Integrand size = 31, antiderivative size = 260 \[ \int \frac {A+B \cosh (d+e x)+C \sinh (d+e x)}{(a+b \cosh (d+e x))^4} \, dx=\frac {\left (2 a^3 A+3 a A b^2-4 a^2 b B-b^3 B\right ) \text {arctanh}\left (\frac {\sqrt {a-b} \tanh \left (\frac {1}{2} (d+e x)\right )}{\sqrt {a+b}}\right )}{(a-b)^{7/2} (a+b)^{7/2} e}-\frac {C}{3 b e (a+b \cosh (d+e x))^3}-\frac {(A b-a B) \sinh (d+e x)}{3 \left (a^2-b^2\right ) e (a+b \cosh (d+e x))^3}-\frac {\left (5 a A b-2 a^2 B-3 b^2 B\right ) \sinh (d+e x)}{6 \left (a^2-b^2\right )^2 e (a+b \cosh (d+e x))^2}-\frac {\left (11 a^2 A b+4 A b^3-2 a^3 B-13 a b^2 B\right ) \sinh (d+e x)}{6 \left (a^2-b^2\right )^3 e (a+b \cosh (d+e x))} \]
(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*e*x+1/2*d )/(a+b)^(1/2))/(a-b)^(7/2)/(a+b)^(7/2)/e-1/3*C/b/e/(a+b*cosh(e*x+d))^3-1/3 *(A*b-B*a)*sinh(e*x+d)/(a^2-b^2)/e/(a+b*cosh(e*x+d))^3-1/6*(5*A*a*b-2*B*a^ 2-3*B*b^2)*sinh(e*x+d)/(a^2-b^2)^2/e/(a+b*cosh(e*x+d))^2-1/6*(11*A*a^2*b+4 *A*b^3-2*B*a^3-13*B*a*b^2)*sinh(e*x+d)/(a^2-b^2)^3/e/(a+b*cosh(e*x+d))
Time = 1.97 (sec) , antiderivative size = 245, normalized size of antiderivative = 0.94 \[ \int \frac {A+B \cosh (d+e x)+C \sinh (d+e x)}{(a+b \cosh (d+e x))^4} \, dx=\frac {\frac {6 \left (2 a^3 A+3 a A b^2-4 a^2 b B-b^3 B\right ) \arctan \left (\frac {(a-b) \tanh \left (\frac {1}{2} (d+e x)\right )}{\sqrt {-a^2+b^2}}\right )}{\left (-a^2+b^2\right )^{7/2}}+\frac {\left (-5 a A b+2 a^2 B+3 b^2 B\right ) \sinh (d+e x)}{(a-b)^2 (a+b)^2 (a+b \cosh (d+e x))^2}+\frac {\left (-11 a^2 A b-4 A b^3+2 a^3 B+13 a b^2 B\right ) \sinh (d+e x)}{(a-b)^3 (a+b)^3 (a+b \cosh (d+e x))}+\frac {2 \left (-a^2+b^2\right ) C-2 b (A b-a B) \sinh (d+e x)}{(a-b) b (a+b) (a+b \cosh (d+e x))^3}}{6 e} \]
((6*(2*a^3*A + 3*a*A*b^2 - 4*a^2*b*B - b^3*B)*ArcTan[((a - b)*Tanh[(d + e* x)/2])/Sqrt[-a^2 + b^2]])/(-a^2 + b^2)^(7/2) + ((-5*a*A*b + 2*a^2*B + 3*b^ 2*B)*Sinh[d + e*x])/((a - b)^2*(a + b)^2*(a + b*Cosh[d + e*x])^2) + ((-11* a^2*A*b - 4*A*b^3 + 2*a^3*B + 13*a*b^2*B)*Sinh[d + e*x])/((a - b)^3*(a + b )^3*(a + b*Cosh[d + e*x])) + (2*(-a^2 + b^2)*C - 2*b*(A*b - a*B)*Sinh[d + e*x])/((a - b)*b*(a + b)*(a + b*Cosh[d + e*x])^3))/(6*e)
Time = 1.14 (sec) , antiderivative size = 302, normalized size of antiderivative = 1.16, number of steps used = 19, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.581, Rules used = {3042, 4877, 26, 3042, 26, 3147, 17, 3233, 25, 3042, 3233, 25, 3042, 3233, 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.
