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

Optimal. Leaf size=9 \[ \frac{\log (\cosh (x)+1)}{a} \]

[Out]

Log[1 + Cosh[x]]/a

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Rubi [A]  time = 0.0266844, antiderivative size = 9, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {3879, 31} \[ \frac{\log (\cosh (x)+1)}{a} \]

Antiderivative was successfully verified.

[In]

Int[Tanh[x]/(a + a*Sech[x]),x]

[Out]

Log[1 + Cosh[x]]/a

Rule 3879

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

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

Mathematica [A]  time = 0.0069035, size = 12, normalized size = 1.33 \[ \frac{2 \log \left (\cosh \left (\frac{x}{2}\right )\right )}{a} \]

Antiderivative was successfully verified.

[In]

Integrate[Tanh[x]/(a + a*Sech[x]),x]

[Out]

(2*Log[Cosh[x/2]])/a

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Maple [A]  time = 0.016, size = 19, normalized size = 2.1 \begin{align*}{\frac{\ln \left ( 1+{\rm sech} \left (x\right ) \right ) }{a}}-{\frac{\ln \left ({\rm sech} \left (x\right ) \right ) }{a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.13244, size = 24, normalized size = 2.67 \begin{align*} \frac{x}{a} + \frac{2 \, \log \left (e^{\left (-x\right )} + 1\right )}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x/a + 2*log(e^(-x) + 1)/a

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Fricas [A]  time = 2.4193, size = 53, normalized size = 5.89 \begin{align*} -\frac{x - 2 \, \log \left (\cosh \left (x\right ) + \sinh \left (x\right ) + 1\right )}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(x - 2*log(cosh(x) + sinh(x) + 1))/a

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Sympy [B]  time = 0.20549, size = 19, normalized size = 2.11 \begin{align*} \frac{x}{a} - \frac{\log{\left (\tanh{\left (x \right )} + 1 \right )}}{a} + \frac{\log{\left (\operatorname{sech}{\left (x \right )} + 1 \right )}}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x/a - log(tanh(x) + 1)/a + log(sech(x) + 1)/a

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Giac [A]  time = 1.1518, size = 23, normalized size = 2.56 \begin{align*} -\frac{x}{a} + \frac{2 \, \log \left (e^{x} + 1\right )}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-x/a + 2*log(e^x + 1)/a