\(\int \frac {A+B \cosh (d+e x)+C \sinh (d+e x)}{a+b \cosh (d+e x)} \, dx\) [206]

Optimal result

Integrand size = 31, antiderivative size = 86 \[ \int \frac {A+B \cosh (d+e x)+C \sinh (d+e x)}{a+b \cosh (d+e x)} \, dx=\frac {B x}{b}+\frac {2 (A b-a B) \text {arctanh}\left (\frac {\sqrt {a-b} \tanh \left (\frac {1}{2} (d+e x)\right )}{\sqrt {a+b}}\right )}{\sqrt {a-b} b \sqrt {a+b} e}+\frac {C \log (a+b \cosh (d+e x))}{b e} \] Output:

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

Mathematica [A] (verified)

Time = 0.44 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.94 \[ \int \frac {A+B \cosh (d+e x)+C \sinh (d+e x)}{a+b \cosh (d+e x)} \, dx=\frac {B (d+e x)+\frac {2 (-A b+a B) \arctan \left (\frac {(a-b) \tanh \left (\frac {1}{2} (d+e x)\right )}{\sqrt {-a^2+b^2}}\right )}{\sqrt {-a^2+b^2}}+C \log (a+b \cosh (d+e x))}{b e} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.59 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.355, Rules used = {3042, 4877, 26, 3042, 26, 3147, 16, 3214, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4877

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 3147

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

\(\Big \downarrow \) 16

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

\(\Big \downarrow \) 3214

\(\displaystyle \frac {(A b-a B) \int \frac {1}{a+b \cosh (d+e x)}dx}{b}+\frac {C \log (a+b \cosh (d+e x))}{b e}+\frac {B x}{b}\)

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3138

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

\(\Big \downarrow \) 218

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 16

Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + 
b*x, x]]/b), x] /; FreeQ[{a, b, c}, 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 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 3214

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])/((c_.) + (d_.)*sin[(e_.) + (f_. 
)*(x_)]), x_Symbol] :> Simp[b*(x/d), x] - Simp[(b*c - a*d)/d   Int[1/(c + d 
*Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0]
 

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 [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(155\) vs. \(2(77)=154\).

