\(\int \coth (x) \sqrt {a+b \tanh ^2(x)} \, dx\) [214]

Optimal result

Integrand size = 15, antiderivative size = 56 \[ \int \coth (x) \sqrt {a+b \tanh ^2(x)} \, dx=-\sqrt {a} \text {arctanh}\left (\frac {\sqrt {a+b \tanh ^2(x)}}{\sqrt {a}}\right )+\sqrt {a+b} \text {arctanh}\left (\frac {\sqrt {a+b \tanh ^2(x)}}{\sqrt {a+b}}\right ) \] Output:

-a^(1/2)*arctanh((a+b*tanh(x)^2)^(1/2)/a^(1/2))+(a+b)^(1/2)*arctanh((a+b*t 
anh(x)^2)^(1/2)/(a+b)^(1/2))
 

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.00 \[ \int \coth (x) \sqrt {a+b \tanh ^2(x)} \, dx=-\sqrt {a} \text {arctanh}\left (\frac {\sqrt {a+b \tanh ^2(x)}}{\sqrt {a}}\right )+\sqrt {a+b} \text {arctanh}\left (\frac {\sqrt {a+b \tanh ^2(x)}}{\sqrt {a+b}}\right ) \] Input:

Integrate[Coth[x]*Sqrt[a + b*Tanh[x]^2],x]
 

Output:

-(Sqrt[a]*ArcTanh[Sqrt[a + b*Tanh[x]^2]/Sqrt[a]]) + Sqrt[a + b]*ArcTanh[Sq 
rt[a + b*Tanh[x]^2]/Sqrt[a + b]]
 

Rubi [A] (verified)

Time = 0.47 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.09, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.533, Rules used = {3042, 26, 4153, 26, 354, 94, 73, 221}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \coth (x) \sqrt {a+b \tanh ^2(x)} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {i \sqrt {a-b \tan (i x)^2}}{\tan (i x)}dx\)

\(\Big \downarrow \) 26

\(\displaystyle i \int \frac {\sqrt {a-b \tan (i x)^2}}{\tan (i x)}dx\)

\(\Big \downarrow \) 4153

\(\displaystyle i \int -\frac {i \coth (x) \sqrt {b \tanh ^2(x)+a}}{1-\tanh ^2(x)}d\tanh (x)\)

\(\Big \downarrow \) 26

\(\displaystyle \int \frac {\coth (x) \sqrt {a+b \tanh ^2(x)}}{1-\tanh ^2(x)}d\tanh (x)\)

\(\Big \downarrow \) 354

\(\displaystyle \frac {1}{2} \int \frac {\coth (x) \sqrt {b \tanh ^2(x)+a}}{1-\tanh ^2(x)}d\tanh ^2(x)\)

\(\Big \downarrow \) 94

\(\displaystyle \frac {1}{2} \left ((a+b) \int \frac {1}{\left (1-\tanh ^2(x)\right ) \sqrt {b \tanh ^2(x)+a}}d\tanh ^2(x)+a \int \frac {\coth (x)}{\sqrt {b \tanh ^2(x)+a}}d\tanh ^2(x)\right )\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {1}{2} \left (\frac {2 (a+b) \int \frac {1}{\frac {a+b}{b}-\frac {\tanh ^4(x)}{b}}d\sqrt {b \tanh ^2(x)+a}}{b}+\frac {2 a \int \frac {1}{\frac {\tanh ^4(x)}{b}-\frac {a}{b}}d\sqrt {b \tanh ^2(x)+a}}{b}\right )\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {1}{2} \left (2 \sqrt {a+b} \text {arctanh}\left (\frac {\sqrt {a+b \tanh ^2(x)}}{\sqrt {a+b}}\right )-2 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a+b \tanh ^2(x)}}{\sqrt {a}}\right )\right )\)

Input:

Int[Coth[x]*Sqrt[a + b*Tanh[x]^2],x]
 

Output:

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

Defintions of rubi rules used

rule 26

Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a])   I 
nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
 

rule 73

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ 
{p = Denominator[m]}, Simp[p/b   Subst[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] && Lt 
Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL 
inearQ[a, b, c, d, m, n, x]
 

rule 94

Int[((e_.) + (f_.)*(x_))^(p_)/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), 
x_] :> Simp[(b*e - a*f)/(b*c - a*d)   Int[(e + f*x)^(p - 1)/(a + b*x), x], 
x] - Simp[(d*e - c*f)/(b*c - a*d)   Int[(e + f*x)^(p - 1)/(c + d*x), x], x] 
 /; FreeQ[{a, b, c, d, e, f}, x] && LtQ[0, p, 1]
 

rule 221

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

rule 354

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.), x_S 
ymbol] :> Simp[1/2   Subst[Int[x^((m - 1)/2)*(a + b*x)^p*(c + d*x)^q, x], x 
, x^2], x] /; FreeQ[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && IntegerQ 
[(m - 1)/2]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 4153

Int[((d_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((a_) + (b_.)*((c_.)*tan[(e_.) + 
(f_.)*(x_)])^(n_))^(p_.), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], 
 x]}, Simp[c*(ff/f)   Subst[Int[(d*ff*(x/c))^m*((a + b*(ff*x)^n)^p/(c^2 + f 
f^2*x^2)), x], x, c*(Tan[e + f*x]/ff)], x]] /; FreeQ[{a, b, c, d, e, f, m, 
n, p}, x] && (IGtQ[p, 0] || EqQ[n, 2] || EqQ[n, 4] || (IntegerQ[p] && Ratio 
nalQ[n]))
 

Maple [F]

\[\int \coth \left (x \right ) \sqrt {a +b \tanh \left (x \right )^{2}}d x\]

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 473 vs. \(2 (44) = 88\).

