\(\int \frac {\tanh ^6(x)}{(a+b \tanh ^2(x))^{5/2}} \, dx\) [246]

Optimal result

Integrand size = 17, antiderivative size = 118 \[ \int \frac {\tanh ^6(x)}{\left (a+b \tanh ^2(x)\right )^{5/2}} \, dx=-\frac {\text {arctanh}\left (\frac {\sqrt {b} \tanh (x)}{\sqrt {a+b \tanh ^2(x)}}\right )}{b^{5/2}}+\frac {\text {arctanh}\left (\frac {\sqrt {a+b} \tanh (x)}{\sqrt {a+b \tanh ^2(x)}}\right )}{(a+b)^{5/2}}+\frac {a \tanh ^3(x)}{3 b (a+b) \left (a+b \tanh ^2(x)\right )^{3/2}}+\frac {a (a+2 b) \tanh (x)}{b^2 (a+b)^2 \sqrt {a+b \tanh ^2(x)}} \] Output:

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

Mathematica [C] (verified)

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

Time = 1.32 (sec) , antiderivative size = 231, normalized size of antiderivative = 1.96 \[ \int \frac {\tanh ^6(x)}{\left (a+b \tanh ^2(x)\right )^{5/2}} \, dx=\frac {\sqrt {(a-b+(a+b) \cosh (2 x)) \text {sech}^2(x)} \left (-\frac {3 \sqrt {2} a \coth (x) \left (\left (a^2+3 a b+2 b^2\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {(a-b+(a+b) \cosh (2 x)) \text {csch}^2(x)}{b}}}{\sqrt {2}}\right ),1\right )+b^2 \operatorname {EllipticPi}\left (\frac {b}{a+b},\arcsin \left (\frac {\sqrt {\frac {(a-b+(a+b) \cosh (2 x)) \text {csch}^2(x)}{b}}}{\sqrt {2}}\right ),1\right )\right )}{b \sqrt {\frac {(a-b+(a+b) \cosh (2 x)) \text {csch}^2(x)}{b}}}+\frac {a (a+b) \left (3 a^2+2 a b-7 b^2+\left (3 a^2+10 a b+7 b^2\right ) \cosh (2 x)\right ) \sinh (2 x)}{(a-b+(a+b) \cosh (2 x))^2}\right )}{3 \sqrt {2} b^2 (a+b)^3} \] Input:

Integrate[Tanh[x]^6/(a + b*Tanh[x]^2)^(5/2),x]
 

Output:

(Sqrt[(a - b + (a + b)*Cosh[2*x])*Sech[x]^2]*((-3*Sqrt[2]*a*Coth[x]*((a^2 
+ 3*a*b + 2*b^2)*EllipticF[ArcSin[Sqrt[((a - b + (a + b)*Cosh[2*x])*Csch[x 
]^2)/b]/Sqrt[2]], 1] + b^2*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)*Cos 
h[2*x])*Csch[x]^2)/b]) + (a*(a + b)*(3*a^2 + 2*a*b - 7*b^2 + (3*a^2 + 10*a 
*b + 7*b^2)*Cosh[2*x])*Sinh[2*x])/(a - b + (a + b)*Cosh[2*x])^2))/(3*Sqrt[ 
2]*b^2*(a + b)^3)
 

Rubi [A] (verified)

Time = 0.68 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.25, 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, 372, 27, 440, 398, 224, 219, 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.

\(\displaystyle \int \frac {\tanh ^6(x)}{\left (a+b \tanh ^2(x)\right )^{5/2}} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int -\frac {\tan (i x)^6}{\left (a-b \tan (i x)^2\right )^{5/2}}dx\)

\(\Big \downarrow \) 25

\(\displaystyle -\int \frac {\tan (i x)^6}{\left (a-b \tan (i x)^2\right )^{5/2}}dx\)

\(\Big \downarrow \) 4153

\(\displaystyle -\int -\frac {\tanh ^6(x)}{\left (1-\tanh ^2(x)\right ) \left (b \tanh ^2(x)+a\right )^{5/2}}d\tanh (x)\)

\(\Big \downarrow \) 25

\(\displaystyle \int \frac {\tanh ^6(x)}{\left (1-\tanh ^2(x)\right ) \left (a+b \tanh ^2(x)\right )^{5/2}}d\tanh (x)\)