| \(\displaystyle \int \frac {A+B \cosh (d+e x)+C \sinh (d+e x)}{(a+b \cosh (d+e x))^4} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {A+B \cos (i d+i e x)-i C \sin (i d+i e x)}{(a+b \cos (i d+i e x))^4}dx\) |
| \(\Big \downarrow \) 4877 |
| \(\displaystyle \int \frac {A+B \cosh (d+e x)}{(a+b \cosh (d+e x))^4}dx-i C \int \frac {i \sinh (d+e x)}{(a+b \cosh (d+e x))^4}dx\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \int \frac {A+B \cosh (d+e x)}{(a+b \cosh (d+e x))^4}dx+C \int \frac {\sinh (d+e x)}{(a+b \cosh (d+e x))^4}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {A+B \sin \left (i d+i e x+\frac {\pi }{2}\right )}{\left (a+b \sin \left (i d+i e x+\frac {\pi }{2}\right )\right )^4}dx+C \int -\frac {i \cos \left (i d+i e x-\frac {\pi }{2}\right )}{\left (a-b \sin \left (i d+i e x-\frac {\pi }{2}\right )\right )^4}dx\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \int \frac {A+B \sin \left (i d+i e x+\frac {\pi }{2}\right )}{\left (a+b \sin \left (i d+i e x+\frac {\pi }{2}\right )\right )^4}dx-i C \int \frac {\cos \left (\frac {1}{2} (2 i d-\pi )+i e x\right )}{\left (a-b \sin \left (\frac {1}{2} (2 i d-\pi )+i e x\right )\right )^4}dx\) |
| \(\Big \downarrow \) 3147 |
| \(\displaystyle \frac {C \int \frac {1}{(a+b \cosh (d+e x))^4}d(b \cosh (d+e x))}{b e}+\int \frac {A+B \sin \left (i d+i e x+\frac {\pi }{2}\right )}{\left (a+b \sin \left (i d+i e x+\frac {\pi }{2}\right )\right )^4}dx\) |
| \(\Big \downarrow \) 17 |
| \(\displaystyle -\frac {C}{3 b e (a+b \cosh (d+e x))^3}+\int \frac {A+B \sin \left (i d+i e x+\frac {\pi }{2}\right )}{\left (a+b \sin \left (i d+i e x+\frac {\pi }{2}\right )\right )^4}dx\) |
| \(\Big \downarrow \) 3233 |
| \(\displaystyle -\frac {\int -\frac {3 (a A-b B)-2 (A b-a B) \cosh (d+e x)}{(a+b \cosh (d+e x))^3}dx}{3 \left (a^2-b^2\right )}-\frac {(A b-a B) \sinh (d+e x)}{3 e \left (a^2-b^2\right ) (a+b \cosh (d+e x))^3}-\frac {C}{3 b e (a+b \cosh (d+e x))^3}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {3 (a A-b B)-2 (A b-a B) \cosh (d+e x)}{(a+b \cosh (d+e x))^3}dx}{3 \left (a^2-b^2\right )}-\frac {(A b-a B) \sinh (d+e x)}{3 e \left (a^2-b^2\right ) (a+b \cosh (d+e x))^3}-\frac {C}{3 b e (a+b \cosh (d+e x))^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {3 (a A-b B)-2 (A b-a B) \sin \left (i d+i e x+\frac {\pi }{2}\right )}{\left (a+b \sin \left (i d+i e x+\frac {\pi }{2}\right )\right )^3}dx}{3 \left (a^2-b^2\right )}-\frac {(A b-a B) \sinh (d+e x)}{3 e \left (a^2-b^2\right ) (a+b \cosh (d+e x))^3}-\frac {C}{3 b e (a+b \cosh (d+e x))^3}\) |
| \(\Big \downarrow \) 3233 |
