\(\int \frac {1}{\text {sech}^{\frac {3}{2}}(a+b x)} \, dx\) [13]

Optimal result

Integrand size = 10, antiderivative size = 66 \[ \int \frac {1}{\text {sech}^{\frac {3}{2}}(a+b x)} \, dx=-\frac {2 i \sqrt {\cosh (a+b x)} \operatorname {EllipticF}\left (\frac {1}{2} i (a+b x),2\right ) \sqrt {\text {sech}(a+b x)}}{3 b}+\frac {2 \sinh (a+b x)}{3 b \sqrt {\text {sech}(a+b x)}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {3854, 3856, 2720} \[ \int \frac {1}{\text {sech}^{\frac {3}{2}}(a+b x)} \, dx=\frac {2 \sinh (a+b x)}{3 b \sqrt {\text {sech}(a+b x)}}-\frac {2 i \sqrt {\cosh (a+b x)} \sqrt {\text {sech}(a+b x)} \operatorname {EllipticF}\left (\frac {1}{2} i (a+b x),2\right )}{3 b} \]

[In]

Int[Sech[a + b*x]^(-3/2),x]

[Out]

(((-2*I)/3)*Sqrt[Cosh[a + b*x]]*EllipticF[(I/2)*(a + b*x), 2]*Sqrt[Sech[a + b*x]])/b + (2*Sinh[a + b*x])/(3*b*
Sqrt[Sech[a + b*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 (a+b x)}{3 b \sqrt {\text {sech}(a+b x)}}+\frac {1}{3} \int \sqrt {\text {sech}(a+b x)} \, dx \\ & = \frac {2 \sinh (a+b x)}{3 b \sqrt {\text {sech}(a+b x)}}+\frac {1}{3} \left (\sqrt {\cosh (a+b x)} \sqrt {\text {sech}(a+b x)}\right ) \int \frac {1}{\sqrt {\cosh (a+b x)}} \, dx \\ & = -\frac {2 i \sqrt {\cosh (a+b x)} \operatorname {EllipticF}\left (\frac {1}{2} i (a+b x),2\right ) \sqrt {\text {sech}(a+b x)}}{3 b}+\frac {2 \sinh (a+b x)}{3 b \sqrt {\text {sech}(a+b x)}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.80 \[ \int \frac {1}{\text {sech}^{\frac {3}{2}}(a+b x)} \, dx=\frac {\sqrt {\text {sech}(a+b x)} \left (-2 i \sqrt {\cosh (a+b x)} \operatorname {EllipticF}\left (\frac {1}{2} i (a+b x),2\right )+\sinh (2 (a+b x))\right )}{3 b} \]

[In]

Integrate[Sech[a + b*x]^(-3/2),x]

[Out]

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(173\) vs. \(2(82)=164\).

Time = 1.23 (sec) , antiderivative size = 174, normalized size of antiderivative = 2.64

method	result	size
default	\(\frac {2 \sqrt {\left (-1+2 \cosh \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}\right ) \sinh \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}}\, \left (4 \cosh \left (\frac {b x}{2}+\frac {a}{2}\right )^{5}-6 \cosh \left (\frac {b x}{2}+\frac {a}{2}\right )^{3}+\sqrt {-\sinh \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}}\, \sqrt {-2 \cosh \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}+1}\, \operatorname {EllipticF}\left (\cosh \left (\frac {b x}{2}+\frac {a}{2}\right ), \sqrt {2}\right )+2 \cosh \left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{3 \sqrt {2 \sinh \left (\frac {b x}{2}+\frac {a}{2}\right )^{4}+\sinh \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}}\, \sinh \left (\frac {b x}{2}+\frac {a}{2}\right ) \sqrt {-1+2 \cosh \left (\frac {b x}{2}+\frac {a}{2}\right )^{2}}\, b}\)	\(174\)

[In]

int(1/sech(b*x+a)^(3/2),x,method=_RETURNVERBOSE)

[Out]

2/3*((-1+2*cosh(1/2*b*x+1/2*a)^2)*sinh(1/2*b*x+1/2*a)^2)^(1/2)*(4*cosh(1/2*b*x+1/2*a)^5-6*cosh(1/2*b*x+1/2*a)^
3+(-sinh(1/2*b*x+1/2*a)^2)^(1/2)*(-2*cosh(1/2*b*x+1/2*a)^2+1)^(1/2)*EllipticF(cosh(1/2*b*x+1/2*a),2^(1/2))+2*c
osh(1/2*b*x+1/2*a))/(2*sinh(1/2*b*x+1/2*a)^4+sinh(1/2*b*x+1/2*a)^2)^(1/2)/sinh(1/2*b*x+1/2*a)/(-1+2*cosh(1/2*b
*x+1/2*a)^2)^(1/2)/b

Fricas [C] (verification not implemented)

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

Time = 0.08 (sec) , antiderivative size = 223, normalized size of antiderivative = 3.38 \[ \int \frac {1}{\text {sech}^{\frac {3}{2}}(a+b x)} \, dx=\frac {\sqrt {2} {\left (\cosh \left (b x + a\right )^{4} + 4 \, \cosh \left (b x + a\right )^{3} \sinh \left (b x + a\right ) + 6 \, \cosh \left (b x + a\right )^{2} \sinh \left (b x + a\right )^{2} + 4 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} + \sinh \left (b x + a\right )^{4} - 1\right )} \sqrt {\frac {\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )}{\cosh \left (b x + a\right )^{2} + 2 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + \sinh \left (b x + a\right )^{2} + 1}} + 4 \, {\left (\sqrt {2} \cosh \left (b x + a\right )^{2} + 2 \, \sqrt {2} \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + \sqrt {2} \sinh \left (b x + a\right )^{2}\right )} {\rm weierstrassPInverse}\left (-4, 0, \cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )}{6 \, {\left (b \cosh \left (b x + a\right )^{2} + 2 \, b \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + b \sinh \left (b x + a\right )^{2}\right )}} \]

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

Integral(sech(a + b*x)**(-3/2), x)

Maxima [F]

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

[In]

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

[Out]

integrate(sech(b*x + a)^(-3/2), x)

Giac [F]

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

[In]

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

[Out]

integrate(sech(b*x + a)^(-3/2), x)

Mupad [F(-1)]

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

[In]

int(1/(1/cosh(a + b*x))^(3/2),x)

[Out]

int(1/(1/cosh(a + b*x))^(3/2), x)