3.119 \(\int \frac{\tanh (x)}{a+b \text{sech}(x)} \, dx\)

Optimal. Leaf size=19 \[ \frac{\log (a+b \text{sech}(x))}{a}+\frac{\log (\cosh (x))}{a} \]

[Out]

Log[Cosh[x]]/a + Log[a + b*Sech[x]]/a

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Rubi [A]  time = 0.0316895, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364, Rules used = {3885, 36, 29, 31} \[ \frac{\log (a+b \text{sech}(x))}{a}+\frac{\log (\cosh (x))}{a} \]

Antiderivative was successfully verified.

[In]

Int[Tanh[x]/(a + b*Sech[x]),x]

[Out]

Log[Cosh[x]]/a + Log[a + b*Sech[x]]/a

Rule 3885

Int[cot[(c_.) + (d_.)*(x_)]^(m_.)*(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_))^(n_), x_Symbol] :> -Dist[(-1)^((m - 1
)/2)/(d*b^(m - 1)), Subst[Int[((b^2 - x^2)^((m - 1)/2)*(a + x)^n)/x, x], x, b*Csc[c + d*x]], x] /; FreeQ[{a, b
, c, d, n}, x] && IntegerQ[(m - 1)/2] && NeQ[a^2 - b^2, 0]

Rule 36

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Dist[b/(b*c - a*d), Int[1/(a + b*x), x], x] -
Dist[d/(b*c - a*d), Int[1/(c + d*x), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

Rule 29

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

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

Mathematica [A]  time = 0.0173727, size = 11, normalized size = 0.58 \[ \frac{\log (a \cosh (x)+b)}{a} \]

Antiderivative was successfully verified.

[In]

Integrate[Tanh[x]/(a + b*Sech[x]),x]

[Out]

Log[b + a*Cosh[x]]/a

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Maple [A]  time = 0.014, size = 21, normalized size = 1.1 \begin{align*} -{\frac{\ln \left ({\rm sech} \left (x\right ) \right ) }{a}}+{\frac{\ln \left ( a+b{\rm sech} \left (x\right ) \right ) }{a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/a*ln(sech(x))+ln(a+b*sech(x))/a

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Maxima [A]  time = 1.13219, size = 35, normalized size = 1.84 \begin{align*} \frac{x}{a} + \frac{\log \left (2 \, b e^{\left (-x\right )} + a e^{\left (-2 \, x\right )} + a\right )}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 2.43681, size = 72, normalized size = 3.79 \begin{align*} -\frac{x - \log \left (\frac{2 \,{\left (a \cosh \left (x\right ) + b\right )}}{\cosh \left (x\right ) - \sinh \left (x\right )}\right )}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.585946, size = 41, normalized size = 2.16 \begin{align*} \begin{cases} \frac{\tilde{\infty }}{\operatorname{sech}{\left (x \right )}} & \text{for}\: a = 0 \wedge b = 0 \\\frac{x - \log{\left (\tanh{\left (x \right )} + 1 \right )}}{a} & \text{for}\: b = 0 \\\frac{1}{b \operatorname{sech}{\left (x \right )}} & \text{for}\: a = 0 \\\frac{x}{a} + \frac{\log{\left (\frac{a}{b} + \operatorname{sech}{\left (x \right )} \right )}}{a} - \frac{\log{\left (\tanh{\left (x \right )} + 1 \right )}}{a} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((zoo/sech(x), Eq(a, 0) & Eq(b, 0)), ((x - log(tanh(x) + 1))/a, Eq(b, 0)), (1/(b*sech(x)), Eq(a, 0)),
(x/a + log(a/b + sech(x))/a - log(tanh(x) + 1)/a, True))

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Giac [A]  time = 1.13428, size = 26, normalized size = 1.37 \begin{align*} \frac{\log \left ({\left | a{\left (e^{\left (-x\right )} + e^{x}\right )} + 2 \, b \right |}\right )}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(abs(a*(e^(-x) + e^x) + 2*b))/a