Integrand size = 31, antiderivative size = 187 \[ \int \frac {A+B \cosh (d+e x)+C \sinh (d+e x)}{(a+b \cosh (d+e x))^3} \, dx=\frac {\left (2 a^2 A+A b^2-3 a b B\right ) \text {arctanh}\left (\frac {\sqrt {a-b} \tanh \left (\frac {1}{2} (d+e x)\right )}{\sqrt {a+b}}\right )}{(a-b)^{5/2} (a+b)^{5/2} e}-\frac {C}{2 b e (a+b \cosh (d+e x))^2}-\frac {(A b-a B) \sinh (d+e x)}{2 \left (a^2-b^2\right ) e (a+b \cosh (d+e x))^2}-\frac {\left (3 a A b-a^2 B-2 b^2 B\right ) \sinh (d+e x)}{2 \left (a^2-b^2\right )^2 e (a+b \cosh (d+e x))} \] Output:
(2*A*a^2+A*b^2-3*B*a*b)*arctanh((a-b)^(1/2)*tanh(1/2*e*x+1/2*d)/(a+b)^(1/2 ))/(a-b)^(5/2)/(a+b)^(5/2)/e-1/2*C/b/e/(a+b*cosh(e*x+d))^2-1/2*(A*b-B*a)*s inh(e*x+d)/(a^2-b^2)/e/(a+b*cosh(e*x+d))^2-1/2*(3*A*a*b-B*a^2-2*B*b^2)*sin h(e*x+d)/(a^2-b^2)^2/e/(a+b*cosh(e*x+d))
Time = 0.78 (sec) , antiderivative size = 175, normalized size of antiderivative = 0.94 \[ \int \frac {A+B \cosh (d+e x)+C \sinh (d+e x)}{(a+b \cosh (d+e x))^3} \, dx=\frac {-\frac {2 \left (2 a^2 A+A b^2-3 a b B\right ) \arctan \left (\frac {(a-b) \tanh \left (\frac {1}{2} (d+e x)\right )}{\sqrt {-a^2+b^2}}\right )}{\left (-a^2+b^2\right )^{5/2}}+\frac {\left (-3 a A b+a^2 B+2 b^2 B\right ) \sinh (d+e x)}{(a-b)^2 (a+b)^2 (a+b \cosh (d+e x))}+\frac {\left (-a^2+b^2\right ) C-b (A b-a B) \sinh (d+e x)}{(a-b) b (a+b) (a+b \cosh (d+e x))^2}}{2 e} \] Input:
Integrate[(A + B*Cosh[d + e*x] + C*Sinh[d + e*x])/(a + b*Cosh[d + e*x])^3, x]
Output:
((-2*(2*a^2*A + A*b^2 - 3*a*b*B)*ArcTan[((a - b)*Tanh[(d + e*x)/2])/Sqrt[- a^2 + b^2]])/(-a^2 + b^2)^(5/2) + ((-3*a*A*b + a^2*B + 2*b^2*B)*Sinh[d + e *x])/((a - b)^2*(a + b)^2*(a + b*Cosh[d + e*x])) + ((-a^2 + b^2)*C - b*(A* b - a*B)*Sinh[d + e*x])/((a - b)*b*(a + b)*(a + b*Cosh[d + e*x])^2))/(2*e)
Time = 0.86 (sec) , antiderivative size = 213, normalized size of antiderivative = 1.14, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.516, Rules used = {3042, 4877, 26, 3042, 26, 3147, 17, 3233, 25, 3042, 3233, 25, 27, 3042, 3138, 218}
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.
