\(\int \frac {A+B \sinh (x)}{(a+b \sinh (x))^4} \, dx\) [132]

Optimal result

Integrand size = 15, antiderivative size = 187 \[ \int \frac {A+B \sinh (x)}{(a+b \sinh (x))^4} \, dx=-\frac {\left (2 a^3 A-3 a A b^2+4 a^2 b B-b^3 B\right ) \text {arctanh}\left (\frac {b-a \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{7/2}}-\frac {(A b-a B) \cosh (x)}{3 \left (a^2+b^2\right ) (a+b \sinh (x))^3}-\frac {\left (5 a A b-2 a^2 B+3 b^2 B\right ) \cosh (x)}{6 \left (a^2+b^2\right )^2 (a+b \sinh (x))^2}-\frac {\left (11 a^2 A b-4 A b^3-2 a^3 B+13 a b^2 B\right ) \cosh (x)}{6 \left (a^2+b^2\right )^3 (a+b \sinh (x))} \] Output:

-(2*A*a^3-3*A*a*b^2+4*B*a^2*b-B*b^3)*arctanh((b-a*tanh(1/2*x))/(a^2+b^2)^( 
1/2))/(a^2+b^2)^(7/2)-1/3*(A*b-B*a)*cosh(x)/(a^2+b^2)/(a+b*sinh(x))^3-1/6* 
(5*A*a*b-2*B*a^2+3*B*b^2)*cosh(x)/(a^2+b^2)^2/(a+b*sinh(x))^2-1/6*(11*A*a^ 
2*b-4*A*b^3-2*B*a^3+13*B*a*b^2)*cosh(x)/(a^2+b^2)^3/(a+b*sinh(x))
 

Mathematica [A] (verified)

Time = 0.31 (sec) , antiderivative size = 189, normalized size of antiderivative = 1.01 \[ \int \frac {A+B \sinh (x)}{(a+b \sinh (x))^4} \, dx=\frac {\frac {6 \left (2 a^3 A-3 a A b^2+4 a^2 b B-b^3 B\right ) \arctan \left (\frac {b-a \tanh \left (\frac {x}{2}\right )}{\sqrt {-a^2-b^2}}\right )}{\sqrt {-a^2-b^2}}+\frac {2 \left (a^2+b^2\right )^2 (-A b+a B) \cosh (x)}{(a+b \sinh (x))^3}+\frac {\left (a^2+b^2\right ) \left (-5 a A b+2 a^2 B-3 b^2 B\right ) \cosh (x)}{(a+b \sinh (x))^2}+\frac {\left (-11 a^2 A b+4 A b^3+2 a^3 B-13 a b^2 B\right ) \cosh (x)}{a+b \sinh (x)}}{6 \left (a^2+b^2\right )^3} \] Input:

Integrate[(A + B*Sinh[x])/(a + b*Sinh[x])^4,x]
 

Output:

((6*(2*a^3*A - 3*a*A*b^2 + 4*a^2*b*B - b^3*B)*ArcTan[(b - a*Tanh[x/2])/Sqr 
t[-a^2 - b^2]])/Sqrt[-a^2 - b^2] + (2*(a^2 + b^2)^2*(-(A*b) + a*B)*Cosh[x] 
)/(a + b*Sinh[x])^3 + ((a^2 + b^2)*(-5*a*A*b + 2*a^2*B - 3*b^2*B)*Cosh[x]) 
/(a + b*Sinh[x])^2 + ((-11*a^2*A*b + 4*A*b^3 + 2*a^3*B - 13*a*b^2*B)*Cosh[ 
x])/(a + b*Sinh[x]))/(6*(a^2 + b^2)^3)
 

Rubi [A] (verified)

Time = 0.86 (sec) , antiderivative size = 218, normalized size of antiderivative = 1.17, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.867, Rules used = {3042, 3233, 25, 3042, 3233, 25, 3042, 3233, 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.

\(\displaystyle \int \frac {A+B \sinh (x)}{(a+b \sinh (x))^4} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {A-i B \sin (i x)}{(a-i b \sin (i x))^4}dx\)

