3.1000 \(\int \frac {\text {sech}^2(x)}{\sqrt {-4+\tanh ^2(x)}} \, dx\)

Optimal. Leaf size=14 \[ \tanh ^{-1}\left (\frac {\tanh (x)}{\sqrt {\tanh ^2(x)-4}}\right ) \]

[Out]

arctanh(tanh(x)/(-4+tanh(x)^2)^(1/2))

________________________________________________________________________________________

Rubi [A]  time = 0.05, antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3675, 217, 206} \[ \tanh ^{-1}\left (\frac {\tanh (x)}{\sqrt {\tanh ^2(x)-4}}\right ) \]

Antiderivative was successfully verified.

[In]

Int[Sech[x]^2/Sqrt[-4 + Tanh[x]^2],x]

[Out]

ArcTanh[Tanh[x]/Sqrt[-4 + Tanh[x]^2]]

Rule 206

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

Rule 217

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 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)}{\sqrt {-4+\tanh ^2(x)}} \, dx &=\operatorname {Subst}\left (\int \frac {1}{\sqrt {-4+x^2}} \, dx,x,\tanh (x)\right )\\ &=\operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {\tanh (x)}{\sqrt {-4+\tanh ^2(x)}}\right )\\ &=\tanh ^{-1}\left (\frac {\tanh (x)}{\sqrt {-4+\tanh ^2(x)}}\right )\\ \end {align*}

________________________________________________________________________________________

Mathematica [B]  time = 0.04, size = 46, normalized size = 3.29 \[ \frac {\sqrt {3 \cosh (2 x)+5} \text {sech}(x) \tan ^{-1}\left (\frac {\sinh (x)}{\sqrt {3 \sinh ^2(x)+4}}\right )}{\sqrt {2} \sqrt {\tanh ^2(x)-4}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sech[x]^2/Sqrt[-4 + Tanh[x]^2],x]

[Out]

(ArcTan[Sinh[x]/Sqrt[4 + 3*Sinh[x]^2]]*Sqrt[5 + 3*Cosh[2*x]]*Sech[x])/(Sqrt[2]*Sqrt[-4 + Tanh[x]^2])

________________________________________________________________________________________

fricas [A]  time = 0.47, size = 1, normalized size = 0.07 \[ 0 \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(x)^2/(-4+tanh(x)^2)^(1/2),x, algorithm="fricas")

[Out]

0

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {sech}\relax (x)^{2}}{\sqrt {\tanh \relax (x)^{2} - 4}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(x)^2/(-4+tanh(x)^2)^(1/2),x, algorithm="giac")

[Out]

integrate(sech(x)^2/sqrt(tanh(x)^2 - 4), x)

________________________________________________________________________________________

maple [F]  time = 0.52, size = 0, normalized size = 0.00 \[ \int \frac {\mathrm {sech}\relax (x )^{2}}{\sqrt {-4+\tanh ^{2}\relax (x )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sech(x)^2/(-4+tanh(x)^2)^(1/2),x)

[Out]

int(sech(x)^2/(-4+tanh(x)^2)^(1/2),x)

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {sech}\relax (x)^{2}}{\sqrt {\tanh \relax (x)^{2} - 4}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(x)^2/(-4+tanh(x)^2)^(1/2),x, algorithm="maxima")

[Out]

integrate(sech(x)^2/sqrt(tanh(x)^2 - 4), x)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.07 \[ \int \frac {1}{{\mathrm {cosh}\relax (x)}^2\,\sqrt {{\mathrm {tanh}\relax (x)}^2-4}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(cosh(x)^2*(tanh(x)^2 - 4)^(1/2)),x)

[Out]

int(1/(cosh(x)^2*(tanh(x)^2 - 4)^(1/2)), x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(x)**2/(-4+tanh(x)**2)**(1/2),x)

[Out]

Integral(sech(x)**2/sqrt((tanh(x) - 2)*(tanh(x) + 2)), x)

________________________________________________________________________________________