\(\int \frac {1}{(a+b \cosh (x)+c \sinh (x))^4} \, dx\) [505]

Optimal result

Integrand size = 12, antiderivative size = 220 \[ \int \frac {1}{(a+b \cosh (x)+c \sinh (x))^4} \, dx=-\frac {a \left (2 a^2+3 b^2-3 c^2\right ) \text {arctanh}\left (\frac {c-(a-b) \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2-b^2+c^2}}\right )}{\left (a^2-b^2+c^2\right )^{7/2}}-\frac {c \cosh (x)+b \sinh (x)}{3 \left (a^2-b^2+c^2\right ) (a+b \cosh (x)+c \sinh (x))^3}-\frac {5 (a c \cosh (x)+a b \sinh (x))}{6 \left (a^2-b^2+c^2\right )^2 (a+b \cosh (x)+c \sinh (x))^2}-\frac {c \left (11 a^2+4 b^2-4 c^2\right ) \cosh (x)+b \left (11 a^2+4 b^2-4 c^2\right ) \sinh (x)}{6 \left (a^2-b^2+c^2\right )^3 (a+b \cosh (x)+c \sinh (x))} \] Output:

-a*(2*a^2+3*b^2-3*c^2)*arctanh((c-(a-b)*tanh(1/2*x))/(a^2-b^2+c^2)^(1/2))/ 
(a^2-b^2+c^2)^(7/2)-1/3*(c*cosh(x)+b*sinh(x))/(a^2-b^2+c^2)/(a+b*cosh(x)+c 
*sinh(x))^3-5/6*(a*c*cosh(x)+a*b*sinh(x))/(a^2-b^2+c^2)^2/(a+b*cosh(x)+c*s 
inh(x))^2-1/6*(c*(11*a^2+4*b^2-4*c^2)*cosh(x)+b*(11*a^2+4*b^2-4*c^2)*sinh( 
x))/(a^2-b^2+c^2)^3/(a+b*cosh(x)+c*sinh(x))
 

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(488\) vs. \(2(220)=440\).

Time = 0.62 (sec) , antiderivative size = 488, normalized size of antiderivative = 2.22 \[ \int \frac {1}{(a+b \cosh (x)+c \sinh (x))^4} \, dx=-\frac {a \left (2 a^2+3 b^2-3 c^2\right ) \arctan \left (\frac {c+(-a+b) \tanh \left (\frac {x}{2}\right )}{\sqrt {-a^2+b^2-c^2}}\right )}{\left (-a^2+b^2-c^2\right )^{7/2}}-\frac {-44 a^5 c-82 a^3 b^2 c-24 a b^4 c+82 a^3 c^3+48 a b^2 c^3-24 a c^5-30 a^2 b c \left (2 a^2+3 b^2-3 c^2\right ) \cosh (x)-6 a c \left (a^2 \left (-7 b^2+11 c^2\right )+2 \left (b^4+b^2 c^2-2 c^4\right )\right ) \cosh (2 x)+22 a^2 b^3 c \cosh (3 x)+8 b^5 c \cosh (3 x)-22 a^2 b c^3 \cosh (3 x)-16 b^3 c^3 \cosh (3 x)+8 b c^5 \cosh (3 x)+72 a^4 b^2 \sinh (x)-9 a^2 b^4 \sinh (x)+12 b^6 \sinh (x)-132 a^4 c^2 \sinh (x)-72 a^2 b^2 c^2 \sinh (x)-36 b^4 c^2 \sinh (x)+81 a^2 c^4 \sinh (x)+36 b^2 c^4 \sinh (x)-12 c^6 \sinh (x)+54 a^3 b^3 \sinh (2 x)+6 a b^5 \sinh (2 x)-78 a^3 b c^2 \sinh (2 x)-48 a b^3 c^2 \sinh (2 x)+42 a b c^4 \sinh (2 x)+11 a^2 b^4 \sinh (3 x)+4 b^6 \sinh (3 x)-4 b^4 c^2 \sinh (3 x)-11 a^2 c^4 \sinh (3 x)-4 b^2 c^4 \sinh (3 x)+4 c^6 \sinh (3 x)}{24 b \left (a^2-b^2+c^2\right )^3 (a+b \cosh (x)+c \sinh (x))^3} \] Input:

