\(\int \frac {1}{(a+b \tanh ^2(x))^{5/2}} \, dx\) [252]

Optimal result

Integrand size = 12, antiderivative size = 93 \[ \int \frac {1}{\left (a+b \tanh ^2(x)\right )^{5/2}} \, dx=\frac {\text {arctanh}\left (\frac {\sqrt {a+b} \tanh (x)}{\sqrt {a+b \tanh ^2(x)}}\right )}{(a+b)^{5/2}}+\frac {b \tanh (x)}{3 a (a+b) \left (a+b \tanh ^2(x)\right )^{3/2}}+\frac {b (5 a+2 b) \tanh (x)}{3 a^2 (a+b)^2 \sqrt {a+b \tanh ^2(x)}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.07 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3742, 425, 541, 12, 385, 212} \[ \int \frac {1}{\left (a+b \tanh ^2(x)\right )^{5/2}} \, dx=\frac {b (5 a+2 b) \tanh (x)}{3 a^2 (a+b)^2 \sqrt {a+b \tanh ^2(x)}}+\frac {\text {arctanh}\left (\frac {\sqrt {a+b} \tanh (x)}{\sqrt {a+b \tanh ^2(x)}}\right )}{(a+b)^{5/2}}+\frac {b \tanh (x)}{3 a (a+b) \left (a+b \tanh ^2(x)\right )^{3/2}} \]

[In]

Int[(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) + (b*Tanh[x])/(3*a*(a + b)*(a + b*Tanh[x]^2
)^(3/2)) + (b*(5*a + 2*b)*Tanh[x])/(3*a^2*(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 425

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(-b)*x*(a + b*x^n)^(p + 1)*
((c + d*x^n)^(q + 1)/(a*n*(p + 1)*(b*c - a*d))), x] + Dist[1/(a*n*(p + 1)*(b*c - a*d)), Int[(a + b*x^n)^(p + 1
)*(c + d*x^n)^q*Simp[b*c + n*(p + 1)*(b*c - a*d) + d*b*(n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c,
d, n, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] &&  !( !IntegerQ[p] && IntegerQ[q] && LtQ[q, -1]) && IntBinomi
alQ[a, b, c, d, 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 3742

Int[((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[(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, e, f, n, p}, x] && (IntegersQ[n, p] || IGtQ[p, 0] || EqQ[n^2, 4] || EqQ[n^2, 16])

Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {1}{\left (1-x^2\right ) \left (a+b x^2\right )^{5/2}} \, dx,x,\tanh (x)\right ) \\ & = \frac {b \tanh (x)}{3 a (a+b) \left (a+b \tanh ^2(x)\right )^{3/2}}-\frac {\text {Subst}\left (\int \frac {b-3 (a+b)+2 b x^2}{\left (1-x^2\right ) \left (a+b x^2\right )^{3/2}} \, dx,x,\tanh (x)\right )}{3 a (a+b)} \\ & = \frac {b \tanh (x)}{3 a (a+b) \left (a+b \tanh ^2(x)\right )^{3/2}}+\frac {b (5 a+2 b) \tanh (x)}{3 a^2 (a+b)^2 \sqrt {a+b \tanh ^2(x)}}+\frac {\text {Subst}\left (\int \frac {3 a^2}{\left (1-x^2\right ) \sqrt {a+b x^2}} \, dx,x,\tanh (x)\right )}{3 a^2 (a+b)^2} \\ & = \frac {b \tanh (x)}{3 a (a+b) \left (a+b \tanh ^2(x)\right )^{3/2}}+\frac {b (5 a+2 b) \tanh (x)}{3 a^2 (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 {b \tanh (x)}{3 a (a+b) \left (a+b \tanh ^2(x)\right )^{3/2}}+\frac {b (5 a+2 b) \tanh (x)}{3 a^2 (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 {b \tanh (x)}{3 a (a+b) \left (a+b \tanh ^2(x)\right )^{3/2}}+\frac {b (5 a+2 b) \tanh (x)}{3 a^2 (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 = 7.82 (sec) , antiderivative size = 976, normalized size of antiderivative = 10.49 \[ \int \frac {1}{\left (a+b \tanh ^2(x)\right )^{5/2}} \, dx=\frac {\cosh (x) \sinh (x) \left (1575 \arcsin \left (\sqrt {-\frac {(a+b) \sinh ^2(x)}{a}}\right )+\frac {3150 (a+b) \arcsin \left (\sqrt {-\frac {(a+b) \sinh ^2(x)}{a}}\right ) \sinh ^2(x)}{a}+\frac {1575 (a+b)^2 \arcsin \left (\sqrt {-\frac {(a+b) \sinh ^2(x)}{a}}\right ) \sinh ^4(x)}{a^2}+\frac {2100 b \arcsin \left (\sqrt {-\frac {(a+b) \sinh ^2(x)}{a}}\right ) \tanh ^2(x)}{a}+\frac {4200 b (a+b) \arcsin \left (\sqrt {-\frac {(a+b) \sinh ^2(x)}{a}}\right ) \sinh ^2(x) \tanh ^2(x)}{a^2}+\frac {2100 b (a+b)^2 \arcsin \left (\sqrt {-\frac {(a+b) \sinh ^2(x)}{a}}\right ) \sinh ^4(x) \tanh ^2(x)}{a^3}+\frac {840 b^2 \arcsin \left (\sqrt {-\frac {(a+b) \sinh ^2(x)}{a}}\right ) \tanh ^4(x)}{a^2}+\frac {1680 b^2 (a+b) \arcsin \left (\sqrt {-\frac {(a+b) \sinh ^2(x)}{a}}\right ) \sinh ^2(x) \tanh ^4(x)}{a^3}+\frac {840 b^2 (a+b)^2 \arcsin \left (\sqrt {-\frac {(a+b) \sinh ^2(x)}{a}}\right ) \sinh ^4(x) \tanh ^4(x)}{a^4}+2100 \left (-\frac {(a+b) \sinh ^2(x)}{a}\right )^{3/2} \sqrt {\frac {\cosh ^2(x) \left (a+b \tanh ^2(x)\right )}{a}}+96 \operatorname {Hypergeometric2F1}\left (2,2,\frac {9}{2},-\frac {(a+b) \sinh ^2(x)}{a}\right ) \left (-\frac {(a+b) \sinh ^2(x)}{a}\right )^{7/2} \sqrt {\frac {\cosh ^2(x) \left (a+b \tanh ^2(x)\right )}{a}}+24 \, _3F_2\left (2,2,2;1,\frac {9}{2};-\frac {(a+b) \sinh ^2(x)}{a}\right ) \left (-\frac {(a+b) \sinh ^2(x)}{a}\right )^{7/2} \sqrt {\frac {\cosh ^2(x) \left (a+b \tanh ^2(x)\right )}{a}}+\frac {2800 b \left (-\frac {(a+b) \sinh ^2(x)}{a}\right )^{3/2} \tanh ^2(x) \sqrt {\frac {\cosh ^2(x) \left (a+b \tanh ^2(x)\right )}{a}}}{a}+\frac {168 b \operatorname {Hypergeometric2F1}\left (2,2,\frac {9}{2},-\frac {(a+b) \sinh ^2(x)}{a}\right ) \left (-\frac {(a+b) \sinh ^2(x)}{a}\right )^{7/2} \tanh ^2(x) \sqrt {\frac {\cosh ^2(x) \left (a+b \tanh ^2(x)\right )}{a}}}{a}+\frac {48 b \, _3F_2\left (2,2,2;1,\frac {9}{2};-\frac {(a+b) \sinh ^2(x)}{a}\right ) \left (-\frac {(a+b) \sinh ^2(x)}{a}\right )^{7/2} \tanh ^2(x) \sqrt {\frac {\cosh ^2(x) \left (a+b \tanh ^2(x)\right )}{a}}}{a}+\frac {1120 b^2 \left (-\frac {(a+b) \sinh ^2(x)}{a}\right )^{3/2} \tanh ^4(x) \sqrt {\frac {\cosh ^2(x) \left (a+b \tanh ^2(x)\right )}{a}}}{a^2}+\frac {72 b^2 \operatorname {Hypergeometric2F1}\left (2,2,\frac {9}{2},-\frac {(a+b) \sinh ^2(x)}{a}\right ) \left (-\frac {(a+b) \sinh ^2(x)}{a}\right )^{7/2} \tanh ^4(x) \sqrt {\frac {\cosh ^2(x) \left (a+b \tanh ^2(x)\right )}{a}}}{a^2}+\frac {24 b^2 \, _3F_2\left (2,2,2;1,\frac {9}{2};-\frac {(a+b) \sinh ^2(x)}{a}\right ) \left (-\frac {(a+b) \sinh ^2(x)}{a}\right )^{7/2} \tanh ^4(x) \sqrt {\frac {\cosh ^2(x) \left (a+b \tanh ^2(x)\right )}{a}}}{a^2}-1575 \sqrt {-\frac {(a+b) \cosh ^2(x) \sinh ^2(x) \left (a+b \tanh ^2(x)\right )}{a^2}}-\frac {2100 b \tanh ^2(x) \sqrt {-\frac {(a+b) \cosh ^2(x) \sinh ^2(x) \left (a+b \tanh ^2(x)\right )}{a^2}}}{a}-\frac {840 b^2 \tanh ^4(x) \sqrt {-\frac {(a+b) \cosh ^2(x) \sinh ^2(x) \left (a+b \tanh ^2(x)\right )}{a^2}}}{a^2}\right )}{315 a^2 \left (-\frac {(a+b) \sinh ^2(x)}{a}\right )^{5/2} \sqrt {a+b \tanh ^2(x)} \sqrt {\frac {\cosh ^2(x) \left (a+b \tanh ^2(x)\right )}{a}} \left (1+\frac {b \tanh ^2(x)}{a}\right )} \]

