\(\int \frac {A+B \cosh (x)}{(a-a \cosh (x))^{3/2}} \, dx\) [105]

Optimal result

Integrand size = 18, 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 = 0.28 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.25 \[ \int \frac {A+B \cosh (x)}{(a-a \cosh (x))^{3/2}} \, dx=\frac {\left ((A+B) \text {csch}^2\left (\frac {x}{4}\right )-4 (A-3 B) \left (\log \left (\cosh \left (\frac {x}{4}\right )\right )-\log \left (\sinh \left (\frac {x}{4}\right )\right )\right )+(A+B) \text {sech}^2\left (\frac {x}{4}\right )\right ) \sinh ^3\left (\frac {x}{2}\right )}{4 a (-1+\cosh (x)) \sqrt {a-a \cosh (x)}} \] Input:

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

Output:

(((A + B)*Csch[x/4]^2 - 4*(A - 3*B)*(Log[Cosh[x/4]] - Log[Sinh[x/4]]) + (A 
 + B)*Sech[x/4]^2)*Sinh[x/2]^3)/(4*a*(-1 + Cosh[x])*Sqrt[a - a*Cosh[x]])
 

Rubi [A] (verified)

Time = 0.32 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, 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.

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

\(\Big \downarrow \) 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

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3128

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

\(\Big \downarrow \) 219

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

Input:

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

Output:

-1/2*((A - 3*B)*ArcTan[(Sqrt[a]*Sinh[x])/(Sqrt[2]*Sqrt[a - a*Cosh[x]])])/( 
Sqrt[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 [A] (verified)

Time = 0.94 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.28

method	result	size
default	\(-\frac {\cosh \left (\frac {x}{2}\right ) \left (-2 A -2 B \right )+\left (\ln \left (\cosh \left (\frac {x}{2}\right )+1\right ) A -\ln \left (\cosh \left (\frac {x}{2}\right )-1\right ) A -3 \ln \left (\cosh \left (\frac {x}{2}\right )+1\right ) B +3 \ln \left (\cosh \left (\frac {x}{2}\right )-1\right ) B \right ) \sinh \left (\frac {x}{2}\right )^{2}}{4 a \sinh \left (\frac {x}{2}\right ) \sqrt {-2 \sinh \left (\frac {x}{2}\right )^{2} a}}\)	\(83\)
parts	\(-\frac {A \left (-2 \cosh \left (\frac {x}{2}\right )+\left (\ln \left (\cosh \left (\frac {x}{2}\right )+1\right )-\ln \left (\cosh \left (\frac {x}{2}\right )-1\right )\right ) \sinh \left (\frac {x}{2}\right )^{2}\right )}{4 a \sinh \left (\frac {x}{2}\right ) \sqrt {-2 \sinh \left (\frac {x}{2}\right )^{2} a}}+\frac {B \left (2 \cosh \left (\frac {x}{2}\right )+\left (3 \ln \left (\cosh \left (\frac {x}{2}\right )+1\right )-3 \ln \left (\cosh \left (\frac {x}{2}\right )-1\right )\right ) \sinh \left (\frac {x}{2}\right )^{2}\right )}{4 a \sinh \left (\frac {x}{2}\right ) \sqrt {-2 \sinh \left (\frac {x}{2}\right )^{2} a}}\)	\(112\)

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 217 vs. \(2 (50) = 100\).

Time = 0.10 (sec) , antiderivative size = 217, normalized size of antiderivative = 3.34 \[ \int \frac {A+B \cosh (x)}{(a-a \cosh (x))^{3/2}} \, dx=\frac {\sqrt {2} {\left ({\left (A - 3 \, B\right )} \cosh \left (x\right )^{2} + {\left (A - 3 \, B\right )} \sinh \left (x\right )^{2} - 2 \, {\left (A - 3 \, B\right )} \cosh \left (x\right ) + 2 \, {\left ({\left (A - 3 \, B\right )} \cosh \left (x\right ) - A + 3 \, B\right )} \sinh \left (x\right ) + A - 3 \, B\right )} \sqrt {-a} \log \left (\frac {2 \, \sqrt {2} \sqrt {\frac {1}{2}} \sqrt {-a} \sqrt {-\frac {a}{\cosh \left (x\right ) + \sinh \left (x\right )}} {\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )} - a \cosh \left (x\right ) - a \sinh \left (x\right ) - a}{\cosh \left (x\right ) + \sinh \left (x\right ) - 1}\right ) - 4 \, \sqrt {\frac {1}{2}} {\left ({\left (A + B\right )} \cosh \left (x\right )^{2} + {\left (A + B\right )} \sinh \left (x\right )^{2} + {\left (A + B\right )} \cosh \left (x\right ) + {\left (2 \, {\left (A + B\right )} \cosh \left (x\right ) + A + B\right )} \sinh \left (x\right )\right )} \sqrt {-\frac {a}{\cosh \left (x\right ) + \sinh \left (x\right )}}}{4 \, {\left (a^{2} \cosh \left (x\right )^{2} + a^{2} \sinh \left (x\right )^{2} - 2 \, a^{2} \cosh \left (x\right ) + a^{2} + 2 \, {\left (a^{2} \cosh \left (x\right ) - a^{2}\right )} \sinh \left (x\right )\right )}} \] Input:

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

Output:

1/4*(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)*log((2*s 
qrt(2)*sqrt(1/2)*sqrt(-a)*sqrt(-a/(cosh(x) + sinh(x)))*(cosh(x) + sinh(x)) 
 - a*cosh(x) - a*sinh(x) - a)/(cosh(x) + sinh(x) - 1)) - 4*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*cosh(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 [F]

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

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

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 111 vs. \(2 (50) = 100\).

Time = 0.12 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.71 \[ \int \frac {A+B \cosh (x)}{(a-a \cosh (x))^{3/2}} \, dx=-\frac {{\left (\sqrt {2} A - 3 \, \sqrt {2} B\right )} \arctan \left (\frac {\sqrt {-a e^{x}}}{\sqrt {a}}\right )}{2 \, a^{\frac {3}{2}} \mathrm {sgn}\left (-e^{x} + 1\right )} + \frac {\sqrt {2} {\left (\sqrt {-a e^{x}} A a e^{x} + \sqrt {-a e^{x}} B a e^{x} + \sqrt {-a e^{x}} A a + \sqrt {-a e^{x}} B a\right )}}{2 \, {\left (a e^{x} - a\right )}^{2} a \mathrm {sgn}\left (-e^{x} + 1\right )} \] 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(sqrt(-a*e^x)/sqrt(a))/(a^(3/2)*sgn(- 
e^x + 1)) + 1/2*sqrt(2)*(sqrt(-a*e^x)*A*a*e^x + sqrt(-a*e^x)*B*a*e^x + sqr 
t(-a*e^x)*A*a + sqrt(-a*e^x)*B*a)/((a*e^x - a)^2*a*sgn(-e^x + 1))
 

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 
((sqrt( - cosh(x) + 1)*cosh(x))/(cosh(x)**2 - 2*cosh(x) + 1),x)*b))/a**2