u], x] - Int[SimplifyIntegrand[(x*D[u, x])/u, x], x] /; InverseFunctionFr
eeQ[u, x]
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 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 3473
Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*(b*Tan[c + d*x])^(n - 1))/(d*(n - 1)), x] - Dis
t[b^2, Int[(b*Tan[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1]
Rule 8
Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]
Rubi steps
\begin{align*} \int x \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 \text{csch}(x) \text{sech}^3(x) \, dx\\ &=x \cosh ^2(x) \log (\tanh (x)) \sqrt{a \text{sech}^4(x)}-\frac{1}{2} x \sqrt{a \text{sech}^4(x)} \sinh ^2(x)-\left (\cosh ^2(x) \sqrt{a \text{sech}^4(x)}\right ) \int \left (\log (\tanh (x))-\frac{\tanh ^2(x)}{2}\right ) \, dx\\ &=x \cosh ^2(x) \log (\tanh (x)) \sqrt{a \text{sech}^4(x)}-\frac{1}{2} x \sqrt{a \text{sech}^4(x)} \sinh ^2(x)+\frac{1}{2} \left (\cosh ^2(x) \sqrt{a \text{sech}^4(x)}\right ) \int \tanh ^2(x) \, dx-\left (\cosh ^2(x) \sqrt{a \text{sech}^4(x)}\right ) \int \log (\tanh (x)) \, dx\\ &=-\frac{1}{2} \cosh (x) \sqrt{a \text{sech}^4(x)} \sinh (x)-\frac{1}{2} x \sqrt{a \text{sech}^4(x)} \sinh ^2(x)+\frac{1}{2} \left (\cosh ^2(x) \sqrt{a \text{sech}^4(x)}\right ) \int 1 \, dx+\left (\cosh ^2(x) \sqrt{a \text{sech}^4(x)}\right ) \int x \text{csch}(x) \text{sech}(x) \, dx\\ &=\frac{1}{2} x \cosh ^2(x) \sqrt{a \text{sech}^4(x)}-\frac{1}{2} \cosh (x) \sqrt{a \text{sech}^4(x)} \sinh (x)-\frac{1}{2} x \sqrt{a \text{sech}^4(x)} \sinh ^2(x)+\left (2 \cosh ^2(x) \sqrt{a \text{sech}^4(x)}\right ) \int x \text{csch}(2 x) \, dx\\ &=\frac{1}{2} x \cosh ^2(x) \sqrt{a \text{sech}^4(x)}-2 x \tanh ^{-1}\left (e^{2 x}\right ) \cosh ^2(x) \sqrt{a \text{sech}^4(x)}-\frac{1}{2} \cosh (x) \sqrt{a \text{sech}^4(x)} \sinh (x)-\frac{1}{2} x \sqrt{a \text{sech}^4(x)} \sinh ^2(x)-\left (\cosh ^2(x) \sqrt{a \text{sech}^4(x)}\right ) \int \log \left (1-e^{2 x}\right ) \, dx+\left (\cosh ^2(x) \sqrt{a \text{sech}^4(x)}\right ) \int \log \left (1+e^{2 x}\right ) \, dx\\ &=\frac{1}{2} x \cosh ^2(x) \sqrt{a \text{sech}^4(x)}-2 x \tanh ^{-1}\left (e^{2 x}\right ) \cosh ^2(x) \sqrt{a \text{sech}^4(x)}-\frac{1}{2} \cosh (x) \sqrt{a \text{sech}^4(x)} \sinh (x)-\frac{1}{2} x \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{\log (1-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{\log (1+x)}{x} \, dx,x,e^{2 x}\right )\\ &=\frac{1}{2} x \cosh ^2(x) \sqrt{a \text{sech}^4(x)}-2 x \tanh ^{-1}\left (e^{2 x}\right ) \cosh ^2(x) \sqrt{a \text{sech}^4(x)}-\frac{1}{2} \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}_2\left (e^{2 x}\right ) \sqrt{a \text{sech}^4(x)}-\frac{1}{2} \cosh (x) \sqrt{a \text{sech}^4(x)} \sinh (x)-\frac{1}{2} x \sqrt{a \text{sech}^4(x)} \sinh ^2(x)\\ \end{align*}
Mathematica [A] time = 0.220251, size = 71, normalized size = 0.54 \[ \frac{1}{2} \cosh ^2(x) \sqrt{a \text{sech}^4(x)} \left (\text{PolyLog}\left (2,-e^{-2 x}\right )-\text{PolyLog}\left (2,e^{-2 x}\right )+2 x \log \left (1-e^{-2 x}\right )-2 x \log \left (e^{-2 x}+1\right )-\tanh (x)+x \text{sech}^2(x)\right ) \]
Antiderivative was successfully verified.
