Integrand size = 15, antiderivative size = 197 \[ \int \frac {A+B \cosh (x)}{(a+b \cosh (x))^4} \, dx=\frac {\left (2 a^3 A+3 a A b^2-4 a^2 b B-b^3 B\right ) \text {arctanh}\left (\frac {\sqrt {a-b} \tanh \left (\frac {x}{2}\right )}{\sqrt {a+b}}\right )}{(a-b)^{7/2} (a+b)^{7/2}}-\frac {(A b-a B) \sinh (x)}{3 \left (a^2-b^2\right ) (a+b \cosh (x))^3}-\frac {\left (5 a A b-2 a^2 B-3 b^2 B\right ) \sinh (x)}{6 \left (a^2-b^2\right )^2 (a+b \cosh (x))^2}-\frac {\left (11 a^2 A b+4 A b^3-2 a^3 B-13 a b^2 B\right ) \sinh (x)}{6 \left (a^2-b^2\right )^3 (a+b \cosh (x))} \] Output:
(2*A*a^3+3*A*a*b^2-4*B*a^2*b-B*b^3)*arctanh((a-b)^(1/2)*tanh(1/2*x)/(a+b)^ (1/2))/(a-b)^(7/2)/(a+b)^(7/2)-1/3*(A*b-B*a)*sinh(x)/(a^2-b^2)/(a+b*cosh(x ))^3-1/6*(5*A*a*b-2*B*a^2-3*B*b^2)*sinh(x)/(a^2-b^2)^2/(a+b*cosh(x))^2-1/6 *(11*A*a^2*b+4*A*b^3-2*B*a^3-13*B*a*b^2)*sinh(x)/(a^2-b^2)^3/(a+b*cosh(x))
Time = 0.49 (sec) , antiderivative size = 196, normalized size of antiderivative = 0.99 \[ \int \frac {A+B \cosh (x)}{(a+b \cosh (x))^4} \, dx=\frac {1}{6} \left (\frac {6 \left (2 a^3 A+3 a A b^2-4 a^2 b B-b^3 B\right ) \arctan \left (\frac {(a-b) \tanh \left (\frac {x}{2}\right )}{\sqrt {-a^2+b^2}}\right )}{\left (-a^2+b^2\right )^{7/2}}+\frac {2 (-A b+a B) \sinh (x)}{(a-b) (a+b) (a+b \cosh (x))^3}+\frac {\left (-5 a A b+2 a^2 B+3 b^2 B\right ) \sinh (x)}{(a-b)^2 (a+b)^2 (a+b \cosh (x))^2}+\frac {\left (-11 a^2 A b-4 A b^3+2 a^3 B+13 a b^2 B\right ) \sinh (x)}{(a-b)^3 (a+b)^3 (a+b \cosh (x))}\right ) \] Input:
Integrate[(A + B*Cosh[x])/(a + b*Cosh[x])^4,x]
Output:
((6*(2*a^3*A + 3*a*A*b^2 - 4*a^2*b*B - b^3*B)*ArcTan[((a - b)*Tanh[x/2])/S qrt[-a^2 + b^2]])/(-a^2 + b^2)^(7/2) + (2*(-(A*b) + a*B)*Sinh[x])/((a - b) *(a + b)*(a + b*Cosh[x])^3) + ((-5*a*A*b + 2*a^2*B + 3*b^2*B)*Sinh[x])/((a - b)^2*(a + b)^2*(a + b*Cosh[x])^2) + ((-11*a^2*A*b - 4*A*b^3 + 2*a^3*B + 13*a*b^2*B)*Sinh[x])/((a - b)^3*(a + b)^3*(a + b*Cosh[x])))/6
Time = 0.95 (sec) , antiderivative size = 239, normalized size of antiderivative = 1.21, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.800, Rules used = {3042, 3233, 25, 3042, 3233, 25, 3042, 3233, 27, 3042, 3138, 221}
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.