\(\Big \downarrow \) 372

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 440

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

\(\Big \downarrow \) 398

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

\(\Big \downarrow \) 224

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

\(\Big \downarrow \) 219

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

\(\Big \downarrow \) 291

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

\(\Big \downarrow \) 219

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

Input:

Int[Tanh[x]^6/(a + b*Tanh[x]^2)^(5/2),x]
 

Output:

(a*Tanh[x]^3)/(3*b*(a + b)*(a + b*Tanh[x]^2)^(3/2)) - ((((a + b)^2*ArcTanh 
[(Sqrt[b]*Tanh[x])/Sqrt[a + b*Tanh[x]^2]])/Sqrt[b] - (b^2*ArcTanh[(Sqrt[a 
+ b]*Tanh[x])/Sqrt[a + b*Tanh[x]^2]])/Sqrt[a + b])/(b*(a + b)) - (a*(a + 2 
*b)*Tanh[x])/(b*(a + b)*Sqrt[a + b*Tanh[x]^2]))/(b*(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 224

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], 
x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b}, x] &&  !GtQ[a, 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 372

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_ 
), x_Symbol] :> Simp[(-a)*e^3*(e*x)^(m - 3)*(a + b*x^2)^(p + 1)*((c + d*x^2 
)^(q + 1)/(2*b*(b*c - a*d)*(p + 1))), x] + Simp[e^4/(2*b*(b*c - a*d)*(p + 1 
))   Int[(e*x)^(m - 4)*(a + b*x^2)^(p + 1)*(c + d*x^2)^q*Simp[a*c*(m - 3) + 
 (a*d*(m + 2*q - 1) + 2*b*c*(p + 1))*x^2, x], x], x] /; FreeQ[{a, b, c, d, 
e, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && GtQ[m, 3] && IntBinomialQ[a 
, b, c, d, e, m, 2, p, q, x]
 

rule 398

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

rule 440

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

Leaf count of result is larger than twice the leaf count of optimal. \(548\) vs. \(2(100)=200\).

Time = 0.08 (sec) , antiderivative size = 549, normalized size of antiderivative = 4.65

