3.70 \(\int \frac{\cosh ^2(x)}{a+a \text{sech}(x)} \, dx\)

Optimal. Leaf size=41 \[ \frac{3 x}{2 a}-\frac{2 \sinh (x)}{a}+\frac{3 \sinh (x) \cosh (x)}{2 a}-\frac{\sinh (x) \cosh (x)}{a \text{sech}(x)+a} \]

[Out]

(3*x)/(2*a) - (2*Sinh[x])/a + (3*Cosh[x]*Sinh[x])/(2*a) - (Cosh[x]*Sinh[x])/(a + a*Sech[x])

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Rubi [A]  time = 0.0796549, antiderivative size = 41, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.385, Rules used = {3819, 3787, 2635, 8, 2637} \[ \frac{3 x}{2 a}-\frac{2 \sinh (x)}{a}+\frac{3 \sinh (x) \cosh (x)}{2 a}-\frac{\sinh (x) \cosh (x)}{a \text{sech}(x)+a} \]

Antiderivative was successfully verified.

[In]

Int[Cosh[x]^2/(a + a*Sech[x]),x]

[Out]

(3*x)/(2*a) - (2*Sinh[x])/a + (3*Cosh[x]*Sinh[x])/(2*a) - (Cosh[x]*Sinh[x])/(a + a*Sech[x])

Rule 3819

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)/(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Simp[(Cot[e + f*
x]*(d*Csc[e + f*x])^n)/(f*(a + b*Csc[e + f*x])), x] - Dist[1/a^2, Int[(d*Csc[e + f*x])^n*(a*(n - 1) - b*n*Csc[
e + f*x]), x], x] /; FreeQ[{a, b, d, e, f}, x] && EqQ[a^2 - b^2, 0] && LtQ[n, 0]

Rule 3787

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)), x_Symbol] :> Dist[a, Int[(d*
Csc[e + f*x])^n, x], x] + Dist[b/d, Int[(d*Csc[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, n}, x]

Rule 2635

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> -Simp[(b*Cos[c + d*x]*(b*Sin[c + d*x])^(n - 1))/(d*n),
x] + Dist[(b^2*(n - 1))/n, Int[(b*Sin[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && Integer
Q[2*n]

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 2637

Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rubi steps

\begin{align*} \int \frac{\cosh ^2(x)}{a+a \text{sech}(x)} \, dx &=-\frac{\cosh (x) \sinh (x)}{a+a \text{sech}(x)}-\frac{\int \cosh ^2(x) (-3 a+2 a \text{sech}(x)) \, dx}{a^2}\\ &=-\frac{\cosh (x) \sinh (x)}{a+a \text{sech}(x)}-\frac{2 \int \cosh (x) \, dx}{a}+\frac{3 \int \cosh ^2(x) \, dx}{a}\\ &=-\frac{2 \sinh (x)}{a}+\frac{3 \cosh (x) \sinh (x)}{2 a}-\frac{\cosh (x) \sinh (x)}{a+a \text{sech}(x)}+\frac{3 \int 1 \, dx}{2 a}\\ &=\frac{3 x}{2 a}-\frac{2 \sinh (x)}{a}+\frac{3 \cosh (x) \sinh (x)}{2 a}-\frac{\cosh (x) \sinh (x)}{a+a \text{sech}(x)}\\ \end{align*}

Mathematica [A]  time = 0.0480794, size = 45, normalized size = 1.1 \[ \frac{\text{sech}\left (\frac{x}{2}\right ) \left (-12 \sinh \left (\frac{x}{2}\right )-3 \sinh \left (\frac{3 x}{2}\right )+\sinh \left (\frac{5 x}{2}\right )+12 x \cosh \left (\frac{x}{2}\right )\right )}{8 a} \]

Antiderivative was successfully verified.

