\(\int \frac {A+B \cosh (d+e x)+C \sinh (d+e x)}{(a+b \cosh (d+e x))^2} \, dx\) [207]

Optimal result

Integrand size = 31, antiderivative size = 121 \[ \int \frac {A+B \cosh (d+e x)+C \sinh (d+e x)}{(a+b \cosh (d+e x))^2} \, dx=\frac {2 (a A-b B) \text {arctanh}\left (\frac {\sqrt {a-b} \tanh \left (\frac {1}{2} (d+e x)\right )}{\sqrt {a+b}}\right )}{(a-b)^{3/2} (a+b)^{3/2} e}-\frac {C}{b e (a+b \cosh (d+e x))}-\frac {(A b-a B) \sinh (d+e x)}{\left (a^2-b^2\right ) e (a+b \cosh (d+e x))} \] Output:

2*(A*a-B*b)*arctanh((a-b)^(1/2)*tanh(1/2*e*x+1/2*d)/(a+b)^(1/2))/(a-b)^(3/ 
2)/(a+b)^(3/2)/e-C/b/e/(a+b*cosh(e*x+d))-(A*b-B*a)*sinh(e*x+d)/(a^2-b^2)/e 
/(a+b*cosh(e*x+d))
 

Mathematica [A] (verified)

Time = 0.55 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.95 \[ \int \frac {A+B \cosh (d+e x)+C \sinh (d+e x)}{(a+b \cosh (d+e x))^2} \, dx=\frac {\frac {2 (a A-b B) \arctan \left (\frac {(a-b) \tanh \left (\frac {1}{2} (d+e x)\right )}{\sqrt {-a^2+b^2}}\right )}{\left (-a^2+b^2\right )^{3/2}}+\frac {\left (-a^2+b^2\right ) C-b (A b-a B) \sinh (d+e x)}{(a-b) b (a+b) (a+b \cosh (d+e x))}}{e} \] Input:

Integrate[(A + B*Cosh[d + e*x] + C*Sinh[d + e*x])/(a + b*Cosh[d + e*x])^2, 
x]
 

Output:

((2*(a*A - b*B)*ArcTan[((a - b)*Tanh[(d + e*x)/2])/Sqrt[-a^2 + b^2]])/(-a^ 
2 + b^2)^(3/2) + ((-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])))/e
 

Rubi [A] (verified)

Time = 0.62 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.09, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.419, Rules used = {3042, 4877, 26, 3042, 26, 3147, 17, 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.

\(\displaystyle \int \frac {A+B \cosh (d+e x)+C \sinh (d+e x)}{(a+b \cosh (d+e x))^2} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {A+B \cos (i d+i e x)-i C \sin (i d+i e x)}{(a+b \cos (i d+i e x))^2}dx\)

\(\Big \downarrow \) 4877

\(\displaystyle \int \frac {A+B \cosh (d+e x)}{(a+b \cosh (d+e x))^2}dx-i C \int \frac {i \sinh (d+e x)}{(a+b \cosh (d+e x))^2}dx\)

\(\Big \downarrow \) 26

\(\displaystyle \int \frac {A+B \cosh (d+e x)}{(a+b \cosh (d+e x))^2}dx+C \int \frac {\sinh (d+e x)}{(a+b \cosh (d+e x))^2}dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {A+B \sin \left (i d+i e x+\frac {\pi }{2}\right )}{\left (a+b \sin \left (i d+i e x+\frac {\pi }{2}\right )\right )^2}dx+C \int -\frac {i \cos \left (i d+i e x-\frac {\pi }{2}\right )}{\left (a-b \sin \left (i d+i e x-\frac {\pi }{2}\right )\right )^2}dx\)

\(\Big \downarrow \) 26

\(\displaystyle \int \frac {A+B \sin \left (i d+i e x+\frac {\pi }{2}\right )}{\left (a+b \sin \left (i d+i e x+\frac {\pi }{2}\right )\right )^2}dx-i C \int \frac {\cos \left (\frac {1}{2} (2 i d-\pi )+i e x\right )}{\left (a-b \sin \left (\frac {1}{2} (2 i d-\pi )+i e x\right )\right )^2}dx\)

