3.276 \(\int \frac{\cosh ^5(x)}{a+a \sinh ^2(x)} \, dx\)

Optimal. Leaf size=18 \[ \frac{\sinh ^3(x)}{3 a}+\frac{\sinh (x)}{a} \]

[Out]

Sinh[x]/a + Sinh[x]^3/(3*a)

________________________________________________________________________________________

Rubi [A]  time = 0.0516122, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {3175, 2633} \[ \frac{\sinh ^3(x)}{3 a}+\frac{\sinh (x)}{a} \]

Antiderivative was successfully verified.

[In]

Int[Cosh[x]^5/(a + a*Sinh[x]^2),x]

[Out]

Sinh[x]/a + Sinh[x]^3/(3*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 2633

Int[sin[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> -Dist[d^(-1), Subst[Int[Expand[(1 - x^2)^((n - 1)/2), x], x], x
, Cos[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[(n - 1)/2, 0]

Rubi steps

\begin{align*} \int \frac{\cosh ^5(x)}{a+a \sinh ^2(x)} \, dx &=\frac{\int \cosh ^3(x) \, dx}{a}\\ &=\frac{i \operatorname{Subst}\left (\int \left (1-x^2\right ) \, dx,x,-i \sinh (x)\right )}{a}\\ &=\frac{\sinh (x)}{a}+\frac{\sinh ^3(x)}{3 a}\\ \end{align*}

Mathematica [A]  time = 0.0034921, size = 19, normalized size = 1.06 \[ \frac{\frac{3 \sinh (x)}{4}+\frac{1}{12} \sinh (3 x)}{a} \]

Antiderivative was successfully verified.

[In]

Integrate[Cosh[x]^5/(a + a*Sinh[x]^2),x]

[Out]

((3*Sinh[x])/4 + Sinh[3*x]/12)/a

________________________________________________________________________________________

Maple [B]  time = 0.025, size = 67, normalized size = 3.7 \begin{align*} 2\,{\frac{1}{a} \left ( -1/6\, \left ( \tanh \left ( x/2 \right ) +1 \right ) ^{-3}+1/4\, \left ( \tanh \left ( x/2 \right ) +1 \right ) ^{-2}-1/2\, \left ( \tanh \left ( x/2 \right ) +1 \right ) ^{-1}-1/6\, \left ( \tanh \left ( x/2 \right ) -1 \right ) ^{-3}-1/4\, \left ( \tanh \left ( x/2 \right ) -1 \right ) ^{-2}-1/2\, \left ( \tanh \left ( x/2 \right ) -1 \right ) ^{-1} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(x)^5/(a+a*sinh(x)^2),x)

[Out]

2/a*(-1/6/(tanh(1/2*x)+1)^3+1/4/(tanh(1/2*x)+1)^2-1/2/(tanh(1/2*x)+1)-1/6/(tanh(1/2*x)-1)^3-1/4/(tanh(1/2*x)-1
)^2-1/2/(tanh(1/2*x)-1))

________________________________________________________________________________________

Maxima [B]  time = 1.03111, size = 46, normalized size = 2.56 \begin{align*} \frac{{\left (9 \, e^{\left (-2 \, x\right )} + 1\right )} e^{\left (3 \, x\right )}}{24 \, a} - \frac{9 \, e^{\left (-x\right )} + e^{\left (-3 \, x\right )}}{24 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(x)^5/(a+a*sinh(x)^2),x, algorithm="maxima")

[Out]

1/24*(9*e^(-2*x) + 1)*e^(3*x)/a - 1/24*(9*e^(-x) + e^(-3*x))/a

________________________________________________________________________________________

Fricas [A]  time = 1.47582, size = 65, normalized size = 3.61 \begin{align*} \frac{\sinh \left (x\right )^{3} + 3 \,{\left (\cosh \left (x\right )^{2} + 3\right )} \sinh \left (x\right )}{12 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(x)^5/(a+a*sinh(x)^2),x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

Sympy [B]  time = 10.0482, size = 124, normalized size = 6.89 \begin{align*} - \frac{6 \tanh ^{5}{\left (\frac{x}{2} \right )}}{3 a \tanh ^{6}{\left (\frac{x}{2} \right )} - 9 a \tanh ^{4}{\left (\frac{x}{2} \right )} + 9 a \tanh ^{2}{\left (\frac{x}{2} \right )} - 3 a} + \frac{4 \tanh ^{3}{\left (\frac{x}{2} \right )}}{3 a \tanh ^{6}{\left (\frac{x}{2} \right )} - 9 a \tanh ^{4}{\left (\frac{x}{2} \right )} + 9 a \tanh ^{2}{\left (\frac{x}{2} \right )} - 3 a} - \frac{6 \tanh{\left (\frac{x}{2} \right )}}{3 a \tanh ^{6}{\left (\frac{x}{2} \right )} - 9 a \tanh ^{4}{\left (\frac{x}{2} \right )} + 9 a \tanh ^{2}{\left (\frac{x}{2} \right )} - 3 a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(x)**5/(a+a*sinh(x)**2),x)

[Out]

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

________________________________________________________________________________________

Giac [A]  time = 1.14123, size = 39, normalized size = 2.17 \begin{align*} -\frac{{\left (9 \, e^{\left (2 \, x\right )} + 1\right )} e^{\left (-3 \, x\right )} - e^{\left (3 \, x\right )} - 9 \, e^{x}}{24 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(x)^5/(a+a*sinh(x)^2),x, algorithm="giac")

[Out]

-1/24*((9*e^(2*x) + 1)*e^(-3*x) - e^(3*x) - 9*e^x)/a