| \(\displaystyle \int \frac {A+B \cosh (x)}{(a+b \cosh (x))^4} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {A+B \sin \left (\frac {\pi }{2}+i x\right )}{\left (a+b \sin \left (\frac {\pi }{2}+i x\right )\right )^4}dx\) |
| \(\Big \downarrow \) 3233 |
| \(\displaystyle -\frac {\int -\frac {3 (a A-b B)-2 (A b-a B) \cosh (x)}{(a+b \cosh (x))^3}dx}{3 \left (a^2-b^2\right )}-\frac {\sinh (x) (A b-a B)}{3 \left (a^2-b^2\right ) (a+b \cosh (x))^3}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {3 (a A-b B)-2 (A b-a B) \cosh (x)}{(a+b \cosh (x))^3}dx}{3 \left (a^2-b^2\right )}-\frac {\sinh (x) (A b-a B)}{3 \left (a^2-b^2\right ) (a+b \cosh (x))^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\sinh (x) (A b-a B)}{3 \left (a^2-b^2\right ) (a+b \cosh (x))^3}+\frac {\int \frac {3 (a A-b B)-2 (A b-a B) \sin \left (i x+\frac {\pi }{2}\right )}{\left (a+b \sin \left (i x+\frac {\pi }{2}\right )\right )^3}dx}{3 \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3233 |
| \(\displaystyle \frac {-\frac {\int -\frac {2 \left (3 A a^2-5 b B a+2 A b^2\right )-\left (-2 B a^2+5 A b a-3 b^2 B\right ) \cosh (x)}{(a+b \cosh (x))^2}dx}{2 \left (a^2-b^2\right )}-\frac {\sinh (x) \left (-2 a^2 B+5 a A b-3 b^2 B\right )}{2 \left (a^2-b^2\right ) (a+b \cosh (x))^2}}{3 \left (a^2-b^2\right )}-\frac {\sinh (x) (A b-a B)}{3 \left (a^2-b^2\right ) (a+b \cosh (x))^3}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\int \frac {2 \left (3 A a^2-5 b B a+2 A b^2\right )-\left (-2 B a^2+5 A b a-3 b^2 B\right ) \cosh (x)}{(a+b \cosh (x))^2}dx}{2 \left (a^2-b^2\right )}-\frac {\sinh (x) \left (-2 a^2 B+5 a A b-3 b^2 B\right )}{2 \left (a^2-b^2\right ) (a+b \cosh (x))^2}}{3 \left (a^2-b^2\right )}-\frac {\sinh (x) (A b-a B)}{3 \left (a^2-b^2\right ) (a+b \cosh (x))^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\sinh (x) (A b-a B)}{3 \left (a^2-b^2\right ) (a+b \cosh (x))^3}+\frac {-\frac {\sinh (x) \left (-2 a^2 B+5 a A b-3 b^2 B\right )}{2 \left (a^2-b^2\right ) (a+b \cosh (x))^2}+\frac {\int \frac {2 \left (3 A a^2-5 b B a+2 A b^2\right )+\left (2 B a^2-5 A b a+3 b^2 B\right ) \sin \left (i x+\frac {\pi }{2}\right )}{\left (a+b \sin \left (i x+\frac {\pi }{2}\right )\right )^2}dx}{2 \left (a^2-b^2\right )}}{3 \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3233 |
