Integrand size = 19, antiderivative size = 194 \[ \int \frac {A+B \cosh (x)}{(a+b \cosh (x)+c \sinh (x))^3} \, dx=-\frac {\left (2 a^2 A-3 a b B+A \left (b^2-c^2\right )\right ) \text {arctanh}\left (\frac {c-(a-b) \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2-b^2+c^2}}\right )}{\left (a^2-b^2+c^2\right )^{5/2}}-\frac {B c+A c \cosh (x)+(A b-a B) \sinh (x)}{2 \left (a^2-b^2+c^2\right ) (a+b \cosh (x)+c \sinh (x))^2}-\frac {a B c+(3 a A-2 b B) c \cosh (x)+\left (3 a A b-a^2 B-2 b^2 B\right ) \sinh (x)}{2 \left (a^2-b^2+c^2\right )^2 (a+b \cosh (x)+c \sinh (x))} \]
-(2*a^2*A-3*a*b*B+A*(b^2-c^2))*arctanh((c-(a-b)*tanh(1/2*x))/(a^2-b^2+c^2) ^(1/2))/(a^2-b^2+c^2)^(5/2)+1/2*(-B*c-A*c*cosh(x)-(A*b-B*a)*sinh(x))/(a^2- b^2+c^2)/(a+b*cosh(x)+c*sinh(x))^2+1/2*(-a*B*c-(3*A*a-2*B*b)*c*cosh(x)-(3* A*a*b-B*a^2-2*B*b^2)*sinh(x))/(a^2-b^2+c^2)^2/(a+b*cosh(x)+c*sinh(x))
Time = 0.50 (sec) , antiderivative size = 336, normalized size of antiderivative = 1.73 \[ \int \frac {A+B \cosh (x)}{(a+b \cosh (x)+c \sinh (x))^3} \, dx=\frac {\left (2 a^2 A-3 a b B+A \left (b^2-c^2\right )\right ) \arctan \left (\frac {c+(-a+b) \tanh \left (\frac {x}{2}\right )}{\sqrt {-a^2+b^2-c^2}}\right )}{\left (-a^2+b^2-c^2\right )^{5/2}}+\frac {6 a^3 A c+3 a A b^2 c-9 a^2 b B c-3 a A c^3+2 b c \left (2 a^2 A-3 a b B+A \left (b^2-c^2\right )\right ) \cosh (x)+c \left (a^2 b B+2 b B \left (b^2-c^2\right )+3 a A \left (-b^2+c^2\right )\right ) \cosh (2 x)-8 a^2 A b^2 \sinh (x)+2 A b^4 \sinh (x)+4 a^3 b B \sinh (x)+2 a b^3 B \sinh (x)+12 a^2 A c^2 \sinh (x)-2 A b^2 c^2 \sinh (x)-8 a b B c^2 \sinh (x)-3 a A b^3 \sinh (2 x)+a^2 b^2 B \sinh (2 x)+2 b^4 B \sinh (2 x)+3 a A b c^2 \sinh (2 x)-2 b^2 B c^2 \sinh (2 x)}{4 b \left (a^2-b^2+c^2\right )^2 (a+b \cosh (x)+c \sinh (x))^2} \]
((2*a^2*A - 3*a*b*B + A*(b^2 - c^2))*ArcTan[(c + (-a + b)*Tanh[x/2])/Sqrt[ -a^2 + b^2 - c^2]])/(-a^2 + b^2 - c^2)^(5/2) + (6*a^3*A*c + 3*a*A*b^2*c - 9*a^2*b*B*c - 3*a*A*c^3 + 2*b*c*(2*a^2*A - 3*a*b*B + A*(b^2 - c^2))*Cosh[x ] + c*(a^2*b*B + 2*b*B*(b^2 - c^2) + 3*a*A*(-b^2 + c^2))*Cosh[2*x] - 8*a^2 *A*b^2*Sinh[x] + 2*A*b^4*Sinh[x] + 4*a^3*b*B*Sinh[x] + 2*a*b^3*B*Sinh[x] + 12*a^2*A*c^2*Sinh[x] - 2*A*b^2*c^2*Sinh[x] - 8*a*b*B*c^2*Sinh[x] - 3*a*A* b^3*Sinh[2*x] + a^2*b^2*B*Sinh[2*x] + 2*b^4*B*Sinh[2*x] + 3*a*A*b*c^2*Sinh [2*x] - 2*b^2*B*c^2*Sinh[2*x])/(4*b*(a^2 - b^2 + c^2)^2*(a + b*Cosh[x] + c *Sinh[x])^2)
Time = 0.71 (sec) , antiderivative size = 216, normalized size of antiderivative = 1.11, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.474, Rules used = {3042, 3637, 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.
