3.1039 \(\int x \cosh (2 x) \text{sech}(x) \, dx\)

Optimal. Leaf size=43 \[ i \text{PolyLog}\left (2,-i e^x\right )-i \text{PolyLog}\left (2,i e^x\right )-2 x \tan ^{-1}\left (e^x\right )+2 x \sinh (x)-2 \cosh (x) \]

[Out]

-2*x*ArcTan[E^x] - 2*Cosh[x] + I*PolyLog[2, (-I)*E^x] - I*PolyLog[2, I*E^x] + 2*x*Sinh[x]

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Rubi [A]  time = 0.0725748, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 7, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.875, Rules used = {5473, 3296, 2638, 5449, 4180, 2279, 2391} \[ i \text{PolyLog}\left (2,-i e^x\right )-i \text{PolyLog}\left (2,i e^x\right )-2 x \tan ^{-1}\left (e^x\right )+2 x \sinh (x)-2 \cosh (x) \]

Antiderivative was successfully verified.

[In]

Int[x*Cosh[2*x]*Sech[x],x]

[Out]

-2*x*ArcTan[E^x] - 2*Cosh[x] + I*PolyLog[2, (-I)*E^x] - I*PolyLog[2, I*E^x] + 2*x*Sinh[x]

Rule 5473

Int[((e_.) + (f_.)*(x_))^(m_.)*(F_)[(a_.) + (b_.)*(x_)]^(p_.)*(G_)[(c_.) + (d_.)*(x_)]^(q_.), x_Symbol] :> Int
[ExpandTrigExpand[(e + f*x)^m*G[c + d*x]^q, F, c + d*x, p, b/d, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && M
emberQ[{Sinh, Cosh}, F] && MemberQ[{Sech, Csch}, G] && IGtQ[p, 0] && IGtQ[q, 0] && EqQ[b*c - a*d, 0] && IGtQ[b
/d, 1]

Rule 3296

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> -Simp[((c + d*x)^m*Cos[e + f*x])/f, x] +
Dist[(d*m)/f, Int[(c + d*x)^(m - 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]

Rule 2638

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

Rule 5449

Int[((c_.) + (d_.)*(x_))^(m_.)*Sinh[(a_.) + (b_.)*(x_)]^(n_.)*Tanh[(a_.) + (b_.)*(x_)]^(p_.), x_Symbol] :> Int
[(c + d*x)^m*Sinh[a + b*x]^n*Tanh[a + b*x]^(p - 2), x] - Int[(c + d*x)^m*Sinh[a + b*x]^(n - 2)*Tanh[a + b*x]^p
, x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && IGtQ[p, 0]

Rule 4180

Int[csc[(e_.) + Pi*(k_.) + (Complex[0, fz_])*(f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[(-2*(c
+ d*x)^m*ArcTanh[E^(-(I*e) + f*fz*x)/E^(I*k*Pi)])/(f*fz*I), x] + (-Dist[(d*m)/(f*fz*I), Int[(c + d*x)^(m - 1)*
Log[1 - E^(-(I*e) + f*fz*x)/E^(I*k*Pi)], x], x] + Dist[(d*m)/(f*fz*I), Int[(c + d*x)^(m - 1)*Log[1 + E^(-(I*e)
+ f*fz*x)/E^(I*k*Pi)], x], x]) /; FreeQ[{c, d, e, f, fz}, x] && IntegerQ[2*k] && 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 x \cosh (2 x) \text{sech}(x) \, dx &=\int (x \cosh (x)+x \sinh (x) \tanh (x)) \, dx\\ &=\int x \cosh (x) \, dx+\int x \sinh (x) \tanh (x) \, dx\\ &=x \sinh (x)+\int x \cosh (x) \, dx-\int x \text{sech}(x) \, dx-\int \sinh (x) \, dx\\ &=-2 x \tan ^{-1}\left (e^x\right )-\cosh (x)+2 x \sinh (x)+i \int \log \left (1-i e^x\right ) \, dx-i \int \log \left (1+i e^x\right ) \, dx-\int \sinh (x) \, dx\\ &=-2 x \tan ^{-1}\left (e^x\right )-2 \cosh (x)+2 x \sinh (x)+i \operatorname{Subst}\left (\int \frac{\log (1-i x)}{x} \, dx,x,e^x\right )-i \operatorname{Subst}\left (\int \frac{\log (1+i x)}{x} \, dx,x,e^x\right )\\ &=-2 x \tan ^{-1}\left (e^x\right )-2 \cosh (x)+i \text{Li}_2\left (-i e^x\right )-i \text{Li}_2\left (i e^x\right )+2 x \sinh (x)\\ \end{align*}

Mathematica [A]  time = 0.0318339, size = 71, normalized size = 1.65 \[ i \left (\text{PolyLog}\left (2,-i e^{-x}\right )-\text{PolyLog}\left (2,i e^{-x}\right )\right )+i x \left (\log \left (1-i e^{-x}\right )-\log \left (1+i e^{-x}\right )\right )+2 x \sinh (x)-2 \cosh (x) \]

Antiderivative was successfully verified.

