3.41 \(\int \sqrt {a \text {sech}^3(x)} \, dx\)

Optimal. Leaf size=46 \[ 2 \sinh (x) \cosh (x) \sqrt {a \text {sech}^3(x)}+2 i \cosh ^{\frac {3}{2}}(x) E\left (\left .\frac {i x}{2}\right |2\right ) \sqrt {a \text {sech}^3(x)} \]

[Out]

2*I*cosh(x)^(3/2)*(cosh(1/2*x)^2)^(1/2)/cosh(1/2*x)*EllipticE(I*sinh(1/2*x),2^(1/2))*(a*sech(x)^3)^(1/2)+2*cos
h(x)*sinh(x)*(a*sech(x)^3)^(1/2)

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Rubi [A]  time = 0.03, antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {4123, 3768, 3771, 2639} \[ 2 \sinh (x) \cosh (x) \sqrt {a \text {sech}^3(x)}+2 i \cosh ^{\frac {3}{2}}(x) E\left (\left .\frac {i x}{2}\right |2\right ) \sqrt {a \text {sech}^3(x)} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[a*Sech[x]^3],x]

[Out]

(2*I)*Cosh[x]^(3/2)*EllipticE[(I/2)*x, 2]*Sqrt[a*Sech[x]^3] + 2*Cosh[x]*Sqrt[a*Sech[x]^3]*Sinh[x]

Rule 2639

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

Rule 3768

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> -Simp[(b*Cos[c + d*x]*(b*Csc[c + d*x])^(n - 1))/(d*(n -
1)), x] + Dist[(b^2*(n - 2))/(n - 1), Int[(b*Csc[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1
] && IntegerQ[2*n]

Rule 3771

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Dist[(b*Csc[c + d*x])^n*Sin[c + d*x]^n, Int[1/Sin[c + d
*x]^n, x], x] /; FreeQ[{b, c, d}, x] && EqQ[n^2, 1/4]

Rule 4123

Int[((b_.)*((c_.)*sec[(e_.) + (f_.)*(x_)])^(n_))^(p_), x_Symbol] :> Dist[(b^IntPart[p]*(b*(c*Sec[e + f*x])^n)^
FracPart[p])/(c*Sec[e + f*x])^(n*FracPart[p]), Int[(c*Sec[e + f*x])^(n*p), x], x] /; FreeQ[{b, c, e, f, n, p},
x] &&  !IntegerQ[p]

Rubi steps

\begin {align*} \int \sqrt {a \text {sech}^3(x)} \, dx &=\frac {\sqrt {a \text {sech}^3(x)} \int \text {sech}^{\frac {3}{2}}(x) \, dx}{\text {sech}^{\frac {3}{2}}(x)}\\ &=2 \cosh (x) \sqrt {a \text {sech}^3(x)} \sinh (x)-\frac {\sqrt {a \text {sech}^3(x)} \int \frac {1}{\sqrt {\text {sech}(x)}} \, dx}{\text {sech}^{\frac {3}{2}}(x)}\\ &=2 \cosh (x) \sqrt {a \text {sech}^3(x)} \sinh (x)-\left (\cosh ^{\frac {3}{2}}(x) \sqrt {a \text {sech}^3(x)}\right ) \int \sqrt {\cosh (x)} \, dx\\ &=2 i \cosh ^{\frac {3}{2}}(x) E\left (\left .\frac {i x}{2}\right |2\right ) \sqrt {a \text {sech}^3(x)}+2 \cosh (x) \sqrt {a \text {sech}^3(x)} \sinh (x)\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 36, normalized size = 0.78 \[ 2 \cosh (x) \sqrt {a \text {sech}^3(x)} \left (\sinh (x)+i \sqrt {\cosh (x)} E\left (\left .\frac {i x}{2}\right |2\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[a*Sech[x]^3],x]

[Out]

2*Cosh[x]*Sqrt[a*Sech[x]^3]*(I*Sqrt[Cosh[x]]*EllipticE[(I/2)*x, 2] + Sinh[x])

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fricas [F]  time = 1.04, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sqrt {a \operatorname {sech}\relax (x)^{3}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*sech(x)^3)^(1/2),x, algorithm="fricas")

[Out]

integral(sqrt(a*sech(x)^3), x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {a \operatorname {sech}\relax (x)^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*sech(x)^3)^(1/2),x, algorithm="giac")

[Out]

integrate(sqrt(a*sech(x)^3), x)

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maple [F]  time = 0.22, size = 0, normalized size = 0.00 \[ \int \sqrt {a \mathrm {sech}\relax (x )^{3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*sech(x)^3)^(1/2),x)

[Out]

int((a*sech(x)^3)^(1/2),x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {a \operatorname {sech}\relax (x)^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*sech(x)^3)^(1/2),x, algorithm="maxima")

[Out]

integrate(sqrt(a*sech(x)^3), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \sqrt {\frac {a}{{\mathrm {cosh}\relax (x)}^3}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a/cosh(x)^3)^(1/2),x)

[Out]

int((a/cosh(x)^3)^(1/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {a \operatorname {sech}^{3}{\relax (x )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*sech(x)**3)**(1/2),x)

[Out]

Integral(sqrt(a*sech(x)**3), x)

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