3.1.25 \(\int \sqrt {\text {sech}(x) \sinh (2 x)} \, dx\) [25]

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

[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))*2^(1/2)*sinh(x)^(
1/2)/(I*sinh(x))^(1/2)

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Rubi [A]
time = 0.05, antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.454, Rules used = {4483, 4485, 4306, 4394, 2719} \begin {gather*} \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)}} \end {gather*}

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]]

Rule 2719

Int[Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticE[(1/2)*(c - Pi/2 + d*x), 2], x] /; FreeQ[{
c, d}, x]

Rule 4306

Int[(u_)*((c_.)*sec[(a_.) + (b_.)*(x_)])^(m_.), x_Symbol] :> Dist[(c*Sec[a + b*x])^m*(c*Cos[a + b*x])^m, Int[A
ctivateTrig[u]/(c*Cos[a + b*x])^m, x], x] /; FreeQ[{a, b, c, m}, x] &&  !IntegerQ[m] && KnownSineIntegrandQ[u,
x]

Rule 4394

Int[(cos[(a_.) + (b_.)*(x_)]*(e_.))^(m_.)*((g_.)*sin[(c_.) + (d_.)*(x_)])^(p_), x_Symbol] :> Dist[(g*Sin[c + d
*x])^p/((e*Cos[a + b*x])^p*Sin[a + b*x]^p), Int[(e*Cos[a + b*x])^(m + p)*Sin[a + b*x]^p, x], x] /; FreeQ[{a, b
, c, d, e, g, m, p}, x] && EqQ[b*c - a*d, 0] && EqQ[d/b, 2] &&  !IntegerQ[p]

Rule 4483

Int[(u_.)*((a_)*(v_))^(p_), x_Symbol] :> With[{uu = ActivateTrig[u], vv = ActivateTrig[v]}, Dist[a^IntPart[p]*
((a*vv)^FracPart[p]/vv^FracPart[p]), Int[uu*vv^p, x], x]] /; FreeQ[{a, p}, x] &&  !IntegerQ[p] &&  !InertTrigF
reeQ[v]

Rule 4485

Int[(u_.)*((v_)^(m_.)*(w_)^(n_.))^(p_), x_Symbol] :> With[{uu = ActivateTrig[u], vv = ActivateTrig[v], ww = Ac
tivateTrig[w]}, Dist[(vv^m*ww^n)^FracPart[p]/(vv^(m*FracPart[p])*ww^(n*FracPart[p])), Int[uu*vv^(m*p)*ww^(n*p)
, x], x]] /; FreeQ[{m, n, p}, x] &&  !IntegerQ[p] && ( !InertTrigFreeQ[v] ||  !InertTrigFreeQ[w])

Rubi steps

\begin {align*} \int \sqrt {\text {sech}(x) \sinh (2 x)} \, dx &=\frac {\sqrt {\text {sech}(x) \sinh (2 x)} \int \sqrt {i \text {sech}(x) \sinh (2 x)} \, dx}{\sqrt {i \text {sech}(x) \sinh (2 x)}}\\ &=\frac {\sqrt {\text {sech}(x) \sinh (2 x)} \int \sqrt {\text {sech}(x)} \sqrt {i \sinh (2 x)} \, dx}{\sqrt {\text {sech}(x)} \sqrt {i \sinh (2 x)}}\\ &=\frac {\left (\sqrt {\cosh (x)} \sqrt {\text {sech}(x) \sinh (2 x)}\right ) \int \frac {\sqrt {i \sinh (2 x)}}{\sqrt {\cosh (x)}} \, dx}{\sqrt {i \sinh (2 x)}}\\ &=\frac {\sqrt {\text {sech}(x) \sinh (2 x)} \int \sqrt {i \sinh (x)} \, dx}{\sqrt {i \sinh (x)}}\\ &=\frac {2 i E\left (\left .\frac {\pi }{4}-\frac {i x}{2}\right |2\right ) \sqrt {\text {sech}(x) \sinh (2 x)}}{\sqrt {i \sinh (x)}}\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
time = 32.14, size = 54, normalized size = 1.35 \begin {gather*} -\frac {2}{3} \left (-3+\, _2F_1\left (\frac {1}{2},\frac {3}{4};\frac {7}{4};\tanh ^2\left (\frac {x}{2}\right )\right ) \sqrt {\text {sech}^2\left (\frac {x}{2}\right )}\right ) \sqrt {\text {sech}(x) \sinh (2 x)} \tanh \left (\frac {x}{2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(-3 + Hypergeometric2F1[1/2, 3/4, 7/4, Tanh[x/2]^2]*Sqrt[Sech[x/2]^2])*Sqrt[Sech[x]*Sinh[2*x]]*Tanh[x/2])/
3