Time = 0.19 (sec) , antiderivative size = 3313, normalized size of antiderivative = 59.16 \[ \int \coth (x) \sqrt {a+b \tanh ^2(x)} \, dx=\text {Too large to display} \] Input:

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

Output:

Too large to include
 

Sympy [F]

\[ \int \coth (x) \sqrt {a+b \tanh ^2(x)} \, dx=\int \sqrt {a + b \tanh ^{2}{\left (x \right )}} \coth {\left (x \right )}\, dx \] Input:

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

Output:

Integral(sqrt(a + b*tanh(x)**2)*coth(x), x)
 

Maxima [F]

\[ \int \coth (x) \sqrt {a+b \tanh ^2(x)} \, dx=\int { \sqrt {b \tanh \left (x\right )^{2} + a} \coth \left (x\right ) \,d x } \] Input:

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

Output:

integrate(sqrt(b*tanh(x)^2 + a)*coth(x), x)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 255 vs. \(2 (44) = 88\).

Time = 0.30 (sec) , antiderivative size = 255, normalized size of antiderivative = 4.55 \[ \int \coth (x) \sqrt {a+b \tanh ^2(x)} \, dx=\frac {2 \, a \arctan \left (-\frac {\sqrt {a + b} e^{\left (2 \, x\right )} - \sqrt {a e^{\left (4 \, x\right )} + b e^{\left (4 \, x\right )} + 2 \, a e^{\left (2 \, x\right )} - 2 \, b e^{\left (2 \, x\right )} + a + b} - \sqrt {a + b}}{2 \, \sqrt {-a}}\right )}{\sqrt {-a}} - \frac {1}{2} \, \sqrt {a + b} \log \left ({\left | -{\left (\sqrt {a + b} e^{\left (2 \, x\right )} - \sqrt {a e^{\left (4 \, x\right )} + b e^{\left (4 \, x\right )} + 2 \, a e^{\left (2 \, x\right )} - 2 \, b e^{\left (2 \, x\right )} + a + b}\right )} {\left (a + b\right )} - \sqrt {a + b} {\left (a - b\right )} \right |}\right ) + \frac {1}{2} \, \sqrt {a + b} \log \left ({\left | -\sqrt {a + b} e^{\left (2 \, x\right )} + \sqrt {a e^{\left (4 \, x\right )} + b e^{\left (4 \, x\right )} + 2 \, a e^{\left (2 \, x\right )} - 2 \, b e^{\left (2 \, x\right )} + a + b} + \sqrt {a + b} \right |}\right ) - \frac {1}{2} \, \sqrt {a + b} \log \left ({\left | -\sqrt {a + b} e^{\left (2 \, x\right )} + \sqrt {a e^{\left (4 \, x\right )} + b e^{\left (4 \, x\right )} + 2 \, a e^{\left (2 \, x\right )} - 2 \, b e^{\left (2 \, x\right )} + a + b} - \sqrt {a + b} \right |}\right ) \] Input:

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

Output:

2*a*arctan(-1/2*(sqrt(a + b)*e^(2*x) - sqrt(a*e^(4*x) + b*e^(4*x) + 2*a*e^ 
(2*x) - 2*b*e^(2*x) + a + b) - sqrt(a + b))/sqrt(-a))/sqrt(-a) - 1/2*sqrt( 
a + b)*log(abs(-(sqrt(a + b)*e^(2*x) - sqrt(a*e^(4*x) + b*e^(4*x) + 2*a*e^ 
(2*x) - 2*b*e^(2*x) + a + b))*(a + b) - sqrt(a + b)*(a - b))) + 1/2*sqrt(a 
 + b)*log(abs(-sqrt(a + b)*e^(2*x) + sqrt(a*e^(4*x) + b*e^(4*x) + 2*a*e^(2 
*x) - 2*b*e^(2*x) + a + b) + sqrt(a + b))) - 1/2*sqrt(a + b)*log(abs(-sqrt 
(a + b)*e^(2*x) + sqrt(a*e^(4*x) + b*e^(4*x) + 2*a*e^(2*x) - 2*b*e^(2*x) + 
 a + b) - sqrt(a + b)))
 

Mupad [F(-1)]

Timed out. \[ \int \coth (x) \sqrt {a+b \tanh ^2(x)} \, dx=\int \mathrm {coth}\left (x\right )\,\sqrt {b\,{\mathrm {tanh}\left (x\right )}^2+a} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int \coth (x) \sqrt {a+b \tanh ^2(x)} \, dx=\int \sqrt {\tanh \left (x \right )^{2} b +a}\, \coth \left (x \right )d x \] Input:

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

Output:

int(sqrt(tanh(x)**2*b + a)*coth(x),x)