\(\int \coth ^4(x) \sqrt {a+b \tanh ^2(x)} \, dx\) [217]

Optimal result

Integrand size = 17, antiderivative size = 78 \[ \int \coth ^4(x) \sqrt {a+b \tanh ^2(x)} \, dx=\sqrt {a+b} \text {arctanh}\left (\frac {\sqrt {a+b} \tanh (x)}{\sqrt {a+b \tanh ^2(x)}}\right )-\frac {(3 a+b) \coth (x) \sqrt {a+b \tanh ^2(x)}}{3 a}-\frac {1}{3} \coth ^3(x) \sqrt {a+b \tanh ^2(x)} \] Output:

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

Mathematica [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.

Time = 4.47 (sec) , antiderivative size = 161, normalized size of antiderivative = 2.06 \[ \int \coth ^4(x) \sqrt {a+b \tanh ^2(x)} \, dx=\frac {\cosh ^4(x) \coth ^3(x) \left (1+\frac {b \tanh ^2(x)}{a}\right ) \left (-\frac {\text {sech}^4(x) \left (\arcsin \left (\sqrt {-\frac {(a+b) \sinh ^2(x)}{a}}\right ) \sqrt {-\frac {(a+b) \sinh ^2(x)}{a}}+\sqrt {\cosh ^2(x)+\frac {b \sinh ^2(x)}{a}}\right ) \left (a-2 b \tanh ^2(x)\right )}{\sqrt {\cosh ^2(x)+\frac {b \sinh ^2(x)}{a}}}-\frac {4 (a+b) \operatorname {Hypergeometric2F1}\left (2,2,\frac {3}{2},-\frac {(a+b) \sinh ^2(x)}{a}\right ) \left (a \tanh (x)+b \tanh ^3(x)\right )^2}{a^2}\right )}{3 \sqrt {a+b \tanh ^2(x)}} \] Input:

Integrate[Coth[x]^4*Sqrt[a + b*Tanh[x]^2],x]
 

Output:

(Cosh[x]^4*Coth[x]^3*(1 + (b*Tanh[x]^2)/a)*(-((Sech[x]^4*(ArcSin[Sqrt[-((( 
a + b)*Sinh[x]^2)/a)]]*Sqrt[-(((a + b)*Sinh[x]^2)/a)] + Sqrt[Cosh[x]^2 + ( 
b*Sinh[x]^2)/a])*(a - 2*b*Tanh[x]^2))/Sqrt[Cosh[x]^2 + (b*Sinh[x]^2)/a]) - 
 (4*(a + b)*Hypergeometric2F1[2, 2, 3/2, -(((a + b)*Sinh[x]^2)/a)]*(a*Tanh 
[x] + b*Tanh[x]^3)^2)/a^2))/(3*Sqrt[a + b*Tanh[x]^2])
 

Rubi [A] (verified)

Time = 0.35 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.05, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.412, Rules used = {3042, 4153, 377, 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.

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

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {\sqrt {a-b \tan (i x)^2}}{\tan (i x)^4}dx\)

\(\Big \downarrow \) 4153

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

\(\Big \downarrow \) 377

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

\(\Big \downarrow \) 445

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 291

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

\(\Big \downarrow \) 219

\(\displaystyle \frac {1}{3} \left (3 \sqrt {a+b} \text {arctanh}\left (\frac {\sqrt {a+b} \tanh (x)}{\sqrt {a+b \tanh ^2(x)}}\right )-\frac {(3 a+b) \coth (x) \sqrt {a+b \tanh ^2(x)}}{a}\right )-\frac {1}{3} \coth ^3(x) \sqrt {a+b \tanh ^2(x)}\)

Input:

Int[Coth[x]^4*Sqrt[a + b*Tanh[x]^2],x]
 

Output:

-1/3*(Coth[x]^3*Sqrt[a + b*Tanh[x]^2]) + (3*Sqrt[a + b]*ArcTanh[(Sqrt[a + 
b]*Tanh[x])/Sqrt[a + b*Tanh[x]^2]] - ((3*a + b)*Coth[x]*Sqrt[a + b*Tanh[x] 
^2])/a)/3
 

Defintions of rubi rules used

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 377

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

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]

\[\int \coth \left (x \right )^{4} \sqrt {a +b \tanh \left (x \right )^{2}}d x\]

Input:

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

Output:

int(coth(x)^4*(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. 828 vs. \(2 (64) = 128\).