Integrate[(a + b*Cosh[x] + c*Sinh[x])^(-4),x]
 

Output:

-((a*(2*a^2 + 3*b^2 - 3*c^2)*ArcTan[(c + (-a + b)*Tanh[x/2])/Sqrt[-a^2 + b 
^2 - c^2]])/(-a^2 + b^2 - c^2)^(7/2)) - (-44*a^5*c - 82*a^3*b^2*c - 24*a*b 
^4*c + 82*a^3*c^3 + 48*a*b^2*c^3 - 24*a*c^5 - 30*a^2*b*c*(2*a^2 + 3*b^2 - 
3*c^2)*Cosh[x] - 6*a*c*(a^2*(-7*b^2 + 11*c^2) + 2*(b^4 + b^2*c^2 - 2*c^4)) 
*Cosh[2*x] + 22*a^2*b^3*c*Cosh[3*x] + 8*b^5*c*Cosh[3*x] - 22*a^2*b*c^3*Cos 
h[3*x] - 16*b^3*c^3*Cosh[3*x] + 8*b*c^5*Cosh[3*x] + 72*a^4*b^2*Sinh[x] - 9 
*a^2*b^4*Sinh[x] + 12*b^6*Sinh[x] - 132*a^4*c^2*Sinh[x] - 72*a^2*b^2*c^2*S 
inh[x] - 36*b^4*c^2*Sinh[x] + 81*a^2*c^4*Sinh[x] + 36*b^2*c^4*Sinh[x] - 12 
*c^6*Sinh[x] + 54*a^3*b^3*Sinh[2*x] + 6*a*b^5*Sinh[2*x] - 78*a^3*b*c^2*Sin 
h[2*x] - 48*a*b^3*c^2*Sinh[2*x] + 42*a*b*c^4*Sinh[2*x] + 11*a^2*b^4*Sinh[3 
*x] + 4*b^6*Sinh[3*x] - 4*b^4*c^2*Sinh[3*x] - 11*a^2*c^4*Sinh[3*x] - 4*b^2 
*c^4*Sinh[3*x] + 4*c^6*Sinh[3*x])/(24*b*(a^2 - b^2 + c^2)^3*(a + b*Cosh[x] 
 + c*Sinh[x])^3)
 

Rubi [A] (verified)

Time = 1.58 (sec) , antiderivative size = 261, normalized size of antiderivative = 1.19, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {3042, 3608, 25, 3042, 3635, 25, 3042, 3632, 3042, 3603, 1083, 219}

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.

\(\displaystyle \int \frac {1}{(a+b \cosh (x)+c \sinh (x))^4} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {1}{(a+b \cos (i x)-i c \sin (i x))^4}dx\)

