3.156 \(\int \frac{\sinh ^4(x)}{a+a \cosh (x)} \, dx\)

Optimal. Leaf size=31 \[ \frac{x}{2 a}+\frac{\sinh ^3(x)}{3 a}-\frac{\sinh (x) \cosh (x)}{2 a} \]

[Out]

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

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Rubi [A]  time = 0.0459738, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {2682, 2635, 8} \[ \frac{x}{2 a}+\frac{\sinh ^3(x)}{3 a}-\frac{\sinh (x) \cosh (x)}{2 a} \]

Antiderivative was successfully verified.

[In]

Int[Sinh[x]^4/(a + a*Cosh[x]),x]

[Out]

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

Rule 2682

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)/((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(g*(g*Cos[e
+ f*x])^(p - 1))/(b*f*(p - 1)), x] + Dist[g^2/a, Int[(g*Cos[e + f*x])^(p - 2), x], x] /; FreeQ[{a, b, e, f, g
}, x] && EqQ[a^2 - b^2, 0] && GtQ[p, 1] && IntegerQ[2*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 (x)} \, dx &=\frac{\sinh ^3(x)}{3 a}-\frac{\int \sinh ^2(x) \, dx}{a}\\ &=-\frac{\cosh (x) \sinh (x)}{2 a}+\frac{\sinh ^3(x)}{3 a}+\frac{\int 1 \, dx}{2 a}\\ &=\frac{x}{2 a}-\frac{\cosh (x) \sinh (x)}{2 a}+\frac{\sinh ^3(x)}{3 a}\\ \end{align*}

Mathematica [A]  time = 0.0345503, size = 25, normalized size = 0.81 \[ \frac{6 x-3 \sinh (x)-3 \sinh (2 x)+\sinh (3 x)}{12 a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(6*x - 3*Sinh[x] - 3*Sinh[2*x] + Sinh[3*x])/(12*a)

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Maple [B]  time = 0.023, size = 103, normalized size = 3.3 \begin{align*} -{\frac{1}{3\,a} \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) ^{-3}}+{\frac{1}{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}{3\,a} \left ( \tanh \left ({\frac{x}{2}} \right ) -1 \right ) ^{-3}}-{\frac{1}{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)),x)

[Out]

-1/3/a/(tanh(1/2*x)+1)^3+1/a/(tanh(1/2*x)+1)^2-1/2/a/(tanh(1/2*x)+1)+1/2/a*ln(tanh(1/2*x)+1)-1/3/a/(tanh(1/2*x
)-1)^3-1/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 [B]  time = 1.1133, size = 73, normalized size = 2.35 \begin{align*} -\frac{{\left (3 \, e^{\left (-x\right )} + 3 \, e^{\left (-2 \, x\right )} - 1\right )} e^{\left (3 \, x\right )}}{24 \, a} + \frac{x}{2 \, a} + \frac{3 \, e^{\left (-x\right )} + 3 \, e^{\left (-2 \, x\right )} - e^{\left (-3 \, x\right )}}{24 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/24*(3*e^(-x) + 3*e^(-2*x) - 1)*e^(3*x)/a + 1/2*x/a + 1/24*(3*e^(-x) + 3*e^(-2*x) - e^(-3*x))/a

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Fricas [A]  time = 1.8617, size = 89, normalized size = 2.87 \begin{align*} \frac{\sinh \left (x\right )^{3} + 3 \,{\left (\cosh \left (x\right )^{2} - 2 \, \cosh \left (x\right ) - 1\right )} \sinh \left (x\right ) + 6 \, x}{12 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*(sinh(x)^3 + 3*(cosh(x)^2 - 2*cosh(x) - 1)*sinh(x) + 6*x)/a