[In]

Integrate[x*Cosh[2*x]*Sech[x],x]

[Out]

-2*Cosh[x] + I*x*(Log[1 - I/E^x] - Log[1 + I/E^x]) + I*(PolyLog[2, (-I)/E^x] - PolyLog[2, I/E^x]) + 2*x*Sinh[x
]

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Maple [A]  time = 0.038, size = 68, normalized size = 1.6 \begin{align*} 2\, \left ( -1/2+x/2 \right ){{\rm e}^{x}}+2\, \left ( -1/2-x/2 \right ){{\rm e}^{-x}}+ix\ln \left ( 1+i{{\rm e}^{x}} \right ) -ix\ln \left ( 1-i{{\rm e}^{x}} \right ) +i{\it dilog} \left ( 1+i{{\rm e}^{x}} \right ) -i{\it dilog} \left ( 1-i{{\rm e}^{x}} \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*cosh(2*x)*sech(x),x)

[Out]

2*(-1/2+1/2*x)*exp(x)+2*(-1/2-1/2*x)*exp(-x)+I*x*ln(1+I*exp(x))-I*x*ln(1-I*exp(x))+I*dilog(1+I*exp(x))-I*dilog
(1-I*exp(x))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -{\left (x + 1\right )} e^{\left (-x\right )} +{\left (x - 1\right )} e^{x} - 2 \, \int \frac{x e^{x}}{e^{\left (2 \, x\right )} + 1}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*cosh(2*x)*sech(x),x, algorithm="maxima")

[Out]

-(x + 1)*e^(-x) + (x - 1)*e^x - 2*integrate(x*e^x/(e^(2*x) + 1), x)

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Fricas [B]  time = 1.835, size = 450, normalized size = 10.47 \begin{align*} \frac{{\left (x - 1\right )} \cosh \left (x\right )^{2} + 2 \,{\left (x - 1\right )} \cosh \left (x\right ) \sinh \left (x\right ) +{\left (x - 1\right )} \sinh \left (x\right )^{2} +{\left (-i \, \cosh \left (x\right ) - i \, \sinh \left (x\right )\right )}{\rm Li}_2\left (i \, \cosh \left (x\right ) + i \, \sinh \left (x\right )\right ) +{\left (i \, \cosh \left (x\right ) + i \, \sinh \left (x\right )\right )}{\rm Li}_2\left (-i \, \cosh \left (x\right ) - i \, \sinh \left (x\right )\right ) +{\left (i \, x \cosh \left (x\right ) + i \, x \sinh \left (x\right )\right )} \log \left (i \, \cosh \left (x\right ) + i \, \sinh \left (x\right ) + 1\right ) +{\left (-i \, x \cosh \left (x\right ) - i \, x \sinh \left (x\right )\right )} \log \left (-i \, \cosh \left (x\right ) - i \, \sinh \left (x\right ) + 1\right ) - x - 1}{\cosh \left (x\right ) + \sinh \left (x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*cosh(2*x)*sech(x),x, algorithm="fricas")

[Out]

((x - 1)*cosh(x)^2 + 2*(x - 1)*cosh(x)*sinh(x) + (x - 1)*sinh(x)^2 + (-I*cosh(x) - I*sinh(x))*dilog(I*cosh(x)
+ I*sinh(x)) + (I*cosh(x) + I*sinh(x))*dilog(-I*cosh(x) - I*sinh(x)) + (I*x*cosh(x) + I*x*sinh(x))*log(I*cosh(
x) + I*sinh(x) + 1) + (-I*x*cosh(x) - I*x*sinh(x))*log(-I*cosh(x) - I*sinh(x) + 1) - x - 1)/(cosh(x) + sinh(x)
)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x \cosh{\left (2 x \right )} \operatorname{sech}{\left (x \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*cosh(2*x)*sech(x),x)

[Out]

Integral(x*cosh(2*x)*sech(x), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x \cosh \left (2 \, x\right ) \operatorname{sech}\left (x\right )\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*cosh(2*x)*sech(x),x, algorithm="giac")

[Out]

integrate(x*cosh(2*x)*sech(x), x)