\(\int \frac {\coth ^2(x)}{(a+b \tanh ^2(x))^{5/2}} \, dx\) [254]

Optimal result

Integrand size = 17, antiderivative size = 131 \[ \int \frac {\coth ^2(x)}{\left (a+b \tanh ^2(x)\right )^{5/2}} \, dx=\frac {\text {arctanh}\left (\frac {\sqrt {a+b} \tanh (x)}{\sqrt {a+b \tanh ^2(x)}}\right )}{(a+b)^{5/2}}+\frac {b \coth (x)}{3 a (a+b) \left (a+b \tanh ^2(x)\right )^{3/2}}+\frac {b (7 a+4 b) \coth (x)}{3 a^2 (a+b)^2 \sqrt {a+b \tanh ^2(x)}}-\frac {(3 a+2 b) (a+4 b) \coth (x) \sqrt {a+b \tanh ^2(x)}}{3 a^3 (a+b)^2} \] Output:

arctanh((a+b)^(1/2)*tanh(x)/(a+b*tanh(x)^2)^(1/2))/(a+b)^(5/2)+1/3*b*coth( 
x)/a/(a+b)/(a+b*tanh(x)^2)^(3/2)+1/3*b*(7*a+4*b)*coth(x)/a^2/(a+b)^2/(a+b* 
tanh(x)^2)^(1/2)-1/3*(3*a+2*b)*(a+4*b)*coth(x)*(a+b*tanh(x)^2)^(1/2)/a^3/( 
a+b)^2
 

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.

Time = 6.69 (sec) , antiderivative size = 246, normalized size of antiderivative = 1.88 \[ \int \frac {\coth ^2(x)}{\left (a+b \tanh ^2(x)\right )^{5/2}} \, dx=\frac {\sqrt {(a-b+(a+b) \cosh (2 x)) \text {sech}^2(x)} \left (\frac {3 \sqrt {2} a^3 \coth (x) \left ((a+b) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {(a-b+(a+b) \cosh (2 x)) \text {csch}^2(x)}{b}}}{\sqrt {2}}\right ),1\right )-a \operatorname {EllipticPi}\left (\frac {b}{a+b},\arcsin \left (\frac {\sqrt {\frac {(a-b+(a+b) \cosh (2 x)) \text {csch}^2(x)}{b}}}{\sqrt {2}}\right ),1\right )\right )}{b \sqrt {\frac {(a-b+(a+b) \cosh (2 x)) \text {csch}^2(x)}{b}}}-\frac {(a+b) \left (3 (a+b)^2 (a-b+(a+b) \cosh (2 x))^2 \coth (x)+2 a b^3 \sinh (2 x)+b^2 (9 a+5 b) (a-b+(a+b) \cosh (2 x)) \sinh (2 x)\right )}{(a-b+(a+b) \cosh (2 x))^2}\right )}{3 \sqrt {2} a^3 (a+b)^3} \] Input:

Integrate[Coth[x]^2/(a + b*Tanh[x]^2)^(5/2),x]
 

Output:

(Sqrt[(a - b + (a + b)*Cosh[2*x])*Sech[x]^2]*((3*Sqrt[2]*a^3*Coth[x]*((a + 
 b)*EllipticF[ArcSin[Sqrt[((a - b + (a + b)*Cosh[2*x])*Csch[x]^2)/b]/Sqrt[ 
2]], 1] - a*EllipticPi[b/(a + b), ArcSin[Sqrt[((a - b + (a + b)*Cosh[2*x]) 
*Csch[x]^2)/b]/Sqrt[2]], 1]))/(b*Sqrt[((a - b + (a + b)*Cosh[2*x])*Csch[x] 
^2)/b]) - ((a + b)*(3*(a + b)^2*(a - b + (a + b)*Cosh[2*x])^2*Coth[x] + 2* 
a*b^3*Sinh[2*x] + b^2*(9*a + 5*b)*(a - b + (a + b)*Cosh[2*x])*Sinh[2*x]))/ 
(a - b + (a + b)*Cosh[2*x])^2))/(3*Sqrt[2]*a^3*(a + b)^3)
 

Rubi [A] (verified)

Time = 0.47 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.13, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.706, Rules used = {3042, 25, 4153, 25, 374, 25, 441, 25, 445, 27, 291, 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.

