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

Optimal. Leaf size=154 \[ -\frac{a x \left (-20 a^2 b^2+8 a^4+15 b^4\right )}{8 b^6}+\frac{\sinh ^3(x) \left (4 \left (a^2-b^2\right )-3 a b \cosh (x)\right )}{12 b^3}+\frac{\sinh (x) \left (8 \left (a^2-b^2\right )^2-a b \left (4 a^2-7 b^2\right ) \cosh (x)\right )}{8 b^5}+\frac{2 (a-b)^{5/2} (a+b)^{5/2} \tanh ^{-1}\left (\frac{\sqrt{a-b} \tanh \left (\frac{x}{2}\right )}{\sqrt{a+b}}\right )}{b^6}+\frac{\sinh ^5(x)}{5 b} \]

[Out]

-(a*(8*a^4 - 20*a^2*b^2 + 15*b^4)*x)/(8*b^6) + (2*(a - b)^(5/2)*(a + b)^(5/2)*ArcTanh[(Sqrt[a - b]*Tanh[x/2])/
Sqrt[a + b]])/b^6 + ((8*(a^2 - b^2)^2 - a*b*(4*a^2 - 7*b^2)*Cosh[x])*Sinh[x])/(8*b^5) + ((4*(a^2 - b^2) - 3*a*
b*Cosh[x])*Sinh[x]^3)/(12*b^3) + Sinh[x]^5/(5*b)

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Rubi [A]  time = 0.425108, antiderivative size = 154, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.385, Rules used = {2695, 2865, 2735, 2659, 208} \[ -\frac{a x \left (-20 a^2 b^2+8 a^4+15 b^4\right )}{8 b^6}+\frac{\sinh ^3(x) \left (4 \left (a^2-b^2\right )-3 a b \cosh (x)\right )}{12 b^3}+\frac{\sinh (x) \left (8 \left (a^2-b^2\right )^2-a b \left (4 a^2-7 b^2\right ) \cosh (x)\right )}{8 b^5}+\frac{2 (a-b)^{5/2} (a+b)^{5/2} \tanh ^{-1}\left (\frac{\sqrt{a-b} \tanh \left (\frac{x}{2}\right )}{\sqrt{a+b}}\right )}{b^6}+\frac{\sinh ^5(x)}{5 b} \]

Antiderivative was successfully verified.

[In]

Int[Sinh[x]^6/(a + b*Cosh[x]),x]

[Out]

-(a*(8*a^4 - 20*a^2*b^2 + 15*b^4)*x)/(8*b^6) + (2*(a - b)^(5/2)*(a + b)^(5/2)*ArcTanh[(Sqrt[a - b]*Tanh[x/2])/
Sqrt[a + b]])/b^6 + ((8*(a^2 - b^2)^2 - a*b*(4*a^2 - 7*b^2)*Cosh[x])*Sinh[x])/(8*b^5) + ((4*(a^2 - b^2) - 3*a*
b*Cosh[x])*Sinh[x]^3)/(12*b^3) + Sinh[x]^5/(5*b)

Rule 2695

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[(g*(g*
Cos[e + f*x])^(p - 1)*(a + b*Sin[e + f*x])^(m + 1))/(b*f*(m + p)), x] + Dist[(g^2*(p - 1))/(b*(m + p)), Int[(g
*Cos[e + f*x])^(p - 2)*(a + b*Sin[e + f*x])^m*(b + a*Sin[e + f*x]), x], x] /; FreeQ[{a, b, e, f, g, m}, x] &&
NeQ[a^2 - b^2, 0] && GtQ[p, 1] && NeQ[m + p, 0] && IntegersQ[2*m, 2*p]

Rule 2865

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

Rule 2735

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

Rule 2659

Int[((a_) + (b_.)*sin[Pi/2 + (c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{e = FreeFactors[Tan[(c + d*x)/2], x
]}, Dist[(2*e)/d, Subst[Int[1/(a + b + (a - b)*e^2*x^2), x], x, Tan[(c + d*x)/2]/e], x]] /; FreeQ[{a, b, c, d}
, x] && NeQ[a^2 - b^2, 0]

