### 3.112 $$\int \coth (a+b x) \text{csch}(a+b x) \, dx$$

Optimal. Leaf size=11 $-\frac{\text{csch}(a+b x)}{b}$

[Out]

-(Csch[a + b*x]/b)

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Rubi [A]  time = 0.0107484, antiderivative size = 11, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.154, Rules used = {2606, 8} $-\frac{\text{csch}(a+b x)}{b}$

Antiderivative was successfully veriﬁed.

[In]

Int[Coth[a + b*x]*Csch[a + b*x],x]

[Out]

-(Csch[a + b*x]/b)

Rule 2606

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])

Rule 8

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

Rubi steps

\begin{align*} \int \coth (a+b x) \text{csch}(a+b x) \, dx &=-\frac{i \operatorname{Subst}(\int 1 \, dx,x,-i \text{csch}(a+b x))}{b}\\ &=-\frac{\text{csch}(a+b x)}{b}\\ \end{align*}

Mathematica [A]  time = 0.0091367, size = 11, normalized size = 1. $-\frac{\text{csch}(a+b x)}{b}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

-(Csch[a + b*x]/b)

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Maple [A]  time = 0.007, size = 12, normalized size = 1.1 \begin{align*} -{\frac{{\rm csch} \left (bx+a\right )}{b}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(csch(b*x+a)*coth(b*x+a),x)

[Out]

-csch(b*x+a)/b

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Maxima [B]  time = 1.04634, size = 34, normalized size = 3.09 \begin{align*} -\frac{2}{b{\left (e^{\left (b x + a\right )} - e^{\left (-b x - a\right )}\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(b*x+a)*csch(b*x+a),x, algorithm="maxima")

[Out]

-2/(b*(e^(b*x + a) - e^(-b*x - a)))

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Fricas [B]  time = 1.73605, size = 154, normalized size = 14. \begin{align*} -\frac{2 \,{\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )}}{b \cosh \left (b x + a\right )^{2} + 2 \, b \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + b \sinh \left (b x + a\right )^{2} - b} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(b*x+a)*csch(b*x+a),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \coth{\left (a + b x \right )} \operatorname{csch}{\left (a + b x \right )}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(b*x+a)*csch(b*x+a),x)

[Out]

Integral(coth(a + b*x)*csch(a + b*x), x)

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Giac [B]  time = 1.20304, size = 32, normalized size = 2.91 \begin{align*} -\frac{2 \, e^{\left (b x + a\right )}}{b{\left (e^{\left (2 \, b x + 2 \, a\right )} - 1\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(b*x+a)*csch(b*x+a),x, algorithm="giac")

[Out]

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