\(\int \frac {A+B \sinh (x)}{\sqrt {a+b \sinh (x)}} \, dx\) [137]

Optimal result

Integrand size = 17, antiderivative size = 136 \[ \int \frac {A+B \sinh (x)}{\sqrt {a+b \sinh (x)}} \, dx=\frac {2 i B E\left (\frac {\pi }{4}-\frac {i x}{2}|\frac {2 b}{i a+b}\right ) \sqrt {a+b \sinh (x)}}{b \sqrt {\frac {a+b \sinh (x)}{a-i b}}}+\frac {2 i (A b-a B) \operatorname {EllipticF}\left (\frac {\pi }{4}-\frac {i x}{2},\frac {2 b}{i a+b}\right ) \sqrt {\frac {a+b \sinh (x)}{a-i b}}}{b \sqrt {a+b \sinh (x)}} \] Output:

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

Mathematica [A] (verified)

Time = 0.38 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.80 \[ \int \frac {A+B \sinh (x)}{\sqrt {a+b \sinh (x)}} \, dx=\frac {2 \left ((i a+b) B E\left (\frac {1}{4} (\pi -2 i x)|-\frac {2 i b}{a-i b}\right )+i (A b-a B) \operatorname {EllipticF}\left (\frac {1}{4} (\pi -2 i x),-\frac {2 i b}{a-i b}\right )\right ) \sqrt {\frac {a+b \sinh (x)}{a-i b}}}{b \sqrt {a+b \sinh (x)}} \] Input:

Integrate[(A + B*Sinh[x])/Sqrt[a + b*Sinh[x]],x]
 

Output:

(2*((I*a + b)*B*EllipticE[(Pi - (2*I)*x)/4, ((-2*I)*b)/(a - I*b)] + I*(A*b 
 - a*B)*EllipticF[(Pi - (2*I)*x)/4, ((-2*I)*b)/(a - I*b)])*Sqrt[(a + b*Sin 
h[x])/(a - I*b)])/(b*Sqrt[a + b*Sinh[x]])
 

Rubi [A] (verified)

Time = 0.61 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.529, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3231

\(\displaystyle \frac {(A b-a B) \int \frac {1}{\sqrt {a+b \sinh (x)}}dx}{b}+\frac {B \int \sqrt {a+b \sinh (x)}dx}{b}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {(A b-a B) \int \frac {1}{\sqrt {a-i b \sin (i x)}}dx}{b}+\frac {B \int \sqrt {a-i b \sin (i x)}dx}{b}\)

\(\Big \downarrow \) 3134

\(\displaystyle \frac {(A b-a B) \int \frac {1}{\sqrt {a-i b \sin (i x)}}dx}{b}+\frac {B \sqrt {a+b \sinh (x)} \int \sqrt {\frac {a}{a-i b}+\frac {b \sinh (x)}{a-i b}}dx}{b \sqrt {\frac {a+b \sinh (x)}{a-i b}}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {(A b-a B) \int \frac {1}{\sqrt {a-i b \sin (i x)}}dx}{b}+\frac {B \sqrt {a+b \sinh (x)} \int \sqrt {\frac {a}{a-i b}-\frac {i b \sin (i x)}{a-i b}}dx}{b \sqrt {\frac {a+b \sinh (x)}{a-i b}}}\)

\(\Big \downarrow \) 3132

\(\displaystyle \frac {(A b-a B) \int \frac {1}{\sqrt {a-i b \sin (i x)}}dx}{b}+\frac {2 i B \sqrt {a+b \sinh (x)} E\left (\frac {\pi }{4}-\frac {i x}{2}|\frac {2 b}{i a+b}\right )}{b \sqrt {\frac {a+b \sinh (x)}{a-i b}}}\)

\(\Big \downarrow \) 3142

\(\displaystyle \frac {(A b-a B) \sqrt {\frac {a+b \sinh (x)}{a-i b}} \int \frac {1}{\sqrt {\frac {a}{a-i b}+\frac {b \sinh (x)}{a-i b}}}dx}{b \sqrt {a+b \sinh (x)}}+\frac {2 i B \sqrt {a+b \sinh (x)} E\left (\frac {\pi }{4}-\frac {i x}{2}|\frac {2 b}{i a+b}\right )}{b \sqrt {\frac {a+b \sinh (x)}{a-i b}}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {(A b-a B) \sqrt {\frac {a+b \sinh (x)}{a-i b}} \int \frac {1}{\sqrt {\frac {a}{a-i b}-\frac {i b \sin (i x)}{a-i b}}}dx}{b \sqrt {a+b \sinh (x)}}+\frac {2 i B \sqrt {a+b \sinh (x)} E\left (\frac {\pi }{4}-\frac {i x}{2}|\frac {2 b}{i a+b}\right )}{b \sqrt {\frac {a+b \sinh (x)}{a-i b}}}\)

\(\Big \downarrow \) 3140

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

Input:

Int[(A + B*Sinh[x])/Sqrt[a + b*Sinh[x]],x]
 

Output:

((2*I)*B*EllipticE[Pi/4 - (I/2)*x, (2*b)/(I*a + b)]*Sqrt[a + b*Sinh[x]])/( 
b*Sqrt[(a + b*Sinh[x])/(a - I*b)]) + ((2*I)*(A*b - a*B)*EllipticF[Pi/4 - ( 
I/2)*x, (2*b)/(I*a + b)]*Sqrt[(a + b*Sinh[x])/(a - I*b)])/(b*Sqrt[a + b*Si 
nh[x]])
 

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]
 

rule 3140

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]
 

rule 3142

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]
 

rule 3231

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]
 

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 265 vs. \(2 (123 ) = 246\).

