3.43 \(\int \frac{1}{(a \text{sech}^3(x))^{3/2}} \, dx\)

Optimal. Leaf size=77 \[ \frac{14 \sinh (x)}{45 a \sqrt{a \text{sech}^3(x)}}+\frac{2 \sinh (x) \cosh ^2(x)}{9 a \sqrt{a \text{sech}^3(x)}}-\frac{14 i E\left (\left .\frac{i x}{2}\right |2\right )}{15 a \cosh ^{\frac{3}{2}}(x) \sqrt{a \text{sech}^3(x)}} \]

[Out]

(((-14*I)/15)*EllipticE[(I/2)*x, 2])/(a*Cosh[x]^(3/2)*Sqrt[a*Sech[x]^3]) + (14*Sinh[x])/(45*a*Sqrt[a*Sech[x]^3
]) + (2*Cosh[x]^2*Sinh[x])/(9*a*Sqrt[a*Sech[x]^3])

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Rubi [A]  time = 0.0441274, antiderivative size = 77, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {4123, 3769, 3771, 2639} \[ \frac{14 \sinh (x)}{45 a \sqrt{a \text{sech}^3(x)}}+\frac{2 \sinh (x) \cosh ^2(x)}{9 a \sqrt{a \text{sech}^3(x)}}-\frac{14 i E\left (\left .\frac{i x}{2}\right |2\right )}{15 a \cosh ^{\frac{3}{2}}(x) \sqrt{a \text{sech}^3(x)}} \]

Antiderivative was successfully verified.

[In]

Int[(a*Sech[x]^3)^(-3/2),x]

[Out]

(((-14*I)/15)*EllipticE[(I/2)*x, 2])/(a*Cosh[x]^(3/2)*Sqrt[a*Sech[x]^3]) + (14*Sinh[x])/(45*a*Sqrt[a*Sech[x]^3
]) + (2*Cosh[x]^2*Sinh[x])/(9*a*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 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]

Rubi steps

\begin{align*} \int \frac{1}{\left (a \text{sech}^3(x)\right )^{3/2}} \, dx &=\frac{\text{sech}^{\frac{3}{2}}(x) \int \frac{1}{\text{sech}^{\frac{9}{2}}(x)} \, dx}{a \sqrt{a \text{sech}^3(x)}}\\ &=\frac{2 \cosh ^2(x) \sinh (x)}{9 a \sqrt{a \text{sech}^3(x)}}+\frac{\left (7 \text{sech}^{\frac{3}{2}}(x)\right ) \int \frac{1}{\text{sech}^{\frac{5}{2}}(x)} \, dx}{9 a \sqrt{a \text{sech}^3(x)}}\\ &=\frac{14 \sinh (x)}{45 a \sqrt{a \text{sech}^3(x)}}+\frac{2 \cosh ^2(x) \sinh (x)}{9 a \sqrt{a \text{sech}^3(x)}}+\frac{\left (7 \text{sech}^{\frac{3}{2}}(x)\right ) \int \frac{1}{\sqrt{\text{sech}(x)}} \, dx}{15 a \sqrt{a \text{sech}^3(x)}}\\ &=\frac{14 \sinh (x)}{45 a \sqrt{a \text{sech}^3(x)}}+\frac{2 \cosh ^2(x) \sinh (x)}{9 a \sqrt{a \text{sech}^3(x)}}+\frac{7 \int \sqrt{\cosh (x)} \, dx}{15 a \cosh ^{\frac{3}{2}}(x) \sqrt{a \text{sech}^3(x)}}\\ &=-\frac{14 i E\left (\left .\frac{i x}{2}\right |2\right )}{15 a \cosh ^{\frac{3}{2}}(x) \sqrt{a \text{sech}^3(x)}}+\frac{14 \sinh (x)}{45 a \sqrt{a \text{sech}^3(x)}}+\frac{2 \cosh ^2(x) \sinh (x)}{9 a \sqrt{a \text{sech}^3(x)}}\\ \end{align*}

Mathematica [A]  time = 0.0890085, size = 47, normalized size = 0.61 \[ \frac{33 \sinh (x)+5 \sinh (3 x)-\frac{84 i E\left (\left .\frac{i x}{2}\right |2\right )}{\cosh ^{\frac{3}{2}}(x)}}{90 a \sqrt{a \text{sech}^3(x)}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a*Sech[x]^3)^(-3/2),x]

[Out]

(((-84*I)*EllipticE[(I/2)*x, 2])/Cosh[x]^(3/2) + 33*Sinh[x] + 5*Sinh[3*x])/(90*a*Sqrt[a*Sech[x]^3])

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Maple [F]  time = 0.058, size = 0, normalized size = 0. \begin{align*} \int \left ( a \left ({\rm sech} \left (x\right ) \right ) ^{3} \right ) ^{-{\frac{3}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (a \operatorname{sech}\left (x\right )^{3}\right )^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((a*sech(x)^3)^(-3/2), 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^{2} \operatorname{sech}\left (x\right )^{6}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(sqrt(a*sech(x)^3)/(a^2*sech(x)^6), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (a \operatorname{sech}^{3}{\left (x \right )}\right )^{\frac{3}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((a*sech(x)**3)**(-3/2), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (a \operatorname{sech}\left (x\right )^{3}\right )^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((a*sech(x)^3)^(-3/2), x)