| \(\displaystyle \int \frac {A+B \cosh (d+e x)+C \sinh (d+e x)}{(a+b \cosh (d+e x))^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {A+B \cos (i d+i e x)-i C \sin (i d+i e x)}{(a+b \cos (i d+i e x))^3}dx\) |
| \(\Big \downarrow \) 4877 |
| \(\displaystyle \int \frac {A+B \cosh (d+e x)}{(a+b \cosh (d+e x))^3}dx-i C \int \frac {i \sinh (d+e x)}{(a+b \cosh (d+e x))^3}dx\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \int \frac {A+B \cosh (d+e x)}{(a+b \cosh (d+e x))^3}dx+C \int \frac {\sinh (d+e x)}{(a+b \cosh (d+e x))^3}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {A+B \sin \left (i d+i e x+\frac {\pi }{2}\right )}{\left (a+b \sin \left (i d+i e x+\frac {\pi }{2}\right )\right )^3}dx+C \int -\frac {i \cos \left (i d+i e x-\frac {\pi }{2}\right )}{\left (a-b \sin \left (i d+i e x-\frac {\pi }{2}\right )\right )^3}dx\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \int \frac {A+B \sin \left (i d+i e x+\frac {\pi }{2}\right )}{\left (a+b \sin \left (i d+i e x+\frac {\pi }{2}\right )\right )^3}dx-i C \int \frac {\cos \left (\frac {1}{2} (2 i d-\pi )+i e x\right )}{\left (a-b \sin \left (\frac {1}{2} (2 i d-\pi )+i e x\right )\right )^3}dx\) |
| \(\Big \downarrow \) 3147 |
| \(\displaystyle \frac {C \int \frac {1}{(a+b \cosh (d+e x))^3}d(b \cosh (d+e x))}{b e}+\int \frac {A+B \sin \left (i d+i e x+\frac {\pi }{2}\right )}{\left (a+b \sin \left (i d+i e x+\frac {\pi }{2}\right )\right )^3}dx\) |
| \(\Big \downarrow \) 17 |
| \(\displaystyle -\frac {C}{2 b e (a+b \cosh (d+e x))^2}+\int \frac {A+B \sin \left (i d+i e x+\frac {\pi }{2}\right )}{\left (a+b \sin \left (i d+i e x+\frac {\pi }{2}\right )\right )^3}dx\) |
| \(\Big \downarrow \) 3233 |
| \(\displaystyle -\frac {\int -\frac {2 (a A-b B)-(A b-a B) \cosh (d+e x)}{(a+b \cosh (d+e x))^2}dx}{2 \left (a^2-b^2\right )}-\frac {(A b-a B) \sinh (d+e x)}{2 e \left (a^2-b^2\right ) (a+b \cosh (d+e x))^2}-\frac {C}{2 b e (a+b \cosh (d+e x))^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {2 (a A-b B)-(A b-a B) \cosh (d+e x)}{(a+b \cosh (d+e x))^2}dx}{2 \left (a^2-b^2\right )}-\frac {(A b-a B) \sinh (d+e x)}{2 e \left (a^2-b^2\right ) (a+b \cosh (d+e x))^2}-\frac {C}{2 b e (a+b \cosh (d+e x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {2 (a A-b B)+(a B-A b) \sin \left (i d+i e x+\frac {\pi }{2}\right )}{\left (a+b \sin \left (i d+i e x+\frac {\pi }{2}\right )\right )^2}dx}{2 \left (a^2-b^2\right )}-\frac {(A b-a B) \sinh (d+e x)}{2 e \left (a^2-b^2\right ) (a+b \cosh (d+e x))^2}-\frac {C}{2 b e (a+b \cosh (d+e x))^2}\) |
| \(\Big \downarrow \) 3233 |
