### 3.979 $$\int \frac{\text{sech}^2(x)}{a+b \tanh (x)} \, dx$$

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

[Out]

Log[a + b*Tanh[x]]/b

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Rubi [A]  time = 0.0401233, 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 = {3506, 31} $\frac{\log (a+b \tanh (x))}{b}$

Antiderivative was successfully veriﬁed.

[In]

Int[Sech[x]^2/(a + b*Tanh[x]),x]

[Out]

Log[a + b*Tanh[x]]/b

Rule 3506

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

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rubi steps

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

Mathematica [A]  time = 0.055458, size = 20, normalized size = 1.82 $\frac{\log (a \cosh (x)+b \sinh (x))-\log (\cosh (x))}{b}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[Sech[x]^2/(a + b*Tanh[x]),x]

[Out]

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

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Maple [A]  time = 0.023, size = 12, normalized size = 1.1 \begin{align*}{\frac{\ln \left ( a+b\tanh \left ( x \right ) \right ) }{b}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.04408, size = 15, normalized size = 1.36 \begin{align*} \frac{\log \left (b \tanh \left (x\right ) + a\right )}{b} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(x)^2/(a+b*tanh(x)),x, algorithm="maxima")

[Out]

log(b*tanh(x) + a)/b

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Fricas [B]  time = 2.06499, size = 126, normalized size = 11.45 \begin{align*} \frac{\log \left (\frac{2 \,{\left (a \cosh \left (x\right ) + b \sinh \left (x\right )\right )}}{\cosh \left (x\right ) - \sinh \left (x\right )}\right ) - \log \left (\frac{2 \, \cosh \left (x\right )}{\cosh \left (x\right ) - \sinh \left (x\right )}\right )}{b} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(x)^2/(a+b*tanh(x)),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{sech}^{2}{\left (x \right )}}{a + b \tanh{\left (x \right )}}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(x)**2/(a+b*tanh(x)),x)

[Out]

Integral(sech(x)**2/(a + b*tanh(x)), x)

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Giac [B]  time = 1.17001, size = 61, normalized size = 5.55 \begin{align*} \frac{{\left (a + b\right )} \log \left ({\left | a e^{\left (2 \, x\right )} + b e^{\left (2 \, x\right )} + a - b \right |}\right )}{a b + b^{2}} - \frac{\log \left (e^{\left (2 \, x\right )} + 1\right )}{b} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(x)^2/(a+b*tanh(x)),x, algorithm="giac")

[Out]

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