\(\int \frac {\tanh ^4(x)}{(a+b \tanh ^2(x))^{5/2}} \, dx\) [248]

Optimal result

Integrand size = 17, antiderivative size = 90 \[ \int \frac {\tanh ^4(x)}{\left (a+b \tanh ^2(x)\right )^{5/2}} \, dx=\frac {\text {arctanh}\left (\frac {\sqrt {a+b} \tanh (x)}{\sqrt {a+b \tanh ^2(x)}}\right )}{(a+b)^{5/2}}+\frac {a \tanh (x)}{3 b (a+b) \left (a+b \tanh ^2(x)\right )^{3/2}}-\frac {(a+4 b) \tanh (x)}{3 b (a+b)^2 \sqrt {a+b \tanh ^2(x)}} \] Output:

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

Mathematica [A] (verified)

Time = 1.79 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.47 \[ \int \frac {\tanh ^4(x)}{\left (a+b \tanh ^2(x)\right )^{5/2}} \, dx=\frac {\tanh ^3(x) \left (3 \text {arctanh}\left (\frac {\sqrt {\frac {(a+b) \tanh ^2(x)}{a}}}{\sqrt {1+\frac {b \tanh ^2(x)}{a}}}\right ) \left (b+a \coth ^2(x)\right )^2 \sqrt {\frac {(a+b) \tanh ^2(x)}{a}}-(a+b) \left (a+4 b+3 a \coth ^2(x)\right ) \sqrt {1+\frac {b \tanh ^2(x)}{a}}\right )}{3 (a+b)^3 \left (a+b \tanh ^2(x)\right )^{3/2} \sqrt {1+\frac {b \tanh ^2(x)}{a}}} \] Input:

Integrate[Tanh[x]^4/(a + b*Tanh[x]^2)^(5/2),x]
 

Output:

(Tanh[x]^3*(3*ArcTanh[Sqrt[((a + b)*Tanh[x]^2)/a]/Sqrt[1 + (b*Tanh[x]^2)/a 
]]*(b + a*Coth[x]^2)^2*Sqrt[((a + b)*Tanh[x]^2)/a] - (a + b)*(a + 4*b + 3* 
a*Coth[x]^2)*Sqrt[1 + (b*Tanh[x]^2)/a]))/(3*(a + b)^3*(a + b*Tanh[x]^2)^(3 
/2)*Sqrt[1 + (b*Tanh[x]^2)/a])
 

Rubi [A] (verified)

Time = 0.52 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.10, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.412, Rules used = {3042, 4153, 372, 402, 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.

\(\displaystyle \int \frac {\tanh ^4(x)}{\left (a+b \tanh ^2(x)\right )^{5/2}} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {\tan (i x)^4}{\left (a-b \tan (i x)^2\right )^{5/2}}dx\)

\(\Big \downarrow \) 4153

\(\displaystyle \int \frac {\tanh ^4(x)}{\left (1-\tanh ^2(x)\right ) \left (a+b \tanh ^2(x)\right )^{5/2}}d\tanh (x)\)

\(\Big \downarrow \) 372

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

\(\Big \downarrow \) 402

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 291

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

\(\Big \downarrow \) 219

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

Input:

Int[Tanh[x]^4/(a + b*Tanh[x]^2)^(5/2),x]
 

Output:

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

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 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 402

Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_.)*((e_) + (f_.)*(x 
_)^2), x_Symbol] :> Simp[(-(b*e - a*f))*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^ 
(q + 1)/(a*2*(b*c - a*d)*(p + 1))), x] + Simp[1/(a*2*(b*c - a*d)*(p + 1)) 
 Int[(a + b*x^2)^(p + 1)*(c + d*x^2)^q*Simp[c*(b*e - a*f) + e*2*(b*c - a*d) 
*(p + 1) + d*(b*e - a*f)*(2*(p + q + 2) + 1)*x^2, x], x], x] /; FreeQ[{a, b 
, c, d, e, f, q}, x] && LtQ[p, -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. \(490\) vs. \(2(76)=152\).

