3.848 \(\int x \text{csch}(x) \text{sech}(x) \sqrt{a \text{sech}^2(x)} \, dx\)

Optimal. Leaf size=88 \[ -\cosh (x) \text{PolyLog}\left (2,-e^x\right ) \sqrt{a \text{sech}^2(x)}+\cosh (x) \text{PolyLog}\left (2,e^x\right ) \sqrt{a \text{sech}^2(x)}+x \sqrt{a \text{sech}^2(x)}-2 x \cosh (x) \tanh ^{-1}\left (e^x\right ) \sqrt{a \text{sech}^2(x)}-\cosh (x) \sqrt{a \text{sech}^2(x)} \tan ^{-1}(\sinh (x)) \]

[Out]

x*Sqrt[a*Sech[x]^2] - ArcTan[Sinh[x]]*Cosh[x]*Sqrt[a*Sech[x]^2] - 2*x*ArcTanh[E^x]*Cosh[x]*Sqrt[a*Sech[x]^2] -
Cosh[x]*PolyLog[2, -E^x]*Sqrt[a*Sech[x]^2] + Cosh[x]*PolyLog[2, E^x]*Sqrt[a*Sech[x]^2]

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Rubi [A]  time = 0.359617, antiderivative size = 88, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 10, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.625, Rules used = {6720, 2622, 321, 207, 5462, 6271, 4182, 2279, 2391, 3770} \[ -\cosh (x) \text{PolyLog}\left (2,-e^x\right ) \sqrt{a \text{sech}^2(x)}+\cosh (x) \text{PolyLog}\left (2,e^x\right ) \sqrt{a \text{sech}^2(x)}+x \sqrt{a \text{sech}^2(x)}-2 x \cosh (x) \tanh ^{-1}\left (e^x\right ) \sqrt{a \text{sech}^2(x)}-\cosh (x) \sqrt{a \text{sech}^2(x)} \tan ^{-1}(\sinh (x)) \]

Antiderivative was successfully verified.

[In]

Int[x*Csch[x]*Sech[x]*Sqrt[a*Sech[x]^2],x]

[Out]

x*Sqrt[a*Sech[x]^2] - ArcTan[Sinh[x]]*Cosh[x]*Sqrt[a*Sech[x]^2] - 2*x*ArcTanh[E^x]*Cosh[x]*Sqrt[a*Sech[x]^2] -
Cosh[x]*PolyLog[2, -E^x]*Sqrt[a*Sech[x]^2] + Cosh[x]*PolyLog[2, E^x]*Sqrt[a*Sech[x]^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 2622

Int[csc[(e_.) + (f_.)*(x_)]^(n_.)*((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Dist[1/(f*a^n), Subst[Int
[x^(m + n - 1)/(-1 + x^2/a^2)^((n + 1)/2), x], x, a*Sec[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n
+ 1)/2] &&  !(IntegerQ[(m + 1)/2] && LtQ[0, m, n])

Rule 321

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n + 1)*(a + b*x^n
)^(p + 1))/(b*(m + n*p + 1)), x] - Dist[(a*c^n*(m - n + 1))/(b*(m + n*p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

Rule 207

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

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 6271

Int[ArcTanh[u_], x_Symbol] :> Simp[x*ArcTanh[u], x] - Int[SimplifyIntegrand[(x*D[u, x])/(1 - u^2), x], x] /; I
nverseFunctionFreeQ[u, x]

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 3770

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]