\(\Big \downarrow \) 3233

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3233

\(\displaystyle \frac {-\frac {\int -\frac {2 \left (3 A a^2+5 b B a-2 A b^2\right )-\left (-2 B a^2+5 A b a+3 b^2 B\right ) \sinh (x)}{(a+b \sinh (x))^2}dx}{2 \left (a^2+b^2\right )}-\frac {\cosh (x) \left (-2 a^2 B+5 a A b+3 b^2 B\right )}{2 \left (a^2+b^2\right ) (a+b \sinh (x))^2}}{3 \left (a^2+b^2\right )}-\frac {\cosh (x) (A b-a B)}{3 \left (a^2+b^2\right ) (a+b \sinh (x))^3}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {\int \frac {2 \left (3 A a^2+5 b B a-2 A b^2\right )-\left (-2 B a^2+5 A b a+3 b^2 B\right ) \sinh (x)}{(a+b \sinh (x))^2}dx}{2 \left (a^2+b^2\right )}-\frac {\cosh (x) \left (-2 a^2 B+5 a A b+3 b^2 B\right )}{2 \left (a^2+b^2\right ) (a+b \sinh (x))^2}}{3 \left (a^2+b^2\right )}-\frac {\cosh (x) (A b-a B)}{3 \left (a^2+b^2\right ) (a+b \sinh (x))^3}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {\cosh (x) (A b-a B)}{3 \left (a^2+b^2\right ) (a+b \sinh (x))^3}+\frac {-\frac {\cosh (x) \left (-2 a^2 B+5 a A b+3 b^2 B\right )}{2 \left (a^2+b^2\right ) (a+b \sinh (x))^2}+\frac {\int \frac {2 \left (3 A a^2+5 b B a-2 A b^2\right )+i \left (-2 B a^2+5 A b a+3 b^2 B\right ) \sin (i x)}{(a-i b \sin (i x))^2}dx}{2 \left (a^2+b^2\right )}}{3 \left (a^2+b^2\right )}\)

\(\Big \downarrow \) 3233

\(\displaystyle \frac {\frac {-\frac {\int -\frac {3 \left (2 A a^3+4 b B a^2-3 A b^2 a-b^3 B\right )}{a+b \sinh (x)}dx}{a^2+b^2}-\frac {\cosh (x) \left (-2 a^3 B+11 a^2 A b+13 a b^2 B-4 A b^3\right )}{\left (a^2+b^2\right ) (a+b \sinh (x))}}{2 \left (a^2+b^2\right )}-\frac {\cosh (x) \left (-2 a^2 B+5 a A b+3 b^2 B\right )}{2 \left (a^2+b^2\right ) (a+b \sinh (x))^2}}{3 \left (a^2+b^2\right )}-\frac {\cosh (x) (A b-a B)}{3 \left (a^2+b^2\right ) (a+b \sinh (x))^3}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\frac {3 \left (2 a^3 A+4 a^2 b B-3 a A b^2-b^3 B\right ) \int \frac {1}{a+b \sinh (x)}dx}{a^2+b^2}-\frac {\cosh (x) \left (-2 a^3 B+11 a^2 A b+13 a b^2 B-4 A b^3\right )}{\left (a^2+b^2\right ) (a+b \sinh (x))}}{2 \left (a^2+b^2\right )}-\frac {\cosh (x) \left (-2 a^2 B+5 a A b+3 b^2 B\right )}{2 \left (a^2+b^2\right ) (a+b \sinh (x))^2}}{3 \left (a^2+b^2\right )}-\frac {\cosh (x) (A b-a B)}{3 \left (a^2+b^2\right ) (a+b \sinh (x))^3}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {\cosh (x) (A b-a B)}{3 \left (a^2+b^2\right ) (a+b \sinh (x))^3}+\frac {-\frac {\cosh (x) \left (-2 a^2 B+5 a A b+3 b^2 B\right )}{2 \left (a^2+b^2\right ) (a+b \sinh (x))^2}+\frac {-\frac {\cosh (x) \left (-2 a^3 B+11 a^2 A b+13 a b^2 B-4 A b^3\right )}{\left (a^2+b^2\right ) (a+b \sinh (x))}+\frac {3 \left (2 a^3 A+4 a^2 b B-3 a A b^2-b^3 B\right ) \int \frac {1}{a-i b \sin (i x)}dx}{a^2+b^2}}{2 \left (a^2+b^2\right )}}{3 \left (a^2+b^2\right )}\)

\(\Big \downarrow \) 3139

\(\displaystyle \frac {\frac {\frac {6 \left (2 a^3 A+4 a^2 b B-3 a A b^2-b^3 B\right ) \int \frac {1}{-a \tanh ^2\left (\frac {x}{2}\right )+2 b \tanh \left (\frac {x}{2}\right )+a}d\tanh \left (\frac {x}{2}\right )}{a^2+b^2}-\frac {\cosh (x) \left (-2 a^3 B+11 a^2 A b+13 a b^2 B-4 A b^3\right )}{\left (a^2+b^2\right ) (a+b \sinh (x))}}{2 \left (a^2+b^2\right )}-\frac {\cosh (x) \left (-2 a^2 B+5 a A b+3 b^2 B\right )}{2 \left (a^2+b^2\right ) (a+b \sinh (x))^2}}{3 \left (a^2+b^2\right )}-\frac {\cosh (x) (A b-a B)}{3 \left (a^2+b^2\right ) (a+b \sinh (x))^3}\)

