Integrand size = 19, antiderivative size = 198 \[ \int \frac {A+C \sinh (x)}{(a+b \cosh (x)+c \sinh (x))^3} \, dx=-\frac {\left (2 a^2 A+A \left (b^2-c^2\right )+3 a c C\right ) \text {arctanh}\left (\frac {c-(a-b) \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2-b^2+c^2}}\right )}{\left (a^2-b^2+c^2\right )^{5/2}}+\frac {b C-(A c-a C) \cosh (x)-A b \sinh (x)}{2 \left (a^2-b^2+c^2\right ) (a+b \cosh (x)+c \sinh (x))^2}+\frac {a b C-\left (3 a A c-a^2 C+2 c^2 C\right ) \cosh (x)-b (3 a A+2 c C) \sinh (x)}{2 \left (a^2-b^2+c^2\right )^2 (a+b \cosh (x)+c \sinh (x))} \]
-(2*a^2*A+A*(b^2-c^2)+3*a*c*C)*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-C*a)*cosh(x)-A*b*sinh(x))/(a^2-b ^2+c^2)/(a+b*cosh(x)+c*sinh(x))^2+1/2*(a*b*C-(3*A*a*c-C*a^2+2*C*c^2)*cosh( x)-b*(3*A*a+2*C*c)*sinh(x))/(a^2-b^2+c^2)^2/(a+b*cosh(x)+c*sinh(x))
Time = 1.00 (sec) , antiderivative size = 373, normalized size of antiderivative = 1.88 \[ \int \frac {A+C \sinh (x)}{(a+b \cosh (x)+c \sinh (x))^3} \, dx=\frac {\left (2 a^2 A+A \left (b^2-c^2\right )+3 a c C\right ) \arctan \left (\frac {c+(-a+b) \tanh \left (\frac {x}{2}\right )}{\sqrt {-a^2+b^2-c^2}}\right )}{\left (-a^2+b^2-c^2\right )^{5/2}}+\frac {6 a^3 A c+3 a A b^2 c-3 a A c^3-2 a^4 C+4 a^2 b^2 C-2 b^4 C+5 a^2 c^2 C+4 b^2 c^2 C-2 c^4 C+2 b c \left (2 a^2 A+A \left (b^2-c^2\right )+3 a c C\right ) \cosh (x)+c \left (3 a A \left (-b^2+c^2\right )-a^2 c C+2 c \left (-b^2+c^2\right ) C\right ) \cosh (2 x)-8 a^2 A b^2 \sinh (x)+2 A b^4 \sinh (x)+12 a^2 A c^2 \sinh (x)-2 A b^2 c^2 \sinh (x)-4 a^3 c C \sinh (x)-2 a b^2 c C \sinh (x)+8 a c^3 C \sinh (x)-3 a A b^3 \sinh (2 x)+3 a A b c^2 \sinh (2 x)-a^2 b c C \sinh (2 x)-2 b^3 c C \sinh (2 x)+2 b c^3 C \sinh (2 x)}{4 b \left (a^2-b^2+c^2\right )^2 (a+b \cosh (x)+c \sinh (x))^2} \]
((2*a^2*A + A*(b^2 - c^2) + 3*a*c*C)*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 - 3*a*A*c^3 - 2*a^4*C + 4*a^2*b^2*C - 2*b^4*C + 5*a^2*c^2*C + 4*b^2*c^2*C - 2*c^4*C + 2*b*c*(2*a^2*A + A*(b^2 - c^2) + 3*a*c*C)*Cosh[x] + c*(3*a*A*(-b ^2 + c^2) - a^2*c*C + 2*c*(-b^2 + c^2)*C)*Cosh[2*x] - 8*a^2*A*b^2*Sinh[x] + 2*A*b^4*Sinh[x] + 12*a^2*A*c^2*Sinh[x] - 2*A*b^2*c^2*Sinh[x] - 4*a^3*c*C *Sinh[x] - 2*a*b^2*c*C*Sinh[x] + 8*a*c^3*C*Sinh[x] - 3*a*A*b^3*Sinh[2*x] + 3*a*A*b*c^2*Sinh[2*x] - a^2*b*c*C*Sinh[2*x] - 2*b^3*c*C*Sinh[2*x] + 2*b*c ^3*C*Sinh[2*x])/(4*b*(a^2 - b^2 + c^2)^2*(a + b*Cosh[x] + c*Sinh[x])^2)
Time = 0.70 (sec) , antiderivative size = 219, 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, 3636, 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.