[In]

Integrate[Cosh[x]^2/(a + a*Sech[x]),x]

[Out]

(Sech[x/2]*(12*x*Cosh[x/2] - 12*Sinh[x/2] - 3*Sinh[(3*x)/2] + Sinh[(5*x)/2]))/(8*a)

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Maple [B]  time = 0.029, size = 87, normalized size = 2.1 \begin{align*} -{\frac{1}{a}\tanh \left ({\frac{x}{2}} \right ) }-{\frac{1}{2\,a} \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) ^{-2}}+{\frac{3}{2\,a} \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) ^{-1}}+{\frac{3}{2\,a}\ln \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) }+{\frac{1}{2\,a} \left ( \tanh \left ({\frac{x}{2}} \right ) -1 \right ) ^{-2}}+{\frac{3}{2\,a} \left ( \tanh \left ({\frac{x}{2}} \right ) -1 \right ) ^{-1}}-{\frac{3}{2\,a}\ln \left ( \tanh \left ({\frac{x}{2}} \right ) -1 \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(x)^2/(a+a*sech(x)),x)

[Out]

-1/a*tanh(1/2*x)-1/2/a/(tanh(1/2*x)+1)^2+3/2/a/(tanh(1/2*x)+1)+3/2/a*ln(tanh(1/2*x)+1)+1/2/a/(tanh(1/2*x)-1)^2
+3/2/a/(tanh(1/2*x)-1)-3/2/a*ln(tanh(1/2*x)-1)

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Maxima [A]  time = 1.1507, size = 76, normalized size = 1.85 \begin{align*} \frac{3 \, x}{2 \, a} + \frac{4 \, e^{\left (-x\right )} - e^{\left (-2 \, x\right )}}{8 \, a} - \frac{3 \, e^{\left (-x\right )} + 20 \, e^{\left (-2 \, x\right )} - 1}{8 \,{\left (a e^{\left (-2 \, x\right )} + a e^{\left (-3 \, x\right )}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/2*x/a + 1/8*(4*e^(-x) - e^(-2*x))/a - 1/8*(3*e^(-x) + 20*e^(-2*x) - 1)/(a*e^(-2*x) + a*e^(-3*x))

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Fricas [A]  time = 2.17091, size = 242, normalized size = 5.9 \begin{align*} \frac{\cosh \left (x\right )^{3} +{\left (3 \, \cosh \left (x\right ) - 4\right )} \sinh \left (x\right )^{2} + \sinh \left (x\right )^{3} +{\left (12 \, x - 1\right )} \cosh \left (x\right ) - 4 \, \cosh \left (x\right )^{2} +{\left (3 \, \cosh \left (x\right )^{2} + 12 \, x - 4 \, \cosh \left (x\right ) - 7\right )} \sinh \left (x\right ) + 12 \, x + 20}{8 \,{\left (a \cosh \left (x\right ) + a \sinh \left (x\right ) + a\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*(cosh(x)^3 + (3*cosh(x) - 4)*sinh(x)^2 + sinh(x)^3 + (12*x - 1)*cosh(x) - 4*cosh(x)^2 + (3*cosh(x)^2 + 12*
x - 4*cosh(x) - 7)*sinh(x) + 12*x + 20)/(a*cosh(x) + a*sinh(x) + a)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{\cosh ^{2}{\left (x \right )}}{\operatorname{sech}{\left (x \right )} + 1}\, dx}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(x)**2/(a+a*sech(x)),x)

[Out]

Integral(cosh(x)**2/(sech(x) + 1), x)/a

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Giac [A]  time = 1.12272, size = 69, normalized size = 1.68 \begin{align*} \frac{3 \, x}{2 \, a} + \frac{{\left (20 \, e^{\left (2 \, x\right )} + 3 \, e^{x} - 1\right )} e^{\left (-2 \, x\right )}}{8 \, a{\left (e^{x} + 1\right )}} + \frac{a e^{\left (2 \, x\right )} - 4 \, a e^{x}}{8 \, a^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/2*x/a + 1/8*(20*e^(2*x) + 3*e^x - 1)*e^(-2*x)/(a*(e^x + 1)) + 1/8*(a*e^(2*x) - 4*a*e^x)/a^2