\(\displaystyle \int \frac {\coth ^2(x)}{\left (a+b \tanh ^2(x)\right )^{5/2}} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int -\frac {1}{\tan (i x)^2 \left (a-b \tan (i x)^2\right )^{5/2}}dx\)

\(\Big \downarrow \) 25

\(\displaystyle -\int \frac {1}{\tan (i x)^2 \left (a-b \tan (i x)^2\right )^{5/2}}dx\)

\(\Big \downarrow \) 4153

\(\displaystyle -\int -\frac {\coth ^2(x)}{\left (1-\tanh ^2(x)\right ) \left (b \tanh ^2(x)+a\right )^{5/2}}d\tanh (x)\)

\(\Big \downarrow \) 25

\(\displaystyle \int \frac {\coth ^2(x)}{\left (1-\tanh ^2(x)\right ) \left (a+b \tanh ^2(x)\right )^{5/2}}d\tanh (x)\)

\(\Big \downarrow \) 374

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 441

\(\displaystyle \frac {\frac {b (7 a+4 b) \coth (x)}{a (a+b) \sqrt {a+b \tanh ^2(x)}}-\frac {\int -\frac {\coth ^2(x) \left ((3 a+2 b) (a+4 b)-2 b (7 a+4 b) \tanh ^2(x)\right )}{\left (1-\tanh ^2(x)\right ) \sqrt {b \tanh ^2(x)+a}}d\tanh (x)}{a (a+b)}}{3 a (a+b)}+\frac {b \coth (x)}{3 a (a+b) \left (a+b \tanh ^2(x)\right )^{3/2}}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {\int \frac {\coth ^2(x) \left ((3 a+2 b) (a+4 b)-2 b (7 a+4 b) \tanh ^2(x)\right )}{\left (1-\tanh ^2(x)\right ) \sqrt {b \tanh ^2(x)+a}}d\tanh (x)}{a (a+b)}+\frac {b (7 a+4 b) \coth (x)}{a (a+b) \sqrt {a+b \tanh ^2(x)}}}{3 a (a+b)}+\frac {b \coth (x)}{3 a (a+b) \left (a+b \tanh ^2(x)\right )^{3/2}}\)

\(\Big \downarrow \) 445

\(\displaystyle \frac {\frac {-\frac {\int -\frac {3 a^3}{\left (1-\tanh ^2(x)\right ) \sqrt {b \tanh ^2(x)+a}}d\tanh (x)}{a}-\frac {(3 a+2 b) (a+4 b) \coth (x) \sqrt {a+b \tanh ^2(x)}}{a}}{a (a+b)}+\frac {b (7 a+4 b) \coth (x)}{a (a+b) \sqrt {a+b \tanh ^2(x)}}}{3 a (a+b)}+\frac {b \coth (x)}{3 a (a+b) \left (a+b \tanh ^2(x)\right )^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {3 a^2 \int \frac {1}{\left (1-\tanh ^2(x)\right ) \sqrt {b \tanh ^2(x)+a}}d\tanh (x)-\frac {(3 a+2 b) (a+4 b) \coth (x) \sqrt {a+b \tanh ^2(x)}}{a}}{a (a+b)}+\frac {b (7 a+4 b) \coth (x)}{a (a+b) \sqrt {a+b \tanh ^2(x)}}}{3 a (a+b)}+\frac {b \coth (x)}{3 a (a+b) \left (a+b \tanh ^2(x)\right )^{3/2}}\)

