\(\int \frac {B \cosh (x)+C \sinh (x)}{(a+b \cosh (x)+c \sinh (x))^3} \, dx\) [557]

Optimal result

Integrand size = 22, antiderivative size = 194 \[ \int \frac {B \cosh (x)+C \sinh (x)}{(a+b \cosh (x)+c \sinh (x))^3} \, dx=\frac {3 a (b B-c C) \text {arctanh}\left (\frac {c-(a-b) \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2-b^2+c^2}}\right )}{\left (a^2-b^2+c^2\right )^{5/2}}-\frac {B c-b C-a C \cosh (x)-a B \sinh (x)}{2 \left (a^2-b^2+c^2\right ) (a+b \cosh (x)+c \sinh (x))^2}-\frac {a (B c-b C)-\left (2 b B c+\left (a^2-2 c^2\right ) C\right ) \cosh (x)-\left (a^2 B+2 b (b B-c C)\right ) \sinh (x)}{2 \left (a^2-b^2+c^2\right )^2 (a+b \cosh (x)+c \sinh (x))} \] Output:

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

Mathematica [A] (verified)

Time = 0.92 (sec) , antiderivative size = 319, normalized size of antiderivative = 1.64 \[ \int \frac {B \cosh (x)+C \sinh (x)}{(a+b \cosh (x)+c \sinh (x))^3} \, dx=-\frac {3 a (b B-c C) \arctan \left (\frac {c+(-a+b) \tanh \left (\frac {x}{2}\right )}{\sqrt {-a^2+b^2-c^2}}\right )}{\left (-a^2+b^2-c^2\right )^{5/2}}+\frac {-9 a^2 b B c-2 a^4 C+4 a^2 b^2 C-2 b^4 C+5 a^2 c^2 C+4 b^2 c^2 C-2 c^4 C-6 a b c (b B-c C) \cosh (x)+c \left (a^2+2 b^2-2 c^2\right ) (b B-c C) \cosh (2 x)+4 a^3 b B \sinh (x)+2 a b^3 B \sinh (x)-8 a b B c^2 \sinh (x)-4 a^3 c C \sinh (x)-2 a b^2 c C \sinh (x)+8 a c^3 C \sinh (x)+a^2 b^2 B \sinh (2 x)+2 b^4 B \sinh (2 x)-2 b^2 B c^2 \sinh (2 x)-a^2 b c C \sinh (2 x)-2 b^3 c C \sinh (2 x)+2 b c^3 C \sinh (2 x)}{4 b \left (a^2-b^2+c^2\right )^2 (a+b \cosh (x)+c \sinh (x))^2} \] Input:

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

Output:

(-3*a*(b*B - c*C)*ArcTan[(c + (-a + b)*Tanh[x/2])/Sqrt[-a^2 + b^2 - c^2]]) 
/(-a^2 + b^2 - c^2)^(5/2) + (-9*a^2*b*B*c - 2*a^4*C + 4*a^2*b^2*C - 2*b^4* 
C + 5*a^2*c^2*C + 4*b^2*c^2*C - 2*c^4*C - 6*a*b*c*(b*B - c*C)*Cosh[x] + c* 
(a^2 + 2*b^2 - 2*c^2)*(b*B - c*C)*Cosh[2*x] + 4*a^3*b*B*Sinh[x] + 2*a*b^3* 
B*Sinh[x] - 8*a*b*B*c^2*Sinh[x] - 4*a^3*c*C*Sinh[x] - 2*a*b^2*c*C*Sinh[x] 
+ 8*a*c^3*C*Sinh[x] + a^2*b^2*B*Sinh[2*x] + 2*b^4*B*Sinh[2*x] - 2*b^2*B*c^ 
2*Sinh[2*x] - a^2*b*c*C*Sinh[2*x] - 2*b^3*c*C*Sinh[2*x] + 2*b*c^3*C*Sinh[2 
*x])/(4*b*(a^2 - b^2 + c^2)^2*(a + b*Cosh[x] + c*Sinh[x])^2)
 

Rubi [A] (verified)

Time = 0.71 (sec) , antiderivative size = 215, normalized size of antiderivative = 1.11, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {3042, 3635, 3042, 3632, 3042, 3603, 1083, 219}

