\(\int \frac {\cosh ^4(x)}{a+b \cosh (x)} \, dx\) [54]

Optimal result

Integrand size = 13, antiderivative size = 112 \[ \int \frac {\cosh ^4(x)}{a+b \cosh (x)} \, dx=-\frac {a \left (2 a^2+b^2\right ) x}{2 b^4}+\frac {2 a^4 \text {arctanh}\left (\frac {\sqrt {a-b} \tanh \left (\frac {x}{2}\right )}{\sqrt {a+b}}\right )}{\sqrt {a-b} b^4 \sqrt {a+b}}+\frac {\left (3 a^2+2 b^2\right ) \sinh (x)}{3 b^3}-\frac {a \cosh (x) \sinh (x)}{2 b^2}+\frac {\cosh ^2(x) \sinh (x)}{3 b} \] Output:

-1/2*a*(2*a^2+b^2)*x/b^4+2*a^4*arctanh((a-b)^(1/2)*tanh(1/2*x)/(a+b)^(1/2) 
)/(a-b)^(1/2)/b^4/(a+b)^(1/2)+1/3*(3*a^2+2*b^2)*sinh(x)/b^3-1/2*a*cosh(x)* 
sinh(x)/b^2+1/3*cosh(x)^2*sinh(x)/b
 

Mathematica [A] (verified)

Time = 0.16 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.88 \[ \int \frac {\cosh ^4(x)}{a+b \cosh (x)} \, dx=\frac {-6 a \left (2 a^2+b^2\right ) x-\frac {24 a^4 \arctan \left (\frac {(a-b) \tanh \left (\frac {x}{2}\right )}{\sqrt {-a^2+b^2}}\right )}{\sqrt {-a^2+b^2}}+3 b \left (4 a^2+3 b^2\right ) \sinh (x)-3 a b^2 \sinh (2 x)+b^3 \sinh (3 x)}{12 b^4} \] Input:

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

Output:

(-6*a*(2*a^2 + b^2)*x - (24*a^4*ArcTan[((a - b)*Tanh[x/2])/Sqrt[-a^2 + b^2 
]])/Sqrt[-a^2 + b^2] + 3*b*(4*a^2 + 3*b^2)*Sinh[x] - 3*a*b^2*Sinh[2*x] + b 
^3*Sinh[3*x])/(12*b^4)
 

Rubi [A] (verified)

Time = 0.81 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.15, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {3042, 3272, 3042, 3528, 25, 3042, 3502, 27, 3042, 3214, 3042, 3138, 221}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {\cosh ^4(x)}{a+b \cosh (x)} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {\sin \left (\frac {\pi }{2}+i x\right )^4}{a+b \sin \left (\frac {\pi }{2}+i x\right )}dx\)

\(\Big \downarrow \) 3272

\(\displaystyle \frac {\int \frac {\cosh (x) \left (-3 a \cosh ^2(x)+2 b \cosh (x)+2 a\right )}{a+b \cosh (x)}dx}{3 b}+\frac {\sinh (x) \cosh ^2(x)}{3 b}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\sinh (x) \cosh ^2(x)}{3 b}+\frac {\int \frac {\sin \left (i x+\frac {\pi }{2}\right ) \left (-3 a \sin \left (i x+\frac {\pi }{2}\right )^2+2 b \sin \left (i x+\frac {\pi }{2}\right )+2 a\right )}{a+b \sin \left (i x+\frac {\pi }{2}\right )}dx}{3 b}\)

\(\Big \downarrow \) 3528

\(\displaystyle \frac {\frac {\int -\frac {3 a^2-b \cosh (x) a-2 \left (3 a^2+2 b^2\right ) \cosh ^2(x)}{a+b \cosh (x)}dx}{2 b}-\frac {3 a \sinh (x) \cosh (x)}{2 b}}{3 b}+\frac {\sinh (x) \cosh ^2(x)}{3 b}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {-\frac {\int \frac {3 a^2-b \cosh (x) a-2 \left (3 a^2+2 b^2\right ) \cosh ^2(x)}{a+b \cosh (x)}dx}{2 b}-\frac {3 a \sinh (x) \cosh (x)}{2 b}}{3 b}+\frac {\sinh (x) \cosh ^2(x)}{3 b}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\sinh (x) \cosh ^2(x)}{3 b}+\frac {-\frac {3 a \sinh (x) \cosh (x)}{2 b}-\frac {\int \frac {3 a^2-b \sin \left (i x+\frac {\pi }{2}\right ) a-2 \left (3 a^2+2 b^2\right ) \sin \left (i x+\frac {\pi }{2}\right )^2}{a+b \sin \left (i x+\frac {\pi }{2}\right )}dx}{2 b}}{3 b}\)