\(\Big \downarrow \) 3147

\(\displaystyle \frac {C \int \frac {1}{(a+b \cosh (d+e x))^2}d(b \cosh (d+e x))}{b e}+\int \frac {A+B \sin \left (i d+i e x+\frac {\pi }{2}\right )}{\left (a+b \sin \left (i d+i e x+\frac {\pi }{2}\right )\right )^2}dx\)

\(\Big \downarrow \) 17

\(\displaystyle -\frac {C}{b e (a+b \cosh (d+e x))}+\int \frac {A+B \sin \left (i d+i e x+\frac {\pi }{2}\right )}{\left (a+b \sin \left (i d+i e x+\frac {\pi }{2}\right )\right )^2}dx\)

\(\Big \downarrow \) 3233

\(\displaystyle -\frac {\int -\frac {a A-b B}{a+b \cosh (d+e x)}dx}{a^2-b^2}-\frac {(A b-a B) \sinh (d+e x)}{e \left (a^2-b^2\right ) (a+b \cosh (d+e x))}-\frac {C}{b e (a+b \cosh (d+e x))}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\int \frac {a A-b B}{a+b \cosh (d+e x)}dx}{a^2-b^2}-\frac {(A b-a B) \sinh (d+e x)}{e \left (a^2-b^2\right ) (a+b \cosh (d+e x))}-\frac {C}{b e (a+b \cosh (d+e x))}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {(a A-b B) \int \frac {1}{a+b \cosh (d+e x)}dx}{a^2-b^2}-\frac {(A b-a B) \sinh (d+e x)}{e \left (a^2-b^2\right ) (a+b \cosh (d+e x))}-\frac {C}{b e (a+b \cosh (d+e x))}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {(a A-b B) \int \frac {1}{a+b \sin \left (i d+i e x+\frac {\pi }{2}\right )}dx}{a^2-b^2}-\frac {(A b-a B) \sinh (d+e x)}{e \left (a^2-b^2\right ) (a+b \cosh (d+e x))}-\frac {C}{b e (a+b \cosh (d+e x))}\)

\(\Big \downarrow \) 3138

\(\displaystyle -\frac {2 i (a A-b B) \int \frac {1}{-\left ((a-b) \tanh ^2\left (\frac {1}{2} (d+e x)\right )\right )+a+b}d\left (i \tanh \left (\frac {1}{2} (d+e x)\right )\right )}{e \left (a^2-b^2\right )}-\frac {(A b-a B) \sinh (d+e x)}{e \left (a^2-b^2\right ) (a+b \cosh (d+e x))}-\frac {C}{b e (a+b \cosh (d+e x))}\)

\(\Big \downarrow \) 218

\(\displaystyle \frac {2 (a A-b B) \text {arctanh}\left (\frac {\sqrt {a-b} \tanh \left (\frac {1}{2} (d+e x)\right )}{\sqrt {a+b}}\right )}{e \sqrt {a-b} \sqrt {a+b} \left (a^2-b^2\right )}-\frac {(A b-a B) \sinh (d+e x)}{e \left (a^2-b^2\right ) (a+b \cosh (d+e x))}-\frac {C}{b e (a+b \cosh (d+e x))}\)

Input:

Int[(A + B*Cosh[d + e*x] + C*Sinh[d + e*x])/(a + b*Cosh[d + e*x])^2,x]
 

Output:

(2*(a*A - b*B)*ArcTanh[(Sqrt[a - b]*Tanh[(d + e*x)/2])/Sqrt[a + b]])/(Sqrt 
[a - b]*Sqrt[a + b]*(a^2 - b^2)*e) - C/(b*e*(a + b*Cosh[d + e*x])) - ((A*b 
 - a*B)*Sinh[d + e*x])/((a^2 - b^2)*e*(a + b*Cosh[d + e*x]))
 

