\(\int \sqrt {1+\coth (x)} \, dx\) [73]

Optimal result

Integrand size = 8, antiderivative size = 21 \[ \int \sqrt {1+\coth (x)} \, dx=\sqrt {2} \text {arctanh}\left (\frac {\sqrt {1+\coth (x)}}{\sqrt {2}}\right ) \]

[Out]

arctanh(1/2*(1+coth(x))^(1/2)*2^(1/2))*2^(1/2)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3561, 212} \[ \int \sqrt {1+\coth (x)} \, dx=\sqrt {2} \text {arctanh}\left (\frac {\sqrt {\coth (x)+1}}{\sqrt {2}}\right ) \]

[In]

Int[Sqrt[1 + Coth[x]],x]

[Out]

Sqrt[2]*ArcTanh[Sqrt[1 + Coth[x]]/Sqrt[2]]

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 3561

Int[Sqrt[(a_) + (b_.)*tan[(c_.) + (d_.)*(x_)]], x_Symbol] :> Dist[-2*(b/d), Subst[Int[1/(2*a - x^2), x], x, Sq
rt[a + b*Tan[c + d*x]]], x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 + b^2, 0]

Rubi steps \begin{align*} \text {integral}& = 2 \text {Subst}\left (\int \frac {1}{2-x^2} \, dx,x,\sqrt {1+\coth (x)}\right ) \\ & = \sqrt {2} \text {arctanh}\left (\frac {\sqrt {1+\coth (x)}}{\sqrt {2}}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.42 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00 \[ \int \sqrt {1+\coth (x)} \, dx=\sqrt {2} \text {arctanh}\left (\frac {\sqrt {1+\coth (x)}}{\sqrt {2}}\right ) \]

[In]

Integrate[Sqrt[1 + Coth[x]],x]

[Out]

Sqrt[2]*ArcTanh[Sqrt[1 + Coth[x]]/Sqrt[2]]

Maple [A] (verified)

Time = 0.09 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.81

method	result	size
derivativedivides	\(\operatorname {arctanh}\left (\frac {\sqrt {1+\coth \left (x \right )}\, \sqrt {2}}{2}\right ) \sqrt {2}\)	\(17\)
default	\(\operatorname {arctanh}\left (\frac {\sqrt {1+\coth \left (x \right )}\, \sqrt {2}}{2}\right ) \sqrt {2}\)	\(17\)

[In]

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

[Out]

arctanh(1/2*(1+coth(x))^(1/2)*2^(1/2))*2^(1/2)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 50 vs. \(2 (16) = 32\).

Time = 0.25 (sec) , antiderivative size = 50, normalized size of antiderivative = 2.38 \[ \int \sqrt {1+\coth (x)} \, dx=\frac {1}{2} \, \sqrt {2} \log \left (2 \, \sqrt {2} \sqrt {\frac {\sinh \left (x\right )}{\cosh \left (x\right ) - \sinh \left (x\right )}} {\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )} + 2 \, \cosh \left (x\right )^{2} + 4 \, \cosh \left (x\right ) \sinh \left (x\right ) + 2 \, \sinh \left (x\right )^{2} - 1\right ) \]

[In]

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

[Out]

1/2*sqrt(2)*log(2*sqrt(2)*sqrt(sinh(x)/(cosh(x) - sinh(x)))*(cosh(x) + sinh(x)) + 2*cosh(x)^2 + 4*cosh(x)*sinh
(x) + 2*sinh(x)^2 - 1)

Sympy [F]

\[ \int \sqrt {1+\coth (x)} \, dx=\int \sqrt {\coth {\left (x \right )} + 1}\, dx \]

[In]

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

[Out]

Integral(sqrt(coth(x) + 1), x)

Maxima [F]

\[ \int \sqrt {1+\coth (x)} \, dx=\int { \sqrt {\coth \left (x\right ) + 1} \,d x } \]

[In]

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

[Out]

integrate(sqrt(coth(x) + 1), x)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 37 vs. \(2 (16) = 32\).

Time = 0.28 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.76 \[ \int \sqrt {1+\coth (x)} \, dx=-\frac {1}{2} \, \sqrt {2} \log \left ({\left | 2 \, \sqrt {e^{\left (4 \, x\right )} - e^{\left (2 \, x\right )}} - 2 \, e^{\left (2 \, x\right )} + 1 \right |}\right ) \mathrm {sgn}\left (e^{\left (2 \, x\right )} - 1\right ) \]

[In]

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

[Out]

-1/2*sqrt(2)*log(abs(2*sqrt(e^(4*x) - e^(2*x)) - 2*e^(2*x) + 1))*sgn(e^(2*x) - 1)

Mupad [B] (verification not implemented)

Time = 1.91 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.76 \[ \int \sqrt {1+\coth (x)} \, dx=\sqrt {2}\,\mathrm {atanh}\left (\frac {\sqrt {2}\,\sqrt {\mathrm {coth}\left (x\right )+1}}{2}\right ) \]

[In]

int((coth(x) + 1)^(1/2),x)

[Out]

2^(1/2)*atanh((2^(1/2)*(coth(x) + 1)^(1/2))/2)