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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3635

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3632

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3603

\(\displaystyle -\frac {\frac {6 a (b B-c C) \int \frac {1}{-\left ((a-b) \tanh ^2\left (\frac {x}{2}\right )\right )+2 c \tanh \left (\frac {x}{2}\right )+a+b}d\tanh \left (\frac {x}{2}\right )}{a^2-b^2+c^2}+\frac {-\cosh (x) \left (C \left (a^2-2 c^2\right )+2 b B c\right )-\sinh (x) \left (a^2 B+2 b (b B-c C)\right )+a (B c-b C)}{\left (a^2-b^2+c^2\right ) (a+b \cosh (x)+c \sinh (x))}}{2 \left (a^2-b^2+c^2\right )}-\frac {-a B \sinh (x)-a C \cosh (x)-b C+B c}{2 \left (a^2-b^2+c^2\right ) (a+b \cosh (x)+c \sinh (x))^2}\)

\(\Big \downarrow \) 1083

\(\displaystyle -\frac {\frac {-\cosh (x) \left (C \left (a^2-2 c^2\right )+2 b B c\right )-\sinh (x) \left (a^2 B+2 b (b B-c C)\right )+a (B c-b C)}{\left (a^2-b^2+c^2\right ) (a+b \cosh (x)+c \sinh (x))}-\frac {12 a (b B-c C) \int \frac {1}{4 \left (a^2-b^2+c^2\right )-\left (2 c-2 (a-b) \tanh \left (\frac {x}{2}\right )\right )^2}d\left (2 c-2 (a-b) \tanh \left (\frac {x}{2}\right )\right )}{a^2-b^2+c^2}}{2 \left (a^2-b^2+c^2\right )}-\frac {-a B \sinh (x)-a C \cosh (x)-b C+B c}{2 \left (a^2-b^2+c^2\right ) (a+b \cosh (x)+c \sinh (x))^2}\)

\(\Big \downarrow \) 219

\(\displaystyle -\frac {\frac {-\cosh (x) \left (C \left (a^2-2 c^2\right )+2 b B c\right )-\sinh (x) \left (a^2 B+2 b (b B-c C)\right )+a (B c-b C)}{\left (a^2-b^2+c^2\right ) (a+b \cosh (x)+c \sinh (x))}-\frac {6 a (b B-c C) \text {arctanh}\left (\frac {2 c-2 (a-b) \tanh \left (\frac {x}{2}\right )}{2 \sqrt {a^2-b^2+c^2}}\right )}{\left (a^2-b^2+c^2\right )^{3/2}}}{2 \left (a^2-b^2+c^2\right )}-\frac {-a B \sinh (x)-a C \cosh (x)-b C+B c}{2 \left (a^2-b^2+c^2\right ) (a+b \cosh (x)+c \sinh (x))^2}\)

Input:

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

Output:

-1/2*(B*c - b*C - a*C*Cosh[x] - a*B*Sinh[x])/((a^2 - b^2 + c^2)*(a + b*Cos 
h[x] + c*Sinh[x])^2) - ((-6*a*(b*B - c*C)*ArcTanh[(2*c - 2*(a - b)*Tanh[x/ 
2])/(2*Sqrt[a^2 - b^2 + c^2])])/(a^2 - b^2 + c^2)^(3/2) + (a*(B*c - b*C) - 
 (2*b*B*c + (a^2 - 2*c^2)*C)*Cosh[x] - (a^2*B + 2*b*(b*B - c*C))*Sinh[x])/ 
((a^2 - b^2 + c^2)*(a + b*Cosh[x] + c*Sinh[x])))/(2*(a^2 - b^2 + c^2))
 

Defintions of rubi rules used

rule 219

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* 
ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt 
Q[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 3603

