3.1.39 \(\int (a \text {sech}^3(x))^{5/2} \, dx\) [39]

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

[Out]

154/195*I*a^2*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)+154/195*a^2*cosh(x)*sinh(x)*(a*sech(x)^3)^(1/2)+154/585*a^2*(a*sech(x)^3)^(1/2)*tanh(x)+22/117*a^2*sech(x
)^2*(a*sech(x)^3)^(1/2)*tanh(x)+2/13*a^2*sech(x)^4*(a*sech(x)^3)^(1/2)*tanh(x)

________________________________________________________________________________________

Rubi [A]
time = 0.04, antiderivative size = 121, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {4208, 3853, 3856, 2719} \begin {gather*} \frac {154}{585} a^2 \tanh (x) \sqrt {a \text {sech}^3(x)}+\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}{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)} \end {gather*}

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 2719

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

Rule 3853

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 3856

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 4208

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 \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 = 0.07, size = 63, normalized size = 0.52 \begin {gather*} \frac {2}{585} a \text {sech}(x) \left (a \text {sech}^3(x)\right )^{3/2} \left (231 i \cosh ^{\frac {11}{2}}(x) E\left (\left .\frac {i x}{2}\right |2\right )+55 \cosh (x) \sinh (x)+77 \cosh ^3(x) \sinh (x)+231 \cosh ^5(x) \sinh (x)+45 \tanh (x)\right ) \end {gather*}

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 = 1.05, size = 0, normalized size = 0.00 \[\int \left (a \mathrm {sech}\left (x \right )^{3}\right )^{\frac {5}{2}}\, dx\]

