\(\int \frac {\sinh (x)}{\sqrt {a+b \sinh (x)}} \, dx\) [111]

Optimal result

Integrand size = 13, antiderivative size = 128 \[ \int \frac {\sinh (x)}{\sqrt {a+b \sinh (x)}} \, dx=\frac {2 i E\left (\frac {\pi }{4}-\frac {i x}{2}|\frac {2 b}{i a+b}\right ) \sqrt {a+b \sinh (x)}}{b \sqrt {\frac {a+b \sinh (x)}{a-i b}}}-\frac {2 i a \operatorname {EllipticF}\left (\frac {\pi }{4}-\frac {i x}{2},\frac {2 b}{i a+b}\right ) \sqrt {\frac {a+b \sinh (x)}{a-i b}}}{b \sqrt {a+b \sinh (x)}} \]

[Out]

2*I*(sin(1/4*Pi+1/2*I*x)^2)^(1/2)/sin(1/4*Pi+1/2*I*x)*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*(sin(1/4*Pi+1/2*I*x)^2)^(1/2)/sin(1/4*Pi+1/2*I*x)*E
llipticF(cos(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)

Rubi [A] (verified)

Time = 0.08 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {2831, 2742, 2740, 2734, 2732} \[ \int \frac {\sinh (x)}{\sqrt {a+b \sinh (x)}} \, dx=\frac {2 i \sqrt {a+b \sinh (x)} E\left (\frac {\pi }{4}-\frac {i x}{2}|\frac {2 b}{i a+b}\right )}{b \sqrt {\frac {a+b \sinh (x)}{a-i b}}}-\frac {2 i a \sqrt {\frac {a+b \sinh (x)}{a-i b}} \operatorname {EllipticF}\left (\frac {\pi }{4}-\frac {i x}{2},\frac {2 b}{i a+b}\right )}{b \sqrt {a+b \sinh (x)}} \]

[In]

Int[Sinh[x]/Sqrt[a + b*Sinh[x]],x]

[Out]

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

Rule 2732

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 2734

Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Dist[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 2740

Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/(d*Sqrt[a + b]))*EllipticF[(1/2)*(c - P
i/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 2742

Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Dist[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 2831

Int[((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])/Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]], x_Symbol] :> Dist[(b*c
- a*d)/b, Int[1/Sqrt[a + b*Sin[e + f*x]], x], x] + Dist[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]

Rubi steps \begin{align*} \text {integral}& = \frac {\int \sqrt {a+b \sinh (x)} \, dx}{b}-\frac {a \int \frac {1}{\sqrt {a+b \sinh (x)}} \, dx}{b} \\ & = \frac {\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}}}-\frac {\left (a \sqrt {\frac {a+b \sinh (x)}{a-i b}}\right ) \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 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 \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)}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.34 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.79 \[ \int \frac {\sinh (x)}{\sqrt {a+b \sinh (x)}} \, dx=\frac {2 \left ((i a+b) E\left (\frac {1}{4} (\pi -2 i x)|-\frac {2 i b}{a-i b}\right )-i a \operatorname {EllipticF}\left (\frac {1}{4} (\pi -2 i x),-\frac {2 i b}{a-i b}\right )\right ) \sqrt {\frac {a+b \sinh (x)}{a-i b}}}{b \sqrt {a+b \sinh (x)}} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 1.54 (sec) , antiderivative size = 218, normalized size of antiderivative = 1.70

