\(\int \frac {1}{(a+b \cosh (x))^{3/2}} \, dx\) [83]

Optimal result

Integrand size = 10, antiderivative size = 84 \[ \int \frac {1}{(a+b \cosh (x))^{3/2}} \, dx=-\frac {2 i \sqrt {a+b \cosh (x)} E\left (\frac {i x}{2}|\frac {2 b}{a+b}\right )}{\left (a^2-b^2\right ) \sqrt {\frac {a+b \cosh (x)}{a+b}}}-\frac {2 b \sinh (x)}{\left (a^2-b^2\right ) \sqrt {a+b \cosh (x)}} \] Output:

-2*I*(a+b*cosh(x))^(1/2)*EllipticE(I*sinh(1/2*x),2^(1/2)*(b/(a+b))^(1/2))/ 
(a^2-b^2)/((a+b*cosh(x))/(a+b))^(1/2)-2*b*sinh(x)/(a^2-b^2)/(a+b*cosh(x))^ 
(1/2)
 

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.81 \[ \int \frac {1}{(a+b \cosh (x))^{3/2}} \, dx=-\frac {2 \left (i (a+b) \sqrt {\frac {a+b \cosh (x)}{a+b}} E\left (\frac {i x}{2}|\frac {2 b}{a+b}\right )+b \sinh (x)\right )}{(a-b) (a+b) \sqrt {a+b \cosh (x)}} \] Input:

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

Output:

(-2*(I*(a + b)*Sqrt[(a + b*Cosh[x])/(a + b)]*EllipticE[(I/2)*x, (2*b)/(a + 
 b)] + b*Sinh[x]))/((a - b)*(a + b)*Sqrt[a + b*Cosh[x]])
 

Rubi [A] (verified)

Time = 0.39 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.700, Rules used = {3042, 3143, 27, 3042, 3134, 3042, 3132}

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.

\(\displaystyle \int \frac {1}{(a+b \cosh (x))^{3/2}} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {1}{\left (a+b \sin \left (\frac {\pi }{2}+i x\right )\right )^{3/2}}dx\)

\(\Big \downarrow \) 3143

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3134

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3132

\(\displaystyle -\frac {2 b \sinh (x)}{\left (a^2-b^2\right ) \sqrt {a+b \cosh (x)}}-\frac {2 i \sqrt {a+b \cosh (x)} E\left (\frac {i x}{2}|\frac {2 b}{a+b}\right )}{\left (a^2-b^2\right ) \sqrt {\frac {a+b \cosh (x)}{a+b}}}\)

Input:

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

Output:

((-2*I)*Sqrt[a + b*Cosh[x]]*EllipticE[(I/2)*x, (2*b)/(a + b)])/((a^2 - b^2 
)*Sqrt[(a + b*Cosh[x])/(a + b)]) - (2*b*Sinh[x])/((a^2 - b^2)*Sqrt[a + b*C 
osh[x]])
 

Defintions of rubi rules used

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 3042

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

rule 3132

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

rule 3134

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

rule 3143

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(297\) vs. \(2(82)=164\).

Time = 1.52 (sec) , antiderivative size = 298, normalized size of antiderivative = 3.55

