3.3.6 \(\int \frac {\coth ^3(x)}{\sqrt {a+b \coth ^2(x)+c \coth ^4(x)}} \, dx\) [206]

3.3.6.1 Optimal result

Integrand size = 23, antiderivative size = 105 \[ \int \frac {\coth ^3(x)}{\sqrt {a+b \coth ^2(x)+c \coth ^4(x)}} \, dx=-\frac {\text {arctanh}\left (\frac {b+2 c \coth ^2(x)}{2 \sqrt {c} \sqrt {a+b \coth ^2(x)+c \coth ^4(x)}}\right )}{2 \sqrt {c}}+\frac {\text {arctanh}\left (\frac {2 a+b+(b+2 c) \coth ^2(x)}{2 \sqrt {a+b+c} \sqrt {a+b \coth ^2(x)+c \coth ^4(x)}}\right )}{2 \sqrt {a+b+c}} \]

output

-1/2*arctanh(1/2*(b+2*c*coth(x)^2)/c^(1/2)/(a+b*coth(x)^2+c*coth(x)^4)^(1/ 
2))/c^(1/2)+1/2*arctanh(1/2*(2*a+b+(b+2*c)*coth(x)^2)/(a+b+c)^(1/2)/(a+b*c 
oth(x)^2+c*coth(x)^4)^(1/2))/(a+b+c)^(1/2)
 

3.3.6.2 Mathematica [A] (verified)

Time = 0.57 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.52 \[ \int \frac {\coth ^3(x)}{\sqrt {a+b \coth ^2(x)+c \coth ^4(x)}} \, dx=-\frac {\left ((a+b+c) \text {arctanh}\left (\frac {2 c+b \tanh ^2(x)}{2 \sqrt {c} \sqrt {c+b \tanh ^2(x)+a \tanh ^4(x)}}\right )-\sqrt {c} \sqrt {a+b+c} \text {arctanh}\left (\frac {b+2 c+(2 a+b) \tanh ^2(x)}{2 \sqrt {a+b+c} \sqrt {c+b \tanh ^2(x)+a \tanh ^4(x)}}\right )\right ) \sqrt {a+b \coth ^2(x)+c \coth ^4(x)} \tanh ^2(x)}{2 \sqrt {c} (a+b+c) \sqrt {c+b \tanh ^2(x)+a \tanh ^4(x)}} \]

input

Integrate[Coth[x]^3/Sqrt[a + b*Coth[x]^2 + c*Coth[x]^4],x]
 

output

-1/2*(((a + b + c)*ArcTanh[(2*c + b*Tanh[x]^2)/(2*Sqrt[c]*Sqrt[c + b*Tanh[ 
x]^2 + a*Tanh[x]^4])] - Sqrt[c]*Sqrt[a + b + c]*ArcTanh[(b + 2*c + (2*a + 
b)*Tanh[x]^2)/(2*Sqrt[a + b + c]*Sqrt[c + b*Tanh[x]^2 + a*Tanh[x]^4])])*Sq 
rt[a + b*Coth[x]^2 + c*Coth[x]^4]*Tanh[x]^2)/(Sqrt[c]*(a + b + c)*Sqrt[c + 
 b*Tanh[x]^2 + a*Tanh[x]^4])
 

3.3.6.3 Rubi [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 0.40 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.50, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.391, Rules used = {3042, 26, 4184, 1578, 1269, 1092, 219, 1154, 219}

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.

