\(\int \frac {\tanh ^2(x)}{(a+b \tanh ^2(x))^{5/2}} \, dx\) [250]

Optimal result

Integrand size = 17, antiderivative size = 88 \[ \int \frac {\tanh ^2(x)}{\left (a+b \tanh ^2(x)\right )^{5/2}} \, dx=\frac {\text {arctanh}\left (\frac {\sqrt {a+b} \tanh (x)}{\sqrt {a+b \tanh ^2(x)}}\right )}{(a+b)^{5/2}}-\frac {\tanh (x)}{3 (a+b) \left (a+b \tanh ^2(x)\right )^{3/2}}-\frac {(2 a-b) \tanh (x)}{3 a (a+b)^2 \sqrt {a+b \tanh ^2(x)}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.10 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.353, Rules used = {3751, 482, 541, 12, 385, 212} \[ \int \frac {\tanh ^2(x)}{\left (a+b \tanh ^2(x)\right )^{5/2}} \, dx=\frac {\text {arctanh}\left (\frac {\sqrt {a+b} \tanh (x)}{\sqrt {a+b \tanh ^2(x)}}\right )}{(a+b)^{5/2}}-\frac {(2 a-b) \tanh (x)}{3 a (a+b)^2 \sqrt {a+b \tanh ^2(x)}}-\frac {\tanh (x)}{3 (a+b) \left (a+b \tanh ^2(x)\right )^{3/2}} \]

[In]

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

[Out]

ArcTanh[(Sqrt[a + b]*Tanh[x])/Sqrt[a + b*Tanh[x]^2]]/(a + b)^(5/2) - Tanh[x]/(3*(a + b)*(a + b*Tanh[x]^2)^(3/2
)) - ((2*a - b)*Tanh[x])/(3*a*(a + b)^2*Sqrt[a + b*Tanh[x]^2])

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 212

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] && (GtQ[a, 0] || LtQ[b, 0])

Rule 385

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

Rule 482

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

Rule 541

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

Rule 3751

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]}, Dist[c*(ff/f), Subst[Int[(d*ff*(x/c))^m*((a + b*(ff*x)^n)^p/(c^2
+ ff^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] && RationalQ[n]))

Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {x^2}{\left (1-x^2\right ) \left (a+b x^2\right )^{5/2}} \, dx,x,\tanh (x)\right ) \\ & = -\frac {\tanh (x)}{3 (a+b) \left (a+b \tanh ^2(x)\right )^{3/2}}+\frac {\text {Subst}\left (\int \frac {1+2 x^2}{\left (1-x^2\right ) \left (a+b x^2\right )^{3/2}} \, dx,x,\tanh (x)\right )}{3 (a+b)} \\ & = -\frac {\tanh (x)}{3 (a+b) \left (a+b \tanh ^2(x)\right )^{3/2}}-\frac {(2 a-b) \tanh (x)}{3 a (a+b)^2 \sqrt {a+b \tanh ^2(x)}}-\frac {\text {Subst}\left (\int -\frac {3 a}{\left (1-x^2\right ) \sqrt {a+b x^2}} \, dx,x,\tanh (x)\right )}{3 a (a+b)^2} \\ & = -\frac {\tanh (x)}{3 (a+b) \left (a+b \tanh ^2(x)\right )^{3/2}}-\frac {(2 a-b) \tanh (x)}{3 a (a+b)^2 \sqrt {a+b \tanh ^2(x)}}+\frac {\text {Subst}\left (\int \frac {1}{\left (1-x^2\right ) \sqrt {a+b x^2}} \, dx,x,\tanh (x)\right )}{(a+b)^2} \\ & = -\frac {\tanh (x)}{3 (a+b) \left (a+b \tanh ^2(x)\right )^{3/2}}-\frac {(2 a-b) \tanh (x)}{3 a (a+b)^2 \sqrt {a+b \tanh ^2(x)}}+\frac {\text {Subst}\left (\int \frac {1}{1-(a+b) x^2} \, dx,x,\frac {\tanh (x)}{\sqrt {a+b \tanh ^2(x)}}\right )}{(a+b)^2} \\ & = \frac {\text {arctanh}\left (\frac {\sqrt {a+b} \tanh (x)}{\sqrt {a+b \tanh ^2(x)}}\right )}{(a+b)^{5/2}}-\frac {\tanh (x)}{3 (a+b) \left (a+b \tanh ^2(x)\right )^{3/2}}-\frac {(2 a-b) \tanh (x)}{3 a (a+b)^2 \sqrt {a+b \tanh ^2(x)}} \\ \end{align*}

Mathematica [C] (warning: unable to verify)

