### 3.202 $$\int \sqrt{1-\tanh ^2(x)} \, dx$$

Optimal. Leaf size=3 $\sin ^{-1}(\tanh (x))$

[Out]

ArcSin[Tanh[x]]

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Rubi [A]  time = 0.0165158, antiderivative size = 3, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 12, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.25, Rules used = {3657, 4122, 216} $\sin ^{-1}(\tanh (x))$

Antiderivative was successfully veriﬁed.

[In]

Int[Sqrt[1 - Tanh[x]^2],x]

[Out]

ArcSin[Tanh[x]]

Rule 3657

Int[(u_.)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> Int[ActivateTrig[u*(a*sec[e + f*x]^2)^p]
, x] /; FreeQ[{a, b, e, f, p}, x] && EqQ[a, b]

Rule 4122

Int[((b_.)*sec[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Dist[(b*ff)
/f, Subst[Int[(b + b*ff^2*x^2)^(p - 1), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{b, e, f, p}, x] &&  !IntegerQ[p
]

Rule 216

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[(Rt[-b, 2]*x)/Sqrt[a]]/Rt[-b, 2], x] /; FreeQ[{a, b}
, x] && GtQ[a, 0] && NegQ[b]

Rubi steps

\begin{align*} \int \sqrt{1-\tanh ^2(x)} \, dx &=\int \sqrt{\text{sech}^2(x)} \, dx\\ &=\operatorname{Subst}\left (\int \frac{1}{\sqrt{1-x^2}} \, dx,x,\tanh (x)\right )\\ &=\sin ^{-1}(\tanh (x))\\ \end{align*}

Mathematica [B]  time = 0.007552, size = 19, normalized size = 6.33 $2 \cosh (x) \sqrt{\text{sech}^2(x)} \tan ^{-1}\left (\tanh \left (\frac{x}{2}\right )\right )$

Antiderivative was successfully veriﬁed.

[In]

Integrate[Sqrt[1 - Tanh[x]^2],x]

[Out]

2*ArcTan[Tanh[x/2]]*Cosh[x]*Sqrt[Sech[x]^2]

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Maple [A]  time = 0.033, size = 4, normalized size = 1.3 \begin{align*} \arcsin \left ( \tanh \left ( x \right ) \right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((1-tanh(x)^2)^(1/2),x)

[Out]

arcsin(tanh(x))

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Maxima [A]  time = 1.68692, size = 7, normalized size = 2.33 \begin{align*} 2 \, \arctan \left (e^{x}\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-tanh(x)^2)^(1/2),x, algorithm="maxima")

[Out]

2*arctan(e^x)

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Fricas [B]  time = 2.16123, size = 39, normalized size = 13. \begin{align*} 2 \, \arctan \left (\cosh \left (x\right ) + \sinh \left (x\right )\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-tanh(x)^2)^(1/2),x, algorithm="fricas")

[Out]

2*arctan(cosh(x) + sinh(x))

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-tanh(x)**2)**(1/2),x)

[Out]

Integral(sqrt(1 - tanh(x)**2), x)

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Giac [A]  time = 1.16934, size = 7, normalized size = 2.33 \begin{align*} 2 \, \arctan \left (e^{x}\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-tanh(x)^2)^(1/2),x, algorithm="giac")

[Out]

2*arctan(e^x)