3.39 \(\int (a \text{sech}^3(x))^{5/2} \, dx\)

Optimal. Leaf size=121 \[ \frac{2}{13} a^2 \tanh (x) \text{sech}^4(x) \sqrt{a \text{sech}^3(x)}+\frac{22}{117} a^2 \tanh (x) \text{sech}^2(x) \sqrt{a \text{sech}^3(x)}+\frac{154}{585} a^2 \tanh (x) \sqrt{a \text{sech}^3(x)}+\frac{154}{195} i a^2 \cosh ^{\frac{3}{2}}(x) E\left (\left .\frac{i x}{2}\right |2\right ) \sqrt{a \text{sech}^3(x)}+\frac{154}{195} a^2 \sinh (x) \cosh (x) \sqrt{a \text{sech}^3(x)} \]

[Out]

((154*I)/195)*a^2*Cosh[x]^(3/2)*EllipticE[(I/2)*x, 2]*Sqrt[a*Sech[x]^3] + (154*a^2*Cosh[x]*Sqrt[a*Sech[x]^3]*S
inh[x])/195 + (154*a^2*Sqrt[a*Sech[x]^3]*Tanh[x])/585 + (22*a^2*Sech[x]^2*Sqrt[a*Sech[x]^3]*Tanh[x])/117 + (2*
a^2*Sech[x]^4*Sqrt[a*Sech[x]^3]*Tanh[x])/13

________________________________________________________________________________________

Rubi [A]  time = 0.0600378, antiderivative size = 121, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {4123, 3768, 3771, 2639} \[ \frac{2}{13} a^2 \tanh (x) \text{sech}^4(x) \sqrt{a \text{sech}^3(x)}+\frac{22}{117} a^2 \tanh (x) \text{sech}^2(x) \sqrt{a \text{sech}^3(x)}+\frac{154}{585} a^2 \tanh (x) \sqrt{a \text{sech}^3(x)}+\frac{154}{195} i a^2 \cosh ^{\frac{3}{2}}(x) E\left (\left .\frac{i x}{2}\right |2\right ) \sqrt{a \text{sech}^3(x)}+\frac{154}{195} a^2 \sinh (x) \cosh (x) \sqrt{a \text{sech}^3(x)} \]

Antiderivative was successfully verified.

[In]

Int[(a*Sech[x]^3)^(5/2),x]

[Out]

((154*I)/195)*a^2*Cosh[x]^(3/2)*EllipticE[(I/2)*x, 2]*Sqrt[a*Sech[x]^3] + (154*a^2*Cosh[x]*Sqrt[a*Sech[x]^3]*S
inh[x])/195 + (154*a^2*Sqrt[a*Sech[x]^3]*Tanh[x])/585 + (22*a^2*Sech[x]^2*Sqrt[a*Sech[x]^3]*Tanh[x])/117 + (2*
a^2*Sech[x]^4*Sqrt[a*Sech[x]^3]*Tanh[x])/13