\(\Big \downarrow \) 3608

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3635

\(\displaystyle \frac {-\frac {\int -\frac {2 \left (3 a^2+2 b^2-2 c^2\right )-5 a b \cosh (x)-5 a c \sinh (x)}{(a+b \cosh (x)+c \sinh (x))^2}dx}{2 \left (a^2-b^2+c^2\right )}-\frac {5 (a b \sinh (x)+a c \cosh (x))}{2 \left (a^2-b^2+c^2\right ) (a+b \cosh (x)+c \sinh (x))^2}}{3 \left (a^2-b^2+c^2\right )}-\frac {b \sinh (x)+c \cosh (x)}{3 \left (a^2-b^2+c^2\right ) (a+b \cosh (x)+c \sinh (x))^3}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {\int \frac {2 \left (3 a^2+2 b^2-2 c^2\right )-5 a b \cosh (x)-5 a c \sinh (x)}{(a+b \cosh (x)+c \sinh (x))^2}dx}{2 \left (a^2-b^2+c^2\right )}-\frac {5 (a b \sinh (x)+a c \cosh (x))}{2 \left (a^2-b^2+c^2\right ) (a+b \cosh (x)+c \sinh (x))^2}}{3 \left (a^2-b^2+c^2\right )}-\frac {b \sinh (x)+c \cosh (x)}{3 \left (a^2-b^2+c^2\right ) (a+b \cosh (x)+c \sinh (x))^3}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {b \sinh (x)+c \cosh (x)}{3 \left (a^2-b^2+c^2\right ) (a+b \cosh (x)+c \sinh (x))^3}+\frac {-\frac {5 (a b \sinh (x)+a c \cosh (x))}{2 \left (a^2-b^2+c^2\right ) (a+b \cosh (x)+c \sinh (x))^2}+\frac {\int \frac {2 \left (3 a^2+2 b^2-2 c^2\right )-5 a b \cos (i x)+5 i a c \sin (i x)}{(a+b \cos (i x)-i c \sin (i x))^2}dx}{2 \left (a^2-b^2+c^2\right )}}{3 \left (a^2-b^2+c^2\right )}\)

\(\Big \downarrow \) 3632

\(\displaystyle \frac {\frac {\frac {3 a \left (2 a^2+3 b^2-3 c^2\right ) \int \frac {1}{a+b \cosh (x)+c \sinh (x)}dx}{a^2-b^2+c^2}-\frac {b \sinh (x) \left (11 a^2+4 b^2-4 c^2\right )+c \cosh (x) \left (11 a^2+4 b^2-4 c^2\right )}{\left (a^2-b^2+c^2\right ) (a+b \cosh (x)+c \sinh (x))}}{2 \left (a^2-b^2+c^2\right )}-\frac {5 (a b \sinh (x)+a c \cosh (x))}{2 \left (a^2-b^2+c^2\right ) (a+b \cosh (x)+c \sinh (x))^2}}{3 \left (a^2-b^2+c^2\right )}-\frac {b \sinh (x)+c \cosh (x)}{3 \left (a^2-b^2+c^2\right ) (a+b \cosh (x)+c \sinh (x))^3}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {b \sinh (x)+c \cosh (x)}{3 \left (a^2-b^2+c^2\right ) (a+b \cosh (x)+c \sinh (x))^3}+\frac {-\frac {5 (a b \sinh (x)+a c \cosh (x))}{2 \left (a^2-b^2+c^2\right ) (a+b \cosh (x)+c \sinh (x))^2}+\frac {-\frac {b \sinh (x) \left (11 a^2+4 b^2-4 c^2\right )+c \cosh (x) \left (11 a^2+4 b^2-4 c^2\right )}{\left (a^2-b^2+c^2\right ) (a+b \cosh (x)+c \sinh (x))}+\frac {3 a \left (2 a^2+3 b^2-3 c^2\right ) \int \frac {1}{a+b \cos (i x)-i c \sin (i x)}dx}{a^2-b^2+c^2}}{2 \left (a^2-b^2+c^2\right )}}{3 \left (a^2-b^2+c^2\right )}\)

\(\Big \downarrow \) 3603

\(\displaystyle \frac {\frac {\frac {6 a \left (2 a^2+3 b^2-3 c^2\right ) \int \frac {1}{-\left ((a-b) \tanh ^2\left (\frac {x}{2}\right )\right )+2 c \tanh \left (\frac {x}{2}\right )+a+b}d\tanh \left (\frac {x}{2}\right )}{a^2-b^2+c^2}-\frac {b \sinh (x) \left (11 a^2+4 b^2-4 c^2\right )+c \cosh (x) \left (11 a^2+4 b^2-4 c^2\right )}{\left (a^2-b^2+c^2\right ) (a+b \cosh (x)+c \sinh (x))}}{2 \left (a^2-b^2+c^2\right )}-\frac {5 (a b \sinh (x)+a c \cosh (x))}{2 \left (a^2-b^2+c^2\right ) (a+b \cosh (x)+c \sinh (x))^2}}{3 \left (a^2-b^2+c^2\right )}-\frac {b \sinh (x)+c \cosh (x)}{3 \left (a^2-b^2+c^2\right ) (a+b \cosh (x)+c \sinh (x))^3}\)

