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

Optimal. Leaf size=7 \[ -\frac{\cosh (x)}{a} \]

[Out]

-(Cosh[x]/a)

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Rubi [A]  time = 0.0439769, antiderivative size = 7, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {3175, 2638} \[ -\frac{\cosh (x)}{a} \]

Antiderivative was successfully verified.

[In]

Int[Sinh[x]^3/(a - a*Cosh[x]^2),x]

[Out]

-(Cosh[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 2638

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

Rubi steps

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

Mathematica [A]  time = 0.0027154, size = 7, normalized size = 1. \[ -\frac{\cosh (x)}{a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(Cosh[x]/a)

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Maple [A]  time = 0.01, size = 8, normalized size = 1.1 \begin{align*} -{\frac{\cosh \left ( x \right ) }{a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-cosh(x)/a

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Maxima [B]  time = 1.07952, size = 23, normalized size = 3.29 \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(sinh(x)^3/(a-a*cosh(x)^2),x, algorithm="maxima")

[Out]

-1/2*e^(-x)/a - 1/2*e^x/a

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Fricas [A]  time = 1.60763, size = 16, normalized size = 2.29 \begin{align*} -\frac{\cosh \left (x\right )}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-cosh(x)/a

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Sympy [B]  time = 1.9182, size = 17, normalized size = 2.43 \begin{align*} \frac{2 \tanh ^{2}{\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(sinh(x)**3/(a-a*cosh(x)**2),x)

[Out]

2*tanh(x/2)**2/(a*tanh(x/2)**2 - a)

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Giac [A]  time = 1.22879, size = 16, normalized size = 2.29 \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(sinh(x)^3/(a-a*cosh(x)^2),x, algorithm="giac")

[Out]

-1/2*(e^(-x) + e^x)/a