\(\int \sqrt {a+b \sinh (x)} \, dx\) [107]

Optimal result

Integrand size = 10, antiderivative size = 60 \[ \int \sqrt {a+b \sinh (x)} \, dx=\frac {2 i E\left (\frac {\pi }{4}-\frac {i x}{2}|\frac {2 b}{i a+b}\right ) \sqrt {a+b \sinh (x)}}{\sqrt {\frac {a+b \sinh (x)}{a-i b}}} \] Output:

2*I*EllipticE(cos(1/4*Pi+1/2*I*x),2^(1/2)*(b/(I*a+b))^(1/2))*(a+b*sinh(x)) 
^(1/2)/((a+b*sinh(x))/(a-I*b))^(1/2)
 

Mathematica [A] (verified)

Time = 0.13 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.08 \[ \int \sqrt {a+b \sinh (x)} \, dx=\frac {2 (i a+b) E\left (\frac {1}{4} (\pi -2 i x)|-\frac {2 i b}{a-i b}\right ) \sqrt {\frac {a+b \sinh (x)}{a-i b}}}{\sqrt {a+b \sinh (x)}} \] Input:

Integrate[Sqrt[a + b*Sinh[x]],x]
 

Output:

(2*(I*a + b)*EllipticE[(Pi - (2*I)*x)/4, ((-2*I)*b)/(a - I*b)]*Sqrt[(a + b 
*Sinh[x])/(a - I*b)])/Sqrt[a + b*Sinh[x]]
 

Rubi [A] (verified)

Time = 0.39 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {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.

\(\displaystyle \int \sqrt {a+b \sinh (x)} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \sqrt {a-i b \sin (i x)}dx\)

\(\Big \downarrow \) 3134

\(\displaystyle \frac {\sqrt {a+b \sinh (x)} \int \sqrt {\frac {a}{a-i b}+\frac {b \sinh (x)}{a-i b}}dx}{\sqrt {\frac {a+b \sinh (x)}{a-i b}}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\sqrt {a+b \sinh (x)} \int \sqrt {\frac {a}{a-i b}-\frac {i b \sin (i x)}{a-i b}}dx}{\sqrt {\frac {a+b \sinh (x)}{a-i b}}}\)

\(\Big \downarrow \) 3132

\(\displaystyle \frac {2 i \sqrt {a+b \sinh (x)} E\left (\frac {\pi }{4}-\frac {i x}{2}|\frac {2 b}{i a+b}\right )}{\sqrt {\frac {a+b \sinh (x)}{a-i b}}}\)

Input:

Int[Sqrt[a + b*Sinh[x]],x]
 

Output:

((2*I)*EllipticE[Pi/4 - (I/2)*x, (2*b)/(I*a + b)]*Sqrt[a + b*Sinh[x]])/Sqr 
t[(a + b*Sinh[x])/(a - I*b)]
 

Defintions of rubi rules used

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]
 

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 261 vs. \(2 (54 ) = 108\).

Time = 0.93 (sec) , antiderivative size = 262, normalized size of antiderivative = 4.37

