### 3.842 $$\int \frac{x \text{csch}(x) \text{sech}(x)}{\sqrt{a \text{sech}^2(x)}} \, dx$$

Optimal. Leaf size=59 $-\frac{\text{sech}(x) \text{PolyLog}\left (2,-e^x\right )}{\sqrt{a \text{sech}^2(x)}}+\frac{\text{sech}(x) \text{PolyLog}\left (2,e^x\right )}{\sqrt{a \text{sech}^2(x)}}-\frac{2 x \tanh ^{-1}\left (e^x\right ) \text{sech}(x)}{\sqrt{a \text{sech}^2(x)}}$

[Out]

(-2*x*ArcTanh[E^x]*Sech[x])/Sqrt[a*Sech[x]^2] - (PolyLog[2, -E^x]*Sech[x])/Sqrt[a*Sech[x]^2] + (PolyLog[2, E^x
]*Sech[x])/Sqrt[a*Sech[x]^2]

________________________________________________________________________________________

Rubi [A]  time = 0.701356, antiderivative size = 59, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 16, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.25, Rules used = {6720, 4182, 2279, 2391} $-\frac{\text{sech}(x) \text{PolyLog}\left (2,-e^x\right )}{\sqrt{a \text{sech}^2(x)}}+\frac{\text{sech}(x) \text{PolyLog}\left (2,e^x\right )}{\sqrt{a \text{sech}^2(x)}}-\frac{2 x \tanh ^{-1}\left (e^x\right ) \text{sech}(x)}{\sqrt{a \text{sech}^2(x)}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(-2*x*ArcTanh[E^x]*Sech[x])/Sqrt[a*Sech[x]^2] - (PolyLog[2, -E^x]*Sech[x])/Sqrt[a*Sech[x]^2] + (PolyLog[2, E^x
]*Sech[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 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]

Rubi steps

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

Mathematica [A]  time = 0.0542851, size = 55, normalized size = 0.93 $\frac{\text{sech}(x) \left (\text{PolyLog}\left (2,-e^{-x}\right )-\text{PolyLog}\left (2,e^{-x}\right )+x \left (\log \left (1-e^{-x}\right )-\log \left (e^{-x}+1\right )\right )\right )}{\sqrt{a \text{sech}^2(x)}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [B]  time = 0.079, size = 136, normalized size = 2.3 \begin{align*} -{\frac{x{{\rm e}^{x}}\ln \left ({{\rm e}^{x}}+1 \right ) }{{{\rm e}^{2\,x}}+1}{\frac{1}{\sqrt{{\frac{a{{\rm e}^{2\,x}}}{ \left ({{\rm e}^{2\,x}}+1 \right ) ^{2}}}}}}}-{\frac{{{\rm e}^{x}}{\it polylog} \left ( 2,-{{\rm e}^{x}} \right ) }{{{\rm e}^{2\,x}}+1}{\frac{1}{\sqrt{{\frac{a{{\rm e}^{2\,x}}}{ \left ({{\rm e}^{2\,x}}+1 \right ) ^{2}}}}}}}+{\frac{x{{\rm e}^{x}}\ln \left ( 1-{{\rm e}^{x}} \right ) }{{{\rm e}^{2\,x}}+1}{\frac{1}{\sqrt{{\frac{a{{\rm e}^{2\,x}}}{ \left ({{\rm e}^{2\,x}}+1 \right ) ^{2}}}}}}}+{\frac{{{\rm e}^{x}}{\it polylog} \left ( 2,{{\rm e}^{x}} \right ) }{{{\rm e}^{2\,x}}+1}{\frac{1}{\sqrt{{\frac{a{{\rm e}^{2\,x}}}{ \left ({{\rm e}^{2\,x}}+1 \right ) ^{2}}}}}}} \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]

-1/(a*exp(2*x)/(exp(2*x)+1)^2)^(1/2)/(exp(2*x)+1)*exp(x)*x*ln(exp(x)+1)-1/(a*exp(2*x)/(exp(2*x)+1)^2)^(1/2)/(e
xp(2*x)+1)*exp(x)*polylog(2,-exp(x))+1/(a*exp(2*x)/(exp(2*x)+1)^2)^(1/2)/(exp(2*x)+1)*exp(x)*x*ln(1-exp(x))+1/
(a*exp(2*x)/(exp(2*x)+1)^2)^(1/2)/(exp(2*x)+1)*exp(x)*polylog(2,exp(x))

________________________________________________________________________________________

Maxima [A]  time = 1.81995, size = 49, normalized size = 0.83 \begin{align*} -\frac{x \log \left (e^{x} + 1\right ) +{\rm Li}_2\left (-e^{x}\right )}{\sqrt{a}} + \frac{x \log \left (-e^{x} + 1\right ) +{\rm Li}_2\left (e^{x}\right )}{\sqrt{a}} \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)

________________________________________________________________________________________

Fricas [A]  time = 2.09782, size = 285, normalized size = 4.83 \begin{align*} \frac{{\left ({\left (e^{\left (2 \, x\right )} + 1\right )}{\rm Li}_2\left (\cosh \left (x\right ) + \sinh \left (x\right )\right ) -{\left (e^{\left (2 \, x\right )} + 1\right )}{\rm Li}_2\left (-\cosh \left (x\right ) - \sinh \left (x\right )\right ) -{\left (x e^{\left (2 \, x\right )} + x\right )} \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) + 1\right ) +{\left (x e^{\left (2 \, x\right )} + x\right )} \log \left (-\cosh \left (x\right ) - \sinh \left (x\right ) + 1\right )\right )} \sqrt{\frac{a}{e^{\left (4 \, x\right )} + 2 \, e^{\left (2 \, x\right )} + 1}}}{a} \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]

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

________________________________________________________________________________________

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

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x \operatorname{csch}\left (x\right ) \operatorname{sech}\left (x\right )}{\sqrt{a \operatorname{sech}\left (x\right )^{2}}}\,{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(x*csch(x)*sech(x)/sqrt(a*sech(x)^2), x)