3.25 \(\int \sqrt{\text{sech}(x) \sinh (2 x)} \, dx\)

Optimal. Leaf size=40 \[ \frac{2 i \sqrt{2} \sqrt{\sinh (x)} E\left (\left .\frac{\pi }{4}-\frac{i x}{2}\right |2\right )}{\sqrt{i \sinh (x)}} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.130855, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.454 \[ \frac{2 i E\left (\left .\frac{\pi }{4}-\frac{i x}{2}\right |2\right ) \sqrt{\sinh (2 x) \text{sech}(x)}}{\sqrt{i \sinh (x)}} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[Sech[x]*Sinh[2*x]],x]

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \sqrt{2} \int \sqrt{\sinh{\left (x \right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((sinh(2*x)/cosh(x))**(1/2),x)

[Out]

sqrt(2)*Integral(sqrt(sinh(x)), x)

_______________________________________________________________________________________

Mathematica [B]  time = 2.26558, size = 84, normalized size = 2.1 \[ \frac{2 \sqrt{\sinh (x)} \sqrt{\frac{1}{\cosh (x)+1}} \left (\sinh (x) \sqrt{\tanh \left (\frac{x}{2}\right )} \sqrt{\text{sech}^2\left (\frac{x}{2}\right )}+2 F\left (\left .\sin ^{-1}\left (\sqrt{\tanh \left (\frac{x}{2}\right )}\right )\right |-1\right )-2 E\left (\left .\sin ^{-1}\left (\sqrt{\tanh \left (\frac{x}{2}\right )}\right )\right |-1\right )\right )}{\sqrt{\tanh \left (\frac{x}{2}\right )}} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[Sech[x]*Sinh[2*x]],x]

[Out]

(2*Sqrt[(1 + Cosh[x])^(-1)]*Sqrt[Sinh[x]]*(-2*EllipticE[ArcSin[Sqrt[Tanh[x/2]]],
-1] + 2*EllipticF[ArcSin[Sqrt[Tanh[x/2]]], -1] + Sqrt[Sech[x/2]^2]*Sinh[x]*Sqrt
[Tanh[x/2]]))/Sqrt[Tanh[x/2]]

_______________________________________________________________________________________

Maple [A]  time = 0.346, size = 75, normalized size = 1.9 \[ 2\,{\frac{\sqrt{-i \left ( \sinh \left ( x \right ) +i \right ) }\sqrt{-i \left ( -\sinh \left ( x \right ) +i \right ) }\sqrt{i\sinh \left ( x \right ) } \left ( 2\,{\it EllipticE} \left ( \sqrt{1-i\sinh \left ( x \right ) },1/2\,\sqrt{2} \right ) -{\it EllipticF} \left ( \sqrt{1-i\sinh \left ( x \right ) },1/2\,\sqrt{2} \right ) \right ) }{\cosh \left ( x \right ) \sqrt{\sinh \left ( x \right ) }}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((sinh(2*x)/cosh(x))^(1/2),x)

[Out]

2*(-I*(sinh(x)+I))^(1/2)*(-I*(-sinh(x)+I))^(1/2)*(I*sinh(x))^(1/2)*(2*EllipticE(
(1-I*sinh(x))^(1/2),1/2*2^(1/2))-EllipticF((1-I*sinh(x))^(1/2),1/2*2^(1/2)))/cos
h(x)/sinh(x)^(1/2)

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \sqrt{\frac{\sinh \left (2 \, x\right )}{\cosh \left (x\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(sinh(2*x)/cosh(x)),x, algorithm="maxima")

[Out]

integrate(sqrt(sinh(2*x)/cosh(x)), x)

_______________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\sqrt{\frac{\sinh \left (2 \, x\right )}{\cosh \left (x\right )}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(sinh(2*x)/cosh(x)),x, algorithm="fricas")

[Out]

integral(sqrt(sinh(2*x)/cosh(x)), x)

_______________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \sqrt{\frac{\sinh{\left (2 x \right )}}{\cosh{\left (x \right )}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((sinh(2*x)/cosh(x))**(1/2),x)

[Out]

Integral(sqrt(sinh(2*x)/cosh(x)), x)

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \sqrt{\frac{\sinh \left (2 \, x\right )}{\cosh \left (x\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(sinh(2*x)/cosh(x)),x, algorithm="giac")

[Out]

integrate(sqrt(sinh(2*x)/cosh(x)), x)