\(\int \sqrt {1+\text {csch}^2(x)} \, dx\) [17]

Optimal result

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

[Out]

ln(sinh(x))*(coth(x)^2)^(1/2)*tanh(x)

Rubi [A] (verified)

Time = 0.01 (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} \[ \int \sqrt {1+\text {csch}^2(x)} \, dx=\tanh (x) \sqrt {\coth ^2(x)} \log (\sinh (x)) \]

[In]

Int[Sqrt[1 + Csch[x]^2],x]

[Out]

Sqrt[Coth[x]^2]*Log[Sinh[x]]*Tanh[x]

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 \sqrt {\coth ^2(x)} \, dx \\ & = \left (\sqrt {\coth ^2(x)} \tanh (x)\right ) \int \coth (x) \, dx \\ & = \sqrt {\coth ^2(x)} \log (\sinh (x)) \tanh (x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.29 \[ \int \sqrt {1+\text {csch}^2(x)} \, dx=\sqrt {\coth ^2(x)} (\log (\cosh (x))+\log (\tanh (x))) \tanh (x) \]

[In]

Integrate[Sqrt[1 + Csch[x]^2],x]

[Out]

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

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 0.40 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.21

method	result	size
default	\(-\frac {\operatorname {csgn}\left (\coth \left (x \right )\right ) \left (\ln \left (\coth \left (x \right )-1\right )+\ln \left (1+\coth \left (x \right )\right )\right )}{2}\)	\(17\)
risch	\(-\frac {\left ({\mathrm e}^{2 x}-1\right ) \sqrt {\frac {\left (1+{\mathrm e}^{2 x}\right )^{2}}{\left ({\mathrm e}^{2 x}-1\right )^{2}}}\, x}{1+{\mathrm e}^{2 x}}+\frac {\left ({\mathrm e}^{2 x}-1\right ) \sqrt {\frac {\left (1+{\mathrm e}^{2 x}\right )^{2}}{\left ({\mathrm e}^{2 x}-1\right )^{2}}}\, \ln \left ({\mathrm e}^{2 x}-1\right )}{1+{\mathrm e}^{2 x}}\)	\(79\)

[In]

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

[Out]

-1/2*csgn(coth(x))*(ln(coth(x)-1)+ln(1+coth(x)))

Fricas [A] (verification not implemented)

none

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

Integral(sqrt(csch(x)**2 + 1), x)

Maxima [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.57 \[ \int \sqrt {1+\text {csch}^2(x)} \, dx=-x - \log \left (e^{\left (-x\right )} + 1\right ) - \log \left (e^{\left (-x\right )} - 1\right ) \]

[In]

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

[Out]

-x - log(e^(-x) + 1) - log(e^(-x) - 1)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 27 vs. \(2 (12) = 24\).

Time = 0.28 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.93 \[ \int \sqrt {1+\text {csch}^2(x)} \, dx=-x \mathrm {sgn}\left (e^{\left (4 \, x\right )} - 1\right ) + \log \left ({\left | e^{\left (2 \, x\right )} - 1 \right |}\right ) \mathrm {sgn}\left (e^{\left (4 \, x\right )} - 1\right ) \]

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

int((1/sinh(x)^2 + 1)^(1/2),x)

[Out]

int((1/sinh(x)^2 + 1)^(1/2), x)