\(\Big \downarrow \) 3502

\(\displaystyle \frac {-\frac {\frac {\int \frac {3 \left (b a^2+\left (2 a^2+b^2\right ) \cosh (x) a\right )}{a+b \cosh (x)}dx}{b}-\frac {2 \left (3 a^2+2 b^2\right ) \sinh (x)}{b}}{2 b}-\frac {3 a \sinh (x) \cosh (x)}{2 b}}{3 b}+\frac {\sinh (x) \cosh ^2(x)}{3 b}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {-\frac {\frac {3 \int \frac {b a^2+\left (2 a^2+b^2\right ) \cosh (x) a}{a+b \cosh (x)}dx}{b}-\frac {2 \left (3 a^2+2 b^2\right ) \sinh (x)}{b}}{2 b}-\frac {3 a \sinh (x) \cosh (x)}{2 b}}{3 b}+\frac {\sinh (x) \cosh ^2(x)}{3 b}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\sinh (x) \cosh ^2(x)}{3 b}+\frac {-\frac {3 a \sinh (x) \cosh (x)}{2 b}-\frac {-\frac {2 \left (3 a^2+2 b^2\right ) \sinh (x)}{b}+\frac {3 \int \frac {b a^2+\left (2 a^2+b^2\right ) \sin \left (i x+\frac {\pi }{2}\right ) a}{a+b \sin \left (i x+\frac {\pi }{2}\right )}dx}{b}}{2 b}}{3 b}\)

\(\Big \downarrow \) 3214

\(\displaystyle \frac {-\frac {\frac {3 \left (\frac {a x \left (2 a^2+b^2\right )}{b}-\frac {2 a^4 \int \frac {1}{a+b \cosh (x)}dx}{b}\right )}{b}-\frac {2 \left (3 a^2+2 b^2\right ) \sinh (x)}{b}}{2 b}-\frac {3 a \sinh (x) \cosh (x)}{2 b}}{3 b}+\frac {\sinh (x) \cosh ^2(x)}{3 b}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\sinh (x) \cosh ^2(x)}{3 b}+\frac {-\frac {3 a \sinh (x) \cosh (x)}{2 b}-\frac {-\frac {2 \left (3 a^2+2 b^2\right ) \sinh (x)}{b}+\frac {3 \left (\frac {a x \left (2 a^2+b^2\right )}{b}-\frac {2 a^4 \int \frac {1}{a+b \sin \left (i x+\frac {\pi }{2}\right )}dx}{b}\right )}{b}}{2 b}}{3 b}\)

\(\Big \downarrow \) 3138

\(\displaystyle \frac {-\frac {\frac {3 \left (\frac {a x \left (2 a^2+b^2\right )}{b}-\frac {4 a^4 \int \frac {1}{-\left ((a-b) \tanh ^2\left (\frac {x}{2}\right )\right )+a+b}d\tanh \left (\frac {x}{2}\right )}{b}\right )}{b}-\frac {2 \left (3 a^2+2 b^2\right ) \sinh (x)}{b}}{2 b}-\frac {3 a \sinh (x) \cosh (x)}{2 b}}{3 b}+\frac {\sinh (x) \cosh ^2(x)}{3 b}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {-\frac {\frac {3 \left (\frac {a x \left (2 a^2+b^2\right )}{b}-\frac {4 a^4 \text {arctanh}\left (\frac {\sqrt {a-b} \tanh \left (\frac {x}{2}\right )}{\sqrt {a+b}}\right )}{b \sqrt {a-b} \sqrt {a+b}}\right )}{b}-\frac {2 \left (3 a^2+2 b^2\right ) \sinh (x)}{b}}{2 b}-\frac {3 a \sinh (x) \cosh (x)}{2 b}}{3 b}+\frac {\sinh (x) \cosh ^2(x)}{3 b}\)