| \(\displaystyle \int \frac {A+C \sinh (x)}{(a+b \cosh (x)+c \sinh (x))^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {A-i C \sin (i x)}{(a+b \cos (i x)-i c \sin (i x))^3}dx\) |
| \(\Big \downarrow \) 3636 |
| \(\displaystyle \frac {-\cosh (x) (A c-a C)-A b \sinh (x)+b C}{2 \left (a^2-b^2+c^2\right ) (a+b \cosh (x)+c \sinh (x))^2}-\frac {\int -\frac {2 (a A+c C)-A b \cosh (x)-(A c-a C) \sinh (x)}{(a+b \cosh (x)+c \sinh (x))^2}dx}{2 \left (a^2-b^2+c^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {2 (a A+c C)-A b \cosh (x)-(A c-a C) \sinh (x)}{(a+b \cosh (x)+c \sinh (x))^2}dx}{2 \left (a^2-b^2+c^2\right )}+\frac {-\cosh (x) (A c-a C)-A b \sinh (x)+b C}{2 \left (a^2-b^2+c^2\right ) (a+b \cosh (x)+c \sinh (x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\cosh (x) (A c-a C)-A b \sinh (x)+b C}{2 \left (a^2-b^2+c^2\right ) (a+b \cosh (x)+c \sinh (x))^2}+\frac {\int \frac {2 (a A+c C)-A b \cos (i x)+i (A c-a C) \sin (i x)}{(a+b \cos (i x)-i c \sin (i x))^2}dx}{2 \left (a^2-b^2+c^2\right )}\) |
| \(\Big \downarrow \) 3632 |
| \(\displaystyle \frac {\frac {\left (2 a^2 A+3 a c C+A \left (b^2-c^2\right )\right ) \int \frac {1}{a+b \cosh (x)+c \sinh (x)}dx}{a^2-b^2+c^2}+\frac {-\cosh (x) \left (a^2 (-C)+3 a A c+2 c^2 C\right )-b \sinh (x) (3 a A+2 c C)+a b C}{\left (a^2-b^2+c^2\right ) (a+b \cosh (x)+c \sinh (x))}}{2 \left (a^2-b^2+c^2\right )}+\frac {-\cosh (x) (A c-a C)-A b \sinh (x)+b C}{2 \left (a^2-b^2+c^2\right ) (a+b \cosh (x)+c \sinh (x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\cosh (x) (A c-a C)-A b \sinh (x)+b C}{2 \left (a^2-b^2+c^2\right ) (a+b \cosh (x)+c \sinh (x))^2}+\frac {\frac {-\cosh (x) \left (a^2 (-C)+3 a A c+2 c^2 C\right )-b \sinh (x) (3 a A+2 c C)+a b C}{\left (a^2-b^2+c^2\right ) (a+b \cosh (x)+c \sinh (x))}+\frac {\left (2 a^2 A+3 a c C+A \left (b^2-c^2\right )\right ) \int \frac {1}{a+b \cos (i x)-i c \sin (i x)}dx}{a^2-b^2+c^2}}{2 \left (a^2-b^2+c^2\right )}\) |
| \(\Big \downarrow \) 3603 |
| \(\displaystyle \frac {\frac {2 \left (2 a^2 A+3 a c C+A \left (b^2-c^2\right )\right ) \int \frac {1}{-\left ((a-b) \tanh ^2\left (\frac {x}{2}\right )\right )+2 c \tanh \left (\frac {x}{2}\right )+a+b}d\tanh \left (\frac {x}{2}\right )}{a^2-b^2+c^2}+\frac {-\cosh (x) \left (a^2 (-C)+3 a A c+2 c^2 C\right )-b \sinh (x) (3 a A+2 c C)+a b C}{\left (a^2-b^2+c^2\right ) (a+b \cosh (x)+c \sinh (x))}}{2 \left (a^2-b^2+c^2\right )}+\frac {-\cosh (x) (A c-a C)-A b \sinh (x)+b C}{2 \left (a^2-b^2+c^2\right ) (a+b \cosh (x)+c \sinh (x))^2}\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle \frac {\frac {-\cosh (x) \left (a^2 (-C)+3 a A c+2 c^2 C\right )-b \sinh (x) (3 a A+2 c C)+a b C}{\left (a^2-b^2+c^2\right ) (a+b \cosh (x)+c \sinh (x))}-\frac {4 \left (2 a^2 A+3 a c C+A \left (b^2-c^2\right )\right ) \int \frac {1}{4 \left (a^2-b^2+c^2\right )-\left (2 c-2 (a-b) \tanh \left (\frac {x}{2}\right )\right )^2}d\left (2 c-2 (a-b) \tanh \left (\frac {x}{2}\right )\right )}{a^2-b^2+c^2}}{2 \left (a^2-b^2+c^2\right )}+\frac {-\cosh (x) (A c-a C)-A b \sinh (x)+b C}{2 \left (a^2-b^2+c^2\right ) (a+b \cosh (x)+c \sinh (x))^2}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\frac {-\cosh (x) \left (a^2 (-C)+3 a A c+2 c^2 C\right )-b \sinh (x) (3 a A+2 c C)+a b C}{\left (a^2-b^2+c^2\right ) (a+b \cosh (x)+c \sinh (x))}-\frac {2 \left (2 a^2 A+3 a c C+A \left (b^2-c^2\right )\right ) \text {arctanh}\left (\frac {2 c-2 (a-b) \tanh \left (\frac {x}{2}\right )}{2 \sqrt {a^2-b^2+c^2}}\right )}{\left (a^2-b^2+c^2\right )^{3/2}}}{2 \left (a^2-b^2+c^2\right )}+\frac {-\cosh (x) (A c-a C)-A b \sinh (x)+b C}{2 \left (a^2-b^2+c^2\right ) (a+b \cosh (x)+c \sinh (x))^2}\) |
