3.152 \(\int \frac{\sinh ^8(x)}{a+a \cosh (x)} \, dx\)

Optimal. Leaf size=57 \[ \frac{5 x}{16 a}+\frac{\sinh ^7(x)}{7 a}-\frac{\sinh ^5(x) \cosh (x)}{6 a}+\frac{5 \sinh ^3(x) \cosh (x)}{24 a}-\frac{5 \sinh (x) \cosh (x)}{16 a} \]

[Out]

(5*x)/(16*a) - (5*Cosh[x]*Sinh[x])/(16*a) + (5*Cosh[x]*Sinh[x]^3)/(24*a) - (Cosh[x]*Sinh[x]^5)/(6*a) + Sinh[x]
^7/(7*a)

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Rubi [A]  time = 0.059228, antiderivative size = 57, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {2682, 2635, 8} \[ \frac{5 x}{16 a}+\frac{\sinh ^7(x)}{7 a}-\frac{\sinh ^5(x) \cosh (x)}{6 a}+\frac{5 \sinh ^3(x) \cosh (x)}{24 a}-\frac{5 \sinh (x) \cosh (x)}{16 a} \]

Antiderivative was successfully verified.

[In]

Int[Sinh[x]^8/(a + a*Cosh[x]),x]

[Out]

(5*x)/(16*a) - (5*Cosh[x]*Sinh[x])/(16*a) + (5*Cosh[x]*Sinh[x]^3)/(24*a) - (Cosh[x]*Sinh[x]^5)/(6*a) + Sinh[x]
^7/(7*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 ^8(x)}{a+a \cosh (x)} \, dx &=\frac{\sinh ^7(x)}{7 a}-\frac{\int \sinh ^6(x) \, dx}{a}\\ &=-\frac{\cosh (x) \sinh ^5(x)}{6 a}+\frac{\sinh ^7(x)}{7 a}+\frac{5 \int \sinh ^4(x) \, dx}{6 a}\\ &=\frac{5 \cosh (x) \sinh ^3(x)}{24 a}-\frac{\cosh (x) \sinh ^5(x)}{6 a}+\frac{\sinh ^7(x)}{7 a}-\frac{5 \int \sinh ^2(x) \, dx}{8 a}\\ &=-\frac{5 \cosh (x) \sinh (x)}{16 a}+\frac{5 \cosh (x) \sinh ^3(x)}{24 a}-\frac{\cosh (x) \sinh ^5(x)}{6 a}+\frac{\sinh ^7(x)}{7 a}+\frac{5 \int 1 \, dx}{16 a}\\ &=\frac{5 x}{16 a}-\frac{5 \cosh (x) \sinh (x)}{16 a}+\frac{5 \cosh (x) \sinh ^3(x)}{24 a}-\frac{\cosh (x) \sinh ^5(x)}{6 a}+\frac{\sinh ^7(x)}{7 a}\\ \end{align*}

Mathematica [A]  time = 0.0671383, size = 51, normalized size = 0.89 \[ \frac{420 x-105 \sinh (x)-315 \sinh (2 x)+63 \sinh (3 x)+63 \sinh (4 x)-21 \sinh (5 x)-7 \sinh (6 x)+3 \sinh (7 x)}{1344 a} \]

Antiderivative was successfully verified.

[In]

Integrate[Sinh[x]^8/(a + a*Cosh[x]),x]

[Out]

(420*x - 105*Sinh[x] - 315*Sinh[2*x] + 63*Sinh[3*x] + 63*Sinh[4*x] - 21*Sinh[5*x] - 7*Sinh[6*x] + 3*Sinh[7*x])
/(1344*a)