Time = 0.19 (sec) , antiderivative size = 2285, normalized size of antiderivative = 29.29 \[ \int \coth ^4(x) \sqrt {a+b \tanh ^2(x)} \, dx=\text {Too large to display} \] Input:

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

Output:

[1/12*(3*(a*cosh(x)^6 + 6*a*cosh(x)*sinh(x)^5 + a*sinh(x)^6 - 3*a*cosh(x)^ 
4 + 3*(5*a*cosh(x)^2 - a)*sinh(x)^4 + 4*(5*a*cosh(x)^3 - 3*a*cosh(x))*sinh 
(x)^3 + 3*a*cosh(x)^2 + 3*(5*a*cosh(x)^4 - 6*a*cosh(x)^2 + a)*sinh(x)^2 + 
6*(a*cosh(x)^5 - 2*a*cosh(x)^3 + a*cosh(x))*sinh(x) - a)*sqrt(a + b)*log(- 
((a*b^2 + b^3)*cosh(x)^8 + 8*(a*b^2 + b^3)*cosh(x)*sinh(x)^7 + (a*b^2 + b^ 
3)*sinh(x)^8 - 2*(a*b^2 + 2*b^3)*cosh(x)^6 - 2*(a*b^2 + 2*b^3 - 14*(a*b^2 
+ b^3)*cosh(x)^2)*sinh(x)^6 + 4*(14*(a*b^2 + b^3)*cosh(x)^3 - 3*(a*b^2 + 2 
*b^3)*cosh(x))*sinh(x)^5 + (a^3 - a^2*b + 4*a*b^2 + 6*b^3)*cosh(x)^4 + (70 
*(a*b^2 + b^3)*cosh(x)^4 + a^3 - a^2*b + 4*a*b^2 + 6*b^3 - 30*(a*b^2 + 2*b 
^3)*cosh(x)^2)*sinh(x)^4 + 4*(14*(a*b^2 + b^3)*cosh(x)^5 - 10*(a*b^2 + 2*b 
^3)*cosh(x)^3 + (a^3 - a^2*b + 4*a*b^2 + 6*b^3)*cosh(x))*sinh(x)^3 + a^3 + 
 3*a^2*b + 3*a*b^2 + b^3 + 2*(a^3 - 3*a*b^2 - 2*b^3)*cosh(x)^2 + 2*(14*(a* 
b^2 + b^3)*cosh(x)^6 - 15*(a*b^2 + 2*b^3)*cosh(x)^4 + a^3 - 3*a*b^2 - 2*b^ 
3 + 3*(a^3 - a^2*b + 4*a*b^2 + 6*b^3)*cosh(x)^2)*sinh(x)^2 + sqrt(2)*(b^2* 
cosh(x)^6 + 6*b^2*cosh(x)*sinh(x)^5 + b^2*sinh(x)^6 - 3*b^2*cosh(x)^4 + 3* 
(5*b^2*cosh(x)^2 - b^2)*sinh(x)^4 + 4*(5*b^2*cosh(x)^3 - 3*b^2*cosh(x))*si 
nh(x)^3 - (a^2 - 2*a*b - 3*b^2)*cosh(x)^2 + (15*b^2*cosh(x)^4 - 18*b^2*cos 
h(x)^2 - a^2 + 2*a*b + 3*b^2)*sinh(x)^2 - a^2 - 2*a*b - b^2 + 2*(3*b^2*cos 
h(x)^5 - 6*b^2*cosh(x)^3 - (a^2 - 2*a*b - 3*b^2)*cosh(x))*sinh(x))*sqrt(a 
+ b)*sqrt(((a + b)*cosh(x)^2 + (a + b)*sinh(x)^2 + a - b)/(cosh(x)^2 - ...
 

Sympy [F]

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

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

Output:

Integral(sqrt(a + b*tanh(x)**2)*coth(x)**4, x)
 

Maxima [F]

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

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

Output:

integrate(sqrt(b*tanh(x)^2 + a)*coth(x)^4, x)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 629 vs. \(2 (64) = 128\).

Time = 0.59 (sec) , antiderivative size = 629, normalized size of antiderivative = 8.06 \[ \int \coth ^4(x) \sqrt {a+b \tanh ^2(x)} \, dx =\text {Too large to display} \] Input:

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

Output:

-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))) + 4/3*(3*(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))^5*(2*a + b) - 3*( 
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))^4*(2*a + 3*b)*sqrt(a + b) - 2*(10*a^2 + 3*a*b - 3*b^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))^3 + 6*(6*a^2 + 3*a*b + b^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))^2*sqrt(a + b) + 3*(26*a^ 
3 + 9*a^2*b - 4*a*b^2 - 3*b^3)*(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)) + (34*a^3 - 17*a^2*b + 3*b^3) 
*sqrt(a + b))/((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))^2 - 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) - 3*a + b)^3
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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