Time = 1.12 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.81

method	result	size
derivativedivides	\(\frac {\frac {\left (-B -C \right ) \ln \left (\tanh \left (\frac {e x}{2}+\frac {d}{2}\right )-1\right )}{b}+\frac {\left (B -C \right ) \ln \left (\tanh \left (\frac {e x}{2}+\frac {d}{2}\right )+1\right )}{b}+\frac {\frac {2 \left (a C -b C \right ) \ln \left (a \tanh \left (\frac {e x}{2}+\frac {d}{2}\right )^{2}-b \tanh \left (\frac {e x}{2}+\frac {d}{2}\right )^{2}-a -b \right )}{2 a -2 b}-\frac {2 \left (-A b +B a \right ) \operatorname {arctanh}\left (\frac {\left (a -b \right ) \tanh \left (\frac {e x}{2}+\frac {d}{2}\right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}}{b}}{e}\)	\(156\)
default	\(\frac {\frac {\left (-B -C \right ) \ln \left (\tanh \left (\frac {e x}{2}+\frac {d}{2}\right )-1\right )}{b}+\frac {\left (B -C \right ) \ln \left (\tanh \left (\frac {e x}{2}+\frac {d}{2}\right )+1\right )}{b}+\frac {\frac {2 \left (a C -b C \right ) \ln \left (a \tanh \left (\frac {e x}{2}+\frac {d}{2}\right )^{2}-b \tanh \left (\frac {e x}{2}+\frac {d}{2}\right )^{2}-a -b \right )}{2 a -2 b}-\frac {2 \left (-A b +B a \right ) \operatorname {arctanh}\left (\frac {\left (a -b \right ) \tanh \left (\frac {e x}{2}+\frac {d}{2}\right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}}{b}}{e}\)	\(156\)
risch	\(\frac {B x}{b}+\frac {x C}{b}+\frac {2 C \,a^{2} b \,e^{2} x}{-a^{2} b^{2} e^{2}+b^{4} e^{2}}-\frac {2 C \,b^{3} e^{2} x}{-a^{2} b^{2} e^{2}+b^{4} e^{2}}+\frac {2 C \,a^{2} b d e}{-a^{2} b^{2} e^{2}+b^{4} e^{2}}-\frac {2 C \,b^{3} d e}{-a^{2} b^{2} e^{2}+b^{4} e^{2}}+\frac {\ln \left ({\mathrm e}^{e x +d}+\frac {A a b -B \,a^{2}-\sqrt {A^{2} a^{2} b^{2}-A^{2} b^{4}-2 A B \,a^{3} b +2 A B a \,b^{3}+B^{2} a^{4}-B^{2} a^{2} b^{2}}}{b \left (A b -B a \right )}\right ) C \,a^{2}}{\left (a^{2}-b^{2}\right ) e b}-\frac {b \ln \left ({\mathrm e}^{e x +d}+\frac {A a b -B \,a^{2}-\sqrt {A^{2} a^{2} b^{2}-A^{2} b^{4}-2 A B \,a^{3} b +2 A B a \,b^{3}+B^{2} a^{4}-B^{2} a^{2} b^{2}}}{b \left (A b -B a \right )}\right ) C}{\left (a^{2}-b^{2}\right ) e}+\frac {\ln \left ({\mathrm e}^{e x +d}+\frac {A a b -B \,a^{2}-\sqrt {A^{2} a^{2} b^{2}-A^{2} b^{4}-2 A B \,a^{3} b +2 A B a \,b^{3}+B^{2} a^{4}-B^{2} a^{2} b^{2}}}{b \left (A b -B a \right )}\right ) \sqrt {A^{2} a^{2} b^{2}-A^{2} b^{4}-2 A B \,a^{3} b +2 A B a \,b^{3}+B^{2} a^{4}-B^{2} a^{2} b^{2}}}{\left (a^{2}-b^{2}\right ) e b}+\frac {\ln \left ({\mathrm e}^{e x +d}+\frac {A a b -B \,a^{2}+\sqrt {A^{2} a^{2} b^{2}-A^{2} b^{4}-2 A B \,a^{3} b +2 A B a \,b^{3}+B^{2} a^{4}-B^{2} a^{2} b^{2}}}{b \left (A b -B a \right )}\right ) C \,a^{2}}{\left (a^{2}-b^{2}\right ) e b}-\frac {b \ln \left ({\mathrm e}^{e x +d}+\frac {A a b -B \,a^{2}+\sqrt {A^{2} a^{2} b^{2}-A^{2} b^{4}-2 A B \,a^{3} b +2 A B a \,b^{3}+B^{2} a^{4}-B^{2} a^{2} b^{2}}}{b \left (A b -B a \right )}\right ) C}{\left (a^{2}-b^{2}\right ) e}-\frac {\ln \left ({\mathrm e}^{e x +d}+\frac {A a b -B \,a^{2}+\sqrt {A^{2} a^{2} b^{2}-A^{2} b^{4}-2 A B \,a^{3} b +2 A B a \,b^{3}+B^{2} a^{4}-B^{2} a^{2} b^{2}}}{b \left (A b -B a \right )}\right ) \sqrt {A^{2} a^{2} b^{2}-A^{2} b^{4}-2 A B \,a^{3} b +2 A B a \,b^{3}+B^{2} a^{4}-B^{2} a^{2} b^{2}}}{\left (a^{2}-b^{2}\right ) e b}\)	\(897\)

Input:

int((A+B*cosh(e*x+d)+C*sinh(e*x+d))/(a+b*cosh(e*x+d)),x,method=_RETURNVERB 
OSE)
 

Output:

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

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 405, normalized size of antiderivative = 4.71 \[ \int \frac {A+B \cosh (d+e x)+C \sinh (d+e x)}{a+b \cosh (d+e x)} \, dx=\left [\frac {{\left ({\left (B - C\right )} a^{2} - {\left (B - C\right )} b^{2}\right )} e x - {\left (B a - A b\right )} \sqrt {a^{2} - b^{2}} \log \left (\frac {b^{2} \cosh \left (e x + d\right )^{2} + b^{2} \sinh \left (e x + d\right )^{2} + 2 \, a b \cosh \left (e x + d\right ) + 2 \, a^{2} - b^{2} + 2 \, {\left (b^{2} \cosh \left (e x + d\right ) + a b\right )} \sinh \left (e x + d\right ) - 2 \, \sqrt {a^{2} - b^{2}} {\left (b \cosh \left (e x + d\right ) + b \sinh \left (e x + d\right ) + a\right )}}{b \cosh \left (e x + d\right )^{2} + b \sinh \left (e x + d\right )^{2} + 2 \, a \cosh \left (e x + d\right ) + 2 \, {\left (b \cosh \left (e x + d\right ) + a\right )} \sinh \left (e x + d\right ) + b}\right ) + {\left (C a^{2} - C b^{2}\right )} \log \left (\frac {2 \, {\left (b \cosh \left (e x + d\right ) + a\right )}}{\cosh \left (e x + d\right ) - \sinh \left (e x + d\right )}\right )}{{\left (a^{2} b - b^{3}\right )} e}, \frac {{\left ({\left (B - C\right )} a^{2} - {\left (B - C\right )} b^{2}\right )} e x + 2 \, {\left (B a - A b\right )} \sqrt {-a^{2} + b^{2}} \arctan \left (-\frac {\sqrt {-a^{2} + b^{2}} {\left (b \cosh \left (e x + d\right ) + b \sinh \left (e x + d\right ) + a\right )}}{a^{2} - b^{2}}\right ) + {\left (C a^{2} - C b^{2}\right )} \log \left (\frac {2 \, {\left (b \cosh \left (e x + d\right ) + a\right )}}{\cosh \left (e x + d\right ) - \sinh \left (e x + d\right )}\right )}{{\left (a^{2} b - b^{3}\right )} e}\right ] \] Input:

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

Output:

[(((B - C)*a^2 - (B - C)*b^2)*e*x - (B*a - A*b)*sqrt(a^2 - b^2)*log((b^2*c 
osh(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)) + (C*a^2 - C* 
b^2)*log(2*(b*cosh(e*x + d) + a)/(cosh(e*x + d) - sinh(e*x + d))))/((a^2*b 
 - b^3)*e), (((B - C)*a^2 - (B - C)*b^2)*e*x + 2*(B*a - A*b)*sqrt(-a^2 + b 
^2)*arctan(-sqrt(-a^2 + b^2)*(b*cosh(e*x + d) + b*sinh(e*x + d) + a)/(a^2 
- b^2)) + (C*a^2 - C*b^2)*log(2*(b*cosh(e*x + d) + a)/(cosh(e*x + d) - sin 
h(e*x + d))))/((a^2*b - b^3)*e)]
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 695 vs. \(2 (73) = 146\).

Time = 15.80 (sec) , antiderivative size = 695, normalized size of antiderivative = 8.08 \[ \int \frac {A+B \cosh (d+e x)+C \sinh (d+e x)}{a+b \cosh (d+e x)} \, dx =\text {Too large to display} \] Input:

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

Output:

Piecewise((zoo*x*(A + B*cosh(d) + C*sinh(d))/cosh(d), Eq(a, 0) & Eq(b, 0) 
& Eq(e, 0)), (A*tanh(d/2 + e*x/2)/(b*e) + B*x/b - B*tanh(d/2 + e*x/2)/(b*e 
) + C*x/b - 2*C*log(tanh(d/2 + e*x/2) + 1)/(b*e), Eq(a, b)), (-A/(b*e*tanh 
(d/2 + e*x/2)) + B*x/b - B/(b*e*tanh(d/2 + e*x/2)) + C*x/b - 2*C*log(tanh( 
d/2 + e*x/2) + 1)/(b*e) + 2*C*log(tanh(d/2 + e*x/2))/(b*e), Eq(a, -b)), (( 
A*x + B*sinh(d + e*x)/e + C*cosh(d + e*x)/e)/a, Eq(b, 0)), (x*(A + B*cosh( 
d) + C*sinh(d))/(a + b*cosh(d)), Eq(e, 0)), (-A*b*sqrt(a/(a - b) + b/(a - 
b))*log(-sqrt(a/(a - b) + b/(a - b)) + tanh(d/2 + e*x/2))/(a*b*e + b**2*e) 
 + A*b*sqrt(a/(a - b) + b/(a - b))*log(sqrt(a/(a - b) + b/(a - b)) + tanh( 
d/2 + e*x/2))/(a*b*e + b**2*e) + B*a*e*x/(a*b*e + b**2*e) + B*a*sqrt(a/(a 
- b) + b/(a - b))*log(-sqrt(a/(a - b) + b/(a - b)) + tanh(d/2 + e*x/2))/(a 
*b*e + b**2*e) - B*a*sqrt(a/(a - b) + b/(a - b))*log(sqrt(a/(a - b) + b/(a 
 - b)) + tanh(d/2 + e*x/2))/(a*b*e + b**2*e) + B*b*e*x/(a*b*e + b**2*e) + 
C*a*e*x/(a*b*e + b**2*e) + C*a*log(-sqrt(a/(a - b) + b/(a - b)) + tanh(d/2 
 + e*x/2))/(a*b*e + b**2*e) + C*a*log(sqrt(a/(a - b) + b/(a - b)) + tanh(d 
/2 + e*x/2))/(a*b*e + b**2*e) - 2*C*a*log(tanh(d/2 + e*x/2) + 1)/(a*b*e + 
b**2*e) + C*b*e*x/(a*b*e + b**2*e) + C*b*log(-sqrt(a/(a - b) + b/(a - b)) 
+ tanh(d/2 + e*x/2))/(a*b*e + b**2*e) + C*b*log(sqrt(a/(a - b) + b/(a - b) 
) + tanh(d/2 + e*x/2))/(a*b*e + b**2*e) - 2*C*b*log(tanh(d/2 + e*x/2) + 1) 
/(a*b*e + b**2*e), True))
 

