Integrand size = 12, antiderivative size = 146 \[ \int \frac {1}{(a+b \cosh (x)+c \sinh (x))^3} \, dx=-\frac {\left (2 a^2+b^2-c^2\right ) \text {arctanh}\left (\frac {c-(a-b) \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2-b^2+c^2}}\right )}{\left (a^2-b^2+c^2\right )^{5/2}}-\frac {c \cosh (x)+b \sinh (x)}{2 \left (a^2-b^2+c^2\right ) (a+b \cosh (x)+c \sinh (x))^2}-\frac {3 (a c \cosh (x)+a b \sinh (x))}{2 \left (a^2-b^2+c^2\right )^2 (a+b \cosh (x)+c \sinh (x))} \]
-(2*a^2+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*(-c*cosh(x)-b*sinh(x))/(a^2-b^2+c^2)/(a+b*cosh(x)+c*sinh (x))^2-3/2*(a*c*cosh(x)+a*b*sinh(x))/(a^2-b^2+c^2)^2/(a+b*cosh(x)+c*sinh(x ))
Time = 0.36 (sec) , antiderivative size = 183, normalized size of antiderivative = 1.25 \[ \int \frac {1}{(a+b \cosh (x)+c \sinh (x))^3} \, dx=\frac {1}{2} \left (\frac {2 \left (2 a^2+b^2-c^2\right ) \arctan \left (\frac {c+(-a+b) \tanh \left (\frac {x}{2}\right )}{\sqrt {-a^2+b^2-c^2}}\right )}{\left (-a^2+b^2-c^2\right )^{5/2}}+\frac {-a c+\left (b^2-c^2\right ) \sinh (x)}{b \left (-a^2+b^2-c^2\right ) (a+b \cosh (x)+c \sinh (x))^2}+\frac {c \left (2 a^2+b^2-c^2\right )-3 a \left (b^2-c^2\right ) \sinh (x)}{b \left (a^2-b^2+c^2\right )^2 (a+b \cosh (x)+c \sinh (x))}\right ) \]
((2*(2*a^2 + 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) + (-(a*c) + (b^2 - c^2)*Sinh[x])/(b*(-a^2 + b^2 - c^2)*(a + b*Cosh[x] + c*Sinh[x])^2) + (c*(2*a^2 + b^2 - c^2) - 3*a *(b^2 - c^2)*Sinh[x])/(b*(a^2 - b^2 + c^2)^2*(a + b*Cosh[x] + c*Sinh[x]))) /2
Time = 0.55 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.15, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {3042, 3608, 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.
| \(\displaystyle \int \frac {1}{(a+b \cosh (x)+c \sinh (x))^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{(a+b \cos (i x)-i c \sin (i x))^3}dx\) |
| \(\Big \downarrow \) 3608 |
| \(\displaystyle -\frac {\int -\frac {2 a-b \cosh (x)-c \sinh (x)}{(a+b \cosh (x)+c \sinh (x))^2}dx}{2 \left (a^2-b^2+c^2\right )}-\frac {b \sinh (x)+c \cosh (x)}{2 \left (a^2-b^2+c^2\right ) (a+b \cosh (x)+c \sinh (x))^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {2 a-b \cosh (x)-c \sinh (x)}{(a+b \cosh (x)+c \sinh (x))^2}dx}{2 \left (a^2-b^2+c^2\right )}-\frac {b \sinh (x)+c \cosh (x)}{2 \left (a^2-b^2+c^2\right ) (a+b \cosh (x)+c \sinh (x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {b \sinh (x)+c \cosh (x)}{2 \left (a^2-b^2+c^2\right ) (a+b \cosh (x)+c \sinh (x))^2}+\frac {\int \frac {2 a-b \cos (i x)+i c \sin (i x)}{(a+b \cos (i x)-i c \sin (i x))^2}dx}{2 \left (a^2-b^2+c^2\right )}\) |
| \(\Big \downarrow \) 3632 |