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

Time = 7.11 (sec) , antiderivative size = 216, normalized size of antiderivative = 2.45 \[ \int \frac {\tanh ^2(x)}{\left (a+b \tanh ^2(x)\right )^{5/2}} \, dx=\frac {\sinh ^2(x) \tanh (x) \left (-\frac {4 (a+b) \operatorname {Hypergeometric2F1}\left (2,2,\frac {9}{2},-\frac {(a+b) \sinh ^2(x)}{a}\right ) \sinh ^2(x) \left (a+b \tanh ^2(x)\right )}{35 a^2}-\frac {\coth ^4(x) \left (-5 a-2 b \tanh ^2(x)\right ) \left (3 \arcsin \left (\sqrt {-\frac {(a+b) \sinh ^2(x)}{a}}\right ) \left (a+b \tanh ^2(x)\right )^2+a \text {sech}^2(x) \sqrt {-\frac {(a+b) \cosh ^2(x) \sinh ^2(x) \left (a+b \tanh ^2(x)\right )}{a^2}} \left (-4 b \tanh ^2(x)+a \left (-3-\tanh ^2(x)\right )\right )\right )}{3 a (a+b)^2 \sqrt {-\frac {(a+b) \cosh ^2(x) \sinh ^2(x) \left (a+b \tanh ^2(x)\right )}{a^2}}}\right )}{3 a^2 \sqrt {a+b \tanh ^2(x)} \left (1+\frac {b \tanh ^2(x)}{a}\right )} \]

[In]

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

[Out]

(Sinh[x]^2*Tanh[x]*((-4*(a + b)*Hypergeometric2F1[2, 2, 9/2, -(((a + b)*Sinh[x]^2)/a)]*Sinh[x]^2*(a + b*Tanh[x
]^2))/(35*a^2) - (Coth[x]^4*(-5*a - 2*b*Tanh[x]^2)*(3*ArcSin[Sqrt[-(((a + b)*Sinh[x]^2)/a)]]*(a + b*Tanh[x]^2)
^2 + a*Sech[x]^2*Sqrt[-(((a + b)*Cosh[x]^2*Sinh[x]^2*(a + b*Tanh[x]^2))/a^2)]*(-4*b*Tanh[x]^2 + a*(-3 - Tanh[x
]^2))))/(3*a*(a + b)^2*Sqrt[-(((a + b)*Cosh[x]^2*Sinh[x]^2*(a + b*Tanh[x]^2))/a^2)])))/(3*a^2*Sqrt[a + b*Tanh[
x]^2]*(1 + (b*Tanh[x]^2)/a))

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(453\) vs. \(2(74)=148\).

Time = 0.08 (sec) , antiderivative size = 454, normalized size of antiderivative = 5.16

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 {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 (1+\tanh \left (x \right )\right )^{2}-2 b \left (1+\tanh \left (x \right )\right )+a +b \right )^{\frac {3}{2}}}+\frac {b \tanh \left (x \right )}{6 \left (a +b \right ) a \left (b \left (1+\tanh \left (x \right )\right )^{2}-2 b \left (1+\tanh \left (x \right )\right )+a +b \right )^{\frac {3}{2}}}+\frac {b \tanh \left (x \right )}{3 \left (a +b \right ) a^{2} \sqrt {b \left (1+\tanh \left (x \right )\right )^{2}-2 b \left (1+\tanh \left (x \right )\right )+a +b}}+\frac {1}{2 \left (a +b \right )^{2} \sqrt {b \left (1+\tanh \left (x \right )\right )^{2}-2 b \left (1+\tanh \left (x \right )\right )+a +b}}+\frac {b \tanh \left (x \right )}{2 \left (a +b \right )^{2} a \sqrt {b \left (1+\tanh \left (x \right )\right )^{2}-2 b \left (1+\tanh \left (x \right )\right )+a +b}}-\frac {\ln \left (\frac {2 a +2 b -2 b \left (1+\tanh \left (x \right )\right )+2 \sqrt {a +b}\, \sqrt {b \left (1+\tanh \left (x \right )\right )^{2}-2 b \left (1+\tanh \left (x \right )\right )+a +b}}{1+\tanh \left (x \right )}\right )}{2 \left (a +b \right )^{\frac {5}{2}}}\)	\(454\)
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 {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 (1+\tanh \left (x \right )\right )^{2}-2 b \left (1+\tanh \left (x \right )\right )+a +b \right )^{\frac {3}{2}}}+\frac {b \tanh \left (x \right )}{6 \left (a +b \right ) a \left (b \left (1+\tanh \left (x \right )\right )^{2}-2 b \left (1+\tanh \left (x \right )\right )+a +b \right )^{\frac {3}{2}}}+\frac {b \tanh \left (x \right )}{3 \left (a +b \right ) a^{2} \sqrt {b \left (1+\tanh \left (x \right )\right )^{2}-2 b \left (1+\tanh \left (x \right )\right )+a +b}}+\frac {1}{2 \left (a +b \right )^{2} \sqrt {b \left (1+\tanh \left (x \right )\right )^{2}-2 b \left (1+\tanh \left (x \right )\right )+a +b}}+\frac {b \tanh \left (x \right )}{2 \left (a +b \right )^{2} a \sqrt {b \left (1+\tanh \left (x \right )\right )^{2}-2 b \left (1+\tanh \left (x \right )\right )+a +b}}-\frac {\ln \left (\frac {2 a +2 b -2 b \left (1+\tanh \left (x \right )\right )+2 \sqrt {a +b}\, \sqrt {b \left (1+\tanh \left (x \right )\right )^{2}-2 b \left (1+\tanh \left (x \right )\right )+a +b}}{1+\tanh \left (x \right )}\right )}{2 \left (a +b \right )^{\frac {5}{2}}}\)	\(454\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2939 vs. \(2 (74) = 148\).

