### 3.1040 $$\int x \cosh (2 x) \text{sech}^2(x) \, dx$$

Optimal. Leaf size=12 $x^2-x \tanh (x)+\log (\cosh (x))$

[Out]

x^2 + Log[Cosh[x]] - x*Tanh[x]

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Rubi [A]  time = 0.0358555, antiderivative size = 12, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 10, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.4, Rules used = {5473, 3720, 3475, 30} $x^2-x \tanh (x)+\log (\cosh (x))$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

x^2 + Log[Cosh[x]] - x*Tanh[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 3720

Int[((c_.) + (d_.)*(x_))^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*(c + d*x)^m*(b*Tan[e
+ f*x])^(n - 1))/(f*(n - 1)), x] + (-Dist[(b*d*m)/(f*(n - 1)), Int[(c + d*x)^(m - 1)*(b*Tan[e + f*x])^(n - 1)
, x], x] - Dist[b^2, Int[(c + d*x)^m*(b*Tan[e + f*x])^(n - 2), x], x]) /; FreeQ[{b, c, d, e, f}, x] && GtQ[n,
1] && GtQ[m, 0]

Rule 3475

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

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rubi steps

\begin{align*} \int x \cosh (2 x) \text{sech}^2(x) \, dx &=\int \left (x+x \tanh ^2(x)\right ) \, dx\\ &=\frac{x^2}{2}+\int x \tanh ^2(x) \, dx\\ &=\frac{x^2}{2}-x \tanh (x)+\int x \, dx+\int \tanh (x) \, dx\\ &=x^2+\log (\cosh (x))-x \tanh (x)\\ \end{align*}

Mathematica [A]  time = 0.0203948, size = 12, normalized size = 1. $x^2-x \tanh (x)+\log (\cosh (x))$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

x^2 + Log[Cosh[x]] - x*Tanh[x]

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Maple [B]  time = 0.032, size = 26, normalized size = 2.2 \begin{align*}{x}^{2}-2\,x+2\,{\frac{x}{{{\rm e}^{2\,x}}+1}}+\ln \left ({{\rm e}^{2\,x}}+1 \right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 1.62947, size = 45, normalized size = 3.75 \begin{align*} \frac{x^{2} +{\left (x^{2} - 2 \, x\right )} e^{\left (2 \, x\right )}}{e^{\left (2 \, x\right )} + 1} + \log \left (e^{\left (2 \, x\right )} + 1\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(x^2 + (x^2 - 2*x)*e^(2*x))/(e^(2*x) + 1) + log(e^(2*x) + 1)

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Fricas [B]  time = 1.82887, size = 297, normalized size = 24.75 \begin{align*} \frac{{\left (x^{2} - 2 \, x\right )} \cosh \left (x\right )^{2} + 2 \,{\left (x^{2} - 2 \, x\right )} \cosh \left (x\right ) \sinh \left (x\right ) +{\left (x^{2} - 2 \, x\right )} \sinh \left (x\right )^{2} + x^{2} +{\left (\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2} + 1\right )} \log \left (\frac{2 \, \cosh \left (x\right )}{\cosh \left (x\right ) - \sinh \left (x\right )}\right )}{\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2} + 1} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x*cosh(2*x)*sech(x)**2, x)

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Giac [B]  time = 1.14985, size = 63, normalized size = 5.25 \begin{align*} \frac{x^{2} e^{\left (2 \, x\right )} + x^{2} - 2 \, x e^{\left (2 \, x\right )} + e^{\left (2 \, x\right )} \log \left (e^{\left (2 \, x\right )} + 1\right ) + \log \left (e^{\left (2 \, x\right )} + 1\right )}{e^{\left (2 \, x\right )} + 1} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(x^2*e^(2*x) + x^2 - 2*x*e^(2*x) + e^(2*x)*log(e^(2*x) + 1) + log(e^(2*x) + 1))/(e^(2*x) + 1)