(b*C - (A*c - a*C)*Cosh[x] - A*b*Sinh[x])/(2*(a^2 - b^2 + c^2)*(a + b*Cosh [x] + c*Sinh[x])^2) + ((-2*(2*a^2*A + A*(b^2 - c^2) + 3*a*c*C)*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*c - a^2*C + 2*c^2*C)*Cosh[x] - b*(3*a*A + 2*c*C)*Sinh [x])/((a^2 - b^2 + c^2)*(a + b*Cosh[x] + c*Sinh[x])))/(2*(a^2 - b^2 + c^2) )
3.8.91.3.1 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_.) + (c_.)*sin[(d_.) + (e_.)*(x_)]) ^(n_)*((A_.) + (C_.)*sin[(d_.) + (e_.)*(x_)]), x_Symbol] :> Simp[(b*C + (a* C - c*A)*Cos[d + e*x] + 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 - c*C) - (n + 2)*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, C}, 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. \(835\) vs. \(2(188)=376\).
Time = 11.62 (sec) , antiderivative size = 836, normalized size of antiderivative = 4.22
| 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}-3 C \,a^{3} c +6 C \,a^{2} b c -3 C a \,b^{2} c \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 {\left (4 A \,a^{4} c -12 A \,a^{3} b c +13 A \,a^{2} b^{2} c -7 A \,a^{2} c^{3}-6 A a \,b^{3} c +6 A a b \,c^{3}+A \,b^{4} c +A \,b^{2} c^{3}-2 A \,c^{5}-2 C \,a^{5}+2 C \,a^{4} b +4 C \,a^{3} b^{2}+5 C \,a^{3} c^{2}-4 C \,a^{2} b^{3}-14 C \,a^{2} b \,c^{2}-2 C a \,b^{4}+13 C a \,b^{2} c^{2}-2 C a \,c^{4}+2 C \,b^{5}-4 C \,b^{3} c^{2}+2 C 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}-5 C \,a^{4} c +5 C \,a^{3} b c +5 C \,a^{2} b^{2} c +4 C \,a^{2} c^{3}-5 C a \,b^{3} c -4 C a b \,c^{3}\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 {4 A \,a^{4} c -3 A \,a^{2} b^{2} c +A \,a^{2} c^{3}-A \,b^{4} c +A \,b^{2} c^{3}-2 C \,a^{5}+4 C \,a^{3} b^{2}+C \,a^{3} c^{2}-2 C a \,b^{4}-C a \,b^{2} c^{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 )}\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 c C \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}}}\) | \(836\) |
| 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^4-3*C*a^3*c+6*C*a^2*b*c-3*C*a*b^2*c)/(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*(4*A*a^4*c-12*A*a^3*b*c+13* A*a^2*b^2*c-7*A*a^2*c^3-6*A*a*b^3*c+6*A*a*b*c^3+A*b^4*c+A*b^2*c^3-2*A*c^5- 2*C*a^5+2*C*a^4*b+4*C*a^3*b^2+5*C*a^3*c^2-4*C*a^2*b^3-14*C*a^2*b*c^2-2*C*a *b^4+13*C*a*b^2*c^2-2*C*a*c^4+2*C*b^5-4*C*b^3*c^2+2*C*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+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^4-5*C*a^4*c+5*C*a^3*b*c+5*C*a^2*b^2*c +4*C*a^2*c^3-5*C*a*b^3*c-4*C*a*b*c^3)/(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*(4*A*a^4*c-3*A*a^2*b^2*c+A*a^2*c^3 -A*b^4*c+A*b^2*c^3-2*C*a^5+4*C*a^3*b^2+C*a^3*c^2-2*C*a*b^4-C*a*b^2*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*C*a*c)/(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. 6084 vs. \(2 (188) = 376\).