| \(\displaystyle \frac {\frac {\left (2 a^2+b^2-c^2\right ) \int \frac {1}{a+b \cosh (x)+c \sinh (x)}dx}{a^2-b^2+c^2}-\frac {3 (a b \sinh (x)+a c \cosh (x))}{\left (a^2-b^2+c^2\right ) (a+b \cosh (x)+c \sinh (x))}}{2 \left (a^2-b^2+c^2\right )}-\frac {b \sinh (x)+c \cosh (x)}{2 \left (a^2-b^2+c^2\right ) (a+b \cosh (x)+c \sinh (x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {b \sinh (x)+c \cosh (x)}{2 \left (a^2-b^2+c^2\right ) (a+b \cosh (x)+c \sinh (x))^2}+\frac {-\frac {3 (a b \sinh (x)+a c \cosh (x))}{\left (a^2-b^2+c^2\right ) (a+b \cosh (x)+c \sinh (x))}+\frac {\left (2 a^2+b^2-c^2\right ) \int \frac {1}{a+b \cos (i x)-i c \sin (i x)}dx}{a^2-b^2+c^2}}{2 \left (a^2-b^2+c^2\right )}\) |
| \(\Big \downarrow \) 3603 |
| \(\displaystyle \frac {\frac {2 \left (2 a^2+b^2-c^2\right ) \int \frac {1}{-\left ((a-b) \tanh ^2\left (\frac {x}{2}\right )\right )+2 c \tanh \left (\frac {x}{2}\right )+a+b}d\tanh \left (\frac {x}{2}\right )}{a^2-b^2+c^2}-\frac {3 (a b \sinh (x)+a c \cosh (x))}{\left (a^2-b^2+c^2\right ) (a+b \cosh (x)+c \sinh (x))}}{2 \left (a^2-b^2+c^2\right )}-\frac {b \sinh (x)+c \cosh (x)}{2 \left (a^2-b^2+c^2\right ) (a+b \cosh (x)+c \sinh (x))^2}\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle \frac {-\frac {4 \left (2 a^2+b^2-c^2\right ) \int \frac {1}{4 \left (a^2-b^2+c^2\right )-\left (2 c-2 (a-b) \tanh \left (\frac {x}{2}\right )\right )^2}d\left (2 c-2 (a-b) \tanh \left (\frac {x}{2}\right )\right )}{a^2-b^2+c^2}-\frac {3 (a b \sinh (x)+a c \cosh (x))}{\left (a^2-b^2+c^2\right ) (a+b \cosh (x)+c \sinh (x))}}{2 \left (a^2-b^2+c^2\right )}-\frac {b \sinh (x)+c \cosh (x)}{2 \left (a^2-b^2+c^2\right ) (a+b \cosh (x)+c \sinh (x))^2}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {-\frac {2 \left (2 a^2+b^2-c^2\right ) \text {arctanh}\left (\frac {2 c-2 (a-b) \tanh \left (\frac {x}{2}\right )}{2 \sqrt {a^2-b^2+c^2}}\right )}{\left (a^2-b^2+c^2\right )^{3/2}}-\frac {3 (a b \sinh (x)+a c \cosh (x))}{\left (a^2-b^2+c^2\right ) (a+b \cosh (x)+c \sinh (x))}}{2 \left (a^2-b^2+c^2\right )}-\frac {b \sinh (x)+c \cosh (x)}{2 \left (a^2-b^2+c^2\right ) (a+b \cosh (x)+c \sinh (x))^2}\) |
-1/2*(c*Cosh[x] + b*Sinh[x])/((a^2 - b^2 + c^2)*(a + b*Cosh[x] + c*Sinh[x] )^2) + ((-2*(2*a^2 + b^2 - c^2)*ArcTanh[(2*c - 2*(a - b)*Tanh[x/2])/(2*Sqr t[a^2 - b^2 + c^2])])/(a^2 - b^2 + c^2)^(3/2) - (3*(a*c*Cosh[x] + a*b*Sinh [x]))/((a^2 - b^2 + c^2)*(a + b*Cosh[x] + c*Sinh[x])))/(2*(a^2 - b^2 + c^2 ))
3.8.44.3.1 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[(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]
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]
Leaf count of result is larger than twice the leaf count of optimal. \(576\) vs. \(2(138)=276\).
