\(\int \frac {A+B \sinh (x)}{(a+b \sinh (x))^3} \, dx\) [131]

Optimal result

Integrand size = 15, antiderivative size = 128 \[ \int \frac {A+B \sinh (x)}{(a+b \sinh (x))^3} \, dx=-\frac {\left (2 a^2 A-A b^2+3 a b B\right ) \text {arctanh}\left (\frac {b-a \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{5/2}}-\frac {(A b-a B) \cosh (x)}{2 \left (a^2+b^2\right ) (a+b \sinh (x))^2}-\frac {\left (3 a A b-a^2 B+2 b^2 B\right ) \cosh (x)}{2 \left (a^2+b^2\right )^2 (a+b \sinh (x))} \] Output:

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

Mathematica [A] (verified)

Time = 0.21 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.02 \[ \int \frac {A+B \sinh (x)}{(a+b \sinh (x))^3} \, dx=\frac {\frac {2 \left (2 a^2 A-A b^2+3 a b B\right ) \arctan \left (\frac {b-a \tanh \left (\frac {x}{2}\right )}{\sqrt {-a^2-b^2}}\right )}{\sqrt {-a^2-b^2}}+\frac {\left (a^2+b^2\right ) (-A b+a B) \cosh (x)}{(a+b \sinh (x))^2}+\frac {\left (-3 a A b+a^2 B-2 b^2 B\right ) \cosh (x)}{a+b \sinh (x)}}{2 \left (a^2+b^2\right )^2} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.56 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.13, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.733, Rules used = {3042, 3233, 25, 3042, 3233, 25, 27, 3042, 3139, 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.

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

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {A-i B \sin (i x)}{(a-i b \sin (i x))^3}dx\)

\(\Big \downarrow \) 3233

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3233

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3139

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

\(\Big \downarrow \) 1083

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

\(\Big \downarrow \) 219

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

Input:

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

Output:

-1/2*((A*b - a*B)*Cosh[x])/((a^2 + b^2)*(a + b*Sinh[x])^2) + ((-2*(2*a^2*A 
 - A*b^2 + 3*a*b*B)*ArcTanh[(2*b - 2*a*Tanh[x/2])/(2*Sqrt[a^2 + b^2])])/(a 
^2 + b^2)^(3/2) - ((3*a*A*b - a^2*B + 2*b^2*B)*Cosh[x])/((a^2 + b^2)*(a + 
b*Sinh[x])))/(2*(a^2 + b^2))
 

Defintions of rubi rules used

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 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 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 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]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(313\) vs. \(2(118)=236\).

Time = 0.31 (sec) , antiderivative size = 314, normalized size of antiderivative = 2.45