\(\Big \downarrow \) 1083

\(\displaystyle \frac {\frac {-\frac {12 a \left (2 a^2+3 b^2-3 c^2\right ) \int \frac {1}{4 \left (a^2-b^2+c^2\right )-\left (2 c-2 (a-b) \tanh \left (\frac {x}{2}\right )\right )^2}d\left (2 c-2 (a-b) \tanh \left (\frac {x}{2}\right )\right )}{a^2-b^2+c^2}-\frac {b \sinh (x) \left (11 a^2+4 b^2-4 c^2\right )+c \cosh (x) \left (11 a^2+4 b^2-4 c^2\right )}{\left (a^2-b^2+c^2\right ) (a+b \cosh (x)+c \sinh (x))}}{2 \left (a^2-b^2+c^2\right )}-\frac {5 (a b \sinh (x)+a c \cosh (x))}{2 \left (a^2-b^2+c^2\right ) (a+b \cosh (x)+c \sinh (x))^2}}{3 \left (a^2-b^2+c^2\right )}-\frac {b \sinh (x)+c \cosh (x)}{3 \left (a^2-b^2+c^2\right ) (a+b \cosh (x)+c \sinh (x))^3}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\frac {-\frac {6 a \left (2 a^2+3 b^2-3 c^2\right ) \text {arctanh}\left (\frac {2 c-2 (a-b) \tanh \left (\frac {x}{2}\right )}{2 \sqrt {a^2-b^2+c^2}}\right )}{\left (a^2-b^2+c^2\right )^{3/2}}-\frac {b \sinh (x) \left (11 a^2+4 b^2-4 c^2\right )+c \cosh (x) \left (11 a^2+4 b^2-4 c^2\right )}{\left (a^2-b^2+c^2\right ) (a+b \cosh (x)+c \sinh (x))}}{2 \left (a^2-b^2+c^2\right )}-\frac {5 (a b \sinh (x)+a c \cosh (x))}{2 \left (a^2-b^2+c^2\right ) (a+b \cosh (x)+c \sinh (x))^2}}{3 \left (a^2-b^2+c^2\right )}-\frac {b \sinh (x)+c \cosh (x)}{3 \left (a^2-b^2+c^2\right ) (a+b \cosh (x)+c \sinh (x))^3}\)

Input:

Int[(a + b*Cosh[x] + c*Sinh[x])^(-4),x]
 

Output:

-1/3*(c*Cosh[x] + b*Sinh[x])/((a^2 - b^2 + c^2)*(a + b*Cosh[x] + c*Sinh[x] 
)^3) + ((-5*(a*c*Cosh[x] + a*b*Sinh[x]))/(2*(a^2 - b^2 + c^2)*(a + b*Cosh[ 
x] + c*Sinh[x])^2) + ((-6*a*(2*a^2 + 3*b^2 - 3*c^2)*ArcTanh[(2*c - 2*(a - 
b)*Tanh[x/2])/(2*Sqrt[a^2 - b^2 + c^2])])/(a^2 - b^2 + c^2)^(3/2) - (c*(11 
*a^2 + 4*b^2 - 4*c^2)*Cosh[x] + b*(11*a^2 + 4*b^2 - 4*c^2)*Sinh[x])/((a^2 
- b^2 + c^2)*(a + b*Cosh[x] + c*Sinh[x])))/(2*(a^2 - b^2 + c^2)))/(3*(a^2 
- b^2 + c^2))
 