\(\Big \downarrow \) 291

\(\displaystyle \frac {\frac {3 a^2 \int \frac {1}{1-\frac {(a+b) \tanh ^2(x)}{b \tanh ^2(x)+a}}d\frac {\tanh (x)}{\sqrt {b \tanh ^2(x)+a}}-\frac {(3 a+2 b) (a+4 b) \coth (x) \sqrt {a+b \tanh ^2(x)}}{a}}{a (a+b)}+\frac {b (7 a+4 b) \coth (x)}{a (a+b) \sqrt {a+b \tanh ^2(x)}}}{3 a (a+b)}+\frac {b \coth (x)}{3 a (a+b) \left (a+b \tanh ^2(x)\right )^{3/2}}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\frac {\frac {3 a^2 \text {arctanh}\left (\frac {\sqrt {a+b} \tanh (x)}{\sqrt {a+b \tanh ^2(x)}}\right )}{\sqrt {a+b}}-\frac {(3 a+2 b) (a+4 b) \coth (x) \sqrt {a+b \tanh ^2(x)}}{a}}{a (a+b)}+\frac {b (7 a+4 b) \coth (x)}{a (a+b) \sqrt {a+b \tanh ^2(x)}}}{3 a (a+b)}+\frac {b \coth (x)}{3 a (a+b) \left (a+b \tanh ^2(x)\right )^{3/2}}\)

Input:

Int[Coth[x]^2/(a + b*Tanh[x]^2)^(5/2),x]
 

Output:

(b*Coth[x])/(3*a*(a + b)*(a + b*Tanh[x]^2)^(3/2)) + ((b*(7*a + 4*b)*Coth[x 
])/(a*(a + b)*Sqrt[a + b*Tanh[x]^2]) + ((3*a^2*ArcTanh[(Sqrt[a + b]*Tanh[x 
])/Sqrt[a + b*Tanh[x]^2]])/Sqrt[a + b] - ((3*a + 2*b)*(a + 4*b)*Coth[x]*Sq 
rt[a + b*Tanh[x]^2])/a)/(a*(a + b)))/(3*a*(a + b))
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

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 291

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*((c_) + (d_.)*(x_)^2)), x_Symbol] :> Subst 
[Int[1/(c - (b*c - a*d)*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b, c, 
d}, x] && NeQ[b*c - a*d, 0]
 

rule 374

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

rule 441

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

rule 445

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

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]

\[\int \frac {\coth \left (x \right )^{2}}{\left (a +b \tanh \left (x \right )^{2}\right )^{\frac {5}{2}}}d x\]

Input:

int(coth(x)^2/(a+b*tanh(x)^2)^(5/2),x)
 

Output:

int(coth(x)^2/(a+b*tanh(x)^2)^(5/2),x)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 5021 vs. \(2 (113) = 226\).

Time = 1.30 (sec) , antiderivative size = 10671, normalized size of antiderivative = 81.46 \[ \int \frac {\coth ^2(x)}{\left (a+b \tanh ^2(x)\right )^{5/2}} \, dx=\text {Too large to display} \] Input:

integrate(coth(x)^2/(a+b*tanh(x)^2)^(5/2),x, algorithm="fricas")
 

Output:

Too large to include
 

Sympy [F]

\[ \int \frac {\coth ^2(x)}{\left (a+b \tanh ^2(x)\right )^{5/2}} \, dx=\int \frac {\coth ^{2}{\left (x \right )}}{\left (a + b \tanh ^{2}{\left (x \right )}\right )^{\frac {5}{2}}}\, dx \] Input:

integrate(coth(x)**2/(a+b*tanh(x)**2)**(5/2),x)
 

Output:

Integral(coth(x)**2/(a + b*tanh(x)**2)**(5/2), x)
 

Maxima [F]

\[ \int \frac {\coth ^2(x)}{\left (a+b \tanh ^2(x)\right )^{5/2}} \, dx=\int { \frac {\coth \left (x\right )^{2}}{{\left (b \tanh \left (x\right )^{2} + a\right )}^{\frac {5}{2}}} \,d x } \] Input:

integrate(coth(x)^2/(a+b*tanh(x)^2)^(5/2),x, algorithm="maxima")
 

Output:

integrate(coth(x)^2/(b*tanh(x)^2 + a)^(5/2), x)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 898 vs. \(2 (113) = 226\).

