\(\int \text {csch}^2(x) \, dx\) [30]

Optimal result

Integrand size = 4, antiderivative size = 4 \[ \int \text {csch}^2(x) \, dx=-\coth (x) \]

[Out]

-coth(x)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 4, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3852, 8} \[ \int \text {csch}^2(x) \, dx=-\coth (x) \]

[In]

Int[Csch[x]^2,x]

[Out]

-Coth[x]

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 3852

Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Dist[-d^(-1), Subst[Int[ExpandIntegrand[(1 + x^2)^(n/2 - 1), x]
, x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 0]

Rubi steps \begin{align*} \text {integral}& = -(i \text {Subst}(\int 1 \, dx,x,-i \coth (x))) \\ & = -\coth (x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 4, normalized size of antiderivative = 1.00 \[ \int \text {csch}^2(x) \, dx=-\coth (x) \]

[In]

Integrate[Csch[x]^2,x]

[Out]

-Coth[x]

Maple [A] (verified)

Time = 0.23 (sec) , antiderivative size = 5, normalized size of antiderivative = 1.25

method	result	size
default	\(-\coth \left (x \right )\)	\(5\)
parallelrisch	\(-\coth \left (x \right )\)	\(5\)
risch	\(-\frac {2}{{\mathrm e}^{2 x}-1}\)	\(11\)

[In]

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

[Out]

-coth(x)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 20 vs. \(2 (4) = 8\).

Time = 0.23 (sec) , antiderivative size = 20, normalized size of antiderivative = 5.00 \[ \int \text {csch}^2(x) \, dx=-\frac {2}{\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2} - 1} \]

[In]

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

[Out]

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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 15 vs. \(2 (3) = 6\).

Time = 0.23 (sec) , antiderivative size = 15, normalized size of antiderivative = 3.75 \[ \int \text {csch}^2(x) \, dx=- \frac {\tanh {\left (\frac {x}{2} \right )}}{2} - \frac {1}{2 \tanh {\left (\frac {x}{2} \right )}} \]

[In]

integrate(1/sinh(x)**2,x)

[Out]

-tanh(x/2)/2 - 1/(2*tanh(x/2))

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 10 vs. \(2 (4) = 8\).

Time = 0.19 (sec) , antiderivative size = 10, normalized size of antiderivative = 2.50 \[ \int \text {csch}^2(x) \, dx=\frac {2}{e^{\left (-2 \, x\right )} - 1} \]

[In]

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

[Out]

2/(e^(-2*x) - 1)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 10 vs. \(2 (4) = 8\).

Time = 0.25 (sec) , antiderivative size = 10, normalized size of antiderivative = 2.50 \[ \int \text {csch}^2(x) \, dx=-\frac {2}{e^{\left (2 \, x\right )} - 1} \]

[In]

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

[Out]

-2/(e^(2*x) - 1)

Mupad [B] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 4, normalized size of antiderivative = 1.00 \[ \int \text {csch}^2(x) \, dx=-\mathrm {coth}\left (x\right ) \]

[In]

int(1/sinh(x)^2,x)

[Out]

-coth(x)