\(\int \frac {\tanh (x)}{\sqrt {a+b \cosh (x)}} \, dx\) [198]

Optimal result

Integrand size = 13, antiderivative size = 24 \[ \int \frac {\tanh (x)}{\sqrt {a+b \cosh (x)}} \, dx=-\frac {2 \text {arctanh}\left (\frac {\sqrt {a+b \cosh (x)}}{\sqrt {a}}\right )}{\sqrt {a}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {2800, 65, 213} \[ \int \frac {\tanh (x)}{\sqrt {a+b \cosh (x)}} \, dx=-\frac {2 \text {arctanh}\left (\frac {\sqrt {a+b \cosh (x)}}{\sqrt {a}}\right )}{\sqrt {a}} \]

[In]

Int[Tanh[x]/Sqrt[a + b*Cosh[x]],x]

[Out]

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

Rule 65

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 213

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

Rule 2800

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

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {\tanh (x)}{\sqrt {a+b \cosh (x)}} \, dx=-\frac {2 \text {arctanh}\left (\frac {\sqrt {a+b \cosh (x)}}{\sqrt {a}}\right )}{\sqrt {a}} \]

[In]

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

[Out]

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(103\) vs. \(2(18)=36\).

Time = 0.21 (sec) , antiderivative size = 104, normalized size of antiderivative = 4.33

method	result	size
default	\(-\frac {\ln \left (\frac {4 \cosh \left (\frac {x}{2}\right ) b \sqrt {2}+4 \sqrt {a}\, \sqrt {2 \sinh \left (\frac {x}{2}\right )^{2} b +a +b}+4 a -4 b}{2 \cosh \left (\frac {x}{2}\right )-\sqrt {2}}\right )+\ln \left (-\frac {4 \left (\cosh \left (\frac {x}{2}\right ) b \sqrt {2}-\sqrt {a}\, \sqrt {2 \sinh \left (\frac {x}{2}\right )^{2} b +a +b}-a +b \right )}{2 \cosh \left (\frac {x}{2}\right )+\sqrt {2}}\right )}{\sqrt {a}}\)	\(104\)

[In]

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

[Out]

-1/a^(1/2)*(ln(4/(2*cosh(1/2*x)-2^(1/2))*(cosh(1/2*x)*b*2^(1/2)+a^(1/2)*(2*sinh(1/2*x)^2*b+a+b)^(1/2)+a-b))+ln
(-4/(2*cosh(1/2*x)+2^(1/2))*(cosh(1/2*x)*b*2^(1/2)-a^(1/2)*(2*sinh(1/2*x)^2*b+a+b)^(1/2)-a+b)))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 96 vs. \(2 (18) = 36\).

Time = 0.29 (sec) , antiderivative size = 356, normalized size of antiderivative = 14.83 \[ \int \frac {\tanh (x)}{\sqrt {a+b \cosh (x)}} \, dx=\left [\frac {\log \left (\frac {b^{2} \cosh \left (x\right )^{4} + b^{2} \sinh \left (x\right )^{4} + 16 \, a b \cosh \left (x\right )^{3} + 4 \, {\left (b^{2} \cosh \left (x\right ) + 4 \, a b\right )} \sinh \left (x\right )^{3} + 16 \, a b \cosh \left (x\right ) + 2 \, {\left (16 \, a^{2} + b^{2}\right )} \cosh \left (x\right )^{2} + 2 \, {\left (3 \, b^{2} \cosh \left (x\right )^{2} + 24 \, a b \cosh \left (x\right ) + 16 \, a^{2} + b^{2}\right )} \sinh \left (x\right )^{2} - 8 \, {\left (b \cosh \left (x\right )^{3} + b \sinh \left (x\right )^{3} + 4 \, a \cosh \left (x\right )^{2} + {\left (3 \, b \cosh \left (x\right ) + 4 \, a\right )} \sinh \left (x\right )^{2} + b \cosh \left (x\right ) + {\left (3 \, b \cosh \left (x\right )^{2} + 8 \, a \cosh \left (x\right ) + b\right )} \sinh \left (x\right )\right )} \sqrt {b \cosh \left (x\right ) + a} \sqrt {a} + b^{2} + 4 \, {\left (b^{2} \cosh \left (x\right )^{3} + 12 \, a b \cosh \left (x\right )^{2} + 4 \, a b + {\left (16 \, a^{2} + b^{2}\right )} \cosh \left (x\right )\right )} \sinh \left (x\right )}{\cosh \left (x\right )^{4} + 4 \, \cosh \left (x\right ) \sinh \left (x\right )^{3} + \sinh \left (x\right )^{4} + 2 \, {\left (3 \, \cosh \left (x\right )^{2} + 1\right )} \sinh \left (x\right )^{2} + 2 \, \cosh \left (x\right )^{2} + 4 \, {\left (\cosh \left (x\right )^{3} + \cosh \left (x\right )\right )} \sinh \left (x\right ) + 1}\right )}{2 \, \sqrt {a}}, \frac {\sqrt {-a} \arctan \left (\frac {{\left (b \cosh \left (x\right )^{2} + b \sinh \left (x\right )^{2} + 4 \, a \cosh \left (x\right ) + 2 \, {\left (b \cosh \left (x\right ) + 2 \, a\right )} \sinh \left (x\right ) + b\right )} \sqrt {b \cosh \left (x\right ) + a} \sqrt {-a}}{2 \, {\left (a b \cosh \left (x\right )^{2} + a b \sinh \left (x\right )^{2} + 2 \, a^{2} \cosh \left (x\right ) + a b + 2 \, {\left (a b \cosh \left (x\right ) + a^{2}\right )} \sinh \left (x\right )\right )}}\right )}{a}\right ] \]

[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]

Sympy [F]

\[ \int \frac {\tanh (x)}{\sqrt {a+b \cosh (x)}} \, dx=\int \frac {\tanh {\left (x \right )}}{\sqrt {a + b \cosh {\left (x \right )}}}\, dx \]

[In]

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

[Out]

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

Maxima [F]

\[ \int \frac {\tanh (x)}{\sqrt {a+b \cosh (x)}} \, dx=\int { \frac {\tanh \left (x\right )}{\sqrt {b \cosh \left (x\right ) + a}} \,d x } \]

[In]

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

[Out]

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

Giac [F]

\[ \int \frac {\tanh (x)}{\sqrt {a+b \cosh (x)}} \, dx=\int { \frac {\tanh \left (x\right )}{\sqrt {b \cosh \left (x\right ) + a}} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {\tanh (x)}{\sqrt {a+b \cosh (x)}} \, dx=\int \frac {\mathrm {tanh}\left (x\right )}{\sqrt {a+b\,\mathrm {cosh}\left (x\right )}} \,d x \]

[In]

int(tanh(x)/(a + b*cosh(x))^(1/2),x)

[Out]

int(tanh(x)/(a + b*cosh(x))^(1/2), x)