\(\int \frac {1}{(a \cosh (x))^{5/2}} \, dx\) [21]

Optimal result

Integrand size = 8, antiderivative size = 50 \[ \int \frac {1}{(a \cosh (x))^{5/2}} \, dx=-\frac {2 i \sqrt {\cosh (x)} \operatorname {EllipticF}\left (\frac {i x}{2},2\right )}{3 a^2 \sqrt {a \cosh (x)}}+\frac {2 \sinh (x)}{3 a (a \cosh (x))^{3/2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 50, 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, 2720} \[ \int \frac {1}{(a \cosh (x))^{5/2}} \, dx=\frac {2 \sinh (x)}{3 a (a \cosh (x))^{3/2}}-\frac {2 i \sqrt {\cosh (x)} \operatorname {EllipticF}\left (\frac {i x}{2},2\right )}{3 a^2 \sqrt {a \cosh (x)}} \]

[In]

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

[Out]

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

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 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 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)}{3 a (a \cosh (x))^{3/2}}+\frac {\int \frac {1}{\sqrt {a \cosh (x)}} \, dx}{3 a^2} \\ & = \frac {2 \sinh (x)}{3 a (a \cosh (x))^{3/2}}+\frac {\sqrt {\cosh (x)} \int \frac {1}{\sqrt {\cosh (x)}} \, dx}{3 a^2 \sqrt {a \cosh (x)}} \\ & = -\frac {2 i \sqrt {\cosh (x)} \operatorname {EllipticF}\left (\frac {i x}{2},2\right )}{3 a^2 \sqrt {a \cosh (x)}}+\frac {2 \sinh (x)}{3 a (a \cosh (x))^{3/2}} \\ \end{align*}

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.

Time = 0.04 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.12 \[ \int \frac {1}{(a \cosh (x))^{5/2}} \, dx=\frac {2 \left (\operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},-\cosh (2 x)-\sinh (2 x)\right ) \sqrt {1+\cosh (2 x)+\sinh (2 x)}+\tanh (x)\right )}{3 a^2 \sqrt {a \cosh (x)}} \]

[In]

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

[Out]

(2*(Hypergeometric2F1[1/4, 1/2, 5/4, -Cosh[2*x] - Sinh[2*x]]*Sqrt[1 + Cosh[2*x] + Sinh[2*x]] + Tanh[x]))/(3*a^
2*Sqrt[a*Cosh[x]])

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(176\) vs. \(2(55)=110\).

Time = 0.42 (sec) , antiderivative size = 177, normalized size of antiderivative = 3.54

method	result	size
default	\(\frac {\left (2 \sqrt {-2 \sinh \left (\frac {x}{2}\right )^{2}-1}\, \sqrt {-\sinh \left (\frac {x}{2}\right )^{2}}\, \operatorname {EllipticF}\left (\sqrt {2}\, \cosh \left (\frac {x}{2}\right ), \frac {\sqrt {2}}{2}\right ) \sqrt {2}\, \sinh \left (\frac {x}{2}\right )^{2}+\sqrt {2}\, \sqrt {-2 \sinh \left (\frac {x}{2}\right )^{2}-1}\, \sqrt {-\sinh \left (\frac {x}{2}\right )^{2}}\, \operatorname {EllipticF}\left (\sqrt {2}\, \cosh \left (\frac {x}{2}\right ), \frac {\sqrt {2}}{2}\right )+4 \sinh \left (\frac {x}{2}\right )^{2} \cosh \left (\frac {x}{2}\right )\right ) \sqrt {a \left (2 \cosh \left (\frac {x}{2}\right )^{2}-1\right ) \sinh \left (\frac {x}{2}\right )^{2}}}{3 a^{2} \sqrt {a \left (2 \sinh \left (\frac {x}{2}\right )^{4}+\sinh \left (\frac {x}{2}\right )^{2}\right )}\, \left (2 \cosh \left (\frac {x}{2}\right )^{2}-1\right ) \sinh \left (\frac {x}{2}\right ) \sqrt {a \left (2 \cosh \left (\frac {x}{2}\right )^{2}-1\right )}}\)	\(177\)

[In]

int(1/(a*cosh(x))^(5/2),x,method=_RETURNVERBOSE)

[Out]