Rule 208

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]

Rubi steps

\begin{align*} \int \frac{\sinh ^6(x)}{a+b \cosh (x)} \, dx &=\frac{\sinh ^5(x)}{5 b}+\frac{\int \frac{(-b-a \cosh (x)) \sinh ^4(x)}{a+b \cosh (x)} \, dx}{b}\\ &=\frac{\left (4 \left (a^2-b^2\right )-3 a b \cosh (x)\right ) \sinh ^3(x)}{12 b^3}+\frac{\sinh ^5(x)}{5 b}-\frac{\int \frac{\left (b \left (a^2-4 b^2\right )+a \left (4 a^2-7 b^2\right ) \cosh (x)\right ) \sinh ^2(x)}{a+b \cosh (x)} \, dx}{4 b^3}\\ &=\frac{\left (8 \left (a^2-b^2\right )^2-a b \left (4 a^2-7 b^2\right ) \cosh (x)\right ) \sinh (x)}{8 b^5}+\frac{\left (4 \left (a^2-b^2\right )-3 a b \cosh (x)\right ) \sinh ^3(x)}{12 b^3}+\frac{\sinh ^5(x)}{5 b}+\frac{\int \frac{-b \left (4 a^4-9 a^2 b^2+8 b^4\right )-a \left (8 a^4-20 a^2 b^2+15 b^4\right ) \cosh (x)}{a+b \cosh (x)} \, dx}{8 b^5}\\ &=-\frac{a \left (8 a^4-20 a^2 b^2+15 b^4\right ) x}{8 b^6}+\frac{\left (8 \left (a^2-b^2\right )^2-a b \left (4 a^2-7 b^2\right ) \cosh (x)\right ) \sinh (x)}{8 b^5}+\frac{\left (4 \left (a^2-b^2\right )-3 a b \cosh (x)\right ) \sinh ^3(x)}{12 b^3}+\frac{\sinh ^5(x)}{5 b}+\frac{\left (a^2-b^2\right )^3 \int \frac{1}{a+b \cosh (x)} \, dx}{b^6}\\ &=-\frac{a \left (8 a^4-20 a^2 b^2+15 b^4\right ) x}{8 b^6}+\frac{\left (8 \left (a^2-b^2\right )^2-a b \left (4 a^2-7 b^2\right ) \cosh (x)\right ) \sinh (x)}{8 b^5}+\frac{\left (4 \left (a^2-b^2\right )-3 a b \cosh (x)\right ) \sinh ^3(x)}{12 b^3}+\frac{\sinh ^5(x)}{5 b}+\frac{\left (2 \left (a^2-b^2\right )^3\right ) \operatorname{Subst}\left (\int \frac{1}{a+b-(a-b) x^2} \, dx,x,\tanh \left (\frac{x}{2}\right )\right )}{b^6}\\ &=-\frac{a \left (8 a^4-20 a^2 b^2+15 b^4\right ) x}{8 b^6}+\frac{2 (a-b)^{5/2} (a+b)^{5/2} \tanh ^{-1}\left (\frac{\sqrt{a-b} \tanh \left (\frac{x}{2}\right )}{\sqrt{a+b}}\right )}{b^6}+\frac{\left (8 \left (a^2-b^2\right )^2-a b \left (4 a^2-7 b^2\right ) \cosh (x)\right ) \sinh (x)}{8 b^5}+\frac{\left (4 \left (a^2-b^2\right )-3 a b \cosh (x)\right ) \sinh ^3(x)}{12 b^3}+\frac{\sinh ^5(x)}{5 b}\\ \end{align*}