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 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 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 \left (a \text{sech}^3(x)\right )^{5/2} \, dx &=\frac{\left (a^2 \sqrt{a \text{sech}^3(x)}\right ) \int \text{sech}^{\frac{15}{2}}(x) \, dx}{\text{sech}^{\frac{3}{2}}(x)}\\ &=\frac{2}{13} a^2 \text{sech}^4(x) \sqrt{a \text{sech}^3(x)} \tanh (x)+\frac{\left (11 a^2 \sqrt{a \text{sech}^3(x)}\right ) \int \text{sech}^{\frac{11}{2}}(x) \, dx}{13 \text{sech}^{\frac{3}{2}}(x)}\\ &=\frac{22}{117} a^2 \text{sech}^2(x) \sqrt{a \text{sech}^3(x)} \tanh (x)+\frac{2}{13} a^2 \text{sech}^4(x) \sqrt{a \text{sech}^3(x)} \tanh (x)+\frac{\left (77 a^2 \sqrt{a \text{sech}^3(x)}\right ) \int \text{sech}^{\frac{7}{2}}(x) \, dx}{117 \text{sech}^{\frac{3}{2}}(x)}\\ &=\frac{154}{585} a^2 \sqrt{a \text{sech}^3(x)} \tanh (x)+\frac{22}{117} a^2 \text{sech}^2(x) \sqrt{a \text{sech}^3(x)} \tanh (x)+\frac{2}{13} a^2 \text{sech}^4(x) \sqrt{a \text{sech}^3(x)} \tanh (x)+\frac{\left (77 a^2 \sqrt{a \text{sech}^3(x)}\right ) \int \text{sech}^{\frac{3}{2}}(x) \, dx}{195 \text{sech}^{\frac{3}{2}}(x)}\\ &=\frac{154}{195} a^2 \cosh (x) \sqrt{a \text{sech}^3(x)} \sinh (x)+\frac{154}{585} a^2 \sqrt{a \text{sech}^3(x)} \tanh (x)+\frac{22}{117} a^2 \text{sech}^2(x) \sqrt{a \text{sech}^3(x)} \tanh (x)+\frac{2}{13} a^2 \text{sech}^4(x) \sqrt{a \text{sech}^3(x)} \tanh (x)-\frac{\left (77 a^2 \sqrt{a \text{sech}^3(x)}\right ) \int \frac{1}{\sqrt{\text{sech}(x)}} \, dx}{195 \text{sech}^{\frac{3}{2}}(x)}\\ &=\frac{154}{195} a^2 \cosh (x) \sqrt{a \text{sech}^3(x)} \sinh (x)+\frac{154}{585} a^2 \sqrt{a \text{sech}^3(x)} \tanh (x)+\frac{22}{117} a^2 \text{sech}^2(x) \sqrt{a \text{sech}^3(x)} \tanh (x)+\frac{2}{13} a^2 \text{sech}^4(x) \sqrt{a \text{sech}^3(x)} \tanh (x)-\frac{1}{195} \left (77 a^2 \cosh ^{\frac{3}{2}}(x) \sqrt{a \text{sech}^3(x)}\right ) \int \sqrt{\cosh (x)} \, dx\\ &=\frac{154}{195} i a^2 \cosh ^{\frac{3}{2}}(x) E\left (\left .\frac{i x}{2}\right |2\right ) \sqrt{a \text{sech}^3(x)}+\frac{154}{195} a^2 \cosh (x) \sqrt{a \text{sech}^3(x)} \sinh (x)+\frac{154}{585} a^2 \sqrt{a \text{sech}^3(x)} \tanh (x)+\frac{22}{117} a^2 \text{sech}^2(x) \sqrt{a \text{sech}^3(x)} \tanh (x)+\frac{2}{13} a^2 \text{sech}^4(x) \sqrt{a \text{sech}^3(x)} \tanh (x)\\ \end{align*}

Mathematica [A]  time = 0.0963402, size = 63, normalized size = 0.52 \[ \frac{2}{585} a \text{sech}(x) \left (a \text{sech}^3(x)\right )^{3/2} \left (45 \tanh (x)+231 \sinh (x) \cosh ^5(x)+77 \sinh (x) \cosh ^3(x)+231 i \cosh ^{\frac{11}{2}}(x) E\left (\left .\frac{i x}{2}\right |2\right )+55 \sinh (x) \cosh (x)\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(a*Sech[x]^3)^(5/2),x]

[Out]

(2*a*Sech[x]*(a*Sech[x]^3)^(3/2)*((231*I)*Cosh[x]^(11/2)*EllipticE[(I/2)*x, 2] + 55*Cosh[x]*Sinh[x] + 77*Cosh[
x]^3*Sinh[x] + 231*Cosh[x]^5*Sinh[x] + 45*Tanh[x]))/585

________________________________________________________________________________________

Maple [F]  time = 0.059, size = 0, normalized size = 0. \begin{align*} \int \left ( a \left ({\rm sech} \left (x\right ) \right ) ^{3} \right ) ^{{\frac{5}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*sech(x)^3)^(5/2),x)

[Out]

int((a*sech(x)^3)^(5/2),x)

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (a \operatorname{sech}\left (x\right )^{3}\right )^{\frac{5}{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*sech(x)^3)^(5/2),x, algorithm="maxima")

[Out]

integrate((a*sech(x)^3)^(5/2), x)

________________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\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((a*sech(x)^3)^(5/2),x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*sech(x)**3)**(5/2),x)

[Out]

Timed out

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (a \operatorname{sech}\left (x\right )^{3}\right )^{\frac{5}{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*sech(x)^3)^(5/2),x, algorithm="giac")

[Out]

integrate((a*sech(x)^3)^(5/2), x)