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

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

[Out]

-arctanh((1+cosh(x)^2)^(1/2))

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Rubi [A]  time = 0.04, antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {3194, 63, 207} \[ -\tanh ^{-1}\left (\sqrt {\cosh ^2(x)+1}\right ) \]

Antiderivative was successfully verified.

[In]

Int[Tanh[x]/Sqrt[1 + Cosh[x]^2],x]

[Out]

-ArcTanh[Sqrt[1 + Cosh[x]^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 207

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

Rule 3194

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

Rubi steps

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

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Mathematica [A]  time = 0.01, size = 13, normalized size = 1.00 \[ -\tanh ^{-1}\left (\sqrt {\cosh ^2(x)+1}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-ArcTanh[Sqrt[1 + Cosh[x]^2]]

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fricas [B]  time = 0.49, size = 63, normalized size = 4.85 \[ \log \left (\frac {\sqrt {2} \sqrt {\frac {\cosh \relax (x)^{2} + \sinh \relax (x)^{2} + 3}{\cosh \relax (x)^{2} - 2 \, \cosh \relax (x) \sinh \relax (x) + \sinh \relax (x)^{2}}} - 2 \, \cosh \relax (x) - 2 \, \sinh \relax (x)}{\cosh \relax (x)^{2} + 2 \, \cosh \relax (x) \sinh \relax (x) + \sinh \relax (x)^{2} + 1}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log((sqrt(2)*sqrt((cosh(x)^2 + sinh(x)^2 + 3)/(cosh(x)^2 - 2*cosh(x)*sinh(x) + sinh(x)^2)) - 2*cosh(x) - 2*sin
h(x))/(cosh(x)^2 + 2*cosh(x)*sinh(x) + sinh(x)^2 + 1))

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\tanh \relax (x)}{\sqrt {\cosh \relax (x)^{2} + 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(tanh(x)/sqrt(cosh(x)^2 + 1), x)

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maple [A]  time = 0.09, size = 12, normalized size = 0.92 \[ -\arctanh \left (\frac {1}{\sqrt {1+\cosh ^{2}\relax (x )}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-arctanh(1/(1+cosh(x)^2)^(1/2))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\tanh \relax (x)}{\sqrt {\cosh \relax (x)^{2} + 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(tanh(x)/sqrt(cosh(x)^2 + 1), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.08 \[ \int \frac {\mathrm {tanh}\relax (x)}{\sqrt {{\mathrm {cosh}\relax (x)}^2+1}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tanh(x)/(cosh(x)^2 + 1)^(1/2),x)

[Out]

int(tanh(x)/(cosh(x)^2 + 1)^(1/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\tanh {\relax (x )}}{\sqrt {\cosh ^{2}{\relax (x )} + 1}}\, dx \]

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)**2 + 1), x)

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