3.154 \(\int \frac{\sinh ^6(x)}{a+a \cosh (x)} \, dx\)

Optimal. Leaf size=44 \[ -\frac{3 x}{8 a}+\frac{\sinh ^5(x)}{5 a}-\frac{\sinh ^3(x) \cosh (x)}{4 a}+\frac{3 \sinh (x) \cosh (x)}{8 a} \]

[Out]

(-3*x)/(8*a) + (3*Cosh[x]*Sinh[x])/(8*a) - (Cosh[x]*Sinh[x]^3)/(4*a) + Sinh[x]^5/(5*a)

________________________________________________________________________________________

Rubi [A]  time = 0.0544319, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {2682, 2635, 8} \[ -\frac{3 x}{8 a}+\frac{\sinh ^5(x)}{5 a}-\frac{\sinh ^3(x) \cosh (x)}{4 a}+\frac{3 \sinh (x) \cosh (x)}{8 a} \]

Antiderivative was successfully verified.

[In]

Int[Sinh[x]^6/(a + a*Cosh[x]),x]

[Out]

(-3*x)/(8*a) + (3*Cosh[x]*Sinh[x])/(8*a) - (Cosh[x]*Sinh[x]^3)/(4*a) + Sinh[x]^5/(5*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 ^6(x)}{a+a \cosh (x)} \, dx &=\frac{\sinh ^5(x)}{5 a}-\frac{\int \sinh ^4(x) \, dx}{a}\\ &=-\frac{\cosh (x) \sinh ^3(x)}{4 a}+\frac{\sinh ^5(x)}{5 a}+\frac{3 \int \sinh ^2(x) \, dx}{4 a}\\ &=\frac{3 \cosh (x) \sinh (x)}{8 a}-\frac{\cosh (x) \sinh ^3(x)}{4 a}+\frac{\sinh ^5(x)}{5 a}-\frac{3 \int 1 \, dx}{8 a}\\ &=-\frac{3 x}{8 a}+\frac{3 \cosh (x) \sinh (x)}{8 a}-\frac{\cosh (x) \sinh ^3(x)}{4 a}+\frac{\sinh ^5(x)}{5 a}\\ \end{align*}

Mathematica [A]  time = 0.051559, size = 39, normalized size = 0.89 \[ \frac{-60 x+20 \sinh (x)+40 \sinh (2 x)-10 \sinh (3 x)-5 \sinh (4 x)+2 \sinh (5 x)}{160 a} \]

Antiderivative was successfully verified.

[In]

Integrate[Sinh[x]^6/(a + a*Cosh[x]),x]

[Out]

(-60*x + 20*Sinh[x] + 40*Sinh[2*x] - 10*Sinh[3*x] - 5*Sinh[4*x] + 2*Sinh[5*x])/(160*a)

________________________________________________________________________________________

Maple [B]  time = 0.033, size = 156, normalized size = 3.6 \begin{align*} -{\frac{1}{5\,a} \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) ^{-5}}+{\frac{3}{4\,a} \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) ^{-4}}-{\frac{3}{4\,a} \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) ^{-3}}-{\frac{1}{4\,a} \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) ^{-2}}+{\frac{3}{8\,a} \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) ^{-1}}-{\frac{3}{8\,a}\ln \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) }-{\frac{1}{5\,a} \left ( \tanh \left ({\frac{x}{2}} \right ) -1 \right ) ^{-5}}-{\frac{3}{4\,a} \left ( \tanh \left ({\frac{x}{2}} \right ) -1 \right ) ^{-4}}-{\frac{3}{4\,a} \left ( \tanh \left ({\frac{x}{2}} \right ) -1 \right ) ^{-3}}+{\frac{1}{4\,a} \left ( \tanh \left ({\frac{x}{2}} \right ) -1 \right ) ^{-2}}+{\frac{3}{8\,a} \left ( \tanh \left ({\frac{x}{2}} \right ) -1 \right ) ^{-1}}+{\frac{3}{8\,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)^6/(a+a*cosh(x)),x)

[Out]

-1/5/a/(tanh(1/2*x)+1)^5+3/4/a/(tanh(1/2*x)+1)^4-3/4/a/(tanh(1/2*x)+1)^3-1/4/a/(tanh(1/2*x)+1)^2+3/8/a/(tanh(1
/2*x)+1)-3/8/a*ln(tanh(1/2*x)+1)-1/5/a/(tanh(1/2*x)-1)^5-3/4/a/(tanh(1/2*x)-1)^4-3/4/a/(tanh(1/2*x)-1)^3+1/4/a
/(tanh(1/2*x)-1)^2+3/8/a/(tanh(1/2*x)-1)+3/8/a*ln(tanh(1/2*x)-1)

________________________________________________________________________________________

Maxima [B]  time = 1.10902, size = 105, normalized size = 2.39 \begin{align*} -\frac{{\left (5 \, e^{\left (-x\right )} + 10 \, e^{\left (-2 \, x\right )} - 40 \, e^{\left (-3 \, x\right )} - 20 \, e^{\left (-4 \, x\right )} - 2\right )} e^{\left (5 \, x\right )}}{320 \, a} - \frac{3 \, x}{8 \, a} - \frac{20 \, e^{\left (-x\right )} + 40 \, e^{\left (-2 \, x\right )} - 10 \, e^{\left (-3 \, x\right )} - 5 \, e^{\left (-4 \, x\right )} + 2 \, e^{\left (-5 \, x\right )}}{320 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(x)^6/(a+a*cosh(x)),x, algorithm="maxima")