| \(\displaystyle \frac {-\frac {\int -\frac {2 A a^2-3 b B a+A b^2}{a+b \cosh (d+e x)}dx}{a^2-b^2}-\frac {\left (a^2 (-B)+3 a A b-2 b^2 B\right ) \sinh (d+e x)}{e \left (a^2-b^2\right ) (a+b \cosh (d+e x))}}{2 \left (a^2-b^2\right )}-\frac {(A b-a B) \sinh (d+e x)}{2 e \left (a^2-b^2\right ) (a+b \cosh (d+e x))^2}-\frac {C}{2 b e (a+b \cosh (d+e x))^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\int \frac {2 A a^2-3 b B a+A b^2}{a+b \cosh (d+e x)}dx}{a^2-b^2}-\frac {\left (a^2 (-B)+3 a A b-2 b^2 B\right ) \sinh (d+e x)}{e \left (a^2-b^2\right ) (a+b \cosh (d+e x))}}{2 \left (a^2-b^2\right )}-\frac {(A b-a B) \sinh (d+e x)}{2 e \left (a^2-b^2\right ) (a+b \cosh (d+e x))^2}-\frac {C}{2 b e (a+b \cosh (d+e x))^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\left (2 a^2 A-3 a b B+A b^2\right ) \int \frac {1}{a+b \cosh (d+e x)}dx}{a^2-b^2}-\frac {\left (a^2 (-B)+3 a A b-2 b^2 B\right ) \sinh (d+e x)}{e \left (a^2-b^2\right ) (a+b \cosh (d+e x))}}{2 \left (a^2-b^2\right )}-\frac {(A b-a B) \sinh (d+e x)}{2 e \left (a^2-b^2\right ) (a+b \cosh (d+e x))^2}-\frac {C}{2 b e (a+b \cosh (d+e x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {\left (a^2 (-B)+3 a A b-2 b^2 B\right ) \sinh (d+e x)}{e \left (a^2-b^2\right ) (a+b \cosh (d+e x))}+\frac {\left (2 a^2 A-3 a b B+A b^2\right ) \int \frac {1}{a+b \sin \left (i d+i e x+\frac {\pi }{2}\right )}dx}{a^2-b^2}}{2 \left (a^2-b^2\right )}-\frac {(A b-a B) \sinh (d+e x)}{2 e \left (a^2-b^2\right ) (a+b \cosh (d+e x))^2}-\frac {C}{2 b e (a+b \cosh (d+e x))^2}\) |
| \(\Big \downarrow \) 3138 |
| \(\displaystyle \frac {-\frac {\left (a^2 (-B)+3 a A b-2 b^2 B\right ) \sinh (d+e x)}{e \left (a^2-b^2\right ) (a+b \cosh (d+e x))}-\frac {2 i \left (2 a^2 A-3 a b B+A b^2\right ) \int \frac {1}{-\left ((a-b) \tanh ^2\left (\frac {1}{2} (d+e x)\right )\right )+a+b}d\left (i \tanh \left (\frac {1}{2} (d+e x)\right )\right )}{e \left (a^2-b^2\right )}}{2 \left (a^2-b^2\right )}-\frac {(A b-a B) \sinh (d+e x)}{2 e \left (a^2-b^2\right ) (a+b \cosh (d+e x))^2}-\frac {C}{2 b e (a+b \cosh (d+e x))^2}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {\frac {2 \left (2 a^2 A-3 a b B+A b^2\right ) \text {arctanh}\left (\frac {\sqrt {a-b} \tanh \left (\frac {1}{2} (d+e x)\right )}{\sqrt {a+b}}\right )}{e \sqrt {a-b} \sqrt {a+b} \left (a^2-b^2\right )}-\frac {\left (a^2 (-B)+3 a A b-2 b^2 B\right ) \sinh (d+e x)}{e \left (a^2-b^2\right ) (a+b \cosh (d+e x))}}{2 \left (a^2-b^2\right )}-\frac {(A b-a B) \sinh (d+e x)}{2 e \left (a^2-b^2\right ) (a+b \cosh (d+e x))^2}-\frac {C}{2 b e (a+b \cosh (d+e x))^2}\) |
Input:
Int[(A + B*Cosh[d + e*x] + C*Sinh[d + e*x])/(a + b*Cosh[d + e*x])^3,x]
Output:
-1/2*C/(b*e*(a + b*Cosh[d + e*x])^2) - ((A*b - a*B)*Sinh[d + e*x])/(2*(a^2 - b^2)*e*(a + b*Cosh[d + e*x])^2) + ((2*(2*a^2*A + A*b^2 - 3*a*b*B)*ArcTa nh[(Sqrt[a - b]*Tanh[(d + e*x)/2])/Sqrt[a + b]])/(Sqrt[a - b]*Sqrt[a + b]* (a^2 - b^2)*e) - ((3*a*A*b - a^2*B - 2*b^2*B)*Sinh[d + e*x])/((a^2 - b^2)* e*(a + b*Cosh[d + e*x])))/(2*(a^2 - b^2))
Int[(c_.)*((a_.) + (b_.)*(x_))^(m_.), x_Symbol] :> Simp[c*((a + b*x)^(m + 1 )/(b*(m + 1))), x] /; FreeQ[{a, b, c, m}, x] && NeQ[m, -1]
Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a]) I nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
