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]

1/3*arctan(1/3*tanh(x))

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Rubi [A]  time = 0.03, antiderivative size = 11, normalized size of antiderivative = 1.00, 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 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])

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])

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*}

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Mathematica [F]  time = 0.02, size = 0, normalized size = 0.00 \[ \int \frac {\text {sech}^2(x)}{9+\tanh ^2(x)} \, dx \]

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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fricas [B]  time = 0.43, size = 21, normalized size = 1.91 \[ -\frac {1}{3} \, \arctan \left (-\frac {9 \, \cosh \relax (x) + \sinh \relax (x)}{3 \, {\left (\cosh \relax (x) - \sinh \relax (x)\right )}}\right ) \]

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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giac [A]  time = 0.13, size = 11, normalized size = 1.00 \[ \frac {1}{3} \, \arctan \left (\frac {5}{3} \, e^{\left (2 \, x\right )} + \frac {4}{3}\right ) \]

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)

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maple [B]  time = 0.30, size = 116, normalized size = 10.55 \[ -\frac {2 \sqrt {10}\, \arctan \left (\frac {18 \tanh \left (\frac {x}{2}\right )}{6 \sqrt {10}+6}\right )}{6 \sqrt {10}+6}-\frac {2 \arctan \left (\frac {18 \tanh \left (\frac {x}{2}\right )}{6 \sqrt {10}+6}\right )}{6 \sqrt {10}+6}+\frac {2 \sqrt {10}\, \arctan \left (\frac {18 \tanh \left (\frac {x}{2}\right )}{6 \sqrt {10}-6}\right )}{6 \sqrt {10}-6}-\frac {2 \arctan \left (\frac {18 \tanh \left (\frac {x}{2}\right )}{6 \sqrt {10}-6}\right )}{6 \sqrt {10}-6} \]

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 = 0.41, size = 11, normalized size = 1.00 \[ -\frac {1}{3} \, \arctan \left (\frac {5}{3} \, e^{\left (-2 \, x\right )} + \frac {4}{3}\right ) \]

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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mupad [B]  time = 1.64, size = 11, normalized size = 1.00 \[ \frac {\mathrm {atan}\left (\frac {5\,{\mathrm {e}}^{2\,x}}{3}+\frac {4}{3}\right )}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(cosh(x)^2*(tanh(x)^2 + 9)),x)

[Out]

atan((5*exp(2*x))/3 + 4/3)/3

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {sech}^{2}{\relax (x )}}{\tanh ^{2}{\relax (x )} + 9}\, dx \]

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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