Mathematica [A]  time = 0.225557, size = 154, normalized size = 1. \[ \frac{-60 a x \left (-20 a^2 b^2+8 a^4+15 b^4\right )-10 b^3 \left (7 b^2-4 a^2\right ) \sinh (3 x)-120 a b^2 \left (a^2-2 b^2\right ) \sinh (2 x)+60 b \left (-18 a^2 b^2+8 a^4+11 b^4\right ) \sinh (x)+960 \left (b^2-a^2\right )^{5/2} \tan ^{-1}\left (\frac{(a-b) \tanh \left (\frac{x}{2}\right )}{\sqrt{b^2-a^2}}\right )-15 a b^4 \sinh (4 x)+6 b^5 \sinh (5 x)}{480 b^6} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-60*a*(8*a^4 - 20*a^2*b^2 + 15*b^4)*x + 960*(-a^2 + b^2)^(5/2)*ArcTan[((a - b)*Tanh[x/2])/Sqrt[-a^2 + b^2]] +
60*b*(8*a^4 - 18*a^2*b^2 + 11*b^4)*Sinh[x] - 120*a*b^2*(a^2 - 2*b^2)*Sinh[2*x] - 10*b^3*(-4*a^2 + 7*b^2)*Sinh
[3*x] - 15*a*b^4*Sinh[4*x] + 6*b^5*Sinh[5*x])/(480*b^6)

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Maple [B]  time = 0.028, size = 679, normalized size = 4.4 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sinh(x)^6/(a+b*cosh(x)),x)

[Out]

-2/((a+b)*(a-b))^(1/2)*arctanh((a-b)*tanh(1/2*x)/((a+b)*(a-b))^(1/2))+1/12/b/(tanh(1/2*x)+1)^3-5/8/b/(tanh(1/2
*x)+1)^2-1/b/(tanh(1/2*x)+1)+1/12/b/(tanh(1/2*x)-1)^3+5/8/b/(tanh(1/2*x)-1)^2-1/b/(tanh(1/2*x)-1)+2/b^6/((a+b)
*(a-b))^(1/2)*arctanh((a-b)*tanh(1/2*x)/((a+b)*(a-b))^(1/2))*a^6-1/5/b/(tanh(1/2*x)-1)^5-1/2/b/(tanh(1/2*x)-1)
^4-1/5/b/(tanh(1/2*x)+1)^5+1/2/b/(tanh(1/2*x)+1)^4-6*a^4/b^4/((a+b)*(a-b))^(1/2)*arctanh((a-b)*tanh(1/2*x)/((a
+b)*(a-b))^(1/2))+6*a^2/b^2/((a+b)*(a-b))^(1/2)*arctanh((a-b)*tanh(1/2*x)/((a+b)*(a-b))^(1/2))-a^5/b^6*ln(tanh
(1/2*x)+1)+a^5/b^6*ln(tanh(1/2*x)-1)-5/8/b^2/(tanh(1/2*x)+1)^2*a+2/b^3/(tanh(1/2*x)+1)*a^2+7/8/b^2/(tanh(1/2*x
)+1)*a+5/2*a^3/b^4*ln(tanh(1/2*x)+1)-15/8*a/b^2*ln(tanh(1/2*x)+1)+5/8/b^2/(tanh(1/2*x)-1)^2*a+2/b^3/(tanh(1/2*
x)-1)*a^2+7/8/b^2/(tanh(1/2*x)-1)*a-5/2*a^3/b^4*ln(tanh(1/2*x)-1)+15/8*a/b^2*ln(tanh(1/2*x)-1)+1/4/b^2/(tanh(1
/2*x)+1)^4*a-1/3/b^3/(tanh(1/2*x)+1)^3*a^2-1/2/b^2/(tanh(1/2*x)+1)^3*a+1/2/b^4/(tanh(1/2*x)+1)^2*a^3+1/2/b^3/(
tanh(1/2*x)+1)^2*a^2-1/b^5/(tanh(1/2*x)+1)*a^4-1/2/b^4/(tanh(1/2*x)+1)*a^3-1/4/b^2/(tanh(1/2*x)-1)^4*a-1/3/b^3
/(tanh(1/2*x)-1)^3*a^2-1/2/b^2/(tanh(1/2*x)-1)^3*a-1/2/b^4/(tanh(1/2*x)-1)^2*a^3-1/2/b^3/(tanh(1/2*x)-1)^2*a^2
-1/b^5/(tanh(1/2*x)-1)*a^4-1/2/b^4/(tanh(1/2*x)-1)*a^3