Rubi steps

\begin{align*} \int x \text{csch}(x) \text{sech}(x) \sqrt{a \text{sech}^2(x)} \, dx &=\left (\cosh (x) \sqrt{a \text{sech}^2(x)}\right ) \int x \text{csch}(x) \text{sech}^2(x) \, dx\\ &=x \sqrt{a \text{sech}^2(x)}-x \tanh ^{-1}(\cosh (x)) \cosh (x) \sqrt{a \text{sech}^2(x)}-\left (\cosh (x) \sqrt{a \text{sech}^2(x)}\right ) \int \left (-\tanh ^{-1}(\cosh (x))+\text{sech}(x)\right ) \, dx\\ &=x \sqrt{a \text{sech}^2(x)}-x \tanh ^{-1}(\cosh (x)) \cosh (x) \sqrt{a \text{sech}^2(x)}+\left (\cosh (x) \sqrt{a \text{sech}^2(x)}\right ) \int \tanh ^{-1}(\cosh (x)) \, dx-\left (\cosh (x) \sqrt{a \text{sech}^2(x)}\right ) \int \text{sech}(x) \, dx\\ &=x \sqrt{a \text{sech}^2(x)}-\tan ^{-1}(\sinh (x)) \cosh (x) \sqrt{a \text{sech}^2(x)}+\left (\cosh (x) \sqrt{a \text{sech}^2(x)}\right ) \int x \text{csch}(x) \, dx\\ &=x \sqrt{a \text{sech}^2(x)}-\tan ^{-1}(\sinh (x)) \cosh (x) \sqrt{a \text{sech}^2(x)}-2 x \tanh ^{-1}\left (e^x\right ) \cosh (x) \sqrt{a \text{sech}^2(x)}-\left (\cosh (x) \sqrt{a \text{sech}^2(x)}\right ) \int \log \left (1-e^x\right ) \, dx+\left (\cosh (x) \sqrt{a \text{sech}^2(x)}\right ) \int \log \left (1+e^x\right ) \, dx\\ &=x \sqrt{a \text{sech}^2(x)}-\tan ^{-1}(\sinh (x)) \cosh (x) \sqrt{a \text{sech}^2(x)}-2 x \tanh ^{-1}\left (e^x\right ) \cosh (x) \sqrt{a \text{sech}^2(x)}-\left (\cosh (x) \sqrt{a \text{sech}^2(x)}\right ) \operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^x\right )+\left (\cosh (x) \sqrt{a \text{sech}^2(x)}\right ) \operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,e^x\right )\\ &=x \sqrt{a \text{sech}^2(x)}-\tan ^{-1}(\sinh (x)) \cosh (x) \sqrt{a \text{sech}^2(x)}-2 x \tanh ^{-1}\left (e^x\right ) \cosh (x) \sqrt{a \text{sech}^2(x)}-\cosh (x) \text{Li}_2\left (-e^x\right ) \sqrt{a \text{sech}^2(x)}+\cosh (x) \text{Li}_2\left (e^x\right ) \sqrt{a \text{sech}^2(x)}\\ \end{align*}

Mathematica [A]  time = 0.056472, size = 74, normalized size = 0.84 \[ \sqrt{a \text{sech}^2(x)} \left (\cosh (x) \text{PolyLog}\left (2,-e^{-x}\right )-\cosh (x) \text{PolyLog}\left (2,e^{-x}\right )+x+x \log \left (1-e^{-x}\right ) \cosh (x)-x \log \left (e^{-x}+1\right ) \cosh (x)-2 \cosh (x) \tan ^{-1}\left (\tanh \left (\frac{x}{2}\right )\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x*Csch[x]*Sech[x]*Sqrt[a*Sech[x]^2],x]

[Out]

(x - 2*ArcTan[Tanh[x/2]]*Cosh[x] + x*Cosh[x]*Log[1 - E^(-x)] - x*Cosh[x]*Log[1 + E^(-x)] + Cosh[x]*PolyLog[2,
-E^(-x)] - Cosh[x]*PolyLog[2, E^(-x)])*Sqrt[a*Sech[x]^2]

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Maple [A]  time = 0.092, size = 150, normalized size = 1.7 \begin{align*} 2\,\sqrt{{\frac{a{{\rm e}^{2\,x}}}{ \left ({{\rm e}^{2\,x}}+1 \right ) ^{2}}}}x-2\,\sqrt{{\frac{a{{\rm e}^{2\,x}}}{ \left ({{\rm e}^{2\,x}}+1 \right ) ^{2}}}}{{\rm e}^{-x}} \left ({{\rm e}^{2\,x}}+1 \right ) \arctan \left ({{\rm e}^{x}} \right ) -\sqrt{{\frac{a{{\rm e}^{2\,x}}}{ \left ({{\rm e}^{2\,x}}+1 \right ) ^{2}}}}{{\rm e}^{-x}} \left ({{\rm e}^{2\,x}}+1 \right ){\it dilog} \left ({{\rm e}^{x}}+1 \right ) -\sqrt{{\frac{a{{\rm e}^{2\,x}}}{ \left ({{\rm e}^{2\,x}}+1 \right ) ^{2}}}}{{\rm e}^{-x}} \left ({{\rm e}^{2\,x}}+1 \right ) x\ln \left ({{\rm e}^{x}}+1 \right ) -\sqrt{{\frac{a{{\rm e}^{2\,x}}}{ \left ({{\rm e}^{2\,x}}+1 \right ) ^{2}}}}{{\rm e}^{-x}} \left ({{\rm e}^{2\,x}}+1 \right ){\it dilog} \left ({{\rm e}^{x}} \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*csch(x)*sech(x)*(a*sech(x)^2)^(1/2),x)