| \(\displaystyle \frac {\frac {-\frac {\int -\frac {3 \left (2 A a^3-4 b B a^2+3 A b^2 a-b^3 B\right )}{a+b \cosh (x)}dx}{a^2-b^2}-\frac {\sinh (x) \left (-2 a^3 B+11 a^2 A b-13 a b^2 B+4 A b^3\right )}{\left (a^2-b^2\right ) (a+b \cosh (x))}}{2 \left (a^2-b^2\right )}-\frac {\sinh (x) \left (-2 a^2 B+5 a A b-3 b^2 B\right )}{2 \left (a^2-b^2\right ) (a+b \cosh (x))^2}}{3 \left (a^2-b^2\right )}-\frac {\sinh (x) (A b-a B)}{3 \left (a^2-b^2\right ) (a+b \cosh (x))^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {3 \left (2 a^3 A-4 a^2 b B+3 a A b^2-b^3 B\right ) \int \frac {1}{a+b \cosh (x)}dx}{a^2-b^2}-\frac {\sinh (x) \left (-2 a^3 B+11 a^2 A b-13 a b^2 B+4 A b^3\right )}{\left (a^2-b^2\right ) (a+b \cosh (x))}}{2 \left (a^2-b^2\right )}-\frac {\sinh (x) \left (-2 a^2 B+5 a A b-3 b^2 B\right )}{2 \left (a^2-b^2\right ) (a+b \cosh (x))^2}}{3 \left (a^2-b^2\right )}-\frac {\sinh (x) (A b-a B)}{3 \left (a^2-b^2\right ) (a+b \cosh (x))^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\sinh (x) (A b-a B)}{3 \left (a^2-b^2\right ) (a+b \cosh (x))^3}+\frac {-\frac {\sinh (x) \left (-2 a^2 B+5 a A b-3 b^2 B\right )}{2 \left (a^2-b^2\right ) (a+b \cosh (x))^2}+\frac {-\frac {\sinh (x) \left (-2 a^3 B+11 a^2 A b-13 a b^2 B+4 A b^3\right )}{\left (a^2-b^2\right ) (a+b \cosh (x))}+\frac {3 \left (2 a^3 A-4 a^2 b B+3 a A b^2-b^3 B\right ) \int \frac {1}{a+b \sin \left (i x+\frac {\pi }{2}\right )}dx}{a^2-b^2}}{2 \left (a^2-b^2\right )}}{3 \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3138 |
| \(\displaystyle \frac {\frac {\frac {6 \left (2 a^3 A-4 a^2 b B+3 a A b^2-b^3 B\right ) \int \frac {1}{-\left ((a-b) \tanh ^2\left (\frac {x}{2}\right )\right )+a+b}d\tanh \left (\frac {x}{2}\right )}{a^2-b^2}-\frac {\sinh (x) \left (-2 a^3 B+11 a^2 A b-13 a b^2 B+4 A b^3\right )}{\left (a^2-b^2\right ) (a+b \cosh (x))}}{2 \left (a^2-b^2\right )}-\frac {\sinh (x) \left (-2 a^2 B+5 a A b-3 b^2 B\right )}{2 \left (a^2-b^2\right ) (a+b \cosh (x))^2}}{3 \left (a^2-b^2\right )}-\frac {\sinh (x) (A b-a B)}{3 \left (a^2-b^2\right ) (a+b \cosh (x))^3}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {\frac {\frac {6 \left (2 a^3 A-4 a^2 b B+3 a A b^2-b^3 B\right ) \text {arctanh}\left (\frac {\sqrt {a-b} \tanh \left (\frac {x}{2}\right )}{\sqrt {a+b}}\right )}{\sqrt {a-b} \sqrt {a+b} \left (a^2-b^2\right )}-\frac {\sinh (x) \left (-2 a^3 B+11 a^2 A b-13 a b^2 B+4 A b^3\right )}{\left (a^2-b^2\right ) (a+b \cosh (x))}}{2 \left (a^2-b^2\right )}-\frac {\sinh (x) \left (-2 a^2 B+5 a A b-3 b^2 B\right )}{2 \left (a^2-b^2\right ) (a+b \cosh (x))^2}}{3 \left (a^2-b^2\right )}-\frac {\sinh (x) (A b-a B)}{3 \left (a^2-b^2\right ) (a+b \cosh (x))^3}\) |
Input:
Int[(A + B*Cosh[x])/(a + b*Cosh[x])^4,x]
Output:
-1/3*((A*b - a*B)*Sinh[x])/((a^2 - b^2)*(a + b*Cosh[x])^3) + (-1/2*((5*a*A *b - 2*a^2*B - 3*b^2*B)*Sinh[x])/((a^2 - b^2)*(a + b*Cosh[x])^2) + ((6*(2* a^3*A + 3*a*A*b^2 - 4*a^2*b*B - b^3*B)*ArcTanh[(Sqrt[a - b]*Tanh[x/2])/Sqr t[a + b]])/(Sqrt[a - b]*Sqrt[a + b]*(a^2 - b^2)) - ((11*a^2*A*b + 4*A*b^3 - 2*a^3*B - 13*a*b^2*B)*Sinh[x])/((a^2 - b^2)*(a + b*Cosh[x])))/(2*(a^2 - b^2)))/(3*(a^2 - b^2))
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)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[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[((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]
Time = 1.07 (sec) , antiderivative size = 342, normalized size of antiderivative = 1.74
| method | result | size |
| default | \(-\frac {2 \left (-\frac {\left (6 A \,a^{2} b +3 A a \,b^{2}+2 A \,b^{3}-2 a^{3} B -2 B \,a^{2} b -6 B a \,b^{2}-B \,b^{3}\right ) \tanh \left (\frac {x}{2}\right )^{5}}{2 \left (a -b \right ) \left (a^{3}+3 a^{2} b +3 b^{2} a +b^{3}\right )}+\frac {2 \left (9 A \,a^{2} b +A \,b^{3}-3 a^{3} B -7 B a \,b^{2}\right ) \tanh \left (\frac {x}{2}\right )^{3}}{3 \left (a^{2}+2 a b +b^{2}\right ) \left (a^{2}-2 a b +b^{2}\right )}-\frac {\left (6 A \,a^{2} b -3 A a \,b^{2}+2 A \,b^{3}-2 a^{3} B +2 B \,a^{2} b -6 B a \,b^{2}+B \,b^{3}\right ) \tanh \left (\frac {x}{2}\right )}{2 \left (a +b \right ) \left (a^{3}-3 a^{2} b +3 b^{2} a -b^{3}\right )}\right )}{\left (\tanh \left (\frac {x}{2}\right )^{2} a -b \tanh \left (\frac {x}{2}\right )^{2}-a -b \right )^{3}}+\frac {\left (2 A \,a^{3}+3 A a \,b^{2}-4 B \,a^{2} b -B \,b^{3}\right ) \operatorname {arctanh}\left (\frac {\left (a -b \right ) \tanh \left (\frac {x}{2}\right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{\left (a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}\right ) \sqrt {\left (a +b \right ) \left (a -b \right )}}\) | \(342\) |
| risch | \(\frac {9 A a \,b^{5} {\mathrm e}^{5 x}-12 B \,a^{2} b^{4} {\mathrm e}^{5 x}+30 A \,a^{4} b^{2} {\mathrm e}^{4 x}+45 A \,a^{2} b^{4} {\mathrm e}^{4 x}-60 B \,a^{3} b^{3} {\mathrm e}^{4 x}-15 B a \,b^{5} {\mathrm e}^{4 x}+44 A \,a^{5} b \,{\mathrm e}^{3 x}+82 A \,a^{3} b^{3} {\mathrm e}^{3 x}+24 A a \,b^{5} {\mathrm e}^{3 x}-64 B \,a^{4} b^{2} {\mathrm e}^{3 x}+3 B \,b^{6} {\mathrm e}^{x}-2 B \,a^{3} b^{3}+4 A \,b^{6}+11 A \,a^{2} b^{4}-13 B a \,b^{5}+6 A \,a^{3} b^{3} {\mathrm e}^{5 x}-8 B \,a^{6} {\mathrm e}^{3 x}+12 A \,b^{6} {\mathrm e}^{2 x}+60 A \,a^{3} b^{3} {\mathrm e}^{x}+15 A a \,b^{5} {\mathrm e}^{x}-12 B \,a^{4} b^{2} {\mathrm e}^{x}-66 B \,a^{2} b^{4} {\mathrm e}^{x}+102 A \,a^{4} b^{2} {\mathrm e}^{2 x}+36 A \,a^{2} b^{4} {\mathrm e}^{2 x}-24 B \,a^{5} b \,{\mathrm e}^{2 x}-102 B \,a^{3} b^{3} {\mathrm e}^{2 x}-24 B a \,b^{5} {\mathrm e}^{2 x}-3 B \,b^{6} {\mathrm e}^{5 x}-78 B \,a^{2} b^{4} {\mathrm e}^{3 x}}{3 b \left (a^{2}-b^{2}\right )^{3} \left ({\mathrm e}^{2 x} b +2 a \,{\mathrm e}^{x}+b \right )^{3}}+\frac {\ln \left ({\mathrm e}^{x}+\frac {a \sqrt {a^{2}-b^{2}}-a^{2}+b^{2}}{b \sqrt {a^{2}-b^{2}}}\right ) A \,a^{3}}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right )^{3} \left (a -b \right )^{3}}+\frac {3 \ln \left ({\mathrm e}^{x}+\frac {a \sqrt {a^{2}-b^{2}}-a^{2}+b^{2}}{b \sqrt {a^{2}-b^{2}}}\right ) A a \,b^{2}}{2 \sqrt {a^{2}-b^{2}}\, \left (a +b \right )^{3} \left (a -b \right )^{3}}-\frac {2 \ln \left ({\mathrm e}^{x}+\frac {a \sqrt {a^{2}-b^{2}}-a^{2}+b^{2}}{b \sqrt {a^{2}-b^{2}}}\right ) B \,a^{2} b}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right )^{3} \left (a -b \right )^{3}}-\frac {\ln \left ({\mathrm e}^{x}+\frac {a \sqrt {a^{2}-b^{2}}-a^{2}+b^{2}}{b \sqrt {a^{2}-b^{2}}}\right ) B \,b^{3}}{2 \sqrt {a^{2}-b^{2}}\, \left (a +b \right )^{3} \left (a -b \right )^{3}}-\frac {\ln \left ({\mathrm e}^{x}+\frac {a \sqrt {a^{2}-b^{2}}+a^{2}-b^{2}}{b \sqrt {a^{2}-b^{2}}}\right ) A \,a^{3}}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right )^{3} \left (a -b \right )^{3}}-\frac {3 \ln \left ({\mathrm e}^{x}+\frac {a \sqrt {a^{2}-b^{2}}+a^{2}-b^{2}}{b \sqrt {a^{2}-b^{2}}}\right ) A a \,b^{2}}{2 \sqrt {a^{2}-b^{2}}\, \left (a +b \right )^{3} \left (a -b \right )^{3}}+\frac {2 \ln \left ({\mathrm e}^{x}+\frac {a \sqrt {a^{2}-b^{2}}+a^{2}-b^{2}}{b \sqrt {a^{2}-b^{2}}}\right ) B \,a^{2} b}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right )^{3} \left (a -b \right )^{3}}+\frac {\ln \left ({\mathrm e}^{x}+\frac {a \sqrt {a^{2}-b^{2}}+a^{2}-b^{2}}{b \sqrt {a^{2}-b^{2}}}\right ) B \,b^{3}}{2 \sqrt {a^{2}-b^{2}}\, \left (a +b \right )^{3} \left (a -b \right )^{3}}\) | \(917\) |
Input:
int((A+B*cosh(x))/(a+b*cosh(x))^4,x,method=_RETURNVERBOSE)
Output:
-2*(-1/2*(6*A*a^2*b+3*A*a*b^2+2*A*b^3-2*B*a^3-2*B*a^2*b-6*B*a*b^2-B*b^3)/( a-b)/(a^3+3*a^2*b+3*a*b^2+b^3)*tanh(1/2*x)^5+2/3*(9*A*a^2*b+A*b^3-3*B*a^3- 7*B*a*b^2)/(a^2+2*a*b+b^2)/(a^2-2*a*b+b^2)*tanh(1/2*x)^3-1/2*(6*A*a^2*b-3* A*a*b^2+2*A*b^3-2*B*a^3+2*B*a^2*b-6*B*a*b^2+B*b^3)/(a+b)/(a^3-3*a^2*b+3*a* b^2-b^3)*tanh(1/2*x))/(tanh(1/2*x)^2*a-b*tanh(1/2*x)^2-a-b)^3+(2*A*a^3+3*A *a*b^2-4*B*a^2*b-B*b^3)/(a^6-3*a^4*b^2+3*a^2*b^4-b^6)/((a+b)*(a-b))^(1/2)* arctanh((a-b)*tanh(1/2*x)/((a+b)*(a-b))^(1/2))
Leaf count of result is larger than twice the leaf count of optimal. 3767 vs. \(2 (181) = 362\).
