3.198 \(\int \frac {\tanh (x)}{\sqrt {a+b \cosh (x)}} \, dx\)

Optimal. Leaf size=24 \[ -\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a+b \cosh (x)}}{\sqrt {a}}\right )}{\sqrt {a}} \]

[Out]

-2*arctanh((a+b*cosh(x))^(1/2)/a^(1/2))/a^(1/2)

________________________________________________________________________________________

Rubi [A]  time = 0.06, antiderivative size = 24, 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 = {2721, 63, 207} \[ -\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a+b \cosh (x)}}{\sqrt {a}}\right )}{\sqrt {a}} \]

Antiderivative was successfully verified.

[In]

Int[Tanh[x]/Sqrt[a + b*Cosh[x]],x]

[Out]

(-2*ArcTanh[Sqrt[a + b*Cosh[x]]/Sqrt[a]])/Sqrt[a]

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 2721

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*tan[(e_.) + (f_.)*(x_)]^(p_.), x_Symbol] :> Dist[1/f, Subst[I
nt[(x^p*(a + x)^m)/(b^2 - x^2)^((p + 1)/2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x] && NeQ[a^2
- b^2, 0] && IntegerQ[(p + 1)/2]

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]  time = 0.01, size = 24, normalized size = 1.00 \[ -\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a+b \cosh (x)}}{\sqrt {a}}\right )}{\sqrt {a}} \]

Antiderivative was successfully verified.

[In]

Integrate[Tanh[x]/Sqrt[a + b*Cosh[x]],x]

[Out]

(-2*ArcTanh[Sqrt[a + b*Cosh[x]]/Sqrt[a]])/Sqrt[a]

________________________________________________________________________________________

fricas [B]  time = 0.65, size = 356, normalized size = 14.83 \[ \left [\frac {\log \left (\frac {b^{2} \cosh \relax (x)^{4} + b^{2} \sinh \relax (x)^{4} + 16 \, a b \cosh \relax (x)^{3} + 4 \, {\left (b^{2} \cosh \relax (x) + 4 \, a b\right )} \sinh \relax (x)^{3} + 16 \, a b \cosh \relax (x) + 2 \, {\left (16 \, a^{2} + b^{2}\right )} \cosh \relax (x)^{2} + 2 \, {\left (3 \, b^{2} \cosh \relax (x)^{2} + 24 \, a b \cosh \relax (x) + 16 \, a^{2} + b^{2}\right )} \sinh \relax (x)^{2} - 8 \, {\left (b \cosh \relax (x)^{3} + b \sinh \relax (x)^{3} + 4 \, a \cosh \relax (x)^{2} + {\left (3 \, b \cosh \relax (x) + 4 \, a\right )} \sinh \relax (x)^{2} + b \cosh \relax (x) + {\left (3 \, b \cosh \relax (x)^{2} + 8 \, a \cosh \relax (x) + b\right )} \sinh \relax (x)\right )} \sqrt {b \cosh \relax (x) + a} \sqrt {a} + b^{2} + 4 \, {\left (b^{2} \cosh \relax (x)^{3} + 12 \, a b \cosh \relax (x)^{2} + 4 \, a b + {\left (16 \, a^{2} + b^{2}\right )} \cosh \relax (x)\right )} \sinh \relax (x)}{\cosh \relax (x)^{4} + 4 \, \cosh \relax (x) \sinh \relax (x)^{3} + \sinh \relax (x)^{4} + 2 \, {\left (3 \, \cosh \relax (x)^{2} + 1\right )} \sinh \relax (x)^{2} + 2 \, \cosh \relax (x)^{2} + 4 \, {\left (\cosh \relax (x)^{3} + \cosh \relax (x)\right )} \sinh \relax (x) + 1}\right )}{2 \, \sqrt {a}}, \frac {\sqrt {-a} \arctan \left (\frac {{\left (b \cosh \relax (x)^{2} + b \sinh \relax (x)^{2} + 4 \, a \cosh \relax (x) + 2 \, {\left (b \cosh \relax (x) + 2 \, a\right )} \sinh \relax (x) + b\right )} \sqrt {b \cosh \relax (x) + a} \sqrt {-a}}{2 \, {\left (a b \cosh \relax (x)^{2} + a b \sinh \relax (x)^{2} + 2 \, a^{2} \cosh \relax (x) + a b + 2 \, {\left (a b \cosh \relax (x) + a^{2}\right )} \sinh \relax (x)\right )}}\right )}{a}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/2*log((b^2*cosh(x)^4 + b^2*sinh(x)^4 + 16*a*b*cosh(x)^3 + 4*(b^2*cosh(x) + 4*a*b)*sinh(x)^3 + 16*a*b*cosh(x
) + 2*(16*a^2 + b^2)*cosh(x)^2 + 2*(3*b^2*cosh(x)^2 + 24*a*b*cosh(x) + 16*a^2 + b^2)*sinh(x)^2 - 8*(b*cosh(x)^
3 + b*sinh(x)^3 + 4*a*cosh(x)^2 + (3*b*cosh(x) + 4*a)*sinh(x)^2 + b*cosh(x) + (3*b*cosh(x)^2 + 8*a*cosh(x) + b
)*sinh(x))*sqrt(b*cosh(x) + a)*sqrt(a) + b^2 + 4*(b^2*cosh(x)^3 + 12*a*b*cosh(x)^2 + 4*a*b + (16*a^2 + b^2)*co
sh(x))*sinh(x))/(cosh(x)^4 + 4*cosh(x)*sinh(x)^3 + sinh(x)^4 + 2*(3*cosh(x)^2 + 1)*sinh(x)^2 + 2*cosh(x)^2 + 4
*(cosh(x)^3 + cosh(x))*sinh(x) + 1))/sqrt(a), sqrt(-a)*arctan(1/2*(b*cosh(x)^2 + b*sinh(x)^2 + 4*a*cosh(x) + 2
*(b*cosh(x) + 2*a)*sinh(x) + b)*sqrt(b*cosh(x) + a)*sqrt(-a)/(a*b*cosh(x)^2 + a*b*sinh(x)^2 + 2*a^2*cosh(x) +
a*b + 2*(a*b*cosh(x) + a^2)*sinh(x)))/a]

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(tanh(x)/sqrt(b*cosh(x) + a), x)

________________________________________________________________________________________

maple [A]  time = 0.07, size = 19, normalized size = 0.79 \[ -\frac {2 \arctanh \left (\frac {\sqrt {a +b \cosh \relax (x )}}{\sqrt {a}}\right )}{\sqrt {a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*arctanh((a+b*cosh(x))^(1/2)/a^(1/2))/a^(1/2)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(tanh(x)/sqrt(b*cosh(x) + a), x)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.04 \[ \int \frac {\mathrm {tanh}\relax (x)}{\sqrt {a+b\,\mathrm {cosh}\relax (x)}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tanh(x)/(a + b*cosh(x))^(1/2),x)

[Out]

int(tanh(x)/(a + b*cosh(x))^(1/2), x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tanh(x)/(a+b*cosh(x))**(1/2),x)

[Out]

Integral(tanh(x)/sqrt(a + b*cosh(x)), x)

________________________________________________________________________________________