Defintions of rubi rules used

rule 17

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]
 

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 26

Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a])   I 
nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 218

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]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3138

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]
 

rule 3147

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]
 

rule 3233

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]
 

rule 4877

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])
 

Maple [A] (verified)

Time = 1.86 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.19

method	result	size
derivativedivides	\(\frac {-\frac {2 \left (-\frac {\left (A b -B a \right ) \tanh \left (\frac {e x}{2}+\frac {d}{2}\right )}{a^{2}-b^{2}}+\frac {C}{a -b}\right )}{a \tanh \left (\frac {e x}{2}+\frac {d}{2}\right )^{2}-b \tanh \left (\frac {e x}{2}+\frac {d}{2}\right )^{2}-a -b}+\frac {2 \left (A a -B b \right ) \operatorname {arctanh}\left (\frac {\left (a -b \right ) \tanh \left (\frac {e x}{2}+\frac {d}{2}\right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{\left (a +b \right ) \left (a -b \right ) \sqrt {\left (a +b \right ) \left (a -b \right )}}}{e}\)	\(144\)
default	\(\frac {-\frac {2 \left (-\frac {\left (A b -B a \right ) \tanh \left (\frac {e x}{2}+\frac {d}{2}\right )}{a^{2}-b^{2}}+\frac {C}{a -b}\right )}{a \tanh \left (\frac {e x}{2}+\frac {d}{2}\right )^{2}-b \tanh \left (\frac {e x}{2}+\frac {d}{2}\right )^{2}-a -b}+\frac {2 \left (A a -B b \right ) \operatorname {arctanh}\left (\frac {\left (a -b \right ) \tanh \left (\frac {e x}{2}+\frac {d}{2}\right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{\left (a +b \right ) \left (a -b \right ) \sqrt {\left (a +b \right ) \left (a -b \right )}}}{e}\)	\(144\)
risch	\(\frac {2 A a b \,{\mathrm e}^{e x +d}-2 B \,a^{2} {\mathrm e}^{e x +d}-2 C \,a^{2} {\mathrm e}^{e x +d}+2 C \,b^{2} {\mathrm e}^{e x +d}+2 A \,b^{2}-2 B a b}{b e \left (a^{2}-b^{2}\right ) \left (b \,{\mathrm e}^{2 e x +2 d}+2 a \,{\mathrm e}^{e x +d}+b \right )}+\frac {\ln \left ({\mathrm e}^{e x +d}+\frac {a \sqrt {a^{2}-b^{2}}-a^{2}+b^{2}}{b \sqrt {a^{2}-b^{2}}}\right ) A a}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right ) \left (a -b \right ) e}-\frac {\ln \left ({\mathrm e}^{e x +d}+\frac {a \sqrt {a^{2}-b^{2}}-a^{2}+b^{2}}{b \sqrt {a^{2}-b^{2}}}\right ) B b}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right ) \left (a -b \right ) e}-\frac {\ln \left ({\mathrm e}^{e x +d}+\frac {a \sqrt {a^{2}-b^{2}}+a^{2}-b^{2}}{b \sqrt {a^{2}-b^{2}}}\right ) A a}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right ) \left (a -b \right ) e}+\frac {\ln \left ({\mathrm e}^{e x +d}+\frac {a \sqrt {a^{2}-b^{2}}+a^{2}-b^{2}}{b \sqrt {a^{2}-b^{2}}}\right ) B b}{\sqrt {a^{2}-b^{2}}\, \left (a +b \right ) \left (a -b \right ) e}\)	\(399\)

Input:

int((A+B*cosh(e*x+d)+C*sinh(e*x+d))/(a+b*cosh(e*x+d))^2,x,method=_RETURNVE 
RBOSE)
                                                                                    
                                                                                    
 

Output:

1/e*(-2*(-(A*b-B*a)/(a^2-b^2)*tanh(1/2*e*x+1/2*d)+C/(a-b))/(a*tanh(1/2*e*x 
+1/2*d)^2-b*tanh(1/2*e*x+1/2*d)^2-a-b)+2*(A*a-B*b)/(a+b)/(a-b)/((a+b)*(a-b 
))^(1/2)*arctanh((a-b)*tanh(1/2*e*x+1/2*d)/((a+b)*(a-b))^(1/2)))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 465 vs. \(2 (111) = 222\).

