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

Optimal. Leaf size=60 \[ \frac{2 i \sqrt{a+b \sinh (x)} E\left (\frac{\pi }{4}-\frac{i x}{2}|\frac{2 b}{i a+b}\right )}{\sqrt{\frac{a+b \sinh (x)}{a-i b}}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0371274, antiderivative size = 60, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {2655, 2653} \[ \frac{2 i \sqrt{a+b \sinh (x)} E\left (\frac{\pi }{4}-\frac{i x}{2}|\frac{2 b}{i a+b}\right )}{\sqrt{\frac{a+b \sinh (x)}{a-i b}}} \]

Antiderivative was successfully verified.

[In]

Int[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]])/Sqrt[(a + b*Sinh[x])/(a - I*b)]

Rule 2655

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*Sin[c + d*x])/(a + b)], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 -
b^2, 0] &&  !GtQ[a + b, 0]

Rule 2653

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

Rubi steps

\begin{align*} \int \sqrt{a+b \sinh (x)} \, dx &=\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}}}\\ &=\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}}}\\ \end{align*}

Mathematica [A]  time = 0.186684, size = 65, normalized size = 1.08 \[ \frac{2 (b+i a) \sqrt{\frac{a+b \sinh (x)}{a-i b}} E\left (\frac{1}{4} (\pi -2 i x)|-\frac{2 i b}{a-i b}\right )}{\sqrt{a+b \sinh (x)}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[a + b*Sinh[x]],x]

[Out]

(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*Sin
h[x]]

________________________________________________________________________________________

Maple [B]  time = 0.078, size = 262, normalized size = 4.4 \begin{align*} 2\,{\frac{ib-a}{b\cosh \left ( x \right ) \sqrt{a+b\sinh \left ( x \right ) }}\sqrt{-{\frac{a+b\sinh \left ( x \right ) }{ib-a}}}\sqrt{{\frac{ \left ( i-\sinh \left ( x \right ) \right ) b}{ib+a}}}\sqrt{{\frac{ \left ( i+\sinh \left ( x \right ) \right ) b}{ib-a}}} \left ( i{\it EllipticE} \left ( \sqrt{-{\frac{a+b\sinh \left ( x \right ) }{ib-a}}},\sqrt{-{\frac{ib-a}{ib+a}}} \right ) b-i{\it EllipticF} \left ( \sqrt{-{\frac{a+b\sinh \left ( x \right ) }{ib-a}}},\sqrt{-{\frac{ib-a}{ib+a}}} \right ) b+{\it EllipticE} \left ( \sqrt{-{\frac{a+b\sinh \left ( x \right ) }{ib-a}}},\sqrt{-{\frac{ib-a}{ib+a}}} \right ) a-a{\it EllipticF} \left ( \sqrt{-{\frac{a+b\sinh \left ( x \right ) }{ib-a}}},\sqrt{-{\frac{ib-a}{ib+a}}} \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{b \sinh \left (x\right ) + a}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(b*sinh(x) + a), x)

________________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\sqrt{b \sinh \left (x\right ) + a}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(sqrt(b*sinh(x) + a), x)

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{a + b \sinh{\left (x \right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(a + b*sinh(x)), x)

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{b \sinh \left (x\right ) + a}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(b*sinh(x) + a), x)