| \(\displaystyle \frac {-\frac {\int -\frac {2 \left (3 A a^2-5 b B a+2 A b^2\right )-\left (-2 B a^2+5 A b a-3 b^2 B\right ) \cosh (d+e x)}{(a+b \cosh (d+e x))^2}dx}{2 \left (a^2-b^2\right )}-\frac {\left (-2 a^2 B+5 a A b-3 b^2 B\right ) \sinh (d+e x)}{2 e \left (a^2-b^2\right ) (a+b \cosh (d+e x))^2}}{3 \left (a^2-b^2\right )}-\frac {(A b-a B) \sinh (d+e x)}{3 e \left (a^2-b^2\right ) (a+b \cosh (d+e x))^3}-\frac {C}{3 b e (a+b \cosh (d+e x))^3}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\int \frac {2 \left (3 A a^2-5 b B a+2 A b^2\right )-\left (-2 B a^2+5 A b a-3 b^2 B\right ) \cosh (d+e x)}{(a+b \cosh (d+e x))^2}dx}{2 \left (a^2-b^2\right )}-\frac {\left (-2 a^2 B+5 a A b-3 b^2 B\right ) \sinh (d+e x)}{2 e \left (a^2-b^2\right ) (a+b \cosh (d+e x))^2}}{3 \left (a^2-b^2\right )}-\frac {(A b-a B) \sinh (d+e x)}{3 e \left (a^2-b^2\right ) (a+b \cosh (d+e x))^3}-\frac {C}{3 b e (a+b \cosh (d+e x))^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {\left (-2 a^2 B+5 a A b-3 b^2 B\right ) \sinh (d+e x)}{2 e \left (a^2-b^2\right ) (a+b \cosh (d+e x))^2}+\frac {\int \frac {2 \left (3 A a^2-5 b B a+2 A b^2\right )+\left (2 B a^2-5 A b a+3 b^2 B\right ) \sin \left (i d+i e x+\frac {\pi }{2}\right )}{\left (a+b \sin \left (i d+i e x+\frac {\pi }{2}\right )\right )^2}dx}{2 \left (a^2-b^2\right )}}{3 \left (a^2-b^2\right )}-\frac {(A b-a B) \sinh (d+e x)}{3 e \left (a^2-b^2\right ) (a+b \cosh (d+e x))^3}-\frac {C}{3 b e (a+b \cosh (d+e x))^3}\) |
| \(\Big \downarrow \) 3233 |
| \(\displaystyle \frac {\frac {-\frac {\int -\frac {3 \left (2 A a^3-4 b B a^2+3 A b^2 a-b^3 B\right )}{a+b \cosh (d+e x)}dx}{a^2-b^2}-\frac {\left (-2 a^3 B+11 a^2 A b-13 a b^2 B+4 A b^3\right ) \sinh (d+e x)}{e \left (a^2-b^2\right ) (a+b \cosh (d+e x))}}{2 \left (a^2-b^2\right )}-\frac {\left (-2 a^2 B+5 a A b-3 b^2 B\right ) \sinh (d+e x)}{2 e \left (a^2-b^2\right ) (a+b \cosh (d+e x))^2}}{3 \left (a^2-b^2\right )}-\frac {(A b-a B) \sinh (d+e x)}{3 e \left (a^2-b^2\right ) (a+b \cosh (d+e x))^3}-\frac {C}{3 b e (a+b \cosh (d+e x))^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {3 \left (2 a^3 A-4 a^2 b B+3 a A b^2-b^3 B\right ) \int \frac {1}{a+b \cosh (d+e x)}dx}{a^2-b^2}-\frac {\left (-2 a^3 B+11 a^2 A b-13 a b^2 B+4 A b^3\right ) \sinh (d+e x)}{e \left (a^2-b^2\right ) (a+b \cosh (d+e x))}}{2 \left (a^2-b^2\right )}-\frac {\left (-2 a^2 B+5 a A b-3 b^2 B\right ) \sinh (d+e x)}{2 e \left (a^2-b^2\right ) (a+b \cosh (d+e x))^2}}{3 \left (a^2-b^2\right )}-\frac {(A b-a B) \sinh (d+e x)}{3 e \left (a^2-b^2\right ) (a+b \cosh (d+e x))^3}-\frac {C}{3 b