\(\displaystyle \int \frac {\coth ^3(x)}{\sqrt {a+b \coth ^2(x)+c \coth ^4(x)}} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int -\frac {i \cot (i x)^3}{\sqrt {a-b \cot (i x)^2+c \cot (i x)^4}}dx\)

\(\Big \downarrow \) 26

\(\displaystyle -i \int \frac {\cot (i x)^3}{\sqrt {c \cot (i x)^4-b \cot (i x)^2+a}}dx\)

\(\Big \downarrow \) 4184

\(\displaystyle \int \frac {i \coth ^3(x)}{\left (1-\coth ^2(x)\right ) \sqrt {a+b \coth ^2(x)+c \coth ^4(x)}}d(-i \coth (x))\)

\(\Big \downarrow \) 1578

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

\(\Big \downarrow \) 1269

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

\(\Big \downarrow \) 1092

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

\(\Big \downarrow \) 219

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

\(\Big \downarrow \) 1154

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

\(\Big \downarrow \) 219

\(\displaystyle \frac {1}{2} \left (-\frac {i \arctan \left (\frac {\coth (x)}{2 \sqrt {a+b+c}}\right )}{\sqrt {a+b+c}}-\frac {i \arctan \left (\frac {\coth (x)}{2 \sqrt {c}}\right )}{\sqrt {c}}\right )\)

input

Int[Coth[x]^3/Sqrt[a + b*Coth[x]^2 + c*Coth[x]^4],x]
 

output

(((-I)*ArcTan[Coth[x]/(2*Sqrt[c])])/Sqrt[c] - (I*ArcTan[Coth[x]/(2*Sqrt[a 
+ b + c])])/Sqrt[a + b + c])/2
 

3.3.6.3.1 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 219

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

rule 1092

Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Simp[2   Subst[I 
nt[1/(4*c - x^2), x], x, (b + 2*c*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a 
, b, c}, x]
 

rule 1154

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Sym 
bol] :> Simp[-2   Subst[Int[1/(4*c*d^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, ( 
2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c 
, d, e}, x]
 

rule 1269

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c 
_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[g/e   Int[(d + e*x)^(m + 1)*(a + b*x + 
 c*x^2)^p, x], x] + Simp[(e*f - d*g)/e   Int[(d + e*x)^m*(a + b*x + c*x^2)^ 
p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] &&  !IGtQ[m, 0]
 

rule 1578

Int[(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_ 
)^4)^(p_.), x_Symbol] :> Simp[1/2   Subst[Int[x^((m - 1)/2)*(d + e*x)^q*(a 
+ b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, d, e, p, q}, x] && Int 
egerQ[(m - 1)/2]
 

rule 3042

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

rule 4184

Int[cot[(d_.) + (e_.)*(x_)]^(m_.)*((a_.) + (b_.)*(cot[(d_.) + (e_.)*(x_)]*( 
f_.))^(n_.) + (c_.)*(cot[(d_.) + (e_.)*(x_)]*(f_.))^(n2_.))^(p_), x_Symbol] 
 :> Simp[-f/e   Subst[Int[(x/f)^m*((a + b*x^n + c*x^(2*n))^p/(f^2 + x^2)), 
x], x, f*Cot[d + e*x]], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && EqQ[ 
n2, 2*n] && NeQ[b^2 - 4*a*c, 0]
 

3.3.6.4 Maple [A] (verified)

Time = 0.73 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.86

method	result	size
derivativedivides	\(-\frac {\ln \left (\frac {\frac {b}{2}+c \coth \left (x \right )^{2}}{\sqrt {c}}+\sqrt {a +b \coth \left (x \right )^{2}+c \coth \left (x \right )^{4}}\right )}{2 \sqrt {c}}+\frac {\operatorname {arctanh}\left (\frac {b \coth \left (x \right )^{2}+2 c \coth \left (x \right )^{2}+2 a +b}{2 \sqrt {a +b +c}\, \sqrt {a +b \coth \left (x \right )^{2}+c \coth \left (x \right )^{4}}}\right )}{2 \sqrt {a +b +c}}\)	\(90\)
default	\(-\frac {\ln \left (\frac {\frac {b}{2}+c \coth \left (x \right )^{2}}{\sqrt {c}}+\sqrt {a +b \coth \left (x \right )^{2}+c \coth \left (x \right )^{4}}\right )}{2 \sqrt {c}}+\frac {\operatorname {arctanh}\left (\frac {b \coth \left (x \right )^{2}+2 c \coth \left (x \right )^{2}+2 a +b}{2 \sqrt {a +b +c}\, \sqrt {a +b \coth \left (x \right )^{2}+c \coth \left (x \right )^{4}}}\right )}{2 \sqrt {a +b +c}}\)	\(90\)