Time = 0.33 (sec) , antiderivative size = 7603, normalized size of antiderivative = 38.59 \[ \int \frac {A+B \cosh (x)}{(a+b \cosh (x))^4} \, dx=\text {Too large to display} \] Input:
integrate((A+B*cosh(x))/(a+b*cosh(x))^4,x, algorithm="fricas")
Output:
Too large to include
Timed out. \[ \int \frac {A+B \cosh (x)}{(a+b \cosh (x))^4} \, dx=\text {Timed out} \] Input:
integrate((A+B*cosh(x))/(a+b*cosh(x))**4,x)
Output:
Timed out
Exception generated. \[ \int \frac {A+B \cosh (x)}{(a+b \cosh (x))^4} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((A+B*cosh(x))/(a+b*cosh(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(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. 453 vs. \(2 (181) = 362\).
Time = 0.12 (sec) , antiderivative size = 453, normalized size of antiderivative = 2.30 \[ \int \frac {A+B \cosh (x)}{(a+b \cosh (x))^4} \, dx=\frac {{\left (2 \, A a^{3} - 4 \, B a^{2} b + 3 \, A a b^{2} - B b^{3}\right )} \arctan \left (\frac {b e^{x} + a}{\sqrt {-a^{2} + b^{2}}}\right )}{{\left (a^{6} - 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} - b^{6}\right )} \sqrt {-a^{2} + b^{2}}} + \frac {6 \, A a^{3} b^{3} e^{\left (5 \, x\right )} - 12 \, B a^{2} b^{4} e^{\left (5 \, x\right )} + 9 \, A a b^{5} e^{\left (5 \, x\right )} - 3 \, B b^{6} e^{\left (5 \, x\right )} + 30 \, A a^{4} b^{2} e^{\left (4 \, x\right )} - 60 \, B a^{3} b^{3} e^{\left (4 \, x\right )} + 45 \, A a^{2} b^{4} e^{\left (4 \, x\right )} - 15 \, B a b^{5} e^{\left (4 \, x\right )} - 8 \, B a^{6} e^{\left (3 \, x\right )} + 44 \, A a^{5} b e^{\left (3 \, x\right )} - 64 \, B a^{4} b^{2} e^{\left (3 \, x\right )} + 82 \, A a^{3} b^{3} e^{\left (3 \, x\right )} - 78 \, B a^{2} b^{4} e^{\left (3 \, x\right )} + 24 \, A a b^{5} e^{\left (3 \, x\right )} - 24 \, B a^{5} b e^{\left (2 \, x\right )} + 102 \, A a^{4} b^{2} e^{\left (2 \, x\right )} - 102 \, B a^{3} b^{3} e^{\left (2 \, x\right )} + 36 \, A a^{2} b^{4} e^{\left (2 \, x\right )} - 24 \, B a b^{5} e^{\left (2 \, x\right )} + 12 \, A b^{6} e^{\left (2 \, x\right )} - 12 \, B a^{4} b^{2} e^{x} + 60 \, A a^{3} b^{3} e^{x} - 66 \, B a^{2} b^{4} e^{x} + 15 \, A a b^{5} e^{x} + 3 \, B b^{6} e^{x} - 2 \, B a^{3} b^{3} + 11 \, A a^{2} b^{4} - 13 \, B a b^{5} + 4 \, A b^{6}}{3 \, {\left (a^{6} b - 3 \, a^{4} b^{3} + 3 \, a^{2} b^{5} - b^{7}\right )} {\left (b e^{\left (2 \, x\right )} + 2 \, a e^{x} + b\right )}^{3}} \] Input:
integrate((A+B*cosh(x))/(a+b*cosh(x))^4,x, algorithm="giac")
Output:
(2*A*a^3 - 4*B*a^2*b + 3*A*a*b^2 - B*b^3)*arctan((b*e^x + a)/sqrt(-a^2 + b ^2))/((a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*sqrt(-a^2 + b^2)) + 1/3*(6*A*a^3 *b^3*e^(5*x) - 12*B*a^2*b^4*e^(5*x) + 9*A*a*b^5*e^(5*x) - 3*B*b^6*e^(5*x) + 30*A*a^4*b^2*e^(4*x) - 60*B*a^3*b^3*e^(4*x) + 45*A*a^2*b^4*e^(4*x) - 15* B*a*b^5*e^(4*x) - 8*B*a^6*e^(3*x) + 44*A*a^5*b*e^(3*x) - 64*B*a^4*b^2*e^(3 *x) + 82*A*a^3*b^3*e^(3*x) - 78*B*a^2*b^4*e^(3*x) + 24*A*a*b^5*e^(3*x) - 2 4*B*a^5*b*e^(2*x) + 102*A*a^4*b^2*e^(2*x) - 102*B*a^3*b^3*e^(2*x) + 36*A*a ^2*b^4*e^(2*x) - 24*B*a*b^5*e^(2*x) + 12*A*b^6*e^(2*x) - 12*B*a^4*b^2*e^x + 60*A*a^3*b^3*e^x - 66*B*a^2*b^4*e^x + 15*A*a*b^5*e^x + 3*B*b^6*e^x - 2*B *a^3*b^3 + 11*A*a^2*b^4 - 13*B*a*b^5 + 4*A*b^6)/((a^6*b - 3*a^4*b^3 + 3*a^ 2*b^5 - b^7)*(b*e^(2*x) + 2*a*e^x + b)^3)
Timed out. \[ \int \frac {A+B \cosh (x)}{(a+b \cosh (x))^4} \, dx=\int \frac {A+B\,\mathrm {cosh}\left (x\right )}{{\left (a+b\,\mathrm {cosh}\left (x\right )\right )}^4} \,d x \] Input:
int((A + B*cosh(x))/(a + b*cosh(x))^4,x)
Output:
int((A + B*cosh(x))/(a + b*cosh(x))^4, x)
Time = 0.25 (sec) , antiderivative size = 817, normalized size of antiderivative = 4.15 \[ \int \frac {A+B \cosh (x)}{(a+b \cosh (x))^4} \, dx=\frac {-8 e^{4 x} \sqrt {-a^{2}+b^{2}}\, \mathit {atan} \left (\frac {e^{x} b +a}{\sqrt {-a^{2}+b^{2}}}\right ) a^{3} b^{2}-4 e^{4 x} \sqrt {-a^{2}+b^{2}}\, \mathit {atan} \left (\frac {e^{x} b +a}{\sqrt {-a^{2}+b^{2}}}\right ) a \,b^{4}-32 e^{3 x} \sqrt {-a^{2}+b^{2}}\, \mathit {atan} \left (\frac {e^{x} b +a}{\sqrt {-a^{2}+b^{2}}}\right ) a^{4} b -16 e^{3 x} \sqrt {-a^{2}+b^{2}}\, \mathit {atan} \left (\frac {e^{x} b +a}{\sqrt {-a^{2}+b^{2}}}\right ) a^{2} b^{3}-32 e^{2 x} \sqrt {-a^{2}+b^{2}}\, \mathit {atan} \left (\frac {e^{x} b +a}{\sqrt {-a^{2}+b^{2}}}\right ) a^{5}-32 e^{2 x} \sqrt {-a^{2}+b^{2}}\, \mathit {atan} \left (\frac {e^{x} b +a}{\sqrt {-a^{2}+b^{2}}}\right ) a^{3} b^{2}-8 e^{2 x} \sqrt {-a^{2}+b^{2}}\, \mathit {atan} \left (\frac {e^{x} b +a}{\sqrt {-a^{2}+b^{2}}}\right ) a \,b^{4}-32 e^{x} \sqrt {-a^{2}+b^{2}}\, \mathit {atan} \left (\frac {e^{x} b +a}{\sqrt {-a^{2}+b^{2}}}\right ) a^{4} b -16 e^{x} \sqrt {-a^{2}+b^{2}}\, \mathit {atan} \left (\frac {e^{x} b +a}{\sqrt {-a^{2}+b^{2}}}\right ) a^{2} b^{3}-8 \sqrt {-a^{2}+b^{2}}\, \mathit {atan} \left (\frac {e^{x} b +a}{\sqrt {-a^{2}+b^{2}}}\right ) a^{3} b^{2}-4 \sqrt {-a^{2}+b^{2}}\, \mathit {atan} \left (\frac {e^{x} b +a}{\sqrt {-a^{2}+b^{2}}}\right ) a \,b^{4}-2 e^{4 x} a^{4} b^{2}+e^{4 x} a^{2} b^{4}+e^{4 x} b^{6}+16 e^{2 x} a^{6}-12 e^{2 x} a^{4} b^{2}-6 e^{2 x} a^{2} b^{4}+2 e^{2 x} b^{6}+32 e^{x} a^{5} b -40 e^{x} a^{3} b^{3}+8 e^{x} a \,b^{5}+10 a^{4} b^{2}-11 a^{2} b^{4}+b^{6}}{4 a \left (e^{4 x} a^{6} b^{2}-3 e^{4 x} a^{4} b^{4}+3 e^{4 x} a^{2} b^{6}-e^{4 x} b^{8}+4 e^{3 x} a^{7} b -12 e^{3 x} a^{5} b^{3}+12 e^{3 x} a^{3} b^{5}-4 e^{3 x} a \,b^{7}+4 e^{2 x} a^{8}-10 e^{2 x} a^{6} b^{2}+6 e^{2 x} a^{4} b^{4}+2 e^{2 x} a^{2} b^{6}-2 e^{2 x} b^{8}+4 e^{x} a^{7} b -12 e^{x} a^{5} b^{3}+12 e^{x} a^{3} b^{5}-4 e^{x} a \,b^{7}+a^{6} b^{2}-3 a^{4} b^{4}+3 a^{2} b^{6}-b^{8}\right )} \] Input:
int((A+B*cosh(x))/(a+b*cosh(x))^4,x)
Output:
( - 8*e**(4*x)*sqrt( - a**2 + b**2)*atan((e**x*b + a)/sqrt( - a**2 + b**2) )*a**3*b**2 - 4*e**(4*x)*sqrt( - a**2 + b**2)*atan((e**x*b + a)/sqrt( - a* *2 + b**2))*a*b**4 - 32*e**(3*x)*sqrt( - a**2 + b**2)*atan((e**x*b + a)/sq rt( - a**2 + b**2))*a**4*b - 16*e**(3*x)*sqrt( - a**2 + b**2)*atan((e**x*b + a)/sqrt( - a**2 + b**2))*a**2*b**3 - 32*e**(2*x)*sqrt( - a**2 + b**2)*a tan((e**x*b + a)/sqrt( - a**2 + b**2))*a**5 - 32*e**(2*x)*sqrt( - a**2 + b **2)*atan((e**x*b + a)/sqrt( - a**2 + b**2))*a**3*b**2 - 8*e**(2*x)*sqrt( - a**2 + b**2)*atan((e**x*b + a)/sqrt( - a**2 + b**2))*a*b**4 - 32*e**x*sq rt( - a**2 + b**2)*atan((e**x*b + a)/sqrt( - a**2 + b**2))*a**4*b - 16*e** x*sqrt( - a**2 + b**2)*atan((e**x*b + a)/sqrt( - a**2 + b**2))*a**2*b**3 - 8*sqrt( - a**2 + b**2)*atan((e**x*b + a)/sqrt( - a**2 + b**2))*a**3*b**2 - 4*sqrt( - a**2 + b**2)*atan((e**x*b + a)/sqrt( - a**2 + b**2))*a*b**4 - 2*e**(4*x)*a**4*b**2 + e**(4*x)*a**2*b**4 + e**(4*x)*b**6 + 16*e**(2*x)*a* *6 - 12*e**(2*x)*a**4*b**2 - 6*e**(2*x)*a**2*b**4 + 2*e**(2*x)*b**6 + 32*e **x*a**5*b - 40*e**x*a**3*b**3 + 8*e**x*a*b**5 + 10*a**4*b**2 - 11*a**2*b* *4 + b**6)/(4*a*(e**(4*x)*a**6*b**2 - 3*e**(4*x)*a**4*b**4 + 3*e**(4*x)*a* *2*b**6 - e**(4*x)*b**8 + 4*e**(3*x)*a**7*b - 12*e**(3*x)*a**5*b**3 + 12*e **(3*x)*a**3*b**5 - 4*e**(3*x)*a*b**7 + 4*e**(2*x)*a**8 - 10*e**(2*x)*a**6 *b**2 + 6*e**(2*x)*a**4*b**4 + 2*e**(2*x)*a**2*b**6 - 2*e**(2*x)*b**8 + 4* e**x*a**7*b - 12*e**x*a**5*b**3 + 12*e**x*a**3*b**5 - 4*e**x*a*b**7 + a...