\(\int \frac {1}{(a \text {sech}^3(x))^{5/2}} \, dx\) [44]

Optimal result

Integrand size = 10, antiderivative size = 121 \[ \int \frac {1}{\left (a \text {sech}^3(x)\right )^{5/2}} \, dx=-\frac {26 i \operatorname {EllipticF}\left (\frac {i x}{2},2\right )}{77 a^2 \cosh ^{\frac {3}{2}}(x) \sqrt {a \text {sech}^3(x)}}+\frac {78 \cosh (x) \sinh (x)}{385 a^2 \sqrt {a \text {sech}^3(x)}}+\frac {26 \cosh ^3(x) \sinh (x)}{165 a^2 \sqrt {a \text {sech}^3(x)}}+\frac {2 \cosh ^5(x) \sinh (x)}{15 a^2 \sqrt {a \text {sech}^3(x)}}+\frac {26 \tanh (x)}{77 a^2 \sqrt {a \text {sech}^3(x)}} \]

[Out]

-26/77*I*(cosh(1/2*x)^2)^(1/2)/cosh(1/2*x)*EllipticF(I*sinh(1/2*x),2^(1/2))/a^2/cosh(x)^(3/2)/(a*sech(x)^3)^(1
/2)+78/385*cosh(x)*sinh(x)/a^2/(a*sech(x)^3)^(1/2)+26/165*cosh(x)^3*sinh(x)/a^2/(a*sech(x)^3)^(1/2)+2/15*cosh(
x)^5*sinh(x)/a^2/(a*sech(x)^3)^(1/2)+26/77*tanh(x)/a^2/(a*sech(x)^3)^(1/2)

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {4208, 3854, 3856, 2720} \[ \int \frac {1}{\left (a \text {sech}^3(x)\right )^{5/2}} \, dx=\frac {26 \tanh (x)}{77 a^2 \sqrt {a \text {sech}^3(x)}}-\frac {26 i \operatorname {EllipticF}\left (\frac {i x}{2},2\right )}{77 a^2 \cosh ^{\frac {3}{2}}(x) \sqrt {a \text {sech}^3(x)}}+\frac {2 \sinh (x) \cosh ^5(x)}{15 a^2 \sqrt {a \text {sech}^3(x)}}+\frac {26 \sinh (x) \cosh ^3(x)}{165 a^2 \sqrt {a \text {sech}^3(x)}}+\frac {78 \sinh (x) \cosh (x)}{385 a^2 \sqrt {a \text {sech}^3(x)}} \]

[In]

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

[Out]

(((-26*I)/77)*EllipticF[(I/2)*x, 2])/(a^2*Cosh[x]^(3/2)*Sqrt[a*Sech[x]^3]) + (78*Cosh[x]*Sinh[x])/(385*a^2*Sqr
t[a*Sech[x]^3]) + (26*Cosh[x]^3*Sinh[x])/(165*a^2*Sqrt[a*Sech[x]^3]) + (2*Cosh[x]^5*Sinh[x])/(15*a^2*Sqrt[a*Se
ch[x]^3]) + (26*Tanh[x])/(77*a^2*Sqrt[a*Sech[x]^3])