[In]
Integrate[x*Csch[x]*Sech[x]*Sqrt[a*Sech[x]^4],x]
[Out]
(Cosh[x]^2*Sqrt[a*Sech[x]^4]*(2*x*Log[1 - E^(-2*x)] - 2*x*Log[1 + E^(-2*x)] + PolyLog[2, -E^(-2*x)] - PolyLog[
2, E^(-2*x)] + x*Sech[x]^2 - Tanh[x]))/2
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Maple [B] time = 0.071, size = 252, normalized size = 1.9 \begin{align*} \sqrt{{\frac{{{\rm e}^{4\,x}}a}{ \left ({{\rm e}^{2\,x}}+1 \right ) ^{4}}}}{{\rm e}^{-2\,x}} \left ( 2\,x{{\rm e}^{2\,x}}+{{\rm e}^{2\,x}}+1 \right ) +\sqrt{{\frac{{{\rm e}^{4\,x}}a}{ \left ({{\rm e}^{2\,x}}+1 \right ) ^{4}}}}{{\rm e}^{-2\,x}} \left ({{\rm e}^{2\,x}}+1 \right ) ^{2}x\ln \left ({{\rm e}^{x}}+1 \right ) +\sqrt{{\frac{{{\rm e}^{4\,x}}a}{ \left ({{\rm e}^{2\,x}}+1 \right ) ^{4}}}}{{\rm e}^{-2\,x}} \left ({{\rm e}^{2\,x}}+1 \right ) ^{2}{\it polylog} \left ( 2,-{{\rm e}^{x}} \right ) +\sqrt{{\frac{{{\rm e}^{4\,x}}a}{ \left ({{\rm e}^{2\,x}}+1 \right ) ^{4}}}}{{\rm e}^{-2\,x}} \left ({{\rm e}^{2\,x}}+1 \right ) ^{2}x\ln \left ( 1-{{\rm e}^{x}} \right ) +\sqrt{{\frac{{{\rm e}^{4\,x}}a}{ \left ({{\rm e}^{2\,x}}+1 \right ) ^{4}}}}{{\rm e}^{-2\,x}} \left ({{\rm e}^{2\,x}}+1 \right ) ^{2}{\it polylog} \left ( 2,{{\rm e}^{x}} \right ) -\sqrt{{\frac{{{\rm e}^{4\,x}}a}{ \left ({{\rm e}^{2\,x}}+1 \right ) ^{4}}}}{{\rm e}^{-2\,x}} \left ({{\rm e}^{2\,x}}+1 \right ) ^{2}x\ln \left ({{\rm e}^{2\,x}}+1 \right ) -{\frac{{{\rm e}^{-2\,x}} \left ({{\rm e}^{2\,x}}+1 \right ) ^{2}{\it polylog} \left ( 2,-{{\rm e}^{2\,x}} \right ) }{2}\sqrt{{\frac{{{\rm e}^{4\,x}}a}{ \left ({{\rm e}^{2\,x}}+1 \right ) ^{4}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(x*csch(x)*sech(x)*(a*sech(x)^4)^(1/2),x)
[Out]
(a*exp(4*x)/(exp(2*x)+1)^4)^(1/2)*exp(-2*x)*(2*x*exp(2*x)+exp(2*x)+1)+(a*exp(4*x)/(exp(2*x)+1)^4)^(1/2)*exp(-2
*x)*(exp(2*x)+1)^2*x*ln(exp(x)+1)+(a*exp(4*x)/(exp(2*x)+1)^4)^(1/2)*exp(-2*x)*(exp(2*x)+1)^2*polylog(2,-exp(x)
)+(a*exp(4*x)/(exp(2*x)+1)^4)^(1/2)*exp(-2*x)*(exp(2*x)+1)^2*x*ln(1-exp(x))+(a*exp(4*x)/(exp(2*x)+1)^4)^(1/2)*
exp(-2*x)*(exp(2*x)+1)^2*polylog(2,exp(x))-(a*exp(4*x)/(exp(2*x)+1)^4)^(1/2)*exp(-2*x)*(exp(2*x)+1)^2*x*ln(exp
(2*x)+1)-1/2*(a*exp(4*x)/(exp(2*x)+1)^4)^(1/2)*exp(-2*x)*(exp(2*x)+1)^2*polylog(2,-exp(2*x))