[Out]

-1/320*(5*e^(-x) + 10*e^(-2*x) - 40*e^(-3*x) - 20*e^(-4*x) - 2)*e^(5*x)/a - 3/8*x/a - 1/320*(20*e^(-x) + 40*e^
(-2*x) - 10*e^(-3*x) - 5*e^(-4*x) + 2*e^(-5*x))/a

________________________________________________________________________________________

Fricas [A]  time = 1.89219, size = 188, normalized size = 4.27 \begin{align*} \frac{\sinh \left (x\right )^{5} + 5 \,{\left (2 \, \cosh \left (x\right )^{2} - 2 \, \cosh \left (x\right ) - 1\right )} \sinh \left (x\right )^{3} + 5 \,{\left (\cosh \left (x\right )^{4} - 2 \, \cosh \left (x\right )^{3} - 3 \, \cosh \left (x\right )^{2} + 8 \, \cosh \left (x\right ) + 2\right )} \sinh \left (x\right ) - 30 \, x}{80 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(x)^6/(a+a*cosh(x)),x, algorithm="fricas")

[Out]

1/80*(sinh(x)^5 + 5*(2*cosh(x)^2 - 2*cosh(x) - 1)*sinh(x)^3 + 5*(cosh(x)^4 - 2*cosh(x)^3 - 3*cosh(x)^2 + 8*cos
h(x) + 2)*sinh(x) - 30*x)/a

________________________________________________________________________________________

Sympy [B]  time = 7.6424, size = 692, normalized size = 15.73 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(x)**6/(a+a*cosh(x)),x)

[Out]

-15*x*tanh(x/2)**10/(40*a*tanh(x/2)**10 - 200*a*tanh(x/2)**8 + 400*a*tanh(x/2)**6 - 400*a*tanh(x/2)**4 + 200*a
*tanh(x/2)**2 - 40*a) + 75*x*tanh(x/2)**8/(40*a*tanh(x/2)**10 - 200*a*tanh(x/2)**8 + 400*a*tanh(x/2)**6 - 400*
a*tanh(x/2)**4 + 200*a*tanh(x/2)**2 - 40*a) - 150*x*tanh(x/2)**6/(40*a*tanh(x/2)**10 - 200*a*tanh(x/2)**8 + 40
0*a*tanh(x/2)**6 - 400*a*tanh(x/2)**4 + 200*a*tanh(x/2)**2 - 40*a) + 150*x*tanh(x/2)**4/(40*a*tanh(x/2)**10 -
200*a*tanh(x/2)**8 + 400*a*tanh(x/2)**6 - 400*a*tanh(x/2)**4 + 200*a*tanh(x/2)**2 - 40*a) - 75*x*tanh(x/2)**2/
(40*a*tanh(x/2)**10 - 200*a*tanh(x/2)**8 + 400*a*tanh(x/2)**6 - 400*a*tanh(x/2)**4 + 200*a*tanh(x/2)**2 - 40*a
) + 15*x/(40*a*tanh(x/2)**10 - 200*a*tanh(x/2)**8 + 400*a*tanh(x/2)**6 - 400*a*tanh(x/2)**4 + 200*a*tanh(x/2)*
*2 - 40*a) + 30*tanh(x/2)**9/(40*a*tanh(x/2)**10 - 200*a*tanh(x/2)**8 + 400*a*tanh(x/2)**6 - 400*a*tanh(x/2)**
4 + 200*a*tanh(x/2)**2 - 40*a) - 140*tanh(x/2)**7/(40*a*tanh(x/2)**10 - 200*a*tanh(x/2)**8 + 400*a*tanh(x/2)**
6 - 400*a*tanh(x/2)**4 + 200*a*tanh(x/2)**2 - 40*a) - 256*tanh(x/2)**5/(40*a*tanh(x/2)**10 - 200*a*tanh(x/2)**
8 + 400*a*tanh(x/2)**6 - 400*a*tanh(x/2)**4 + 200*a*tanh(x/2)**2 - 40*a) + 140*tanh(x/2)**3/(40*a*tanh(x/2)**1
0 - 200*a*tanh(x/2)**8 + 400*a*tanh(x/2)**6 - 400*a*tanh(x/2)**4 + 200*a*tanh(x/2)**2 - 40*a) - 30*tanh(x/2)/(
40*a*tanh(x/2)**10 - 200*a*tanh(x/2)**8 + 400*a*tanh(x/2)**6 - 400*a*tanh(x/2)**4 + 200*a*tanh(x/2)**2 - 40*a)

________________________________________________________________________________________

Giac [A]  time = 1.25476, size = 89, normalized size = 2.02 \begin{align*} -\frac{{\left (20 \, e^{\left (4 \, x\right )} + 40 \, e^{\left (3 \, x\right )} - 10 \, e^{\left (2 \, x\right )} - 5 \, e^{x} + 2\right )} e^{\left (-5 \, x\right )} + 120 \, x - 2 \, e^{\left (5 \, x\right )} + 5 \, e^{\left (4 \, x\right )} + 10 \, e^{\left (3 \, x\right )} - 40 \, e^{\left (2 \, x\right )} - 20 \, e^{x}}{320 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(x)^6/(a+a*cosh(x)),x, algorithm="giac")

[Out]

-1/320*((20*e^(4*x) + 40*e^(3*x) - 10*e^(2*x) - 5*e^x + 2)*e^(-5*x) + 120*x - 2*e^(5*x) + 5*e^(4*x) + 10*e^(3*
x) - 40*e^(2*x) - 20*e^x)/a