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

Optimal. Leaf size=6 \[ -\frac {x}{a} \]

[Out]

-x/a

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Rubi [A]
time = 0.03, antiderivative size = 6, normalized size of antiderivative = 1.00, 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 = {3254, 8} \begin {gather*} -\frac {x}{a} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sinh[x]^2/(a - a*Cosh[x]^2),x]

[Out]

-(x/a)

Rule 8

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

Rule 3254

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]

Rubi steps

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

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Mathematica [A]
time = 0.00, size = 6, normalized size = 1.00 \begin {gather*} -\frac {x}{a} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-(x/a)

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Maple [C] Result contains higher order function than in optimal. Order 3 vs. order 1.
time = 0.60, size = 11, normalized size = 1.83

method	result	size
risch	\(-\frac {x}{a}\)	\(7\)
default	\(-\frac {2 \arctanh \left (\tanh \left (\frac {x}{2}\right )\right )}{a}\)	\(11\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sinh(x)^2/(a-a*cosh(x)^2),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.26, size = 6, normalized size = 1.00 \begin {gather*} -\frac {x}{a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x/a

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Fricas [A]
time = 0.38, size = 6, normalized size = 1.00 \begin {gather*} -\frac {x}{a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x/a

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Sympy [A]
time = 0.28, size = 3, normalized size = 0.50 \begin {gather*} - \frac {x}{a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x/a

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Giac [A]
time = 0.41, size = 6, normalized size = 1.00 \begin {gather*} -\frac {x}{a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x/a

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Mupad [B]
time = 0.03, size = 6, normalized size = 1.00 \begin {gather*} -\frac {x}{a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sinh(x)^2/(a - a*cosh(x)^2),x)

[Out]

-x/a

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