| \(\displaystyle \int \frac {A+B \cosh (x)}{(a+b \cosh (x)+c \sinh (x))^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {A+B \cos (i x)}{(a+b \cos (i x)-i c \sin (i x))^3}dx\) |
| \(\Big \downarrow \) 3637 |
| \(\displaystyle -\frac {\int -\frac {2 (a A-b B)-(A b-a B) \cosh (x)-A c \sinh (x)}{(a+b \cosh (x)+c \sinh (x))^2}dx}{2 \left (a^2-b^2+c^2\right )}-\frac {\sinh (x) (A b-a B)+A c \cosh (x)+B c}{2 \left (a^2-b^2+c^2\right ) (a+b \cosh (x)+c \sinh (x))^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {2 (a A-b B)-(A b-a B) \cosh (x)-A c \sinh (x)}{(a+b \cosh (x)+c \sinh (x))^2}dx}{2 \left (a^2-b^2+c^2\right )}-\frac {\sinh (x) (A b-a B)+A c \cosh (x)+B c}{2 \left (a^2-b^2+c^2\right ) (a+b \cosh (x)+c \sinh (x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\sinh (x) (A b-a B)+A c \cosh (x)+B c}{2 \left (a^2-b^2+c^2\right ) (a+b \cosh (x)+c \sinh (x))^2}+\frac {\int \frac {2 (a A-b B)-(A b-a B) \cos (i x)+i A c \sin (i x)}{(a+b \cos (i x)-i c \sin (i x))^2}dx}{2 \left (a^2-b^2+c^2\right )}\) |
| \(\Big \downarrow \) 3632 |
| \(\displaystyle \frac {\frac {\left (2 a^2 A-3 a b B+A \left (b^2-c^2\right )\right ) \int \frac {1}{a+b \cosh (x)+c \sinh (x)}dx}{a^2-b^2+c^2}-\frac {\sinh (x) \left (a^2 (-B)+3 a A b-2 b^2 B\right )+c \cosh (x) (3 a A-2 b B)+a B c}{\left (a^2-b^2+c^2\right ) (a+b \cosh (x)+c \sinh (x))}}{2 \left (a^2-b^2+c^2\right )}-\frac {\sinh (x) (A b-a B)+A c \cosh (x)+B c}{2 \left (a^2-b^2+c^2\right ) (a+b \cosh (x)+c \sinh (x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\sinh (x) (A b-a B)+A c \cosh (x)+B c}{2 \left (a^2-b^2+c^2\right ) (a+b \cosh (x)+c \sinh (x))^2}+\frac {-\frac {\sinh (x) \left (a^2 (-B)+3 a A b-2 b^2 B\right )+c \cosh (x) (3 a A-2 b B)+a B c}{\left (a^2-b^2+c^2\right ) (a+b \cosh (x)+c \sinh (x))}+\frac {\left (2 a^2 A-3 a b B+A \left (b^2-c^2\right )\right ) \int \frac {1}{a+b \cos (i x)-i c \sin (i x)}dx}{a^2-b^2+c^2}}{2 \left (a^2-b^2+c^2\right )}\) |
| \(\Big \downarrow \) 3603 |
| \(\displaystyle \frac {\frac {2 \left (2 a^2 A-3 a b B+A \left (b^2-c^2\right )\right ) \int \frac {1}{-\left ((a-b) \tanh ^2\left (\frac {x}{2}\right )\right )+2 c \tanh \left (\frac {x}{2}\right )+a+b}d\tanh \left (\frac {x}{2}\right )}{a^2-b^2+c^2}-\frac {\sinh (x) \left (a^2 (-B)+3 a A b-2 b^2 B\right )+c \cosh (x) (3 a A-2 b B)+a B c}{\left (a^2-b^2+c^2\right ) (a+b \cosh (x)+c \sinh (x))}}{2 \left (a^2-b^2+c^2\right )}-\frac {\sinh (x) (A b-a B)+A c \cosh (x)+B c}{2 \left (a^2-b^2+c^2\right ) (a+b \cosh (x)+c \sinh (x))^2}\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle \frac {-\frac {4 \left (2 a^2 A-3 a b B+A \left (b^2-c^2\right )\right ) \int \frac {1}{4 \left (a^2-b^2+c^2\right )-\left (2 c-2 (a-b) \tanh \left (\frac {x}{2}\right )\right )^2}d\left (2 c-2 (a-b) \tanh \left (\frac {x}{2}\right )\right )}{a^2-b^2+c^2}-\frac {\sinh (x) \left (a^2 (-B)+3 a A b-2 b^2 B\right )+c \cosh (x) (3 a A-2 b B)+a B c}{\left (a^2-b^2+c^2\right ) (a+b \cosh (x)+c \sinh (x))}}{2 \left (a^2-b^2+c^2\right )}-\frac {\sinh (x) (A b-a B)+A c \cosh (x)+B c}{2 \left (a^2-b^2+c^2\right ) (a+b \cosh (x)+c \sinh (x))^2}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {-\frac {2 \left (2 a^2 A-3 a b B+A \left (b^2-c^2\right )\right ) \text {arctanh}\left (\frac {2 c-2 (a-b) \tanh \left (\frac {x}{2}\right )}{2 \sqrt {a^2-b^2+c^2}}\right )}{\left (a^2-b^2+c^2\right )^{3/2}}-\frac {\sinh (x) \left (a^2 (-B)+3 a A b-2 b^2 B\right )+c \cosh (x) (3 a A-2 b B)+a B c}{\left (a^2-b^2+c^2\right ) (a+b \cosh (x)+c \sinh (x))}}{2 \left (a^2-b^2+c^2\right )}-\frac {\sinh (x) (A b-a B)+A c \cosh (x)+B c}{2 \left (a^2-b^2+c^2\right ) (a+b \cosh (x)+c \sinh (x))^2}\) |
-1/2*(B*c + A*c*Cosh[x] + (A*b - a*B)*Sinh[x])/((a^2 - b^2 + c^2)*(a + b*C osh[x] + c*Sinh[x])^2) + ((-2*(2*a^2*A - 3*a*b*B + A*(b^2 - 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) - (a*B*c + (3*a*A - 2*b*B)*c*Cosh[x] + (3*a*A*b - a^2*B - 2*b^2*B)*S inh[x])/((a^2 - b^2 + c^2)*(a + b*Cosh[x] + c*Sinh[x])))/(2*(a^2 - b^2 + c ^2))
3.8.94.3.1 Defintions of rubi rules used
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])
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]
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]
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]
Int[((A_.) + cos[(d_.) + (e_.)*(x_)]*(B_.))*((a_.) + cos[(d_.) + (e_.)*(x_) ]*(b_.) + (c_.)*sin[(d_.) + (e_.)*(x_)])^(n_), x_Symbol] :> Simp[(-(c*B + 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*Cos[d + e*x] + c*Sin[d + e*x])^(n + 1)*Simp[(n + 1)*(a*A - b*B) + (n + 2)*(a*B - b*A)*Cos[d + e*x] - (n + 2)*c*A*Sin[d + e* x], x], x], x] /; FreeQ[{a, b, c, d, e, A, B}, x] && LtQ[n, -1] && NeQ[a^2 - b^2 - c^2, 0] && NeQ[n, -2]
Leaf count of result is larger than twice the leaf count of optimal. \(856\) vs. \(2(190)=380\).
