3.80 \(\int \frac{\tanh (x)}{\sqrt{1-\cosh ^2(x)}} \, dx\)

Optimal. Leaf size=13 \[ -\tanh ^{-1}\left (\sqrt{-\sinh ^2(x)}\right ) \]

[Out]

-ArcTanh[Sqrt[-Sinh[x]^2]]

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Rubi [A]  time = 0.0607551, antiderivative size = 13, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267, Rules used = {3176, 3205, 63, 206} \[ -\tanh ^{-1}\left (\sqrt{-\sinh ^2(x)}\right ) \]

Antiderivative was successfully verified.

[In]

Int[Tanh[x]/Sqrt[1 - Cosh[x]^2],x]

[Out]

-ArcTanh[Sqrt[-Sinh[x]^2]]

Rule 3176

Int[(u_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> Int[ActivateTrig[u*(a*cos[e + f*x]^2)^p]
, x] /; FreeQ[{a, b, e, f, p}, x] && EqQ[a + b, 0]

Rule 3205

Int[((b_.)*sin[(e_.) + (f_.)*(x_)]^(n_))^(p_.)*tan[(e_.) + (f_.)*(x_)]^(m_.), x_Symbol] :> With[{ff = FreeFact
ors[Sin[e + f*x]^2, x]}, Dist[ff^((m + 1)/2)/(2*f), Subst[Int[(x^((m - 1)/2)*(b*ff^(n/2)*x^(n/2))^p)/(1 - ff*x
)^((m + 1)/2), x], x, Sin[e + f*x]^2/ff], x]] /; FreeQ[{b, e, f, p}, x] && IntegerQ[(m - 1)/2] && IntegerQ[n/2
]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

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])

Rubi steps

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

Mathematica [A]  time = 0.0129443, size = 21, normalized size = 1.62 \[ \frac{2 \sinh (x) \tan ^{-1}\left (\tanh \left (\frac{x}{2}\right )\right )}{\sqrt{-\sinh ^2(x)}} \]

Antiderivative was successfully verified.

[In]

Integrate[Tanh[x]/Sqrt[1 - Cosh[x]^2],x]

[Out]

(2*ArcTan[Tanh[x/2]]*Sinh[x])/Sqrt[-Sinh[x]^2]

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Maple [A]  time = 0.075, size = 12, normalized size = 0.9 \begin{align*} -{\it Artanh} \left ({\frac{1}{\sqrt{- \left ( \sinh \left ( x \right ) \right ) ^{2}}}} \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-arctanh(1/(-sinh(x)^2)^(1/2))

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Maxima [C]  time = 1.617, size = 9, normalized size = 0.69 \begin{align*} -2 i \, \arctan \left (e^{\left (-x\right )}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*I*arctan(e^(-x))

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Fricas [A]  time = 2.21386, size = 4, normalized size = 0.31 \begin{align*} 0 \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

0

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(tanh(x)/sqrt(-(cosh(x) - 1)*(cosh(x) + 1)), x)

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Giac [C]  time = 1.31964, size = 51, normalized size = 3.92 \begin{align*} -\frac{\log \left (e^{x} + i\right )}{\mathrm{sgn}\left (-e^{\left (3 \, x\right )} + e^{x}\right )} + \frac{\log \left (e^{x} - i\right )}{\mathrm{sgn}\left (-e^{\left (3 \, x\right )} + e^{x}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-log(e^x + I)/sgn(-e^(3*x) + e^x) + log(e^x - I)/sgn(-e^(3*x) + e^x)