Time = 0.10 (sec) , antiderivative size = 1044, normalized size of antiderivative = 8.63 \[ \int \frac {A+B \cosh (d+e x)+C \sinh (d+e x)}{(a+b \cosh (d+e x))^2} \, dx=\text {Too large to display} \] Input:

integrate((A+B*cosh(e*x+d)+C*sinh(e*x+d))/(a+b*cosh(e*x+d))^2,x, algorithm 
="fricas")
 

Output:

[-(2*B*a^3*b - 2*A*a^2*b^2 - 2*B*a*b^3 + 2*A*b^4 - (A*a*b^2 - B*b^3 + (A*a 
*b^2 - B*b^3)*cosh(e*x + d)^2 + (A*a*b^2 - B*b^3)*sinh(e*x + d)^2 + 2*(A*a 
^2*b - B*a*b^2)*cosh(e*x + d) + 2*(A*a^2*b - B*a*b^2 + (A*a*b^2 - B*b^3)*c 
osh(e*x + d))*sinh(e*x + d))*sqrt(a^2 - b^2)*log((b^2*cosh(e*x + d)^2 + b^ 
2*sinh(e*x + d)^2 + 2*a*b*cosh(e*x + d) + 2*a^2 - b^2 + 2*(b^2*cosh(e*x + 
d) + a*b)*sinh(e*x + d) - 2*sqrt(a^2 - b^2)*(b*cosh(e*x + d) + b*sinh(e*x 
+ d) + a))/(b*cosh(e*x + d)^2 + b*sinh(e*x + d)^2 + 2*a*cosh(e*x + d) + 2* 
(b*cosh(e*x + d) + a)*sinh(e*x + d) + b)) + 2*((B + C)*a^4 - A*a^3*b - (B 
+ 2*C)*a^2*b^2 + A*a*b^3 + C*b^4)*cosh(e*x + d) + 2*((B + C)*a^4 - A*a^3*b 
 - (B + 2*C)*a^2*b^2 + A*a*b^3 + C*b^4)*sinh(e*x + d))/((a^4*b^2 - 2*a^2*b 
^4 + b^6)*e*cosh(e*x + d)^2 + (a^4*b^2 - 2*a^2*b^4 + b^6)*e*sinh(e*x + d)^ 
2 + 2*(a^5*b - 2*a^3*b^3 + a*b^5)*e*cosh(e*x + d) + (a^4*b^2 - 2*a^2*b^4 + 
 b^6)*e + 2*((a^4*b^2 - 2*a^2*b^4 + b^6)*e*cosh(e*x + d) + (a^5*b - 2*a^3* 
b^3 + a*b^5)*e)*sinh(e*x + d)), -2*(B*a^3*b - A*a^2*b^2 - B*a*b^3 + A*b^4 
+ (A*a*b^2 - B*b^3 + (A*a*b^2 - B*b^3)*cosh(e*x + d)^2 + (A*a*b^2 - B*b^3) 
*sinh(e*x + d)^2 + 2*(A*a^2*b - B*a*b^2)*cosh(e*x + d) + 2*(A*a^2*b - B*a* 
b^2 + (A*a*b^2 - B*b^3)*cosh(e*x + d))*sinh(e*x + d))*sqrt(-a^2 + b^2)*arc 
tan(-sqrt(-a^2 + b^2)*(b*cosh(e*x + d) + b*sinh(e*x + d) + a)/(a^2 - b^2)) 
 + ((B + C)*a^4 - A*a^3*b - (B + 2*C)*a^2*b^2 + A*a*b^3 + C*b^4)*cosh(e*x 
+ d) + ((B + C)*a^4 - A*a^3*b - (B + 2*C)*a^2*b^2 + A*a*b^3 + C*b^4)*si...
 