e (a+b \cosh (d+e x))^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {-\frac {\left (-2 a^2 B+5 a A b-3 b^2 B\right ) \sinh (d+e x)}{2 e \left (a^2-b^2\right ) (a+b \cosh (d+e x))^2}+\frac {-\frac {\left (-2 a^3 B+11 a^2 A b-13 a b^2 B+4 A b^3\right ) \sinh (d+e x)}{e \left (a^2-b^2\right ) (a+b \cosh (d+e x))}+\frac {3 \left (2 a^3 A-4 a^2 b B+3 a A b^2-b^3 B\right ) \int \frac {1}{a+b \sin \left (i d+i e x+\frac {\pi }{2}\right )}dx}{a^2-b^2}}{2 \left (a^2-b^2\right )}}{3 \left (a^2-b^2\right )}-\frac {(A b-a B) \sinh (d+e x)}{3 e \left (a^2-b^2\right ) (a+b \cosh (d+e x))^3}-\frac {C}{3 b e (a+b \cosh (d+e x))^3}\) |
| \(\Big \downarrow \) 3138 |
| \(\displaystyle \frac {-\frac {\left (-2 a^2 B+5 a A b-3 b^2 B\right ) \sinh (d+e x)}{2 e \left (a^2-b^2\right ) (a+b \cosh (d+e x))^2}+\frac {-\frac {\left (-2 a^3 B+11 a^2 A b-13 a b^2 B+4 A b^3\right ) \sinh (d+e x)}{e \left (a^2-b^2\right ) (a+b \cosh (d+e x))}-\frac {6 i \left (2 a^3 A-4 a^2 b B+3 a A b^2-b^3 B\right ) \int \frac {1}{-\left ((a-b) \tanh ^2\left (\frac {1}{2} (d+e x)\right )\right )+a+b}d\left (i \tanh \left (\frac {1}{2} (d+e x)\right )\right )}{e \left (a^2-b^2\right )}}{2 \left (a^2-b^2\right )}}{3 \left (a^2-b^2\right )}-\frac {(A b-a B) \sinh (d+e x)}{3 e \left (a^2-b^2\right ) (a+b \cosh (d+e x))^3}-\frac {C}{3 b e (a+b \cosh (d+e x))^3}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle -\frac {(A b-a B) \sinh (d+e x)}{3 e \left (a^2-b^2\right ) (a+b \cosh (d+e x))^3}+\frac {\frac {\frac {6 \left (2 a^3 A-4 a^2 b B+3 a A b^2-b^3 B\right ) \text {arctanh}\left (\frac {\sqrt {a-b} \tanh \left (\frac {1}{2} (d+e x)\right )}{\sqrt {a+b}}\right )}{e \sqrt {a-b} \sqrt {a+b} \left (a^2-b^2\right )}-\frac {\left (-2 a^3 B+11 a^2 A b-13 a b^2 B+4 A b^3\right ) \sinh (d+e x)}{e \left (a^2-b^2\right ) (a+b \cosh (d+e x))}}{2 \left (a^2-b^2\right )}-\frac {\left (-2 a^2 B+5 a A b-3 b^2 B\right ) \sinh (d+e x)}{2 e \left (a^2-b^2\right ) (a+b \cosh (d+e x))^2}}{3 \left (a^2-b^2\right )}-\frac {C}{3 b e (a+b \cosh (d+e x))^3}\) |
-1/3*C/(b*e*(a + b*Cosh[d + e*x])^3) - ((A*b - a*B)*Sinh[d + e*x])/(3*(a^2 - b^2)*e*(a + b*Cosh[d + e*x])^3) + (-1/2*((5*a*A*b - 2*a^2*B - 3*b^2*B)* Sinh[d + e*x])/((a^2 - b^2)*e*(a + b*Cosh[d + e*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[(d + e*x)/2])/Sqrt[ a + b]])/(Sqrt[a - b]*Sqrt[a + b]*(a^2 - b^2)*e) - ((11*a^2*A*b + 4*A*b^3 - 2*a^3*B - 13*a*b^2*B)*Sinh[d + e*x])/((a^2 - b^2)*e*(a + b*Cosh[d + e*x] )))/(2*(a^2 - b^2)))/(3*(a^2 - b^2))
3.3.9.3.1 Defintions of rubi rules used