Time = 11.59 (sec) , antiderivative size = 857, normalized size of antiderivative = 4.42
| method | result | size |
| default | \(-\frac {2 \left (-\frac {\left (4 A \,a^{3} b -7 A \,a^{2} b^{2}+5 A \,a^{2} c^{2}+2 A a \,b^{3}-2 A a b \,c^{2}+A \,b^{4}-3 A \,b^{2} c^{2}+2 A \,c^{4}-2 B \,a^{4}+3 B \,a^{3} b -2 B \,a^{2} b^{2}-4 B \,a^{2} c^{2}+3 B a \,b^{3}-2 B \,b^{4}+4 B \,b^{2} c^{2}-2 B \,c^{4}\right ) \tanh \left (\frac {x}{2}\right )^{3}}{2 \left (a^{4}-2 a^{2} b^{2}+2 a^{2} c^{2}+b^{4}-2 b^{2} c^{2}+c^{4}\right ) \left (a -b \right )}-\frac {c \left (4 A \,a^{4}-12 A \,a^{3} b +13 A \,a^{2} b^{2}-7 A \,a^{2} c^{2}-6 A a \,b^{3}+6 A a b \,c^{2}+A \,b^{4}+A \,b^{2} c^{2}-2 A \,c^{4}+2 B \,a^{4}-9 B \,a^{3} b +14 B \,a^{2} b^{2}+4 B \,a^{2} c^{2}-9 B a \,b^{3}+2 B \,b^{4}-4 B \,b^{2} c^{2}+2 B \,c^{4}\right ) \tanh \left (\frac {x}{2}\right )^{2}}{2 \left (a^{4}-2 a^{2} b^{2}+2 a^{2} c^{2}+b^{4}-2 b^{2} c^{2}+c^{4}\right ) \left (a^{2}-2 a b +b^{2}\right )}+\frac {\left (4 A \,a^{4} b -5 A \,a^{3} b^{2}+11 A \,a^{3} c^{2}-3 A \,a^{2} b^{3}-3 A \,a^{2} b \,c^{2}+5 A a \,b^{4}-7 A a \,b^{2} c^{2}+2 A a \,c^{4}-A \,b^{5}-A \,b^{3} c^{2}+2 A b \,c^{4}-2 B \,a^{5}+3 B \,a^{4} b -B \,a^{3} b^{2}-4 B \,a^{3} c^{2}-B \,a^{2} b^{3}-8 B \,a^{2} b \,c^{2}+3 B a \,b^{4}+8 B a \,b^{2} c^{2}-2 B a \,c^{4}-2 B \,b^{5}+4 B \,b^{3} c^{2}-2 B b \,c^{4}\right ) \tanh \left (\frac {x}{2}\right )}{2 \left (a^{4}-2 a^{2} b^{2}+2 a^{2} c^{2}+b^{4}-2 b^{2} c^{2}+c^{4}\right ) \left (a^{2}-2 a b +b^{2}\right )}+\frac {c \left (4 A \,a^{4}-3 A \,a^{2} b^{2}+A \,a^{2} c^{2}-A \,b^{4}+A \,b^{2} c^{2}-5 B \,a^{3} b +5 B a \,b^{3}-2 B a b \,c^{2}\right )}{2 \left (a^{4}-2 a^{2} b^{2}+2 a^{2} c^{2}+b^{4}-2 b^{2} c^{2}+c^{4}\right ) \left (a^{2}-2 a b +b^{2}\right )}\right )}{\left (a \tanh \left (\frac {x}{2}\right )^{2}-\tanh \left (\frac {x}{2}\right )^{2} b -2 c \tanh \left (\frac {x}{2}\right )-a -b \right )^{2}}-\frac {\left (2 a^{2} A +A \,b^{2}-A \,c^{2}-3 a b B \right ) \arctan \left (\frac {2 \left (a -b \right ) \tanh \left (\frac {x}{2}\right )-2 c}{2 \sqrt {-a^{2}+b^{2}-c^{2}}}\right )}{\left (a^{4}-2 a^{2} b^{2}+2 a^{2} c^{2}+b^{4}-2 b^{2} c^{2}+c^{4}\right ) \sqrt {-a^{2}+b^{2}-c^{2}}}\) | \(857\) |
| risch | \(\text {Expression too large to display}\) | \(1515\) |
