\(\int \frac {A+B \cosh (d+e x)+C \sinh (d+e x)}{(a+c \sinh (d+e x))^3} \, dx\) [255]

Optimal result

Integrand size = 31, antiderivative size = 180 \[ \int \frac {A+B \cosh (d+e x)+C \sinh (d+e x)}{(a+c \sinh (d+e x))^3} \, dx=-\frac {\left (2 a^2 A-A c^2+3 a c C\right ) \text {arctanh}\left (\frac {c-a \tanh \left (\frac {1}{2} (d+e x)\right )}{\sqrt {a^2+c^2}}\right )}{\left (a^2+c^2\right )^{5/2} e}-\frac {B}{2 c e (a+c \sinh (d+e x))^2}-\frac {(A c-a C) \cosh (d+e x)}{2 \left (a^2+c^2\right ) e (a+c \sinh (d+e x))^2}-\frac {\left (3 a A c-a^2 C+2 c^2 C\right ) \cosh (d+e x)}{2 \left (a^2+c^2\right )^2 e (a+c \sinh (d+e x))} \] Output:

-(2*A*a^2-A*c^2+3*C*a*c)*arctanh((c-a*tanh(1/2*e*x+1/2*d))/(a^2+c^2)^(1/2) 
)/(a^2+c^2)^(5/2)/e-1/2*B/c/e/(a+c*sinh(e*x+d))^2-1/2*(A*c-C*a)*cosh(e*x+d 
)/(a^2+c^2)/e/(a+c*sinh(e*x+d))^2-1/2*(3*A*a*c-C*a^2+2*C*c^2)*cosh(e*x+d)/ 
(a^2+c^2)^2/e/(a+c*sinh(e*x+d))
 

Mathematica [A] (verified)

Time = 1.44 (sec) , antiderivative size = 170, normalized size of antiderivative = 0.94 \[ \int \frac {A+B \cosh (d+e x)+C \sinh (d+e x)}{(a+c \sinh (d+e x))^3} \, dx=\frac {\frac {2 \left (2 a^2 A-A c^2+3 a c C\right ) \arctan \left (\frac {c-a \tanh \left (\frac {1}{2} (d+e x)\right )}{\sqrt {-a^2-c^2}}\right )}{\sqrt {-a^2-c^2}}-\frac {\left (a^2+c^2\right ) \left (B \left (a^2+c^2\right )+c (A c-a C) \cosh (d+e x)\right )}{c (a+c \sinh (d+e x))^2}+\frac {\left (-3 a A c+a^2 C-2 c^2 C\right ) \cosh (d+e x)}{a+c \sinh (d+e x)}}{2 \left (a^2+c^2\right )^2 e} \] Input:

Integrate[(A + B*Cosh[d + e*x] + C*Sinh[d + e*x])/(a + c*Sinh[d + e*x])^3, 
x]
 

Output:

((2*(2*a^2*A - A*c^2 + 3*a*c*C)*ArcTan[(c - a*Tanh[(d + e*x)/2])/Sqrt[-a^2 
 - c^2]])/Sqrt[-a^2 - c^2] - ((a^2 + c^2)*(B*(a^2 + c^2) + c*(A*c - a*C)*C 
osh[d + e*x]))/(c*(a + c*Sinh[d + e*x])^2) + ((-3*a*A*c + a^2*C - 2*c^2*C) 
*Cosh[d + e*x])/(a + c*Sinh[d + e*x]))/(2*(a^2 + c^2)^2*e)
 

Rubi [A] (warning: unable to verify)

Time = 0.90 (sec) , antiderivative size = 190, normalized size of antiderivative = 1.06, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.484, Rules used = {3042, 4876, 3042, 3147, 17, 3233, 25, 3042, 3233, 25, 27, 3042, 3139, 1083, 217}

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.

\(\displaystyle \int \frac {A+B \cosh (d+e x)+C \sinh (d+e x)}{(a+c \sinh (d+e x))^3} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {A+B \cos (i d+i e x)-i C \sin (i d+i e x)}{(a-i c \sin (i d+i e x))^3}dx\)

\(\Big \downarrow \) 4876

\(\displaystyle \int \frac {A+C \sinh (d+e x)}{(a+c \sinh (d+e x))^3}dx+B \int \frac {\cosh (d+e x)}{(a+c \sinh (d+e x))^3}dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {A-i C \sin (i d+i e x)}{(a-i c \sin (i d+i e x))^3}dx+B \int \frac {\cos (i d+i e x)}{(a-i c \sin (i d+i e x))^3}dx\)

\(\Big \downarrow \) 3147

\(\displaystyle \frac {B \int \frac {1}{(a+c \sinh (d+e x))^3}d(c \sinh (d+e x))}{c e}+\int \frac {A-i C \sin (i d+i e x)}{(a-i c \sin (i d+i e x))^3}dx\)

\(\Big \downarrow \) 17

\(\displaystyle -\frac {B}{2 c e (a+c \sinh (d+e x))^2}+\int \frac {A-i C \sin (i d+i e x)}{(a-i c \sin (i d+i e x))^3}dx\)