Int[(cos[(d_.) + (e_.)*(x_)]*(b_.) + (a_) + (c_.)*sin[(d_.) + (e_.)*(x_)])^ 
(-1), x_Symbol] :> Module[{f = FreeFactors[Tan[(d + e*x)/2], x]}, Simp[2*(f 
/e)   Subst[Int[1/(a + b + 2*c*f*x + (a - b)*f^2*x^2), x], x, Tan[(d + e*x) 
/2]/f], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[a^2 - b^2 - c^2, 0]
 

rule 3632

Int[((A_.) + cos[(d_.) + (e_.)*(x_)]*(B_.) + (C_.)*sin[(d_.) + (e_.)*(x_)]) 
/((a_.) + cos[(d_.) + (e_.)*(x_)]*(b_.) + (c_.)*sin[(d_.) + (e_.)*(x_)])^2, 
 x_Symbol] :> Simp[(c*B - b*C - (a*C - c*A)*Cos[d + e*x] + (a*B - b*A)*Sin[ 
d + e*x])/(e*(a^2 - b^2 - c^2)*(a + b*Cos[d + e*x] + c*Sin[d + e*x])), x] + 
 Simp[(a*A - b*B - c*C)/(a^2 - b^2 - c^2)   Int[1/(a + b*Cos[d + e*x] + c*S 
in[d + e*x]), x], x] /; FreeQ[{a, b, c, d, e, A, B, C}, x] && NeQ[a^2 - b^2 
 - c^2, 0] && NeQ[a*A - b*B - c*C, 0]
 

rule 3635

Int[((a_.) + cos[(d_.) + (e_.)*(x_)]*(b_.) + (c_.)*sin[(d_.) + (e_.)*(x_)]) 
^(n_)*((A_.) + cos[(d_.) + (e_.)*(x_)]*(B_.) + (C_.)*sin[(d_.) + (e_.)*(x_) 
]), x_Symbol] :> Simp[(-(c*B - b*C - (a*C - c*A)*Cos[d + e*x] + (a*B - b*A) 
*Sin[d + e*x]))*((a + b*Cos[d + e*x] + c*Sin[d + e*x])^(n + 1)/(e*(n + 1)*( 
a^2 - b^2 - c^2))), x] + Simp[1/((n + 1)*(a^2 - b^2 - c^2))   Int[(a + b*Co 
s[d + e*x] + c*Sin[d + e*x])^(n + 1)*Simp[(n + 1)*(a*A - b*B - c*C) + (n + 
2)*(a*B - b*A)*Cos[d + e*x] + (n + 2)*(a*C - c*A)*Sin[d + e*x], x], x], x] 
/; FreeQ[{a, b, c, d, e, A, B, C}, x] && LtQ[n, -1] && NeQ[a^2 - b^2 - c^2, 
 0] && NeQ[n, -2]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(799\) vs. \(2(184)=368\).

Time = 11.78 (sec) , antiderivative size = 800, normalized size of antiderivative = 4.12