Defintions of rubi rules used

rule 25

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

rule 219

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* 
ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

rule 1083

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Simp[-2   Subst[I 
nt[1/Simp[b^2 - 4*a*c - x^2, x], x], x, b + 2*c*x], x] /; FreeQ[{a, b, c}, 
x]
 

rule 3042

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

rule 3603

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

rule 3608

Int[(cos[(d_.) + (e_.)*(x_)]*(b_.) + (a_) + (c_.)*sin[(d_.) + (e_.)*(x_)])^ 
(n_), x_Symbol] :> Simp[((-c)*Cos[d + e*x] + b*Sin[d + e*x])*((a + b*Cos[d 
+ e*x] + c*Sin[d + e*x])^(n + 1)/(e*(n + 1)*(a^2 - b^2 - c^2))), x] + Simp[ 
1/((n + 1)*(a^2 - b^2 - c^2))   Int[(a*(n + 1) - b*(n + 2)*Cos[d + e*x] - c 
*(n + 2)*Sin[d + e*x])*(a + b*Cos[d + e*x] + c*Sin[d + e*x])^(n + 1), x], x 
] /; FreeQ[{a, b, c, d, e}, x] && NeQ[a^2 - b^2 - c^2, 0] && LtQ[n, -1] && 
NeQ[n, -3/2]
 

rule 3632

Int[((A_.) + cos[(d_.) + (e_.)*(x_)]*(B_.) + (C_.)*sin[(d_.) + (e_.)*(x_)]) 
/((a_.) + cos[(d_.) + (e_.)*(x_)]*(b_.) + (c_.)*sin[(d_.) + (e_.)*(x_)])^2, 
 x_Symbol] :> Simp[(c*B - b*C - (a*C - c*A)*Cos[d + e*x] + (a*B - b*A)*Sin[ 
d + e*x])/(e*(a^2 - b^2 - c^2)*(a + b*Cos[d + e*x] + c*Sin[d + e*x])), x] + 
 Simp[(a*A - b*B - c*C)/(a^2 - b^2 - c^2)   Int[1/(a + b*Cos[d + e*x] + c*S 
in[d + e*x]), x], x] /; FreeQ[{a, b, c, d, e, A, B, C}, x] && NeQ[a^2 - b^2 
 - c^2, 0] && NeQ[a*A - b*B - c*C, 0]
 

rule 3635

Int[((a_.) + cos[(d_.) + (e_.)*(x_)]*(b_.) + (c_.)*sin[(d_.) + (e_.)*(x_)]) 
^(n_)*((A_.) + cos[(d_.) + (e_.)*(x_)]*(B_.) + (C_.)*sin[(d_.) + (e_.)*(x_) 
]), x_Symbol] :> Simp[(-(c*B - b*C - (a*C - c*A)*Cos[d + e*x] + (a*B - b*A) 
*Sin[d + e*x]))*((a + b*Cos[d + e*x] + c*Sin[d + e*x])^(n + 1)/(e*(n + 1)*( 
a^2 - b^2 - c^2))), x] + Simp[1/((n + 1)*(a^2 - b^2 - c^2))   Int[(a + b*Co 
s[d + e*x] + c*Sin[d + e*x])^(n + 1)*Simp[(n + 1)*(a*A - b*B - c*C) + (n + 
2)*(a*B - b*A)*Cos[d + e*x] + (n + 2)*(a*C - c*A)*Sin[d + e*x], x], x], x] 
/; FreeQ[{a, b, c, d, e, A, B, C}, x] && LtQ[n, -1] && NeQ[a^2 - b^2 - c^2, 
 0] && NeQ[n, -2]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1587\) vs. \(2(208)=416\).