method	result	size
derivativedivides	\(-\frac {\tanh \left (x \right )}{3 a \left (a +b \tanh \left (x \right )^{2}\right )^{\frac {3}{2}}}-\frac {2 \tanh \left (x \right )}{3 a^{2} \sqrt {a +b \tanh \left (x \right )^{2}}}+\frac {\tanh \left (x \right )}{3 b \left (a +b \tanh \left (x \right )^{2}\right )^{\frac {3}{2}}}-\frac {\tanh \left (x \right )}{3 a b \sqrt {a +b \tanh \left (x \right )^{2}}}+\frac {\tanh \left (x \right )^{3}}{3 b \left (a +b \tanh \left (x \right )^{2}\right )^{\frac {3}{2}}}+\frac {\tanh \left (x \right )}{b^{2} \sqrt {a +b \tanh \left (x \right )^{2}}}-\frac {\ln \left (\sqrt {b}\, \tanh \left (x \right )+\sqrt {a +b \tanh \left (x \right )^{2}}\right )}{b^{\frac {5}{2}}}-\frac {1}{6 \left (a +b \right ) \left (b \left (\tanh \left (x \right )-1\right )^{2}+2 b \left (\tanh \left (x \right )-1\right )+a +b \right )^{\frac {3}{2}}}+\frac {b \tanh \left (x \right )}{6 \left (a +b \right ) a \left (b \left (\tanh \left (x \right )-1\right )^{2}+2 b \left (\tanh \left (x \right )-1\right )+a +b \right )^{\frac {3}{2}}}+\frac {b \tanh \left (x \right )}{3 \left (a +b \right ) a^{2} \sqrt {b \left (\tanh \left (x \right )-1\right )^{2}+2 b \left (\tanh \left (x \right )-1\right )+a +b}}-\frac {1}{2 \left (a +b \right )^{2} \sqrt {b \left (\tanh \left (x \right )-1\right )^{2}+2 b \left (\tanh \left (x \right )-1\right )+a +b}}+\frac {b \tanh \left (x \right )}{2 \left (a +b \right )^{2} a \sqrt {b \left (\tanh \left (x \right )-1\right )^{2}+2 b \left (\tanh \left (x \right )-1\right )+a +b}}+\frac {\ln \left (\frac {2 a +2 b +2 b \left (\tanh \left (x \right )-1\right )+2 \sqrt {a +b}\, \sqrt {b \left (\tanh \left (x \right )-1\right )^{2}+2 b \left (\tanh \left (x \right )-1\right )+a +b}}{\tanh \left (x \right )-1}\right )}{2 \left (a +b \right )^{\frac {5}{2}}}+\frac {1}{6 \left (a +b \right ) \left (b \left (\tanh \left (x \right )+1\right )^{2}-2 b \left (\tanh \left (x \right )+1\right )+a +b \right )^{\frac {3}{2}}}+\frac {b \tanh \left (x \right )}{6 \left (a +b \right ) a \left (b \left (\tanh \left (x \right )+1\right )^{2}-2 b \left (\tanh \left (x \right )+1\right )+a +b \right )^{\frac {3}{2}}}+\frac {b \tanh \left (x \right )}{3 \left (a +b \right ) a^{2} \sqrt {b \left (\tanh \left (x \right )+1\right )^{2}-2 b \left (\tanh \left (x \right )+1\right )+a +b}}+\frac {1}{2 \left (a +b \right )^{2} \sqrt {b \left (\tanh \left (x \right )+1\right )^{2}-2 b \left (\tanh \left (x \right )+1\right )+a +b}}+\frac {b \tanh \left (x \right )}{2 \left (a +b \right )^{2} a \sqrt {b \left (\tanh \left (x \right )+1\right )^{2}-2 b \left (\tanh \left (x \right )+1\right )+a +b}}-\frac {\ln \left (\frac {2 a +2 b -2 b \left (\tanh \left (x \right )+1\right )+2 \sqrt {a +b}\, \sqrt {b \left (\tanh \left (x \right )+1\right )^{2}-2 b \left (\tanh \left (x \right )+1\right )+a +b}}{\tanh \left (x \right )+1}\right )}{2 \left (a +b \right )^{\frac {5}{2}}}\)	\(549\)
default	\(-\frac {\tanh \left (x \right )}{3 a \left (a +b \tanh \left (x \right )^{2}\right )^{\frac {3}{2}}}-\frac {2 \tanh \left (x \right )}{3 a^{2} \sqrt {a +b \tanh \left (x \right )^{2}}}+\frac {\tanh \left (x \right )}{3 b \left (a +b \tanh \left (x \right )^{2}\right )^{\frac {3}{2}}}-\frac {\tanh \left (x \right )}{3 a b \sqrt {a +b \tanh \left (x \right )^{2}}}+\frac {\tanh \left (x \right )^{3}}{3 b \left (a +b \tanh \left (x \right )^{2}\right )^{\frac {3}{2}}}+\frac {\tanh \left (x \right )}{b^{2} \sqrt {a +b \tanh \left (x \right )^{2}}}-\frac {\ln \left (\sqrt {b}\, \tanh \left (x \right )+\sqrt {a +b \tanh \left (x \right )^{2}}\right )}{b^{\frac {5}{2}}}-\frac {1}{6 \left (a +b \right ) \left (b \left (\tanh \left (x \right )-1\right )^{2}+2 b \left (\tanh \left (x \right )-1\right )+a +b \right )^{\frac {3}{2}}}+\frac {b \tanh \left (x \right )}{6 \left (a +b \right ) a \left (b \left (\tanh \left (x \right )-1\right )^{2}+2 b \left (\tanh \left (x \right )-1\right )+a +b \right )^{\frac {3}{2}}}+\frac {b \tanh \left (x \right )}{3 \left (a +b \right ) a^{2} \sqrt {b \left (\tanh \left (x \right )-1\right )^{2}+2 b \left (\tanh \left (x \right )-1\right )+a +b}}-\frac {1}{2 \left (a +b \right )^{2} \sqrt {b \left (\tanh \left (x \right )-1\right )^{2}+2 b \left (\tanh \left (x \right )-1\right )+a +b}}+\frac {b \tanh \left (x \right )}{2 \left (a +b \right )^{2} a \sqrt {b \left (\tanh \left (x \right )-1\right )^{2}+2 b \left (\tanh \left (x \right )-1\right )+a +b}}+\frac {\ln \left (\frac {2 a +2 b +2 b \left (\tanh \left (x \right )-1\right )+2 \sqrt {a +b}\, \sqrt {b \left (\tanh \left (x \right )-1\right )^{2}+2 b \left (\tanh \left (x \right )-1\right )+a +b}}{\tanh \left (x \right )-1}\right )}{2 \left (a +b \right )^{\frac {5}{2}}}+\frac {1}{6 \left (a +b \right ) \left (b \left (\tanh \left (x \right )+1\right )^{2}-2 b \left (\tanh \left (x \right )+1\right )+a +b \right )^{\frac {3}{2}}}+\frac {b \tanh \left (x \right )}{6 \left (a +b \right ) a \left (b \left (\tanh \left (x \right )+1\right )^{2}-2 b \left (\tanh \left (x \right )+1\right )+a +b \right )^{\frac {3}{2}}}+\frac {b \tanh \left (x \right )}{3 \left (a +b \right ) a^{2} \sqrt {b \left (\tanh \left (x \right )+1\right )^{2}-2 b \left (\tanh \left (x \right )+1\right )+a +b}}+\frac {1}{2 \left (a +b \right )^{2} \sqrt {b \left (\tanh \left (x \right )+1\right )^{2}-2 b \left (\tanh \left (x \right )+1\right )+a +b}}+\frac {b \tanh \left (x \right )}{2 \left (a +b \right )^{2} a \sqrt {b \left (\tanh \left (x \right )+1\right )^{2}-2 b \left (\tanh \left (x \right )+1\right )+a +b}}-\frac {\ln \left (\frac {2 a +2 b -2 b \left (\tanh \left (x \right )+1\right )+2 \sqrt {a +b}\, \sqrt {b \left (\tanh \left (x \right )+1\right )^{2}-2 b \left (\tanh \left (x \right )+1\right )+a +b}}{\tanh \left (x \right )+1}\right )}{2 \left (a +b \right )^{\frac {5}{2}}}\)	\(549\)

