\(\int \coth (a+b x) \text {csch}^2(a+b x) \, dx\) [449]

Optimal result

Integrand size = 15, antiderivative size = 15 \[ \int \coth (a+b x) \text {csch}^2(a+b x) \, dx=-\frac {\text {csch}^2(a+b x)}{2 b} \]

[Out]

-1/2*csch(b*x+a)^2/b

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {2686, 30} \[ \int \coth (a+b x) \text {csch}^2(a+b x) \, dx=-\frac {\text {csch}^2(a+b x)}{2 b} \]

[In]

Int[Coth[a + b*x]*Csch[a + b*x]^2,x]

[Out]

-1/2*Csch[a + b*x]^2/b

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 2686

Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[a/f, Subst[
Int[(a*x)^(m - 1)*(-1 + x^2)^((n - 1)/2), x], x, Sec[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n -
1)/2] &&  !(IntegerQ[m/2] && LtQ[0, m, n + 1])

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}(\int x \, dx,x,-i \text {csch}(a+b x))}{b} \\ & = -\frac {\text {csch}^2(a+b x)}{2 b} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00 \[ \int \coth (a+b x) \text {csch}^2(a+b x) \, dx=-\frac {\text {csch}^2(a+b x)}{2 b} \]

[In]

Integrate[Coth[a + b*x]*Csch[a + b*x]^2,x]

[Out]

-1/2*Csch[a + b*x]^2/b

Maple [A] (verified)

Time = 0.17 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.93

method	result	size
derivativedivides	\(-\frac {\operatorname {csch}\left (b x +a \right )^{2}}{2 b}\)	\(14\)
default	\(-\frac {\operatorname {csch}\left (b x +a \right )^{2}}{2 b}\)	\(14\)
risch	\(-\frac {2 \,{\mathrm e}^{2 b x +2 a}}{b \left ({\mathrm e}^{2 b x +2 a}-1\right )^{2}}\)	\(28\)

[In]

int(cosh(b*x+a)*csch(b*x+a)^3,x,method=_RETURNVERBOSE)

[Out]

-1/2*csch(b*x+a)^2/b

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 86 vs. \(2 (13) = 26\).

Time = 0.25 (sec) , antiderivative size = 86, normalized size of antiderivative = 5.73 \[ \int \coth (a+b x) \text {csch}^2(a+b x) \, dx=-\frac {2 \, {\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )}}{b \cosh \left (b x + a\right )^{3} + 3 \, b \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{2} + b \sinh \left (b x + a\right )^{3} - b \cosh \left (b x + a\right ) + 3 \, {\left (b \cosh \left (b x + a\right )^{2} - b\right )} \sinh \left (b x + a\right )} \]

[In]

integrate(cosh(b*x+a)*csch(b*x+a)^3,x, algorithm="fricas")

[Out]

-2*(cosh(b*x + a) + sinh(b*x + a))/(b*cosh(b*x + a)^3 + 3*b*cosh(b*x + a)*sinh(b*x + a)^2 + b*sinh(b*x + a)^3
- b*cosh(b*x + a) + 3*(b*cosh(b*x + a)^2 - b)*sinh(b*x + a))

Sympy [F]

\[ \int \coth (a+b x) \text {csch}^2(a+b x) \, dx=\int \cosh {\left (a + b x \right )} \operatorname {csch}^{3}{\left (a + b x \right )}\, dx \]

[In]

integrate(cosh(b*x+a)*csch(b*x+a)**3,x)

[Out]

Integral(cosh(a + b*x)*csch(a + b*x)**3, x)

Maxima [A] (verification not implemented)

none

Time = 0.18 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.67 \[ \int \coth (a+b x) \text {csch}^2(a+b x) \, dx=-\frac {2}{b {\left (e^{\left (b x + a\right )} - e^{\left (-b x - a\right )}\right )}^{2}} \]

[In]

integrate(cosh(b*x+a)*csch(b*x+a)^3,x, algorithm="maxima")

[Out]

-2/(b*(e^(b*x + a) - e^(-b*x - a))^2)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 27 vs. \(2 (13) = 26\).

Time = 0.27 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.80 \[ \int \coth (a+b x) \text {csch}^2(a+b x) \, dx=-\frac {2 \, e^{\left (2 \, b x + 2 \, a\right )}}{b {\left (e^{\left (2 \, b x + 2 \, a\right )} - 1\right )}^{2}} \]

[In]

integrate(cosh(b*x+a)*csch(b*x+a)^3,x, algorithm="giac")

[Out]

-2*e^(2*b*x + 2*a)/(b*(e^(2*b*x + 2*a) - 1)^2)

Mupad [B] (verification not implemented)

Time = 2.38 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.87 \[ \int \coth (a+b x) \text {csch}^2(a+b x) \, dx=-\frac {1}{2\,b\,{\mathrm {sinh}\left (a+b\,x\right )}^2} \]

[In]

int(cosh(a + b*x)/sinh(a + b*x)^3,x)

[Out]

-1/(2*b*sinh(a + b*x)^2)