method	result	size
default	\(-\frac {2 \left (i b -a \right ) \sqrt {-\frac {a +b \sinh \left (x \right )}{i b -a}}\, \sqrt {\frac {\left (i-\sinh \left (x \right )\right ) b}{i b +a}}\, \sqrt {\frac {b \left (i+\sinh \left (x \right )\right )}{i b -a}}\, \left (i \operatorname {EllipticF}\left (\sqrt {-\frac {a +b \sinh \left (x \right )}{i b -a}}, \sqrt {-\frac {i b -a}{i b +a}}\right ) b -i \operatorname {EllipticE}\left (\sqrt {-\frac {a +b \sinh \left (x \right )}{i b -a}}, \sqrt {-\frac {i b -a}{i b +a}}\right ) b +\operatorname {EllipticF}\left (\sqrt {-\frac {a +b \sinh \left (x \right )}{i b -a}}, \sqrt {-\frac {i b -a}{i b +a}}\right ) a -\operatorname {EllipticE}\left (\sqrt {-\frac {a +b \sinh \left (x \right )}{i b -a}}, \sqrt {-\frac {i b -a}{i b +a}}\right ) a \right )}{b \cosh \left (x \right ) \sqrt {a +b \sinh \left (x \right )}}\)	\(262\)
risch	\(\sqrt {2}\, \sqrt {\left (b \,{\mathrm e}^{2 x}+2 \,{\mathrm e}^{x} a -b \right ) {\mathrm e}^{-x}}+\frac {\left (\frac {4 a \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}}}}\, \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 \sqrt {{\mathrm e}^{3 x} b +2 a \,{\mathrm e}^{2 x}-{\mathrm e}^{x} b}}-2 b \left (\frac {2 b \,{\mathrm e}^{2 x}+4 \,{\mathrm e}^{x} a -2 b}{b \sqrt {\left (b \,{\mathrm e}^{2 x}+2 \,{\mathrm e}^{x} a -b \right ) {\mathrm e}^{x}}}-\frac {2 \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 )\right ) \sqrt {2}\, \sqrt {\left (b \,{\mathrm e}^{2 x}+2 \,{\mathrm e}^{x} a -b \right ) {\mathrm e}^{-x}}\, \sqrt {\left (b \,{\mathrm e}^{2 x}+2 \,{\mathrm e}^{x} a -b \right ) {\mathrm e}^{x}}}{2 b \,{\mathrm e}^{2 x}+4 \,{\mathrm e}^{x} a -2 b}\)	\(771\)

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 174 vs. \(2 (50) = 100\).

Time = 0.09 (sec) , antiderivative size = 174, normalized size of antiderivative = 2.90 \[ \int \sqrt {a+b \sinh (x)} \, dx=\frac {2 \, {\left (2 \, \sqrt {\frac {1}{2}} a \sqrt {b} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (4 \, a^{2} + 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 \, a^{3} + 9 \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cosh \left (x\right ) + 3 \, b \sinh \left (x\right ) + 2 \, a}{3 \, b}\right ) - 6 \, \sqrt {\frac {1}{2}} b^{\frac {3}{2}} {\rm weierstrassZeta}\left (\frac {4 \, {\left (4 \, a^{2} + 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 \, a^{3} + 9 \, a b^{2}\right )}}{27 \, b^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (4 \, a^{2} + 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 \, a^{3} + 9 \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cosh \left (x\right ) + 3 \, b \sinh \left (x\right ) + 2 \, a}{3 \, b}\right )\right ) - 3 \, \sqrt {b \sinh \left (x\right ) + a} b\right )}}{3 \, b} \] Input:

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

Output:

2/3*(2*sqrt(1/2)*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) - 6*sq 
rt(1/2)*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*sinh(x) + 
 a)*b)/b
 

Sympy [F]

\[ \int \sqrt {a+b \sinh (x)} \, dx=\int \sqrt {a + b \sinh {\left (x \right )}}\, dx \] Input:

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

Output:

Integral(sqrt(a + b*sinh(x)), x)
 

Maxima [F]

\[ \int \sqrt {a+b \sinh (x)} \, dx=\int { \sqrt {b \sinh \left (x\right ) + a} \,d x } \] Input:

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

Output:

integrate(sqrt(b*sinh(x) + a), x)
 

Giac [F]

\[ \int \sqrt {a+b \sinh (x)} \, dx=\int { \sqrt {b \sinh \left (x\right ) + a} \,d x } \] Input:

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

Output:

integrate(sqrt(b*sinh(x) + a), x)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \sqrt {a+b \sinh (x)} \, dx=\int \sqrt {\sinh \left (x \right ) b +a}d x \] Input:

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

Output:

int(sqrt(sinh(x)*b + a),x)