Time = 0.53 (sec) , antiderivative size = 898, normalized size of antiderivative = 6.85 \[ \int \frac {\coth ^2(x)}{\left (a+b \tanh ^2(x)\right )^{5/2}} \, dx=\text {Too large to display} \] Input:

integrate(coth(x)^2/(a+b*tanh(x)^2)^(5/2),x, algorithm="giac")
 

Output:

-1/3*((((9*a^13*b^4 + 50*a^12*b^5 + 115*a^11*b^6 + 140*a^10*b^7 + 95*a^9*b 
^8 + 34*a^8*b^9 + 5*a^7*b^10)*e^(2*x)/(a^16*b^2 + 6*a^15*b^3 + 15*a^14*b^4 
 + 20*a^13*b^5 + 15*a^12*b^6 + 6*a^11*b^7 + a^10*b^8) + 3*(3*a^13*b^4 + 6* 
a^12*b^5 - 11*a^11*b^6 - 44*a^10*b^7 - 51*a^9*b^8 - 26*a^8*b^9 - 5*a^7*b^1 
0)/(a^16*b^2 + 6*a^15*b^3 + 15*a^14*b^4 + 20*a^13*b^5 + 15*a^12*b^6 + 6*a^ 
11*b^7 + a^10*b^8))*e^(2*x) - 3*(3*a^13*b^4 + 6*a^12*b^5 - 11*a^11*b^6 - 4 
4*a^10*b^7 - 51*a^9*b^8 - 26*a^8*b^9 - 5*a^7*b^10)/(a^16*b^2 + 6*a^15*b^3 
+ 15*a^14*b^4 + 20*a^13*b^5 + 15*a^12*b^6 + 6*a^11*b^7 + a^10*b^8))*e^(2*x 
) - (9*a^13*b^4 + 50*a^12*b^5 + 115*a^11*b^6 + 140*a^10*b^7 + 95*a^9*b^8 + 
 34*a^8*b^9 + 5*a^7*b^10)/(a^16*b^2 + 6*a^15*b^3 + 15*a^14*b^4 + 20*a^13*b 
^5 + 15*a^12*b^6 + 6*a^11*b^7 + a^10*b^8))/(a*e^(4*x) + b*e^(4*x) + 2*a*e^ 
(2*x) - 2*b*e^(2*x) + a + b)^(3/2) - 1/2*log(abs(-(sqrt(a + b)*e^(2*x) - s 
qrt(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)))/((a^2 + 2*a*b + b^2)*sqrt(a + b)) - 1/2*log(abs(-sqr 
t(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)))/((a^2 + 2*a*b + b^2)*sqrt(a + b)) + 1/2*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)))/((a^2 + 2*a*b + b^2)*sqrt(a + b)) + 4*(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 + b)*e^(2*x) - sqrt(a*e^(4*x) + b*e^(4*x)...
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\coth ^2(x)}{\left (a+b \tanh ^2(x)\right )^{5/2}} \, dx=\int \frac {{\mathrm {coth}\left (x\right )}^2}{{\left (b\,{\mathrm {tanh}\left (x\right )}^2+a\right )}^{5/2}} \,d x \] Input:

int(coth(x)^2/(a + b*tanh(x)^2)^(5/2),x)
                                                                                    
                                                                                    
 

Output:

int(coth(x)^2/(a + b*tanh(x)^2)^(5/2), x)
 

Reduce [F]

\[ \int \frac {\coth ^2(x)}{\left (a+b \tanh ^2(x)\right )^{5/2}} \, dx=\int \frac {\sqrt {\tanh \left (x \right )^{2} b +a}\, \coth \left (x \right )^{2}}{\tanh \left (x \right )^{6} b^{3}+3 \tanh \left (x \right )^{4} a \,b^{2}+3 \tanh \left (x \right )^{2} a^{2} b +a^{3}}d x \] Input:

int(coth(x)^2/(a+b*tanh(x)^2)^(5/2),x)
 

Output:

int((sqrt(tanh(x)**2*b + a)*coth(x)**2)/(tanh(x)**6*b**3 + 3*tanh(x)**4*a* 
b**2 + 3*tanh(x)**2*a**2*b + a**3),x)