Input:

int(tanh(x)^6/(a+b*tanh(x)^2)^(5/2),x,method=_RETURNVERBOSE)
 

Output:

-1/3*tanh(x)/a/(a+b*tanh(x)^2)^(3/2)-2/3/a^2*tanh(x)/(a+b*tanh(x)^2)^(1/2) 
+1/3*tanh(x)/b/(a+b*tanh(x)^2)^(3/2)-1/3/a/b*tanh(x)/(a+b*tanh(x)^2)^(1/2) 
+1/3*tanh(x)^3/b/(a+b*tanh(x)^2)^(3/2)+1/b^2*tanh(x)/(a+b*tanh(x)^2)^(1/2) 
-1/b^(5/2)*ln(b^(1/2)*tanh(x)+(a+b*tanh(x)^2)^(1/2))-1/6/(a+b)/(b*(tanh(x) 
-1)^2+2*b*(tanh(x)-1)+a+b)^(3/2)+1/6*b/(a+b)/a/(b*(tanh(x)-1)^2+2*b*(tanh( 
x)-1)+a+b)^(3/2)*tanh(x)+1/3*b/(a+b)/a^2/(b*(tanh(x)-1)^2+2*b*(tanh(x)-1)+ 
a+b)^(1/2)*tanh(x)-1/2/(a+b)^2/(b*(tanh(x)-1)^2+2*b*(tanh(x)-1)+a+b)^(1/2) 
+1/2/(a+b)^2/a/(b*(tanh(x)-1)^2+2*b*(tanh(x)-1)+a+b)^(1/2)*b*tanh(x)+1/2/( 
a+b)^(5/2)*ln((2*a+2*b+2*b*(tanh(x)-1)+2*(a+b)^(1/2)*(b*(tanh(x)-1)^2+2*b* 
(tanh(x)-1)+a+b)^(1/2))/(tanh(x)-1))+1/6/(a+b)/(b*(tanh(x)+1)^2-2*b*(tanh( 
x)+1)+a+b)^(3/2)+1/6*b/(a+b)/a/(b*(tanh(x)+1)^2-2*b*(tanh(x)+1)+a+b)^(3/2) 
*tanh(x)+1/3*b/(a+b)/a^2/(b*(tanh(x)+1)^2-2*b*(tanh(x)+1)+a+b)^(1/2)*tanh( 
x)+1/2/(a+b)^2/(b*(tanh(x)+1)^2-2*b*(tanh(x)+1)+a+b)^(1/2)+1/2/(a+b)^2/a/( 
b*(tanh(x)+1)^2-2*b*(tanh(x)+1)+a+b)^(1/2)*b*tanh(x)-1/2/(a+b)^(5/2)*ln((2 
*a+2*b-2*b*(tanh(x)+1)+2*(a+b)^(1/2)*(b*(tanh(x)+1)^2-2*b*(tanh(x)+1)+a+b) 
^(1/2))/(tanh(x)+1))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 4472 vs. \(2 (100) = 200\).