Maxima [F(-2)]

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

integrate((A+B*cosh(e*x+d)+C*sinh(e*x+d))/(a+b*cosh(e*x+d)),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 = 0.12 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.09 \[ \int \frac {A+B \cosh (d+e x)+C \sinh (d+e x)}{a+b \cosh (d+e x)} \, dx=\frac {\frac {{\left (e x + d\right )} {\left (B - C\right )}}{b} + \frac {C \log \left (b e^{\left (2 \, e x + 2 \, d\right )} + 2 \, a e^{\left (e x + d\right )} + b\right )}{b} - \frac {2 \, {\left (B a - A b\right )} \arctan \left (\frac {b e^{\left (e x + d\right )} + a}{\sqrt {-a^{2} + b^{2}}}\right )}{\sqrt {-a^{2} + b^{2}} b}}{e} \] Input:

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

Output:

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

Mupad [B] (verification not implemented)

Time = 3.27 (sec) , antiderivative size = 653, normalized size of antiderivative = 7.59 \[ \int \frac {A+B \cosh (d+e x)+C \sinh (d+e x)}{a+b \cosh (d+e x)} \, dx=\frac {2\,\mathrm {atan}\left (\frac {a\,\sqrt {b^4\,e^2-a^2\,b^2\,e^2}\,\sqrt {A^2\,b^2-2\,A\,B\,a\,b+B^2\,a^2}}{B\,e\,a^3\,b-A\,e\,a^2\,b^2-B\,e\,a\,b^3+A\,e\,b^4}+\frac {a^2\,b^2\,{\mathrm {e}}^{e\,x}\,{\mathrm {e}}^d\,\sqrt {b^4\,e^2-a^2\,b^2\,e^2}\,\sqrt {A^2\,b^2-2\,A\,B\,a\,b+B^2\,a^2}}{B\,e\,a^3\,b^4-A\,e\,a^2\,b^5-B\,e\,a\,b^6+A\,e\,b^7}+\frac {A\,{\mathrm {e}}^{e\,x}\,{\mathrm {e}}^d\,\sqrt {b^4\,e^2-a^2\,b^2\,e^2}}{b\,e\,\sqrt {A^2\,b^2-2\,A\,B\,a\,b+B^2\,a^2}}-\frac {B\,a\,{\mathrm {e}}^{e\,x}\,{\mathrm {e}}^d\,\sqrt {b^4\,e^2-a^2\,b^2\,e^2}}{b^2\,e\,\sqrt {A^2\,b^2-2\,A\,B\,a\,b+B^2\,a^2}}\right )\,\sqrt {A^2\,b^2-2\,A\,B\,a\,b+B^2\,a^2}}{\sqrt {b^4\,e^2-a^2\,b^2\,e^2}}+\frac {B\,x}{b}-\frac {C\,x}{b}+\frac {C\,b^3\,e\,\ln \left (4\,A^2\,b^3+4\,B^2\,a^2\,b-8\,A\,B\,a\,b^2+8\,B^2\,a^3\,{\mathrm {e}}^{e\,x}\,{\mathrm {e}}^d+4\,A^2\,b^3\,{\mathrm {e}}^{2\,d}\,{\mathrm {e}}^{2\,e\,x}+8\,A^2\,a\,b^2\,{\mathrm {e}}^{e\,x}\,{\mathrm {e}}^d+4\,B^2\,a^2\,b\,{\mathrm {e}}^{2\,d}\,{\mathrm {e}}^{2\,e\,x}-16\,A\,B\,a^2\,b\,{\mathrm {e}}^{e\,x}\,{\mathrm {e}}^d-8\,A\,B\,a\,b^2\,{\mathrm {e}}^{2\,d}\,{\mathrm {e}}^{2\,e\,x}\right )}{b^4\,e^2-a^2\,b^2\,e^2}-\frac {C\,a^2\,b\,e\,\ln \left (4\,A^2\,b^3+4\,B^2\,a^2\,b-8\,A\,B\,a\,b^2+8\,B^2\,a^3\,{\mathrm {e}}^{e\,x}\,{\mathrm {e}}^d+4\,A^2\,b^3\,{\mathrm {e}}^{2\,d}\,{\mathrm {e}}^{2\,e\,x}+8\,A^2\,a\,b^2\,{\mathrm {e}}^{e\,x}\,{\mathrm {e}}^d+4\,B^2\,a^2\,b\,{\mathrm {e}}^{2\,d}\,{\mathrm {e}}^{2\,e\,x}-16\,A\,B\,a^2\,b\,{\mathrm {e}}^{e\,x}\,{\mathrm {e}}^d-8\,A\,B\,a\,b^2\,{\mathrm {e}}^{2\,d}\,{\mathrm {e}}^{2\,e\,x}\right )}{b^4\,e^2-a^2\,b^2\,e^2} \] Input:

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

Output:

(2*atan((a*(b^4*e^2 - a^2*b^2*e^2)^(1/2)*(A^2*b^2 + B^2*a^2 - 2*A*B*a*b)^( 
1/2))/(A*b^4*e - B*a*b^3*e + B*a^3*b*e - A*a^2*b^2*e) + (a^2*b^2*exp(e*x)* 
exp(d)*(b^4*e^2 - a^2*b^2*e^2)^(1/2)*(A^2*b^2 + B^2*a^2 - 2*A*B*a*b)^(1/2) 
)/(A*b^7*e - B*a*b^6*e - A*a^2*b^5*e + B*a^3*b^4*e) + (A*exp(e*x)*exp(d)*( 
b^4*e^2 - a^2*b^2*e^2)^(1/2))/(b*e*(A^2*b^2 + B^2*a^2 - 2*A*B*a*b)^(1/2)) 
- (B*a*exp(e*x)*exp(d)*(b^4*e^2 - a^2*b^2*e^2)^(1/2))/(b^2*e*(A^2*b^2 + B^ 
2*a^2 - 2*A*B*a*b)^(1/2)))*(A^2*b^2 + B^2*a^2 - 2*A*B*a*b)^(1/2))/(b^4*e^2 
 - a^2*b^2*e^2)^(1/2) + (B*x)/b - (C*x)/b + (C*b^3*e*log(4*A^2*b^3 + 4*B^2 
*a^2*b - 8*A*B*a*b^2 + 8*B^2*a^3*exp(e*x)*exp(d) + 4*A^2*b^3*exp(2*d)*exp( 
2*e*x) + 8*A^2*a*b^2*exp(e*x)*exp(d) + 4*B^2*a^2*b*exp(2*d)*exp(2*e*x) - 1 
6*A*B*a^2*b*exp(e*x)*exp(d) - 8*A*B*a*b^2*exp(2*d)*exp(2*e*x)))/(b^4*e^2 - 
 a^2*b^2*e^2) - (C*a^2*b*e*log(4*A^2*b^3 + 4*B^2*a^2*b - 8*A*B*a*b^2 + 8*B 
^2*a^3*exp(e*x)*exp(d) + 4*A^2*b^3*exp(2*d)*exp(2*e*x) + 8*A^2*a*b^2*exp(e 
*x)*exp(d) + 4*B^2*a^2*b*exp(2*d)*exp(2*e*x) - 16*A*B*a^2*b*exp(e*x)*exp(d 
) - 8*A*B*a*b^2*exp(2*d)*exp(2*e*x)))/(b^4*e^2 - a^2*b^2*e^2)
 

Reduce [B] (verification not implemented)

Time = 0.28 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.29 \[ \int \frac {A+B \cosh (d+e x)+C \sinh (d+e x)}{a+b \cosh (d+e x)} \, dx=\frac {\mathrm {log}\left (\cosh \left (e x +d \right ) b +a \right ) c +b e x}{b e} \] Input:

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

Output:

(log(cosh(d + e*x)*b + a)*c + b*e*x)/(b*e)