Time = 0.66 (sec) , antiderivative size = 6507, normalized size of antiderivative = 73.94 \[ \int \frac {\tanh ^2(x)}{\left (a+b \tanh ^2(x)\right )^{5/2}} \, dx=\text {Too large to display} \]

[In]

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

[Out]

Too large to include

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

integrate(tanh(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. 728 vs. \(2 (74) = 148\).

Time = 0.52 (sec) , antiderivative size = 728, normalized size of antiderivative = 8.27 \[ \int \frac {\tanh ^2(x)}{\left (a+b \tanh ^2(x)\right )^{5/2}} \, dx=-\frac {{\left ({\left (\frac {{\left (3 \, a^{7} b^{2} + 14 \, a^{6} b^{3} + 25 \, a^{5} b^{4} + 20 \, a^{4} b^{5} + 5 \, a^{3} b^{6} - 2 \, a^{2} b^{7} - a b^{8}\right )} e^{\left (2 \, x\right )}}{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}} + \frac {3 \, {\left (a^{7} b^{2} + 2 \, a^{6} b^{3} - a^{5} b^{4} - 4 \, a^{4} b^{5} - a^{3} b^{6} + 2 \, a^{2} b^{7} + a b^{8}\right )}}{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}}\right )} e^{\left (2 \, x\right )} - \frac {3 \, {\left (a^{7} b^{2} + 2 \, a^{6} b^{3} - a^{5} b^{4} - 4 \, a^{4} b^{5} - a^{3} b^{6} + 2 \, a^{2} b^{7} + a b^{8}\right )}}{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}}\right )} e^{\left (2 \, x\right )} - \frac {3 \, a^{7} b^{2} + 14 \, a^{6} b^{3} + 25 \, a^{5} b^{4} + 20 \, a^{4} b^{5} + 5 \, a^{3} b^{6} - 2 \, a^{2} b^{7} - a b^{8}}{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 \, {\left (a e^{\left (4 \, x\right )} + b e^{\left (4 \, x\right )} + 2 \, a e^{\left (2 \, x\right )} - 2 \, b e^{\left (2 \, x\right )} + a + b\right )}^{\frac {3}{2}}} - \frac {\log \left ({\left | -{\left (\sqrt {a + b} e^{\left (2 \, x\right )} - \sqrt {a e^{\left (4 \, x\right )} + b e^{\left (4 \, x\right )} + 2 \, a e^{\left (2 \, x\right )} - 2 \, b e^{\left (2 \, x\right )} + a + b}\right )} {\left (a + b\right )} - \sqrt {a + b} {\left (a - b\right )} \right |}\right )}{2 \, {\left (a^{2} + 2 \, a b + b^{2}\right )} \sqrt {a + b}} - \frac {\log \left ({\left | -\sqrt {a + b} e^{\left (2 \, x\right )} + \sqrt {a e^{\left (4 \, x\right )} + b e^{\left (4 \, x\right )} + 2 \, a e^{\left (2 \, x\right )} - 2 \, b e^{\left (2 \, x\right )} + a + b} + \sqrt {a + b} \right |}\right )}{2 \, {\left (a^{2} + 2 \, a b + b^{2}\right )} \sqrt {a + b}} + \frac {\log \left ({\left | -\sqrt {a + b} e^{\left (2 \, x\right )} + \sqrt {a e^{\left (4 \, x\right )} + b e^{\left (4 \, x\right )} + 2 \, a e^{\left (2 \, x\right )} - 2 \, b e^{\left (2 \, x\right )} + a + b} - \sqrt {a + b} \right |}\right )}{2 \, {\left (a^{2} + 2 \, a b + b^{2}\right )} \sqrt {a + b}} \]

[In]

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

[Out]

-1/3*((((3*a^7*b^2 + 14*a^6*b^3 + 25*a^5*b^4 + 20*a^4*b^5 + 5*a^3*b^6 - 2*a^2*b^7 - a*b^8)*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^7*b^2 + 2*a^6*b^3 - a^5*b^4 - 4
*a^4*b^5 - a^3*b^6 + 2*a^2*b^7 + a*b^8)/(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^7*b^2 + 2*a^6*b^3 - a^5*b^4 - 4*a^4*b^5 - a^3*b^6 + 2*a^2*b^7 + a*b^8)/(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^7*b^2 + 14*a^6*b^3 +
25*a^5*b^4 + 20*a^4*b^5 + 5*a^3*b^6 - 2*a^2*b^7 - a*b^8)/(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. \[ \int \frac {\tanh ^2(x)}{\left (a+b \tanh ^2(x)\right )^{5/2}} \, dx=\int \frac {{\mathrm {tanh}\left (x\right )}^2}{{\left (b\,{\mathrm {tanh}\left (x\right )}^2+a\right )}^{5/2}} \,d x \]

[In]

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

[Out]

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