\(\Big \downarrow \) 3233

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3233

\(\displaystyle \frac {-\frac {\int -\frac {2 A a^2+3 c C a-A c^2}{a+c \sinh (d+e x)}dx}{a^2+c^2}-\frac {\left (a^2 (-C)+3 a A c+2 c^2 C\right ) \cosh (d+e x)}{e \left (a^2+c^2\right ) (a+c \sinh (d+e x))}}{2 \left (a^2+c^2\right )}-\frac {(A c-a C) \cosh (d+e x)}{2 e \left (a^2+c^2\right ) (a+c \sinh (d+e x))^2}-\frac {B}{2 c e (a+c \sinh (d+e x))^2}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {\int \frac {2 A a^2+3 c C a-A c^2}{a+c \sinh (d+e x)}dx}{a^2+c^2}-\frac {\left (a^2 (-C)+3 a A c+2 c^2 C\right ) \cosh (d+e x)}{e \left (a^2+c^2\right ) (a+c \sinh (d+e x))}}{2 \left (a^2+c^2\right )}-\frac {(A c-a C) \cosh (d+e x)}{2 e \left (a^2+c^2\right ) (a+c \sinh (d+e x))^2}-\frac {B}{2 c e (a+c \sinh (d+e x))^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\left (2 a^2 A+3 a c C-A c^2\right ) \int \frac {1}{a+c \sinh (d+e x)}dx}{a^2+c^2}-\frac {\left (a^2 (-C)+3 a A c+2 c^2 C\right ) \cosh (d+e x)}{e \left (a^2+c^2\right ) (a+c \sinh (d+e x))}}{2 \left (a^2+c^2\right )}-\frac {(A c-a C) \cosh (d+e x)}{2 e \left (a^2+c^2\right ) (a+c \sinh (d+e x))^2}-\frac {B}{2 c e (a+c \sinh (d+e x))^2}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {-\frac {\left (a^2 (-C)+3 a A c+2 c^2 C\right ) \cosh (d+e x)}{e \left (a^2+c^2\right ) (a+c \sinh (d+e x))}+\frac {\left (2 a^2 A+3 a c C-A c^2\right ) \int \frac {1}{a-i c \sin (i d+i e x)}dx}{a^2+c^2}}{2 \left (a^2+c^2\right )}-\frac {(A c-a C) \cosh (d+e x)}{2 e \left (a^2+c^2\right ) (a+c \sinh (d+e x))^2}-\frac {B}{2 c e (a+c \sinh (d+e x))^2}\)

\(\Big \downarrow \) 3139

\(\displaystyle \frac {-\frac {\left (a^2 (-C)+3 a A c+2 c^2 C\right ) \cosh (d+e x)}{e \left (a^2+c^2\right ) (a+c \sinh (d+e x))}-\frac {2 i \left (2 a^2 A+3 a c C-A c^2\right ) \int \frac {1}{-a \tanh ^2\left (\frac {1}{2} (d+e x)\right )+2 c \tanh \left (\frac {1}{2} (d+e x)\right )+a}d\left (i \tanh \left (\frac {1}{2} (d+e x)\right )\right )}{e \left (a^2+c^2\right )}}{2 \left (a^2+c^2\right )}-\frac {(A c-a C) \cosh (d+e x)}{2 e \left (a^2+c^2\right ) (a+c \sinh (d+e x))^2}-\frac {B}{2 c e (a+c \sinh (d+e x))^2}\)

\(\Big \downarrow \) 1083

\(\displaystyle \frac {-\frac {\left (a^2 (-C)+3 a A c+2 c^2 C\right ) \cosh (d+e x)}{e \left (a^2+c^2\right ) (a+c \sinh (d+e x))}+\frac {4 i \left (2 a^2 A+3 a c C-A c^2\right ) \int \frac {1}{\tanh ^2\left (\frac {1}{2} (d+e x)\right )-4 \left (a^2+c^2\right )}d\left (2 i a \tanh \left (\frac {1}{2} (d+e x)\right )-2 i c\right )}{e \left (a^2+c^2\right )}}{2 \left (a^2+c^2\right )}-\frac {(A c-a C) \cosh (d+e x)}{2 e \left (a^2+c^2\right ) (a+c \sinh (d+e x))^2}-\frac {B}{2 c e (a+c \sinh (d+e x))^2}\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {\frac {2 \left (2 a^2 A+3 a c C-A c^2\right ) \text {arctanh}\left (\frac {\tanh \left (\frac {1}{2} (d+e x)\right )}{2 \sqrt {a^2+c^2}}\right )}{e \left (a^2+c^2\right )^{3/2}}-\frac {\left (a^2 (-C)+3 a A c+2 c^2 C\right ) \cosh (d+e x)}{e \left (a^2+c^2\right ) (a+c \sinh (d+e x))}}{2 \left (a^2+c^2\right )}-\frac {(A c-a C) \cosh (d+e x)}{2 e \left (a^2+c^2\right ) (a+c \sinh (d+e x))^2}-\frac {B}{2 c e (a+c \sinh (d+e x))^2}\)

Input:

Int[(A + B*Cosh[d + e*x] + C*Sinh[d + e*x])/(a + c*Sinh[d + e*x])^3,x]
 

Output:

-1/2*B/(c*e*(a + c*Sinh[d + e*x])^2) - ((A*c - a*C)*Cosh[d + e*x])/(2*(a^2 
 + c^2)*e*(a + c*Sinh[d + e*x])^2) + ((2*(2*a^2*A - A*c^2 + 3*a*c*C)*ArcTa 
nh[Tanh[(d + e*x)/2]/(2*Sqrt[a^2 + c^2])])/((a^2 + c^2)^(3/2)*e) - ((3*a*A 
*c - a^2*C + 2*c^2*C)*Cosh[d + e*x])/((a^2 + c^2)*e*(a + c*Sinh[d + e*x])) 
)/(2*(a^2 + c^2))
 

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 27

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

rule 217

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( 
-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & 
& (LtQ[a, 0] || LtQ[b, 0])
 

rule 1083

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Simp[-2   Subst[I 
nt[1/Simp[b^2 - 4*a*c - x^2, x], x], x, b + 2*c*x], x] /; FreeQ[{a, b, c}, 
x]
 

rule 3042

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

rule 3139

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{e = Fre 
eFactors[Tan[(c + d*x)/2], x]}, Simp[2*(e/d)   Subst[Int[1/(a + 2*b*e*x + a 
*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 4876

Int[(u_)*((v_) + (d_.)*(F_)[(c_.)*((a_.) + (b_.)*(x_))]^(n_.)), x_Symbol] : 
> With[{e = FreeFactors[Sin[c*(a + b*x)], x]}, Int[ActivateTrig[u*v], x] + 
Simp[d   Int[ActivateTrig[u]*Cos[c*(a + b*x)]^n, x], x] /; FunctionOfQ[Sin[ 
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, Cos] || EqQ[F, cos])
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(369\) vs. \(2(169)=338\).

Time = 24.94 (sec) , antiderivative size = 370, normalized size of antiderivative = 2.06

