\(\int (a \cosh (x))^{3/2} \, dx\) [17]

Optimal result

Integrand size = 8, antiderivative size = 48 \[ \int (a \cosh (x))^{3/2} \, dx=-\frac {2 i a^2 \sqrt {\cosh (x)} \operatorname {EllipticF}\left (\frac {i x}{2},2\right )}{3 \sqrt {a \cosh (x)}}+\frac {2}{3} a \sqrt {a \cosh (x)} \sinh (x) \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 48, 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 = {2715, 2721, 2720} \[ \int (a \cosh (x))^{3/2} \, dx=\frac {2}{3} a \sinh (x) \sqrt {a \cosh (x)}-\frac {2 i a^2 \sqrt {\cosh (x)} \operatorname {EllipticF}\left (\frac {i x}{2},2\right )}{3 \sqrt {a \cosh (x)}} \]

[In]

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

[Out]

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

Rule 2715

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d*x]*((b*Sin[c + d*x])^(n - 1)/(d*n))
, x] + Dist[b^2*((n - 1)/n), Int[(b*Sin[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && Integ
erQ[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}{3} a \sqrt {a \cosh (x)} \sinh (x)+\frac {1}{3} a^2 \int \frac {1}{\sqrt {a \cosh (x)}} \, dx \\ & = \frac {2}{3} a \sqrt {a \cosh (x)} \sinh (x)+\frac {\left (a^2 \sqrt {\cosh (x)}\right ) \int \frac {1}{\sqrt {\cosh (x)}} \, dx}{3 \sqrt {a \cosh (x)}} \\ & = -\frac {2 i a^2 \sqrt {\cosh (x)} \operatorname {EllipticF}\left (\frac {i x}{2},2\right )}{3 \sqrt {a \cosh (x)}}+\frac {2}{3} a \sqrt {a \cosh (x)} \sinh (x) \\ \end{align*}

Mathematica [C] (verified)

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

Time = 0.05 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.19 \[ \int (a \cosh (x))^{3/2} \, dx=\frac {2}{3} (a \cosh (x))^{3/2} \left (\operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},-\cosh (2 x)-\sinh (2 x)\right ) \text {sech}^2(x) \sqrt {1+\cosh (2 x)+\sinh (2 x)}+\tanh (x)\right ) \]

[In]

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

[Out]

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(129\) vs. \(2(53)=106\).

Time = 0.59 (sec) , antiderivative size = 130, normalized size of antiderivative = 2.71

method	result	size
default	\(\frac {\sqrt {a \left (2 \cosh \left (\frac {x}{2}\right )^{2}-1\right ) \sinh \left (\frac {x}{2}\right )^{2}}\, a^{2} \left (8 \sinh \left (\frac {x}{2}\right )^{4} \cosh \left (\frac {x}{2}\right )+\sqrt {2}\, \sqrt {-2 \sinh \left (\frac {x}{2}\right )^{2}-1}\, \sqrt {-\sinh \left (\frac {x}{2}\right )^{2}}\, \operatorname {EllipticF}\left (\sqrt {2}\, \cosh \left (\frac {x}{2}\right ), \frac {\sqrt {2}}{2}\right )+4 \sinh \left (\frac {x}{2}\right )^{2} \cosh \left (\frac {x}{2}\right )\right )}{3 \sqrt {a \left (2 \sinh \left (\frac {x}{2}\right )^{4}+\sinh \left (\frac {x}{2}\right )^{2}\right )}\, \sinh \left (\frac {x}{2}\right ) \sqrt {a \left (2 \cosh \left (\frac {x}{2}\right )^{2}-1\right )}}\)	\(130\)

[In]

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

[Out]

1/3*(a*(2*cosh(1/2*x)^2-1)*sinh(1/2*x)^2)^(1/2)*a^2*(8*sinh(1/2*x)^4*cosh(1/2*x)+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*sin
h(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 = 0.07 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.42 \[ \int (a \cosh (x))^{3/2} \, dx=\frac {2 \, {\left (\sqrt {2} a \cosh \left (x\right ) + \sqrt {2} a \sinh \left (x\right )\right )} \sqrt {a} {\rm weierstrassPInverse}\left (-4, 0, \cosh \left (x\right ) + \sinh \left (x\right )\right ) + {\left (a \cosh \left (x\right )^{2} + 2 \, a \cosh \left (x\right ) \sinh \left (x\right ) + a \sinh \left (x\right )^{2} - a\right )} \sqrt {a \cosh \left (x\right )}}{3 \, {\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )}} \]

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

Integral((a*cosh(x))**(3/2), x)

Maxima [F]

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

[In]

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

[Out]

integrate((a*cosh(x))^(3/2), x)

Giac [F]

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

[In]

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

[Out]

integrate((a*cosh(x))^(3/2), x)

Mupad [F(-1)]

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

[In]

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

[Out]

int((a*cosh(x))^(3/2), x)