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]
Int[cos[(e_.) + (f_.)*(x_)]^(p_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m _.), x_Symbol] :> Simp[1/(b^p*f) Subst[Int[(a + x)^m*(b^2 - x^2)^((p - 1) /2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x] && IntegerQ[(p - 1)/2] && NeQ[a^2 - b^2, 0]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(-(b*c - a*d))*Cos[e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(f*(m + 1)*(a^2 - b^2))), x] + Simp[1/((m + 1)*(a^2 - b^2)) Int[(a + b*Sin[e + f*x])^(m + 1)*Simp[(a*c - b*d)*(m + 1) - (b*c - a*d)*( m + 2)*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && LtQ[m, -1] && IntegerQ[2*m]
Int[(u_)*((v_) + (d_.)*(F_)[(c_.)*((a_.) + (b_.)*(x_))]^(n_.)), x_Symbol] : > With[{e = FreeFactors[Cos[c*(a + b*x)], x]}, Int[ActivateTrig[u*v], x] + Simp[d Int[ActivateTrig[u]*Sin[c*(a + b*x)]^n, x], x] /; FunctionOfQ[Cos[ c*(a + b*x)]/e, u, x]] /; FreeQ[{a, b, c, d}, x] && !FreeQ[v, x] && Intege rQ[(n - 1)/2] && NonsumQ[u] && (EqQ[F, Sin] || EqQ[F, sin])
Time = 6.41 (sec) , antiderivative size = 273, normalized size of antiderivative = 1.46
| method | result | size |
| derivativedivides | \(\frac {-\frac {2 \left (-\frac {\left (4 A a b +A \,b^{2}-2 B \,a^{2}-B a b -2 B \,b^{2}\right ) \tanh \left (\frac {e x}{2}+\frac {d}{2}\right )^{3}}{2 \left (a -b \right ) \left (a^{2}+2 a b +b^{2}\right )}+\frac {C \tanh \left (\frac {e x}{2}+\frac {d}{2}\right )^{2}}{a -b}+\frac {\left (4 A a b -A \,b^{2}-2 B \,a^{2}+B a b -2 B \,b^{2}\right ) \tanh \left (\frac {e x}{2}+\frac {d}{2}\right )}{2 \left (a +b \right ) \left (a^{2}-2 a b +b^{2}\right )}-\frac {a C}{a^{2}-2 a b +b^{2}}\right )}{\left (a \tanh \left (\frac {e x}{2}+\frac {d}{2}\right )^{2}-b \tanh \left (\frac {e x}{2}+\frac {d}{2}\right )^{2}-a -b \right )^{2}}+\frac {\left (2 A \,a^{2}+A \,b^{2}-3 B a b \right ) \operatorname {arctanh}\left (\frac {\left (a -b \right ) \tanh \left (\frac {e x}{2}+\frac {d}{2}\right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{\left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) \sqrt {\left (a +b \right ) \left (a -b \right )}}}{e}\) | \(273\) |
| default | \(\frac {-\frac {2 \left (-\frac {\left (4 A a b +A \,b^{2}-2 B \,a^{2}-B a b -2 B \,b^{2}\right ) \tanh \left (\frac {e x}{2}+\frac {d}{2}\right )^{3}}{2 \left (a -b \right ) \left (a^{2}+2 a b +b^{2}\right )}+\frac {C \tanh \left (\frac {e x}{2}+\frac {d}{2}\right )^{2}}{a -b}+\frac {\left (4 A a b -A \,b^{2}-2 B \,a^{2}+B a b -2 B \,b^{2}\right ) \tanh \left (\frac {e x}{2}+\frac {d}{2}\right )}{2 \left (a +b \right ) \left (a^{2}-2 a b +b^{2}\right )}-\frac {a C}{a^{2}-2 a b +b^{2}}\right )}{\left (a \tanh \left (\frac {e x}{2}+\frac {d}{2}\right )^{2}-b \tanh \left (\frac {e x}{2}+\frac {d}{2}\right )^{2}-a -b \right )^{2}}+\frac {\left (2 A \,a^{2}+A \,b^{2}-3 B a b \right ) \operatorname {arctanh}\left (\frac {\left (a -b \right ) \tanh \left (\frac {e x}{2}+\frac {d}{2}\right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{\left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) \sqrt {\left (a +b \right ) \left (a -b \right )}}}{e}\) | \(273\) |