method	result	size
default	\(\frac {-\frac {\left (2 B \,a^{4}-3 B \,a^{3} b +2 B \,a^{2} b^{2}+4 B \,a^{2} c^{2}-3 B a \,b^{3}+2 b^{4} B -4 c^{2} b^{2} B +2 B \,c^{4}+3 C \,a^{3} c -6 C \,a^{2} b c +3 C a \,b^{2} c \right ) \tanh \left (\frac {x}{2}\right )^{3}}{\left (a^{4}-2 b^{2} a^{2}+2 a^{2} c^{2}+b^{4}-2 b^{2} c^{2}+c^{4}\right ) \left (a -b \right )}+\frac {\left (2 B \,a^{4} c -9 B \,a^{3} b c +14 B \,a^{2} b^{2} c +4 B \,a^{2} c^{3}-9 B a \,b^{3} c +2 B \,b^{4} c -4 B \,b^{2} c^{3}+2 B \,c^{5}-2 C \,a^{5}+2 C \,a^{4} b +4 C \,a^{3} b^{2}+5 C \,a^{3} c^{2}-4 C \,a^{2} b^{3}-14 C \,a^{2} b \,c^{2}-2 C a \,b^{4}+13 C a \,b^{2} c^{2}-2 C a \,c^{4}+2 C \,b^{5}-4 C \,b^{3} c^{2}+2 C b \,c^{4}\right ) \tanh \left (\frac {x}{2}\right )^{2}}{\left (a^{4}-2 b^{2} a^{2}+2 a^{2} c^{2}+b^{4}-2 b^{2} c^{2}+c^{4}\right ) \left (a^{2}-2 a b +b^{2}\right )}+\frac {\left (2 B \,a^{5}-3 B \,a^{4} b +B \,a^{3} b^{2}+4 B \,a^{3} c^{2}+B \,a^{2} b^{3}+8 B \,a^{2} b \,c^{2}-3 B a \,b^{4}-8 B a \,b^{2} c^{2}+2 B a \,c^{4}+2 B \,b^{5}-4 B \,b^{3} c^{2}+2 B b \,c^{4}+5 C \,a^{4} c -5 C \,a^{3} b c -5 C \,a^{2} b^{2} c -4 C \,a^{2} c^{3}+5 C a \,b^{3} c +4 C a b \,c^{3}\right ) \tanh \left (\frac {x}{2}\right )}{\left (a^{4}-2 b^{2} a^{2}+2 a^{2} c^{2}+b^{4}-2 b^{2} c^{2}+c^{4}\right ) \left (a^{2}-2 a b +b^{2}\right )}+\frac {a \left (5 B \,a^{2} b c -5 B \,b^{3} c +2 B b \,c^{3}+2 C \,a^{4}-4 C \,a^{2} b^{2}-C \,a^{2} c^{2}+2 C \,b^{4}+C \,b^{2} c^{2}\right )}{\left (a^{4}-2 b^{2} a^{2}+2 a^{2} c^{2}+b^{4}-2 b^{2} c^{2}+c^{4}\right ) \left (a^{2}-2 a b +b^{2}\right )}}{\left (a \tanh \left (\frac {x}{2}\right )^{2}-b \tanh \left (\frac {x}{2}\right )^{2}-2 c \tanh \left (\frac {x}{2}\right )-a -b \right )^{2}}+\frac {3 a \left (b B -C c \right ) \arctan \left (\frac {2 \left (a -b \right ) \tanh \left (\frac {x}{2}\right )-2 c}{2 \sqrt {-a^{2}+b^{2}-c^{2}}}\right )}{\left (a^{4}-2 b^{2} a^{2}+2 a^{2} c^{2}+b^{4}-2 b^{2} c^{2}+c^{4}\right ) \sqrt {-a^{2}+b^{2}-c^{2}}}\)	\(800\)
