\(\int \frac {1}{(a \text {sech}^3(x))^{3/2}} \, dx\) [43]

Optimal result

Integrand size = 10, antiderivative size = 77 \[ \int \frac {1}{\left (a \text {sech}^3(x)\right )^{3/2}} \, dx=-\frac {14 i E\left (\left .\frac {i x}{2}\right |2\right )}{15 a \cosh ^{\frac {3}{2}}(x) \sqrt {a \text {sech}^3(x)}}+\frac {14 \sinh (x)}{45 a \sqrt {a \text {sech}^3(x)}}+\frac {2 \cosh ^2(x) \sinh (x)}{9 a \sqrt {a \text {sech}^3(x)}} \]

[Out]

-14/15*I*(cosh(1/2*x)^2)^(1/2)/cosh(1/2*x)*EllipticE(I*sinh(1/2*x),2^(1/2))/a/cosh(x)^(3/2)/(a*sech(x)^3)^(1/2
)+14/45*sinh(x)/a/(a*sech(x)^3)^(1/2)+2/9*cosh(x)^2*sinh(x)/a/(a*sech(x)^3)^(1/2)

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {4208, 3854, 3856, 2719} \[ \int \frac {1}{\left (a \text {sech}^3(x)\right )^{3/2}} \, dx=\frac {14 \sinh (x)}{45 a \sqrt {a \text {sech}^3(x)}}-\frac {14 i E\left (\left .\frac {i x}{2}\right |2\right )}{15 a \cosh ^{\frac {3}{2}}(x) \sqrt {a \text {sech}^3(x)}}+\frac {2 \sinh (x) \cosh ^2(x)}{9 a \sqrt {a \text {sech}^3(x)}} \]

[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 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 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 {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] (verified)

Time = 0.12 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.61 \[ \int \frac {1}{\left (a \text {sech}^3(x)\right )^{3/2}} \, dx=\frac {-\frac {84 i E\left (\left .\frac {i x}{2}\right |2\right )}{\cosh ^{\frac {3}{2}}(x)}+33 \sinh (x)+5 \sinh (3 x)}{90 a \sqrt {a \text {sech}^3(x)}} \]

[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])

Maple [F]

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

[In]

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

[Out]

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

Fricas [C] (verification not implemented)

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

Time = 0.09 (sec) , antiderivative size = 407, normalized size of antiderivative = 5.29 \[ \int \frac {1}{\left (a \text {sech}^3(x)\right )^{3/2}} \, dx=-\frac {672 \, \sqrt {2} {\left (\cosh \left (x\right )^{5} + 5 \, \cosh \left (x\right )^{4} \sinh \left (x\right ) + 10 \, \cosh \left (x\right )^{3} \sinh \left (x\right )^{2} + 10 \, \cosh \left (x\right )^{2} \sinh \left (x\right )^{3} + 5 \, \cosh \left (x\right ) \sinh \left (x\right )^{4} + \sinh \left (x\right )^{5}\right )} \sqrt {a} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cosh \left (x\right ) + \sinh \left (x\right )\right )\right ) - \sqrt {2} {\left (5 \, \cosh \left (x\right )^{10} + 50 \, \cosh \left (x\right ) \sinh \left (x\right )^{9} + 5 \, \sinh \left (x\right )^{10} + {\left (225 \, \cosh \left (x\right )^{2} + 43\right )} \sinh \left (x\right )^{8} + 43 \, \cosh \left (x\right )^{8} + 8 \, {\left (75 \, \cosh \left (x\right )^{3} + 43 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{7} + 2 \, {\left (525 \, \cosh \left (x\right )^{4} + 602 \, \cosh \left (x\right )^{2} - 149\right )} \sinh \left (x\right )^{6} - 298 \, \cosh \left (x\right )^{6} + 4 \, {\left (315 \, \cosh \left (x\right )^{5} + 602 \, \cosh \left (x\right )^{3} - 447 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{5} + 2 \, {\left (525 \, \cosh \left (x\right )^{6} + 1505 \, \cosh \left (x\right )^{4} - 2235 \, \cosh \left (x\right )^{2} - 187\right )} \sinh \left (x\right )^{4} - 374 \, \cosh \left (x\right )^{4} + 8 \, {\left (75 \, \cosh \left (x\right )^{7} + 301 \, \cosh \left (x\right )^{5} - 745 \, \cosh \left (x\right )^{3} - 187 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{3} + {\left (225 \, \cosh \left (x\right )^{8} + 1204 \, \cosh \left (x\right )^{6} - 4470 \, \cosh \left (x\right )^{4} - 2244 \, \cosh \left (x\right )^{2} - 43\right )} \sinh \left (x\right )^{2} - 43 \, \cosh \left (x\right )^{2} + 2 \, {\left (25 \, \cosh \left (x\right )^{9} + 172 \, \cosh \left (x\right )^{7} - 894 \, \cosh \left (x\right )^{5} - 748 \, \cosh \left (x\right )^{3} - 43 \, \cosh \left (x\right )\right )} \sinh \left (x\right ) - 5\right )} \sqrt {\frac {a \cosh \left (x\right ) + a \sinh \left (x\right )}{\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2} + 1}}}{720 \, {\left (a^{2} \cosh \left (x\right )^{5} + 5 \, a^{2} \cosh \left (x\right )^{4} \sinh \left (x\right ) + 10 \, a^{2} \cosh \left (x\right )^{3} \sinh \left (x\right )^{2} + 10 \, a^{2} \cosh \left (x\right )^{2} \sinh \left (x\right )^{3} + 5 \, a^{2} \cosh \left (x\right ) \sinh \left (x\right )^{4} + a^{2} \sinh \left (x\right )^{5}\right )}} \]