-2*(-1/2*(4*A*a^3*b-7*A*a^2*b^2+5*A*a^2*c^2+2*A*a*b^3-2*A*a*b*c^2+A*b^4-3* A*b^2*c^2+2*A*c^4-2*B*a^4+3*B*a^3*b-2*B*a^2*b^2-4*B*a^2*c^2+3*B*a*b^3-2*B* b^4+4*B*b^2*c^2-2*B*c^4)/(a^4-2*a^2*b^2+2*a^2*c^2+b^4-2*b^2*c^2+c^4)/(a-b) *tanh(1/2*x)^3-1/2*c*(4*A*a^4-12*A*a^3*b+13*A*a^2*b^2-7*A*a^2*c^2-6*A*a*b^ 3+6*A*a*b*c^2+A*b^4+A*b^2*c^2-2*A*c^4+2*B*a^4-9*B*a^3*b+14*B*a^2*b^2+4*B*a ^2*c^2-9*B*a*b^3+2*B*b^4-4*B*b^2*c^2+2*B*c^4)/(a^4-2*a^2*b^2+2*a^2*c^2+b^4 -2*b^2*c^2+c^4)/(a^2-2*a*b+b^2)*tanh(1/2*x)^2+1/2*(4*A*a^4*b-5*A*a^3*b^2+1 1*A*a^3*c^2-3*A*a^2*b^3-3*A*a^2*b*c^2+5*A*a*b^4-7*A*a*b^2*c^2+2*A*a*c^4-A* b^5-A*b^3*c^2+2*A*b*c^4-2*B*a^5+3*B*a^4*b-B*a^3*b^2-4*B*a^3*c^2-B*a^2*b^3- 8*B*a^2*b*c^2+3*B*a*b^4+8*B*a*b^2*c^2-2*B*a*c^4-2*B*b^5+4*B*b^3*c^2-2*B*b* c^4)/(a^4-2*a^2*b^2+2*a^2*c^2+b^4-2*b^2*c^2+c^4)/(a^2-2*a*b+b^2)*tanh(1/2* x)+1/2*c*(4*A*a^4-3*A*a^2*b^2+A*a^2*c^2-A*b^4+A*b^2*c^2-5*B*a^3*b+5*B*a*b^ 3-2*B*a*b*c^2)/(a^4-2*a^2*b^2+2*a^2*c^2+b^4-2*b^2*c^2+c^4)/(a^2-2*a*b+b^2) )/(a*tanh(1/2*x)^2-tanh(1/2*x)^2*b-2*c*tanh(1/2*x)-a-b)^2-(2*A*a^2+A*b^2-A *c^2-3*B*a*b)/(a^4-2*a^2*b^2+2*a^2*c^2+b^4-2*b^2*c^2+c^4)/(-a^2+b^2-c^2)^( 1/2)*arctan(1/2*(2*(a-b)*tanh(1/2*x)-2*c)/(-a^2+b^2-c^2)^(1/2))
Leaf count of result is larger than twice the leaf count of optimal. 6126 vs. \(2 (187) = 374\).
Time = 0.60 (sec) , antiderivative size = 12366, normalized size of antiderivative = 63.74 \[ \int \frac {A+B \cosh (x)}{(a+b \cosh (x)+c \sinh (x))^3} \, dx=\text {Too large to display} \]
Timed out. \[ \int \frac {A+B \cosh (x)}{(a+b \cosh (x)+c \sinh (x))^3} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {A+B \cosh (x)}{(a+b \cosh (x)+c \sinh (x))^3} \, dx=\text {Exception raised: ValueError} \]
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
Leaf count of result is larger than twice the leaf count of optimal. 625 vs. \(2 (187) = 374\).
Time = 0.28 (sec) , antiderivative size = 625, normalized size of antiderivative = 3.22 \[ \int \frac {A+B \cosh (x)}{(a+b \cosh (x)+c \sinh (x))^3} \, dx=\frac {{\left (2 \, A a^{2} - 3 \, B a b + A b^{2} - A c^{2}\right )} \arctan \left (\frac {b e^{x} + c e^{x} + a}{\sqrt {-a^{2} + b^{2} - c^{2}}}\right )}{{\left (a^{4} - 2 \, a^{2} b^{2} + b^{4} + 2 \, a^{2} c^{2} - 2 \, b^{2} c^{2} + c^{4}\right )} \sqrt {-a^{2} + b^{2} - c^{2}}} + \frac {2 \, A a^{2} b^{2} e^{\left (3 \, x\right )} - 3 \, B a b^{3} e^{\left (3 \, x\right )} + A b^{4} e^{\left (3 \, x\right )} + 4 \, A a^{2} b c e^{\left (3 \, x\right )} - 6 \, B a b^{2} c e^{\left (3 \, x\right )} + 2 \, A b^{3} c e^{\left (3 \, x\right )} + 2 \, A a^{2} c^{2} e^{\left (3 \, x\right )} - 3 \, B a b c^{2} e^{\left (3 \, x\right )} - 2 \, A b c^{3} e^{\left (3 \, x\right )} - A c^{4} e^{\left (3 \, x\right )} - 2 \, B