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 = 11.28 (sec) , antiderivative size = 459, normalized size of antiderivative = 1.77
| method | result | size |
| derivativedivides | \(\frac {-\frac {2 \left (-\frac {\left (6 A \,a^{2} b +3 A a \,b^{2}+2 A \,b^{3}-2 a^{3} B -2 B \,a^{2} b -6 B a \,b^{2}-B \,b^{3}\right ) \tanh \left (\frac {e x}{2}+\frac {d}{2}\right )^{5}}{2 \left (a -b \right ) \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right )}+\frac {C \tanh \left (\frac {e x}{2}+\frac {d}{2}\right )^{4}}{a -b}+\frac {2 \left (9 A \,a^{2} b +A \,b^{3}-3 a^{3} B -7 B a \,b^{2}\right ) \tanh \left (\frac {e x}{2}+\frac {d}{2}\right )^{3}}{3 \left (a^{2}+2 a b +b^{2}\right ) \left (a^{2}-2 a b +b^{2}\right )}-\frac {2 a C \tanh \left (\frac {e x}{2}+\frac {d}{2}\right )^{2}}{a^{2}-2 a b +b^{2}}-\frac {\left (6 A \,a^{2} b -3 A a \,b^{2}+2 A \,b^{3}-2 a^{3} B +2 B \,a^{2} b -6 B a \,b^{2}+B \,b^{3}\right ) \tanh \left (\frac {e x}{2}+\frac {d}{2}\right )}{2 \left (a +b \right ) \left (a^{3}-3 a^{2} b +3 a \,b^{2}-b^{3}\right )}+\frac {C \left (3 a^{2}+b^{2}\right )}{3 a^{3}-9 a^{2} b +9 a \,b^{2}-3 b^{3}}\right )}{\left (a \tanh \left (\frac {e x}{2}+\frac {d}{2}\right )^{2}-b \tanh \left (\frac {e x}{2}+\frac {d}{2}\right )^{2}-a -b \right )^{3}}+\frac {\left (2 A \,a^{3}+3 A a \,b^{2}-4 B \,a^{2} b -B \,b^{3}\right ) \operatorname {arctanh}\left (\frac {\left (a -b \right ) \tanh \left (\frac {e x}{2}+\frac {d}{2}\right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{\left (a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}\right ) \sqrt {\left (a +b \right ) \left (a -b \right )}}}{e}\) | \(459\) |
| default | \(\frac {-\frac {2 \left (-\frac {\left (6 A \,a^{2} b +3 A a \,b^{2}+2 A \,b^{3}-2 a^{3} B -2 B \,a^{2} b -6 B a \,b^{2}-B \,b^{3}\right ) \tanh \left (\frac {e x}{2}+\frac {d}{2}\right )^{5}}{2 \left (a -b \right ) \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right )}+\frac {C \tanh \left (\frac {e x}{2}+\frac {d}{2}\right )^{4}}{a -b}+\frac {2 \left (9 A \,a^{2} b +A \,b^{3}-3 a^{3} B -7 B a \,b^{2}\right ) \tanh \left (\frac {e x}{2}+\frac {d}{2}\right )^{3}}{3 \left (a^{2}+2 a b +b^{2}\right ) \left (a^{2}-2 a b +b^{2}\right )}-\frac {2 a C \tanh \left (\frac {e x}{2}+\frac {d}{2}\right )^{2}}{a^{2}-2 a b +b^{2}}-\frac {\left (6 A \,a^{2} b -3 A a \,b^{2}+2 A \,b^{3}-2 a^{3} B +2 B \,a^{2} b -6 B a \,b^{2}+B \,b^{3}\right ) \tanh \left (\frac {e x}{2}+\frac {d}{2}\right )}{2 \left (a +b \right ) \left (a^{3}-3 a^{2} b +3 a \,b^{2}-b^{3}\right )}+\frac {C \left (3 a^{2}+b^{2}\right )}{3 a^{3}-9 a^{2} b +9 a \,b^{2}-3 b^{3}}\right )}{\left (a \tanh \left (\frac {e x}{2}+\frac {d}{2}\right )^{2}-b \tanh \left (\frac {e x}{2}+\frac {d}{2}\right )^{2}-a -b \right )^{3}}+\frac {\left (2 A \,a^{3}+3 A a \,b^{2}-4 B \,a^{2} b -B \,b^{3}\right ) \operatorname {arctanh}\left (\frac {\left (a -b \right ) \tanh \left (\frac {e x}{2}+\frac {d}{2}\right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{\left (a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}\right ) \sqrt {\left (a +b \right ) \left (a -b \right )}}}{e}\) | \(459\) |
| risch | \(\text {Expression too large to display}\) | \(1171\) |
1/e*(-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*e*x+1/2*d)^5+C/(a-b)*tanh(1/2 *e*x+1/2*d)^4+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*e*x+1/2*d)^3-2*a*C/(a^2-2*a*b+b^2)*tanh(1/2*e*x+1/2*d )^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*e*x+1/2*d)+1/3*C*(3*a^2+b^2)/(a^3-3 *a^2*b+3*a*b^2-b^3))/(a*tanh(1/2*e*x+1/2*d)^2-b*tanh(1/2*e*x+1/2*d)^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*e*x+1/2*d)/((a+b)*(a-b))^(1/2)))
Leaf count of result is larger than twice the leaf count of optimal. 4211 vs. \(2 (243) = 486\).
Time = 0.52 (sec) , antiderivative size = 8531, normalized size of antiderivative = 32.81 \[ \int \frac {A+B \cosh (d+e x)+C \sinh (d+e x)}{(a+b \cosh (d+e x))^4} \, dx=\text {Too large to display} \]
Timed out. \[ \int \frac {A+B \cosh (d+e x)+C \sinh (d+e x)}{(a+b \cosh (d+e x))^4} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {A+B \cosh (d+e x)+C \sinh (d+e x)}{(a+b \cosh (d+e x))^4} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(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. 657 vs. \(2 (243) = 486\).
Time = 0.30 (sec) , antiderivative size = 657, normalized size of antiderivative = 2.53 \[ \int \frac {A+B \cosh (d+e x)+C \sinh (d+e x)}{(a+b \cosh (d+e x))^4} \, dx=\frac {\frac {3 \, {\left (2 \, A a^{3} - 4 \, B a^{2} b + 3 \, A a b^{2} - B b^{3}\right )} \arctan \left (\frac {b e^{\left (e x + d\right )} + 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 \, e x + 5 \, d\right )} - 12 \, B a^{2} b^{4} e^{\left (5 \, e x + 5 \, d\right )} + 9 \, A a b^{5} e^{\left (5 \, e x + 5 \, d\right )} - 3 \, B b^{6} e^{\left (5 \, e x + 5 \, d\right )} + 30 \, A a^{4} b^{2} e^{\left (4 \, e x + 4 \, d\right )} - 60 \, B a^{3} b^{3} e^{\left (4 \, e x + 4 \, d\right )} + 45 \, A a^{2} b^{4} e^{\left (4 \, e x + 4 \, d\right )} - 15 \, B a b^{5} e^{\left (4 \, e x + 4 \, d\right )} - 8 \, B a^{6} e^{\left (3 \, e x + 3 \, d\right )} - 8 \, C a^{6} e^{\left (3 \, e x + 3 \, d\right )} + 44 \, A a^{5} b e^{\left (3 \, e