Sympy [F(-1)]

Timed out. \[ \int \frac {A+B \cosh (d+e x)+C \sinh (d+e x)}{(a+b \cosh (d+e x))^2} \, dx=\text {Timed out} \] Input:

integrate((A+B*cosh(e*x+d)+C*sinh(e*x+d))/(a+b*cosh(e*x+d))**2,x)
 

Output:

Timed out
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {A+B \cosh (d+e x)+C \sinh (d+e x)}{(a+b \cosh (d+e x))^2} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((A+B*cosh(e*x+d)+C*sinh(e*x+d))/(a+b*cosh(e*x+d))^2,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
 

Giac [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.28 \[ \int \frac {A+B \cosh (d+e x)+C \sinh (d+e x)}{(a+b \cosh (d+e x))^2} \, dx=\frac {2 \, {\left (\frac {{\left (A a - B b\right )} \arctan \left (\frac {b e^{\left (e x + d\right )} + a}{\sqrt {-a^{2} + b^{2}}}\right )}{{\left (a^{2} - b^{2}\right )} \sqrt {-a^{2} + b^{2}}} - \frac {B a^{2} e^{\left (e x + d\right )} + C a^{2} e^{\left (e x + d\right )} - A a b e^{\left (e x + d\right )} - C b^{2} e^{\left (e x + d\right )} + B a b - A b^{2}}{{\left (a^{2} b - b^{3}\right )} {\left (b e^{\left (2 \, e x + 2 \, d\right )} + 2 \, a e^{\left (e x + d\right )} + b\right )}}\right )}}{e} \] Input:

integrate((A+B*cosh(e*x+d)+C*sinh(e*x+d))/(a+b*cosh(e*x+d))^2,x, algorithm 
="giac")
 

Output:

2*((A*a - B*b)*arctan((b*e^(e*x + d) + a)/sqrt(-a^2 + b^2))/((a^2 - b^2)*s 
qrt(-a^2 + b^2)) - (B*a^2*e^(e*x + d) + C*a^2*e^(e*x + d) - A*a*b*e^(e*x + 
 d) - C*b^2*e^(e*x + d) + B*a*b - A*b^2)/((a^2*b - b^3)*(b*e^(2*e*x + 2*d) 
 + 2*a*e^(e*x + d) + b)))/e
 

Mupad [B] (verification not implemented)

Time = 2.69 (sec) , antiderivative size = 301, normalized size of antiderivative = 2.49 \[ \int \frac {A+B \cosh (d+e x)+C \sinh (d+e x)}{(a+b \cosh (d+e x))^2} \, dx=\frac {\frac {2\,\left (A\,b^3-B\,a\,b^2\right )}{b\,e\,\left (a^2\,b-b^3\right )}+\frac {2\,{\mathrm {e}}^{d+e\,x}\,\left (C\,b^4-B\,a^2\,b^2-C\,a^2\,b^2+A\,a\,b^3\right )}{b^2\,e\,\left (a^2\,b-b^3\right )}}{b+2\,a\,{\mathrm {e}}^{d+e\,x}+b\,{\mathrm {e}}^{2\,d+2\,e\,x}}+\frac {\ln \left (-\frac {2\,{\mathrm {e}}^{d+e\,x}\,\left (A\,a-B\,b\right )}{b\,\left (a^2-b^2\right )}-\frac {2\,\left (A\,a-B\,b\right )\,\left (b+a\,{\mathrm {e}}^{d+e\,x}\right )}{b\,{\left (a+b\right )}^{3/2}\,{\left (a-b\right )}^{3/2}}\right )\,\left (A\,a-B\,b\right )}{e\,{\left (a+b\right )}^{3/2}\,{\left (a-b\right )}^{3/2}}-\frac {\ln \left (\frac {2\,\left (A\,a-B\,b\right )\,\left (b+a\,{\mathrm {e}}^{d+e\,x}\right )}{b\,{\left (a+b\right )}^{3/2}\,{\left (a-b\right )}^{3/2}}-\frac {2\,{\mathrm {e}}^{d+e\,x}\,\left (A\,a-B\,b\right )}{b\,\left (a^2-b^2\right )}\right )\,\left (A\,a-B\,b\right )}{e\,{\left (a+b\right )}^{3/2}\,{\left (a-b\right )}^{3/2}} \] Input:

int((A + B*cosh(d + e*x) + C*sinh(d + e*x))/(a + b*cosh(d + e*x))^2,x)
 

Output:

((2*(A*b^3 - B*a*b^2))/(b*e*(a^2*b - b^3)) + (2*exp(d + e*x)*(C*b^4 - B*a^ 
2*b^2 - C*a^2*b^2 + A*a*b^3))/(b^2*e*(a^2*b - b^3)))/(b + 2*a*exp(d + e*x) 
 + b*exp(2*d + 2*e*x)) + (log(- (2*exp(d + e*x)*(A*a - B*b))/(b*(a^2 - b^2 
)) - (2*(A*a - B*b)*(b + a*exp(d + e*x)))/(b*(a + b)^(3/2)*(a - b)^(3/2))) 
*(A*a - B*b))/(e*(a + b)^(3/2)*(a - b)^(3/2)) - (log((2*(A*a - B*b)*(b + a 
*exp(d + e*x)))/(b*(a + b)^(3/2)*(a - b)^(3/2)) - (2*exp(d + e*x)*(A*a - B 
*b))/(b*(a^2 - b^2)))*(A*a - B*b))/(e*(a + b)^(3/2)*(a - b)^(3/2))
 

Reduce [B] (verification not implemented)

Time = 0.26 (sec) , antiderivative size = 253, normalized size of antiderivative = 2.09 \[ \int \frac {A+B \cosh (d+e x)+C \sinh (d+e x)}{(a+b \cosh (d+e x))^2} \, dx=\frac {-2 e^{2 e x +2 d} \sqrt {-a^{2}+b^{2}}\, \mathit {atan} \left (\frac {e^{e x +d} b +a}{\sqrt {-a^{2}+b^{2}}}\right ) a b -4 e^{e x +d} \sqrt {-a^{2}+b^{2}}\, \mathit {atan} \left (\frac {e^{e x +d} b +a}{\sqrt {-a^{2}+b^{2}}}\right ) a^{2}-2 \sqrt {-a^{2}+b^{2}}\, \mathit {atan} \left (\frac {e^{e x +d} b +a}{\sqrt {-a^{2}+b^{2}}}\right ) a b +e^{2 e x +2 d} a^{2} c -e^{2 e x +2 d} b^{2} c +a^{2} c -b^{2} c}{a e \left (e^{2 e x +2 d} a^{2} b -e^{2 e x +2 d} b^{3}+2 e^{e x +d} a^{3}-2 e^{e x +d} a \,b^{2}+a^{2} b -b^{3}\right )} \] Input:

int((A+B*cosh(e*x+d)+C*sinh(e*x+d))/(a+b*cosh(e*x+d))^2,x)
 

Output:

( - 2*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 - 4*e**(d + e*x)*sqrt( - a**2 + b**2)*atan((e**(d + 
e*x)*b + a)/sqrt( - a**2 + b**2))*a**2 - 2*sqrt( - a**2 + b**2)*atan((e**( 
d + e*x)*b + a)/sqrt( - a**2 + b**2))*a*b + e**(2*d + 2*e*x)*a**2*c - e**( 
2*d + 2*e*x)*b**2*c + a**2*c - b**2*c)/(a*e*(e**(2*d + 2*e*x)*a**2*b - e** 
(2*d + 2*e*x)*b**3 + 2*e**(d + e*x)*a**3 - 2*e**(d + e*x)*a*b**2 + a**2*b 
- b**3))