\(\Big \downarrow \) 1083

\(\displaystyle \frac {\frac {-\frac {12 \left (2 a^3 A+4 a^2 b B-3 a A b^2-b^3 B\right ) \int \frac {1}{4 \left (a^2+b^2\right )-\left (2 b-2 a \tanh \left (\frac {x}{2}\right )\right )^2}d\left (2 b-2 a \tanh \left (\frac {x}{2}\right )\right )}{a^2+b^2}-\frac {\cosh (x) \left (-2 a^3 B+11 a^2 A b+13 a b^2 B-4 A b^3\right )}{\left (a^2+b^2\right ) (a+b \sinh (x))}}{2 \left (a^2+b^2\right )}-\frac {\cosh (x) \left (-2 a^2 B+5 a A b+3 b^2 B\right )}{2 \left (a^2+b^2\right ) (a+b \sinh (x))^2}}{3 \left (a^2+b^2\right )}-\frac {\cosh (x) (A b-a B)}{3 \left (a^2+b^2\right ) (a+b \sinh (x))^3}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\frac {-\frac {6 \left (2 a^3 A+4 a^2 b B-3 a A b^2-b^3 B\right ) \text {arctanh}\left (\frac {2 b-2 a \tanh \left (\frac {x}{2}\right )}{2 \sqrt {a^2+b^2}}\right )}{\left (a^2+b^2\right )^{3/2}}-\frac {\cosh (x) \left (-2 a^3 B+11 a^2 A b+13 a b^2 B-4 A b^3\right )}{\left (a^2+b^2\right ) (a+b \sinh (x))}}{2 \left (a^2+b^2\right )}-\frac {\cosh (x) \left (-2 a^2 B+5 a A b+3 b^2 B\right )}{2 \left (a^2+b^2\right ) (a+b \sinh (x))^2}}{3 \left (a^2+b^2\right )}-\frac {\cosh (x) (A b-a B)}{3 \left (a^2+b^2\right ) (a+b \sinh (x))^3}\)

Input:

Int[(A + B*Sinh[x])/(a + b*Sinh[x])^4,x]
 

Output:

-1/3*((A*b - a*B)*Cosh[x])/((a^2 + b^2)*(a + b*Sinh[x])^3) + (-1/2*((5*a*A 
*b - 2*a^2*B + 3*b^2*B)*Cosh[x])/((a^2 + b^2)*(a + b*Sinh[x])^2) + ((-6*(2 
*a^3*A - 3*a*A*b^2 + 4*a^2*b*B - b^3*B)*ArcTanh[(2*b - 2*a*Tanh[x/2])/(2*S 
qrt[a^2 + b^2])])/(a^2 + b^2)^(3/2) - ((11*a^2*A*b - 4*A*b^3 - 2*a^3*B + 1 
3*a*b^2*B)*Cosh[x])/((a^2 + b^2)*(a + b*Sinh[x])))/(2*(a^2 + b^2)))/(3*(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. \(632\) vs. \(2(175)=350\).

Time = 0.52 (sec) , antiderivative size = 633, normalized size of antiderivative = 3.39