1/3*(2*(-2*sinh(1/2*x)^2-1)^(1/2)*(-sinh(1/2*x)^2)^(1/2)*EllipticF(2^(1/2)*cosh(1/2*x),1/2*2^(1/2))*2^(1/2)*si
nh(1/2*x)^2+2^(1/2)*(-2*sinh(1/2*x)^2-1)^(1/2)*(-sinh(1/2*x)^2)^(1/2)*EllipticF(2^(1/2)*cosh(1/2*x),1/2*2^(1/2
))+4*sinh(1/2*x)^2*cosh(1/2*x))/a^2*(a*(2*cosh(1/2*x)^2-1)*sinh(1/2*x)^2)^(1/2)/(a*(2*sinh(1/2*x)^4+sinh(1/2*x
)^2))^(1/2)/(2*cosh(1/2*x)^2-1)/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 = 213, normalized size of antiderivative = 4.26 \[ \int \frac {1}{(a \cosh (x))^{5/2}} \, dx=\frac {2 \, {\left ({\left (\sqrt {2} \cosh \left (x\right )^{4} + 4 \, \sqrt {2} \cosh \left (x\right ) \sinh \left (x\right )^{3} + \sqrt {2} \sinh \left (x\right )^{4} + 2 \, {\left (3 \, \sqrt {2} \cosh \left (x\right )^{2} + \sqrt {2}\right )} \sinh \left (x\right )^{2} + 2 \, \sqrt {2} \cosh \left (x\right )^{2} + 4 \, {\left (\sqrt {2} \cosh \left (x\right )^{3} + \sqrt {2} \cosh \left (x\right )\right )} \sinh \left (x\right ) + \sqrt {2}\right )} \sqrt {a} {\rm weierstrassPInverse}\left (-4, 0, \cosh \left (x\right ) + \sinh \left (x\right )\right ) + 2 \, {\left (\cosh \left (x\right )^{3} + 3 \, \cosh \left (x\right ) \sinh \left (x\right )^{2} + \sinh \left (x\right )^{3} + {\left (3 \, \cosh \left (x\right )^{2} - 1\right )} \sinh \left (x\right ) - \cosh \left (x\right )\right )} \sqrt {a \cosh \left (x\right )}\right )}}{3 \, {\left (a^{3} \cosh \left (x\right )^{4} + 4 \, a^{3} \cosh \left (x\right ) \sinh \left (x\right )^{3} + a^{3} \sinh \left (x\right )^{4} + 2 \, a^{3} \cosh \left (x\right )^{2} + a^{3} + 2 \, {\left (3 \, a^{3} \cosh \left (x\right )^{2} + a^{3}\right )} \sinh \left (x\right )^{2} + 4 \, {\left (a^{3} \cosh \left (x\right )^{3} + a^{3} \cosh \left (x\right )\right )} \sinh \left (x\right )\right )}} \]

[In]

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

[Out]

2/3*((sqrt(2)*cosh(x)^4 + 4*sqrt(2)*cosh(x)*sinh(x)^3 + sqrt(2)*sinh(x)^4 + 2*(3*sqrt(2)*cosh(x)^2 + sqrt(2))*
sinh(x)^2 + 2*sqrt(2)*cosh(x)^2 + 4*(sqrt(2)*cosh(x)^3 + sqrt(2)*cosh(x))*sinh(x) + sqrt(2))*sqrt(a)*weierstra
ssPInverse(-4, 0, cosh(x) + sinh(x)) + 2*(cosh(x)^3 + 3*cosh(x)*sinh(x)^2 + sinh(x)^3 + (3*cosh(x)^2 - 1)*sinh
(x) - cosh(x))*sqrt(a*cosh(x)))/(a^3*cosh(x)^4 + 4*a^3*cosh(x)*sinh(x)^3 + a^3*sinh(x)^4 + 2*a^3*cosh(x)^2 + a
^3 + 2*(3*a^3*cosh(x)^2 + a^3)*sinh(x)^2 + 4*(a^3*cosh(x)^3 + a^3*cosh(x))*sinh(x))

Sympy [F]

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

[In]

integrate(1/(a*cosh(x))**(5/2),x)

[Out]

Integral((a*cosh(x))**(-5/2), x)

Maxima [F]

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

[In]

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

[Out]

integrate((a*cosh(x))^(-5/2), x)

Giac [F]

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

[In]

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

[Out]

integrate((a*cosh(x))^(-5/2), x)

Mupad [F(-1)]

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

[In]

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

[Out]

int(1/(a*cosh(x))^(5/2), x)