[In]

Integrate[(a + b*Tanh[x]^2)^(-5/2),x]

[Out]

(Cosh[x]*Sinh[x]*(1575*ArcSin[Sqrt[-(((a + b)*Sinh[x]^2)/a)]] + (3150*(a + b)*ArcSin[Sqrt[-(((a + b)*Sinh[x]^2
)/a)]]*Sinh[x]^2)/a + (1575*(a + b)^2*ArcSin[Sqrt[-(((a + b)*Sinh[x]^2)/a)]]*Sinh[x]^4)/a^2 + (2100*b*ArcSin[S
qrt[-(((a + b)*Sinh[x]^2)/a)]]*Tanh[x]^2)/a + (4200*b*(a + b)*ArcSin[Sqrt[-(((a + b)*Sinh[x]^2)/a)]]*Sinh[x]^2
*Tanh[x]^2)/a^2 + (2100*b*(a + b)^2*ArcSin[Sqrt[-(((a + b)*Sinh[x]^2)/a)]]*Sinh[x]^4*Tanh[x]^2)/a^3 + (840*b^2
*ArcSin[Sqrt[-(((a + b)*Sinh[x]^2)/a)]]*Tanh[x]^4)/a^2 + (1680*b^2*(a + b)*ArcSin[Sqrt[-(((a + b)*Sinh[x]^2)/a
)]]*Sinh[x]^2*Tanh[x]^4)/a^3 + (840*b^2*(a + b)^2*ArcSin[Sqrt[-(((a + b)*Sinh[x]^2)/a)]]*Sinh[x]^4*Tanh[x]^4)/
a^4 + 2100*(-(((a + b)*Sinh[x]^2)/a))^(3/2)*Sqrt[(Cosh[x]^2*(a + b*Tanh[x]^2))/a] + 96*Hypergeometric2F1[2, 2,
9/2, -(((a + b)*Sinh[x]^2)/a)]*(-(((a + b)*Sinh[x]^2)/a))^(7/2)*Sqrt[(Cosh[x]^2*(a + b*Tanh[x]^2))/a] + 24*Hy
pergeometricPFQ[{2, 2, 2}, {1, 9/2}, -(((a + b)*Sinh[x]^2)/a)]*(-(((a + b)*Sinh[x]^2)/a))^(7/2)*Sqrt[(Cosh[x]^
2*(a + b*Tanh[x]^2))/a] + (2800*b*(-(((a + b)*Sinh[x]^2)/a))^(3/2)*Tanh[x]^2*Sqrt[(Cosh[x]^2*(a + b*Tanh[x]^2)
)/a])/a + (168*b*Hypergeometric2F1[2, 2, 9/2, -(((a + b)*Sinh[x]^2)/a)]*(-(((a + b)*Sinh[x]^2)/a))^(7/2)*Tanh[
x]^2*Sqrt[(Cosh[x]^2*(a + b*Tanh[x]^2))/a])/a + (48*b*HypergeometricPFQ[{2, 2, 2}, {1, 9/2}, -(((a + b)*Sinh[x
]^2)/a)]*(-(((a + b)*Sinh[x]^2)/a))^(7/2)*Tanh[x]^2*Sqrt[(Cosh[x]^2*(a + b*Tanh[x]^2))/a])/a + (1120*b^2*(-(((
a + b)*Sinh[x]^2)/a))^(3/2)*Tanh[x]^4*Sqrt[(Cosh[x]^2*(a + b*Tanh[x]^2))/a])/a^2 + (72*b^2*Hypergeometric2F1[2
, 2, 9/2, -(((a + b)*Sinh[x]^2)/a)]*(-(((a + b)*Sinh[x]^2)/a))^(7/2)*Tanh[x]^4*Sqrt[(Cosh[x]^2*(a + b*Tanh[x]^
2))/a])/a^2 + (24*b^2*HypergeometricPFQ[{2, 2, 2}, {1, 9/2}, -(((a + b)*Sinh[x]^2)/a)]*(-(((a + b)*Sinh[x]^2)/
a))^(7/2)*Tanh[x]^4*Sqrt[(Cosh[x]^2*(a + b*Tanh[x]^2))/a])/a^2 - 1575*Sqrt[-(((a + b)*Cosh[x]^2*Sinh[x]^2*(a +
b*Tanh[x]^2))/a^2)] - (2100*b*Tanh[x]^2*Sqrt[-(((a + b)*Cosh[x]^2*Sinh[x]^2*(a + b*Tanh[x]^2))/a^2)])/a - (84
0*b^2*Tanh[x]^4*Sqrt[-(((a + b)*Cosh[x]^2*Sinh[x]^2*(a + b*Tanh[x]^2))/a^2)])/a^2))/(315*a^2*(-(((a + b)*Sinh[
x]^2)/a))^(5/2)*Sqrt[a + b*Tanh[x]^2]*Sqrt[(Cosh[x]^2*(a + b*Tanh[x]^2))/a]*(1 + (b*Tanh[x]^2)/a))

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(419\) vs. \(2(79)=158\).