Time = 21.91 (sec) , antiderivative size = 577, normalized size of antiderivative = 3.95
| method | result | size |
| default | \(-\frac {2 \left (-\frac {\left (4 a^{3} b -7 a^{2} b^{2}+5 c^{2} a^{2}+2 a \,b^{3}-2 c^{2} a b +b^{4}-3 b^{2} c^{2}+2 c^{4}\right ) \tanh \left (\frac {x}{2}\right )^{3}}{2 \left (a -b \right ) \left (a^{4}-2 a^{2} b^{2}+2 c^{2} a^{2}+b^{4}-2 b^{2} c^{2}+c^{4}\right )}-\frac {c \left (4 a^{4}-12 a^{3} b +13 a^{2} b^{2}-7 c^{2} a^{2}-6 a \,b^{3}+6 c^{2} a b +b^{4}+b^{2} c^{2}-2 c^{4}\right ) \tanh \left (\frac {x}{2}\right )^{2}}{2 \left (a^{4}-2 a^{2} b^{2}+2 c^{2} a^{2}+b^{4}-2 b^{2} c^{2}+c^{4}\right ) \left (a^{2}-2 a b +b^{2}\right )}+\frac {\left (4 a^{4} b -5 a^{3} b^{2}+11 a^{3} c^{2}-3 a^{2} b^{3}-3 a^{2} b \,c^{2}+5 a \,b^{4}-7 a \,b^{2} c^{2}+2 a \,c^{4}-b^{5}-b^{3} c^{2}+2 b \,c^{4}\right ) \tanh \left (\frac {x}{2}\right )}{2 \left (a^{4}-2 a^{2} b^{2}+2 c^{2} a^{2}+b^{4}-2 b^{2} c^{2}+c^{4}\right ) \left (a^{2}-2 a b +b^{2}\right )}+\frac {c \left (4 a^{4}-3 a^{2} b^{2}+c^{2} a^{2}-b^{4}+b^{2} c^{2}\right )}{2 \left (a^{4}-2 a^{2} b^{2}+2 c^{2} a^{2}+b^{4}-2 b^{2} c^{2}+c^{4}\right ) \left (a^{2}-2 a b +b^{2}\right )}\right )}{\left (\tanh \left (\frac {x}{2}\right )^{2} a -\tanh \left (\frac {x}{2}\right )^{2} b -2 c \tanh \left (\frac {x}{2}\right )-a -b \right )^{2}}-\frac {\left (2 a^{2}+b^{2}-c^{2}\right ) \arctan \left (\frac {2 \left (a -b \right ) \tanh \left (\frac {x}{2}\right )-2 c}{2 \sqrt {-a^{2}+b^{2}-c^{2}}}\right )}{\left (a^{4}-2 a^{2} b^{2}+2 c^{2} a^{2}+b^{4}-2 b^{2} c^{2}+c^{4}\right ) \sqrt {-a^{2}+b^{2}-c^{2}}}\) | \(577\) |
| risch | \(\frac {2 \,{\mathrm e}^{3 x} a^{2} b +2 a^{2} c \,{\mathrm e}^{3 x}+{\mathrm e}^{3 x} b^{3}+{\mathrm e}^{3 x} c \,b^{2}-{\mathrm e}^{3 x} c^{2} b -{\mathrm e}^{3 x} c^{3}+6 a^{3} {\mathrm e}^{2 x}+3 a \,b^{2} {\mathrm e}^{2 x}-3 \,{\mathrm e}^{2 x} a \,c^{2}+10 a^{2} b \,{\mathrm e}^{x}-10 \,{\mathrm e}^{x} c \,a^{2}-b^{3} {\mathrm e}^{x}+{\mathrm e}^{x} c \,b^{2}+{\mathrm e}^{x} c^{2} b -{\mathrm e}^{x} c^{3}+3 a \,b^{2}-6 c a b +3 a \,c^{2}}{\left (a^{2}-b^{2}+c^{2}\right )^{2} \left (b \,{\mathrm e}^{2 x}+{\mathrm e}^{2 x} c +2 a \,{\mathrm e}^{x}+b -c \right )^{2}}+\frac {\ln \left ({\mathrm e}^{x}+\frac {\left (a^{2}-b^{2}+c^{2}\right )^{\frac {5}{2}} a -a^{6}+3 a^{4} b^{2}-3 a^{4} c^{2}-3 a^{2} b^{4}+6 a^{2} b^{2} c^{2}-3 a^{2} c^{4}+b^{6}-3 b^{4} c^{2}+3 b^{2} c^{4}-c^{6}}{\left (a^{2}-b^{2}+c^{2}\right )^{\frac {5}{2}} \left (b +c \right )}\right ) a^{2}}{\left (a^{2}-b^{2}+c^{2}\right )^{\frac {5}{2}}}+\frac {\ln \left ({\mathrm e}^{x}+\frac {\left (a^{2}-b^{2}+c^{2}\right )^{\frac {5}{2}} a -a^{6}+3 a^{4} b^{2}-3 a^{4} c^{2}-3 a^{2} b^{4}+6 a^{2} b^{2} c^{2}-3 a^{2} c^{4}+b^{6}-3 b^{4} c^{2}+3 b^{2} c^{4}-c^{6}}{\left (a^{2}-b^{2}+c^{2}\right )^{\frac {5}{2}} \left (b +c \right )}\right ) b^{2}}{2 \left (a^{2}-b^{2}+c^{2}\right )^{\frac {5}{2}}}-\frac {\ln \left ({\mathrm e}^{x}+\frac {\left (a^{2}-b^{2}+c^{2}\right )^{\frac {5}{2}} a -a^{6}+3 a^{4} b^{2}-3 a^{4} c^{2}-3 a^{2} b^{4}+6 a^{2} b^{2} c^{2}-3 a^{2} c^{4}+b^{6}-3 b^{4} c^{2}+3 b^{2} c^{4}-c^{6}}{\left (a^{2}-b^{2}+c^{2}\right )^{\frac {5}{2}} \left (b +c \right )}\right ) c^{2}}{2 \left (a^{2}-b^{2}+c^{2}\right )^{\frac {5}{2}}}-\frac {\ln \left ({\mathrm e}^{x}+\frac {\left (a^{2}-b^{2}+c^{2}\right )^{\frac {5}{2}} a +a^{6}-3 a^{4} b^{2}+3 a^{4} c^{2}+3 a^{2} b^{4}-6 a^{2} b^{2} c^{2}+3 a^{2} c^{4}-b^{6}+3 b^{4} c^{2}-3 b^{2} c^{4}+c^{6}}{\left (a^{2}-b^{2}+c^{2}\right )^{\frac {5}{2}} \left (b +c \right )}\right ) a^{2}}{\left (a^{2}-b^{2}+c^{2}\right )^{\frac {5}{2}}}-\frac {\ln \left ({\mathrm e}^{x}+\frac {\left (a^{2}-b^{2}+c^{2}\right )^{\frac {5}{2}} a +a^{6}-3 a^{4} b^{2}+3 a^{4} c^{2}+3 a^{2} b^{4}-6 a^{2} b^{2} c^{2}+3 a^{2} c^{4}-b^{6}+3 b^{4} c^{2}-3 b^{2} c^{4}+c^{6}}{\left (a^{2}-b^{2}+c^{2}\right )^{\frac {5}{2}} \left (b +c \right )}\right ) b^{2}}{2 \left (a^{2}-b^{2}+c^{2}\right )^{\frac {5}{2}}}+\frac {\ln \left ({\mathrm e}^{x}+\frac {\left (a^{2}-b^{2}+c^{2}\right )^{\frac {5}{2}} a +a^{6}-3 a^{4} b^{2}+3 a^{4} c^{2}+3 a^{2} b^{4}-6 a^{2} b^{2} c^{2}+3 a^{2} c^{4}-b^{6}+3 b^{4} c^{2}-3 b^{2} c^{4}+c^{6}}{\left (a^{2}-b^{2}+c^{2}\right )^{\frac {5}{2}} \left (b +c \right )}\right ) c^{2}}{2 \left (a^{2}-b^{2}+c^{2}\right )^{\frac {5}{2}}}\) | \(973\) |
-2*(-1/2*(4*a^3*b-7*a^2*b^2+5*a^2*c^2+2*a*b^3-2*a*b*c^2+b^4-3*b^2*c^2+2*c^ 4)/(a-b)/(a^4-2*a^2*b^2+2*a^2*c^2+b^4-2*b^2*c^2+c^4)*tanh(1/2*x)^3-1/2*c*( 4*a^4-12*a^3*b+13*a^2*b^2-7*a^2*c^2-6*a*b^3+6*a*b*c^2+b^4+b^2*c^2-2*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^4*b-5*a^3*b^2+11*a^3*c^2-3*a^2*b^3-3*a^2*b*c^2+5*a*b^4-7*a*b^2*c^2 +2*a*c^4-b^5-b^3*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)+1/2*c*(4*a^4-3*a^2*b^2+a^2*c^2-b^4+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))/(tanh(1/2*x)^2* a-tanh(1/2*x)^2*b-2*c*tanh(1/2*x)-a-b)^2-(2*a^2+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+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. 3633 vs. \(2 (138) = 276\).