method	result	size
default	\(-\frac {2 \left (-\frac {b \left (5 A \,a^{2} b +2 A \,b^{3}-3 a^{3} B \right ) \tanh \left (\frac {x}{2}\right )^{3}}{2 a \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}-\frac {\left (4 A \,a^{4} b -7 A \,b^{3} a^{2}-2 A \,b^{5}-2 B \,a^{5}+5 B \,a^{3} b^{2}-2 B a \,b^{4}\right ) \tanh \left (\frac {x}{2}\right )^{2}}{2 \left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) a^{2}}+\frac {b \left (11 A \,a^{2} b +2 A \,b^{3}-5 a^{3} B +4 B a \,b^{2}\right ) \tanh \left (\frac {x}{2}\right )}{2 \left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) a}+\frac {4 A \,a^{2} b +A \,b^{3}-2 a^{3} B +B a \,b^{2}}{2 a^{4}+4 a^{2} b^{2}+2 b^{4}}\right )}{\left (\tanh \left (\frac {x}{2}\right )^{2} a -2 b \tanh \left (\frac {x}{2}\right )-a \right )^{2}}+\frac {\left (2 a^{2} A -A \,b^{2}+3 a B b \right ) \operatorname {arctanh}\left (\frac {2 a \tanh \left (\frac {x}{2}\right )-2 b}{2 \sqrt {a^{2}+b^{2}}}\right )}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) \sqrt {a^{2}+b^{2}}}\)	\(314\)
risch	\(\frac {2 A \,a^{2} b^{2} {\mathrm e}^{3 x}-A \,b^{4} {\mathrm e}^{3 x}+3 B a \,b^{3} {\mathrm e}^{3 x}+6 A \,a^{3} b \,{\mathrm e}^{2 x}-3 A a \,b^{3} {\mathrm e}^{2 x}-2 B \,a^{4} {\mathrm e}^{2 x}+5 B \,a^{2} b^{2} {\mathrm e}^{2 x}-2 B \,b^{4} {\mathrm e}^{2 x}-10 A \,a^{2} b^{2} {\mathrm e}^{x}-A \,b^{4} {\mathrm e}^{x}+4 B \,a^{3} b \,{\mathrm e}^{x}-5 B a \,b^{3} {\mathrm e}^{x}+3 A \,b^{3} a -B \,a^{2} b^{2}+2 B \,b^{4}}{b \left (a^{2}+b^{2}\right )^{2} \left (b \,{\mathrm e}^{2 x}+2 \,{\mathrm e}^{x} a -b \right )^{2}}+\frac {\ln \left ({\mathrm e}^{x}+\frac {\left (a^{2}+b^{2}\right )^{\frac {5}{2}} a -a^{6}-3 a^{4} b^{2}-3 a^{2} b^{4}-b^{6}}{b \left (a^{2}+b^{2}\right )^{\frac {5}{2}}}\right ) a^{2} A}{\left (a^{2}+b^{2}\right )^{\frac {5}{2}}}-\frac {\ln \left ({\mathrm e}^{x}+\frac {\left (a^{2}+b^{2}\right )^{\frac {5}{2}} a -a^{6}-3 a^{4} b^{2}-3 a^{2} b^{4}-b^{6}}{b \left (a^{2}+b^{2}\right )^{\frac {5}{2}}}\right ) A \,b^{2}}{2 \left (a^{2}+b^{2}\right )^{\frac {5}{2}}}+\frac {3 \ln \left ({\mathrm e}^{x}+\frac {\left (a^{2}+b^{2}\right )^{\frac {5}{2}} a -a^{6}-3 a^{4} b^{2}-3 a^{2} b^{4}-b^{6}}{b \left (a^{2}+b^{2}\right )^{\frac {5}{2}}}\right ) a B b}{2 \left (a^{2}+b^{2}\right )^{\frac {5}{2}}}-\frac {\ln \left ({\mathrm e}^{x}+\frac {\left (a^{2}+b^{2}\right )^{\frac {5}{2}} a +a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}}{b \left (a^{2}+b^{2}\right )^{\frac {5}{2}}}\right ) a^{2} A}{\left (a^{2}+b^{2}\right )^{\frac {5}{2}}}+\frac {\ln \left ({\mathrm e}^{x}+\frac {\left (a^{2}+b^{2}\right )^{\frac {5}{2}} a +a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}}{b \left (a^{2}+b^{2}\right )^{\frac {5}{2}}}\right ) A \,b^{2}}{2 \left (a^{2}+b^{2}\right )^{\frac {5}{2}}}-\frac {3 \ln \left ({\mathrm e}^{x}+\frac {\left (a^{2}+b^{2}\right )^{\frac {5}{2}} a +a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}}{b \left (a^{2}+b^{2}\right )^{\frac {5}{2}}}\right ) a B b}{2 \left (a^{2}+b^{2}\right )^{\frac {5}{2}}}\)	\(586\)

Input:

int((A+B*sinh(x))/(a+b*sinh(x))^3,x,method=_RETURNVERBOSE)
 

Output:

-2*(-1/2*b*(5*A*a^2*b+2*A*b^3-3*B*a^3)/a/(a^4+2*a^2*b^2+b^4)*tanh(1/2*x)^3 
-1/2*(4*A*a^4*b-7*A*a^2*b^3-2*A*b^5-2*B*a^5+5*B*a^3*b^2-2*B*a*b^4)/(a^4+2* 
a^2*b^2+b^4)/a^2*tanh(1/2*x)^2+1/2*b*(11*A*a^2*b+2*A*b^3-5*B*a^3+4*B*a*b^2 
)/(a^4+2*a^2*b^2+b^4)/a*tanh(1/2*x)+1/2*(4*A*a^2*b+A*b^3-2*B*a^3+B*a*b^2)/ 
(a^4+2*a^2*b^2+b^4))/(tanh(1/2*x)^2*a-2*b*tanh(1/2*x)-a)^2+(2*A*a^2-A*b^2+ 
3*B*a*b)/(a^4+2*a^2*b^2+b^4)/(a^2+b^2)^(1/2)*arctanh(1/2*(2*a*tanh(1/2*x)- 
2*b)/(a^2+b^2)^(1/2))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1614 vs. \(2 (119) = 238\).