method	result	size
default	\(\frac {2 \left (i b -a \right ) \sqrt {-\frac {a +b \sinh \left (x \right )}{i b -a}}\, \sqrt {\frac {\left (i-\sinh \left (x \right )\right ) b}{i b +a}}\, \sqrt {\frac {\left (i+\sinh \left (x \right )\right ) b}{i b -a}}\, \left (i \operatorname {EllipticE}\left (\sqrt {-\frac {a +b \sinh \left (x \right )}{i b -a}}, \sqrt {-\frac {i b -a}{i b +a}}\right ) b -i \operatorname {EllipticF}\left (\sqrt {-\frac {a +b \sinh \left (x \right )}{i b -a}}, \sqrt {-\frac {i b -a}{i b +a}}\right ) b +\operatorname {EllipticE}\left (\sqrt {-\frac {a +b \sinh \left (x \right )}{i b -a}}, \sqrt {-\frac {i b -a}{i b +a}}\right ) a \right )}{b^{2} \cosh \left (x \right ) \sqrt {a +b \sinh \left (x \right )}}\)	\(218\)
risch	\(\frac {\left (b \,{\mathrm e}^{2 x}+2 \,{\mathrm e}^{x} a -b \right ) \sqrt {2}\, {\mathrm e}^{-x}}{b \sqrt {\left (b \,{\mathrm e}^{2 x}+2 \,{\mathrm e}^{x} a -b \right ) {\mathrm e}^{-x}}}+\frac {\left (-\frac {4 \left (b \,{\mathrm e}^{2 x}+2 \,{\mathrm e}^{x} a -b \right )}{b \sqrt {\left (b \,{\mathrm e}^{2 x}+2 \,{\mathrm e}^{x} a -b \right ) {\mathrm e}^{x}}}+\frac {4 \left (a +\sqrt {a^{2}+b^{2}}\right ) \sqrt {\frac {\left ({\mathrm e}^{x}+\frac {a +\sqrt {a^{2}+b^{2}}}{b}\right ) b}{a +\sqrt {a^{2}+b^{2}}}}\, \sqrt {\frac {{\mathrm e}^{x}-\frac {-a +\sqrt {a^{2}+b^{2}}}{b}}{-\frac {a +\sqrt {a^{2}+b^{2}}}{b}-\frac {-a +\sqrt {a^{2}+b^{2}}}{b}}}\, \sqrt {-\frac {{\mathrm e}^{x} b}{a +\sqrt {a^{2}+b^{2}}}}\, \left (\left (-\frac {a +\sqrt {a^{2}+b^{2}}}{b}-\frac {-a +\sqrt {a^{2}+b^{2}}}{b}\right ) \operatorname {EllipticE}\left (\sqrt {\frac {\left ({\mathrm e}^{x}+\frac {a +\sqrt {a^{2}+b^{2}}}{b}\right ) b}{a +\sqrt {a^{2}+b^{2}}}}, \sqrt {-\frac {a +\sqrt {a^{2}+b^{2}}}{b \left (-\frac {a +\sqrt {a^{2}+b^{2}}}{b}-\frac {-a +\sqrt {a^{2}+b^{2}}}{b}\right )}}\right )+\frac {\left (-a +\sqrt {a^{2}+b^{2}}\right ) \operatorname {EllipticF}\left (\sqrt {\frac {\left ({\mathrm e}^{x}+\frac {a +\sqrt {a^{2}+b^{2}}}{b}\right ) b}{a +\sqrt {a^{2}+b^{2}}}}, \sqrt {-\frac {a +\sqrt {a^{2}+b^{2}}}{b \left (-\frac {a +\sqrt {a^{2}+b^{2}}}{b}-\frac {-a +\sqrt {a^{2}+b^{2}}}{b}\right )}}\right )}{b}\right )}{b \sqrt {{\mathrm e}^{3 x} b +2 \,{\mathrm e}^{2 x} a -{\mathrm e}^{x} b}}\right ) \sqrt {2}\, \sqrt {\left (b \,{\mathrm e}^{2 x}+2 \,{\mathrm e}^{x} a -b \right ) {\mathrm e}^{x}}\, {\mathrm e}^{-x}}{2 \sqrt {\left (b \,{\mathrm e}^{2 x}+2 \,{\mathrm e}^{x} a -b \right ) {\mathrm e}^{-x}}}\)	\(531\)

[In]

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

[Out]

2*(I*b-a)*(-(a+b*sinh(x))/(I*b-a))^(1/2)*((I-sinh(x))*b/(I*b+a))^(1/2)*((I+sinh(x))*b/(I*b-a))^(1/2)*(I*Ellipt
icE((-(a+b*sinh(x))/(I*b-a))^(1/2),(-(I*b-a)/(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+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 [C] (verification not implemented)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.12 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.36 \[ \int \frac {\sinh (x)}{\sqrt {a+b \sinh (x)}} \, dx=-\frac {2 \, {\left (2 \, \sqrt {2} a \sqrt {b} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (4 \, a^{2} + 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 \, a^{3} + 9 \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cosh \left (x\right ) + 3 \, b \sinh \left (x\right ) + 2 \, a}{3 \, b}\right ) + 3 \, \sqrt {2} b^{\frac {3}{2}} {\rm weierstrassZeta}\left (\frac {4 \, {\left (4 \, a^{2} + 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 \, a^{3} + 9 \, a b^{2}\right )}}{27 \, b^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (4 \, a^{2} + 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 \, a^{3} + 9 \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cosh \left (x\right ) + 3 \, b \sinh \left (x\right ) + 2 \, a}{3 \, b}\right )\right ) + 3 \, \sqrt {b \sinh \left (x\right ) + a} b\right )}}{3 \, b^{2}} \]

[In]

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

[Out]

-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*c
osh(x) + 3*b*sinh(x) + 2*a)/b) + 3*sqrt(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*si
nh(x) + 2*a)/b)) + 3*sqrt(b*sinh(x) + a)*b)/b^2

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

int(sinh(x)/(a + b*sinh(x))^(1/2),x)

[Out]

int(sinh(x)/(a + b*sinh(x))^(1/2), x)