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Maple [B]  time = 0.04, size = 208, normalized size = 3.7 \begin{align*} -{\frac{1}{7\,a} \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) ^{-7}}+{\frac{2}{3\,a} \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) ^{-6}}-{\frac{1}{a} \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) ^{-5}}+{\frac{1}{4\,a} \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) ^{-4}}+{\frac{11}{24\,a} \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) ^{-3}}+{\frac{1}{8\,a} \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) ^{-2}}-{\frac{5}{16\,a} \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) ^{-1}}+{\frac{5}{16\,a}\ln \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) }-{\frac{1}{7\,a} \left ( \tanh \left ({\frac{x}{2}} \right ) -1 \right ) ^{-7}}-{\frac{2}{3\,a} \left ( \tanh \left ({\frac{x}{2}} \right ) -1 \right ) ^{-6}}-{\frac{1}{a} \left ( \tanh \left ({\frac{x}{2}} \right ) -1 \right ) ^{-5}}-{\frac{1}{4\,a} \left ( \tanh \left ({\frac{x}{2}} \right ) -1 \right ) ^{-4}}+{\frac{11}{24\,a} \left ( \tanh \left ({\frac{x}{2}} \right ) -1 \right ) ^{-3}}-{\frac{1}{8\,a} \left ( \tanh \left ({\frac{x}{2}} \right ) -1 \right ) ^{-2}}-{\frac{5}{16\,a} \left ( \tanh \left ({\frac{x}{2}} \right ) -1 \right ) ^{-1}}-{\frac{5}{16\,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)^8/(a+a*cosh(x)),x)

[Out]

-1/7/a/(tanh(1/2*x)+1)^7+2/3/a/(tanh(1/2*x)+1)^6-1/a/(tanh(1/2*x)+1)^5+1/4/a/(tanh(1/2*x)+1)^4+11/24/a/(tanh(1
/2*x)+1)^3+1/8/a/(tanh(1/2*x)+1)^2-5/16/a/(tanh(1/2*x)+1)+5/16/a*ln(tanh(1/2*x)+1)-1/7/a/(tanh(1/2*x)-1)^7-2/3
/a/(tanh(1/2*x)-1)^6-1/a/(tanh(1/2*x)-1)^5-1/4/a/(tanh(1/2*x)-1)^4+11/24/a/(tanh(1/2*x)-1)^3-1/8/a/(tanh(1/2*x
)-1)^2-5/16/a/(tanh(1/2*x)-1)-5/16/a*ln(tanh(1/2*x)-1)

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Maxima [B]  time = 1.11207, size = 138, normalized size = 2.42 \begin{align*} -\frac{{\left (7 \, e^{\left (-x\right )} + 21 \, e^{\left (-2 \, x\right )} - 63 \, e^{\left (-3 \, x\right )} - 63 \, e^{\left (-4 \, x\right )} + 315 \, e^{\left (-5 \, x\right )} + 105 \, e^{\left (-6 \, x\right )} - 3\right )} e^{\left (7 \, x\right )}}{2688 \, a} + \frac{5 \, x}{16 \, a} + \frac{105 \, e^{\left (-x\right )} + 315 \, e^{\left (-2 \, x\right )} - 63 \, e^{\left (-3 \, x\right )} - 63 \, e^{\left (-4 \, x\right )} + 21 \, e^{\left (-5 \, x\right )} + 7 \, e^{\left (-6 \, x\right )} - 3 \, e^{\left (-7 \, x\right )}}{2688 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(x)^8/(a+a*cosh(x)),x, algorithm="maxima")

[Out]

-1/2688*(7*e^(-x) + 21*e^(-2*x) - 63*e^(-3*x) - 63*e^(-4*x) + 315*e^(-5*x) + 105*e^(-6*x) - 3)*e^(7*x)/a + 5/1
6*x/a + 1/2688*(105*e^(-x) + 315*e^(-2*x) - 63*e^(-3*x) - 63*e^(-4*x) + 21*e^(-5*x) + 7*e^(-6*x) - 3*e^(-7*x))
/a

