\(\int \frac {1}{(a \cosh (x))^{3/2}} \, dx\) [20]

Optimal result

Integrand size = 8, antiderivative size = 46 \[ \int \frac {1}{(a \cosh (x))^{3/2}} \, dx=\frac {2 i \sqrt {a \cosh (x)} E\left (\left .\frac {i x}{2}\right |2\right )}{a^2 \sqrt {\cosh (x)}}+\frac {2 \sinh (x)}{a \sqrt {a \cosh (x)}} \]

[Out]

2*sinh(x)/a/(a*cosh(x))^(1/2)+2*I*(cosh(1/2*x)^2)^(1/2)/cosh(1/2*x)*EllipticE(I*sinh(1/2*x),2^(1/2))*(a*cosh(x
))^(1/2)/a^2/cosh(x)^(1/2)

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {2716, 2721, 2719} \[ \int \frac {1}{(a \cosh (x))^{3/2}} \, dx=\frac {2 \sinh (x)}{a \sqrt {a \cosh (x)}}+\frac {2 i E\left (\left .\frac {i x}{2}\right |2\right ) \sqrt {a \cosh (x)}}{a^2 \sqrt {\cosh (x)}} \]

[In]

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

[Out]

((2*I)*Sqrt[a*Cosh[x]]*EllipticE[(I/2)*x, 2])/(a^2*Sqrt[Cosh[x]]) + (2*Sinh[x])/(a*Sqrt[a*Cosh[x]])

Rule 2716

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[Cos[c + d*x]*((b*Sin[c + d*x])^(n + 1)/(b*d*(n + 1
))), x] + Dist[(n + 2)/(b^2*(n + 1)), Int[(b*Sin[c + d*x])^(n + 2), x], x] /; FreeQ[{b, c, d}, x] && LtQ[n, -1
] && IntegerQ[2*n]

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 2721

Int[((b_)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Dist[(b*Sin[c + d*x])^n/Sin[c + d*x]^n, Int[Sin[c + d*x]
^n, x], x] /; FreeQ[{b, c, d}, x] && LtQ[-1, n, 1] && IntegerQ[2*n]

Rubi steps \begin{align*} \text {integral}& = \frac {2 \sinh (x)}{a \sqrt {a \cosh (x)}}-\frac {\int \sqrt {a \cosh (x)} \, dx}{a^2} \\ & = \frac {2 \sinh (x)}{a \sqrt {a \cosh (x)}}-\frac {\sqrt {a \cosh (x)} \int \sqrt {\cosh (x)} \, dx}{a^2 \sqrt {\cosh (x)}} \\ & = \frac {2 i \sqrt {a \cosh (x)} E\left (\left .\frac {i x}{2}\right |2\right )}{a^2 \sqrt {\cosh (x)}}+\frac {2 \sinh (x)}{a \sqrt {a \cosh (x)}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.74 \[ \int \frac {1}{(a \cosh (x))^{3/2}} \, dx=\frac {2 \cosh (x) \left (i \sqrt {\cosh (x)} E\left (\left .\frac {i x}{2}\right |2\right )+\sinh (x)\right )}{(a \cosh (x))^{3/2}} \]

[In]

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

[Out]

(2*Cosh[x]*(I*Sqrt[Cosh[x]]*EllipticE[(I/2)*x, 2] + Sinh[x]))/(a*Cosh[x])^(3/2)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(158\) vs. \(2(55)=110\).

