\(\int \frac {1}{\sqrt {1+\text {csch}^2(x)}} \, dx\) [18]

Optimal result

Integrand size = 10, antiderivative size = 14 \[ \int \frac {1}{\sqrt {1+\text {csch}^2(x)}} \, dx=\frac {\coth (x) \log (\cosh (x))}{\sqrt {\coth ^2(x)}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {4206, 3739, 3556} \[ \int \frac {1}{\sqrt {1+\text {csch}^2(x)}} \, dx=\frac {\coth (x) \log (\cosh (x))}{\sqrt {\coth ^2(x)}} \]

[In]

Int[1/Sqrt[1 + Csch[x]^2],x]

[Out]

(Coth[x]*Log[Cosh[x]])/Sqrt[Coth[x]^2]

Rule 3556

Int[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Log[RemoveContent[Cos[c + d*x], x]]/d, x] /; FreeQ[{c, d}, x]

Rule 3739

Int[(u_.)*((b_.)*tan[(e_.) + (f_.)*(x_)]^(n_))^(p_), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Di
st[(b*ff^n)^IntPart[p]*((b*Tan[e + f*x]^n)^FracPart[p]/(Tan[e + f*x]/ff)^(n*FracPart[p])), Int[ActivateTrig[u]
*(Tan[e + f*x]/ff)^(n*p), x], x]] /; FreeQ[{b, e, f, n, p}, x] &&  !IntegerQ[p] && IntegerQ[n] && (EqQ[u, 1] |
| MatchQ[u, ((d_.)*(trig_)[e + f*x])^(m_.) /; FreeQ[{d, m}, x] && MemberQ[{sin, cos, tan, cot, sec, csc}, trig
]])

Rule 4206

Int[(u_.)*((a_) + (b_.)*sec[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> Int[ActivateTrig[u*(b*tan[e + f*x]^2)^p]
, x] /; FreeQ[{a, b, e, f, p}, x] && EqQ[a + b, 0]

Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{\sqrt {\coth ^2(x)}} \, dx \\ & = \frac {\coth (x) \int \tanh (x) \, dx}{\sqrt {\coth ^2(x)}} \\ & = \frac {\coth (x) \log (\cosh (x))}{\sqrt {\coth ^2(x)}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\sqrt {1+\text {csch}^2(x)}} \, dx=\frac {\coth (x) \log (\cosh (x))}{\sqrt {\coth ^2(x)}} \]

[In]

Integrate[1/Sqrt[1 + Csch[x]^2],x]

[Out]

(Coth[x]*Log[Cosh[x]])/Sqrt[Coth[x]^2]

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(78\) vs. \(2(12)=24\).

Time = 0.23 (sec) , antiderivative size = 79, normalized size of antiderivative = 5.64

method	result	size
risch	\(-\frac {\left (1+{\mathrm e}^{2 x}\right ) x}{\sqrt {\frac {\left (1+{\mathrm e}^{2 x}\right )^{2}}{\left ({\mathrm e}^{2 x}-1\right )^{2}}}\, \left ({\mathrm e}^{2 x}-1\right )}+\frac {\left (1+{\mathrm e}^{2 x}\right ) \ln \left (1+{\mathrm e}^{2 x}\right )}{\sqrt {\frac {\left (1+{\mathrm e}^{2 x}\right )^{2}}{\left ({\mathrm e}^{2 x}-1\right )^{2}}}\, \left ({\mathrm e}^{2 x}-1\right )}\)	\(79\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.29 \[ \int \frac {1}{\sqrt {1+\text {csch}^2(x)}} \, dx=-x + \log \left (\frac {2 \, \cosh \left (x\right )}{\cosh \left (x\right ) - \sinh \left (x\right )}\right ) \]

[In]

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

[Out]

-x + log(2*cosh(x)/(cosh(x) - sinh(x)))

Sympy [F]

\[ \int \frac {1}{\sqrt {1+\text {csch}^2(x)}} \, dx=\int \frac {1}{\sqrt {\operatorname {csch}^{2}{\left (x \right )} + 1}}\, dx \]

[In]

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

[Out]

Integral(1/sqrt(csch(x)**2 + 1), x)

Maxima [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.93 \[ \int \frac {1}{\sqrt {1+\text {csch}^2(x)}} \, dx=-x - \log \left (e^{\left (-2 \, x\right )} + 1\right ) \]

[In]

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

[Out]

-x - log(e^(-2*x) + 1)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 30 vs. \(2 (12) = 24\).

Time = 0.26 (sec) , antiderivative size = 30, normalized size of antiderivative = 2.14 \[ \int \frac {1}{\sqrt {1+\text {csch}^2(x)}} \, dx=-\frac {x}{\mathrm {sgn}\left (e^{\left (4 \, x\right )} - 1\right )} + \frac {\log \left (e^{\left (2 \, x\right )} + 1\right )}{\mathrm {sgn}\left (e^{\left (4 \, x\right )} - 1\right )} \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\sqrt {1+\text {csch}^2(x)}} \, dx=\int \frac {1}{\sqrt {\frac {1}{{\mathrm {sinh}\left (x\right )}^2}+1}} \,d x \]

[In]

int(1/(1/sinh(x)^2 + 1)^(1/2),x)

[Out]

int(1/(1/sinh(x)^2 + 1)^(1/2), x)