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 2.34401, size = 7443, normalized size = 48.33 \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+b*cosh(x)),x, algorithm="fricas")

[Out]

[1/960*(6*b^5*cosh(x)^10 + 6*b^5*sinh(x)^10 - 15*a*b^4*cosh(x)^9 + 15*(4*b^5*cosh(x) - a*b^4)*sinh(x)^9 + 10*(
4*a^2*b^3 - 7*b^5)*cosh(x)^8 + 5*(54*b^5*cosh(x)^2 - 27*a*b^4*cosh(x) + 8*a^2*b^3 - 14*b^5)*sinh(x)^8 - 120*(a
^3*b^2 - 2*a*b^4)*cosh(x)^7 + 20*(36*b^5*cosh(x)^3 - 27*a*b^4*cosh(x)^2 - 6*a^3*b^2 + 12*a*b^4 + 4*(4*a^2*b^3
- 7*b^5)*cosh(x))*sinh(x)^7 - 120*(8*a^5 - 20*a^3*b^2 + 15*a*b^4)*x*cosh(x)^5 + 60*(8*a^4*b - 18*a^2*b^3 + 11*
b^5)*cosh(x)^6 + 20*(63*b^5*cosh(x)^4 - 63*a*b^4*cosh(x)^3 + 24*a^4*b - 54*a^2*b^3 + 33*b^5 + 14*(4*a^2*b^3 -
7*b^5)*cosh(x)^2 - 42*(a^3*b^2 - 2*a*b^4)*cosh(x))*sinh(x)^6 + 15*a*b^4*cosh(x) + 2*(756*b^5*cosh(x)^5 - 945*a
*b^4*cosh(x)^4 + 280*(4*a^2*b^3 - 7*b^5)*cosh(x)^3 - 1260*(a^3*b^2 - 2*a*b^4)*cosh(x)^2 - 60*(8*a^5 - 20*a^3*b
^2 + 15*a*b^4)*x + 180*(8*a^4*b - 18*a^2*b^3 + 11*b^5)*cosh(x))*sinh(x)^5 - 6*b^5 - 60*(8*a^4*b - 18*a^2*b^3 +
11*b^5)*cosh(x)^4 + 10*(126*b^5*cosh(x)^6 - 189*a*b^4*cosh(x)^5 - 48*a^4*b + 108*a^2*b^3 - 66*b^5 + 70*(4*a^2
*b^3 - 7*b^5)*cosh(x)^4 - 420*(a^3*b^2 - 2*a*b^4)*cosh(x)^3 - 60*(8*a^5 - 20*a^3*b^2 + 15*a*b^4)*x*cosh(x) + 9
0*(8*a^4*b - 18*a^2*b^3 + 11*b^5)*cosh(x)^2)*sinh(x)^4 + 120*(a^3*b^2 - 2*a*b^4)*cosh(x)^3 + 20*(36*b^5*cosh(x
)^7 - 63*a*b^4*cosh(x)^6 + 28*(4*a^2*b^3 - 7*b^5)*cosh(x)^5 + 6*a^3*b^2 - 12*a*b^4 - 210*(a^3*b^2 - 2*a*b^4)*c
osh(x)^4 - 60*(8*a^5 - 20*a^3*b^2 + 15*a*b^4)*x*cosh(x)^2 + 60*(8*a^4*b - 18*a^2*b^3 + 11*b^5)*cosh(x)^3 - 12*
(8*a^4*b - 18*a^2*b^3 + 11*b^5)*cosh(x))*sinh(x)^3 - 10*(4*a^2*b^3 - 7*b^5)*cosh(x)^2 + 10*(27*b^5*cosh(x)^8 -
54*a*b^4*cosh(x)^7 + 28*(4*a^2*b^3 - 7*b^5)*cosh(x)^6 - 252*(a^3*b^2 - 2*a*b^4)*cosh(x)^5 - 4*a^2*b^3 + 7*b^5
- 120*(8*a^5 - 20*a^3*b^2 + 15*a*b^4)*x*cosh(x)^3 + 90*(8*a^4*b - 18*a^2*b^3 + 11*b^5)*cosh(x)^4 - 36*(8*a^4*