Rule 2720

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

Rule 3854

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 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*} \text {integral}& = \frac {\text {sech}^{\frac {3}{2}}(x) \int \frac {1}{\text {sech}^{\frac {15}{2}}(x)} \, dx}{a^2 \sqrt {a \text {sech}^3(x)}} \\ & = \frac {2 \cosh ^5(x) \sinh (x)}{15 a^2 \sqrt {a \text {sech}^3(x)}}+\frac {\left (13 \text {sech}^{\frac {3}{2}}(x)\right ) \int \frac {1}{\text {sech}^{\frac {11}{2}}(x)} \, dx}{15 a^2 \sqrt {a \text {sech}^3(x)}} \\ & = \frac {26 \cosh ^3(x) \sinh (x)}{165 a^2 \sqrt {a \text {sech}^3(x)}}+\frac {2 \cosh ^5(x) \sinh (x)}{15 a^2 \sqrt {a \text {sech}^3(x)}}+\frac {\left (39 \text {sech}^{\frac {3}{2}}(x)\right ) \int \frac {1}{\text {sech}^{\frac {7}{2}}(x)} \, dx}{55 a^2 \sqrt {a \text {sech}^3(x)}} \\ & = \frac {78 \cosh (x) \sinh (x)}{385 a^2 \sqrt {a \text {sech}^3(x)}}+\frac {26 \cosh ^3(x) \sinh (x)}{165 a^2 \sqrt {a \text {sech}^3(x)}}+\frac {2 \cosh ^5(x) \sinh (x)}{15 a^2 \sqrt {a \text {sech}^3(x)}}+\frac {\left (39 \text {sech}^{\frac {3}{2}}(x)\right ) \int \frac {1}{\text {sech}^{\frac {3}{2}}(x)} \, dx}{77 a^2 \sqrt {a \text {sech}^3(x)}} \\ & = \frac {78 \cosh (x) \sinh (x)}{385 a^2 \sqrt {a \text {sech}^3(x)}}+\frac {26 \cosh ^3(x) \sinh (x)}{165 a^2 \sqrt {a \text {sech}^3(x)}}+\frac {2 \cosh ^5(x) \sinh (x)}{15 a^2 \sqrt {a \text {sech}^3(x)}}+\frac {26 \tanh (x)}{77 a^2 \sqrt {a \text {sech}^3(x)}}+\frac {\left (13 \text {sech}^{\frac {3}{2}}(x)\right ) \int \sqrt {\text {sech}(x)} \, dx}{77 a^2 \sqrt {a \text {sech}^3(x)}} \\ & = \frac {78 \cosh (x) \sinh (x)}{385 a^2 \sqrt {a \text {sech}^3(x)}}+\frac {26 \cosh ^3(x) \sinh (x)}{165 a^2 \sqrt {a \text {sech}^3(x)}}+\frac {2 \cosh ^5(x) \sinh (x)}{15 a^2 \sqrt {a \text {sech}^3(x)}}+\frac {26 \tanh (x)}{77 a^2 \sqrt {a \text {sech}^3(x)}}+\frac {13 \int \frac {1}{\sqrt {\cosh (x)}} \, dx}{77 a^2 \cosh ^{\frac {3}{2}}(x) \sqrt {a \text {sech}^3(x)}} \\ & = -\frac {26 i \operatorname {EllipticF}\left (\frac {i x}{2},2\right )}{77 a^2 \cosh ^{\frac {3}{2}}(x) \sqrt {a \text {sech}^3(x)}}+\frac {78 \cosh (x) \sinh (x)}{385 a^2 \sqrt {a \text {sech}^3(x)}}+\frac {26 \cosh ^3(x) \sinh (x)}{165 a^2 \sqrt {a \text {sech}^3(x)}}+\frac {2 \cosh ^5(x) \sinh (x)}{15 a^2 \sqrt {a \text {sech}^3(x)}}+\frac {26 \tanh (x)}{77 a^2 \sqrt {a \text {sech}^3(x)}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.12 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.52 \[ \int \frac {1}{\left (a \text {sech}^3(x)\right )^{5/2}} \, dx=\frac {\cosh (x) \sqrt {a \text {sech}^3(x)} \left (-24960 i \sqrt {\cosh (x)} \operatorname {EllipticF}\left (\frac {i x}{2},2\right )+19122 \sinh (2 x)+4406 \sinh (4 x)+826 \sinh (6 x)+77 \sinh (8 x)\right )}{73920 a^3} \]

[In]

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

[Out]

(Cosh[x]*Sqrt[a*Sech[x]^3]*((-24960*I)*Sqrt[Cosh[x]]*EllipticF[(I/2)*x, 2] + 19122*Sinh[2*x] + 4406*Sinh[4*x]
+ 826*Sinh[6*x] + 77*Sinh[8*x]))/(73920*a^3)

Maple [F]

\[\int \frac {1}{\left (a \operatorname {sech}\left (x \right )^{3}\right )^{\frac {5}{2}}}d x\]

[In]

int(1/(a*sech(x)^3)^(5/2),x)

[Out]

int(1/(a*sech(x)^3)^(5/2),x)

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.10 (sec) , antiderivative size = 718, normalized size of antiderivative = 5.93 \[ \int \frac {1}{\left (a \text {sech}^3(x)\right )^{5/2}} \, dx=\text {Too large to display} \]

[In]

integrate(1/(a*sech(x)^3)^(5/2),x, algorithm="fricas")

[Out]

