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

Optimal. Leaf size=22 \[ \frac{\tan ^{-1}(\sinh (x))}{2 a}+\frac{\tanh (x) \text{sech}(x)}{2 a} \]

[Out]

ArcTan[Sinh[x]]/(2*a) + (Sech[x]*Tanh[x])/(2*a)

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Rubi [A]  time = 0.0454984, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {3175, 3768, 3770} \[ \frac{\tan ^{-1}(\sinh (x))}{2 a}+\frac{\tanh (x) \text{sech}(x)}{2 a} \]

Antiderivative was successfully verified.

[In]

Int[Sech[x]/(a + a*Sinh[x]^2),x]

[Out]

ArcTan[Sinh[x]]/(2*a) + (Sech[x]*Tanh[x])/(2*a)

Rule 3175

Int[(u_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> Dist[a^p, Int[ActivateTrig[u*cos[e + f*x
]^(2*p)], x], x] /; FreeQ[{a, b, e, f, p}, x] && EqQ[a + b, 0] && IntegerQ[p]

Rule 3768

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

Rule 3770

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

Rubi steps

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

Mathematica [A]  time = 0.0037607, size = 20, normalized size = 0.91 \[ \frac{\tan ^{-1}\left (\tanh \left (\frac{x}{2}\right )\right )+\frac{1}{2} \tanh (x) \text{sech}(x)}{a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(ArcTan[Tanh[x/2]] + (Sech[x]*Tanh[x])/2)/a

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Maple [B]  time = 0.021, size = 50, normalized size = 2.3 \begin{align*} -{\frac{1}{a} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{3} \left ( \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{2}+1 \right ) ^{-2}}+{\frac{1}{a}\tanh \left ({\frac{x}{2}} \right ) \left ( \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{2}+1 \right ) ^{-2}}+{\frac{1}{a}\arctan \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 1.52255, size = 54, normalized size = 2.45 \begin{align*} \frac{e^{\left (-x\right )} - e^{\left (-3 \, x\right )}}{2 \, a e^{\left (-2 \, x\right )} + a e^{\left (-4 \, x\right )} + a} - \frac{\arctan \left (e^{\left (-x\right )}\right )}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(e^(-x) - e^(-3*x))/(2*a*e^(-2*x) + a*e^(-4*x) + a) - arctan(e^(-x))/a

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Fricas [B]  time = 1.48705, size = 522, normalized size = 23.73 \begin{align*} \frac{\cosh \left (x\right )^{3} + 3 \, \cosh \left (x\right ) \sinh \left (x\right )^{2} + \sinh \left (x\right )^{3} +{\left (\cosh \left (x\right )^{4} + 4 \, \cosh \left (x\right ) \sinh \left (x\right )^{3} + \sinh \left (x\right )^{4} + 2 \,{\left (3 \, \cosh \left (x\right )^{2} + 1\right )} \sinh \left (x\right )^{2} + 2 \, \cosh \left (x\right )^{2} + 4 \,{\left (\cosh \left (x\right )^{3} + \cosh \left (x\right )\right )} \sinh \left (x\right ) + 1\right )} \arctan \left (\cosh \left (x\right ) + \sinh \left (x\right )\right ) +{\left (3 \, \cosh \left (x\right )^{2} - 1\right )} \sinh \left (x\right ) - \cosh \left (x\right )}{a \cosh \left (x\right )^{4} + 4 \, a \cosh \left (x\right ) \sinh \left (x\right )^{3} + a \sinh \left (x\right )^{4} + 2 \, a \cosh \left (x\right )^{2} + 2 \,{\left (3 \, a \cosh \left (x\right )^{2} + a\right )} \sinh \left (x\right )^{2} + 4 \,{\left (a \cosh \left (x\right )^{3} + a \cosh \left (x\right )\right )} \sinh \left (x\right ) + a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(cosh(x)^3 + 3*cosh(x)*sinh(x)^2 + sinh(x)^3 + (cosh(x)^4 + 4*cosh(x)*sinh(x)^3 + sinh(x)^4 + 2*(3*cosh(x)^2 +
 1)*sinh(x)^2 + 2*cosh(x)^2 + 4*(cosh(x)^3 + cosh(x))*sinh(x) + 1)*arctan(cosh(x) + sinh(x)) + (3*cosh(x)^2 -
1)*sinh(x) - cosh(x))/(a*cosh(x)^4 + 4*a*cosh(x)*sinh(x)^3 + a*sinh(x)^4 + 2*a*cosh(x)^2 + 2*(3*a*cosh(x)^2 +
a)*sinh(x)^2 + 4*(a*cosh(x)^3 + a*cosh(x))*sinh(x) + a)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B]  time = 1.13744, size = 70, normalized size = 3.18 \begin{align*} \frac{\pi + 2 \, \arctan \left (\frac{1}{2} \,{\left (e^{\left (2 \, x\right )} - 1\right )} e^{\left (-x\right )}\right )}{4 \, a} - \frac{e^{\left (-x\right )} - e^{x}}{{\left ({\left (e^{\left (-x\right )} - e^{x}\right )}^{2} + 4\right )} a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(pi + 2*arctan(1/2*(e^(2*x) - 1)*e^(-x)))/a - (e^(-x) - e^x)/(((e^(-x) - e^x)^2 + 4)*a)