Time = 70.39 (sec) , antiderivative size = 1588, normalized size of antiderivative = 7.22

method	result	size
risch	\(\text {Expression too large to display}\)	\(1588\)
default	\(\text {Expression too large to display}\)	\(1598\)

Input:

int(1/(a+b*cosh(x)+c*sinh(x))^4,x,method=_RETURNVERBOSE)
 

Output:

1/3*(-12*b^4*c+33*a^2*b*c^2+44*exp(x)^3*a^5+12*b^5*exp(x)^2-12*c^5*exp(x)^ 
2+8*b^2*c^3+4*c^5-33*a^2*b^2*c+30*a*b*c^3*exp(x)+12*a^3*b*c*exp(x)^5+18*a* 
b^3*c*exp(x)^5-18*a*b*c^3*exp(x)^5+45*a^2*b^2*c*exp(x)^4-45*a^2*b*c^2*exp( 
x)^4-48*a*b^2*c^2*exp(x)^3-36*a^2*b^2*c*exp(x)^2-36*a^2*b*c^2*exp(x)^2-120 
*a^3*b*c*exp(x)-30*a*b^3*c*exp(x)-12*b*c^4+11*a^2*b^3+8*b^3*c^2+24*a*c^4*e 
xp(x)^3+82*a^3*b^2*exp(x)^3-102*a^4*c*exp(x)^2+24*a*b^4*exp(x)^3-82*a^3*c^ 
2*exp(x)^3+45*a^2*b^3*exp(x)^4-45*a^2*c^3*exp(x)^4+9*a*b^4*exp(x)^5+6*a^3* 
c^2*exp(x)^5+30*a^4*b*exp(x)^4+102*a^4*b*exp(x)^2-9*a*c^4*exp(x)^5+6*a^3*b 
^2*exp(x)^5+30*a^4*c*exp(x)^4+36*a^2*b^3*exp(x)^2+36*a^2*c^3*exp(x)^2-12*b 
^4*c*exp(x)^2-24*b^3*c^2*exp(x)^2+24*b^2*c^3*exp(x)^2+12*b*c^4*exp(x)^2+60 
*a^3*b^2*exp(x)+60*a^3*c^2*exp(x)+15*a*b^4*exp(x)-15*a*c^4*exp(x)-11*a^2*c 
^3+4*b^5)/(a^2-b^2+c^2)^3/(b*exp(x)^2+c*exp(x)^2+2*a*exp(x)+b-c)^3+1/(a^2- 
b^2+c^2)^(7/2)*a^3*ln(exp(x)+((a^2-b^2+c^2)^(7/2)*a-a^8+4*a^6*b^2-4*a^6*c^ 
2-6*b^4*a^4+12*a^4*b^2*c^2-6*a^4*c^4+4*b^6*a^2-12*a^2*b^4*c^2+12*a^2*b^2*c 
^4-4*a^2*c^6-b^8+4*b^6*c^2-6*b^4*c^4+4*b^2*c^6-c^8)/(a^2-b^2+c^2)^(7/2)/(b 
+c))+3/2/(a^2-b^2+c^2)^(7/2)*a*ln(exp(x)+((a^2-b^2+c^2)^(7/2)*a-a^8+4*a^6* 
b^2-4*a^6*c^2-6*b^4*a^4+12*a^4*b^2*c^2-6*a^4*c^4+4*b^6*a^2-12*a^2*b^4*c^2+ 
12*a^2*b^2*c^4-4*a^2*c^6-b^8+4*b^6*c^2-6*b^4*c^4+4*b^2*c^6-c^8)/(a^2-b^2+c 
^2)^(7/2)/(b+c))*b^2-3/2/(a^2-b^2+c^2)^(7/2)*a*ln(exp(x)+((a^2-b^2+c^2)^(7 
/2)*a-a^8+4*a^6*b^2-4*a^6*c^2-6*b^4*a^4+12*a^4*b^2*c^2-6*a^4*c^4+4*b^6*...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 11492 vs. \(2 (210) = 420\).