Input:

Int[Cosh[x]^4/(a + b*Cosh[x]),x]
 

Output:

(Cosh[x]^2*Sinh[x])/(3*b) + ((-3*a*Cosh[x]*Sinh[x])/(2*b) - ((3*((a*(2*a^2 
 + b^2)*x)/b - (4*a^4*ArcTanh[(Sqrt[a - b]*Tanh[x/2])/Sqrt[a + b]])/(Sqrt[ 
a - b]*b*Sqrt[a + b])))/b - (2*(3*a^2 + 2*b^2)*Sinh[x])/b)/(2*b))/(3*b)
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 221

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

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3138

Int[((a_) + (b_.)*sin[Pi/2 + (c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{ 
e = FreeFactors[Tan[(c + d*x)/2], x]}, Simp[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 3214

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])/((c_.) + (d_.)*sin[(e_.) + (f_. 
)*(x_)]), x_Symbol] :> Simp[b*(x/d), x] - Simp[(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 3272

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + 
 (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b^2)*Cos[e + f*x]*(a + b*Sin[e + f* 
x])^(m - 2)*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(m + n))), x] + Simp[1/(d*(m 
 + n))   Int[(a + b*Sin[e + f*x])^(m - 3)*(c + d*Sin[e + f*x])^n*Simp[a^3*d 
*(m + n) + b^2*(b*c*(m - 2) + a*d*(n + 1)) - b*(a*b*c - b^2*d*(m + n - 1) - 
 3*a^2*d*(m + n))*Sin[e + f*x] - b^2*(b*c*(m - 1) - a*d*(3*m + 2*n - 2))*Si 
n[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a* 
d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 2] && (IntegerQ[m 
] || IntegersQ[2*m, 2*n]) &&  !(IGtQ[n, 2] && ( !IntegerQ[m] || (EqQ[a, 0] 
&& NeQ[c, 0])))
 

rule 3502

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) 
+ (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-C)*Co 
s[e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(b*f*(m + 2))), x] + Simp[1/(b*(m 
+ 2))   Int[(a + b*Sin[e + f*x])^m*Simp[A*b*(m + 2) + b*C*(m + 1) + (b*B*(m 
 + 2) - a*C)*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] 
 &&  !LtQ[m, -1]
 

rule 3528

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.) 
+ (f_.)*(x_)])^(n_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_ 
.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-C)*Cos[e + f*x]*(a + b*Sin[e + f*x 
])^m*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(m + n + 2))), x] + Simp[1/(d*(m + 
n + 2))   Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*x])^n*Simp[a*A* 
d*(m + n + 2) + C*(b*c*m + a*d*(n + 1)) + (d*(A*b + a*B)*(m + n + 2) - C*(a 
*c - b*d*(m + n + 1)))*Sin[e + f*x] + (C*(a*d*m - b*c*(m + 1)) + b*B*d*(m + 
 n + 2))*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, n} 
, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[ 
m, 0] &&  !(IGtQ[n, 0] && ( !IntegerQ[m] || (EqQ[a, 0] && NeQ[c, 0])))
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(202\) vs. \(2(94)=188\).

Time = 0.86 (sec) , antiderivative size = 203, normalized size of antiderivative = 1.81