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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 [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.11, size = 1382, normalized size = 11.42 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/585*(231*sqrt(2)*(a^2*cosh(x)^12 + 12*a^2*cosh(x)*sinh(x)^11 + a^2*sinh(x)^12 + 6*a^2*cosh(x)^10 + 6*(11*a^2
*cosh(x)^2 + a^2)*sinh(x)^10 + 15*a^2*cosh(x)^8 + 20*(11*a^2*cosh(x)^3 + 3*a^2*cosh(x))*sinh(x)^9 + 15*(33*a^2
*cosh(x)^4 + 18*a^2*cosh(x)^2 + a^2)*sinh(x)^8 + 20*a^2*cosh(x)^6 + 24*(33*a^2*cosh(x)^5 + 30*a^2*cosh(x)^3 +
5*a^2*cosh(x))*sinh(x)^7 + 4*(231*a^2*cosh(x)^6 + 315*a^2*cosh(x)^4 + 105*a^2*cosh(x)^2 + 5*a^2)*sinh(x)^6 + 1
5*a^2*cosh(x)^4 + 24*(33*a^2*cosh(x)^7 + 63*a^2*cosh(x)^5 + 35*a^2*cosh(x)^3 + 5*a^2*cosh(x))*sinh(x)^5 + 15*(
33*a^2*cosh(x)^8 + 84*a^2*cosh(x)^6 + 70*a^2*cosh(x)^4 + 20*a^2*cosh(x)^2 + a^2)*sinh(x)^4 + 6*a^2*cosh(x)^2 +
20*(11*a^2*cosh(x)^9 + 36*a^2*cosh(x)^7 + 42*a^2*cosh(x)^5 + 20*a^2*cosh(x)^3 + 3*a^2*cosh(x))*sinh(x)^3 + 6*
(11*a^2*cosh(x)^10 + 45*a^2*cosh(x)^8 + 70*a^2*cosh(x)^6 + 50*a^2*cosh(x)^4 + 15*a^2*cosh(x)^2 + a^2)*sinh(x)^
2 + a^2 + 12*(a^2*cosh(x)^11 + 5*a^2*cosh(x)^9 + 10*a^2*cosh(x)^7 + 10*a^2*cosh(x)^5 + 5*a^2*cosh(x)^3 + a^2*c
osh(x))*sinh(x))*sqrt(a)*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cosh(x) + sinh(x))) + sqrt(2)*(231*
a^2*cosh(x)^13 + 3003*a^2*cosh(x)*sinh(x)^12 + 231*a^2*sinh(x)^13 + 1540*a^2*cosh(x)^11 + 154*(117*a^2*cosh(x)
^2 + 10*a^2)*sinh(x)^11 + 4367*a^2*cosh(x)^9 + 1694*(39*a^2*cosh(x)^3 + 10*a^2*cosh(x))*sinh(x)^10 + 11*(15015
*a^2*cosh(x)^4 + 7700*a^2*cosh(x)^2 + 397*a^2)*sinh(x)^9 + 6808*a^2*cosh(x)^7 + 33*(9009*a^2*cosh(x)^5 + 7700*
a^2*cosh(x)^3 + 1191*a^2*cosh(x))*sinh(x)^8 + 4*(99099*a^2*cosh(x)^6 + 127050*a^2*cosh(x)^4 + 39303*a^2*cosh(x
)^2 + 1702*a^2)*sinh(x)^7 + 1277*a^2*cosh(x)^5 + 28*(14157*a^2*cosh(x)^7 + 25410*a^2*cosh(x)^5 + 13101*a^2*cos
h(x)^3 + 1702*a^2*cosh(x))*sinh(x)^6 + (297297*a^2*cosh(x)^8 + 711480*a^2*cosh(x)^6 + 550242*a^2*cosh(x)^4 + 1
42968*a^2*cosh(x)^2 + 1277*a^2)*sinh(x)^5 + 484*a^2*cosh(x)^3 + (165165*a^2*cosh(x)^9 + 508200*a^2*cosh(x)^7 +
550242*a^2*cosh(x)^5 + 238280*a^2*cosh(x)^3 + 6385*a^2*cosh(x))*sinh(x)^4 + 2*(33033*a^2*cosh(x)^10 + 127050*
a^2*cosh(x)^8 + 183414*a^2*cosh(x)^6 + 119140*a^2*cosh(x)^4 + 6385*a^2*cosh(x)^2 + 242*a^2)*sinh(x)^3 + 77*a^2
*cosh(x) + 2*(9009*a^2*cosh(x)^11 + 42350*a^2*cosh(x)^9 + 78606*a^2*cosh(x)^7 + 71484*a^2*cosh(x)^5 + 6385*a^2
*cosh(x)^3 + 726*a^2*cosh(x))*sinh(x)^2 + (3003*a^2*cosh(x)^12 + 16940*a^2*cosh(x)^10 + 39303*a^2*cosh(x)^8 +
47656*a^2*cosh(x)^6 + 6385*a^2*cosh(x)^4 + 1452*a^2*cosh(x)^2 + 77*a^2)*sinh(x))*sqrt((a*cosh(x) + a*sinh(x))/
(cosh(x)^2 + 2*cosh(x)*sinh(x) + sinh(x)^2 + 1)))/(cosh(x)^12 + 12*cosh(x)*sinh(x)^11 + sinh(x)^12 + 6*(11*cos
h(x)^2 + 1)*sinh(x)^10 + 6*cosh(x)^10 + 20*(11*cosh(x)^3 + 3*cosh(x))*sinh(x)^9 + 15*(33*cosh(x)^4 + 18*cosh(x
)^2 + 1)*sinh(x)^8 + 15*cosh(x)^8 + 24*(33*cosh(x)^5 + 30*cosh(x)^3 + 5*cosh(x))*sinh(x)^7 + 4*(231*cosh(x)^6
+ 315*cosh(x)^4 + 105*cosh(x)^2 + 5)*sinh(x)^6 + 20*cosh(x)^6 + 24*(33*cosh(x)^7 + 63*cosh(x)^5 + 35*cosh(x)^3
+ 5*cosh(x))*sinh(x)^5 + 15*(33*cosh(x)^8 + 84*cosh(x)^6 + 70*cosh(x)^4 + 20*cosh(x)^2 + 1)*sinh(x)^4 + 15*co
sh(x)^4 + 20*(11*cosh(x)^9 + 36*cosh(x)^7 + 42*cosh(x)^5 + 20*cosh(x)^3 + 3*cosh(x))*sinh(x)^3 + 6*(11*cosh(x)
^10 + 45*cosh(x)^8 + 70*cosh(x)^6 + 50*cosh(x)^4 + 15*cosh(x)^2 + 1)*sinh(x)^2 + 6*cosh(x)^2 + 12*(cosh(x)^11
+ 5*cosh(x)^9 + 10*cosh(x)^7 + 10*cosh(x)^5 + 5*cosh(x)^3 + cosh(x))*sinh(x) + 1)

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (a \operatorname {sech}^{3}{\left (x \right )}\right )^{\frac {5}{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((a*sech(x)**3)**(5/2), x)

________________________________________________________________________________________

Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int {\left (\frac {a}{{\mathrm {cosh}\left (x\right )}^3}\right )}^{5/2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a/cosh(x)^3)^(5/2),x)

[Out]

int((a/cosh(x)^3)^(5/2), x)

________________________________________________________________________________________