[In]

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

[Out]

-1/720*(672*sqrt(2)*(cosh(x)^5 + 5*cosh(x)^4*sinh(x) + 10*cosh(x)^3*sinh(x)^2 + 10*cosh(x)^2*sinh(x)^3 + 5*cos
h(x)*sinh(x)^4 + sinh(x)^5)*sqrt(a)*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cosh(x) + sinh(x))) - sq
rt(2)*(5*cosh(x)^10 + 50*cosh(x)*sinh(x)^9 + 5*sinh(x)^10 + (225*cosh(x)^2 + 43)*sinh(x)^8 + 43*cosh(x)^8 + 8*
(75*cosh(x)^3 + 43*cosh(x))*sinh(x)^7 + 2*(525*cosh(x)^4 + 602*cosh(x)^2 - 149)*sinh(x)^6 - 298*cosh(x)^6 + 4*
(315*cosh(x)^5 + 602*cosh(x)^3 - 447*cosh(x))*sinh(x)^5 + 2*(525*cosh(x)^6 + 1505*cosh(x)^4 - 2235*cosh(x)^2 -
187)*sinh(x)^4 - 374*cosh(x)^4 + 8*(75*cosh(x)^7 + 301*cosh(x)^5 - 745*cosh(x)^3 - 187*cosh(x))*sinh(x)^3 + (
225*cosh(x)^8 + 1204*cosh(x)^6 - 4470*cosh(x)^4 - 2244*cosh(x)^2 - 43)*sinh(x)^2 - 43*cosh(x)^2 + 2*(25*cosh(x
)^9 + 172*cosh(x)^7 - 894*cosh(x)^5 - 748*cosh(x)^3 - 43*cosh(x))*sinh(x) - 5)*sqrt((a*cosh(x) + a*sinh(x))/(c
osh(x)^2 + 2*cosh(x)*sinh(x) + sinh(x)^2 + 1)))/(a^2*cosh(x)^5 + 5*a^2*cosh(x)^4*sinh(x) + 10*a^2*cosh(x)^3*si
nh(x)^2 + 10*a^2*cosh(x)^2*sinh(x)^3 + 5*a^2*cosh(x)*sinh(x)^4 + a^2*sinh(x)^5)

Sympy [F]

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

[In]

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

[Out]

Integral((a*sech(x)**3)**(-3/2), x)

Maxima [F]

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

[In]

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

[Out]

integrate((a*sech(x)^3)^(-3/2), x)

Giac [F]

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

[In]

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

[Out]

integrate((a*sech(x)^3)^(-3/2), x)

Mupad [F(-1)]

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

[In]

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

[Out]

int(1/(a/cosh(x)^3)^(3/2), x)