1/147840*(49920*sqrt(2)*(cosh(x)^8 + 8*cosh(x)^7*sinh(x) + 28*cosh(x)^6*sinh(x)^2 + 56*cosh(x)^5*sinh(x)^3 + 7
0*cosh(x)^4*sinh(x)^4 + 56*cosh(x)^3*sinh(x)^5 + 28*cosh(x)^2*sinh(x)^6 + 8*cosh(x)*sinh(x)^7 + sinh(x)^8)*sqr
t(a)*weierstrassPInverse(-4, 0, cosh(x) + sinh(x)) + sqrt(2)*(77*cosh(x)^16 + 1232*cosh(x)*sinh(x)^15 + 77*sin
h(x)^16 + 14*(660*cosh(x)^2 + 59)*sinh(x)^14 + 826*cosh(x)^14 + 196*(220*cosh(x)^3 + 59*cosh(x))*sinh(x)^13 +
2*(70070*cosh(x)^4 + 37583*cosh(x)^2 + 2203)*sinh(x)^12 + 4406*cosh(x)^12 + 8*(42042*cosh(x)^5 + 37583*cosh(x)
^3 + 6609*cosh(x))*sinh(x)^11 + 2*(308308*cosh(x)^6 + 413413*cosh(x)^4 + 145398*cosh(x)^2 + 9561)*sinh(x)^10 +
19122*cosh(x)^10 + 4*(220220*cosh(x)^7 + 413413*cosh(x)^5 + 242330*cosh(x)^3 + 47805*cosh(x))*sinh(x)^9 + 6*(
165165*cosh(x)^8 + 413413*cosh(x)^6 + 363495*cosh(x)^4 + 143415*cosh(x)^2)*sinh(x)^8 + 16*(55055*cosh(x)^9 + 1
77177*cosh(x)^7 + 218097*cosh(x)^5 + 143415*cosh(x)^3)*sinh(x)^7 + 2*(308308*cosh(x)^10 + 1240239*cosh(x)^8 +
2035572*cosh(x)^6 + 2007810*cosh(x)^4 - 9561)*sinh(x)^6 - 19122*cosh(x)^6 + 4*(84084*cosh(x)^11 + 413413*cosh(
x)^9 + 872388*cosh(x)^7 + 1204686*cosh(x)^5 - 28683*cosh(x))*sinh(x)^5 + 2*(70070*cosh(x)^12 + 413413*cosh(x)^
10 + 1090485*cosh(x)^8 + 2007810*cosh(x)^6 - 143415*cosh(x)^2 - 2203)*sinh(x)^4 - 4406*cosh(x)^4 + 8*(5390*cos
h(x)^13 + 37583*cosh(x)^11 + 121165*cosh(x)^9 + 286830*cosh(x)^7 - 47805*cosh(x)^3 - 2203*cosh(x))*sinh(x)^3 +
2*(4620*cosh(x)^14 + 37583*cosh(x)^12 + 145398*cosh(x)^10 + 430245*cosh(x)^8 - 143415*cosh(x)^4 - 13218*cosh(
x)^2 - 413)*sinh(x)^2 - 826*cosh(x)^2 + 4*(308*cosh(x)^15 + 2891*cosh(x)^13 + 13218*cosh(x)^11 + 47805*cosh(x)
^9 - 28683*cosh(x)^5 - 4406*cosh(x)^3 - 413*cosh(x))*sinh(x) - 77)*sqrt((a*cosh(x) + a*sinh(x))/(cosh(x)^2 + 2
*cosh(x)*sinh(x) + sinh(x)^2 + 1)))/(a^3*cosh(x)^8 + 8*a^3*cosh(x)^7*sinh(x) + 28*a^3*cosh(x)^6*sinh(x)^2 + 56
*a^3*cosh(x)^5*sinh(x)^3 + 70*a^3*cosh(x)^4*sinh(x)^4 + 56*a^3*cosh(x)^3*sinh(x)^5 + 28*a^3*cosh(x)^2*sinh(x)^
6 + 8*a^3*cosh(x)*sinh(x)^7 + a^3*sinh(x)^8)

Sympy [F]

\[ \int \frac {1}{\left (a \text {sech}^3(x)\right )^{5/2}} \, dx=\int \frac {1}{\left (a \operatorname {sech}^{3}{\left (x \right )}\right )^{\frac {5}{2}}}\, dx \]

[In]

integrate(1/(a*sech(x)**3)**(5/2),x)

[Out]

Integral((a*sech(x)**3)**(-5/2), x)

Maxima [F]

\[ \int \frac {1}{\left (a \text {sech}^3(x)\right )^{5/2}} \, dx=\int { \frac {1}{\left (a \operatorname {sech}\left (x\right )^{3}\right )^{\frac {5}{2}}} \,d x } \]

[In]

integrate(1/(a*sech(x)^3)^(5/2),x, algorithm="maxima")

[Out]

integrate((a*sech(x)^3)^(-5/2), x)

Giac [F]

\[ \int \frac {1}{\left (a \text {sech}^3(x)\right )^{5/2}} \, dx=\int { \frac {1}{\left (a \operatorname {sech}\left (x\right )^{3}\right )^{\frac {5}{2}}} \,d x } \]

[In]

integrate(1/(a*sech(x)^3)^(5/2),x, algorithm="giac")

[Out]

integrate((a*sech(x)^3)^(-5/2), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\left (a \text {sech}^3(x)\right )^{5/2}} \, dx=\int \frac {1}{{\left (\frac {a}{{\mathrm {cosh}\left (x\right )}^3}\right )}^{5/2}} \,d x \]

[In]

int(1/(a/cosh(x)^3)^(5/2),x)

[Out]

int(1/(a/cosh(x)^3)^(5/2), x)