risch	\(-\frac {2 b^{4} B -2 c^{2} b^{2} B +C \,a^{2} c^{2}+2 C \,b^{2} c^{2}-C \,a^{2} b c -B \,a^{2} b c +2 B b \,c^{3}+2 \,{\mathrm e}^{2 x} a^{4} C -2 C \,b^{3} c -4 C \,a^{3} c \,{\mathrm e}^{x}+5 C a \,c^{3} {\mathrm e}^{x}+4 B \,a^{3} b \,{\mathrm e}^{x}+5 B a \,b^{3} {\mathrm e}^{x}+4 B \,a^{2} c^{2} {\mathrm e}^{2 x}-4 B \,b^{2} c^{2} {\mathrm e}^{2 x}-4 C \,a^{2} b^{2} {\mathrm e}^{2 x}-5 C \,a^{2} c^{2} {\mathrm e}^{2 x}-4 C \,b^{2} c^{2} {\mathrm e}^{2 x}+3 B a \,b^{3} {\mathrm e}^{3 x}+5 B \,a^{2} b^{2} {\mathrm e}^{2 x}-3 C a \,c^{3} {\mathrm e}^{3 x}+B \,a^{2} b^{2}-5 B a b \,c^{2} {\mathrm e}^{x}+9 B \,a^{2} b c \,{\mathrm e}^{2 x}-3 C a \,b^{2} c \,{\mathrm e}^{3 x}-6 C a b \,c^{2} {\mathrm e}^{3 x}-9 C \,a^{2} b c \,{\mathrm e}^{2 x}+6 B a \,b^{2} c \,{\mathrm e}^{3 x}+3 B a b \,c^{2} {\mathrm e}^{3 x}+2 C \,b^{4} {\mathrm e}^{2 x}+2 C \,c^{4} {\mathrm e}^{2 x}-2 B \,b^{3} c +2 B \,c^{4} {\mathrm e}^{2 x}+2 \,{\mathrm e}^{2 x} a^{4} B +2 B \,b^{4} {\mathrm e}^{2 x}-5 C a \,b^{2} c \,{\mathrm e}^{x}+2 C b \,c^{3}-2 C \,c^{4}}{\left (b +c \right ) \left (a^{2}-b^{2}+c^{2}\right )^{2} \left (b \,{\mathrm e}^{2 x}+c \,{\mathrm e}^{2 x}+2 a \,{\mathrm e}^{x}+b -c \right )^{2}}+\frac {3 \ln \left ({\mathrm e}^{x}+\frac {\left (a^{2}-b^{2}+c^{2}\right )^{\frac {5}{2}} a +a^{6}-3 a^{4} b^{2}+3 a^{4} c^{2}+3 a^{2} b^{4}-6 a^{2} b^{2} c^{2}+3 a^{2} c^{4}-b^{6}+3 c^{2} b^{4}-3 c^{4} b^{2}+c^{6}}{\left (a^{2}-b^{2}+c^{2}\right )^{\frac {5}{2}} \left (b +c \right )}\right ) a b B}{2 \left (a^{2}-b^{2}+c^{2}\right )^{\frac {5}{2}}}-\frac {3 a \ln \left ({\mathrm e}^{x}+\frac {\left (a^{2}-b^{2}+c^{2}\right )^{\frac {5}{2}} a +a^{6}-3 a^{4} b^{2}+3 a^{4} c^{2}+3 a^{2} b^{4}-6 a^{2} b^{2} c^{2}+3 a^{2} c^{4}-b^{6}+3 c^{2} b^{4}-3 c^{4} b^{2}+c^{6}}{\left (a^{2}-b^{2}+c^{2}\right )^{\frac {5}{2}} \left (b +c \right )}\right ) C c}{2 \left (a^{2}-b^{2}+c^{2}\right )^{\frac {5}{2}}}-\frac {3 \ln \left ({\mathrm e}^{x}+\frac {\left (a^{2}-b^{2}+c^{2}\right )^{\frac {5}{2}} a -a^{6}+3 a^{4} b^{2}-3 a^{4} c^{2}-3 a^{2} b^{4}+6 a^{2} b^{2} c^{2}-3 a^{2} c^{4}+b^{6}-3 c^{2} b^{4}+3 c^{4} b^{2}-c^{6}}{\left (a^{2}-b^{2}+c^{2}\right )^{\frac {5}{2}} \left (b +c \right )}\right ) a b B}{2 \left (a^{2}-b^{2}+c^{2}\right )^{\frac {5}{2}}}+\frac {3 a \ln \left ({\mathrm e}^{x}+\frac {\left (a^{2}-b^{2}+c^{2}\right )^{\frac {5}{2}} a -a^{6}+3 a^{4} b^{2}-3 a^{4} c^{2}-3 a^{2} b^{4}+6 a^{2} b^{2} c^{2}-3 a^{2} c^{4}+b^{6}-3 c^{2} b^{4}+3 c^{4} b^{2}-c^{6}}{\left (a^{2}-b^{2}+c^{2}\right )^{\frac {5}{2}} \left (b +c \right )}\right ) C c}{2 \left (a^{2}-b^{2}+c^{2}\right )^{\frac {5}{2}}}\)	\(950\)