method	result	size
default	\(-\frac {2 \left (-\frac {b \left (9 A \,a^{4} b +6 A \,b^{3} a^{2}+2 A \,b^{5}-4 B \,a^{5}+B \,a^{3} b^{2}\right ) \tanh \left (\frac {x}{2}\right )^{5}}{2 a \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}-\frac {\left (6 A \,a^{6} b -27 A \,a^{4} b^{3}-12 A \,a^{2} b^{5}-4 A \,b^{7}-2 B \,a^{7}+14 B \,a^{5} b^{2}-11 B \,a^{3} b^{4}-2 B a \,b^{6}\right ) \tanh \left (\frac {x}{2}\right )^{4}}{2 \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right ) a^{2}}+\frac {b \left (54 A \,a^{6} b -21 A \,a^{4} b^{3}-4 A \,a^{2} b^{5}-4 A \,b^{7}-18 B \,a^{7}+42 B \,a^{5} b^{2}-17 B \,a^{3} b^{4}-2 B a \,b^{6}\right ) \tanh \left (\frac {x}{2}\right )^{3}}{3 a^{3} \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}+\frac {\left (6 A \,a^{6} b -20 A \,a^{4} b^{3}-3 A \,a^{2} b^{5}-2 A \,b^{7}-2 B \,a^{7}+10 B \,a^{5} b^{2}-14 B \,a^{3} b^{4}-B a \,b^{6}\right ) \tanh \left (\frac {x}{2}\right )^{2}}{a^{2} \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}-\frac {b \left (27 A \,a^{4} b +4 A \,b^{3} a^{2}+2 A \,b^{5}-8 B \,a^{5}+19 B \,a^{3} b^{2}+2 B a \,b^{4}\right ) \tanh \left (\frac {x}{2}\right )}{2 a \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}-\frac {18 A \,a^{4} b +5 A \,b^{3} a^{2}+2 A \,b^{5}-6 B \,a^{5}+10 B \,a^{3} b^{2}+B a \,b^{4}}{6 \left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right )}\right )}{\left (\tanh \left (\frac {x}{2}\right )^{2} a -2 b \tanh \left (\frac {x}{2}\right )-a \right )^{3}}+\frac {\left (2 a^{3} A -3 A a \,b^{2}+4 B \,a^{2} b -B \,b^{3}\right ) \operatorname {arctanh}\left (\frac {2 a \tanh \left (\frac {x}{2}\right )-2 b}{2 \sqrt {a^{2}+b^{2}}}\right )}{\left (a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}\right ) \sqrt {a^{2}+b^{2}}}\)	\(633\)
risch	\(\frac {-11 A \,a^{2} b^{4}+2 B \,a^{3} b^{3}+4 A \,b^{6}+60 B \,a^{3} b^{3} {\mathrm e}^{4 x}-15 B a \,b^{5} {\mathrm e}^{4 x}+44 A \,a^{5} b \,{\mathrm e}^{3 x}-82 A \,a^{3} b^{3} {\mathrm e}^{3 x}+24 A a \,b^{5} {\mathrm e}^{3 x}+64 B \,a^{4} b^{2} {\mathrm e}^{3 x}-78 B \,a^{2} b^{4} {\mathrm e}^{3 x}-102 A \,a^{4} b^{2} {\mathrm e}^{2 x}+6 A \,a^{3} b^{3} {\mathrm e}^{5 x}-9 A a \,b^{5} {\mathrm e}^{5 x}+12 B \,a^{2} b^{4} {\mathrm e}^{5 x}+30 A \,a^{4} b^{2} {\mathrm e}^{4 x}-45 A \,a^{2} b^{4} {\mathrm e}^{4 x}-15 A a \,b^{5} {\mathrm e}^{x}-12 B \,a^{4} b^{2} {\mathrm e}^{x}+66 B \,a^{2} b^{4} {\mathrm e}^{x}+60 A \,a^{3} b^{3} {\mathrm e}^{x}+3 B \,b^{6} {\mathrm e}^{x}-3 B \,b^{6} {\mathrm e}^{5 x}-8 B \,a^{6} {\mathrm e}^{3 x}-12 A \,b^{6} {\mathrm e}^{2 x}+36 A \,a^{2} b^{4} {\mathrm e}^{2 x}+24 B \,a^{5} b \,{\mathrm e}^{2 x}-102 B \,a^{3} b^{3} {\mathrm e}^{2 x}+24 B a \,b^{5} {\mathrm e}^{2 x}-13 B a \,b^{5}}{3 b \left (a^{2}+b^{2}\right )^{3} \left (b \,{\mathrm e}^{2 x}+2 \,{\mathrm e}^{x} a -b \right )^{3}}+\frac {\ln \left ({\mathrm e}^{x}+\frac {\left (a^{2}+b^{2}\right )^{\frac {7}{2}} a -a^{8}-4 a^{6} b^{2}-6 b^{4} a^{4}-4 a^{2} b^{6}-b^{8}}{b \left (a^{2}+b^{2}\right )^{\frac {7}{2}}}\right ) a^{3} A}{\left (a^{2}+b^{2}\right )^{\frac {7}{2}}}-\frac {3 \ln \left ({\mathrm e}^{x}+\frac {\left (a^{2}+b^{2}\right )^{\frac {7}{2}} a -a^{8}-4 a^{6} b^{2}-6 b^{4} a^{4}-4 a^{2} b^{6}-b^{8}}{b \left (a^{2}+b^{2}\right )^{\frac {7}{2}}}\right ) A a \,b^{2}}{2 \left (a^{2}+b^{2}\right )^{\frac {7}{2}}}+\frac {2 \ln \left ({\mathrm e}^{x}+\frac {\left (a^{2}+b^{2}\right )^{\frac {7}{2}} a -a^{8}-4 a^{6} b^{2}-6 b^{4} a^{4}-4 a^{2} b^{6}-b^{8}}{b \left (a^{2}+b^{2}\right )^{\frac {7}{2}}}\right ) B \,a^{2} b}{\left (a^{2}+b^{2}\right )^{\frac {7}{2}}}-\frac {\ln \left ({\mathrm e}^{x}+\frac {\left (a^{2}+b^{2}\right )^{\frac {7}{2}} a -a^{8}-4 a^{6} b^{2}-6 b^{4} a^{4}-4 a^{2} b^{6}-b^{8}}{b \left (a^{2}+b^{2}\right )^{\frac {7}{2}}}\right ) B \,b^{3}}{2 \left (a^{2}+b^{2}\right )^{\frac {7}{2}}}-\frac {\ln \left ({\mathrm e}^{x}+\frac {\left (a^{2}+b^{2}\right )^{\frac {7}{2}} a +a^{8}+4 a^{6} b^{2}+6 b^{4} a^{4}+4 a^{2} b^{6}+b^{8}}{b \left (a^{2}+b^{2}\right )^{\frac {7}{2}}}\right ) a^{3} A}{\left (a^{2}+b^{2}\right )^{\frac {7}{2}}}+\frac {3 \ln \left ({\mathrm e}^{x}+\frac {\left (a^{2}+b^{2}\right )^{\frac {7}{2}} a +a^{8}+4 a^{6} b^{2}+6 b^{4} a^{4}+4 a^{2} b^{6}+b^{8}}{b \left (a^{2}+b^{2}\right )^{\frac {7}{2}}}\right ) A a \,b^{2}}{2 \left (a^{2}+b^{2}\right )^{\frac {7}{2}}}-\frac {2 \ln \left ({\mathrm e}^{x}+\frac {\left (a^{2}+b^{2}\right )^{\frac {7}{2}} a +a^{8}+4 a^{6} b^{2}+6 b^{4} a^{4}+4 a^{2} b^{6}+b^{8}}{b \left (a^{2}+b^{2}\right )^{\frac {7}{2}}}\right ) B \,a^{2} b}{\left (a^{2}+b^{2}\right )^{\frac {7}{2}}}+\frac {\ln \left ({\mathrm e}^{x}+\frac {\left (a^{2}+b^{2}\right )^{\frac {7}{2}} a +a^{8}+4 a^{6} b^{2}+6 b^{4} a^{4}+4 a^{2} b^{6}+b^{8}}{b \left (a^{2}+b^{2}\right )^{\frac {7}{2}}}\right ) B \,b^{3}}{2 \left (a^{2}+b^{2}\right )^{\frac {7}{2}}}\)	\(965\)