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Sympy [B]  time = 2.14946, size = 294, normalized size = 9.48 \begin{align*} \frac{3 x \tanh ^{6}{\left (\frac{x}{2} \right )}}{6 a \tanh ^{6}{\left (\frac{x}{2} \right )} - 18 a \tanh ^{4}{\left (\frac{x}{2} \right )} + 18 a \tanh ^{2}{\left (\frac{x}{2} \right )} - 6 a} - \frac{9 x \tanh ^{4}{\left (\frac{x}{2} \right )}}{6 a \tanh ^{6}{\left (\frac{x}{2} \right )} - 18 a \tanh ^{4}{\left (\frac{x}{2} \right )} + 18 a \tanh ^{2}{\left (\frac{x}{2} \right )} - 6 a} + \frac{9 x \tanh ^{2}{\left (\frac{x}{2} \right )}}{6 a \tanh ^{6}{\left (\frac{x}{2} \right )} - 18 a \tanh ^{4}{\left (\frac{x}{2} \right )} + 18 a \tanh ^{2}{\left (\frac{x}{2} \right )} - 6 a} - \frac{3 x}{6 a \tanh ^{6}{\left (\frac{x}{2} \right )} - 18 a \tanh ^{4}{\left (\frac{x}{2} \right )} + 18 a \tanh ^{2}{\left (\frac{x}{2} \right )} - 6 a} - \frac{6 \tanh ^{5}{\left (\frac{x}{2} \right )}}{6 a \tanh ^{6}{\left (\frac{x}{2} \right )} - 18 a \tanh ^{4}{\left (\frac{x}{2} \right )} + 18 a \tanh ^{2}{\left (\frac{x}{2} \right )} - 6 a} - \frac{16 \tanh ^{3}{\left (\frac{x}{2} \right )}}{6 a \tanh ^{6}{\left (\frac{x}{2} \right )} - 18 a \tanh ^{4}{\left (\frac{x}{2} \right )} + 18 a \tanh ^{2}{\left (\frac{x}{2} \right )} - 6 a} + \frac{6 \tanh{\left (\frac{x}{2} \right )}}{6 a \tanh ^{6}{\left (\frac{x}{2} \right )} - 18 a \tanh ^{4}{\left (\frac{x}{2} \right )} + 18 a \tanh ^{2}{\left (\frac{x}{2} \right )} - 6 a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(x)**4/(a+a*cosh(x)),x)

[Out]

3*x*tanh(x/2)**6/(6*a*tanh(x/2)**6 - 18*a*tanh(x/2)**4 + 18*a*tanh(x/2)**2 - 6*a) - 9*x*tanh(x/2)**4/(6*a*tanh
(x/2)**6 - 18*a*tanh(x/2)**4 + 18*a*tanh(x/2)**2 - 6*a) + 9*x*tanh(x/2)**2/(6*a*tanh(x/2)**6 - 18*a*tanh(x/2)*
*4 + 18*a*tanh(x/2)**2 - 6*a) - 3*x/(6*a*tanh(x/2)**6 - 18*a*tanh(x/2)**4 + 18*a*tanh(x/2)**2 - 6*a) - 6*tanh(
x/2)**5/(6*a*tanh(x/2)**6 - 18*a*tanh(x/2)**4 + 18*a*tanh(x/2)**2 - 6*a) - 16*tanh(x/2)**3/(6*a*tanh(x/2)**6 -
18*a*tanh(x/2)**4 + 18*a*tanh(x/2)**2 - 6*a) + 6*tanh(x/2)/(6*a*tanh(x/2)**6 - 18*a*tanh(x/2)**4 + 18*a*tanh(
x/2)**2 - 6*a)

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Giac [A]  time = 1.2793, size = 54, normalized size = 1.74 \begin{align*} \frac{{\left (3 \, e^{\left (2 \, x\right )} + 3 \, e^{x} - 1\right )} e^{\left (-3 \, x\right )} + 12 \, x + e^{\left (3 \, x\right )} - 3 \, e^{\left (2 \, x\right )} - 3 \, e^{x}}{24 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/24*((3*e^(2*x) + 3*e^x - 1)*e^(-3*x) + 12*x + e^(3*x) - 3*e^(2*x) - 3*e^x)/a