b - 18*a^2*b^3 + 11*b^5)*cosh(x)^2 + 36*(a^3*b^2 - 2*a*b^4)*cosh(x))*sinh(x)^2 + 960*((a^4 - 2*a^2*b^2 + b^4)*
cosh(x)^5 + 5*(a^4 - 2*a^2*b^2 + b^4)*cosh(x)^4*sinh(x) + 10*(a^4 - 2*a^2*b^2 + b^4)*cosh(x)^3*sinh(x)^2 + 10*
(a^4 - 2*a^2*b^2 + b^4)*cosh(x)^2*sinh(x)^3 + 5*(a^4 - 2*a^2*b^2 + b^4)*cosh(x)*sinh(x)^4 + (a^4 - 2*a^2*b^2 +
b^4)*sinh(x)^5)*sqrt(a^2 - b^2)*log((b^2*cosh(x)^2 + b^2*sinh(x)^2 + 2*a*b*cosh(x) + 2*a^2 - b^2 + 2*(b^2*cos
h(x) + a*b)*sinh(x) - 2*sqrt(a^2 - b^2)*(b*cosh(x) + b*sinh(x) + a))/(b*cosh(x)^2 + b*sinh(x)^2 + 2*a*cosh(x)
+ 2*(b*cosh(x) + a)*sinh(x) + b)) + 5*(12*b^5*cosh(x)^9 - 27*a*b^4*cosh(x)^8 + 16*(4*a^2*b^3 - 7*b^5)*cosh(x)^
7 - 168*(a^3*b^2 - 2*a*b^4)*cosh(x)^6 - 120*(8*a^5 - 20*a^3*b^2 + 15*a*b^4)*x*cosh(x)^4 + 72*(8*a^4*b - 18*a^2
*b^3 + 11*b^5)*cosh(x)^5 + 3*a*b^4 - 48*(8*a^4*b - 18*a^2*b^3 + 11*b^5)*cosh(x)^3 + 72*(a^3*b^2 - 2*a*b^4)*cos
h(x)^2 - 4*(4*a^2*b^3 - 7*b^5)*cosh(x))*sinh(x))/(b^6*cosh(x)^5 + 5*b^6*cosh(x)^4*sinh(x) + 10*b^6*cosh(x)^3*s
inh(x)^2 + 10*b^6*cosh(x)^2*sinh(x)^3 + 5*b^6*cosh(x)*sinh(x)^4 + b^6*sinh(x)^5), 1/960*(6*b^5*cosh(x)^10 + 6*
b^5*sinh(x)^10 - 15*a*b^4*cosh(x)^9 + 15*(4*b^5*cosh(x) - a*b^4)*sinh(x)^9 + 10*(4*a^2*b^3 - 7*b^5)*cosh(x)^8
+ 5*(54*b^5*cosh(x)^2 - 27*a*b^4*cosh(x) + 8*a^2*b^3 - 14*b^5)*sinh(x)^8 - 120*(a^3*b^2 - 2*a*b^4)*cosh(x)^7 +
20*(36*b^5*cosh(x)^3 - 27*a*b^4*cosh(x)^2 - 6*a^3*b^2 + 12*a*b^4 + 4*(4*a^2*b^3 - 7*b^5)*cosh(x))*sinh(x)^7 -
120*(8*a^5 - 20*a^3*b^2 + 15*a*b^4)*x*cosh(x)^5 + 60*(8*a^4*b - 18*a^2*b^3 + 11*b^5)*cosh(x)^6 + 20*(63*b^5*c
osh(x)^4 - 63*a*b^4*cosh(x)^3 + 24*a^4*b - 54*a^2*b^3 + 33*b^5 + 14*(4*a^2*b^3 - 7*b^5)*cosh(x)^2 - 42*(a^3*b^
2 - 2*a*b^4)*cosh(x))*sinh(x)^6 + 15*a*b^4*cosh(x) + 2*(756*b^5*cosh(x)^5 - 945*a*b^4*cosh(x)^4 + 280*(4*a^2*b
^3 - 7*b^5)*cosh(x)^3 - 1260*(a^3*b^2 - 2*a*b^4)*cosh(x)^2 - 60*(8*a^5 - 20*a^3*b^2 + 15*a*b^4)*x + 180*(8*a^4