| risch | \(\frac {2 A \,a^{2} b^{2} {\mathrm e}^{3 e x +3 d}+A \,b^{4} {\mathrm e}^{3 e x +3 d}-3 B a \,b^{3} {\mathrm e}^{3 e x +3 d}+6 A \,a^{3} b \,{\mathrm e}^{2 e x +2 d}+3 A a \,b^{3} {\mathrm e}^{2 e x +2 d}-2 B \,a^{4} {\mathrm e}^{2 e x +2 d}-5 B \,a^{2} b^{2} {\mathrm e}^{2 e x +2 d}-2 B \,b^{4} {\mathrm e}^{2 e x +2 d}-2 C \,a^{4} {\mathrm e}^{2 e x +2 d}+4 C \,a^{2} b^{2} {\mathrm e}^{2 e x +2 d}-2 C \,b^{4} {\mathrm e}^{2 e x +2 d}+10 A \,a^{2} b^{2} {\mathrm e}^{e x +d}-A \,b^{4} {\mathrm e}^{e x +d}-4 B \,a^{3} b \,{\mathrm e}^{e x +d}-5 B a \,b^{3} {\mathrm e}^{e x +d}+3 A a \,b^{3}-B \,a^{2} b^{2}-2 B \,b^{4}}{b e \left (a^{2}-b^{2}\right )^{2} \left (b \,{\mathrm e}^{2 e x +2 d}+2 a \,{\mathrm e}^{e x +d}+b \right )^{2}}+\frac {\ln \left ({\mathrm e}^{e x +d}+\frac {a \sqrt {a^{2}-b^{2}}-a^{2}+b^{2}}{b \sqrt {a^{2}-b^{2}}}\right ) A \,a^{2}}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right )^{2} \left (a -b \right )^{2} e}+\frac {\ln \left ({\mathrm e}^{e x +d}+\frac {a \sqrt {a^{2}-b^{2}}-a^{2}+b^{2}}{b \sqrt {a^{2}-b^{2}}}\right ) A \,b^{2}}{2 \sqrt {a^{2}-b^{2}}\, \left (a +b \right )^{2} \left (a -b \right )^{2} e}-\frac {3 \ln \left ({\mathrm e}^{e x +d}+\frac {a \sqrt {a^{2}-b^{2}}-a^{2}+b^{2}}{b \sqrt {a^{2}-b^{2}}}\right ) B a b}{2 \sqrt {a^{2}-b^{2}}\, \left (a +b \right )^{2} \left (a -b \right )^{2} e}-\frac {\ln \left ({\mathrm e}^{e x +d}+\frac {a \sqrt {a^{2}-b^{2}}+a^{2}-b^{2}}{b \sqrt {a^{2}-b^{2}}}\right ) A \,a^{2}}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right )^{2} \left (a -b \right )^{2} e}-\frac {\ln \left ({\mathrm e}^{e x +d}+\frac {a \sqrt {a^{2}-b^{2}}+a^{2}-b^{2}}{b \sqrt {a^{2}-b^{2}}}\right ) A \,b^{2}}{2 \sqrt {a^{2}-b^{2}}\, \left (a +b \right )^{2} \left (a -b \right )^{2} e}+\frac {3 \ln \left ({\mathrm e}^{e x +d}+\frac {a \sqrt {a^{2}-b^{2}}+a^{2}-b^{2}}{b \sqrt {a^{2}-b^{2}}}\right ) B a b}{2 \sqrt {a^{2}-b^{2}}\, \left (a +b \right )^{2} \left (a -b \right )^{2} e}\) | \(755\) |
Input:
int((A+B*cosh(e*x+d)+C*sinh(e*x+d))/(a+b*cosh(e*x+d))^3,x,method=_RETURNVE RBOSE)
Output:
1/e*(-2*(-1/2*(4*A*a*b+A*b^2-2*B*a^2-B*a*b-2*B*b^2)/(a-b)/(a^2+2*a*b+b^2)* tanh(1/2*e*x+1/2*d)^3+C/(a-b)*tanh(1/2*e*x+1/2*d)^2+1/2*(4*A*a*b-A*b^2-2*B *a^2+B*a*b-2*B*b^2)/(a+b)/(a^2-2*a*b+b^2)*tanh(1/2*e*x+1/2*d)-a*C/(a^2-2*a *b+b^2))/(a*tanh(1/2*e*x+1/2*d)^2-b*tanh(1/2*e*x+1/2*d)^2-a-b)^2+(2*A*a^2+ A*b^2-3*B*a*b)/(a^4-2*a^2*b^2+b^4)/((a+b)*(a-b))^(1/2)*arctanh((a-b)*tanh( 1/2*e*x+1/2*d)/((a+b)*(a-b))^(1/2)))
Leaf count of result is larger than twice the leaf count of optimal. 1760 vs. \(2 (171) = 342\).
