\(\int \frac {A+B \cosh (x)}{(a+a \cosh (x))^{3/2}} \, dx\) [102]

Optimal result

Integrand size = 17, antiderivative size = 65 \[ \int \frac {A+B \cosh (x)}{(a+a \cosh (x))^{3/2}} \, dx=\frac {(A+3 B) \arctan \left (\frac {\sqrt {a} \sinh (x)}{\sqrt {2} \sqrt {a+a \cosh (x)}}\right )}{2 \sqrt {2} a^{3/2}}+\frac {(A-B) \sinh (x)}{2 (a+a \cosh (x))^{3/2}} \] Output:

1/4*(A+3*B)*arctan(1/2*a^(1/2)*sinh(x)*2^(1/2)/(a+a*cosh(x))^(1/2))*2^(1/2 
)/a^(3/2)+1/2*(A-B)*sinh(x)/(a+a*cosh(x))^(3/2)
 

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.68 \[ \int \frac {A+B \cosh (x)}{(a+a \cosh (x))^{3/2}} \, dx=\frac {(A+3 B) \arctan \left (\sinh \left (\frac {x}{2}\right )\right ) \cosh ^3\left (\frac {x}{2}\right )+\frac {1}{2} (A-B) \sinh (x)}{(a (1+\cosh (x)))^{3/2}} \] Input:

Integrate[(A + B*Cosh[x])/(a + a*Cosh[x])^(3/2),x]
 

Output:

((A + 3*B)*ArcTan[Sinh[x/2]]*Cosh[x/2]^3 + ((A - B)*Sinh[x])/2)/(a*(1 + Co 
sh[x]))^(3/2)
 

Rubi [A] (verified)

Time = 0.34 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.294, Rules used = {3042, 3229, 3042, 3128, 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.

\(\displaystyle \int \frac {A+B \cosh (x)}{(a \cosh (x)+a)^{3/2}} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {A+B \sin \left (\frac {\pi }{2}+i x\right )}{\left (a+a \sin \left (\frac {\pi }{2}+i x\right )\right )^{3/2}}dx\)

\(\Big \downarrow \) 3229

\(\displaystyle \frac {(A+3 B) \int \frac {1}{\sqrt {\cosh (x) a+a}}dx}{4 a}+\frac {(A-B) \sinh (x)}{2 (a \cosh (x)+a)^{3/2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {(A-B) \sinh (x)}{2 (a \cosh (x)+a)^{3/2}}+\frac {(A+3 B) \int \frac {1}{\sqrt {\sin \left (i x+\frac {\pi }{2}\right ) a+a}}dx}{4 a}\)

\(\Big \downarrow \) 3128

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

\(\Big \downarrow \) 219

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

Input:

Int[(A + B*Cosh[x])/(a + a*Cosh[x])^(3/2),x]
 

Output:

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

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 3042

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

rule 3128

Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[-2/d 
Subst[Int[1/(2*a - x^2), x], x, b*(Cos[c + d*x]/Sqrt[a + b*Sin[c + d*x]])], 
 x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0]
 

rule 3229

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/(a*f*(2*m + 1))), x] + Simp[(a*d*m + b*c*(m + 1))/(a*b*(2*m + 1))   I 
nt[(a + b*Sin[e + f*x])^(m + 1), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && 
NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && LtQ[m, -2^(-1)]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(158\) vs. \(2(50)=100\).

Time = 0.99 (sec) , antiderivative size = 159, normalized size of antiderivative = 2.45

