3.278 \(\int \frac{\cosh ^3(x)}{a+a \sinh ^2(x)} \, dx\)

Optimal. Leaf size=6 \[ \frac{\sinh (x)}{a} \]

[Out]

Sinh[x]/a

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Rubi [A]  time = 0.0439533, antiderivative size = 6, 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, 2637} \[ \frac{\sinh (x)}{a} \]

Antiderivative was successfully verified.

[In]

Int[Cosh[x]^3/(a + a*Sinh[x]^2),x]

[Out]

Sinh[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 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 ^3(x)}{a+a \sinh ^2(x)} \, dx &=\frac{\int \cosh (x) \, dx}{a}\\ &=\frac{\sinh (x)}{a}\\ \end{align*}

Mathematica [A]  time = 0.0016884, size = 6, normalized size = 1. \[ \frac{\sinh (x)}{a} \]

Antiderivative was successfully verified.

[In]

Integrate[Cosh[x]^3/(a + a*Sinh[x]^2),x]

[Out]

Sinh[x]/a

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Maple [A]  time = 0.012, size = 7, normalized size = 1.2 \begin{align*}{\frac{\sinh \left ( x \right ) }{a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(x)^3/(a+a*sinh(x)^2),x)

[Out]

sinh(x)/a

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Maxima [B]  time = 1.17094, size = 23, normalized size = 3.83 \begin{align*} -\frac{e^{\left (-x\right )}}{2 \, a} + \frac{e^{x}}{2 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(x)^3/(a+a*sinh(x)^2),x, algorithm="maxima")

[Out]

-1/2*e^(-x)/a + 1/2*e^x/a

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Fricas [A]  time = 1.44102, size = 15, normalized size = 2.5 \begin{align*} \frac{\sinh \left (x\right )}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(x)^3/(a+a*sinh(x)^2),x, algorithm="fricas")

[Out]

sinh(x)/a

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Sympy [B]  time = 3.29542, size = 17, normalized size = 2.83 \begin{align*} - \frac{2 \tanh{\left (\frac{x}{2} \right )}}{a \tanh ^{2}{\left (\frac{x}{2} \right )} - a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(x)**3/(a+a*sinh(x)**2),x)

[Out]

-2*tanh(x/2)/(a*tanh(x/2)**2 - a)

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Giac [B]  time = 1.10998, size = 19, normalized size = 3.17 \begin{align*} -\frac{e^{\left (-x\right )} - e^{x}}{2 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(x)^3/(a+a*sinh(x)^2),x, algorithm="giac")

[Out]

-1/2*(e^(-x) - e^x)/a