3.1 \(\int \frac{\sinh ^4(x)}{a-a \cosh ^2(x)} \, dx\)

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

[Out]

x/(2*a) - (Cosh[x]*Sinh[x])/(2*a)

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Rubi [A]  time = 0.0460336, antiderivative size = 20, normalized size of antiderivative = 1., 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 = {3175, 2635, 8} \[ \frac{x}{2 a}-\frac{\sinh (x) \cosh (x)}{2 a} \]

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 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 2635

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] && Integer
Q[2*n]

Rule 8

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

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*}

Mathematica [A]  time = 0.0041697, size = 19, normalized size = 0.95 \[ -\frac{\frac{1}{4} \sinh (2 x)-\frac{x}{2}}{a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((-x/2 + Sinh[2*x]/4)/a)

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Maple [B]  time = 0.024, size = 78, normalized size = 3.9 \begin{align*}{\frac{1}{2\,a} \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) ^{-2}}-{\frac{1}{2\,a} \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) ^{-1}}+{\frac{1}{2\,a}\ln \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) }-{\frac{1}{2\,a} \left ( \tanh \left ({\frac{x}{2}} \right ) -1 \right ) ^{-2}}-{\frac{1}{2\,a} \left ( \tanh \left ({\frac{x}{2}} \right ) -1 \right ) ^{-1}}-{\frac{1}{2\,a}\ln \left ( \tanh \left ({\frac{x}{2}} \right ) -1 \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.07239, size = 34, normalized size = 1.7 \begin{align*} \frac{x}{2 \, a} - \frac{e^{\left (2 \, x\right )}}{8 \, a} + \frac{e^{\left (-2 \, x\right )}}{8 \, a} \end{align*}

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 = 1.73236, size = 41, normalized size = 2.05 \begin{align*} -\frac{\cosh \left (x\right ) \sinh \left (x\right ) - x}{2 \, a} \end{align*}

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]  time = 3.66779, size = 153, normalized size = 7.65 \begin{align*} \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{align*}

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 = 1.28488, size = 35, normalized size = 1.75 \begin{align*} -\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{align*}

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