method	result	size
default	\(-\frac {\sqrt {\sinh \left (\frac {x}{2}\right )^{2} a}\, \left (A \ln \left (\frac {2 \sqrt {\sinh \left (\frac {x}{2}\right )^{2} a}\, \sqrt {-a}-2 a}{\cosh \left (\frac {x}{2}\right )}\right ) \cosh \left (\frac {x}{2}\right )^{2} a +3 B \ln \left (\frac {2 \sqrt {\sinh \left (\frac {x}{2}\right )^{2} a}\, \sqrt {-a}-2 a}{\cosh \left (\frac {x}{2}\right )}\right ) \cosh \left (\frac {x}{2}\right )^{2} a -A \sqrt {\sinh \left (\frac {x}{2}\right )^{2} a}\, \sqrt {-a}+B \sqrt {\sinh \left (\frac {x}{2}\right )^{2} a}\, \sqrt {-a}\right ) \sqrt {2}}{4 \cosh \left (\frac {x}{2}\right ) a^{2} \sqrt {-a}\, \sinh \left (\frac {x}{2}\right ) \sqrt {a \cosh \left (\frac {x}{2}\right )^{2}}}\)	\(159\)
parts	\(-\frac {A \sqrt {\sinh \left (\frac {x}{2}\right )^{2} a}\, \left (\ln \left (\frac {2 \sqrt {\sinh \left (\frac {x}{2}\right )^{2} a}\, \sqrt {-a}-2 a}{\cosh \left (\frac {x}{2}\right )}\right ) \cosh \left (\frac {x}{2}\right )^{2} a -\sqrt {\sinh \left (\frac {x}{2}\right )^{2} a}\, \sqrt {-a}\right ) \sqrt {2}}{4 a^{2} \cosh \left (\frac {x}{2}\right ) \sqrt {-a}\, \sinh \left (\frac {x}{2}\right ) \sqrt {a \cosh \left (\frac {x}{2}\right )^{2}}}-\frac {B \sqrt {\sinh \left (\frac {x}{2}\right )^{2} a}\, \left (3 \ln \left (\frac {2 \sqrt {\sinh \left (\frac {x}{2}\right )^{2} a}\, \sqrt {-a}-2 a}{\cosh \left (\frac {x}{2}\right )}\right ) \cosh \left (\frac {x}{2}\right )^{2} a +\sqrt {\sinh \left (\frac {x}{2}\right )^{2} a}\, \sqrt {-a}\right ) \sqrt {2}}{4 \cosh \left (\frac {x}{2}\right ) a^{2} \sqrt {-a}\, \sinh \left (\frac {x}{2}\right ) \sqrt {a \cosh \left (\frac {x}{2}\right )^{2}}}\)	\(204\)

Input:

int((A+B*cosh(x))/(a+cosh(x)*a)^(3/2),x,method=_RETURNVERBOSE)
 

Output:

-1/4*(sinh(1/2*x)^2*a)^(1/2)*(A*ln(2/cosh(1/2*x)*((sinh(1/2*x)^2*a)^(1/2)* 
(-a)^(1/2)-a))*cosh(1/2*x)^2*a+3*B*ln(2/cosh(1/2*x)*((sinh(1/2*x)^2*a)^(1/ 
2)*(-a)^(1/2)-a))*cosh(1/2*x)^2*a-A*(sinh(1/2*x)^2*a)^(1/2)*(-a)^(1/2)+B*( 
sinh(1/2*x)^2*a)^(1/2)*(-a)^(1/2))/cosh(1/2*x)/a^2/(-a)^(1/2)/sinh(1/2*x)* 
2^(1/2)/(a*cosh(1/2*x)^2)^(1/2)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 197 vs. \(2 (50) = 100\).

Time = 0.11 (sec) , antiderivative size = 197, normalized size of antiderivative = 3.03 \[ \int \frac {A+B \cosh (x)}{(a+a \cosh (x))^{3/2}} \, dx=\frac {\sqrt {2} {\left ({\left (A + 3 \, B\right )} \cosh \left (x\right )^{2} + {\left (A + 3 \, B\right )} \sinh \left (x\right )^{2} + 2 \, {\left (A + 3 \, B\right )} \cosh \left (x\right ) + 2 \, {\left ({\left (A + 3 \, B\right )} \cosh \left (x\right ) + A + 3 \, B\right )} \sinh \left (x\right ) + A + 3 \, B\right )} \sqrt {a} \arctan \left (\frac {\sqrt {2} \sqrt {\frac {1}{2}} \sqrt {a} \sqrt {\frac {a}{\cosh \left (x\right ) + \sinh \left (x\right )}} {\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )}}{a}\right ) + 2 \, \sqrt {\frac {1}{2}} {\left ({\left (A - B\right )} \cosh \left (x\right )^{2} + {\left (A - B\right )} \sinh \left (x\right )^{2} - {\left (A - B\right )} \cosh \left (x\right ) + {\left (2 \, {\left (A - B\right )} \cosh \left (x\right ) - A + B\right )} \sinh \left (x\right )\right )} \sqrt {\frac {a}{\cosh \left (x\right ) + \sinh \left (x\right )}}}{2 \, {\left (a^{2} \cosh \left (x\right )^{2} + a^{2} \sinh \left (x\right )^{2} + 2 \, a^{2} \cosh \left (x\right ) + a^{2} + 2 \, {\left (a^{2} \cosh \left (x\right ) + a^{2}\right )} \sinh \left (x\right )\right )}} \] Input:

integrate((A+B*cosh(x))/(a+a*cosh(x))^(3/2),x, algorithm="fricas")
 

Output:

1/2*(sqrt(2)*((A + 3*B)*cosh(x)^2 + (A + 3*B)*sinh(x)^2 + 2*(A + 3*B)*cosh 
(x) + 2*((A + 3*B)*cosh(x) + A + 3*B)*sinh(x) + A + 3*B)*sqrt(a)*arctan(sq 
rt(2)*sqrt(1/2)*sqrt(a)*sqrt(a/(cosh(x) + sinh(x)))*(cosh(x) + sinh(x))/a) 
 + 2*sqrt(1/2)*((A - B)*cosh(x)^2 + (A - B)*sinh(x)^2 - (A - B)*cosh(x) + 
(2*(A - B)*cosh(x) - A + B)*sinh(x))*sqrt(a/(cosh(x) + sinh(x))))/(a^2*cos 
h(x)^2 + a^2*sinh(x)^2 + 2*a^2*cosh(x) + a^2 + 2*(a^2*cosh(x) + a^2)*sinh( 
x))
 

Sympy [F]

\[ \int \frac {A+B \cosh (x)}{(a+a \cosh (x))^{3/2}} \, dx=\int \frac {A + B \cosh {\left (x \right )}}{\left (a \left (\cosh {\left (x \right )} + 1\right )\right )^{\frac {3}{2}}}\, dx \] Input:

integrate((A+B*cosh(x))/(a+a*cosh(x))**(3/2),x)
 

Output:

Integral((A + B*cosh(x))/(a*(cosh(x) + 1))**(3/2), x)
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 300 vs. \(2 (50) = 100\).

Time = 0.25 (sec) , antiderivative size = 300, normalized size of antiderivative = 4.62 \[ \int \frac {A+B \cosh (x)}{(a+a \cosh (x))^{3/2}} \, dx=\frac {1}{6} \, {\left (\sqrt {2} {\left (\frac {3 \, e^{\left (\frac {5}{2} \, x\right )} + 8 \, e^{\left (\frac {3}{2} \, x\right )} - 3 \, e^{\left (\frac {1}{2} \, x\right )}}{a^{\frac {3}{2}} e^{\left (3 \, x\right )} + 3 \, a^{\frac {3}{2}} e^{\left (2 \, x\right )} + 3 \, a^{\frac {3}{2}} e^{x} + a^{\frac {3}{2}}} + \frac {3 \, \arctan \left (e^{\left (\frac {1}{2} \, x\right )}\right )}{a^{\frac {3}{2}}}\right )} - \frac {8 \, \sqrt {2} e^{\left (\frac {3}{2} \, x\right )}}{a^{\frac {3}{2}} e^{\left (3 \, x\right )} + 3 \, a^{\frac {3}{2}} e^{\left (2 \, x\right )} + 3 \, a^{\frac {3}{2}} e^{x} + a^{\frac {3}{2}}}\right )} A + \frac {1}{20} \, {\left (\sqrt {2} {\left (\frac {15 \, e^{\left (\frac {5}{2} \, x\right )} + 40 \, e^{\left (\frac {3}{2} \, x\right )} + 33 \, e^{\left (\frac {1}{2} \, x\right )}}{a^{\frac {3}{2}} e^{\left (3 \, x\right )} + 3 \, a^{\frac {3}{2}} e^{\left (2 \, x\right )} + 3 \, a^{\frac {3}{2}} e^{x} + a^{\frac {3}{2}}} + \frac {15 \, \arctan \left (e^{\left (\frac {1}{2} \, x\right )}\right )}{a^{\frac {3}{2}}}\right )} + 5 \, \sqrt {2} {\left (\frac {3 \, e^{\left (\frac {5}{2} \, x\right )} - 8 \, e^{\left (\frac {3}{2} \, x\right )} - 3 \, e^{\left (\frac {1}{2} \, x\right )}}{a^{\frac {3}{2}} e^{\left (3 \, x\right )} + 3 \, a^{\frac {3}{2}} e^{\left (2 \, x\right )} + 3 \, a^{\frac {3}{2}} e^{x} + a^{\frac {3}{2}}} + \frac {3 \, \arctan \left (e^{\left (\frac {1}{2} \, x\right )}\right )}{a^{\frac {3}{2}}}\right )} - \frac {8 \, {\left (5 \, \sqrt {2} \sqrt {a} e^{\left (\frac {5}{2} \, x\right )} + \sqrt {2} \sqrt {a} e^{\left (\frac {1}{2} \, x\right )}\right )}}{a^{2} e^{\left (3 \, x\right )} + 3 \, a^{2} e^{\left (2 \, x\right )} + 3 \, a^{2} e^{x} + a^{2}}\right )} B \] Input:

integrate((A+B*cosh(x))/(a+a*cosh(x))^(3/2),x, algorithm="maxima")
 