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Maxima [A] time = 1.70466, size = 124, normalized size = 0.94 \begin{align*} -\frac{1}{2} \,{\left (2 \, x \log \left (e^{\left (2 \, x\right )} + 1\right ) +{\rm Li}_2\left (-e^{\left (2 \, x\right )}\right )\right )} \sqrt{a} +{\left (x \log \left (e^{x} + 1\right ) +{\rm Li}_2\left (-e^{x}\right )\right )} \sqrt{a} +{\left (x \log \left (-e^{x} + 1\right ) +{\rm Li}_2\left (e^{x}\right )\right )} \sqrt{a} + \frac{{\left (2 \, \sqrt{a} x + \sqrt{a}\right )} e^{\left (2 \, x\right )} + \sqrt{a}}{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*csch(x)*sech(x)*(a*sech(x)^4)^(1/2),x, algorithm="maxima")
[Out]
-1/2*(2*x*log(e^(2*x) + 1) + dilog(-e^(2*x)))*sqrt(a) + (x*log(e^x + 1) + dilog(-e^x))*sqrt(a) + (x*log(-e^x +
1) + dilog(e^x))*sqrt(a) + ((2*sqrt(a)*x + sqrt(a))*e^(2*x) + sqrt(a))/(e^(4*x) + 2*e^(2*x) + 1)
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Fricas [C] time = 2.53856, size = 5553, normalized size = 42.07 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(x*csch(x)*sech(x)*(a*sech(x)^4)^(1/2),x, algorithm="fricas")
[Out]
((2*x + 1)*cosh(x)^2 + ((2*x + 1)*e^(4*x) + 2*(2*x + 1)*e^(2*x) + 2*x + 1)*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)*dilog(cosh(x) + sinh(x)) - ((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 + cos
h(x))*e^(2*x) + cosh(x))*sinh(x) + 1)*dilog(I*cosh(x) + I*sinh(x)) - ((e^(4*x) + 2*e^(2*x) + 1)*sinh(x)^4 + co
sh(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*(c
osh(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)*dilog(-I*cosh(x) - I*sinh(x)) + ((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)*dilog(-cosh(x) - sinh(x)) + ((2*x + 1)*cosh(x)^2 + 1)*e^(4*x) + 2*((2*x + 1)*cosh(x)^
2 + 1)*e^(2*x) + (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)*log(cosh(x) + sinh(x) + 1) - (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)*log(I*cosh(x) + I*sinh(x) + 1) - (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)*log(-I*cosh(x) - I*sinh(x
) + 1) + (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))*si
nh(x) + x)*log(-cosh(x) - sinh(x) + 1) + 2*((2*x + 1)*cosh(x)*e^(4*x) + 2*(2*x + 1)*cosh(x)*e^(2*x) + (2*x + 1
)*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)/(4*cosh(x)*e^(2*x)*s
inh(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 \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*csch(x)*sech(x)*(a*sech(x)**4)**(1/2),x)
[Out]
Integral(x*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 \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*csch(x)*sech(x)*(a*sech(x)^4)^(1/2),x, algorithm="giac")
[Out]
integrate(sqrt(a*sech(x)^4)*x*csch(x)*sech(x), x)