### 3.24 $$\int \text{csch}(a+b x) \text{sech}(a+b x) \, dx$$

Optimal. Leaf size=11 $\frac{\log (\tanh (a+b x))}{b}$

[Out]

Log[Tanh[a + b*x]]/b

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Rubi [A]  time = 0.0122369, 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 = {2620, 29} $\frac{\log (\tanh (a+b x))}{b}$

Antiderivative was successfully veriﬁed.

[In]

Int[Csch[a + b*x]*Sech[a + b*x],x]

[Out]

Log[Tanh[a + b*x]]/b

Rule 2620

Int[csc[(e_.) + (f_.)*(x_)]^(m_.)*sec[(e_.) + (f_.)*(x_)]^(n_.), x_Symbol] :> Dist[1/f, Subst[Int[(1 + x^2)^((
m + n)/2 - 1)/x^m, x], x, Tan[e + f*x]], x] /; FreeQ[{e, f}, x] && IntegersQ[m, n, (m + n)/2]

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rubi steps

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

Mathematica [B]  time = 0.0202292, size = 31, normalized size = 2.82 $2 \left (\frac{\log (\sinh (a+b x))}{2 b}-\frac{\log (\cosh (a+b x))}{2 b}\right )$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

2*(-Log[Cosh[a + b*x]]/(2*b) + Log[Sinh[a + b*x]]/(2*b))

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Maple [A]  time = 0.013, size = 12, normalized size = 1.1 \begin{align*}{\frac{\ln \left ( \tanh \left ( bx+a \right ) \right ) }{b}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

ln(tanh(b*x+a))/b

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Maxima [B]  time = 1.56052, size = 68, normalized size = 6.18 \begin{align*} \frac{\log \left (e^{\left (-b x - a\right )} + 1\right )}{b} + \frac{\log \left (e^{\left (-b x - a\right )} - 1\right )}{b} - \frac{\log \left (e^{\left (-2 \, b x - 2 \, a\right )} + 1\right )}{b} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 2.16458, size = 154, normalized size = 14. \begin{align*} -\frac{\log \left (\frac{2 \, \cosh \left (b x + a\right )}{\cosh \left (b x + a\right ) - \sinh \left (b x + a\right )}\right ) - \log \left (\frac{2 \, \sinh \left (b x + a\right )}{\cosh \left (b x + a\right ) - \sinh \left (b x + a\right )}\right )}{b} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

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

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Giac [B]  time = 1.16251, size = 47, normalized size = 4.27 \begin{align*} -\frac{\log \left (e^{\left (2 \, b x + 2 \, a\right )} + 1\right )}{b} + \frac{\log \left ({\left | e^{\left (2 \, b x + 2 \, a\right )} - 1 \right |}\right )}{b} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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