Time = 1.13 (sec) , antiderivative size = 19265, normalized size of antiderivative = 163.26 \[ \int \frac {\tanh ^6(x)}{\left (a+b \tanh ^2(x)\right )^{5/2}} \, dx=\text {Too large to display} \] Input:

integrate(tanh(x)^6/(a+b*tanh(x)^2)^(5/2),x, algorithm="fricas")
 

Output:

Too large to include
 

Sympy [F]

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

integrate(tanh(x)**6/(a+b*tanh(x)**2)**(5/2),x)
 

Output:

Integral(tanh(x)**6/(a + b*tanh(x)**2)**(5/2), x)
 

Maxima [F]

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

integrate(tanh(x)^6/(a+b*tanh(x)^2)^(5/2),x, algorithm="maxima")
 

Output:

integrate(tanh(x)^6/(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. 805 vs. \(2 (100) = 200\).

Time = 0.43 (sec) , antiderivative size = 805, normalized size of antiderivative = 6.82 \[ \int \frac {\tanh ^6(x)}{\left (a+b \tanh ^2(x)\right )^{5/2}} \, dx =\text {Too large to display} \] Input:

integrate(tanh(x)^6/(a+b*tanh(x)^2)^(5/2),x, algorithm="giac")
 

Output:

1/3*((((3*a^9*b^8 + 22*a^8*b^9 + 65*a^7*b^10 + 100*a^6*b^11 + 85*a^5*b^12 
+ 38*a^4*b^13 + 7*a^3*b^14)*e^(2*x)/(a^8*b^10 + 6*a^7*b^11 + 15*a^6*b^12 + 
 20*a^5*b^13 + 15*a^4*b^14 + 6*a^3*b^15 + a^2*b^16) + 3*(a^9*b^8 + 2*a^8*b 
^9 - 9*a^7*b^10 - 36*a^6*b^11 - 49*a^5*b^12 - 30*a^4*b^13 - 7*a^3*b^14)/(a 
^8*b^10 + 6*a^7*b^11 + 15*a^6*b^12 + 20*a^5*b^13 + 15*a^4*b^14 + 6*a^3*b^1 
5 + a^2*b^16))*e^(2*x) - 3*(a^9*b^8 + 2*a^8*b^9 - 9*a^7*b^10 - 36*a^6*b^11 
 - 49*a^5*b^12 - 30*a^4*b^13 - 7*a^3*b^14)/(a^8*b^10 + 6*a^7*b^11 + 15*a^6 
*b^12 + 20*a^5*b^13 + 15*a^4*b^14 + 6*a^3*b^15 + a^2*b^16))*e^(2*x) - (3*a 
^9*b^8 + 22*a^8*b^9 + 65*a^7*b^10 + 100*a^6*b^11 + 85*a^5*b^12 + 38*a^4*b^ 
13 + 7*a^3*b^14)/(a^8*b^10 + 6*a^7*b^11 + 15*a^6*b^12 + 20*a^5*b^13 + 15*a 
^4*b^14 + 6*a^3*b^15 + a^2*b^16))/(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) - 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)))/((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)) + 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)) - 2*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(-b))/(sqrt(-b)*b^2)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\tanh ^6(x)}{\left (a+b \tanh ^2(x)\right )^{5/2}} \, dx=\int \frac {{\mathrm {tanh}\left (x\right )}^6}{{\left (b\,{\mathrm {tanh}\left (x\right )}^2+a\right )}^{5/2}} \,d x \] Input:

int(tanh(x)^6/(a + b*tanh(x)^2)^(5/2),x)
 

Output:

int(tanh(x)^6/(a + b*tanh(x)^2)^(5/2), x)
 

Reduce [F]

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

int(tanh(x)^6/(a+b*tanh(x)^2)^(5/2),x)
 

Output:

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