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

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

[Out]

-cosh(x)/a

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Rubi [A]
time = 0.03, antiderivative size = 7, 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, 2718} \begin {gather*} -\frac {\cosh (x)}{a} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sinh[x]^3/(a - a*Cosh[x]^2),x]

[Out]

-(Cosh[x]/a)

Rule 2718

Int[sin[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Cos[c + d*x]/d, x] /; FreeQ[{c, d}, 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 ^3(x)}{a-a \cosh ^2(x)} \, dx &=-\frac {\int \sinh (x) \, dx}{a}\\ &=-\frac {\cosh (x)}{a}\\ \end {align*}

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Mathematica [A]
time = 0.00, size = 7, normalized size = 1.00 \begin {gather*} -\frac {\cosh (x)}{a} \end {gather*}

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 = 0.41, size = 8, normalized size = 1.14

method	result	size
derivativedivides	\(-\frac {\cosh \left (x \right )}{a}\)	\(8\)
default	\(-\frac {\cosh \left (x \right )}{a}\)	\(8\)
risch	\(-\frac {{\mathrm e}^{x}}{2 a}-\frac {{\mathrm e}^{-x}}{2 a}\)	\(18\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-cosh(x)/a

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 17 vs. \(2 (7) = 14\).
time = 0.26, size = 17, normalized size = 2.43 \begin {gather*} -\frac {e^{\left (-x\right )}}{2 \, a} - \frac {e^{x}}{2 \, a} \end {gather*}

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 = 0.38, size = 7, normalized size = 1.00 \begin {gather*} -\frac {\cosh \left (x\right )}{a} \end {gather*}

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 [A]
time = 0.53, size = 10, normalized size = 1.43 \begin {gather*} \frac {2}{a \tanh ^{2}{\left (\frac {x}{2} \right )} - a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/(a*tanh(x/2)**2 - a)

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Giac [A]
time = 0.44, size = 12, normalized size = 1.71 \begin {gather*} -\frac {e^{\left (-x\right )} + e^{x}}{2 \, a} \end {gather*}

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

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Mupad [B]
time = 0.91, size = 7, normalized size = 1.00 \begin {gather*} -\frac {\mathrm {cosh}\left (x\right )}{a} \end {gather*}

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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