x + 3 \, d\right )} - 64 \, B a^{4} b^{2} e^{\left (3 \, e x + 3 \, d\right )} + 24 \, C a^{4} b^{2} e^{\left (3 \, e x + 3 \, d\right )} + 82 \, A a^{3} b^{3} e^{\left (3 \, e x + 3 \, d\right )} - 78 \, B a^{2} b^{4} e^{\left (3 \, e x + 3 \, d\right )} - 24 \, C a^{2} b^{4} e^{\left (3 \, e x + 3 \, d\right )} + 24 \, A a b^{5} e^{\left (3 \, e x + 3 \, d\right )} + 8 \, C b^{6} e^{\left (3 \, e x + 3 \, d\right )} - 24 \, B a^{5} b e^{\left (2 \, e x + 2 \, d\right )} + 102 \, A a^{4} b^{2} e^{\left (2 \, e x + 2 \, d\right )} - 102 \, B a^{3} b^{3} e^{\left (2 \, e x + 2 \, d\right )} + 36 \, A a^{2} b^{4} e^{\left (2 \, e x + 2 \, d\right )} - 24 \, B a b^{5} e^{\left (2 \, e x + 2 \, d\right )} + 12 \, A b^{6} e^{\left (2 \, e x + 2 \, d\right )} - 12 \, B a^{4} b^{2} e^{\left (e x + d\right )} + 60 \, A a^{3} b^{3} e^{\left (e x + d\right )} - 66 \, B a^{2} b^{4} e^{\left (e x + d\right )} + 15 \, A a b^{5} e^{\left (e x + d\right )} + 3 \, B b^{6} e^{\left (e x + d\right )} - 2 \, B a^{3} b^{3} + 11 \, A a^{2} b^{4} - 13 \, B a b^{5} + 4 \, A b^{6}}{{\left (a^{6} b - 3 \, a^{4} b^{3} + 3 \, a^{2} b^{5} - b^{7}\right )} {\left (b e^{\left (2 \, e x + 2 \, d\right )} + 2 \, a e^{\left (e x + d\right )} + b\right )}^{3}}}{3 \, e} \]
1/3*(3*(2*A*a^3 - 4*B*a^2*b + 3*A*a*b^2 - B*b^3)*arctan((b*e^(e*x + d) + 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)) + (6*A*a^3*b^3*e^(5*e*x + 5*d) - 12*B*a^2*b^4*e^(5*e*x + 5*d) + 9*A*a*b^5 *e^(5*e*x + 5*d) - 3*B*b^6*e^(5*e*x + 5*d) + 30*A*a^4*b^2*e^(4*e*x + 4*d) - 60*B*a^3*b^3*e^(4*e*x + 4*d) + 45*A*a^2*b^4*e^(4*e*x + 4*d) - 15*B*a*b^5 *e^(4*e*x + 4*d) - 8*B*a^6*e^(3*e*x + 3*d) - 8*C*a^6*e^(3*e*x + 3*d) + 44* A*a^5*b*e^(3*e*x + 3*d) - 64*B*a^4*b^2*e^(3*e*x + 3*d) + 24*C*a^4*b^2*e^(3 *e*x + 3*d) + 82*A*a^3*b^3*e^(3*e*x + 3*d) - 78*B*a^2*b^4*e^(3*e*x + 3*d) - 24*C*a^2*b^4*e^(3*e*x + 3*d) + 24*A*a*b^5*e^(3*e*x + 3*d) + 8*C*b^6*e^(3 *e*x + 3*d) - 24*B*a^5*b*e^(2*e*x + 2*d) + 102*A*a^4*b^2*e^(2*e*x + 2*d) - 102*B*a^3*b^3*e^(2*e*x + 2*d) + 36*A*a^2*b^4*e^(2*e*x + 2*d) - 24*B*a*b^5 *e^(2*e*x + 2*d) + 12*A*b^6*e^(2*e*x + 2*d) - 12*B*a^4*b^2*e^(e*x + d) + 6 0*A*a^3*b^3*e^(e*x + d) - 66*B*a^2*b^4*e^(e*x + d) + 15*A*a*b^5*e^(e*x + d ) + 3*B*b^6*e^(e*x + d) - 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*e*x + 2*d) + 2*a*e^(e*x + d) + b)^3))/e
Timed out. \[ \int \frac {A+B \cosh (d+e x)+C \sinh (d+e x)}{(a+b \cosh (d+e x))^4} \, dx=\int \frac {A+B\,\mathrm {cosh}\left (d+e\,x\right )+C\,\mathrm {sinh}\left (d+e\,x\right )}{{\left (a+b\,\mathrm {cosh}\left (d+e\,x\right )\right )}^4} \,d x \]