Time = 0.38 (sec) , antiderivative size = 23093, normalized size of antiderivative = 104.97 \[ \int \frac {1}{(a+b \cosh (x)+c \sinh (x))^4} \, dx=\text {Too large to display} \] Input:

integrate(1/(a+b*cosh(x)+c*sinh(x))^4,x, algorithm="fricas")
 

Output:

Too large to include
 

Sympy [F(-1)]

Timed out. \[ \int \frac {1}{(a+b \cosh (x)+c \sinh (x))^4} \, dx=\text {Timed out} \] Input:

integrate(1/(a+b*cosh(x)+c*sinh(x))**4,x)
 

Output:

Timed out
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {1}{(a+b \cosh (x)+c \sinh (x))^4} \, dx=\text {Exception raised: ValueError} \] Input:

integrate(1/(a+b*cosh(x)+c*sinh(x))^4,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(c^2-b^2+a^2>0)', see `assume?` f 
or more de
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 717 vs. \(2 (210) = 420\).

Time = 0.17 (sec) , antiderivative size = 717, normalized size of antiderivative = 3.26 \[ \int \frac {1}{(a+b \cosh (x)+c \sinh (x))^4} \, dx =\text {Too large to display} \] Input:

integrate(1/(a+b*cosh(x)+c*sinh(x))^4,x, algorithm="giac")
 

Output:

(2*a^3 + 3*a*b^2 - 3*a*c^2)*arctan((b*e^x + c*e^x + a)/sqrt(-a^2 + b^2 - c 
^2))/((a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6 + 3*a^4*c^2 - 6*a^2*b^2*c^2 + 3*b 
^4*c^2 + 3*a^2*c^4 - 3*b^2*c^4 + c^6)*sqrt(-a^2 + b^2 - c^2)) + 1/3*(6*a^3 
*b^2*e^(5*x) + 9*a*b^4*e^(5*x) + 12*a^3*b*c*e^(5*x) + 18*a*b^3*c*e^(5*x) + 
 6*a^3*c^2*e^(5*x) - 18*a*b*c^3*e^(5*x) - 9*a*c^4*e^(5*x) + 30*a^4*b*e^(4* 
x) + 45*a^2*b^3*e^(4*x) + 30*a^4*c*e^(4*x) + 45*a^2*b^2*c*e^(4*x) - 45*a^2 
*b*c^2*e^(4*x) - 45*a^2*c^3*e^(4*x) + 44*a^5*e^(3*x) + 82*a^3*b^2*e^(3*x) 
+ 24*a*b^4*e^(3*x) - 82*a^3*c^2*e^(3*x) - 48*a*b^2*c^2*e^(3*x) + 24*a*c^4* 
e^(3*x) + 102*a^4*b*e^(2*x) + 36*a^2*b^3*e^(2*x) + 12*b^5*e^(2*x) - 102*a^ 
4*c*e^(2*x) - 36*a^2*b^2*c*e^(2*x) - 12*b^4*c*e^(2*x) - 36*a^2*b*c^2*e^(2* 
x) - 24*b^3*c^2*e^(2*x) + 36*a^2*c^3*e^(2*x) + 24*b^2*c^3*e^(2*x) + 12*b*c 
^4*e^(2*x) - 12*c^5*e^(2*x) + 60*a^3*b^2*e^x + 15*a*b^4*e^x - 120*a^3*b*c* 
e^x - 30*a*b^3*c*e^x + 60*a^3*c^2*e^x + 30*a*b*c^3*e^x - 15*a*c^4*e^x + 11 
*a^2*b^3 + 4*b^5 - 33*a^2*b^2*c - 12*b^4*c + 33*a^2*b*c^2 + 8*b^3*c^2 - 11 
*a^2*c^3 + 8*b^2*c^3 - 12*b*c^4 + 4*c^5)/((a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b 
^6 + 3*a^4*c^2 - 6*a^2*b^2*c^2 + 3*b^4*c^2 + 3*a^2*c^4 - 3*b^2*c^4 + c^6)* 
(b*e^(2*x) + c*e^(2*x) + 2*a*e^x + b - c)^3)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{(a+b \cosh (x)+c \sinh (x))^4} \, dx=\int \frac {1}{{\left (a+b\,\mathrm {cosh}\left (x\right )+c\,\mathrm {sinh}\left (x\right )\right )}^4} \,d x \] Input:

int(1/(a + b*cosh(x) + c*sinh(x))^4,x)
 