method	result	size
default	\(-\frac {2 \left (2 \cosh \left (\frac {x}{2}\right ) \sqrt {-\frac {2 b}{a -b}}\, \sinh \left (\frac {x}{2}\right )^{2} b -\sqrt {-\sinh \left (\frac {x}{2}\right )^{2}}\, \sqrt {\frac {2 b \sinh \left (\frac {x}{2}\right )^{2}}{a -b}+\frac {a +b}{a -b}}\, \operatorname {EllipticF}\left (\cosh \left (\frac {x}{2}\right ) \sqrt {-\frac {2 b}{a -b}}, \frac {\sqrt {-\frac {2 \left (a -b \right )}{b}}}{2}\right ) a -\sqrt {-\sinh \left (\frac {x}{2}\right )^{2}}\, \sqrt {\frac {2 b \sinh \left (\frac {x}{2}\right )^{2}}{a -b}+\frac {a +b}{a -b}}\, \operatorname {EllipticF}\left (\cosh \left (\frac {x}{2}\right ) \sqrt {-\frac {2 b}{a -b}}, \frac {\sqrt {-\frac {2 \left (a -b \right )}{b}}}{2}\right ) b +2 \sqrt {-\sinh \left (\frac {x}{2}\right )^{2}}\, \sqrt {\frac {2 b \sinh \left (\frac {x}{2}\right )^{2}}{a -b}+\frac {a +b}{a -b}}\, \operatorname {EllipticE}\left (\cosh \left (\frac {x}{2}\right ) \sqrt {-\frac {2 b}{a -b}}, \frac {\sqrt {-\frac {2 \left (a -b \right )}{b}}}{2}\right ) b \right )}{\sqrt {-\frac {2 b}{a -b}}\, \left (a -b \right ) \left (a +b \right ) \sinh \left (\frac {x}{2}\right ) \sqrt {2 b \sinh \left (\frac {x}{2}\right )^{2}+a +b}}\)	\(298\)

Input:

int(1/(a+b*cosh(x))^(3/2),x,method=_RETURNVERBOSE)
 

Output:

-2*(2*cosh(1/2*x)*(-2*b/(a-b))^(1/2)*sinh(1/2*x)^2*b-(-sinh(1/2*x)^2)^(1/2 
)*(2*b/(a-b)*sinh(1/2*x)^2+(a+b)/(a-b))^(1/2)*EllipticF(cosh(1/2*x)*(-2*b/ 
(a-b))^(1/2),1/2*(-2/b*(a-b))^(1/2))*a-(-sinh(1/2*x)^2)^(1/2)*(2*b/(a-b)*s 
inh(1/2*x)^2+(a+b)/(a-b))^(1/2)*EllipticF(cosh(1/2*x)*(-2*b/(a-b))^(1/2),1 
/2*(-2/b*(a-b))^(1/2))*b+2*(-sinh(1/2*x)^2)^(1/2)*(2*b/(a-b)*sinh(1/2*x)^2 
+(a+b)/(a-b))^(1/2)*EllipticE(cosh(1/2*x)*(-2*b/(a-b))^(1/2),1/2*(-2/b*(a- 
b))^(1/2))*b)/(-2*b/(a-b))^(1/2)/(a-b)/(a+b)/sinh(1/2*x)/(2*b*sinh(1/2*x)^ 
2+a+b)^(1/2)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 381 vs. \(2 (80) = 160\).

Time = 0.10 (sec) , antiderivative size = 381, normalized size of antiderivative = 4.54 \[ \int \frac {1}{(a+b \cosh (x))^{3/2}} \, dx=\frac {4 \, {\left (\sqrt {\frac {1}{2}} {\left (a b \cosh \left (x\right )^{2} + a b \sinh \left (x\right )^{2} + 2 \, a^{2} \cosh \left (x\right ) + a b + 2 \, {\left (a b \cosh \left (x\right ) + a^{2}\right )} \sinh \left (x\right )\right )} \sqrt {b} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 \, a^{3} - 9 \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cosh \left (x\right ) + 3 \, b \sinh \left (x\right ) + 2 \, a}{3 \, b}\right ) - 3 \, \sqrt {\frac {1}{2}} {\left (b^{2} \cosh \left (x\right )^{2} + b^{2} \sinh \left (x\right )^{2} + 2 \, a b \cosh \left (x\right ) + b^{2} + 2 \, {\left (b^{2} \cosh \left (x\right ) + a b\right )} \sinh \left (x\right )\right )} \sqrt {b} {\rm weierstrassZeta}\left (\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 \, a^{3} - 9 \, a b^{2}\right )}}{27 \, b^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 \, a^{3} - 9 \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cosh \left (x\right ) + 3 \, b \sinh \left (x\right ) + 2 \, a}{3 \, b}\right )\right ) - 3 \, {\left (b^{2} \cosh \left (x\right )^{2} + b^{2} \sinh \left (x\right )^{2} + a b \cosh \left (x\right ) + {\left (2 \, b^{2} \cosh \left (x\right ) + a b\right )} \sinh \left (x\right )\right )} \sqrt {b \cosh \left (x\right ) + a}\right )}}{3 \, {\left (a^{2} b^{2} - b^{4} + {\left (a^{2} b^{2} - b^{4}\right )} \cosh \left (x\right )^{2} + {\left (a^{2} b^{2} - b^{4}\right )} \sinh \left (x\right )^{2} + 2 \, {\left (a^{3} b - a b^{3}\right )} \cosh \left (x\right ) + 2 \, {\left (a^{3} b - a b^{3} + {\left (a^{2} b^{2} - b^{4}\right )} \cosh \left (x\right )\right )} \sinh \left (x\right )\right )}} \] Input:

integrate(1/(a+b*cosh(x))^(3/2),x, algorithm="fricas")
 

Output:

4/3*(sqrt(1/2)*(a*b*cosh(x)^2 + a*b*sinh(x)^2 + 2*a^2*cosh(x) + a*b + 2*(a 
*b*cosh(x) + a^2)*sinh(x))*sqrt(b)*weierstrassPInverse(4/3*(4*a^2 - 3*b^2) 
/b^2, -8/27*(8*a^3 - 9*a*b^2)/b^3, 1/3*(3*b*cosh(x) + 3*b*sinh(x) + 2*a)/b 
) - 3*sqrt(1/2)*(b^2*cosh(x)^2 + b^2*sinh(x)^2 + 2*a*b*cosh(x) + b^2 + 2*( 
b^2*cosh(x) + a*b)*sinh(x))*sqrt(b)*weierstrassZeta(4/3*(4*a^2 - 3*b^2)/b^ 
2, -8/27*(8*a^3 - 9*a*b^2)/b^3, weierstrassPInverse(4/3*(4*a^2 - 3*b^2)/b^ 
2, -8/27*(8*a^3 - 9*a*b^2)/b^3, 1/3*(3*b*cosh(x) + 3*b*sinh(x) + 2*a)/b)) 
- 3*(b^2*cosh(x)^2 + b^2*sinh(x)^2 + a*b*cosh(x) + (2*b^2*cosh(x) + a*b)*s 
inh(x))*sqrt(b*cosh(x) + a))/(a^2*b^2 - b^4 + (a^2*b^2 - b^4)*cosh(x)^2 + 
(a^2*b^2 - b^4)*sinh(x)^2 + 2*(a^3*b - a*b^3)*cosh(x) + 2*(a^3*b - a*b^3 + 
 (a^2*b^2 - b^4)*cosh(x))*sinh(x))
 

Sympy [F]

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

integrate(1/(a+b*cosh(x))**(3/2),x)
 

Output:

Integral((a + b*cosh(x))**(-3/2), x)
 

Maxima [F]

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

integrate(1/(a+b*cosh(x))^(3/2),x, algorithm="maxima")
 

Output:

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

Giac [F]

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

integrate(1/(a+b*cosh(x))^(3/2),x, algorithm="giac")
 

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{(a+b \cosh (x))^{3/2}} \, dx=\int \frac {1}{{\left (a+b\,\mathrm {cosh}\left (x\right )\right )}^{3/2}} \,d x \] Input:

int(1/(a + b*cosh(x))^(3/2),x)
 

Output:

int(1/(a + b*cosh(x))^(3/2), x)
 

Reduce [F]

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

int(1/(a+b*cosh(x))^(3/2),x)
 

Output:

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