Time = 0.34 (sec) , antiderivative size = 159, normalized size of antiderivative = 3.46

method	result	size
default	\(\frac {\sqrt {2 \sinh \left (\frac {x}{2}\right )^{4} a +\sinh \left (\frac {x}{2}\right )^{2} a}\, \left (-\sqrt {2}\, \sqrt {-2 \sinh \left (\frac {x}{2}\right )^{2}-1}\, \sqrt {-\sinh \left (\frac {x}{2}\right )^{2}}\, \operatorname {EllipticF}\left (\sqrt {2}\, \cosh \left (\frac {x}{2}\right ), \frac {\sqrt {2}}{2}\right )+2 \sqrt {2}\, \sqrt {-2 \sinh \left (\frac {x}{2}\right )^{2}-1}\, \sqrt {-\sinh \left (\frac {x}{2}\right )^{2}}\, \operatorname {EllipticE}\left (\sqrt {2}\, \cosh \left (\frac {x}{2}\right ), \frac {\sqrt {2}}{2}\right )+4 \sinh \left (\frac {x}{2}\right )^{2} \cosh \left (\frac {x}{2}\right )\right )}{a \sqrt {a \left (2 \sinh \left (\frac {x}{2}\right )^{4}+\sinh \left (\frac {x}{2}\right )^{2}\right )}\, \sinh \left (\frac {x}{2}\right ) \sqrt {a \left (2 \cosh \left (\frac {x}{2}\right )^{2}-1\right )}}\)	\(159\)

[In]

int(1/(a*cosh(x))^(3/2),x,method=_RETURNVERBOSE)

[Out]

1/a*(2*sinh(1/2*x)^4*a+sinh(1/2*x)^2*a)^(1/2)*(-2^(1/2)*(-2*sinh(1/2*x)^2-1)^(1/2)*(-sinh(1/2*x)^2)^(1/2)*Elli
pticF(2^(1/2)*cosh(1/2*x),1/2*2^(1/2))+2*2^(1/2)*(-2*sinh(1/2*x)^2-1)^(1/2)*(-sinh(1/2*x)^2)^(1/2)*EllipticE(2
^(1/2)*cosh(1/2*x),1/2*2^(1/2))+4*sinh(1/2*x)^2*cosh(1/2*x))/(a*(2*sinh(1/2*x)^4+sinh(1/2*x)^2))^(1/2)/sinh(1/
2*x)/(a*(2*cosh(1/2*x)^2-1))^(1/2)

Fricas [C] (verification not implemented)

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

Time = 0.08 (sec) , antiderivative size = 101, normalized size of antiderivative = 2.20 \[ \int \frac {1}{(a \cosh (x))^{3/2}} \, dx=\frac {2 \, {\left ({\left (\sqrt {2} \cosh \left (x\right )^{2} + 2 \, \sqrt {2} \cosh \left (x\right ) \sinh \left (x\right ) + \sqrt {2} \sinh \left (x\right )^{2} + \sqrt {2}\right )} \sqrt {a} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cosh \left (x\right ) + \sinh \left (x\right )\right )\right ) + 2 \, \sqrt {a \cosh \left (x\right )} {\left (\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2}\right )}\right )}}{a^{2} \cosh \left (x\right )^{2} + 2 \, a^{2} \cosh \left (x\right ) \sinh \left (x\right ) + a^{2} \sinh \left (x\right )^{2} + a^{2}} \]

[In]

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

[Out]

2*((sqrt(2)*cosh(x)^2 + 2*sqrt(2)*cosh(x)*sinh(x) + sqrt(2)*sinh(x)^2 + sqrt(2))*sqrt(a)*weierstrassZeta(-4, 0
, weierstrassPInverse(-4, 0, cosh(x) + sinh(x))) + 2*sqrt(a*cosh(x))*(cosh(x)^2 + 2*cosh(x)*sinh(x) + sinh(x)^
2))/(a^2*cosh(x)^2 + 2*a^2*cosh(x)*sinh(x) + a^2*sinh(x)^2 + a^2)

Sympy [F]

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

[In]

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

[Out]

Integral((a*cosh(x))**(-3/2), x)

Maxima [F]

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

[In]

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

[Out]

integrate((a*cosh(x))^(-3/2), x)

Giac [F]

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

[In]

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

[Out]

integrate((a*cosh(x))^(-3/2), x)

Mupad [F(-1)]

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

[In]

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

[Out]

int(1/(a*cosh(x))^(3/2), x)