Input:

int((B*cosh(x)+C*sinh(x))/(a+b*cosh(x)+c*sinh(x))^3,x,method=_RETURNVERBOS 
E)
 

Output:

2*(-1/2*(2*B*a^4-3*B*a^3*b+2*B*a^2*b^2+4*B*a^2*c^2-3*B*a*b^3+2*B*b^4-4*B*b 
^2*c^2+2*B*c^4+3*C*a^3*c-6*C*a^2*b*c+3*C*a*b^2*c)/(a^4-2*a^2*b^2+2*a^2*c^2 
+b^4-2*b^2*c^2+c^4)/(a-b)*tanh(1/2*x)^3+1/2*(2*B*a^4*c-9*B*a^3*b*c+14*B*a^ 
2*b^2*c+4*B*a^2*c^3-9*B*a*b^3*c+2*B*b^4*c-4*B*b^2*c^3+2*B*c^5-2*C*a^5+2*C* 
a^4*b+4*C*a^3*b^2+5*C*a^3*c^2-4*C*a^2*b^3-14*C*a^2*b*c^2-2*C*a*b^4+13*C*a* 
b^2*c^2-2*C*a*c^4+2*C*b^5-4*C*b^3*c^2+2*C*b*c^4)/(a^4-2*a^2*b^2+2*a^2*c^2+ 
b^4-2*b^2*c^2+c^4)/(a^2-2*a*b+b^2)*tanh(1/2*x)^2+1/2*(2*B*a^5-3*B*a^4*b+B* 
a^3*b^2+4*B*a^3*c^2+B*a^2*b^3+8*B*a^2*b*c^2-3*B*a*b^4-8*B*a*b^2*c^2+2*B*a* 
c^4+2*B*b^5-4*B*b^3*c^2+2*B*b*c^4+5*C*a^4*c-5*C*a^3*b*c-5*C*a^2*b^2*c-4*C* 
a^2*c^3+5*C*a*b^3*c+4*C*a*b*c^3)/(a^4-2*a^2*b^2+2*a^2*c^2+b^4-2*b^2*c^2+c^ 
4)/(a^2-2*a*b+b^2)*tanh(1/2*x)+1/2*a*(5*B*a^2*b*c-5*B*b^3*c+2*B*b*c^3+2*C* 
a^4-4*C*a^2*b^2-C*a^2*c^2+2*C*b^4+C*b^2*c^2)/(a^4-2*a^2*b^2+2*a^2*c^2+b^4- 
2*b^2*c^2+c^4)/(a^2-2*a*b+b^2))/(a*tanh(1/2*x)^2-b*tanh(1/2*x)^2-2*c*tanh( 
1/2*x)-a-b)^2+3*a*(B*b-C*c)/(a^4-2*a^2*b^2+2*a^2*c^2+b^4-2*b^2*c^2+c^4)/(- 
a^2+b^2-c^2)^(1/2)*arctan(1/2*(2*(a-b)*tanh(1/2*x)-2*c)/(-a^2+b^2-c^2)^(1/ 
2))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 4996 vs. \(2 (182) = 364\).

Time = 0.42 (sec) , antiderivative size = 10107, normalized size of antiderivative = 52.10 \[ \int \frac {B \cosh (x)+C \sinh (x)}{(a+b \cosh (x)+c \sinh (x))^3} \, dx=\text {Too large to display} \] Input:

integrate((B*cosh(x)+C*sinh(x))/(a+b*cosh(x)+c*sinh(x))^3,x, algorithm="fr 
icas")
 

Output:

Too large to include
 

Sympy [F(-1)]

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

integrate((B*cosh(x)+C*sinh(x))/(a+b*cosh(x)+c*sinh(x))**3,x)
 

Output:

Timed out
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {B \cosh (x)+C \sinh (x)}{(a+b \cosh (x)+c \sinh (x))^3} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((B*cosh(x)+C*sinh(x))/(a+b*cosh(x)+c*sinh(x))^3,x, algorithm="ma 
xima")
                                                                                    
                                                                                    
 

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(c^2-b^2+a^2>0)', see `assume?` f 
or more de
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 577 vs. \(2 (182) = 364\).