*b - 18*a^2*b^3 + 11*b^5)*cosh(x))*sinh(x)^5 - 6*b^5 - 60*(8*a^4*b - 18*a^2*b^3 + 11*b^5)*cosh(x)^4 + 10*(126*
b^5*cosh(x)^6 - 189*a*b^4*cosh(x)^5 - 48*a^4*b + 108*a^2*b^3 - 66*b^5 + 70*(4*a^2*b^3 - 7*b^5)*cosh(x)^4 - 420
*(a^3*b^2 - 2*a*b^4)*cosh(x)^3 - 60*(8*a^5 - 20*a^3*b^2 + 15*a*b^4)*x*cosh(x) + 90*(8*a^4*b - 18*a^2*b^3 + 11*
b^5)*cosh(x)^2)*sinh(x)^4 + 120*(a^3*b^2 - 2*a*b^4)*cosh(x)^3 + 20*(36*b^5*cosh(x)^7 - 63*a*b^4*cosh(x)^6 + 28
*(4*a^2*b^3 - 7*b^5)*cosh(x)^5 + 6*a^3*b^2 - 12*a*b^4 - 210*(a^3*b^2 - 2*a*b^4)*cosh(x)^4 - 60*(8*a^5 - 20*a^3
*b^2 + 15*a*b^4)*x*cosh(x)^2 + 60*(8*a^4*b - 18*a^2*b^3 + 11*b^5)*cosh(x)^3 - 12*(8*a^4*b - 18*a^2*b^3 + 11*b^
5)*cosh(x))*sinh(x)^3 - 10*(4*a^2*b^3 - 7*b^5)*cosh(x)^2 + 10*(27*b^5*cosh(x)^8 - 54*a*b^4*cosh(x)^7 + 28*(4*a
^2*b^3 - 7*b^5)*cosh(x)^6 - 252*(a^3*b^2 - 2*a*b^4)*cosh(x)^5 - 4*a^2*b^3 + 7*b^5 - 120*(8*a^5 - 20*a^3*b^2 +
15*a*b^4)*x*cosh(x)^3 + 90*(8*a^4*b - 18*a^2*b^3 + 11*b^5)*cosh(x)^4 - 36*(8*a^4*b - 18*a^2*b^3 + 11*b^5)*cosh
(x)^2 + 36*(a^3*b^2 - 2*a*b^4)*cosh(x))*sinh(x)^2 - 1920*((a^4 - 2*a^2*b^2 + b^4)*cosh(x)^5 + 5*(a^4 - 2*a^2*b
^2 + b^4)*cosh(x)^4*sinh(x) + 10*(a^4 - 2*a^2*b^2 + b^4)*cosh(x)^3*sinh(x)^2 + 10*(a^4 - 2*a^2*b^2 + b^4)*cosh
(x)^2*sinh(x)^3 + 5*(a^4 - 2*a^2*b^2 + b^4)*cosh(x)*sinh(x)^4 + (a^4 - 2*a^2*b^2 + b^4)*sinh(x)^5)*sqrt(-a^2 +
b^2)*arctan(-sqrt(-a^2 + b^2)*(b*cosh(x) + b*sinh(x) + a)/(a^2 - b^2)) + 5*(12*b^5*cosh(x)^9 - 27*a*b^4*cosh(
x)^8 + 16*(4*a^2*b^3 - 7*b^5)*cosh(x)^7 - 168*(a^3*b^2 - 2*a*b^4)*cosh(x)^6 - 120*(8*a^5 - 20*a^3*b^2 + 15*a*b
^4)*x*cosh(x)^4 + 72*(8*a^4*b - 18*a^2*b^3 + 11*b^5)*cosh(x)^5 + 3*a*b^4 - 48*(8*a^4*b - 18*a^2*b^3 + 11*b^5)*
cosh(x)^3 + 72*(a^3*b^2 - 2*a*b^4)*cosh(x)^2 - 4*(4*a^2*b^3 - 7*b^5)*cosh(x))*sinh(x))/(b^6*cosh(x)^5 + 5*b^6*
cosh(x)^4*sinh(x) + 10*b^6*cosh(x)^3*sinh(x)^2 + 10*b^6*cosh(x)^2*sinh(x)^3 + 5*b^6*cosh(x)*sinh(x)^4 + b^6*si
nh(x)^5)]

