\(\int \frac {1}{(b \text {sech}(c+d x))^{3/2}} \, dx\) [20]

Optimal result

Integrand size = 12, antiderivative size = 76 \[ \int \frac {1}{(b \text {sech}(c+d x))^{3/2}} \, dx=-\frac {2 i \sqrt {\cosh (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} i (c+d x),2\right ) \sqrt {b \text {sech}(c+d x)}}{3 b^2 d}+\frac {2 \sinh (c+d x)}{3 b d \sqrt {b \text {sech}(c+d x)}} \]

[Out]

2/3*sinh(d*x+c)/b/d/(b*sech(d*x+c))^(1/2)-2/3*I*(cosh(1/2*d*x+1/2*c)^2)^(1/2)/cosh(1/2*d*x+1/2*c)*EllipticF(I*
sinh(1/2*d*x+1/2*c),2^(1/2))*cosh(d*x+c)^(1/2)*(b*sech(d*x+c))^(1/2)/b^2/d

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3854, 3856, 2720} \[ \int \frac {1}{(b \text {sech}(c+d x))^{3/2}} \, dx=\frac {2 \sinh (c+d x)}{3 b d \sqrt {b \text {sech}(c+d x)}}-\frac {2 i \sqrt {\cosh (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} i (c+d x),2\right ) \sqrt {b \text {sech}(c+d x)}}{3 b^2 d} \]

[In]

Int[(b*Sech[c + d*x])^(-3/2),x]

[Out]

(((-2*I)/3)*Sqrt[Cosh[c + d*x]]*EllipticF[(I/2)*(c + d*x), 2]*Sqrt[b*Sech[c + d*x]])/(b^2*d) + (2*Sinh[c + d*x
])/(3*b*d*Sqrt[b*Sech[c + d*x]])

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]

Rubi steps \begin{align*} \text {integral}& = \frac {2 \sinh (c+d x)}{3 b d \sqrt {b \text {sech}(c+d x)}}+\frac {\int \sqrt {b \text {sech}(c+d x)} \, dx}{3 b^2} \\ & = \frac {2 \sinh (c+d x)}{3 b d \sqrt {b \text {sech}(c+d x)}}+\frac {\left (\sqrt {\cosh (c+d x)} \sqrt {b \text {sech}(c+d x)}\right ) \int \frac {1}{\sqrt {\cosh (c+d x)}} \, dx}{3 b^2} \\ & = -\frac {2 i \sqrt {\cosh (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} i (c+d x),2\right ) \sqrt {b \text {sech}(c+d x)}}{3 b^2 d}+\frac {2 \sinh (c+d x)}{3 b d \sqrt {b \text {sech}(c+d x)}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.83 \[ \int \frac {1}{(b \text {sech}(c+d x))^{3/2}} \, dx=\frac {\text {sech}^2(c+d x) \left (-2 i \sqrt {\cosh (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} i (c+d x),2\right )+\sinh (2 (c+d x))\right )}{3 d (b \text {sech}(c+d x))^{3/2}} \]

[In]

Integrate[(b*Sech[c + d*x])^(-3/2),x]

[Out]

(Sech[c + d*x]^2*((-2*I)*Sqrt[Cosh[c + d*x]]*EllipticF[(I/2)*(c + d*x), 2] + Sinh[2*(c + d*x)]))/(3*d*(b*Sech[
c + d*x])^(3/2))

Maple [F]

\[\int \frac {1}{\left (b \,\operatorname {sech}\left (d x +c \right )\right )^{\frac {3}{2}}}d x\]

[In]

int(1/(b*sech(d*x+c))^(3/2),x)

[Out]

int(1/(b*sech(d*x+c))^(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 = 231, normalized size of antiderivative = 3.04 \[ \int \frac {1}{(b \text {sech}(c+d x))^{3/2}} \, dx=\frac {4 \, \sqrt {2} {\left (\cosh \left (d x + c\right )^{2} + 2 \, \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + \sinh \left (d x + c\right )^{2}\right )} \sqrt {b} {\rm weierstrassPInverse}\left (-4, 0, \cosh \left (d x + c\right ) + \sinh \left (d x + c\right )\right ) + \sqrt {2} {\left (\cosh \left (d x + c\right )^{4} + 4 \, \cosh \left (d x + c\right )^{3} \sinh \left (d x + c\right ) + 6 \, \cosh \left (d x + c\right )^{2} \sinh \left (d x + c\right )^{2} + 4 \, \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + \sinh \left (d x + c\right )^{4} - 1\right )} \sqrt {\frac {b \cosh \left (d x + c\right ) + b \sinh \left (d x + c\right )}{\cosh \left (d x + c\right )^{2} + 2 \, \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + \sinh \left (d x + c\right )^{2} + 1}}}{6 \, {\left (b^{2} d \cosh \left (d x + c\right )^{2} + 2 \, b^{2} d \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + b^{2} d \sinh \left (d x + c\right )^{2}\right )}} \]

[In]

integrate(1/(b*sech(d*x+c))^(3/2),x, algorithm="fricas")

[Out]

1/6*(4*sqrt(2)*(cosh(d*x + c)^2 + 2*cosh(d*x + c)*sinh(d*x + c) + sinh(d*x + c)^2)*sqrt(b)*weierstrassPInverse
(-4, 0, cosh(d*x + c) + sinh(d*x + c)) + sqrt(2)*(cosh(d*x + c)^4 + 4*cosh(d*x + c)^3*sinh(d*x + c) + 6*cosh(d
*x + c)^2*sinh(d*x + c)^2 + 4*cosh(d*x + c)*sinh(d*x + c)^3 + sinh(d*x + c)^4 - 1)*sqrt((b*cosh(d*x + c) + b*s
inh(d*x + c))/(cosh(d*x + c)^2 + 2*cosh(d*x + c)*sinh(d*x + c) + sinh(d*x + c)^2 + 1)))/(b^2*d*cosh(d*x + c)^2
+ 2*b^2*d*cosh(d*x + c)*sinh(d*x + c) + b^2*d*sinh(d*x + c)^2)

Sympy [F]

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

[In]

integrate(1/(b*sech(d*x+c))**(3/2),x)

[Out]

Integral((b*sech(c + d*x))**(-3/2), x)

Maxima [F]

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

[In]

integrate(1/(b*sech(d*x+c))^(3/2),x, algorithm="maxima")

[Out]

integrate((b*sech(d*x + c))^(-3/2), x)

Giac [F]

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

[In]

integrate(1/(b*sech(d*x+c))^(3/2),x, algorithm="giac")

[Out]

integrate((b*sech(d*x + c))^(-3/2), x)

Mupad [F(-1)]

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

[In]

int(1/(b/cosh(c + d*x))^(3/2),x)

[Out]

int(1/(b/cosh(c + d*x))^(3/2), x)