Time = 0.10 (sec) , antiderivative size = 420, normalized size of antiderivative = 4.52

method	result	size
derivativedivides	\(\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}}}-\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}}}\)	\(420\)
default	\(\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}}}-\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}}}\)	\(420\)

[In]

int(1/(a+b*tanh(x)^2)^(5/2),x,method=_RETURNVERBOSE)

[Out]

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+tanh(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+tanh(x))^2-2*b*(1+tanh(x))+a+b)^(1/2))/(1+tanh(x)))-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. 3152 vs. \(2 (79) = 158\).

Time = 0.68 (sec) , antiderivative size = 6933, normalized size of antiderivative = 74.55 \[ \int \frac {1}{\left (a+b \tanh ^2(x)\right )^{5/2}} \, dx=\text {Too large to display} \]

[In]

integrate(1/(a+b*tanh(x)^2)^(5/2),x, algorithm="fricas")

[Out]

Too large to include

Sympy [F]

\[ \int \frac {1}{\left (a+b \tanh ^2(x)\right )^{5/2}} \, dx=\int \frac {1}{\left (a + b \tanh ^{2}{\left (x \right )}\right )^{\frac {5}{2}}}\, dx \]

[In]

integrate(1/(a+b*tanh(x)**2)**(5/2),x)

[Out]

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

Maxima [F]

\[ \int \frac {1}{\left (a+b \tanh ^2(x)\right )^{5/2}} \, dx=\int { \frac {1}{{\left (b \tanh \left (x\right )^{2} + a\right )}^{\frac {5}{2}}} \,d x } \]

[In]

integrate(1/(a+b*tanh(x)^2)^(5/2),x, algorithm="maxima")

[Out]

integrate((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. 714 vs. \(2 (79) = 158\).

Time = 0.52 (sec) , antiderivative size = 714, normalized size of antiderivative = 7.68 \[ \int \frac {1}{\left (a+b \tanh ^2(x)\right )^{5/2}} \, dx=\frac {2 \, {\left ({\left ({\left (\frac {{\left (3 \, a^{6} b^{3} + 16 \, a^{5} b^{4} + 35 \, a^{4} b^{5} + 40 \, a^{3} b^{6} + 25 \, a^{2} b^{7} + 8 \, a b^{8} + b^{9}\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^{6} b^{3} + 2 \, a^{5} b^{4} - 3 \, a^{4} b^{5} - 12 \, a^{3} b^{6} - 13 \, a^{2} b^{7} - 6 \, a b^{8} - b^{9}\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^{6} b^{3} + 2 \, a^{5} b^{4} - 3 \, a^{4} b^{5} - 12 \, a^{3} b^{6} - 13 \, a^{2} b^{7} - 6 \, a b^{8} - b^{9}\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^{6} b^{3} + 16 \, a^{5} b^{4} + 35 \, a^{4} b^{5} + 40 \, a^{3} b^{6} + 25 \, a^{2} b^{7} + 8 \, a b^{8} + b^{9}}{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 )}}{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(1/(a+b*tanh(x)^2)^(5/2),x, algorithm="giac")

[Out]

2/3*((((3*a^6*b^3 + 16*a^5*b^4 + 35*a^4*b^5 + 40*a^3*b^6 + 25*a^2*b^7 + 8*a*b^8 + b^9)*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 + 2*a^5*b^4 - 3*a^4*b^5 - 12*
a^3*b^6 - 13*a^2*b^7 - 6*a*b^8 - b^9)/(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 + 2*a^5*b^4 - 3*a^4*b^5 - 12*a^3*b^6 - 13*a^2*b^7 - 6*a*b^8 - b^9)/(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 + 16*a^5*b^4 +
35*a^4*b^5 + 40*a^3*b^6 + 25*a^2*b^7 + 8*a*b^8 + b^9)/(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. \[ \int \frac {1}{\left (a+b \tanh ^2(x)\right )^{5/2}} \, dx=\int \frac {1}{{\left (b\,{\mathrm {tanh}\left (x\right )}^2+a\right )}^{5/2}} \,d x \]

[In]

int(1/(a + b*tanh(x)^2)^(5/2),x)

[Out]

int(1/(a + b*tanh(x)^2)^(5/2), x)