### 3.29 $$\int \text{csch}^2(a+b x) \text{sech}(a+b x) \, dx$$

Optimal. Leaf size=24 $-\frac{\text{csch}(a+b x)}{b}-\frac{\tan ^{-1}(\sinh (a+b x))}{b}$

[Out]

-(ArcTan[Sinh[a + b*x]]/b) - Csch[a + b*x]/b

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Rubi [A]  time = 0.0222952, antiderivative size = 24, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.2, Rules used = {2621, 321, 207} $-\frac{\text{csch}(a+b x)}{b}-\frac{\tan ^{-1}(\sinh (a+b x))}{b}$

Antiderivative was successfully veriﬁed.

[In]

Int[Csch[a + b*x]^2*Sech[a + b*x],x]

[Out]

-(ArcTan[Sinh[a + b*x]]/b) - Csch[a + b*x]/b

Rule 2621

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

Rule 321

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n + 1)*(a + b*x^n
)^(p + 1))/(b*(m + n*p + 1)), x] - Dist[(a*c^n*(m - n + 1))/(b*(m + n*p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

Rule 207

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTanh[(Rt[b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[a, 0] || GtQ[b, 0])

Rubi steps

\begin{align*} \int \text{csch}^2(a+b x) \text{sech}(a+b x) \, dx &=-\frac{i \operatorname{Subst}\left (\int \frac{x^2}{-1+x^2} \, dx,x,-i \text{csch}(a+b x)\right )}{b}\\ &=-\frac{\text{csch}(a+b x)}{b}-\frac{i \operatorname{Subst}\left (\int \frac{1}{-1+x^2} \, dx,x,-i \text{csch}(a+b x)\right )}{b}\\ &=-\frac{\tan ^{-1}(\sinh (a+b x))}{b}-\frac{\text{csch}(a+b x)}{b}\\ \end{align*}

Mathematica [C]  time = 0.0185736, size = 29, normalized size = 1.21 $-\frac{\text{csch}(a+b x) \, _2F_1\left (-\frac{1}{2},1;\frac{1}{2};-\sinh ^2(a+b x)\right )}{b}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

-((Csch[a + b*x]*Hypergeometric2F1[-1/2, 1, 1/2, -Sinh[a + b*x]^2])/b)

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Maple [A]  time = 0.015, size = 27, normalized size = 1.1 \begin{align*} -{\frac{1}{b\sinh \left ( bx+a \right ) }}-2\,{\frac{\arctan \left ({{\rm e}^{bx+a}} \right ) }{b}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(csch(b*x+a)^2*sech(b*x+a),x)

[Out]

-1/b/sinh(b*x+a)-2*arctan(exp(b*x+a))/b

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Maxima [A]  time = 1.50841, size = 58, normalized size = 2.42 \begin{align*} \frac{2 \, \arctan \left (e^{\left (-b x - a\right )}\right )}{b} + \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(csch(b*x+a)^2*sech(b*x+a),x, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 2.13367, size = 305, normalized size = 12.71 \begin{align*} -\frac{2 \,{\left ({\left (\cosh \left (b x + a\right )^{2} + 2 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + \sinh \left (b x + a\right )^{2} - 1\right )} \arctan \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right ) + \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(csch(b*x+a)^2*sech(b*x+a),x, algorithm="fricas")

[Out]

-2*((cosh(b*x + a)^2 + 2*cosh(b*x + a)*sinh(b*x + a) + sinh(b*x + a)^2 - 1)*arctan(cosh(b*x + a) + sinh(b*x +
a)) + 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 \operatorname{csch}^{2}{\left (a + b x \right )} \operatorname{sech}{\left (a + b x \right )}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(b*x+a)**2*sech(b*x+a),x)

[Out]

Integral(csch(a + b*x)**2*sech(a + b*x), x)

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Giac [B]  time = 1.21202, size = 78, normalized size = 3.25 \begin{align*} -\frac{\pi + 2 \, \arctan \left (\frac{1}{2} \,{\left (e^{\left (2 \, b x + 2 \, a\right )} - 1\right )} e^{\left (-b x - a\right )}\right )}{2 \, b} - \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(csch(b*x+a)^2*sech(b*x+a),x, algorithm="giac")

[Out]

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