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

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

[Out]

-2/3*I*(cosh(1/2*x)^2)^(1/2)/cosh(1/2*x)*EllipticF(I*sinh(1/2*x),2^(1/2))/cosh(x)^(3/2)/(a*sech(x)^3)^(1/2)+2/
3*tanh(x)/(a*sech(x)^3)^(1/2)

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Rubi [A]  time = 0.03, antiderivative size = 48, 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, 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 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]

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 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 \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*}

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Mathematica [A]  time = 0.04, size = 38, normalized size = 0.79 \[ \frac {2 \tanh (x)-\frac {2 i F\left (\left .\frac {i x}{2}\right |2\right )}{\cosh ^{\frac {3}{2}}(x)}}{3 \sqrt {a \text {sech}^3(x)}} \]

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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fricas [F]  time = 0.53, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {a \operatorname {sech}\relax (x)^{3}}}{a \operatorname {sech}\relax (x)^{3}}, x\right ) \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {a \operatorname {sech}\relax (x)^{3}}}\,{d x} \]

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)

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maple [F]  time = 0.22, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {a \mathrm {sech}\relax (x )^{3}}}\, dx \]

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 = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {a \operatorname {sech}\relax (x)^{3}}}\,{d x} \]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(1/(a/cosh(x)^3)^(1/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {a \operatorname {sech}^{3}{\relax (x )}}}\, dx \]

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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