method	result	size
default	\(\frac {2 a^{4} \operatorname {arctanh}\left (\frac {\left (a -b \right ) \tanh \left (\frac {x}{2}\right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{b^{4} \sqrt {\left (a +b \right ) \left (a -b \right )}}-\frac {1}{3 b \left (\tanh \left (\frac {x}{2}\right )-1\right )^{3}}-\frac {a +b}{2 b^{2} \left (\tanh \left (\frac {x}{2}\right )-1\right )^{2}}-\frac {2 a^{2}+a b +2 b^{2}}{2 b^{3} \left (\tanh \left (\frac {x}{2}\right )-1\right )}+\frac {a \left (2 a^{2}+b^{2}\right ) \ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{2 b^{4}}-\frac {1}{3 b \left (1+\tanh \left (\frac {x}{2}\right )\right )^{3}}-\frac {-a -b}{2 b^{2} \left (1+\tanh \left (\frac {x}{2}\right )\right )^{2}}-\frac {2 a^{2}+a b +2 b^{2}}{2 b^{3} \left (1+\tanh \left (\frac {x}{2}\right )\right )}-\frac {a \left (2 a^{2}+b^{2}\right ) \ln \left (1+\tanh \left (\frac {x}{2}\right )\right )}{2 b^{4}}\)	\(203\)
risch	\(-\frac {a^{3} x}{b^{4}}-\frac {a x}{2 b^{2}}+\frac {{\mathrm e}^{3 x}}{24 b}-\frac {a \,{\mathrm e}^{2 x}}{8 b^{2}}+\frac {{\mathrm e}^{x} a^{2}}{2 b^{3}}+\frac {3 \,{\mathrm e}^{x}}{8 b}-\frac {{\mathrm e}^{-x} a^{2}}{2 b^{3}}-\frac {3 \,{\mathrm e}^{-x}}{8 b}+\frac {a \,{\mathrm e}^{-2 x}}{8 b^{2}}-\frac {{\mathrm e}^{-3 x}}{24 b}+\frac {a^{4} \ln \left ({\mathrm e}^{x}+\frac {a \sqrt {a^{2}-b^{2}}-a^{2}+b^{2}}{b \sqrt {a^{2}-b^{2}}}\right )}{\sqrt {a^{2}-b^{2}}\, b^{4}}-\frac {a^{4} \ln \left ({\mathrm e}^{x}+\frac {a \sqrt {a^{2}-b^{2}}+a^{2}-b^{2}}{b \sqrt {a^{2}-b^{2}}}\right )}{\sqrt {a^{2}-b^{2}}\, b^{4}}\)	\(213\)

Input:

int(cosh(x)^4/(a+b*cosh(x)),x,method=_RETURNVERBOSE)
 

Output:

2*a^4/b^4/((a+b)*(a-b))^(1/2)*arctanh((a-b)*tanh(1/2*x)/((a+b)*(a-b))^(1/2 
))-1/3/b/(tanh(1/2*x)-1)^3-1/2*(a+b)/b^2/(tanh(1/2*x)-1)^2-1/2*(2*a^2+a*b+ 
2*b^2)/b^3/(tanh(1/2*x)-1)+1/2*a*(2*a^2+b^2)/b^4*ln(tanh(1/2*x)-1)-1/3/b/( 
1+tanh(1/2*x))^3-1/2*(-a-b)/b^2/(1+tanh(1/2*x))^2-1/2*(2*a^2+a*b+2*b^2)/b^ 
3/(1+tanh(1/2*x))-1/2*a*(2*a^2+b^2)/b^4*ln(1+tanh(1/2*x))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 778 vs. \(2 (94) = 188\).

Time = 0.13 (sec) , antiderivative size = 1625, normalized size of antiderivative = 14.51 \[ \int \frac {\cosh ^4(x)}{a+b \cosh (x)} \, dx=\text {Too large to display} \] Input:

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

Output:

[1/24*((a^2*b^3 - b^5)*cosh(x)^6 + (a^2*b^3 - b^5)*sinh(x)^6 - 3*(a^3*b^2 
- a*b^4)*cosh(x)^5 - 3*(a^3*b^2 - a*b^4 - 2*(a^2*b^3 - b^5)*cosh(x))*sinh( 
x)^5 - a^2*b^3 + b^5 - 12*(2*a^5 - a^3*b^2 - a*b^4)*x*cosh(x)^3 + 3*(4*a^4 
*b - a^2*b^3 - 3*b^5)*cosh(x)^4 + 3*(4*a^4*b - a^2*b^3 - 3*b^5 + 5*(a^2*b^ 
3 - b^5)*cosh(x)^2 - 5*(a^3*b^2 - a*b^4)*cosh(x))*sinh(x)^4 + 2*(10*(a^2*b 
^3 - b^5)*cosh(x)^3 - 15*(a^3*b^2 - a*b^4)*cosh(x)^2 - 6*(2*a^5 - a^3*b^2 
- a*b^4)*x + 6*(4*a^4*b - a^2*b^3 - 3*b^5)*cosh(x))*sinh(x)^3 - 3*(4*a^4*b 
 - a^2*b^3 - 3*b^5)*cosh(x)^2 - 3*(4*a^4*b - a^2*b^3 - 3*b^5 - 5*(a^2*b^3 
- b^5)*cosh(x)^4 + 10*(a^3*b^2 - a*b^4)*cosh(x)^3 + 12*(2*a^5 - a^3*b^2 - 
a*b^4)*x*cosh(x) - 6*(4*a^4*b - a^2*b^3 - 3*b^5)*cosh(x)^2)*sinh(x)^2 + 24 
*(a^4*cosh(x)^3 + 3*a^4*cosh(x)^2*sinh(x) + 3*a^4*cosh(x)*sinh(x)^2 + a^4* 
sinh(x)^3)*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*cosh(x) + a*b)*sinh(x) - 2*sqrt(a^2 - b^2)*(b*c 
osh(x) + b*sinh(x) + a))/(b*cosh(x)^2 + b*sinh(x)^2 + 2*a*cosh(x) + 2*(b*c 
osh(x) + a)*sinh(x) + b)) + 3*(a^3*b^2 - a*b^4)*cosh(x) + 3*(2*(a^2*b^3 - 
b^5)*cosh(x)^5 + a^3*b^2 - a*b^4 - 5*(a^3*b^2 - a*b^4)*cosh(x)^4 - 12*(2*a 
^5 - a^3*b^2 - a*b^4)*x*cosh(x)^2 + 4*(4*a^4*b - a^2*b^3 - 3*b^5)*cosh(x)^ 
3 - 2*(4*a^4*b - a^2*b^3 - 3*b^5)*cosh(x))*sinh(x))/((a^2*b^4 - b^6)*cosh( 
x)^3 + 3*(a^2*b^4 - b^6)*cosh(x)^2*sinh(x) + 3*(a^2*b^4 - b^6)*cosh(x)*sin 
h(x)^2 + (a^2*b^4 - b^6)*sinh(x)^3), 1/24*((a^2*b^3 - b^5)*cosh(x)^6 + ...
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\cosh ^4(x)}{a+b \cosh (x)} \, dx=\text {Timed out} \] Input:

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

Output:

Timed out
                                                                                    
                                                                                    
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {\cosh ^4(x)}{a+b \cosh (x)} \, dx=\text {Exception raised: ValueError} \] Input:

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

Output:

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(4*a^2-4*b^2>0)', see `assume?` f 
or more de
 

Giac [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.19 \[ \int \frac {\cosh ^4(x)}{a+b \cosh (x)} \, dx=\frac {2 \, a^{4} \arctan \left (\frac {b e^{x} + a}{\sqrt {-a^{2} + b^{2}}}\right )}{\sqrt {-a^{2} + b^{2}} b^{4}} + \frac {b^{2} e^{\left (3 \, x\right )} - 3 \, a b e^{\left (2 \, x\right )} + 12 \, a^{2} e^{x} + 9 \, b^{2} e^{x}}{24 \, b^{3}} - \frac {{\left (2 \, a^{3} + a b^{2}\right )} x}{2 \, b^{4}} + \frac {{\left (3 \, a b^{2} e^{x} - b^{3} - 3 \, {\left (4 \, a^{2} b + 3 \, b^{3}\right )} e^{\left (2 \, x\right )}\right )} e^{\left (-3 \, x\right )}}{24 \, b^{4}} \] Input:

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

Output:

2*a^4*arctan((b*e^x + a)/sqrt(-a^2 + b^2))/(sqrt(-a^2 + b^2)*b^4) + 1/24*( 
b^2*e^(3*x) - 3*a*b*e^(2*x) + 12*a^2*e^x + 9*b^2*e^x)/b^3 - 1/2*(2*a^3 + a 
*b^2)*x/b^4 + 1/24*(3*a*b^2*e^x - b^3 - 3*(4*a^2*b + 3*b^3)*e^(2*x))*e^(-3 
*x)/b^4
 

Mupad [B] (verification not implemented)