Time = 0.06 (sec) , antiderivative size = 491, normalized size of antiderivative = 5.46

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 {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}}}\)	\(491\)
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 {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}}}\)	\(491\)

Input:

int(tanh(x)^4/(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) 
-1/6/(a+b)/(b*(tanh(x)-1)^2+2*b*(tanh(x)-1)+a+b)^(3/2)+1/6*b/(a+b)/a/(b*(t 
anh(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*(t 
anh(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. 2545 vs. \(2 (76) = 152\).

Time = 0.51 (sec) , antiderivative size = 5719, normalized size of antiderivative = 63.54 \[ \int \frac {\tanh ^4(x)}{\left (a+b \tanh ^2(x)\right )^{5/2}} \, dx=\text {Too large to display} \] Input:

integrate(tanh(x)^4/(a+b*tanh(x)^2)^(5/2),x, algorithm="fricas")
 

Output:

Too large to include
 

Sympy [F]

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

integrate(tanh(x)**4/(a+b*tanh(x)**2)**(5/2),x)
 

Output:

Integral(tanh(x)**4/(a + b*tanh(x)**2)**(5/2), x)
 

Maxima [F]

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

integrate(tanh(x)^4/(a+b*tanh(x)^2)^(5/2),x, algorithm="maxima")
 

Output:

integrate(tanh(x)^4/(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. 684 vs. \(2 (76) = 152\).

Time = 0.32 (sec) , antiderivative size = 684, normalized size of antiderivative = 7.60 \[ \int \frac {\tanh ^4(x)}{\left (a+b \tanh ^2(x)\right )^{5/2}} \, dx =\text {Too large to display} \] Input:

integrate(tanh(x)^4/(a+b*tanh(x)^2)^(5/2),x, algorithm="giac")
 

Output:

-4/3*((((a^7*b^2 + 5*a^6*b^3 + 10*a^5*b^4 + 10*a^4*b^5 + 5*a^3*b^6 + a^2*b 
^7)*e^(2*x)/(a^8*b^2 + 6*a^7*b^3 + 15*a^6*b^4 + 20*a^5*b^5 + 15*a^4*b^6 + 
6*a^3*b^7 + a^2*b^8) - 3*(a^6*b^3 + 4*a^5*b^4 + 6*a^4*b^5 + 4*a^3*b^6 + a^ 
2*b^7)/(a^8*b^2 + 6*a^7*b^3 + 15*a^6*b^4 + 20*a^5*b^5 + 15*a^4*b^6 + 6*a^3 
*b^7 + a^2*b^8))*e^(2*x) + 3*(a^6*b^3 + 4*a^5*b^4 + 6*a^4*b^5 + 4*a^3*b^6 
+ a^2*b^7)/(a^8*b^2 + 6*a^7*b^3 + 15*a^6*b^4 + 20*a^5*b^5 + 15*a^4*b^6 + 6 
*a^3*b^7 + a^2*b^8))*e^(2*x) - (a^7*b^2 + 5*a^6*b^3 + 10*a^5*b^4 + 10*a^4* 
b^5 + 5*a^3*b^6 + a^2*b^7)/(a^8*b^2 + 6*a^7*b^3 + 15*a^6*b^4 + 20*a^5*b^5 
+ 15*a^4*b^6 + 6*a^3*b^7 + a^2*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) - 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))
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\tanh ^4(x)}{\left (a+b \tanh ^2(x)\right )^{5/2}} \, dx=\int \frac {{\mathrm {tanh}\left (x\right )}^4}{{\left (b\,{\mathrm {tanh}\left (x\right )}^2+a\right )}^{5/2}} \,d x \] Input:

int(tanh(x)^4/(a + b*tanh(x)^2)^(5/2),x)
 

Output:

int(tanh(x)^4/(a + b*tanh(x)^2)^(5/2), x)
 

Reduce [F]

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

int(tanh(x)^4/(a+b*tanh(x)^2)^(5/2),x)
 

Output:

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