3.601 \(\int e^{-2 x} \text{sech}^4(x) \, dx\)

Optimal. Leaf size=13 \[ -\frac{8}{3 \left (e^{2 x}+1\right )^3} \]

[Out]

-8/(3*(1 + E^(2*x))^3)

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Rubi [A]  time = 0.01701, antiderivative size = 13, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {2282, 12, 261} \[ -\frac{8}{3 \left (e^{2 x}+1\right )^3} \]

Antiderivative was successfully verified.

[In]

Int[Sech[x]^4/E^(2*x),x]

[Out]

-8/(3*(1 + E^(2*x))^3)

Rule 2282

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[x]]]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 261

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

Rubi steps

\begin{align*} \int e^{-2 x} \text{sech}^4(x) \, dx &=\operatorname{Subst}\left (\int \frac{16 x}{\left (1+x^2\right )^4} \, dx,x,e^x\right )\\ &=16 \operatorname{Subst}\left (\int \frac{x}{\left (1+x^2\right )^4} \, dx,x,e^x\right )\\ &=-\frac{8}{3 \left (1+e^{2 x}\right )^3}\\ \end{align*}

Mathematica [A]  time = 0.0095893, size = 13, normalized size = 1. \[ -\frac{8}{3 \left (e^{2 x}+1\right )^3} \]

Antiderivative was successfully verified.

[In]

Integrate[Sech[x]^4/E^(2*x),x]

[Out]

-8/(3*(1 + E^(2*x))^3)

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Maple [B]  time = 0.043, size = 52, normalized size = 4. \begin{align*} -2\,{\frac{- \left ( \tanh \left ( x/2 \right ) \right ) ^{5}+2\, \left ( \tanh \left ( x/2 \right ) \right ) ^{4}-10/3\, \left ( \tanh \left ( x/2 \right ) \right ) ^{3}+2\, \left ( \tanh \left ( x/2 \right ) \right ) ^{2}-\tanh \left ( x/2 \right ) }{ \left ( \left ( \tanh \left ( x/2 \right ) \right ) ^{2}+1 \right ) ^{3}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/exp(2*x)/cosh(x)^4,x)

[Out]

-2*(-tanh(1/2*x)^5+2*tanh(1/2*x)^4-10/3*tanh(1/2*x)^3+2*tanh(1/2*x)^2-tanh(1/2*x))/(tanh(1/2*x)^2+1)^3

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Maxima [B]  time = 0.976321, size = 101, normalized size = 7.77 \begin{align*} \frac{8 \, e^{\left (-2 \, x\right )}}{3 \, e^{\left (-2 \, x\right )} + 3 \, e^{\left (-4 \, x\right )} + e^{\left (-6 \, x\right )} + 1} + \frac{8 \, e^{\left (-4 \, x\right )}}{3 \, e^{\left (-2 \, x\right )} + 3 \, e^{\left (-4 \, x\right )} + e^{\left (-6 \, x\right )} + 1} + \frac{8}{3 \,{\left (3 \, e^{\left (-2 \, x\right )} + 3 \, e^{\left (-4 \, x\right )} + e^{\left (-6 \, x\right )} + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/exp(2*x)/cosh(x)^4,x, algorithm="maxima")

[Out]

8*e^(-2*x)/(3*e^(-2*x) + 3*e^(-4*x) + e^(-6*x) + 1) + 8*e^(-4*x)/(3*e^(-2*x) + 3*e^(-4*x) + e^(-6*x) + 1) + 8/
3/(3*e^(-2*x) + 3*e^(-4*x) + e^(-6*x) + 1)

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Fricas [B]  time = 2.11093, size = 338, normalized size = 26. \begin{align*} -\frac{8}{3 \,{\left (\cosh \left (x\right )^{6} + 6 \, \cosh \left (x\right ) \sinh \left (x\right )^{5} + \sinh \left (x\right )^{6} + 3 \,{\left (5 \, \cosh \left (x\right )^{2} + 1\right )} \sinh \left (x\right )^{4} + 3 \, \cosh \left (x\right )^{4} + 4 \,{\left (5 \, \cosh \left (x\right )^{3} + 3 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{3} + 3 \,{\left (5 \, \cosh \left (x\right )^{4} + 6 \, \cosh \left (x\right )^{2} + 1\right )} \sinh \left (x\right )^{2} + 3 \, \cosh \left (x\right )^{2} + 6 \,{\left (\cosh \left (x\right )^{5} + 2 \, \cosh \left (x\right )^{3} + \cosh \left (x\right )\right )} \sinh \left (x\right ) + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/exp(2*x)/cosh(x)^4,x, algorithm="fricas")

[Out]

-8/3/(cosh(x)^6 + 6*cosh(x)*sinh(x)^5 + sinh(x)^6 + 3*(5*cosh(x)^2 + 1)*sinh(x)^4 + 3*cosh(x)^4 + 4*(5*cosh(x)
^3 + 3*cosh(x))*sinh(x)^3 + 3*(5*cosh(x)^4 + 6*cosh(x)^2 + 1)*sinh(x)^2 + 3*cosh(x)^2 + 6*(cosh(x)^5 + 2*cosh(
x)^3 + cosh(x))*sinh(x) + 1)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{e^{- 2 x}}{\cosh ^{4}{\left (x \right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/exp(2*x)/cosh(x)**4,x)

[Out]

Integral(exp(-2*x)/cosh(x)**4, x)

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Giac [A]  time = 1.08337, size = 14, normalized size = 1.08 \begin{align*} -\frac{8}{3 \,{\left (e^{\left (2 \, x\right )} + 1\right )}^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/exp(2*x)/cosh(x)^4,x, algorithm="giac")

[Out]

-8/3/(e^(2*x) + 1)^3