Time = 0.11 (sec) , antiderivative size = 1614, normalized size of antiderivative = 12.61 \[ \int \frac {A+B \sinh (x)}{(a+b \sinh (x))^3} \, dx=\text {Too large to display} \] Input:

integrate((A+B*sinh(x))/(a+b*sinh(x))^3,x, algorithm="fricas")
 

Output:

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

Sympy [F(-1)]

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

integrate((A+B*sinh(x))/(a+b*sinh(x))**3,x)
 

Output:

Timed out
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 537 vs. \(2 (119) = 238\).

Time = 0.14 (sec) , antiderivative size = 537, normalized size of antiderivative = 4.20 \[ \int \frac {A+B \sinh (x)}{(a+b \sinh (x))^3} \, dx=\frac {1}{2} \, {\left (\frac {3 \, a b \log \left (\frac {b e^{\left (-x\right )} - a - \sqrt {a^{2} + b^{2}}}{b e^{\left (-x\right )} - a + \sqrt {a^{2} + b^{2}}}\right )}{{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} \sqrt {a^{2} + b^{2}}} + \frac {2 \, {\left (3 \, a b^{3} e^{\left (-3 \, x\right )} + a^{2} b^{2} - 2 \, b^{4} + {\left (4 \, a^{3} b - 5 \, a b^{3}\right )} e^{\left (-x\right )} + {\left (2 \, a^{4} - 5 \, a^{2} b^{2} + 2 \, b^{4}\right )} e^{\left (-2 \, x\right )}\right )}}{a^{4} b^{3} + 2 \, a^{2} b^{5} + b^{7} + 4 \, {\left (a^{5} b^{2} + 2 \, a^{3} b^{4} + a b^{6}\right )} e^{\left (-x\right )} + 2 \, {\left (2 \, a^{6} b + 3 \, a^{4} b^{3} - b^{7}\right )} e^{\left (-2 \, x\right )} - 4 \, {\left (a^{5} b^{2} + 2 \, a^{3} b^{4} + a b^{6}\right )} e^{\left (-3 \, x\right )} + {\left (a^{4} b^{3} + 2 \, a^{2} b^{5} + b^{7}\right )} e^{\left (-4 \, x\right )}}\right )} B + \frac {1}{2} \, A {\left (\frac {{\left (2 \, a^{2} - b^{2}\right )} \log \left (\frac {b e^{\left (-x\right )} - a - \sqrt {a^{2} + b^{2}}}{b e^{\left (-x\right )} - a + \sqrt {a^{2} + b^{2}}}\right )}{{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} \sqrt {a^{2} + b^{2}}} - \frac {2 \, {\left (3 \, a b^{2} + {\left (10 \, a^{2} b + b^{3}\right )} e^{\left (-x\right )} + 3 \, {\left (2 \, a^{3} - a b^{2}\right )} e^{\left (-2 \, x\right )} - {\left (2 \, a^{2} b - b^{3}\right )} e^{\left (-3 \, x\right )}\right )}}{a^{4} b^{2} + 2 \, a^{2} b^{4} + b^{6} + 4 \, {\left (a^{5} b + 2 \, a^{3} b^{3} + a b^{5}\right )} e^{\left (-x\right )} + 2 \, {\left (2 \, a^{6} + 3 \, a^{4} b^{2} - b^{6}\right )} e^{\left (-2 \, x\right )} - 4 \, {\left (a^{5} b + 2 \, a^{3} b^{3} + a b^{5}\right )} e^{\left (-3 \, x\right )} + {\left (a^{4} b^{2} + 2 \, a^{2} b^{4} + b^{6}\right )} e^{\left (-4 \, x\right )}}\right )} \] Input:

integrate((A+B*sinh(x))/(a+b*sinh(x))^3,x, algorithm="maxima")
 

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 279 vs. \(2 (119) = 238\).

