3.279 \(\int \frac{\cosh ^2(x)}{a+a \sinh ^2(x)} \, dx\)

Optimal. Leaf size=5 \[ \frac{x}{a} \]

[Out]

x/a

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Rubi [A]  time = 0.0415225, antiderivative size = 5, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {3175, 8} \[ \frac{x}{a} \]

Antiderivative was successfully verified.

[In]

Int[Cosh[x]^2/(a + a*Sinh[x]^2),x]

[Out]

x/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 8

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

Rubi steps

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

Mathematica [A]  time = 0.0003036, size = 5, normalized size = 1. \[ \frac{x}{a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

x/a

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Maple [C]  time = 0.012, size = 11, normalized size = 2.2 \begin{align*} 2\,{\frac{{\it Artanh} \left ( \tanh \left ( x/2 \right ) \right ) }{a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/a*arctanh(tanh(1/2*x))

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Maxima [A]  time = 1.11412, size = 7, normalized size = 1.4 \begin{align*} \frac{x}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x/a

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Fricas [A]  time = 1.42275, size = 7, normalized size = 1.4 \begin{align*} \frac{x}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x/a

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Sympy [A]  time = 1.77234, size = 2, normalized size = 0.4 \begin{align*} \frac{x}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x/a

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Giac [A]  time = 1.15493, size = 7, normalized size = 1.4 \begin{align*} \frac{x}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x/a