Time = 0.33 (sec) , antiderivative size = 7379, normalized size of antiderivative = 50.54 \[ \int \frac {1}{(a+b \cosh (x)+c \sinh (x))^3} \, dx=\text {Too large to display} \]
Timed out. \[ \int \frac {1}{(a+b \cosh (x)+c \sinh (x))^3} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {1}{(a+b \cosh (x)+c \sinh (x))^3} \, dx=\text {Exception raised: ValueError} \]
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. 304 vs. \(2 (138) = 276\).
Time = 0.27 (sec) , antiderivative size = 304, normalized size of antiderivative = 2.08 \[ \int \frac {1}{(a+b \cosh (x)+c \sinh (x))^3} \, dx=\frac {{\left (2 \, a^{2} + b^{2} - c^{2}\right )} \arctan \left (\frac {b e^{x} + c e^{x} + a}{\sqrt {-a^{2} + b^{2} - c^{2}}}\right )}{{\left (a^{4} - 2 \, a^{2} b^{2} + b^{4} + 2 \, a^{2} c^{2} - 2 \, b^{2} c^{2} + c^{4}\right )} \sqrt {-a^{2} + b^{2} - c^{2}}} + \frac {2 \, a^{2} b e^{\left (3 \, x\right )} + b^{3} e^{\left (3 \, x\right )} + 2 \, a^{2} c e^{\left (3 \, x\right )} + b^{2} c e^{\left (3 \, x\right )} - b c^{2} e^{\left (3 \, x\right )} - c^{3} e^{\left (3 \, x\right )} + 6 \, a^{3} e^{\left (2 \, x\right )} + 3 \, a b^{2} e^{\left (2 \, x\right )} - 3 \, a c^{2} e^{\left (2 \, x\right )} + 10 \, a^{2} b e^{x} - b^{3} e^{x} - 10 \, a^{2} c e^{x} + b^{2} c e^{x} + b c^{2} e^{x} - c^{3} e^{x} + 3 \, a b^{2} - 6 \, a b c + 3 \, a c^{2}}{{\left (a^{4} - 2 \, a^{2} b^{2} + b^{4} + 2 \, a^{2} c^{2} - 2 \, b^{2} c^{2} + c^{4}\right )} {\left (b e^{\left (2 \, x\right )} + c e^{\left (2 \, x\right )} + 2 \, a e^{x} + b - c\right )}^{2}} \]
(2*a^2 + b^2 - 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^2*b*e^(3*x) + b^3*e^(3*x) + 2*a^2*c*e^(3*x) + b^2*c*e^(3*x) - b*c ^2*e^(3*x) - c^3*e^(3*x) + 6*a^3*e^(2*x) + 3*a*b^2*e^(2*x) - 3*a*c^2*e^(2* x) + 10*a^2*b*e^x - b^3*e^x - 10*a^2*c*e^x + b^2*c*e^x + b*c^2*e^x - c^3*e ^x + 3*a*b^2 - 6*a*b*c + 3*a*c^2)/((a^4 - 2*a^2*b^2 + b^4 + 2*a^2*c^2 - 2* b^2*c^2 + c^4)*(b*e^(2*x) + c*e^(2*x) + 2*a*e^x + b - c)^2)
Timed out. \[ \int \frac {1}{(a+b \cosh (x)+c \sinh (x))^3} \, dx=\int \frac {1}{{\left (a+b\,\mathrm {cosh}\left (x\right )+c\,\mathrm {sinh}\left (x\right )\right )}^3} \,d x \]