Output:

1/6*(sqrt(2)*((3*e^(5/2*x) + 8*e^(3/2*x) - 3*e^(1/2*x))/(a^(3/2)*e^(3*x) + 
 3*a^(3/2)*e^(2*x) + 3*a^(3/2)*e^x + a^(3/2)) + 3*arctan(e^(1/2*x))/a^(3/2 
)) - 8*sqrt(2)*e^(3/2*x)/(a^(3/2)*e^(3*x) + 3*a^(3/2)*e^(2*x) + 3*a^(3/2)* 
e^x + a^(3/2)))*A + 1/20*(sqrt(2)*((15*e^(5/2*x) + 40*e^(3/2*x) + 33*e^(1/ 
2*x))/(a^(3/2)*e^(3*x) + 3*a^(3/2)*e^(2*x) + 3*a^(3/2)*e^x + a^(3/2)) + 15 
*arctan(e^(1/2*x))/a^(3/2)) + 5*sqrt(2)*((3*e^(5/2*x) - 8*e^(3/2*x) - 3*e^ 
(1/2*x))/(a^(3/2)*e^(3*x) + 3*a^(3/2)*e^(2*x) + 3*a^(3/2)*e^x + a^(3/2)) + 
 3*arctan(e^(1/2*x))/a^(3/2)) - 8*(5*sqrt(2)*sqrt(a)*e^(5/2*x) + sqrt(2)*s 
qrt(a)*e^(1/2*x))/(a^2*e^(3*x) + 3*a^2*e^(2*x) + 3*a^2*e^x + a^2))*B
 

Giac [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.20 \[ \int \frac {A+B \cosh (x)}{(a+a \cosh (x))^{3/2}} \, dx=\frac {{\left (\sqrt {2} A + 3 \, \sqrt {2} B\right )} \arctan \left (e^{\left (\frac {1}{2} \, x\right )}\right )}{2 \, a^{\frac {3}{2}}} + \frac {\sqrt {2} {\left (A a^{\frac {3}{2}} e^{\left (\frac {3}{2} \, x\right )} - B a^{\frac {3}{2}} e^{\left (\frac {3}{2} \, x\right )} - A a^{\frac {3}{2}} e^{\left (\frac {1}{2} \, x\right )} + B a^{\frac {3}{2}} e^{\left (\frac {1}{2} \, x\right )}\right )}}{2 \, {\left (a e^{x} + a\right )}^{2} a} \] Input:

integrate((A+B*cosh(x))/(a+a*cosh(x))^(3/2),x, algorithm="giac")
 

Output:

1/2*(sqrt(2)*A + 3*sqrt(2)*B)*arctan(e^(1/2*x))/a^(3/2) + 1/2*sqrt(2)*(A*a 
^(3/2)*e^(3/2*x) - B*a^(3/2)*e^(3/2*x) - A*a^(3/2)*e^(1/2*x) + B*a^(3/2)*e 
^(1/2*x))/((a*e^x + a)^2*a)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {A+B \cosh (x)}{(a+a \cosh (x))^{3/2}} \, dx=\int \frac {A+B\,\mathrm {cosh}\left (x\right )}{{\left (a+a\,\mathrm {cosh}\left (x\right )\right )}^{3/2}} \,d x \] Input:

int((A + B*cosh(x))/(a + a*cosh(x))^(3/2),x)
 

Output:

int((A + B*cosh(x))/(a + a*cosh(x))^(3/2), x)
 

Reduce [F]

\[ \int \frac {A+B \cosh (x)}{(a+a \cosh (x))^{3/2}} \, dx=\frac {\sqrt {a}\, \left (\left (\int \frac {\sqrt {\cosh \left (x \right )+1}}{\cosh \left (x \right )^{2}+2 \cosh \left (x \right )+1}d x \right ) a +\left (\int \frac {\sqrt {\cosh \left (x \right )+1}\, \cosh \left (x \right )}{\cosh \left (x \right )^{2}+2 \cosh \left (x \right )+1}d x \right ) b \right )}{a^{2}} \] Input:

int((A+B*cosh(x))/(a+a*cosh(x))^(3/2),x)
 

Output:

(sqrt(a)*(int(sqrt(cosh(x) + 1)/(cosh(x)**2 + 2*cosh(x) + 1),x)*a + int((s 
qrt(cosh(x) + 1)*cosh(x))/(cosh(x)**2 + 2*cosh(x) + 1),x)*b))/a**2