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Fricas [B]  time = 1.99905, size = 340, normalized size = 5.96 \begin{align*} \frac{3 \, \sinh \left (x\right )^{7} + 21 \,{\left (3 \, \cosh \left (x\right )^{2} - 2 \, \cosh \left (x\right ) - 1\right )} \sinh \left (x\right )^{5} + 7 \,{\left (15 \, \cosh \left (x\right )^{4} - 20 \, \cosh \left (x\right )^{3} - 30 \, \cosh \left (x\right )^{2} + 36 \, \cosh \left (x\right ) + 9\right )} \sinh \left (x\right )^{3} + 21 \,{\left (\cosh \left (x\right )^{6} - 2 \, \cosh \left (x\right )^{5} - 5 \, \cosh \left (x\right )^{4} + 12 \, \cosh \left (x\right )^{3} + 9 \, \cosh \left (x\right )^{2} - 30 \, \cosh \left (x\right ) - 5\right )} \sinh \left (x\right ) + 420 \, x}{1344 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(x)^8/(a+a*cosh(x)),x, algorithm="fricas")

[Out]

1/1344*(3*sinh(x)^7 + 21*(3*cosh(x)^2 - 2*cosh(x) - 1)*sinh(x)^5 + 7*(15*cosh(x)^4 - 20*cosh(x)^3 - 30*cosh(x)
^2 + 36*cosh(x) + 9)*sinh(x)^3 + 21*(cosh(x)^6 - 2*cosh(x)^5 - 5*cosh(x)^4 + 12*cosh(x)^3 + 9*cosh(x)^2 - 30*c
osh(x) - 5)*sinh(x) + 420*x)/a

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Sympy [B]  time = 22.9578, size = 1253, normalized size = 21.98 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(x)**8/(a+a*cosh(x)),x)

[Out]

105*x*tanh(x/2)**14/(336*a*tanh(x/2)**14 - 2352*a*tanh(x/2)**12 + 7056*a*tanh(x/2)**10 - 11760*a*tanh(x/2)**8
+ 11760*a*tanh(x/2)**6 - 7056*a*tanh(x/2)**4 + 2352*a*tanh(x/2)**2 - 336*a) - 735*x*tanh(x/2)**12/(336*a*tanh(
x/2)**14 - 2352*a*tanh(x/2)**12 + 7056*a*tanh(x/2)**10 - 11760*a*tanh(x/2)**8 + 11760*a*tanh(x/2)**6 - 7056*a*
tanh(x/2)**4 + 2352*a*tanh(x/2)**2 - 336*a) + 2205*x*tanh(x/2)**10/(336*a*tanh(x/2)**14 - 2352*a*tanh(x/2)**12
+ 7056*a*tanh(x/2)**10 - 11760*a*tanh(x/2)**8 + 11760*a*tanh(x/2)**6 - 7056*a*tanh(x/2)**4 + 2352*a*tanh(x/2)
**2 - 336*a) - 3675*x*tanh(x/2)**8/(336*a*tanh(x/2)**14 - 2352*a*tanh(x/2)**12 + 7056*a*tanh(x/2)**10 - 11760*
a*tanh(x/2)**8 + 11760*a*tanh(x/2)**6 - 7056*a*tanh(x/2)**4 + 2352*a*tanh(x/2)**2 - 336*a) + 3675*x*tanh(x/2)*
*6/(336*a*tanh(x/2)**14 - 2352*a*tanh(x/2)**12 + 7056*a*tanh(x/2)**10 - 11760*a*tanh(x/2)**8 + 11760*a*tanh(x/
2)**6 - 7056*a*tanh(x/2)**4 + 2352*a*tanh(x/2)**2 - 336*a) - 2205*x*tanh(x/2)**4/(336*a*tanh(x/2)**14 - 2352*a
*tanh(x/2)**12 + 7056*a*tanh(x/2)**10 - 11760*a*tanh(x/2)**8 + 11760*a*tanh(x/2)**6 - 7056*a*tanh(x/2)**4 + 23
52*a*tanh(x/2)**2 - 336*a) + 735*x*tanh(x/2)**2/(336*a*tanh(x/2)**14 - 2352*a*tanh(x/2)**12 + 7056*a*tanh(x/2)
**10 - 11760*a*tanh(x/2)**8 + 11760*a*tanh(x/2)**6 - 7056*a*tanh(x/2)**4 + 2352*a*tanh(x/2)**2 - 336*a) - 105*
x/(336*a*tanh(x/2)**14 - 2352*a*tanh(x/2)**12 + 7056*a*tanh(x/2)**10 - 11760*a*tanh(x/2)**8 + 11760*a*tanh(x/2
)**6 - 7056*a*tanh(x/2)**4 + 2352*a*tanh(x/2)**2 - 336*a) - 210*tanh(x/2)**13/(336*a*tanh(x/2)**14 - 2352*a*ta
nh(x/2)**12 + 7056*a*tanh(x/2)**10 - 11760*a*tanh(x/2)**8 + 11760*a*tanh(x/2)**6 - 7056*a*tanh(x/2)**4 + 2352*
a*tanh(x/2)**2 - 336*a) + 1400*tanh(x/2)**11/(336*a*tanh(x/2)**14 - 2352*a*tanh(x/2)**12 + 7056*a*tanh(x/2)**1
0 - 11760*a*tanh(x/2)**8 + 11760*a*tanh(x/2)**6 - 7056*a*tanh(x/2)**4 + 2352*a*tanh(x/2)**2 - 336*a) - 3962*ta
nh(x/2)**9/(336*a*tanh(x/2)**14 - 2352*a*tanh(x/2)**12 + 7056*a*tanh(x/2)**10 - 11760*a*tanh(x/2)**8 + 11760*a
*tanh(x/2)**6 - 7056*a*tanh(x/2)**4 + 2352*a*tanh(x/2)**2 - 336*a) - 6144*tanh(x/2)**7/(336*a*tanh(x/2)**14 -
2352*a*tanh(x/2)**12 + 7056*a*tanh(x/2)**10 - 11760*a*tanh(x/2)**8 + 11760*a*tanh(x/2)**6 - 7056*a*tanh(x/2)**
4 + 2352*a*tanh(x/2)**2 - 336*a) + 3962*tanh(x/2)**5/(336*a*tanh(x/2)**14 - 2352*a*tanh(x/2)**12 + 7056*a*tanh
(x/2)**10 - 11760*a*tanh(x/2)**8 + 11760*a*tanh(x/2)**6 - 7056*a*tanh(x/2)**4 + 2352*a*tanh(x/2)**2 - 336*a) -
1400*tanh(x/2)**3/(336*a*tanh(x/2)**14 - 2352*a*tanh(x/2)**12 + 7056*a*tanh(x/2)**10 - 11760*a*tanh(x/2)**8 +
11760*a*tanh(x/2)**6 - 7056*a*tanh(x/2)**4 + 2352*a*tanh(x/2)**2 - 336*a) + 210*tanh(x/2)/(336*a*tanh(x/2)**1
4 - 2352*a*tanh(x/2)**12 + 7056*a*tanh(x/2)**10 - 11760*a*tanh(x/2)**8 + 11760*a*tanh(x/2)**6 - 7056*a*tanh(x/
2)**4 + 2352*a*tanh(x/2)**2 - 336*a)