Input:

int((A+B*sinh(x))/(a+b*sinh(x))^4,x,method=_RETURNVERBOSE)
 

Output:

-2*(-1/2*b*(9*A*a^4*b+6*A*a^2*b^3+2*A*b^5-4*B*a^5+B*a^3*b^2)/a/(a^6+3*a^4* 
b^2+3*a^2*b^4+b^6)*tanh(1/2*x)^5-1/2*(6*A*a^6*b-27*A*a^4*b^3-12*A*a^2*b^5- 
4*A*b^7-2*B*a^7+14*B*a^5*b^2-11*B*a^3*b^4-2*B*a*b^6)/(a^6+3*a^4*b^2+3*a^2* 
b^4+b^6)/a^2*tanh(1/2*x)^4+1/3/a^3*b*(54*A*a^6*b-21*A*a^4*b^3-4*A*a^2*b^5- 
4*A*b^7-18*B*a^7+42*B*a^5*b^2-17*B*a^3*b^4-2*B*a*b^6)/(a^6+3*a^4*b^2+3*a^2 
*b^4+b^6)*tanh(1/2*x)^3+1/a^2*(6*A*a^6*b-20*A*a^4*b^3-3*A*a^2*b^5-2*A*b^7- 
2*B*a^7+10*B*a^5*b^2-14*B*a^3*b^4-B*a*b^6)/(a^6+3*a^4*b^2+3*a^2*b^4+b^6)*t 
anh(1/2*x)^2-1/2/a*b*(27*A*a^4*b+4*A*a^2*b^3+2*A*b^5-8*B*a^5+19*B*a^3*b^2+ 
2*B*a*b^4)/(a^6+3*a^4*b^2+3*a^2*b^4+b^6)*tanh(1/2*x)-1/6*(18*A*a^4*b+5*A*a 
^2*b^3+2*A*b^5-6*B*a^5+10*B*a^3*b^2+B*a*b^4)/(a^6+3*a^4*b^2+3*a^2*b^4+b^6) 
)/(tanh(1/2*x)^2*a-2*b*tanh(1/2*x)-a)^3+(2*A*a^3-3*A*a*b^2+4*B*a^2*b-B*b^3 
)/(a^6+3*a^4*b^2+3*a^2*b^4+b^6)/(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. 3870 vs. \(2 (177) = 354\).

Time = 0.24 (sec) , antiderivative size = 3870, normalized size of antiderivative = 20.70 \[ \int \frac {A+B \sinh (x)}{(a+b \sinh (x))^4} \, dx=\text {Too large to display} \] Input:

integrate((A+B*sinh(x))/(a+b*sinh(x))^4,x, algorithm="fricas")
 

Output:

Too large to include
 

Sympy [F(-1)]

Timed out. \[ \int \frac {A+B \sinh (x)}{(a+b \sinh (x))^4} \, dx=\text {Timed out} \] Input:

integrate((A+B*sinh(x))/(a+b*sinh(x))**4,x)
 

Output:

Timed out
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 982 vs. \(2 (177) = 354\).

Time = 0.18 (sec) , antiderivative size = 982, normalized size of antiderivative = 5.25 \[ \int \frac {A+B \sinh (x)}{(a+b \sinh (x))^4} \, dx=\text {Too large to display} \] Input:

integrate((A+B*sinh(x))/(a+b*sinh(x))^4,x, algorithm="maxima")
 

Output:

1/6*(3*(2*a^2 - 3*b^2)*a*log((b*e^(-x) - a - sqrt(a^2 + b^2))/(b*e^(-x) - 
a + sqrt(a^2 + b^2)))/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*sqrt(a^2 + b^2) 
) - 2*(11*a^2*b^3 - 4*b^5 + 15*(4*a^3*b^2 - a*b^4)*e^(-x) + 6*(17*a^4*b - 
6*a^2*b^3 + 2*b^5)*e^(-2*x) + 2*(22*a^5 - 41*a^3*b^2 + 12*a*b^4)*e^(-3*x) 
- 15*(2*a^4*b - 3*a^2*b^3)*e^(-4*x) + 3*(2*a^3*b^2 - 3*a*b^4)*e^(-5*x))/(a 
^6*b^3 + 3*a^4*b^5 + 3*a^2*b^7 + b^9 + 6*(a^7*b^2 + 3*a^5*b^4 + 3*a^3*b^6 
+ a*b^8)*e^(-x) + 3*(4*a^8*b + 11*a^6*b^3 + 9*a^4*b^5 + a^2*b^7 - b^9)*e^( 
-2*x) + 4*(2*a^9 + 3*a^7*b^2 - 3*a^5*b^4 - 7*a^3*b^6 - 3*a*b^8)*e^(-3*x) - 
 3*(4*a^8*b + 11*a^6*b^3 + 9*a^4*b^5 + a^2*b^7 - b^9)*e^(-4*x) + 6*(a^7*b^ 
2 + 3*a^5*b^4 + 3*a^3*b^6 + a*b^8)*e^(-5*x) - (a^6*b^3 + 3*a^4*b^5 + 3*a^2 
*b^7 + b^9)*e^(-6*x)))*A + 1/6*B*(3*(4*a^2*b - b^3)*log((b*e^(-x) - a - sq 
rt(a^2 + b^2))/(b*e^(-x) - a + sqrt(a^2 + b^2)))/((a^6 + 3*a^4*b^2 + 3*a^2 
*b^4 + b^6)*sqrt(a^2 + b^2)) + 2*(2*a^3*b^3 - 13*a*b^5 + 3*(4*a^4*b^2 - 22 
*a^2*b^4 - b^6)*e^(-x) + 6*(4*a^5*b - 17*a^3*b^3 + 4*a*b^5)*e^(-2*x) + 2*( 
4*a^6 - 32*a^4*b^2 + 39*a^2*b^4)*e^(-3*x) + 15*(4*a^3*b^3 - a*b^5)*e^(-4*x 
) - 3*(4*a^2*b^4 - b^6)*e^(-5*x))/(a^6*b^4 + 3*a^4*b^6 + 3*a^2*b^8 + b^10 
+ 6*(a^7*b^3 + 3*a^5*b^5 + 3*a^3*b^7 + a*b^9)*e^(-x) + 3*(4*a^8*b^2 + 11*a 
^6*b^4 + 9*a^4*b^6 + a^2*b^8 - b^10)*e^(-2*x) + 4*(2*a^9*b + 3*a^7*b^3 - 3 
*a^5*b^5 - 7*a^3*b^7 - 3*a*b^9)*e^(-3*x) - 3*(4*a^8*b^2 + 11*a^6*b^4 + 9*a 
^4*b^6 + a^2*b^8 - b^10)*e^(-4*x) + 6*(a^7*b^3 + 3*a^5*b^5 + 3*a^3*b^7 ...
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 477 vs. \(2 (177) = 354\).

Time = 0.15 (sec) , antiderivative size = 477, normalized size of antiderivative = 2.55 \[ \int \frac {A+B \sinh (x)}{(a+b \sinh (x))^4} \, dx=\frac {{\left (2 \, A a^{3} + 4 \, B a^{2} b - 3 \, A a b^{2} - B b^{3}\right )} \log \left (\frac {{\left | 2 \, b e^{x} + 2 \, a - 2 \, \sqrt {a^{2} + b^{2}} \right |}}{{\left | 2 \, b e^{x} + 2 \, a + 2 \, \sqrt {a^{2} + b^{2}} \right |}}\right )}{2 \, {\left (a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}\right )} \sqrt {a^{2} + b^{2}}} + \frac {6 \, A a^{3} b^{3} e^{\left (5 \, x\right )} + 12 \, B a^{2} b^{4} e^{\left (5 \, x\right )} - 9 \, A a b^{5} e^{\left (5 \, x\right )} - 3 \, B b^{6} e^{\left (5 \, x\right )} + 30 \, A a^{4} b^{2} e^{\left (4 \, x\right )} + 60 \, B a^{3} b^{3} e^{\left (4 \, x\right )} - 45 \, A a^{2} b^{4} e^{\left (4 \, x\right )} - 15 \, B a b^{5} e^{\left (4 \, x\right )} - 8 \, B a^{6} e^{\left (3 \, x\right )} + 44 \, A a^{5} b e^{\left (3 \, x\right )} + 64 \, B a^{4} b^{2} e^{\left (3 \, x\right )} - 82 \, A a^{3} b^{3} e^{\left (3 \, x\right )} - 78 \, B a^{2} b^{4} e^{\left (3 \, x\right )} + 24 \, A a b^{5} e^{\left (3 \, x\right )} + 24 \, B a^{5} b e^{\left (2 \, x\right )} - 102 \, A a^{4} b^{2} e^{\left (2 \, x\right )} - 102 \, B a^{3} b^{3} e^{\left (2 \, x\right )} + 36 \, A a^{2} b^{4} e^{\left (2 \, x\right )} + 24 \, B a b^{5} e^{\left (2 \, x\right )} - 12 \, A b^{6} e^{\left (2 \, x\right )} - 12 \, B a^{4} b^{2} e^{x} + 60 \, A a^{3} b^{3} e^{x} + 66 \, B a^{2} b^{4} e^{x} - 15 \, A a b^{5} e^{x} + 3 \, B b^{6} e^{x} + 2 \, B a^{3} b^{3} - 11 \, A a^{2} b^{4} - 13 \, B a b^{5} + 4 \, A b^{6}}{3 \, {\left (a^{6} b + 3 \, a^{4} b^{3} + 3 \, a^{2} b^{5} + b^{7}\right )} {\left (b e^{\left (2 \, x\right )} + 2 \, a e^{x} - b\right )}^{3}} \] Input:

integrate((A+B*sinh(x))/(a+b*sinh(x))^4,x, algorithm="giac")
 