Time = 1.13 (sec) , antiderivative size = 266, normalized size of antiderivative = 1.96

method	result	size
default	\(-\frac {2 \left (i b -a \right ) \sqrt {-\frac {a +b \sinh \left (x \right )}{i b -a}}\, \sqrt {\frac {\left (i-\sinh \left (x \right )\right ) b}{i b +a}}\, \sqrt {\frac {b \left (i+\sinh \left (x \right )\right )}{i b -a}}\, \left (i B \operatorname {EllipticF}\left (\sqrt {-\frac {a +b \sinh \left (x \right )}{i b -a}}, \sqrt {-\frac {i b -a}{i b +a}}\right ) b -i B \operatorname {EllipticE}\left (\sqrt {-\frac {a +b \sinh \left (x \right )}{i b -a}}, \sqrt {-\frac {i b -a}{i b +a}}\right ) b +A \operatorname {EllipticF}\left (\sqrt {-\frac {a +b \sinh \left (x \right )}{i b -a}}, \sqrt {-\frac {i b -a}{i b +a}}\right ) b -B \operatorname {EllipticE}\left (\sqrt {-\frac {a +b \sinh \left (x \right )}{i b -a}}, \sqrt {-\frac {i b -a}{i b +a}}\right ) a \right )}{b^{2} \cosh \left (x \right ) \sqrt {a +b \sinh \left (x \right )}}\)	\(266\)
parts	\(-\frac {2 A \left (i b -a \right ) \sqrt {-\frac {a +b \sinh \left (x \right )}{i b -a}}\, \sqrt {\frac {\left (i-\sinh \left (x \right )\right ) b}{i b +a}}\, \sqrt {\frac {b \left (i+\sinh \left (x \right )\right )}{i b -a}}\, \operatorname {EllipticF}\left (\sqrt {-\frac {a +b \sinh \left (x \right )}{i b -a}}, \sqrt {-\frac {i b -a}{i b +a}}\right )}{b \cosh \left (x \right ) \sqrt {a +b \sinh \left (x \right )}}+\frac {2 B \left (i b -a \right ) \sqrt {-\frac {a +b \sinh \left (x \right )}{i b -a}}\, \sqrt {\frac {\left (i-\sinh \left (x \right )\right ) b}{i b +a}}\, \sqrt {\frac {b \left (i+\sinh \left (x \right )\right )}{i b -a}}\, \left (i \operatorname {EllipticE}\left (\sqrt {-\frac {a +b \sinh \left (x \right )}{i b -a}}, \sqrt {-\frac {i b -a}{i b +a}}\right ) b -i \operatorname {EllipticF}\left (\sqrt {-\frac {a +b \sinh \left (x \right )}{i b -a}}, \sqrt {-\frac {i b -a}{i b +a}}\right ) b +\operatorname {EllipticE}\left (\sqrt {-\frac {a +b \sinh \left (x \right )}{i b -a}}, \sqrt {-\frac {i b -a}{i b +a}}\right ) a \right )}{b^{2} \cosh \left (x \right ) \sqrt {a +b \sinh \left (x \right )}}\)	\(345\)
risch	\(\frac {B \left (b \,{\mathrm e}^{2 x}+2 \,{\mathrm e}^{x} a -b \right ) \sqrt {2}\, {\mathrm e}^{-x}}{b \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}}\, {\mathrm e}^{-x}}{2 \sqrt {\left (b \,{\mathrm e}^{2 x}+2 \,{\mathrm e}^{x} a -b \right ) {\mathrm e}^{-x}}}\)	\(781\)

Input:

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

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 184, normalized size of antiderivative = 1.35 \[ \int \frac {A+B \sinh (x)}{\sqrt {a+b \sinh (x)}} \, dx=-\frac {2 \, {\left (6 \, \sqrt {\frac {1}{2}} B b^{\frac {3}{2}} {\rm weierstrassZeta}\left (\frac {4 \, {\left (4 \, a^{2} + 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 \, a^{3} + 9 \, a b^{2}\right )}}{27 \, b^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (4 \, a^{2} + 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 \, a^{3} + 9 \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cosh \left (x\right ) + 3 \, b \sinh \left (x\right ) + 2 \, a}{3 \, b}\right )\right ) + 2 \, \sqrt {\frac {1}{2}} {\left (2 \, B a - 3 \, A b\right )} \sqrt {b} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (4 \, a^{2} + 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 \, a^{3} + 9 \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cosh \left (x\right ) + 3 \, b \sinh \left (x\right ) + 2 \, a}{3 \, b}\right ) + 3 \, \sqrt {b \sinh \left (x\right ) + a} B b\right )}}{3 \, b^{2}} \] Input:

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

Output:

-2/3*(6*sqrt(1/2)*B*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)) + 2*sqrt 
(1/2)*(2*B*a - 3*A*b)*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(b*sinh(x) + a)*B*b)/b^2
 

Sympy [F]

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

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

Output:

Integral((A + B*sinh(x))/sqrt(a + b*sinh(x)), x)
 

Maxima [F]

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

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

Output:

integrate((B*sinh(x) + A)/sqrt(b*sinh(x) + a), x)
 

Giac [F]

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

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

Output:

integrate((B*sinh(x) + A)/sqrt(b*sinh(x) + a), x)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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