### 3.981 $$\int \frac{\text{sech}^2(x)}{9+\tanh ^2(x)} \, dx$$

Optimal. Leaf size=11 $\frac{1}{3} \tan ^{-1}\left (\frac{\tanh (x)}{3}\right )$

[Out]

ArcTan[Tanh[x]/3]/3

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Rubi [A]  time = 0.0340017, 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 = {3675, 203} $\frac{1}{3} \tan ^{-1}\left (\frac{\tanh (x)}{3}\right )$

Antiderivative was successfully veriﬁed.

[In]

Int[Sech[x]^2/(9 + Tanh[x]^2),x]

[Out]

ArcTan[Tanh[x]/3]/3

Rule 3675

Int[sec[(e_.) + (f_.)*(x_)]^(m_)*((a_) + (b_.)*((c_.)*tan[(e_.) + (f_.)*(x_)])^(n_))^(p_.), x_Symbol] :> With[
{ff = FreeFactors[Tan[e + f*x], x]}, Dist[ff/(c^(m - 1)*f), Subst[Int[(c^2 + ff^2*x^2)^(m/2 - 1)*(a + b*(ff*x)
^n)^p, x], x, (c*Tan[e + f*x])/ff], x]] /; FreeQ[{a, b, c, e, f, n, p}, x] && IntegerQ[m/2] && (IntegersQ[n, p
] || IGtQ[m, 0] || IGtQ[p, 0] || EqQ[n^2, 4] || EqQ[n^2, 16])

Rule 203

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

Rubi steps

\begin{align*} \int \frac{\text{sech}^2(x)}{9+\tanh ^2(x)} \, dx &=\operatorname{Subst}\left (\int \frac{1}{9+x^2} \, dx,x,\tanh (x)\right )\\ &=\frac{1}{3} \tan ^{-1}\left (\frac{\tanh (x)}{3}\right )\\ \end{align*}

Mathematica [F]  time = 0.0196953, size = 0, normalized size = 0. $\int \frac{\text{sech}^2(x)}{9+\tanh ^2(x)} \, dx$

Veriﬁcation is Not applicable to the result.

[In]

Integrate[Sech[x]^2/(9 + Tanh[x]^2),x]

[Out]

Integrate[Sech[x]^2/(9 + Tanh[x]^2), x]

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Maple [B]  time = 0.071, size = 116, normalized size = 10.6 \begin{align*} -2\,{\frac{\sqrt{10}}{6\,\sqrt{10}+6}\arctan \left ( 18\,{\frac{\tanh \left ( x/2 \right ) }{6\,\sqrt{10}+6}} \right ) }-2\,{\frac{1}{6\,\sqrt{10}+6}\arctan \left ( 18\,{\frac{\tanh \left ( x/2 \right ) }{6\,\sqrt{10}+6}} \right ) }+2\,{\frac{\sqrt{10}}{6\,\sqrt{10}-6}\arctan \left ( 18\,{\frac{\tanh \left ( x/2 \right ) }{6\,\sqrt{10}-6}} \right ) }-2\,{\frac{1}{6\,\sqrt{10}-6}\arctan \left ( 18\,{\frac{\tanh \left ( x/2 \right ) }{6\,\sqrt{10}-6}} \right ) } \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(sech(x)^2/(9+tanh(x)^2),x)

[Out]

-2*10^(1/2)/(6*10^(1/2)+6)*arctan(18*tanh(1/2*x)/(6*10^(1/2)+6))-2/(6*10^(1/2)+6)*arctan(18*tanh(1/2*x)/(6*10^
(1/2)+6))+2*10^(1/2)/(6*10^(1/2)-6)*arctan(18*tanh(1/2*x)/(6*10^(1/2)-6))-2/(6*10^(1/2)-6)*arctan(18*tanh(1/2*
x)/(6*10^(1/2)-6))

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Maxima [A]  time = 1.57431, size = 15, normalized size = 1.36 \begin{align*} -\frac{1}{3} \, \arctan \left (\frac{5}{3} \, e^{\left (-2 \, x\right )} + \frac{4}{3}\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(x)^2/(9+tanh(x)^2),x, algorithm="maxima")

[Out]

-1/3*arctan(5/3*e^(-2*x) + 4/3)

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Fricas [B]  time = 2.33005, size = 82, normalized size = 7.45 \begin{align*} -\frac{1}{3} \, \arctan \left (-\frac{9 \, \cosh \left (x\right ) + \sinh \left (x\right )}{3 \,{\left (\cosh \left (x\right ) - \sinh \left (x\right )\right )}}\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(x)^2/(9+tanh(x)^2),x, algorithm="fricas")

[Out]

-1/3*arctan(-1/3*(9*cosh(x) + sinh(x))/(cosh(x) - sinh(x)))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{sech}^{2}{\left (x \right )}}{\tanh ^{2}{\left (x \right )} + 9}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(x)**2/(9+tanh(x)**2),x)

[Out]

Integral(sech(x)**2/(tanh(x)**2 + 9), x)

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Giac [A]  time = 1.1588, size = 15, normalized size = 1.36 \begin{align*} \frac{1}{3} \, \arctan \left (\frac{5}{3} \, e^{\left (2 \, x\right )} + \frac{4}{3}\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(x)^2/(9+tanh(x)^2),x, algorithm="giac")

[Out]

1/3*arctan(5/3*e^(2*x) + 4/3)