3.42 \(\int \frac{1}{\sqrt{a \text{sech}^3(x)}} \, dx\)

Optimal. Leaf size=48 \[ \frac{2 \tanh (x)}{3 \sqrt{a \text{sech}^3(x)}}-\frac{2 i \text{EllipticF}\left (\frac{i x}{2},2\right )}{3 \cosh ^{\frac{3}{2}}(x) \sqrt{a \text{sech}^3(x)}} \]

[Out]

(((-2*I)/3)*EllipticF[(I/2)*x, 2])/(Cosh[x]^(3/2)*Sqrt[a*Sech[x]^3]) + (2*Tanh[x])/(3*Sqrt[a*Sech[x]^3])

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Rubi [A]  time = 0.031463, antiderivative size = 48, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {4123, 3769, 3771, 2641} \[ \frac{2 \tanh (x)}{3 \sqrt{a \text{sech}^3(x)}}-\frac{2 i F\left (\left .\frac{i x}{2}\right |2\right )}{3 \cosh ^{\frac{3}{2}}(x) \sqrt{a \text{sech}^3(x)}} \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[a*Sech[x]^3],x]

[Out]

(((-2*I)/3)*EllipticF[(I/2)*x, 2])/(Cosh[x]^(3/2)*Sqrt[a*Sech[x]^3]) + (2*Tanh[x])/(3*Sqrt[a*Sech[x]^3])

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]

Rule 3769

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[(Cos[c + d*x]*(b*Csc[c + d*x])^(n + 1))/(b*d*n), x
] + Dist[(n + 1)/(b^2*n), Int[(b*Csc[c + d*x])^(n + 2), x], x] /; FreeQ[{b, c, d}, x] && LtQ[n, -1] && Integer
Q[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 2641

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

Rubi steps

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

Mathematica [A]  time = 0.0401381, size = 38, normalized size = 0.79 \[ \frac{2 \tanh (x)-\frac{2 i \text{EllipticF}\left (\frac{i x}{2},2\right )}{\cosh ^{\frac{3}{2}}(x)}}{3 \sqrt{a \text{sech}^3(x)}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[a*Sech[x]^3],x]

[Out]

(((-2*I)*EllipticF[(I/2)*x, 2])/Cosh[x]^(3/2) + 2*Tanh[x])/(3*Sqrt[a*Sech[x]^3])

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Maple [F]  time = 0.079, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{\sqrt{a \left ({\rm sech} \left (x\right ) \right ) ^{3}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{a \operatorname{sech}\left (x\right )^{3}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/sqrt(a*sech(x)^3), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{a \operatorname{sech}\left (x\right )^{3}}}{a \operatorname{sech}\left (x\right )^{3}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(sqrt(a*sech(x)^3)/(a*sech(x)^3), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{a \operatorname{sech}^{3}{\left (x \right )}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/sqrt(a*sech(x)**3), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{a \operatorname{sech}\left (x\right )^{3}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/sqrt(a*sech(x)^3), x)