Time = 0.14 (sec) , antiderivative size = 279, normalized size of antiderivative = 2.18 \[ \int \frac {A+B \sinh (x)}{(a+b \sinh (x))^3} \, dx=-\frac {{\left (2 \, A a^{2} + 3 \, B a b - A b^{2}\right )} \log \left (\frac {{\left | -2 \, b e^{x} - 2 \, a - 2 \, \sqrt {a^{2} + b^{2}} \right |}}{{\left | -2 \, b e^{x} - 2 \, a + 2 \, \sqrt {a^{2} + b^{2}} \right |}}\right )}{2 \, {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} \sqrt {a^{2} + b^{2}}} + \frac {2 \, A a^{2} b^{2} e^{\left (3 \, x\right )} + 3 \, B a b^{3} e^{\left (3 \, x\right )} - A b^{4} e^{\left (3 \, x\right )} - 2 \, B a^{4} e^{\left (2 \, x\right )} + 6 \, A a^{3} b e^{\left (2 \, x\right )} + 5 \, B a^{2} b^{2} e^{\left (2 \, x\right )} - 3 \, A a b^{3} e^{\left (2 \, x\right )} - 2 \, B b^{4} e^{\left (2 \, x\right )} + 4 \, B a^{3} b e^{x} - 10 \, A a^{2} b^{2} e^{x} - 5 \, B a b^{3} e^{x} - A b^{4} e^{x} - B a^{2} b^{2} + 3 \, A a b^{3} + 2 \, B b^{4}}{{\left (a^{4} b + 2 \, a^{2} b^{3} + b^{5}\right )} {\left (b e^{\left (2 \, x\right )} + 2 \, a e^{x} - b\right )}^{2}} \] Input:

integrate((A+B*sinh(x))/(a+b*sinh(x))^3,x, algorithm="giac")
 

Output:

-1/2*(2*A*a^2 + 3*B*a*b - A*b^2)*log(abs(-2*b*e^x - 2*a - 2*sqrt(a^2 + b^2 
))/abs(-2*b*e^x - 2*a + 2*sqrt(a^2 + b^2)))/((a^4 + 2*a^2*b^2 + b^4)*sqrt( 
a^2 + b^2)) + (2*A*a^2*b^2*e^(3*x) + 3*B*a*b^3*e^(3*x) - A*b^4*e^(3*x) - 2 
*B*a^4*e^(2*x) + 6*A*a^3*b*e^(2*x) + 5*B*a^2*b^2*e^(2*x) - 3*A*a*b^3*e^(2* 
x) - 2*B*b^4*e^(2*x) + 4*B*a^3*b*e^x - 10*A*a^2*b^2*e^x - 5*B*a*b^3*e^x - 
A*b^4*e^x - B*a^2*b^2 + 3*A*a*b^3 + 2*B*b^4)/((a^4*b + 2*a^2*b^3 + b^5)*(b 
*e^(2*x) + 2*a*e^x - b)^2)
 

Mupad [F(-1)]

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

int((A + B*sinh(x))/(a + b*sinh(x))^3,x)
 

Output:

int((A + B*sinh(x))/(a + b*sinh(x))^3, x)
 

Reduce [B] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 230, normalized size of antiderivative = 1.80 \[ \int \frac {A+B \sinh (x)}{(a+b \sinh (x))^3} \, dx=\frac {2 e^{2 x} \sqrt {a^{2}+b^{2}}\, \mathit {atan} \left (\frac {e^{x} b i +a i}{\sqrt {a^{2}+b^{2}}}\right ) a b i +4 e^{x} \sqrt {a^{2}+b^{2}}\, \mathit {atan} \left (\frac {e^{x} b i +a i}{\sqrt {a^{2}+b^{2}}}\right ) a^{2} i -2 \sqrt {a^{2}+b^{2}}\, \mathit {atan} \left (\frac {e^{x} b i +a i}{\sqrt {a^{2}+b^{2}}}\right ) a b i -e^{2 x} a^{2} b -e^{2 x} b^{3}-a^{2} b -b^{3}}{e^{2 x} a^{4} b +2 e^{2 x} a^{2} b^{3}+e^{2 x} b^{5}+2 e^{x} a^{5}+4 e^{x} a^{3} b^{2}+2 e^{x} a \,b^{4}-a^{4} b -2 a^{2} b^{3}-b^{5}} \] Input:

int((A+B*sinh(x))/(a+b*sinh(x))^3,x)
 

Output:

(2*e**(2*x)*sqrt(a**2 + b**2)*atan((e**x*b*i + a*i)/sqrt(a**2 + b**2))*a*b 
*i + 4*e**x*sqrt(a**2 + b**2)*atan((e**x*b*i + a*i)/sqrt(a**2 + b**2))*a** 
2*i - 2*sqrt(a**2 + b**2)*atan((e**x*b*i + a*i)/sqrt(a**2 + b**2))*a*b*i - 
 e**(2*x)*a**2*b - e**(2*x)*b**3 - a**2*b - b**3)/(e**(2*x)*a**4*b + 2*e** 
(2*x)*a**2*b**3 + e**(2*x)*b**5 + 2*e**x*a**5 + 4*e**x*a**3*b**2 + 2*e**x* 
a*b**4 - a**4*b - 2*a**2*b**3 - b**5)