Time = 0.16 (sec) , antiderivative size = 577, normalized size of antiderivative = 2.97 \[ \int \frac {B \cosh (x)+C \sinh (x)}{(a+b \cosh (x)+c \sinh (x))^3} \, dx=-\frac {3 \, {\left (B a b - C a c\right )} \arctan \left (\frac {b e^{x} + c e^{x} + a}{\sqrt {-a^{2} + b^{2} - c^{2}}}\right )}{{\left (a^{4} - 2 \, a^{2} b^{2} + b^{4} + 2 \, a^{2} c^{2} - 2 \, b^{2} c^{2} + c^{4}\right )} \sqrt {-a^{2} + b^{2} - c^{2}}} - \frac {3 \, B a b^{3} e^{\left (3 \, x\right )} + 6 \, B a b^{2} c e^{\left (3 \, x\right )} - 3 \, C a b^{2} c e^{\left (3 \, x\right )} + 3 \, B a b c^{2} e^{\left (3 \, x\right )} - 6 \, C a b c^{2} e^{\left (3 \, x\right )} - 3 \, C a c^{3} e^{\left (3 \, x\right )} + 2 \, B a^{4} e^{\left (2 \, x\right )} + 2 \, C a^{4} e^{\left (2 \, x\right )} + 5 \, B a^{2} b^{2} e^{\left (2 \, x\right )} - 4 \, C a^{2} b^{2} e^{\left (2 \, x\right )} + 2 \, B b^{4} e^{\left (2 \, x\right )} + 2 \, C b^{4} e^{\left (2 \, x\right )} + 9 \, B a^{2} b c e^{\left (2 \, x\right )} - 9 \, C a^{2} b c e^{\left (2 \, x\right )} + 4 \, B a^{2} c^{2} e^{\left (2 \, x\right )} - 5 \, C a^{2} c^{2} e^{\left (2 \, x\right )} - 4 \, B b^{2} c^{2} e^{\left (2 \, x\right )} - 4 \, C b^{2} c^{2} e^{\left (2 \, x\right )} + 2 \, B c^{4} e^{\left (2 \, x\right )} + 2 \, C c^{4} e^{\left (2 \, x\right )} + 4 \, B a^{3} b e^{x} + 5 \, B a b^{3} e^{x} - 4 \, C a^{3} c e^{x} - 5 \, C a b^{2} c e^{x} - 5 \, B a b c^{2} e^{x} + 5 \, C a c^{3} e^{x} + B a^{2} b^{2} + 2 \, B b^{4} - B a^{2} b c - C a^{2} b c - 2 \, B b^{3} c - 2 \, C b^{3} c + C a^{2} c^{2} - 2 \, B b^{2} c^{2} + 2 \, C b^{2} c^{2} + 2 \, B b c^{3} + 2 \, C b c^{3} - 2 \, C c^{4}}{{\left (a^{4} b - 2 \, a^{2} b^{3} + b^{5} + a^{4} c - 2 \, a^{2} b^{2} c + b^{4} c + 2 \, a^{2} b c^{2} - 2 \, b^{3} c^{2} + 2 \, a^{2} c^{3} - 2 \, b^{2} c^{3} + b c^{4} + c^{5}\right )} {\left (b e^{\left (2 \, x\right )} + c e^{\left (2 \, x\right )} + 2 \, a e^{x} + b - c\right )}^{2}} \] Input:

integrate((B*cosh(x)+C*sinh(x))/(a+b*cosh(x)+c*sinh(x))^3,x, algorithm="gi 
ac")
 

Output:

-3*(B*a*b - C*a*c)*arctan((b*e^x + c*e^x + a)/sqrt(-a^2 + b^2 - c^2))/((a^ 
4 - 2*a^2*b^2 + b^4 + 2*a^2*c^2 - 2*b^2*c^2 + c^4)*sqrt(-a^2 + b^2 - c^2)) 
 - (3*B*a*b^3*e^(3*x) + 6*B*a*b^2*c*e^(3*x) - 3*C*a*b^2*c*e^(3*x) + 3*B*a* 
b*c^2*e^(3*x) - 6*C*a*b*c^2*e^(3*x) - 3*C*a*c^3*e^(3*x) + 2*B*a^4*e^(2*x) 
+ 2*C*a^4*e^(2*x) + 5*B*a^2*b^2*e^(2*x) - 4*C*a^2*b^2*e^(2*x) + 2*B*b^4*e^ 
(2*x) + 2*C*b^4*e^(2*x) + 9*B*a^2*b*c*e^(2*x) - 9*C*a^2*b*c*e^(2*x) + 4*B* 
a^2*c^2*e^(2*x) - 5*C*a^2*c^2*e^(2*x) - 4*B*b^2*c^2*e^(2*x) - 4*C*b^2*c^2* 
e^(2*x) + 2*B*c^4*e^(2*x) + 2*C*c^4*e^(2*x) + 4*B*a^3*b*e^x + 5*B*a*b^3*e^ 
x - 4*C*a^3*c*e^x - 5*C*a*b^2*c*e^x - 5*B*a*b*c^2*e^x + 5*C*a*c^3*e^x + B* 
a^2*b^2 + 2*B*b^4 - B*a^2*b*c - C*a^2*b*c - 2*B*b^3*c - 2*C*b^3*c + C*a^2* 
c^2 - 2*B*b^2*c^2 + 2*C*b^2*c^2 + 2*B*b*c^3 + 2*C*b*c^3 - 2*C*c^4)/((a^4*b 
 - 2*a^2*b^3 + b^5 + a^4*c - 2*a^2*b^2*c + b^4*c + 2*a^2*b*c^2 - 2*b^3*c^2 
 + 2*a^2*c^3 - 2*b^2*c^3 + b*c^4 + c^5)*(b*e^(2*x) + c*e^(2*x) + 2*a*e^x + 
 b - c)^2)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.29 (sec) , antiderivative size = 22480, normalized size of antiderivative = 115.88 \[ \int \frac {B \cosh (x)+C \sinh (x)}{(a+b \cosh (x)+c \sinh (x))^3} \, dx =\text {Too large to display} \] Input:

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

Output:

(12*e**(4*x)*sqrt( - a**2 + b**2 - c**2)*atan((e**x*b + e**x*c + a)/sqrt( 
- a**2 + b**2 - c**2))*cosh(x)**2*a*b**6*c + 24*e**(4*x)*sqrt( - a**2 + b* 
*2 - c**2)*atan((e**x*b + e**x*c + a)/sqrt( - a**2 + b**2 - c**2))*cosh(x) 
**2*a*b**5*c**2 - 24*e**(4*x)*sqrt( - a**2 + b**2 - c**2)*atan((e**x*b + e 
**x*c + a)/sqrt( - a**2 + b**2 - c**2))*cosh(x)**2*a*b**3*c**4 - 12*e**(4* 
x)*sqrt( - a**2 + b**2 - c**2)*atan((e**x*b + e**x*c + a)/sqrt( - a**2 + b 
**2 - c**2))*cosh(x)**2*a*b**2*c**5 + 48*e**(3*x)*sqrt( - a**2 + b**2 - c* 
*2)*atan((e**x*b + e**x*c + a)/sqrt( - a**2 + b**2 - c**2))*cosh(x)**2*a** 
2*b**5*c + 48*e**(3*x)*sqrt( - a**2 + b**2 - c**2)*atan((e**x*b + e**x*c + 
 a)/sqrt( - a**2 + b**2 - c**2))*cosh(x)**2*a**2*b**4*c**2 - 48*e**(3*x)*s 
qrt( - a**2 + b**2 - c**2)*atan((e**x*b + e**x*c + a)/sqrt( - a**2 + b**2 
- c**2))*cosh(x)**2*a**2*b**3*c**3 - 48*e**(3*x)*sqrt( - a**2 + b**2 - c** 
2)*atan((e**x*b + e**x*c + a)/sqrt( - a**2 + b**2 - c**2))*cosh(x)**2*a**2 
*b**2*c**4 + 48*e**(2*x)*sqrt( - a**2 + b**2 - c**2)*atan((e**x*b + e**x*c 
 + a)/sqrt( - a**2 + b**2 - c**2))*cosh(x)**2*a**3*b**4*c - 48*e**(2*x)*sq 
rt( - a**2 + b**2 - c**2)*atan((e**x*b + e**x*c + a)/sqrt( - a**2 + b**2 - 
 c**2))*cosh(x)**2*a**3*b**2*c**3 + 24*e**(2*x)*sqrt( - a**2 + b**2 - c**2 
)*atan((e**x*b + e**x*c + a)/sqrt( - a**2 + b**2 - c**2))*cosh(x)**2*a*b** 
6*c - 48*e**(2*x)*sqrt( - a**2 + b**2 - c**2)*atan((e**x*b + e**x*c + a)/s 
qrt( - a**2 + b**2 - c**2))*cosh(x)**2*a*b**4*c**3 + 24*e**(2*x)*sqrt( ...