[Out]

2*(a*exp(2*x)/(exp(2*x)+1)^2)^(1/2)*x-2*(a*exp(2*x)/(exp(2*x)+1)^2)^(1/2)*exp(-x)*(exp(2*x)+1)*arctan(exp(x))-
(a*exp(2*x)/(exp(2*x)+1)^2)^(1/2)*exp(-x)*(exp(2*x)+1)*dilog(exp(x)+1)-(a*exp(2*x)/(exp(2*x)+1)^2)^(1/2)*exp(-
x)*(exp(2*x)+1)*x*ln(exp(x)+1)-(a*exp(2*x)/(exp(2*x)+1)^2)^(1/2)*exp(-x)*(exp(2*x)+1)*dilog(exp(x))

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Maxima [A]  time = 1.78432, size = 81, normalized size = 0.92 \begin{align*} -{\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} - 2 \, \sqrt{a} \arctan \left (e^{x}\right ) + \frac{2 \, \sqrt{a} x e^{x}}{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)^2)^(1/2),x, algorithm="maxima")

[Out]

-(x*log(e^x + 1) + dilog(-e^x))*sqrt(a) + (x*log(-e^x + 1) + dilog(e^x))*sqrt(a) - 2*sqrt(a)*arctan(e^x) + 2*s
qrt(a)*x*e^x/(e^(2*x) + 1)

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Fricas [B]  time = 2.12159, size = 1173, normalized size = 13.33 \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)^2)^(1/2),x, algorithm="fricas")

[Out]

(2*x*cosh(x)*e^(2*x) - 2*((e^(2*x) + 1)*sinh(x)^2 + cosh(x)^2 + (cosh(x)^2 + 1)*e^(2*x) + 2*(cosh(x)*e^(2*x) +
cosh(x))*sinh(x) + 1)*arctan(cosh(x) + sinh(x)) + 2*x*cosh(x) + ((e^(2*x) + 1)*sinh(x)^2 + cosh(x)^2 + (cosh(
x)^2 + 1)*e^(2*x) + 2*(cosh(x)*e^(2*x) + cosh(x))*sinh(x) + 1)*dilog(cosh(x) + sinh(x)) - ((e^(2*x) + 1)*sinh(
x)^2 + cosh(x)^2 + (cosh(x)^2 + 1)*e^(2*x) + 2*(cosh(x)*e^(2*x) + cosh(x))*sinh(x) + 1)*dilog(-cosh(x) - sinh(
x)) - (x*cosh(x)^2 + (x*e^(2*x) + x)*sinh(x)^2 + (x*cosh(x)^2 + x)*e^(2*x) + 2*(x*cosh(x)*e^(2*x) + x*cosh(x))
*sinh(x) + x)*log(cosh(x) + sinh(x) + 1) + (x*cosh(x)^2 + (x*e^(2*x) + x)*sinh(x)^2 + (x*cosh(x)^2 + x)*e^(2*x
) + 2*(x*cosh(x)*e^(2*x) + x*cosh(x))*sinh(x) + x)*log(-cosh(x) - sinh(x) + 1) + 2*(x*e^(2*x) + x)*sinh(x))*sq
rt(a/(e^(4*x) + 2*e^(2*x) + 1))*e^x/(2*cosh(x)*e^x*sinh(x) + e^x*sinh(x)^2 + (cosh(x)^2 + 1)*e^x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x \sqrt{a \operatorname{sech}^{2}{\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)**2)**(1/2),x)

[Out]

Integral(x*sqrt(a*sech(x)**2)*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 )^{2}} 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)^2)^(1/2),x, algorithm="giac")

[Out]

integrate(sqrt(a*sech(x)^2)*x*csch(x)*sech(x), x)