3.25 \(\int \frac{\cosh ^3(x)}{a+a \cosh (x)} \, dx\)

Optimal. Leaf size=43 \[ \frac{3 x}{2 a}-\frac{2 \sinh (x)}{a}-\frac{\sinh (x) \cosh ^2(x)}{a \cosh (x)+a}+\frac{3 \sinh (x) \cosh (x)}{2 a} \]

[Out]

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

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Rubi [A]  time = 0.0512014, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {2767, 2734} \[ \frac{3 x}{2 a}-\frac{2 \sinh (x)}{a}-\frac{\sinh (x) \cosh ^2(x)}{a \cosh (x)+a}+\frac{3 \sinh (x) \cosh (x)}{2 a} \]

Antiderivative was successfully verified.

[In]

Int[Cosh[x]^3/(a + a*Cosh[x]),x]

[Out]

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

Rule 2767

Int[((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)/((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> -Simp[((
b*c - a*d)*Cos[e + f*x]*(c + d*Sin[e + f*x])^(n - 1))/(a*f*(a + b*Sin[e + f*x])), x] - Dist[d/(a*b), Int[(c +
d*Sin[e + f*x])^(n - 2)*Simp[b*d*(n - 1) - a*c*n + (b*c*(n - 1) - a*d*n)*Sin[e + f*x], x], x], x] /; FreeQ[{a,
b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[n, 1] && (IntegerQ[2
*n] || EqQ[c, 0])

Rule 2734

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[((2*a*c
+ b*d)*x)/2, x] + (-Simp[((b*c + a*d)*Cos[e + f*x])/f, x] - Simp[(b*d*Cos[e + f*x]*Sin[e + f*x])/(2*f), x]) /;
FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0]

Rubi steps

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

Mathematica [A]  time = 0.048308, size = 45, normalized size = 1.05 \[ \frac{\text{sech}\left (\frac{x}{2}\right ) \left (-12 \sinh \left (\frac{x}{2}\right )-3 \sinh \left (\frac{3 x}{2}\right )+\sinh \left (\frac{5 x}{2}\right )+12 x \cosh \left (\frac{x}{2}\right )\right )}{8 a} \]

Antiderivative was successfully verified.

[In]

Integrate[Cosh[x]^3/(a + a*Cosh[x]),x]

[Out]

(Sech[x/2]*(12*x*Cosh[x/2] - 12*Sinh[x/2] - 3*Sinh[(3*x)/2] + Sinh[(5*x)/2]))/(8*a)

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Maple [B]  time = 0.021, size = 87, normalized size = 2. \begin{align*} -{\frac{1}{a}\tanh \left ({\frac{x}{2}} \right ) }-{\frac{1}{2\,a} \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) ^{-2}}+{\frac{3}{2\,a} \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) ^{-1}}+{\frac{3}{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{3}{2\,a} \left ( \tanh \left ({\frac{x}{2}} \right ) -1 \right ) ^{-1}}-{\frac{3}{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(cosh(x)^3/(a+a*cosh(x)),x)

[Out]

-1/a*tanh(1/2*x)-1/2/a/(tanh(1/2*x)+1)^2+3/2/a/(tanh(1/2*x)+1)+3/2/a*ln(tanh(1/2*x)+1)+1/2/a/(tanh(1/2*x)-1)^2
+3/2/a/(tanh(1/2*x)-1)-3/2/a*ln(tanh(1/2*x)-1)

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Maxima [A]  time = 1.1151, size = 76, normalized size = 1.77 \begin{align*} \frac{3 \, x}{2 \, a} + \frac{4 \, e^{\left (-x\right )} - e^{\left (-2 \, x\right )}}{8 \, a} - \frac{3 \, e^{\left (-x\right )} + 20 \, e^{\left (-2 \, x\right )} - 1}{8 \,{\left (a e^{\left (-2 \, x\right )} + a e^{\left (-3 \, x\right )}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(x)^3/(a+a*cosh(x)),x, algorithm="maxima")

[Out]

3/2*x/a + 1/8*(4*e^(-x) - e^(-2*x))/a - 1/8*(3*e^(-x) + 20*e^(-2*x) - 1)/(a*e^(-2*x) + a*e^(-3*x))

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Fricas [A]  time = 1.87436, size = 242, normalized size = 5.63 \begin{align*} \frac{\cosh \left (x\right )^{3} +{\left (3 \, \cosh \left (x\right ) - 4\right )} \sinh \left (x\right )^{2} + \sinh \left (x\right )^{3} +{\left (12 \, x - 1\right )} \cosh \left (x\right ) - 4 \, \cosh \left (x\right )^{2} +{\left (3 \, \cosh \left (x\right )^{2} + 12 \, x - 4 \, \cosh \left (x\right ) - 7\right )} \sinh \left (x\right ) + 12 \, x + 20}{8 \,{\left (a \cosh \left (x\right ) + a \sinh \left (x\right ) + a\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(x)^3/(a+a*cosh(x)),x, algorithm="fricas")

[Out]

1/8*(cosh(x)^3 + (3*cosh(x) - 4)*sinh(x)^2 + sinh(x)^3 + (12*x - 1)*cosh(x) - 4*cosh(x)^2 + (3*cosh(x)^2 + 12*
x - 4*cosh(x) - 7)*sinh(x) + 12*x + 20)/(a*cosh(x) + a*sinh(x) + a)

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Sympy [B]  time = 1.7469, size = 189, normalized size = 4.4 \begin{align*} \frac{3 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{6 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{3 x}{2 a \tanh ^{4}{\left (\frac{x}{2} \right )} - 4 a \tanh ^{2}{\left (\frac{x}{2} \right )} + 2 a} - \frac{2 \tanh ^{5}{\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{10 \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{4 \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(cosh(x)**3/(a+a*cosh(x)),x)

[Out]

3*x*tanh(x/2)**4/(2*a*tanh(x/2)**4 - 4*a*tanh(x/2)**2 + 2*a) - 6*x*tanh(x/2)**2/(2*a*tanh(x/2)**4 - 4*a*tanh(x
/2)**2 + 2*a) + 3*x/(2*a*tanh(x/2)**4 - 4*a*tanh(x/2)**2 + 2*a) - 2*tanh(x/2)**5/(2*a*tanh(x/2)**4 - 4*a*tanh(
x/2)**2 + 2*a) + 10*tanh(x/2)**3/(2*a*tanh(x/2)**4 - 4*a*tanh(x/2)**2 + 2*a) - 4*tanh(x/2)/(2*a*tanh(x/2)**4 -
4*a*tanh(x/2)**2 + 2*a)

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Giac [A]  time = 1.15494, size = 69, normalized size = 1.6 \begin{align*} \frac{3 \, x}{2 \, a} + \frac{{\left (20 \, e^{\left (2 \, x\right )} + 3 \, e^{x} - 1\right )} e^{\left (-2 \, x\right )}}{8 \, a{\left (e^{x} + 1\right )}} + \frac{a e^{\left (2 \, x\right )} - 4 \, a e^{x}}{8 \, a^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(x)^3/(a+a*cosh(x)),x, algorithm="giac")

[Out]

3/2*x/a + 1/8*(20*e^(2*x) + 3*e^x - 1)*e^(-2*x)/(a*(e^x + 1)) + 1/8*(a*e^(2*x) - 4*a*e^x)/a^2