a^{4} e^{\left (2 \, x\right )} + 6 \, A a^{3} b e^{\left (2 \, x\right )} - 5 \, B a^{2} b^{2} e^{\left (2 \, x\right )} + 3 \, A a b^{3} e^{\left (2 \, x\right )} - 2 \, B b^{4} e^{\left (2 \, x\right )} + 6 \, A a^{3} c e^{\left (2 \, x\right )} - 9 \, B a^{2} b c e^{\left (2 \, x\right )} + 3 \, A a b^{2} c e^{\left (2 \, x\right )} - 4 \, B a^{2} c^{2} e^{\left (2 \, x\right )} - 3 \, A a b c^{2} e^{\left (2 \, x\right )} + 4 \, B b^{2} c^{2} e^{\left (2 \, x\right )} - 3 \, A a c^{3} e^{\left (2 \, x\right )} - 2 \, B c^{4} e^{\left (2 \, x\right )} - 4 \, B a^{3} b e^{x} + 10 \, A a^{2} b^{2} e^{x} - 5 \, B a b^{3} e^{x} - A b^{4} e^{x} - 10 \, A a^{2} c^{2} e^{x} + 5 \, B a b c^{2} e^{x} + 2 \, A b^{2} c^{2} e^{x} - A c^{4} e^{x} - B a^{2} b^{2} + 3 \, A a b^{3} - 2 \, B b^{4} + B a^{2} b c - 3 \, A a b^{2} c + 2 \, B b^{3} c - 3 \, A a b c^{2} + 2 \, B b^{2} c^{2} + 3 \, A a c^{3} - 2 \, B b c^{3}}{{\left (a^{4} b - 2 \, a^{2} b^{3} + b^{5} + a^{4} c - 2 \, a^{2} b^{2} c + b^{4} c + 2 \, a^{2} b c^{2} - 2 \, b^{3} c^{2} + 2 \, a^{2} c^{3} - 2 \, b^{2} c^{3} + b c^{4} + c^{5}\right )} {\left (b e^{\left (2 \, x\right )} + c e^{\left (2 \, x\right )} + 2 \, a e^{x} + b - c\right )}^{2}} \]
(2*A*a^2 - 3*B*a*b + A*b^2 - A*c^2)*arctan((b*e^x + c*e^x + a)/sqrt(-a^2 + b^2 - c^2))/((a^4 - 2*a^2*b^2 + b^4 + 2*a^2*c^2 - 2*b^2*c^2 + c^4)*sqrt(- a^2 + b^2 - c^2)) + (2*A*a^2*b^2*e^(3*x) - 3*B*a*b^3*e^(3*x) + A*b^4*e^(3* x) + 4*A*a^2*b*c*e^(3*x) - 6*B*a*b^2*c*e^(3*x) + 2*A*b^3*c*e^(3*x) + 2*A*a ^2*c^2*e^(3*x) - 3*B*a*b*c^2*e^(3*x) - 2*A*b*c^3*e^(3*x) - A*c^4*e^(3*x) - 2*B*a^4*e^(2*x) + 6*A*a^3*b*e^(2*x) - 5*B*a^2*b^2*e^(2*x) + 3*A*a*b^3*e^( 2*x) - 2*B*b^4*e^(2*x) + 6*A*a^3*c*e^(2*x) - 9*B*a^2*b*c*e^(2*x) + 3*A*a*b ^2*c*e^(2*x) - 4*B*a^2*c^2*e^(2*x) - 3*A*a*b*c^2*e^(2*x) + 4*B*b^2*c^2*e^( 2*x) - 3*A*a*c^3*e^(2*x) - 2*B*c^4*e^(2*x) - 4*B*a^3*b*e^x + 10*A*a^2*b^2* e^x - 5*B*a*b^3*e^x - A*b^4*e^x - 10*A*a^2*c^2*e^x + 5*B*a*b*c^2*e^x + 2*A *b^2*c^2*e^x - A*c^4*e^x - B*a^2*b^2 + 3*A*a*b^3 - 2*B*b^4 + B*a^2*b*c - 3 *A*a*b^2*c + 2*B*b^3*c - 3*A*a*b*c^2 + 2*B*b^2*c^2 + 3*A*a*c^3 - 2*B*b*c^3 )/((a^4*b - 2*a^2*b^3 + b^5 + a^4*c - 2*a^2*b^2*c + b^4*c + 2*a^2*b*c^2 - 2*b^3*c^2 + 2*a^2*c^3 - 2*b^2*c^3 + b*c^4 + c^5)*(b*e^(2*x) + c*e^(2*x) + 2*a*e^x + b - c)^2)
Timed out. \[ \int \frac {A+B \cosh (x)}{(a+b \cosh (x)+c \sinh (x))^3} \, dx=\int \frac {A+B\,\mathrm {cosh}\left (x\right )}{{\left (a+b\,\mathrm {cosh}\left (x\right )+c\,\mathrm {sinh}\left (x\right )\right )}^3} \,d x \]