Time = 0.15 (sec) , antiderivative size = 3636, normalized size of antiderivative = 19.44 \[ \int \frac {A+B \cosh (d+e x)+C \sinh (d+e x)}{(a+b \cosh (d+e x))^3} \, dx=\text {Too large to display} \] Input:
integrate((A+B*cosh(e*x+d)+C*sinh(e*x+d))/(a+b*cosh(e*x+d))^3,x, algorithm ="fricas")
Output:
Too large to include
Timed out. \[ \int \frac {A+B \cosh (d+e x)+C \sinh (d+e x)}{(a+b \cosh (d+e x))^3} \, dx=\text {Timed out} \] Input:
integrate((A+B*cosh(e*x+d)+C*sinh(e*x+d))/(a+b*cosh(e*x+d))**3,x)
Output:
Timed out
Exception generated. \[ \int \frac {A+B \cosh (d+e x)+C \sinh (d+e x)}{(a+b \cosh (d+e x))^3} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((A+B*cosh(e*x+d)+C*sinh(e*x+d))/(a+b*cosh(e*x+d))^3,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
Leaf count of result is larger than twice the leaf count of optimal. 370 vs. \(2 (171) = 342\).
Time = 0.15 (sec) , antiderivative size = 370, normalized size of antiderivative = 1.98 \[ \int \frac {A+B \cosh (d+e x)+C \sinh (d+e x)}{(a+b \cosh (d+e x))^3} \, dx=\frac {\frac {{\left (2 \, A a^{2} - 3 \, B a b + A b^{2}\right )} \arctan \left (\frac {b e^{\left (e x + d\right )} + a}{\sqrt {-a^{2} + b^{2}}}\right )}{{\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} \sqrt {-a^{2} + b^{2}}} + \frac {2 \, A a^{2} b^{2} e^{\left (3 \, e x + 3 \, d\right )} - 3 \, B a b^{3} e^{\left (3 \, e x + 3 \, d\right )} + A b^{4} e^{\left (3 \, e x + 3 \, d\right )} - 2 \, B a^{4} e^{\left (2 \, e x + 2 \, d\right )} - 2 \, C a^{4} e^{\left (2 \, e x + 2 \, d\right )} + 6 \, A a^{3} b e^{\left (2 \, e x + 2 \, d\right )} - 5 \, B a^{2} b^{2} e^{\left (2 \, e x + 2 \, d\right )} + 4 \, C a^{2} b^{2} e^{\left (2 \, e x + 2 \, d\right )} + 3 \, A a b^{3} e^{\left (2 \, e x + 2 \, d\right )} - 2 \, B b^{4} e^{\left (2 \, e x + 2 \, d\right )} - 2 \, C b^{4} e^{\left (2 \, e x + 2 \, d\right )} - 4 \, B a^{3} b e^{\left (e x + d\right )} + 10 \, A a^{2} b^{2} e^{\left (e x + d\right )} - 5 \, B a b^{3} e^{\left (e x + d\right )} - A b^{4} e^{\left (e x + d\right )} - B a^{2} b^{2} + 3 \, A a b^{3} - 2 \, B b^{4}}{{\left (a^{4} b - 2 \, a^{2} b^{3} + b^{5}\right )} {\left (b e^{\left (2 \, e x + 2 \, d\right )} + 2 \, a e^{\left (e x + d\right )} + b\right )}^{2}}}{e} \] Input:
integrate((A+B*cosh(e*x+d)+C*sinh(e*x+d))/(a+b*cosh(e*x+d))^3,x, algorithm ="giac")
Output:
((2*A*a^2 - 3*B*a*b + A*b^2)*arctan((b*e^(e*x + d) + a)/sqrt(-a^2 + b^2))/ ((a^4 - 2*a^2*b^2 + b^4)*sqrt(-a^2 + b^2)) + (2*A*a^2*b^2*e^(3*e*x + 3*d) - 3*B*a*b^3*e^(3*e*x + 3*d) + A*b^4*e^(3*e*x + 3*d) - 2*B*a^4*e^(2*e*x + 2 *d) - 2*C*a^4*e^(2*e*x + 2*d) + 6*A*a^3*b*e^(2*e*x + 2*d) - 5*B*a^2*b^2*e^ (2*e*x + 2*d) + 4*C*a^2*b^2*e^(2*e*x + 2*d) + 3*A*a*b^3*e^(2*e*x + 2*d) - 2*B*b^4*e^(2*e*x + 2*d) - 2*C*b^4*e^(2*e*x + 2*d) - 4*B*a^3*b*e^(e*x + d) + 10*A*a^2*b^2*e^(e*x + d) - 5*B*a*b^3*e^(e*x + d) - A*b^4*e^(e*x + d) - B *a^2*b^2 + 3*A*a*b^3 - 2*B*b^4)/((a^4*b - 2*a^2*b^3 + b^5)*(b*e^(2*e*x + 2 *d) + 2*a*e^(e*x + d) + b)^2))/e
Timed out. \[ \int \frac {A+B \cosh (d+e x)+C \sinh (d+e x)}{(a+b \cosh (d+e x))^3} \, dx=\int \frac {A+B\,\mathrm {cosh}\left (d+e\,x\right )+C\,\mathrm {sinh}\left (d+e\,x\right )}{{\left (a+b\,\mathrm {cosh}\left (d+e\,x\right )\right )}^3} \,d x \] Input:
int((A + B*cosh(d + e*x) + C*sinh(d + e*x))/(a + b*cosh(d + e*x))^3,x)
Output:
int((A + B*cosh(d + e*x) + C*sinh(d + e*x))/(a + b*cosh(d + e*x))^3, x)
Time = 0.26 (sec) , antiderivative size = 688, normalized size of antiderivative = 3.68 \[ \int \frac {A+B \cosh (d+e x)+C \sinh (d+e x)}{(a+b \cosh (d+e x))^3} \, dx=\frac {-4 e^{4 e x +4 d} \sqrt {-a^{2}+b^{2}}\, \mathit {atan} \left (\frac {e^{e x +d} b +a}{\sqrt {-a^{2}+b^{2}}}\right ) a \,b^{3}-16 e^{3 e x +3 d} \sqrt {-a^{2}+b^{2}}\, \mathit {atan} \left (\frac {e^{e x +d} b +a}{\sqrt {-a^{2}+b^{2}}}\right ) a^{2} b^{2}-16 e^{2 e x +2 d} \sqrt {-a^{2}+b^{2}}\, \mathit {atan} \left (\frac {e^{e x +d} b +a}{\sqrt {-a^{2}+b^{2}}}\right ) a^{3} b -8 e^{2 e x +2 d} \sqrt {-a^{2}+b^{2}}\, \mathit {atan} \left (\frac {e^{e x +d} b +a}{\sqrt {-a^{2}+b^{2}}}\right ) a \,b^{3}-16 e^{e x +d} \sqrt {-a^{2}+b^{2}}\, \mathit {atan} \left (\frac {e^{e x +d} b +a}{\sqrt {-a^{2}+b^{2}}}\right ) a^{2} b^{2}-4 \sqrt {-a^{2}+b^{2}}\, \mathit {atan} \left (\frac {e^{e x +d} b +a}{\sqrt {-a^{2}+b^{2}}}\right ) a \,b^{3}-e^{4 e x +4 d} a^{2} b^{3}+e^{4 e x +4 d} b^{5}+4 e^{2 e x +2 d} a^{4} b -4 e^{2 e x +2 d} a^{4} c -2 e^{2 e x +2 d} a^{2} b^{3}+8 e^{2 e x +2 d} a^{2} b^{2} c -2 e^{2 e x +2 d} b^{5}-4 e^{2 e x +2 d} b^{4} c +8 e^{e x +d} a^{3} b^{2}-8 e^{e x +d} a \,b^{4}+3 a^{2} b^{3}-3 b^{5}}{2 b e \left (e^{4 e x +4 d} a^{4} b^{2}-2 e^{4 e x +4 d} a^{2} b^{4}+e^{4 e x +4 d} b^{6}+4 e^{3 e x +3 d} a^{5} b -8 e^{3 e x +3 d} a^{3} b^{3}+4 e^{3 e x +3 d} a \,b^{5}+4 e^{2 e x +2 d} a^{6}-6 e^{2 e x +2 d} a^{4} b^{2}+2 e^{2 e x +2 d} b^{6}+4 e^{e x +d} a^{5} b -8 e^{e x +d} a^{3} b^{3}+4 e^{e x +d} a \,b^{5}+a^{4} b^{2}-2 a^{2} b^{4}+b^{6}\right )} \] Input:
int((A+B*cosh(e*x+d)+C*sinh(e*x+d))/(a+b*cosh(e*x+d))^3,x)
Output:
( - 4*e**(4*d + 4*e*x)*sqrt( - a**2 + b**2)*atan((e**(d + e*x)*b + a)/sqrt ( - a**2 + b**2))*a*b**3 - 16*e**(3*d + 3*e*x)*sqrt( - a**2 + b**2)*atan(( e**(d + e*x)*b + a)/sqrt( - a**2 + b**2))*a**2*b**2 - 16*e**(2*d + 2*e*x)* sqrt( - a**2 + b**2)*atan((e**(d + e*x)*b + a)/sqrt( - a**2 + b**2))*a**3* b - 8*e**(2*d + 2*e*x)*sqrt( - a**2 + b**2)*atan((e**(d + e*x)*b + a)/sqrt ( - a**2 + b**2))*a*b**3 - 16*e**(d + e*x)*sqrt( - a**2 + b**2)*atan((e**( d + e*x)*b + a)/sqrt( - a**2 + b**2))*a**2*b**2 - 4*sqrt( - a**2 + b**2)*a tan((e**(d + e*x)*b + a)/sqrt( - a**2 + b**2))*a*b**3 - e**(4*d + 4*e*x)*a **2*b**3 + e**(4*d + 4*e*x)*b**5 + 4*e**(2*d + 2*e*x)*a**4*b - 4*e**(2*d + 2*e*x)*a**4*c - 2*e**(2*d + 2*e*x)*a**2*b**3 + 8*e**(2*d + 2*e*x)*a**2*b* *2*c - 2*e**(2*d + 2*e*x)*b**5 - 4*e**(2*d + 2*e*x)*b**4*c + 8*e**(d + e*x )*a**3*b**2 - 8*e**(d + e*x)*a*b**4 + 3*a**2*b**3 - 3*b**5)/(2*b*e*(e**(4* d + 4*e*x)*a**4*b**2 - 2*e**(4*d + 4*e*x)*a**2*b**4 + e**(4*d + 4*e*x)*b** 6 + 4*e**(3*d + 3*e*x)*a**5*b - 8*e**(3*d + 3*e*x)*a**3*b**3 + 4*e**(3*d + 3*e*x)*a*b**5 + 4*e**(2*d + 2*e*x)*a**6 - 6*e**(2*d + 2*e*x)*a**4*b**2 + 2*e**(2*d + 2*e*x)*b**6 + 4*e**(d + e*x)*a**5*b - 8*e**(d + e*x)*a**3*b**3 + 4*e**(d + e*x)*a*b**5 + a**4*b**2 - 2*a**2*b**4 + b**6))