method	result	size
parts	\(\frac {-\frac {2 \left (-\frac {c \left (5 A \,a^{2} c +2 A \,c^{3}-3 C \,a^{3}\right ) \tanh \left (\frac {e x}{2}+\frac {d}{2}\right )^{3}}{2 a \left (a^{4}+2 a^{2} c^{2}+c^{4}\right )}-\frac {\left (4 A \,a^{4} c -7 A \,a^{2} c^{3}-2 A \,c^{5}-2 C \,a^{5}+5 C \,a^{3} c^{2}-2 C a \,c^{4}\right ) \tanh \left (\frac {e x}{2}+\frac {d}{2}\right )^{2}}{2 \left (a^{4}+2 a^{2} c^{2}+c^{4}\right ) a^{2}}+\frac {c \left (11 A \,a^{2} c +2 A \,c^{3}-5 C \,a^{3}+4 C a \,c^{2}\right ) \tanh \left (\frac {e x}{2}+\frac {d}{2}\right )}{2 \left (a^{4}+2 a^{2} c^{2}+c^{4}\right ) a}+\frac {4 A \,a^{2} c +A \,c^{3}-2 C \,a^{3}+C a \,c^{2}}{2 a^{4}+4 a^{2} c^{2}+2 c^{4}}\right )}{\left (a \tanh \left (\frac {e x}{2}+\frac {d}{2}\right )^{2}-2 c \tanh \left (\frac {e x}{2}+\frac {d}{2}\right )-a \right )^{2}}+\frac {\left (2 a^{2} A -A \,c^{2}+3 C a c \right ) \operatorname {arctanh}\left (\frac {2 a \tanh \left (\frac {e x}{2}+\frac {d}{2}\right )-2 c}{2 \sqrt {a^{2}+c^{2}}}\right )}{\left (a^{4}+2 a^{2} c^{2}+c^{4}\right ) \sqrt {a^{2}+c^{2}}}}{e}-\frac {B}{2 c e \left (a +c \sinh \left (e x +d \right )\right )^{2}}\)	\(370\)
derivativedivides	\(\frac {-\frac {2 \left (-\frac {\left (5 A \,c^{2} a^{2}+2 A \,c^{4}-2 B \,a^{4}-4 B \,a^{2} c^{2}-2 B \,c^{4}-3 C \,a^{3} c \right ) \tanh \left (\frac {e x}{2}+\frac {d}{2}\right )^{3}}{2 a \left (a^{4}+2 a^{2} c^{2}+c^{4}\right )}-\frac {\left (4 A \,a^{4} c -7 A \,a^{2} c^{3}-2 A \,c^{5}+2 B \,a^{4} c +4 B \,a^{2} c^{3}+2 B \,c^{5}-2 C \,a^{5}+5 C \,a^{3} c^{2}-2 C a \,c^{4}\right ) \tanh \left (\frac {e x}{2}+\frac {d}{2}\right )^{2}}{2 \left (a^{4}+2 a^{2} c^{2}+c^{4}\right ) a^{2}}+\frac {\left (11 A \,c^{2} a^{2}+2 A \,c^{4}-2 B \,a^{4}-4 B \,a^{2} c^{2}-2 B \,c^{4}-5 C \,a^{3} c +4 C a \,c^{3}\right ) \tanh \left (\frac {e x}{2}+\frac {d}{2}\right )}{2 \left (a^{4}+2 a^{2} c^{2}+c^{4}\right ) a}+\frac {4 A \,a^{2} c +A \,c^{3}-2 C \,a^{3}+C a \,c^{2}}{2 a^{4}+4 a^{2} c^{2}+2 c^{4}}\right )}{\left (a \tanh \left (\frac {e x}{2}+\frac {d}{2}\right )^{2}-2 c \tanh \left (\frac {e x}{2}+\frac {d}{2}\right )-a \right )^{2}}+\frac {\left (2 a^{2} A -A \,c^{2}+3 C a c \right ) \operatorname {arctanh}\left (\frac {2 a \tanh \left (\frac {e x}{2}+\frac {d}{2}\right )-2 c}{2 \sqrt {a^{2}+c^{2}}}\right )}{\left (a^{4}+2 a^{2} c^{2}+c^{4}\right ) \sqrt {a^{2}+c^{2}}}}{e}\)	\(416\)
default	\(\frac {-\frac {2 \left (-\frac {\left (5 A \,c^{2} a^{2}+2 A \,c^{4}-2 B \,a^{4}-4 B \,a^{2} c^{2}-2 B \,c^{4}-3 C \,a^{3} c \right ) \tanh \left (\frac {e x}{2}+\frac {d}{2}\right )^{3}}{2 a \left (a^{4}+2 a^{2} c^{2}+c^{4}\right )}-\frac {\left (4 A \,a^{4} c -7 A \,a^{2} c^{3}-2 A \,c^{5}+2 B \,a^{4} c +4 B \,a^{2} c^{3}+2 B \,c^{5}-2 C \,a^{5}+5 C \,a^{3} c^{2}-2 C a \,c^{4}\right ) \tanh \left (\frac {e x}{2}+\frac {d}{2}\right )^{2}}{2 \left (a^{4}+2 a^{2} c^{2}+c^{4}\right ) a^{2}}+\frac {\left (11 A \,c^{2} a^{2}+2 A \,c^{4}-2 B \,a^{4}-4 B \,a^{2} c^{2}-2 B \,c^{4}-5 C \,a^{3} c +4 C a \,c^{3}\right ) \tanh \left (\frac {e x}{2}+\frac {d}{2}\right )}{2 \left (a^{4}+2 a^{2} c^{2}+c^{4}\right ) a}+\frac {4 A \,a^{2} c +A \,c^{3}-2 C \,a^{3}+C a \,c^{2}}{2 a^{4}+4 a^{2} c^{2}+2 c^{4}}\right )}{\left (a \tanh \left (\frac {e x}{2}+\frac {d}{2}\right )^{2}-2 c \tanh \left (\frac {e x}{2}+\frac {d}{2}\right )-a \right )^{2}}+\frac {\left (2 a^{2} A -A \,c^{2}+3 C a c \right ) \operatorname {arctanh}\left (\frac {2 a \tanh \left (\frac {e x}{2}+\frac {d}{2}\right )-2 c}{2 \sqrt {a^{2}+c^{2}}}\right )}{\left (a^{4}+2 a^{2} c^{2}+c^{4}\right ) \sqrt {a^{2}+c^{2}}}}{e}\)	\(416\)
risch	\(\frac {2 A \,a^{2} c^{2} {\mathrm e}^{3 e x +3 d}-A \,c^{4} {\mathrm e}^{3 e x +3 d}+3 C a \,c^{3} {\mathrm e}^{3 e x +3 d}+6 A \,a^{3} c \,{\mathrm e}^{2 e x +2 d}-3 A a \,c^{3} {\mathrm e}^{2 e x +2 d}-2 B \,a^{4} {\mathrm e}^{2 e x +2 d}-4 B \,a^{2} c^{2} {\mathrm e}^{2 e x +2 d}-2 B \,c^{4} {\mathrm e}^{2 e x +2 d}-2 C \,a^{4} {\mathrm e}^{2 e x +2 d}+5 C \,a^{2} c^{2} {\mathrm e}^{2 e x +2 d}-2 C \,c^{4} {\mathrm e}^{2 e x +2 d}-10 A \,a^{2} c^{2} {\mathrm e}^{e x +d}-A \,c^{4} {\mathrm e}^{e x +d}+4 C \,a^{3} c \,{\mathrm e}^{e x +d}-5 C a \,c^{3} {\mathrm e}^{e x +d}+3 A a \,c^{3}-C \,a^{2} c^{2}+2 C \,c^{4}}{c e \left (a^{2}+c^{2}\right )^{2} \left (c \,{\mathrm e}^{2 e x +2 d}+2 a \,{\mathrm e}^{e x +d}-c \right )^{2}}+\frac {\ln \left ({\mathrm e}^{e x +d}+\frac {\left (a^{2}+c^{2}\right )^{\frac {5}{2}} a -a^{6}-3 a^{4} c^{2}-3 a^{2} c^{4}-c^{6}}{c \left (a^{2}+c^{2}\right )^{\frac {5}{2}}}\right ) a^{2} A}{\left (a^{2}+c^{2}\right )^{\frac {5}{2}} e}-\frac {\ln \left ({\mathrm e}^{e x +d}+\frac {\left (a^{2}+c^{2}\right )^{\frac {5}{2}} a -a^{6}-3 a^{4} c^{2}-3 a^{2} c^{4}-c^{6}}{c \left (a^{2}+c^{2}\right )^{\frac {5}{2}}}\right ) A \,c^{2}}{2 \left (a^{2}+c^{2}\right )^{\frac {5}{2}} e}+\frac {3 \ln \left ({\mathrm e}^{e x +d}+\frac {\left (a^{2}+c^{2}\right )^{\frac {5}{2}} a -a^{6}-3 a^{4} c^{2}-3 a^{2} c^{4}-c^{6}}{c \left (a^{2}+c^{2}\right )^{\frac {5}{2}}}\right ) C a c}{2 \left (a^{2}+c^{2}\right )^{\frac {5}{2}} e}-\frac {\ln \left ({\mathrm e}^{e x +d}+\frac {\left (a^{2}+c^{2}\right )^{\frac {5}{2}} a +a^{6}+3 a^{4} c^{2}+3 a^{2} c^{4}+c^{6}}{c \left (a^{2}+c^{2}\right )^{\frac {5}{2}}}\right ) a^{2} A}{\left (a^{2}+c^{2}\right )^{\frac {5}{2}} e}+\frac {\ln \left ({\mathrm e}^{e x +d}+\frac {\left (a^{2}+c^{2}\right )^{\frac {5}{2}} a +a^{6}+3 a^{4} c^{2}+3 a^{2} c^{4}+c^{6}}{c \left (a^{2}+c^{2}\right )^{\frac {5}{2}}}\right ) A \,c^{2}}{2 \left (a^{2}+c^{2}\right )^{\frac {5}{2}} e}-\frac {3 \ln \left ({\mathrm e}^{e x +d}+\frac {\left (a^{2}+c^{2}\right )^{\frac {5}{2}} a +a^{6}+3 a^{4} c^{2}+3 a^{2} c^{4}+c^{6}}{c \left (a^{2}+c^{2}\right )^{\frac {5}{2}}}\right ) C a c}{2 \left (a^{2}+c^{2}\right )^{\frac {5}{2}} e}\)	\(744\)