input

int(coth(x)^3/(a+b*coth(x)^2+c*coth(x)^4)^(1/2),x,method=_RETURNVERBOSE)
 

output

-1/2*ln((1/2*b+c*coth(x)^2)/c^(1/2)+(a+b*coth(x)^2+c*coth(x)^4)^(1/2))/c^( 
1/2)+1/2/(a+b+c)^(1/2)*arctanh(1/2*(b*coth(x)^2+2*c*coth(x)^2+2*a+b)/(a+b+ 
c)^(1/2)/(a+b*coth(x)^2+c*coth(x)^4)^(1/2))
 

3.3.6.5 Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1522 vs. \(2 (85) = 170\).

Time = 0.90 (sec) , antiderivative size = 6695, normalized size of antiderivative = 63.76 \[ \int \frac {\coth ^3(x)}{\sqrt {a+b \coth ^2(x)+c \coth ^4(x)}} \, dx=\text {Too large to display} \]

input

integrate(coth(x)^3/(a+b*coth(x)^2+c*coth(x)^4)^(1/2),x, algorithm="fricas 
")
 

output

Too large to include
 

3.3.6.6 Sympy [F]

\[ \int \frac {\coth ^3(x)}{\sqrt {a+b \coth ^2(x)+c \coth ^4(x)}} \, dx=\int \frac {\coth ^{3}{\left (x \right )}}{\sqrt {a + b \coth ^{2}{\left (x \right )} + c \coth ^{4}{\left (x \right )}}}\, dx \]

input

integrate(coth(x)**3/(a+b*coth(x)**2+c*coth(x)**4)**(1/2),x)
 

output

Integral(coth(x)**3/sqrt(a + b*coth(x)**2 + c*coth(x)**4), x)
 

3.3.6.7 Maxima [F]

\[ \int \frac {\coth ^3(x)}{\sqrt {a+b \coth ^2(x)+c \coth ^4(x)}} \, dx=\int { \frac {\coth \left (x\right )^{3}}{\sqrt {c \coth \left (x\right )^{4} + b \coth \left (x\right )^{2} + a}} \,d x } \]

input

integrate(coth(x)^3/(a+b*coth(x)^2+c*coth(x)^4)^(1/2),x, algorithm="maxima 
")
 

output

integrate(coth(x)^3/sqrt(c*coth(x)^4 + b*coth(x)^2 + a), x)
 

3.3.6.8 Giac [F]

\[ \int \frac {\coth ^3(x)}{\sqrt {a+b \coth ^2(x)+c \coth ^4(x)}} \, dx=\int { \frac {\coth \left (x\right )^{3}}{\sqrt {c \coth \left (x\right )^{4} + b \coth \left (x\right )^{2} + a}} \,d x } \]

input

integrate(coth(x)^3/(a+b*coth(x)^2+c*coth(x)^4)^(1/2),x, algorithm="giac")
 

output

integrate(coth(x)^3/sqrt(c*coth(x)^4 + b*coth(x)^2 + a), x)
 

3.3.6.9 Mupad [F(-1)]

Timed out. \[ \int \frac {\coth ^3(x)}{\sqrt {a+b \coth ^2(x)+c \coth ^4(x)}} \, dx=\int \frac {{\mathrm {coth}\left (x\right )}^3}{\sqrt {c\,{\mathrm {coth}\left (x\right )}^4+b\,{\mathrm {coth}\left (x\right )}^2+a}} \,d x \]

input

int(coth(x)^3/(a + b*coth(x)^2 + c*coth(x)^4)^(1/2),x)
 

output

int(coth(x)^3/(a + b*coth(x)^2 + c*coth(x)^4)^(1/2), x)