Time = 0.59 (sec) , antiderivative size = 12285, normalized size of antiderivative = 62.05 \[ \int \frac {A+C \sinh (x)}{(a+b \cosh (x)+c \sinh (x))^3} \, dx=\text {Too large to display} \]
Timed out. \[ \int \frac {A+C \sinh (x)}{(a+b \cosh (x)+c \sinh (x))^3} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {A+C \sinh (x)}{(a+b \cosh (x)+c \sinh (x))^3} \, dx=\text {Exception raised: ValueError} \]
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. \(2 (188) = 376\).
Time = 0.28 (sec) , antiderivative size = 625, normalized size of antiderivative = 3.16 \[ \int \frac {A+C \sinh (x)}{(a+b \cosh (x)+c \sinh (x))^3} \, dx=\frac {{\left (2 \, A a^{2} + A b^{2} + 3 \, C a c - 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 )} + A b^{4} e^{\left (3 \, x\right )} + 4 \, A a^{2} b c e^{\left (3 \, x\right )} + 3 \, C 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 )} + 6 \, C a b c^{2} e^{\left (3 \, x\right )} + 3 \, C a c^{3} e^{\left (3 \, x\right )} - 2 \, A b c^{3} e^{\left (3 \, x\right )} - A c^{4} e^{\left (3 \, x\right )} - 2 \, C a^{4} e^{\left (2 \, x\right )} + 6 \, A a^{3} b e^{\left (2 \, x\right )} + 4 \, C a^{2} b^{2} e^{\left (2 \, x\right )} + 3 \, A a b^{3} e^{\left (2 \, x\right )} - 2 \, C b^{4} e^{\left (2 \, x\right )} + 6 \, A a^{3} c e^{\left (2 \, x\right )} + 9 \, C a^{2} b c e^{\left (2 \, x\right )} + 3 \, A a b^{2} c e^{\left (2 \, x\right )} + 5 \, C a^{2} c^{2} e^{\left (2 \, x\right )} - 3 \, A a b c^{2} e^{\left (2 \, x\right )} + 4 \, C b^{2} c^{2} e^{\left (2 \, x\right )} - 3 \, A a c^{3} e^{\left (2 \, x\right )} - 2 \, C c^{4} e^{\left (2 \, x\right )} + 10 \, A a^{2} b^{2} e^{x} - A b^{4} e^{x} + 4 \, C a^{3} c e^{x} + 5 \, C a b^{2} c e^{x} - 10 \, A a^{2} c^{2} e^{x} + 2 \, A b^{2} c^{2} e^{x} - 5 \, C a c^{3} e^{x} - A c^{4} e^{x} + 3 \, A a b^{3} + C a^{2} b c - 3 \, A a b^{2} c + 2 \, C b^{3} c - C a^{2} c^{2} - 3 \, A a b c^{2} - 2 \, C b^{2} c^{2} + 3 \, A a c^{3} - 2 \, C b c^{3} + 2 \, C c^{4}}{{\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 + A*b^2 + 3*C*a*c - 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) + A*b^4*e^(3*x) + 4*A*a^2*b*c*e^( 3*x) + 3*C*a*b^2*c*e^(3*x) + 2*A*b^3*c*e^(3*x) + 2*A*a^2*c^2*e^(3*x) + 6*C *a*b*c^2*e^(3*x) + 3*C*a*c^3*e^(3*x) - 2*A*b*c^3*e^(3*x) - A*c^4*e^(3*x) - 2*C*a^4*e^(2*x) + 6*A*a^3*b*e^(2*x) + 4*C*a^2*b^2*e^(2*x) + 3*A*a*b^3*e^( 2*x) - 2*C*b^4*e^(2*x) + 6*A*a^3*c*e^(2*x) + 9*C*a^2*b*c*e^(2*x) + 3*A*a*b ^2*c*e^(2*x) + 5*C*a^2*c^2*e^(2*x) - 3*A*a*b*c^2*e^(2*x) + 4*C*b^2*c^2*e^( 2*x) - 3*A*a*c^3*e^(2*x) - 2*C*c^4*e^(2*x) + 10*A*a^2*b^2*e^x - A*b^4*e^x + 4*C*a^3*c*e^x + 5*C*a*b^2*c*e^x - 10*A*a^2*c^2*e^x + 2*A*b^2*c^2*e^x - 5 *C*a*c^3*e^x - A*c^4*e^x + 3*A*a*b^3 + C*a^2*b*c - 3*A*a*b^2*c + 2*C*b^3*c - C*a^2*c^2 - 3*A*a*b*c^2 - 2*C*b^2*c^2 + 3*A*a*c^3 - 2*C*b*c^3 + 2*C*c^4 )/((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. \[ \int \frac {A+C \sinh (x)}{(a+b \cosh (x)+c \sinh (x))^3} \, dx=\int \frac {A+C\,\mathrm {sinh}\left (x\right )}{{\left (a+b\,\mathrm {cosh}\left (x\right )+c\,\mathrm {sinh}\left (x\right )\right )}^3} \,d x \]