Integrand size = 13, antiderivative size = 100 \[ \int \frac {\cosh (x)}{\sqrt {a+b \cosh (x)}} \, dx=-\frac {2 i \sqrt {a+b \cosh (x)} E\left (\frac {i x}{2}|\frac {2 b}{a+b}\right )}{b \sqrt {\frac {a+b \cosh (x)}{a+b}}}+\frac {2 i a \sqrt {\frac {a+b \cosh (x)}{a+b}} \operatorname {EllipticF}\left (\frac {i x}{2},\frac {2 b}{a+b}\right )}{b \sqrt {a+b \cosh (x)}} \]
-2*I*(cosh(1/2*x)^2)^(1/2)/cosh(1/2*x)*EllipticE(I*sinh(1/2*x),2^(1/2)*(b/ (a+b))^(1/2))*(a+b*cosh(x))^(1/2)/b/((a+b*cosh(x))/(a+b))^(1/2)+2*I*a*(cos h(1/2*x)^2)^(1/2)/cosh(1/2*x)*EllipticF(I*sinh(1/2*x),2^(1/2)*(b/(a+b))^(1 /2))*((a+b*cosh(x))/(a+b))^(1/2)/b/(a+b*cosh(x))^(1/2)
Time = 0.28 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.73 \[ \int \frac {\cosh (x)}{\sqrt {a+b \cosh (x)}} \, dx=-\frac {2 i \sqrt {\frac {a+b \cosh (x)}{a+b}} \left ((a+b) E\left (\frac {i x}{2}|\frac {2 b}{a+b}\right )-a \operatorname {EllipticF}\left (\frac {i x}{2},\frac {2 b}{a+b}\right )\right )}{b \sqrt {a+b \cosh (x)}} \]
((-2*I)*Sqrt[(a + b*Cosh[x])/(a + b)]*((a + b)*EllipticE[(I/2)*x, (2*b)/(a + b)] - a*EllipticF[(I/2)*x, (2*b)/(a + b)]))/(b*Sqrt[a + b*Cosh[x]])
Time = 0.54 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.692, Rules used = {3042, 3231, 3042, 3134, 3042, 3132, 3142, 3042, 3140}
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.
| \(\displaystyle \int \frac {\cosh (x)}{\sqrt {a+b \cosh (x)}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sin \left (\frac {\pi }{2}+i x\right )}{\sqrt {a+b \sin \left (\frac {\pi }{2}+i x\right )}}dx\) |
| \(\Big \downarrow \) 3231 |
| \(\displaystyle \frac {\int \sqrt {a+b \cosh (x)}dx}{b}-\frac {a \int \frac {1}{\sqrt {a+b \cosh (x)}}dx}{b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \sqrt {a+b \sin \left (i x+\frac {\pi }{2}\right )}dx}{b}-\frac {a \int \frac {1}{\sqrt {a+b \sin \left (i x+\frac {\pi }{2}\right )}}dx}{b}\) |
| \(\Big \downarrow \) 3134 |
| \(\displaystyle \frac {\sqrt {a+b \cosh (x)} \int \sqrt {\frac {a}{a+b}+\frac {b \cosh (x)}{a+b}}dx}{b \sqrt {\frac {a+b \cosh (x)}{a+b}}}-\frac {a \int \frac {1}{\sqrt {a+b \sin \left (i x+\frac {\pi }{2}\right )}}dx}{b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\sqrt {a+b \cosh (x)} \int \sqrt {\frac {a}{a+b}+\frac {b \sin \left (i x+\frac {\pi }{2}\right )}{a+b}}dx}{b \sqrt {\frac {a+b \cosh (x)}{a+b}}}-\frac {a \int \frac {1}{\sqrt {a+b \sin \left (i x+\frac {\pi }{2}\right )}}dx}{b}\) |
| \(\Big \downarrow \) 3132 |
| \(\displaystyle -\frac {a \int \frac {1}{\sqrt {a+b \sin \left (i x+\frac {\pi }{2}\right )}}dx}{b}-\frac {2 i \sqrt {a+b \cosh (x)} E\left (\frac {i x}{2}|\frac {2 b}{a+b}\right )}{b \sqrt {\frac {a+b \cosh (x)}{a+b}}}\) |
| \(\Big \downarrow \) 3142 |
| \(\displaystyle -\frac {a \sqrt {\frac {a+b \cosh (x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \cosh (x)}{a+b}}}dx}{b \sqrt {a+b \cosh (x)}}-\frac {2 i \sqrt {a+b \cosh (x)} E\left (\frac {i x}{2}|\frac {2 b}{a+b}\right )}{b \sqrt {\frac {a+b \cosh (x)}{a+b}}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {a \sqrt {\frac {a+b \cosh (x)}{a+b}} \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \sin \left (i x+\frac {\pi }{2}\right )}{a+b}}}dx}{b \sqrt {a+b \cosh (x)}}-\frac {2 i \sqrt {a+b \cosh (x)} E\left (\frac {i x}{2}|\frac {2 b}{a+b}\right )}{b \sqrt {\frac {a+b \cosh (x)}{a+b}}}\) |
