\(\int \sqrt {\cosh (x) \coth (x)} \, dx\) [563]

Optimal result

Integrand size = 9, antiderivative size = 13 \[ \int \sqrt {\cosh (x) \coth (x)} \, dx=2 \sqrt {\cosh (x) \coth (x)} \tanh (x) \]

[Out]

2*(cosh(x)*coth(x))^(1/2)*tanh(x)

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {4483, 4485, 2669} \[ \int \sqrt {\cosh (x) \coth (x)} \, dx=2 \tanh (x) \sqrt {\cosh (x) \coth (x)} \]

[In]

Int[Sqrt[Cosh[x]*Coth[x]],x]

[Out]

2*Sqrt[Cosh[x]*Coth[x]]*Tanh[x]

Rule 2669

Int[((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*(a*Sin[e
+ f*x])^m*((b*Tan[e + f*x])^(n - 1)/(f*m)), x] /; FreeQ[{a, b, e, f, m, n}, x] && EqQ[m + n - 1, 0]

Rule 4483

Int[(u_.)*((a_)*(v_))^(p_), x_Symbol] :> With[{uu = ActivateTrig[u], vv = ActivateTrig[v]}, Dist[a^IntPart[p]*
((a*vv)^FracPart[p]/vv^FracPart[p]), Int[uu*vv^p, x], x]] /; FreeQ[{a, p}, x] &&  !IntegerQ[p] &&  !InertTrigF
reeQ[v]

Rule 4485

Int[(u_.)*((v_)^(m_.)*(w_)^(n_.))^(p_), x_Symbol] :> With[{uu = ActivateTrig[u], vv = ActivateTrig[v], ww = Ac
tivateTrig[w]}, Dist[(vv^m*ww^n)^FracPart[p]/(vv^(m*FracPart[p])*ww^(n*FracPart[p])), Int[uu*vv^(m*p)*ww^(n*p)
, x], x]] /; FreeQ[{m, n, p}, x] &&  !IntegerQ[p] && ( !InertTrigFreeQ[v] ||  !InertTrigFreeQ[w])

Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {\cosh (x) \coth (x)} \int \sqrt {-i \cosh (x) \coth (x)} \, dx}{\sqrt {-i \cosh (x) \coth (x)}} \\ & = \frac {\sqrt {\cosh (x) \coth (x)} \int \sqrt {\cosh (x)} \sqrt {-i \coth (x)} \, dx}{\sqrt {\cosh (x)} \sqrt {-i \coth (x)}} \\ & = 2 \sqrt {\cosh (x) \coth (x)} \tanh (x) \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(35\) vs. \(2(13)=26\).

Time = 0.11 (sec) , antiderivative size = 35, normalized size of antiderivative = 2.69 \[ \int \sqrt {\cosh (x) \coth (x)} \, dx=\frac {2 \sqrt {\cosh (x) \coth (x)} \left (-1+\sqrt [4]{-\sinh ^2(x)}\right ) \tanh (x)}{\sqrt [4]{-\sinh ^2(x)}} \]

[In]

Integrate[Sqrt[Cosh[x]*Coth[x]],x]

[Out]

(2*Sqrt[Cosh[x]*Coth[x]]*(-1 + (-Sinh[x]^2)^(1/4))*Tanh[x])/(-Sinh[x]^2)^(1/4)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(41\) vs. \(2(11)=22\).

Time = 0.64 (sec) , antiderivative size = 42, normalized size of antiderivative = 3.23

method	result	size
risch	\(\frac {\sqrt {2}\, \sqrt {\frac {\left (1+{\mathrm e}^{2 x}\right )^{2} {\mathrm e}^{-x}}{{\mathrm e}^{2 x}-1}}\, \left ({\mathrm e}^{2 x}-1\right )}{1+{\mathrm e}^{2 x}}\)	\(42\)

[In]

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

[Out]

2^(1/2)*((1+exp(2*x))^2*exp(-x)/(exp(2*x)-1))^(1/2)/(1+exp(2*x))*(exp(2*x)-1)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 55 vs. \(2 (11) = 22\).

Time = 0.24 (sec) , antiderivative size = 55, normalized size of antiderivative = 4.23 \[ \int \sqrt {\cosh (x) \coth (x)} \, dx=\frac {2 \, \sqrt {\frac {1}{2}} {\left (\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2} - 1\right )}}{\sqrt {\cosh \left (x\right )^{3} + 3 \, \cosh \left (x\right ) \sinh \left (x\right )^{2} + \sinh \left (x\right )^{3} + {\left (3 \, \cosh \left (x\right )^{2} - 1\right )} \sinh \left (x\right ) - \cosh \left (x\right )}} \]

[In]

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

[Out]

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

Sympy [F]

\[ \int \sqrt {\cosh (x) \coth (x)} \, dx=\int \sqrt {\cosh {\left (x \right )} \coth {\left (x \right )}}\, dx \]

[In]

integrate((cosh(x)*coth(x))**(1/2),x)

[Out]

Integral(sqrt(cosh(x)*coth(x)), x)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 54 vs. \(2 (11) = 22\).

Time = 0.30 (sec) , antiderivative size = 54, normalized size of antiderivative = 4.15 \[ \int \sqrt {\cosh (x) \coth (x)} \, dx=\frac {\sqrt {2} e^{\left (\frac {1}{2} \, x\right )}}{\sqrt {e^{\left (-x\right )} + 1} \sqrt {-e^{\left (-x\right )} + 1}} - \frac {\sqrt {2} e^{\left (-\frac {3}{2} \, x\right )}}{\sqrt {e^{\left (-x\right )} + 1} \sqrt {-e^{\left (-x\right )} + 1}} \]

[In]

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

[Out]

sqrt(2)*e^(1/2*x)/(sqrt(e^(-x) + 1)*sqrt(-e^(-x) + 1)) - sqrt(2)*e^(-3/2*x)/(sqrt(e^(-x) + 1)*sqrt(-e^(-x) + 1
))

Giac [F]

\[ \int \sqrt {\cosh (x) \coth (x)} \, dx=\int { \sqrt {\cosh \left (x\right ) \coth \left (x\right )} \,d x } \]

[In]

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

[Out]

integrate(sqrt(cosh(x)*coth(x)), x)

Mupad [B] (verification not implemented)

Time = 2.23 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.77 \[ \int \sqrt {\cosh (x) \coth (x)} \, dx=4\,{\mathrm {e}}^x\,\mathrm {sinh}\left (x\right )\,\sqrt {\frac {{\mathrm {e}}^{-x}}{2\,\left ({\mathrm {e}}^{2\,x}-1\right )}} \]

[In]

int((cosh(x)*coth(x))^(1/2),x)

[Out]

4*exp(x)*sinh(x)*(exp(-x)/(2*(exp(2*x) - 1)))^(1/2)