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Mathics [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {cought exception: maximum recursion depth exceeded} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

mathics('Integrate[Sqrt[Sinh[2*x]/Cosh[x]],x]')

[Out]

cought exception: maximum recursion depth exceeded

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Maple [A]
time = 0.15, size = 75, normalized size = 1.88

method	result	size
default	\(\frac {2 \sqrt {-i \left (\sinh \left (x \right )+i\right )}\, \sqrt {-i \left (-\sinh \left (x \right )+i\right )}\, \sqrt {i \sinh \left (x \right )}\, \left (2 \EllipticE \left (\sqrt {1-i \sinh \left (x \right )}, \frac {\sqrt {2}}{2}\right )-\EllipticF \left (\sqrt {1-i \sinh \left (x \right )}, \frac {\sqrt {2}}{2}\right )\right )}{\cosh \left (x \right ) \sqrt {\sinh \left (x \right )}}\)	\(75\)
risch	\(2 \sqrt {{\mathrm e}^{-x} \left ({\mathrm e}^{2 x}-1\right )}+\frac {\left (-\frac {4 \left ({\mathrm e}^{2 x}-1\right )}{\sqrt {{\mathrm e}^{x} \left ({\mathrm e}^{2 x}-1\right )}}+\frac {2 \sqrt {1+{\mathrm e}^{x}}\, \sqrt {2-2 \,{\mathrm e}^{x}}\, \sqrt {-{\mathrm e}^{x}}\, \left (-2 \EllipticE \left (\sqrt {1+{\mathrm e}^{x}}, \frac {\sqrt {2}}{2}\right )+\EllipticF \left (\sqrt {1+{\mathrm e}^{x}}, \frac {\sqrt {2}}{2}\right )\right )}{\sqrt {{\mathrm e}^{3 x}-{\mathrm e}^{x}}}\right ) \sqrt {{\mathrm e}^{-x} \left ({\mathrm e}^{2 x}-1\right )}\, \sqrt {{\mathrm e}^{x} \left ({\mathrm e}^{2 x}-1\right )}}{{\mathrm e}^{2 x}-1}\)	\(130\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((sinh(2*x)/cosh(x))^(1/2),x,method=_RETURNVERBOSE)

[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)))/cosh(x)/sinh(x)^(1/2)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((sinh(2*x)/cosh(x))^(1/2),x, algorithm="maxima")

[Out]

integrate(sqrt(sinh(2*x)/cosh(x)), x)

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Fricas [F]
time = 0.37, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((sinh(2*x)/cosh(x))^(1/2),x, algorithm="fricas")

[Out]

integral(sqrt(sinh(2*x)/cosh(x)), x)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt {\frac {\sinh {\left (2 x \right )}}{\cosh {\left (x \right )}}}\, dx \end {gather*}

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)

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Giac [F] N/A
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((sinh(2*x)/cosh(x))^(1/2),x)

[Out]

Could not integrate

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \sqrt {\frac {\mathrm {sinh}\left (2\,x\right )}{\mathrm {cosh}\left (x\right )}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int((sinh(2*x)/cosh(x))^(1/2), x)

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