3.9.42 \(\int \frac {x \text {csch}(x) \text {sech}(x)}{\sqrt {a \text {sech}^2(x)}} \, dx\) [842]

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

[Out]

-2*x*arctanh(exp(x))*sech(x)/(a*sech(x)^2)^(1/2)-polylog(2,-exp(x))*sech(x)/(a*sech(x)^2)^(1/2)+polylog(2,exp(
x))*sech(x)/(a*sech(x)^2)^(1/2)

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Rubi [A]
time = 0.40, antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6852, 4267, 2317, 2438} \begin {gather*} -\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)}}-\frac {2 x \tanh ^{-1}\left (e^x\right ) \text {sech}(x)}{\sqrt {a \text {sech}^2(x)}} \end {gather*}

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 2317

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 2438

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 4267

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 6852

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])

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) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^x\right )}{\sqrt {a \text {sech}^2(x)}}+\frac {\text {sech}(x) \text {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*}

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Mathematica [A]
time = 0.04, size = 55, normalized size = 0.93 \begin {gather*} \frac {\left (x \left (\log \left (1-e^{-x}\right )-\log \left (1+e^{-x}\right )\right )+\text {PolyLog}\left (2,-e^{-x}\right )-\text {PolyLog}\left (2,e^{-x}\right )\right ) \text {sech}(x)}{\sqrt {a \text {sech}^2(x)}} \end {gather*}

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]

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(135\) vs. \(2(50)=100\).
time = 1.70, size = 136, normalized size = 2.31

method	result	size
risch	\(-\frac {{\mathrm e}^{x} x \ln \left ({\mathrm e}^{x}+1\right )}{\sqrt {\frac {a \,{\mathrm e}^{2 x}}{\left (1+{\mathrm e}^{2 x}\right )^{2}}}\, \left (1+{\mathrm e}^{2 x}\right )}-\frac {{\mathrm e}^{x} \polylog \left (2, -{\mathrm e}^{x}\right )}{\sqrt {\frac {a \,{\mathrm e}^{2 x}}{\left (1+{\mathrm e}^{2 x}\right )^{2}}}\, \left (1+{\mathrm e}^{2 x}\right )}+\frac {{\mathrm e}^{x} x \ln \left (1-{\mathrm e}^{x}\right )}{\sqrt {\frac {a \,{\mathrm e}^{2 x}}{\left (1+{\mathrm e}^{2 x}\right )^{2}}}\, \left (1+{\mathrm e}^{2 x}\right )}+\frac {{\mathrm e}^{x} \polylog \left (2, {\mathrm e}^{x}\right )}{\sqrt {\frac {a \,{\mathrm e}^{2 x}}{\left (1+{\mathrm e}^{2 x}\right )^{2}}}\, \left (1+{\mathrm e}^{2 x}\right )}\)	\(136\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.49, size = 36, normalized size = 0.61 \begin {gather*} -\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 {gather*}

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)

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Fricas [A]
time = 0.39, size = 91, normalized size = 1.54 \begin {gather*} \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 {gather*}

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x \operatorname {csch}{\left (x \right )} \operatorname {sech}{\left (x \right )}}{\sqrt {a \operatorname {sech}^{2}{\left (x \right )}}}\, dx \end {gather*}

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)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {x}{\mathrm {cosh}\left (x\right )\,\mathrm {sinh}\left (x\right )\,\sqrt {\frac {a}{{\mathrm {cosh}\left (x\right )}^2}}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x/(cosh(x)*sinh(x)*(a/cosh(x)^2)^(1/2)),x)

[Out]

int(x/(cosh(x)*sinh(x)*(a/cosh(x)^2)^(1/2)), x)

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