Input:

int((A+B*cosh(e*x+d)+C*sinh(e*x+d))/(a+c*sinh(e*x+d))^3,x,method=_RETURNVE 
RBOSE)
 

Output:

1/e*(-2*(-1/2*c*(5*A*a^2*c+2*A*c^3-3*C*a^3)/a/(a^4+2*a^2*c^2+c^4)*tanh(1/2 
*e*x+1/2*d)^3-1/2*(4*A*a^4*c-7*A*a^2*c^3-2*A*c^5-2*C*a^5+5*C*a^3*c^2-2*C*a 
*c^4)/(a^4+2*a^2*c^2+c^4)/a^2*tanh(1/2*e*x+1/2*d)^2+1/2*c*(11*A*a^2*c+2*A* 
c^3-5*C*a^3+4*C*a*c^2)/(a^4+2*a^2*c^2+c^4)/a*tanh(1/2*e*x+1/2*d)+1/2*(4*A* 
a^2*c+A*c^3-2*C*a^3+C*a*c^2)/(a^4+2*a^2*c^2+c^4))/(a*tanh(1/2*e*x+1/2*d)^2 
-2*c*tanh(1/2*e*x+1/2*d)-a)^2+(2*A*a^2-A*c^2+3*C*a*c)/(a^4+2*a^2*c^2+c^4)/ 
(a^2+c^2)^(1/2)*arctanh(1/2*(2*a*tanh(1/2*e*x+1/2*d)-2*c)/(a^2+c^2)^(1/2)) 
)-1/2*B/c/e/(a+c*sinh(e*x+d))^2
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1880 vs. \(2 (170) = 340\).

Time = 0.13 (sec) , antiderivative size = 1880, normalized size of antiderivative = 10.44 \[ \int \frac {A+B \cosh (d+e x)+C \sinh (d+e x)}{(a+c \sinh (d+e x))^3} \, dx=\text {Too large to display} \] Input:

integrate((A+B*cosh(e*x+d)+C*sinh(e*x+d))/(a+c*sinh(e*x+d))^3,x, algorithm 
="fricas")
 

Output:

-1/2*(2*C*a^4*c^2 - 6*A*a^3*c^3 - 2*C*a^2*c^4 - 6*A*a*c^5 - 4*C*c^6 - 2*(2 
*A*a^4*c^2 + 3*C*a^3*c^3 + A*a^2*c^4 + 3*C*a*c^5 - A*c^6)*cosh(e*x + d)^3 
- 2*(2*A*a^4*c^2 + 3*C*a^3*c^3 + A*a^2*c^4 + 3*C*a*c^5 - A*c^6)*sinh(e*x + 
 d)^3 + 2*(2*(B + C)*a^6 - 6*A*a^5*c + 3*(2*B - C)*a^4*c^2 - 3*A*a^3*c^3 + 
 3*(2*B - C)*a^2*c^4 + 3*A*a*c^5 + 2*(B + C)*c^6)*cosh(e*x + d)^2 + 2*(2*( 
B + C)*a^6 - 6*A*a^5*c + 3*(2*B - C)*a^4*c^2 - 3*A*a^3*c^3 + 3*(2*B - C)*a 
^2*c^4 + 3*A*a*c^5 + 2*(B + C)*c^6 - 3*(2*A*a^4*c^2 + 3*C*a^3*c^3 + A*a^2* 
c^4 + 3*C*a*c^5 - A*c^6)*cosh(e*x + d))*sinh(e*x + d)^2 + (2*A*a^2*c^3 + 3 
*C*a*c^4 - A*c^5 + (2*A*a^2*c^3 + 3*C*a*c^4 - A*c^5)*cosh(e*x + d)^4 + (2* 
A*a^2*c^3 + 3*C*a*c^4 - A*c^5)*sinh(e*x + d)^4 + 4*(2*A*a^3*c^2 + 3*C*a^2* 
c^3 - A*a*c^4)*cosh(e*x + d)^3 + 4*(2*A*a^3*c^2 + 3*C*a^2*c^3 - A*a*c^4 + 
(2*A*a^2*c^3 + 3*C*a*c^4 - A*c^5)*cosh(e*x + d))*sinh(e*x + d)^3 + 2*(4*A* 
a^4*c + 6*C*a^3*c^2 - 4*A*a^2*c^3 - 3*C*a*c^4 + A*c^5)*cosh(e*x + d)^2 + 2 
*(4*A*a^4*c + 6*C*a^3*c^2 - 4*A*a^2*c^3 - 3*C*a*c^4 + A*c^5 + 3*(2*A*a^2*c 
^3 + 3*C*a*c^4 - A*c^5)*cosh(e*x + d)^2 + 6*(2*A*a^3*c^2 + 3*C*a^2*c^3 - A 
*a*c^4)*cosh(e*x + d))*sinh(e*x + d)^2 - 4*(2*A*a^3*c^2 + 3*C*a^2*c^3 - A* 
a*c^4)*cosh(e*x + d) - 4*(2*A*a^3*c^2 + 3*C*a^2*c^3 - A*a*c^4 - (2*A*a^2*c 
^3 + 3*C*a*c^4 - A*c^5)*cosh(e*x + d)^3 - 3*(2*A*a^3*c^2 + 3*C*a^2*c^3 - A 
*a*c^4)*cosh(e*x + d)^2 - (4*A*a^4*c + 6*C*a^3*c^2 - 4*A*a^2*c^3 - 3*C*a*c 
^4 + A*c^5)*cosh(e*x + d))*sinh(e*x + d))*sqrt(a^2 + c^2)*log((c^2*cosh...
 

Sympy [F(-1)]

Timed out. \[ \int \frac {A+B \cosh (d+e x)+C \sinh (d+e x)}{(a+c \sinh (d+e x))^3} \, dx=\text {Timed out} \] Input:

integrate((A+B*cosh(e*x+d)+C*sinh(e*x+d))/(a+c*sinh(e*x+d))**3,x)
                                                                                    
                                                                                    
 

Output:

Timed out
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 726 vs. \(2 (170) = 340\).

Time = 0.17 (sec) , antiderivative size = 726, normalized size of antiderivative = 4.03 \[ \int \frac {A+B \cosh (d+e x)+C \sinh (d+e x)}{(a+c \sinh (d+e x))^3} \, dx =\text {Too large to display} \] Input:

integrate((A+B*cosh(e*x+d)+C*sinh(e*x+d))/(a+c*sinh(e*x+d))^3,x, algorithm 
="maxima")
 

Output:

1/2*C*(3*a*c*log((c*e^(-e*x - d) - a - sqrt(a^2 + c^2))/(c*e^(-e*x - d) - 
a + sqrt(a^2 + c^2)))/((a^4 + 2*a^2*c^2 + c^4)*sqrt(a^2 + c^2)*e) + 2*(3*a 
*c^3*e^(-3*e*x - 3*d) + a^2*c^2 - 2*c^4 + (4*a^3*c - 5*a*c^3)*e^(-e*x - d) 
 + (2*a^4 - 5*a^2*c^2 + 2*c^4)*e^(-2*e*x - 2*d))/((a^4*c^3 + 2*a^2*c^5 + c 
^7 + 4*(a^5*c^2 + 2*a^3*c^4 + a*c^6)*e^(-e*x - d) + 2*(2*a^6*c + 3*a^4*c^3 
 - c^7)*e^(-2*e*x - 2*d) - 4*(a^5*c^2 + 2*a^3*c^4 + a*c^6)*e^(-3*e*x - 3*d 
) + (a^4*c^3 + 2*a^2*c^5 + c^7)*e^(-4*e*x - 4*d))*e)) + 1/2*A*((2*a^2 - c^ 
2)*log((c*e^(-e*x - d) - a - sqrt(a^2 + c^2))/(c*e^(-e*x - d) - a + sqrt(a 
^2 + c^2)))/((a^4 + 2*a^2*c^2 + c^4)*sqrt(a^2 + c^2)*e) - 2*(3*a*c^2 + (10 
*a^2*c + c^3)*e^(-e*x - d) + 3*(2*a^3 - a*c^2)*e^(-2*e*x - 2*d) - (2*a^2*c 
 - c^3)*e^(-3*e*x - 3*d))/((a^4*c^2 + 2*a^2*c^4 + c^6 + 4*(a^5*c + 2*a^3*c 
^3 + a*c^5)*e^(-e*x - d) + 2*(2*a^6 + 3*a^4*c^2 - c^6)*e^(-2*e*x - 2*d) - 
4*(a^5*c + 2*a^3*c^3 + a*c^5)*e^(-3*e*x - 3*d) + (a^4*c^2 + 2*a^2*c^4 + c^ 
6)*e^(-4*e*x - 4*d))*e)) - 2*B*e^(-2*e*x - 2*d)/((4*a*c^2*e^(-e*x - d) - 4 
*a*c^2*e^(-3*e*x - 3*d) + c^3*e^(-4*e*x - 4*d) + c^3 + 2*(2*a^2*c - c^3)*e 
^(-2*e*x - 2*d))*e)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 405 vs. \(2 (170) = 340\).