Output:

1/2*(2*A*a^3 + 4*B*a^2*b - 3*A*a*b^2 - B*b^3)*log(abs(2*b*e^x + 2*a - 2*sq 
rt(a^2 + b^2))/abs(2*b*e^x + 2*a + 2*sqrt(a^2 + b^2)))/((a^6 + 3*a^4*b^2 + 
 3*a^2*b^4 + b^6)*sqrt(a^2 + b^2)) + 1/3*(6*A*a^3*b^3*e^(5*x) + 12*B*a^2*b 
^4*e^(5*x) - 9*A*a*b^5*e^(5*x) - 3*B*b^6*e^(5*x) + 30*A*a^4*b^2*e^(4*x) + 
60*B*a^3*b^3*e^(4*x) - 45*A*a^2*b^4*e^(4*x) - 15*B*a*b^5*e^(4*x) - 8*B*a^6 
*e^(3*x) + 44*A*a^5*b*e^(3*x) + 64*B*a^4*b^2*e^(3*x) - 82*A*a^3*b^3*e^(3*x 
) - 78*B*a^2*b^4*e^(3*x) + 24*A*a*b^5*e^(3*x) + 24*B*a^5*b*e^(2*x) - 102*A 
*a^4*b^2*e^(2*x) - 102*B*a^3*b^3*e^(2*x) + 36*A*a^2*b^4*e^(2*x) + 24*B*a*b 
^5*e^(2*x) - 12*A*b^6*e^(2*x) - 12*B*a^4*b^2*e^x + 60*A*a^3*b^3*e^x + 66*B 
*a^2*b^4*e^x - 15*A*a*b^5*e^x + 3*B*b^6*e^x + 2*B*a^3*b^3 - 11*A*a^2*b^4 - 
 13*B*a*b^5 + 4*A*b^6)/((a^6*b + 3*a^4*b^3 + 3*a^2*b^5 + b^7)*(b*e^(2*x) + 
 2*a*e^x - b)^3)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {A+B \sinh (x)}{(a+b \sinh (x))^4} \, dx=\int \frac {A+B\,\mathrm {sinh}\left (x\right )}{{\left (a+b\,\mathrm {sinh}\left (x\right )\right )}^4} \,d x \] Input:

int((A + B*sinh(x))/(a + b*sinh(x))^4,x)
 

Output:

int((A + B*sinh(x))/(a + b*sinh(x))^4, x)
 

Reduce [B] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 815, normalized size of antiderivative = 4.36 \[ \int \frac {A+B \sinh (x)}{(a+b \sinh (x))^4} \, dx=\frac {8 e^{4 x} \sqrt {a^{2}+b^{2}}\, \mathit {atan} \left (\frac {e^{x} b i +a i}{\sqrt {a^{2}+b^{2}}}\right ) a^{3} b^{2} i -4 e^{4 x} \sqrt {a^{2}+b^{2}}\, \mathit {atan} \left (\frac {e^{x} b i +a i}{\sqrt {a^{2}+b^{2}}}\right ) a \,b^{4} i +32 e^{3 x} \sqrt {a^{2}+b^{2}}\, \mathit {atan} \left (\frac {e^{x} b i +a i}{\sqrt {a^{2}+b^{2}}}\right ) a^{4} b i -16 e^{3 x} \sqrt {a^{2}+b^{2}}\, \mathit {atan} \left (\frac {e^{x} b i +a i}{\sqrt {a^{2}+b^{2}}}\right ) a^{2} b^{3} i +32 e^{2 x} \sqrt {a^{2}+b^{2}}\, \mathit {atan} \left (\frac {e^{x} b i +a i}{\sqrt {a^{2}+b^{2}}}\right ) a^{5} i -32 e^{2 x} \sqrt {a^{2}+b^{2}}\, \mathit {atan} \left (\frac {e^{x} b i +a i}{\sqrt {a^{2}+b^{2}}}\right ) a^{3} b^{2} i +8 e^{2 x} \sqrt {a^{2}+b^{2}}\, \mathit {atan} \left (\frac {e^{x} b i +a i}{\sqrt {a^{2}+b^{2}}}\right ) a \,b^{4} i -32 e^{x} \sqrt {a^{2}+b^{2}}\, \mathit {atan} \left (\frac {e^{x} b i +a i}{\sqrt {a^{2}+b^{2}}}\right ) a^{4} b i +16 e^{x} \sqrt {a^{2}+b^{2}}\, \mathit {atan} \left (\frac {e^{x} b i +a i}{\sqrt {a^{2}+b^{2}}}\right ) a^{2} b^{3} i +8 \sqrt {a^{2}+b^{2}}\, \mathit {atan} \left (\frac {e^{x} b i +a i}{\sqrt {a^{2}+b^{2}}}\right ) a^{3} b^{2} i -4 \sqrt {a^{2}+b^{2}}\, \mathit {atan} \left (\frac {e^{x} b i +a i}{\sqrt {a^{2}+b^{2}}}\right ) a \,b^{4} i -2 e^{4 x} a^{4} b^{2}-e^{4 x} a^{2} b^{4}+e^{4 x} b^{6}+16 e^{2 x} a^{6}+12 e^{2 x} a^{4} b^{2}-6 e^{2 x} a^{2} b^{4}-2 e^{2 x} b^{6}-32 e^{x} a^{5} b -40 e^{x} a^{3} b^{3}-8 e^{x} a \,b^{5}+10 a^{4} b^{2}+11 a^{2} b^{4}+b^{6}}{4 a \left (e^{4 x} a^{6} b^{2}+3 e^{4 x} a^{4} b^{4}+3 e^{4 x} a^{2} b^{6}+e^{4 x} b^{8}+4 e^{3 x} a^{7} b +12 e^{3 x} a^{5} b^{3}+12 e^{3 x} a^{3} b^{5}+4 e^{3 x} a \,b^{7}+4 e^{2 x} a^{8}+10 e^{2 x} a^{6} b^{2}+6 e^{2 x} a^{4} b^{4}-2 e^{2 x} a^{2} b^{6}-2 e^{2 x} b^{8}-4 e^{x} a^{7} b -12 e^{x} a^{5} b^{3}-12 e^{x} a^{3} b^{5}-4 e^{x} a \,b^{7}+a^{6} b^{2}+3 a^{4} b^{4}+3 a^{2} b^{6}+b^{8}\right )} \] Input:

int((A+B*sinh(x))/(a+b*sinh(x))^4,x)
 

Output:

(8*e**(4*x)*sqrt(a**2 + b**2)*atan((e**x*b*i + a*i)/sqrt(a**2 + b**2))*a** 
3*b**2*i - 4*e**(4*x)*sqrt(a**2 + b**2)*atan((e**x*b*i + a*i)/sqrt(a**2 + 
b**2))*a*b**4*i + 32*e**(3*x)*sqrt(a**2 + b**2)*atan((e**x*b*i + a*i)/sqrt 
(a**2 + b**2))*a**4*b*i - 16*e**(3*x)*sqrt(a**2 + b**2)*atan((e**x*b*i + a 
*i)/sqrt(a**2 + b**2))*a**2*b**3*i + 32*e**(2*x)*sqrt(a**2 + b**2)*atan((e 
**x*b*i + a*i)/sqrt(a**2 + b**2))*a**5*i - 32*e**(2*x)*sqrt(a**2 + b**2)*a 
tan((e**x*b*i + a*i)/sqrt(a**2 + b**2))*a**3*b**2*i + 8*e**(2*x)*sqrt(a**2 
 + b**2)*atan((e**x*b*i + a*i)/sqrt(a**2 + b**2))*a*b**4*i - 32*e**x*sqrt( 
a**2 + b**2)*atan((e**x*b*i + a*i)/sqrt(a**2 + b**2))*a**4*b*i + 16*e**x*s 
qrt(a**2 + b**2)*atan((e**x*b*i + a*i)/sqrt(a**2 + b**2))*a**2*b**3*i + 8* 
sqrt(a**2 + b**2)*atan((e**x*b*i + a*i)/sqrt(a**2 + b**2))*a**3*b**2*i - 4 
*sqrt(a**2 + b**2)*atan((e**x*b*i + a*i)/sqrt(a**2 + b**2))*a*b**4*i - 2*e 
**(4*x)*a**4*b**2 - e**(4*x)*a**2*b**4 + e**(4*x)*b**6 + 16*e**(2*x)*a**6 
+ 12*e**(2*x)*a**4*b**2 - 6*e**(2*x)*a**2*b**4 - 2*e**(2*x)*b**6 - 32*e**x 
*a**5*b - 40*e**x*a**3*b**3 - 8*e**x*a*b**5 + 10*a**4*b**2 + 11*a**2*b**4 
+ b**6)/(4*a*(e**(4*x)*a**6*b**2 + 3*e**(4*x)*a**4*b**4 + 3*e**(4*x)*a**2* 
b**6 + e**(4*x)*b**8 + 4*e**(3*x)*a**7*b + 12*e**(3*x)*a**5*b**3 + 12*e**( 
3*x)*a**3*b**5 + 4*e**(3*x)*a*b**7 + 4*e**(2*x)*a**8 + 10*e**(2*x)*a**6*b* 
*2 + 6*e**(2*x)*a**4*b**4 - 2*e**(2*x)*a**2*b**6 - 2*e**(2*x)*b**8 - 4*e** 
x*a**7*b - 12*e**x*a**5*b**3 - 12*e**x*a**3*b**5 - 4*e**x*a*b**7 + a**6...