3.107 \(\int \frac{\tanh ^2(x)}{a+a \text{sech}(x)} \, dx\)

Optimal. Leaf size=14 \[ \frac{x}{a}-\frac{\tan ^{-1}(\sinh (x))}{a} \]

[Out]

x/a - ArcTan[Sinh[x]]/a

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Rubi [A]  time = 0.0457572, antiderivative size = 14, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {3888, 3770} \[ \frac{x}{a}-\frac{\tan ^{-1}(\sinh (x))}{a} \]

Antiderivative was successfully verified.

[In]

Int[Tanh[x]^2/(a + a*Sech[x]),x]

[Out]

x/a - ArcTan[Sinh[x]]/a

Rule 3888

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

Rule 3770

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]

Rubi steps

\begin{align*} \int \frac{\tanh ^2(x)}{a+a \text{sech}(x)} \, dx &=-\frac{\int (-a+a \text{sech}(x)) \, dx}{a^2}\\ &=\frac{x}{a}-\frac{\int \text{sech}(x) \, dx}{a}\\ &=\frac{x}{a}-\frac{\tan ^{-1}(\sinh (x))}{a}\\ \end{align*}

Mathematica [A]  time = 0.0295558, size = 15, normalized size = 1.07 \[ \frac{x-2 \tan ^{-1}\left (\tanh \left (\frac{x}{2}\right )\right )}{a} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x - 2*ArcTan[Tanh[x/2]])/a

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Maple [B]  time = 0.019, size = 35, normalized size = 2.5 \begin{align*}{\frac{1}{a}\ln \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) }-{\frac{1}{a}\ln \left ( \tanh \left ({\frac{x}{2}} \right ) -1 \right ) }-2\,{\frac{\arctan \left ( \tanh \left ( x/2 \right ) \right ) }{a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/a*ln(tanh(1/2*x)+1)-1/a*ln(tanh(1/2*x)-1)-2/a*arctan(tanh(1/2*x))

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Maxima [A]  time = 1.65944, size = 22, normalized size = 1.57 \begin{align*} \frac{x}{a} + \frac{2 \, \arctan \left (e^{\left (-x\right )}\right )}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x/a + 2*arctan(e^(-x))/a

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Fricas [A]  time = 2.42832, size = 50, normalized size = 3.57 \begin{align*} \frac{x - 2 \, \arctan \left (\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)^2/(a+a*sech(x)),x, algorithm="fricas")

[Out]

(x - 2*arctan(cosh(x) + sinh(x)))/a

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{\tanh ^{2}{\left (x \right )}}{\operatorname{sech}{\left (x \right )} + 1}\, dx}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(tanh(x)**2/(sech(x) + 1), x)/a

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Giac [A]  time = 1.15041, size = 19, normalized size = 1.36 \begin{align*} \frac{x}{a} - \frac{2 \, \arctan \left (e^{x}\right )}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x/a - 2*arctan(e^x)/a