| \(\Big \downarrow \) 3140 |
| \(\displaystyle \frac {2 i a \sqrt {\frac {a+b \cosh (x)}{a+b}} \operatorname {EllipticF}\left (\frac {i x}{2},\frac {2 b}{a+b}\right )}{b \sqrt {a+b \cosh (x)}}-\frac {2 i \sqrt {a+b \cosh (x)} E\left (\frac {i x}{2}|\frac {2 b}{a+b}\right )}{b \sqrt {\frac {a+b \cosh (x)}{a+b}}}\) |
((-2*I)*Sqrt[a + b*Cosh[x]]*EllipticE[(I/2)*x, (2*b)/(a + b)])/(b*Sqrt[(a + b*Cosh[x])/(a + b)]) + ((2*I)*a*Sqrt[(a + b*Cosh[x])/(a + b)]*EllipticF[ (I/2)*x, (2*b)/(a + b)])/(b*Sqrt[a + b*Cosh[x]])
3.1.86.3.1 Defintions of rubi rules used
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]
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]
Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/(d*S qrt[a + b]))*EllipticF[(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]
Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[Sqrt[(a + b*Sin[c + d*x])/(a + b)]/Sqrt[a + b*Sin[c + d*x]] Int[1/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]
Int[((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])/Sqrt[(a_) + (b_.)*sin[(e_.) + ( f_.)*(x_)]], x_Symbol] :> Simp[(b*c - a*d)/b Int[1/Sqrt[a + b*Sin[e + f*x ]], x], x] + Simp[d/b Int[Sqrt[a + b*Sin[e + f*x]], x], x] /; FreeQ[{a, b , c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0]
Time = 0.99 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.81
| method | result | size |
| default | \(\frac {2 \left (\operatorname {EllipticF}\left (\cosh \left (\frac {x}{2}\right ) \sqrt {-\frac {2 b}{a -b}}, \frac {\sqrt {-\frac {2 \left (a -b \right )}{b}}}{2}\right )-2 \operatorname {EllipticE}\left (\cosh \left (\frac {x}{2}\right ) \sqrt {-\frac {2 b}{a -b}}, \frac {\sqrt {-\frac {2 \left (a -b \right )}{b}}}{2}\right )\right ) \sqrt {-\sinh \left (\frac {x}{2}\right )^{2}}\, \sqrt {\frac {2 \cosh \left (\frac {x}{2}\right )^{2} b +a -b}{a -b}}\, \sqrt {\left (2 \cosh \left (\frac {x}{2}\right )^{2} b +a -b \right ) \sinh \left (\frac {x}{2}\right )^{2}}}{\sqrt {-\frac {2 b}{a -b}}\, \sqrt {2 \sinh \left (\frac {x}{2}\right )^{4} b +\left (a +b \right ) \sinh \left (\frac {x}{2}\right )^{2}}\, \sinh \left (\frac {x}{2}\right ) \sqrt {2 \sinh \left (\frac {x}{2}\right )^{2} b +a +b}}\) | \(181\) |
| risch | \(\frac {\left (b \,{\mathrm e}^{2 x}+2 a \,{\mathrm e}^{x}+b \right ) \sqrt {2}\, {\mathrm e}^{-x}}{b \sqrt {\left (b \,{\mathrm e}^{2 x}+2 a \,{\mathrm e}^{x}+b \right ) {\mathrm e}^{-x}}}+\frac {\left (-\frac {4 \left (b \,{\mathrm e}^{2 x}+2 a \,{\mathrm e}^{x}+b \right )}{b \sqrt {\left (b \,{\mathrm e}^{2 x}+2 a \,{\mathrm e}^{x}+b \right ) {\mathrm e}^{x}}}+\frac {4 \left (a +\sqrt {a^{2}-b^{2}}\right ) \sqrt {\frac {\left ({\mathrm e}^{x}+\frac {a +\sqrt {a^{2}-b^{2}}}{b}\right ) b}{a +\sqrt {a^{2}-b^{2}}}}\, \sqrt {\frac {{\mathrm e}^{x}-\frac {-a +\sqrt {a^{2}-b^{2}}}{b}}{-\frac {a +\sqrt {a^{2}-b^{2}}}{b}-\frac {-a +\sqrt {a^{2}-b^{2}}}{b}}}\, \sqrt {-\frac {{\mathrm e}^{x} b}{a +\sqrt {a^{2}-b^{2}}}}\, \left (\left (-\frac {a +\sqrt {a^{2}-b^{2}}}{b}-\frac {-a +\sqrt {a^{2}-b^{2}}}{b}\right ) \operatorname {EllipticE}\left (\sqrt {\frac {\left ({\mathrm e}^{x}+\frac {a +\sqrt {a^{2}-b^{2}}}{b}\right ) b}{a +\sqrt {a^{2}-b^{2}}}}, \sqrt {-\frac {a +\sqrt {a^{2}-b^{2}}}{b \left (-\frac {a +\sqrt {a^{2}-b^{2}}}{b}-\frac {-a +\sqrt {a^{2}-b^{2}}}{b}\right )}}\right )+\frac {\left (-a +\sqrt {a^{2}-b^{2}}\right ) \operatorname {EllipticF}\left (\sqrt {\frac {\left ({\mathrm e}^{x}+\frac {a +\sqrt {a^{2}-b^{2}}}{b}\right ) b}{a +\sqrt {a^{2}-b^{2}}}}, \sqrt {-\frac {a +\sqrt {a^{2}-b^{2}}}{b \left (-\frac {a +\sqrt {a^{2}-b^{2}}}{b}-\frac {-a +\sqrt {a^{2}-b^{2}}}{b}\right )}}\right )}{b}\right )}{b \sqrt {{\mathrm e}^{3 x} b +2 a \,{\mathrm e}^{2 x}+{\mathrm e}^{x} b}}\right ) \sqrt {2}\, \sqrt {\left (b \,{\mathrm e}^{2 x}+2 a \,{\mathrm e}^{x}+b \right ) {\mathrm e}^{x}}\, {\mathrm e}^{-x}}{2 \sqrt {\left (b \,{\mathrm e}^{2 x}+2 a \,{\mathrm e}^{x}+b \right ) {\mathrm e}^{-x}}}\) | \(558\) |
2*(EllipticF(cosh(1/2*x)*(-2*b/(a-b))^(1/2),1/2*(-2*(a-b)/b)^(1/2))-2*Elli pticE(cosh(1/2*x)*(-2*b/(a-b))^(1/2),1/2*(-2*(a-b)/b)^(1/2)))*(-sinh(1/2*x )^2)^(1/2)*((2*cosh(1/2*x)^2*b+a-b)/(a-b))^(1/2)*((2*cosh(1/2*x)^2*b+a-b)* sinh(1/2*x)^2)^(1/2)/(-2*b/(a-b))^(1/2)/(2*sinh(1/2*x)^4*b+(a+b)*sinh(1/2* x)^2)^(1/2)/sinh(1/2*x)/(2*sinh(1/2*x)^2*b+a+b)^(1/2)
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.08 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.74 \[ \int \frac {\cosh (x)}{\sqrt {a+b \cosh (x)}} \, dx=-\frac {2 \, {\left (2 \, \sqrt {2} a \sqrt {b} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 \, a^{3} - 9 \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cosh \left (x\right ) + 3 \, b \sinh \left (x\right ) + 2 \, a}{3 \, b}\right ) + 3 \, \sqrt {2} b^{\frac {3}{2}} {\rm weierstrassZeta}\left (\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 \, a^{3} - 9 \, a b^{2}\right )}}{27 \, b^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 \, a^{3} - 9 \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cosh \left (x\right ) + 3 \, b \sinh \left (x\right ) + 2 \, a}{3 \, b}\right )\right ) + 3 \, \sqrt {b \cosh \left (x\right ) + a} b\right )}}{3 \, b^{2}} \]
-2/3*(2*sqrt(2)*a*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*sqr t(2)*b^(3/2)*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*sqrt(b*cosh(x) + a) *b)/b^2
\[ \int \frac {\cosh (x)}{\sqrt {a+b \cosh (x)}} \, dx=\int \frac {\cosh {\left (x \right )}}{\sqrt {a + b \cosh {\left (x \right )}}}\, dx \]
\[ \int \frac {\cosh (x)}{\sqrt {a+b \cosh (x)}} \, dx=\int { \frac {\cosh \left (x\right )}{\sqrt {b \cosh \left (x\right ) + a}} \,d x } \]
\[ \int \frac {\cosh (x)}{\sqrt {a+b \cosh (x)}} \, dx=\int { \frac {\cosh \left (x\right )}{\sqrt {b \cosh \left (x\right ) + a}} \,d x } \]
Timed out. \[ \int \frac {\cosh (x)}{\sqrt {a+b \cosh (x)}} \, dx=\int \frac {\mathrm {cosh}\left (x\right )}{\sqrt {a+b\,\mathrm {cosh}\left (x\right )}} \,d x \]