3.1.1 \(\int \frac {\sinh ^4(x)}{a-a \cosh ^2(x)} \, dx\) [1]

Optimal. Leaf size=20 \[ \frac {x}{2 a}-\frac {\cosh (x) \sinh (x)}{2 a} \]

[Out]

1/2*x/a-1/2*cosh(x)*sinh(x)/a

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Rubi [A]
time = 0.03, antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {3254, 2715, 8} \begin {gather*} \frac {x}{2 a}-\frac {\sinh (x) \cosh (x)}{2 a} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sinh[x]^4/(a - a*Cosh[x]^2),x]

[Out]

x/(2*a) - (Cosh[x]*Sinh[x])/(2*a)

Rule 8

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

Rule 2715

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d*x]*((b*Sin[c + d*x])^(n - 1)/(d*n))
, x] + Dist[b^2*((n - 1)/n), Int[(b*Sin[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && Integ
erQ[2*n]

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 ^4(x)}{a-a \cosh ^2(x)} \, dx &=-\frac {\int \sinh ^2(x) \, dx}{a}\\ &=-\frac {\cosh (x) \sinh (x)}{2 a}+\frac {\int 1 \, dx}{2 a}\\ &=\frac {x}{2 a}-\frac {\cosh (x) \sinh (x)}{2 a}\\ \end {align*}

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Mathematica [A]
time = 0.00, size = 19, normalized size = 0.95 \begin {gather*} -\frac {-\frac {x}{2}+\frac {1}{4} \sinh (2 x)}{a} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sinh[x]^4/(a - a*Cosh[x]^2),x]

[Out]

-((-1/2*x + Sinh[2*x]/4)/a)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(64\) vs. \(2(16)=32\).
time = 0.50, size = 65, normalized size = 3.25

method	result	size
risch	\(\frac {x}{2 a}-\frac {{\mathrm e}^{2 x}}{8 a}+\frac {{\mathrm e}^{-2 x}}{8 a}\)	\(26\)
default	\(\frac {\frac {1}{2 \left (\tanh \left (\frac {x}{2}\right )+1\right )^{2}}-\frac {1}{2 \left (\tanh \left (\frac {x}{2}\right )+1\right )}+\frac {\ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{2}-\frac {1}{2 \left (\tanh \left (\frac {x}{2}\right )-1\right )^{2}}-\frac {1}{2 \left (\tanh \left (\frac {x}{2}\right )-1\right )}-\frac {\ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{2}}{a}\)	\(65\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sinh(x)^4/(a-a*cosh(x)^2),x,method=_RETURNVERBOSE)

[Out]

8/a*(1/16/(tanh(1/2*x)+1)^2-1/16/(tanh(1/2*x)+1)+1/16*ln(tanh(1/2*x)+1)-1/16/(tanh(1/2*x)-1)^2-1/16/(tanh(1/2*
x)-1)-1/16*ln(tanh(1/2*x)-1))

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Maxima [A]
time = 0.26, size = 25, normalized size = 1.25 \begin {gather*} \frac {x}{2 \, a} - \frac {e^{\left (2 \, x\right )}}{8 \, a} + \frac {e^{\left (-2 \, x\right )}}{8 \, a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(x)^4/(a-a*cosh(x)^2),x, algorithm="maxima")

[Out]

1/2*x/a - 1/8*e^(2*x)/a + 1/8*e^(-2*x)/a

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Fricas [A]
time = 0.37, size = 14, normalized size = 0.70 \begin {gather*} -\frac {\cosh \left (x\right ) \sinh \left (x\right ) - x}{2 \, a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(x)^4/(a-a*cosh(x)^2),x, algorithm="fricas")

[Out]

-1/2*(cosh(x)*sinh(x) - x)/a

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 153 vs. \(2 (14) = 28\).
time = 0.88, size = 153, normalized size = 7.65 \begin {gather*} \frac {x \tanh ^{4}{\left (\frac {x}{2} \right )}}{2 a \tanh ^{4}{\left (\frac {x}{2} \right )} - 4 a \tanh ^{2}{\left (\frac {x}{2} \right )} + 2 a} - \frac {2 x \tanh ^{2}{\left (\frac {x}{2} \right )}}{2 a \tanh ^{4}{\left (\frac {x}{2} \right )} - 4 a \tanh ^{2}{\left (\frac {x}{2} \right )} + 2 a} + \frac {x}{2 a \tanh ^{4}{\left (\frac {x}{2} \right )} - 4 a \tanh ^{2}{\left (\frac {x}{2} \right )} + 2 a} - \frac {2 \tanh ^{3}{\left (\frac {x}{2} \right )}}{2 a \tanh ^{4}{\left (\frac {x}{2} \right )} - 4 a \tanh ^{2}{\left (\frac {x}{2} \right )} + 2 a} - \frac {2 \tanh {\left (\frac {x}{2} \right )}}{2 a \tanh ^{4}{\left (\frac {x}{2} \right )} - 4 a \tanh ^{2}{\left (\frac {x}{2} \right )} + 2 a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(x)**4/(a-a*cosh(x)**2),x)

[Out]

x*tanh(x/2)**4/(2*a*tanh(x/2)**4 - 4*a*tanh(x/2)**2 + 2*a) - 2*x*tanh(x/2)**2/(2*a*tanh(x/2)**4 - 4*a*tanh(x/2
)**2 + 2*a) + x/(2*a*tanh(x/2)**4 - 4*a*tanh(x/2)**2 + 2*a) - 2*tanh(x/2)**3/(2*a*tanh(x/2)**4 - 4*a*tanh(x/2)
**2 + 2*a) - 2*tanh(x/2)/(2*a*tanh(x/2)**4 - 4*a*tanh(x/2)**2 + 2*a)

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Giac [A]
time = 0.42, size = 26, normalized size = 1.30 \begin {gather*} -\frac {{\left (2 \, e^{\left (2 \, x\right )} - 1\right )} e^{\left (-2 \, x\right )} - 4 \, x + e^{\left (2 \, x\right )}}{8 \, a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(x)^4/(a-a*cosh(x)^2),x, algorithm="giac")

[Out]

-1/8*((2*e^(2*x) - 1)*e^(-2*x) - 4*x + e^(2*x))/a

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Mupad [B]
time = 0.94, size = 25, normalized size = 1.25 \begin {gather*} \frac {{\mathrm {e}}^{-2\,x}}{8\,a}-\frac {{\mathrm {e}}^{2\,x}}{8\,a}+\frac {x}{2\,a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sinh(x)^4/(a - a*cosh(x)^2),x)

[Out]

exp(-2*x)/(8*a) - exp(2*x)/(8*a) + x/(2*a)

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