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Giac [A]  time = 1.17844, size = 122, normalized size = 2.14 \begin{align*} \frac{{\left (105 \, e^{\left (6 \, x\right )} + 315 \, e^{\left (5 \, x\right )} - 63 \, e^{\left (4 \, x\right )} - 63 \, e^{\left (3 \, x\right )} + 21 \, e^{\left (2 \, x\right )} + 7 \, e^{x} - 3\right )} e^{\left (-7 \, x\right )} + 840 \, x + 3 \, e^{\left (7 \, x\right )} - 7 \, e^{\left (6 \, x\right )} - 21 \, e^{\left (5 \, x\right )} + 63 \, e^{\left (4 \, x\right )} + 63 \, e^{\left (3 \, x\right )} - 315 \, e^{\left (2 \, x\right )} - 105 \, e^{x}}{2688 \, a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(x)^8/(a+a*cosh(x)),x, algorithm="giac")

[Out]

1/2688*((105*e^(6*x) + 315*e^(5*x) - 63*e^(4*x) - 63*e^(3*x) + 21*e^(2*x) + 7*e^x - 3)*e^(-7*x) + 840*x + 3*e^
(7*x) - 7*e^(6*x) - 21*e^(5*x) + 63*e^(4*x) + 63*e^(3*x) - 315*e^(2*x) - 105*e^x)/a