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [A]  time = 1.18367, size = 359, normalized size = 2.33 \begin{align*} \frac{6 \, b^{4} e^{\left (5 \, x\right )} - 15 \, a b^{3} e^{\left (4 \, x\right )} + 40 \, a^{2} b^{2} e^{\left (3 \, x\right )} - 70 \, b^{4} e^{\left (3 \, x\right )} - 120 \, a^{3} b e^{\left (2 \, x\right )} + 240 \, a b^{3} e^{\left (2 \, x\right )} + 480 \, a^{4} e^{x} - 1080 \, a^{2} b^{2} e^{x} + 660 \, b^{4} e^{x}}{960 \, b^{5}} - \frac{{\left (8 \, a^{5} - 20 \, a^{3} b^{2} + 15 \, a b^{4}\right )} x}{8 \, b^{6}} + \frac{{\left (15 \, a b^{4} e^{x} - 6 \, b^{5} - 60 \,{\left (8 \, a^{4} b - 18 \, a^{2} b^{3} + 11 \, b^{5}\right )} e^{\left (4 \, x\right )} + 120 \,{\left (a^{3} b^{2} - 2 \, a b^{4}\right )} e^{\left (3 \, x\right )} - 10 \,{\left (4 \, a^{2} b^{3} - 7 \, b^{5}\right )} e^{\left (2 \, x\right )}\right )} e^{\left (-5 \, x\right )}}{960 \, b^{6}} + \frac{2 \,{\left (a^{6} - 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} - b^{6}\right )} \arctan \left (\frac{b e^{x} + a}{\sqrt{-a^{2} + b^{2}}}\right )}{\sqrt{-a^{2} + b^{2}} b^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/960*(6*b^4*e^(5*x) - 15*a*b^3*e^(4*x) + 40*a^2*b^2*e^(3*x) - 70*b^4*e^(3*x) - 120*a^3*b*e^(2*x) + 240*a*b^3*
e^(2*x) + 480*a^4*e^x - 1080*a^2*b^2*e^x + 660*b^4*e^x)/b^5 - 1/8*(8*a^5 - 20*a^3*b^2 + 15*a*b^4)*x/b^6 + 1/96
0*(15*a*b^4*e^x - 6*b^5 - 60*(8*a^4*b - 18*a^2*b^3 + 11*b^5)*e^(4*x) + 120*(a^3*b^2 - 2*a*b^4)*e^(3*x) - 10*(4
*a^2*b^3 - 7*b^5)*e^(2*x))*e^(-5*x)/b^6 + 2*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*arctan((b*e^x + a)/sqrt(-a^2 +
b^2))/(sqrt(-a^2 + b^2)*b^6)