Time = 0.17 (sec) , antiderivative size = 405, normalized size of antiderivative = 2.25 \[ \int \frac {A+B \cosh (d+e x)+C \sinh (d+e x)}{(a+c \sinh (d+e x))^3} \, dx=-\frac {\frac {{\left (2 \, A a^{2} + 3 \, C a c - A c^{2}\right )} \log \left (\frac {{\left | -2 \, c e^{\left (e x + d\right )} - 2 \, a - 2 \, \sqrt {a^{2} + c^{2}} \right |}}{{\left | -2 \, c e^{\left (e x + d\right )} - 2 \, a + 2 \, \sqrt {a^{2} + c^{2}} \right |}}\right )}{{\left (a^{4} + 2 \, a^{2} c^{2} + c^{4}\right )} \sqrt {a^{2} + c^{2}}} - \frac {2 \, {\left (2 \, A a^{2} c^{2} e^{\left (3 \, e x + 3 \, d\right )} + 3 \, C a c^{3} e^{\left (3 \, e x + 3 \, d\right )} - A c^{4} e^{\left (3 \, e x + 3 \, d\right )} - 2 \, B a^{4} e^{\left (2 \, e x + 2 \, d\right )} - 2 \, C a^{4} e^{\left (2 \, e x + 2 \, d\right )} + 6 \, A a^{3} c e^{\left (2 \, e x + 2 \, d\right )} - 4 \, B a^{2} c^{2} e^{\left (2 \, e x + 2 \, d\right )} + 5 \, C a^{2} c^{2} e^{\left (2 \, e x + 2 \, d\right )} - 3 \, A a c^{3} e^{\left (2 \, e x + 2 \, d\right )} - 2 \, B c^{4} e^{\left (2 \, e x + 2 \, d\right )} - 2 \, C c^{4} e^{\left (2 \, e x + 2 \, d\right )} + 4 \, C a^{3} c e^{\left (e x + d\right )} - 10 \, A a^{2} c^{2} e^{\left (e x + d\right )} - 5 \, C a c^{3} e^{\left (e x + d\right )} - A c^{4} e^{\left (e x + d\right )} - C a^{2} c^{2} + 3 \, A a c^{3} + 2 \, C c^{4}\right )}}{{\left (a^{4} c + 2 \, a^{2} c^{3} + c^{5}\right )} {\left (c e^{\left (2 \, e x + 2 \, d\right )} + 2 \, a e^{\left (e x + d\right )} - c\right )}^{2}}}{2 \, e} \] Input:

integrate((A+B*cosh(e*x+d)+C*sinh(e*x+d))/(a+c*sinh(e*x+d))^3,x, algorithm 
="giac")
 

Output:

-1/2*((2*A*a^2 + 3*C*a*c - A*c^2)*log(abs(-2*c*e^(e*x + d) - 2*a - 2*sqrt( 
a^2 + c^2))/abs(-2*c*e^(e*x + d) - 2*a + 2*sqrt(a^2 + c^2)))/((a^4 + 2*a^2 
*c^2 + c^4)*sqrt(a^2 + c^2)) - 2*(2*A*a^2*c^2*e^(3*e*x + 3*d) + 3*C*a*c^3* 
e^(3*e*x + 3*d) - A*c^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*c*e^(2*e*x + 2*d) - 4*B*a^2*c^2*e^(2*e*x + 2*d 
) + 5*C*a^2*c^2*e^(2*e*x + 2*d) - 3*A*a*c^3*e^(2*e*x + 2*d) - 2*B*c^4*e^(2 
*e*x + 2*d) - 2*C*c^4*e^(2*e*x + 2*d) + 4*C*a^3*c*e^(e*x + d) - 10*A*a^2*c 
^2*e^(e*x + d) - 5*C*a*c^3*e^(e*x + d) - A*c^4*e^(e*x + d) - C*a^2*c^2 + 3 
*A*a*c^3 + 2*C*c^4)/((a^4*c + 2*a^2*c^3 + c^5)*(c*e^(2*e*x + 2*d) + 2*a*e^ 
(e*x + d) - c)^2))/e
 

Mupad [F(-1)]

Timed out. \[ \int \frac {A+B \cosh (d+e x)+C \sinh (d+e x)}{(a+c \sinh (d+e x))^3} \, dx=\int \frac {A+B\,\mathrm {cosh}\left (d+e\,x\right )+C\,\mathrm {sinh}\left (d+e\,x\right )}{{\left (a+c\,\mathrm {sinh}\left (d+e\,x\right )\right )}^3} \,d x \] Input:

int((A + B*cosh(d + e*x) + C*sinh(d + e*x))/(a + c*sinh(d + e*x))^3,x)
 

Output:

int((A + B*cosh(d + e*x) + C*sinh(d + e*x))/(a + c*sinh(d + e*x))^3, x)
 