Output:

int(1/(a + b*cosh(x) + c*sinh(x))^4, x)
 

Reduce [B] (verification not implemented)

Time = 0.30 (sec) , antiderivative size = 7670, normalized size of antiderivative = 34.86 \[ \int \frac {1}{(a+b \cosh (x)+c \sinh (x))^4} \, dx =\text {Too large to display} \] Input:

int(1/(a+b*cosh(x)+c*sinh(x))^4,x)
 

Output:

( - 12*e**(6*x)*sqrt( - a**2 + b**2 - c**2)*atan((e**x*b + e**x*c + a)/sqr 
t( - a**2 + b**2 - c**2))*a**3*b**3 - 36*e**(6*x)*sqrt( - a**2 + b**2 - c* 
*2)*atan((e**x*b + e**x*c + a)/sqrt( - a**2 + b**2 - c**2))*a**3*b**2*c - 
36*e**(6*x)*sqrt( - a**2 + b**2 - c**2)*atan((e**x*b + e**x*c + a)/sqrt( - 
 a**2 + b**2 - c**2))*a**3*b*c**2 - 12*e**(6*x)*sqrt( - a**2 + b**2 - c**2 
)*atan((e**x*b + e**x*c + a)/sqrt( - a**2 + b**2 - c**2))*a**3*c**3 - 18*e 
**(6*x)*sqrt( - a**2 + b**2 - c**2)*atan((e**x*b + e**x*c + a)/sqrt( - a** 
2 + b**2 - c**2))*a*b**5 - 54*e**(6*x)*sqrt( - a**2 + b**2 - c**2)*atan((e 
**x*b + e**x*c + a)/sqrt( - a**2 + b**2 - c**2))*a*b**4*c - 36*e**(6*x)*sq 
rt( - a**2 + b**2 - c**2)*atan((e**x*b + e**x*c + a)/sqrt( - a**2 + b**2 - 
 c**2))*a*b**3*c**2 + 36*e**(6*x)*sqrt( - a**2 + b**2 - c**2)*atan((e**x*b 
 + e**x*c + a)/sqrt( - a**2 + b**2 - c**2))*a*b**2*c**3 + 54*e**(6*x)*sqrt 
( - a**2 + b**2 - c**2)*atan((e**x*b + e**x*c + a)/sqrt( - a**2 + b**2 - c 
**2))*a*b*c**4 + 18*e**(6*x)*sqrt( - a**2 + b**2 - c**2)*atan((e**x*b + e* 
*x*c + a)/sqrt( - a**2 + b**2 - c**2))*a*c**5 - 72*e**(5*x)*sqrt( - a**2 + 
 b**2 - c**2)*atan((e**x*b + e**x*c + a)/sqrt( - a**2 + b**2 - c**2))*a**4 
*b**2 - 144*e**(5*x)*sqrt( - a**2 + b**2 - c**2)*atan((e**x*b + e**x*c + a 
)/sqrt( - a**2 + b**2 - c**2))*a**4*b*c - 72*e**(5*x)*sqrt( - a**2 + b**2 
- c**2)*atan((e**x*b + e**x*c + a)/sqrt( - a**2 + b**2 - c**2))*a**4*c**2 
- 108*e**(5*x)*sqrt( - a**2 + b**2 - c**2)*atan((e**x*b + e**x*c + a)/s...