Time = 2.38 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.87 \[ \int \frac {\cosh ^4(x)}{a+b \cosh (x)} \, dx=\frac {{\mathrm {e}}^{3\,x}}{24\,b}-\frac {{\mathrm {e}}^{-3\,x}}{24\,b}-\frac {x\,\left (2\,a^3+a\,b^2\right )}{2\,b^4}+\frac {{\mathrm {e}}^x\,\left (4\,a^2+3\,b^2\right )}{8\,b^3}+\frac {a\,{\mathrm {e}}^{-2\,x}}{8\,b^2}-\frac {a\,{\mathrm {e}}^{2\,x}}{8\,b^2}-\frac {{\mathrm {e}}^{-x}\,\left (4\,a^2+3\,b^2\right )}{8\,b^3}+\frac {a^4\,\ln \left (-\frac {2\,a^4\,{\mathrm {e}}^x}{b^5}-\frac {2\,a^4\,\left (b+a\,{\mathrm {e}}^x\right )}{b^5\,\sqrt {a+b}\,\sqrt {a-b}}\right )}{b^4\,\sqrt {a+b}\,\sqrt {a-b}}-\frac {a^4\,\ln \left (\frac {2\,a^4\,\left (b+a\,{\mathrm {e}}^x\right )}{b^5\,\sqrt {a+b}\,\sqrt {a-b}}-\frac {2\,a^4\,{\mathrm {e}}^x}{b^5}\right )}{b^4\,\sqrt {a+b}\,\sqrt {a-b}} \] Input:

int(cosh(x)^4/(a + b*cosh(x)),x)
 

Output:

exp(3*x)/(24*b) - exp(-3*x)/(24*b) - (x*(a*b^2 + 2*a^3))/(2*b^4) + (exp(x) 
*(4*a^2 + 3*b^2))/(8*b^3) + (a*exp(-2*x))/(8*b^2) - (a*exp(2*x))/(8*b^2) - 
 (exp(-x)*(4*a^2 + 3*b^2))/(8*b^3) + (a^4*log(- (2*a^4*exp(x))/b^5 - (2*a^ 
4*(b + a*exp(x)))/(b^5*(a + b)^(1/2)*(a - b)^(1/2))))/(b^4*(a + b)^(1/2)*( 
a - b)^(1/2)) - (a^4*log((2*a^4*(b + a*exp(x)))/(b^5*(a + b)^(1/2)*(a - b) 
^(1/2)) - (2*a^4*exp(x))/b^5))/(b^4*(a + b)^(1/2)*(a - b)^(1/2))
 

Reduce [B] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 247, normalized size of antiderivative = 2.21 \[ \int \frac {\cosh ^4(x)}{a+b \cosh (x)} \, dx=\frac {-48 e^{3 x} \sqrt {-a^{2}+b^{2}}\, \mathit {atan} \left (\frac {e^{x} b +a}{\sqrt {-a^{2}+b^{2}}}\right ) a^{4}+e^{6 x} a^{2} b^{3}-e^{6 x} b^{5}-3 e^{5 x} a^{3} b^{2}+3 e^{5 x} a \,b^{4}+12 e^{4 x} a^{4} b -3 e^{4 x} a^{2} b^{3}-9 e^{4 x} b^{5}-24 e^{3 x} a^{5} x +12 e^{3 x} a^{3} b^{2} x +12 e^{3 x} a \,b^{4} x -12 e^{2 x} a^{4} b +3 e^{2 x} a^{2} b^{3}+9 e^{2 x} b^{5}+3 e^{x} a^{3} b^{2}-3 e^{x} a \,b^{4}-a^{2} b^{3}+b^{5}}{24 e^{3 x} b^{4} \left (a^{2}-b^{2}\right )} \] Input:

int(cosh(x)^4/(a+b*cosh(x)),x)
 

Output:

( - 48*e**(3*x)*sqrt( - a**2 + b**2)*atan((e**x*b + a)/sqrt( - a**2 + b**2 
))*a**4 + e**(6*x)*a**2*b**3 - e**(6*x)*b**5 - 3*e**(5*x)*a**3*b**2 + 3*e* 
*(5*x)*a*b**4 + 12*e**(4*x)*a**4*b - 3*e**(4*x)*a**2*b**3 - 9*e**(4*x)*b** 
5 - 24*e**(3*x)*a**5*x + 12*e**(3*x)*a**3*b**2*x + 12*e**(3*x)*a*b**4*x - 
12*e**(2*x)*a**4*b + 3*e**(2*x)*a**2*b**3 + 9*e**(2*x)*b**5 + 3*e**x*a**3* 
b**2 - 3*e**x*a*b**4 - a**2*b**3 + b**5)/(24*e**(3*x)*b**4*(a**2 - b**2))