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 689, normalized size of antiderivative = 3.83 \[ \int \frac {A+B \cosh (d+e x)+C \sinh (d+e x)}{(a+c \sinh (d+e x))^3} \, dx=\frac {4 e^{4 e x +4 d} \sqrt {a^{2}+c^{2}}\, \mathit {atan} \left (\frac {e^{e x +d} c i +a i}{\sqrt {a^{2}+c^{2}}}\right ) a \,c^{3} i +16 e^{3 e x +3 d} \sqrt {a^{2}+c^{2}}\, \mathit {atan} \left (\frac {e^{e x +d} c i +a i}{\sqrt {a^{2}+c^{2}}}\right ) a^{2} c^{2} i +16 e^{2 e x +2 d} \sqrt {a^{2}+c^{2}}\, \mathit {atan} \left (\frac {e^{e x +d} c i +a i}{\sqrt {a^{2}+c^{2}}}\right ) a^{3} c i -8 e^{2 e x +2 d} \sqrt {a^{2}+c^{2}}\, \mathit {atan} \left (\frac {e^{e x +d} c i +a i}{\sqrt {a^{2}+c^{2}}}\right ) a \,c^{3} i -16 e^{e x +d} \sqrt {a^{2}+c^{2}}\, \mathit {atan} \left (\frac {e^{e x +d} c i +a i}{\sqrt {a^{2}+c^{2}}}\right ) a^{2} c^{2} i +4 \sqrt {a^{2}+c^{2}}\, \mathit {atan} \left (\frac {e^{e x +d} c i +a i}{\sqrt {a^{2}+c^{2}}}\right ) a \,c^{3} i -e^{4 e x +4 d} a^{2} c^{3}-e^{4 e x +4 d} c^{5}-4 e^{2 e x +2 d} a^{4} b +4 e^{2 e x +2 d} a^{4} c -8 e^{2 e x +2 d} a^{2} b \,c^{2}+2 e^{2 e x +2 d} a^{2} c^{3}-4 e^{2 e x +2 d} b \,c^{4}-2 e^{2 e x +2 d} c^{5}-8 e^{e x +d} a^{3} c^{2}-8 e^{e x +d} a \,c^{4}+3 a^{2} c^{3}+3 c^{5}}{2 c e \left (e^{4 e x +4 d} a^{4} c^{2}+2 e^{4 e x +4 d} a^{2} c^{4}+e^{4 e x +4 d} c^{6}+4 e^{3 e x +3 d} a^{5} c +8 e^{3 e x +3 d} a^{3} c^{3}+4 e^{3 e x +3 d} a \,c^{5}+4 e^{2 e x +2 d} a^{6}+6 e^{2 e x +2 d} a^{4} c^{2}-2 e^{2 e x +2 d} c^{6}-4 e^{e x +d} a^{5} c -8 e^{e x +d} a^{3} c^{3}-4 e^{e x +d} a \,c^{5}+a^{4} c^{2}+2 a^{2} c^{4}+c^{6}\right )} \] Input:

int((A+B*cosh(e*x+d)+C*sinh(e*x+d))/(a+c*sinh(e*x+d))^3,x)
 

Output:

(4*e**(4*d + 4*e*x)*sqrt(a**2 + c**2)*atan((e**(d + e*x)*c*i + a*i)/sqrt(a 
**2 + c**2))*a*c**3*i + 16*e**(3*d + 3*e*x)*sqrt(a**2 + c**2)*atan((e**(d 
+ e*x)*c*i + a*i)/sqrt(a**2 + c**2))*a**2*c**2*i + 16*e**(2*d + 2*e*x)*sqr 
t(a**2 + c**2)*atan((e**(d + e*x)*c*i + a*i)/sqrt(a**2 + c**2))*a**3*c*i - 
 8*e**(2*d + 2*e*x)*sqrt(a**2 + c**2)*atan((e**(d + e*x)*c*i + a*i)/sqrt(a 
**2 + c**2))*a*c**3*i - 16*e**(d + e*x)*sqrt(a**2 + c**2)*atan((e**(d + e* 
x)*c*i + a*i)/sqrt(a**2 + c**2))*a**2*c**2*i + 4*sqrt(a**2 + c**2)*atan((e 
**(d + e*x)*c*i + a*i)/sqrt(a**2 + c**2))*a*c**3*i - e**(4*d + 4*e*x)*a**2 
*c**3 - e**(4*d + 4*e*x)*c**5 - 4*e**(2*d + 2*e*x)*a**4*b + 4*e**(2*d + 2* 
e*x)*a**4*c - 8*e**(2*d + 2*e*x)*a**2*b*c**2 + 2*e**(2*d + 2*e*x)*a**2*c** 
3 - 4*e**(2*d + 2*e*x)*b*c**4 - 2*e**(2*d + 2*e*x)*c**5 - 8*e**(d + e*x)*a 
**3*c**2 - 8*e**(d + e*x)*a*c**4 + 3*a**2*c**3 + 3*c**5)/(2*c*e*(e**(4*d + 
 4*e*x)*a**4*c**2 + 2*e**(4*d + 4*e*x)*a**2*c**4 + e**(4*d + 4*e*x)*c**6 + 
 4*e**(3*d + 3*e*x)*a**5*c + 8*e**(3*d + 3*e*x)*a**3*c**3 + 4*e**(3*d + 3* 
e*x)*a*c**5 + 4*e**(2*d + 2*e*x)*a**6 + 6*e**(2*d + 2*e*x)*a**4*c**2 - 2*e 
**(2*d + 2*e*x)*c**6 - 4*e**(d + e*x)*a**5*c - 8*e**(d + e*x)*a**3*c**3 - 
4*e**(d + e*x)*a*c**5 + a**4*c**2 + 2*a**2*c**4 + c**6))