\(\int \frac {\coth ^2(x)}{(a+b \tanh ^2(x))^{5/2}} \, dx\) [254]

Optimal result

Integrand size = 17, antiderivative size = 131 \[ \int \frac {\coth ^2(x)}{\left (a+b \tanh ^2(x)\right )^{5/2}} \, dx=\frac {\text {arctanh}\left (\frac {\sqrt {a+b} \tanh (x)}{\sqrt {a+b \tanh ^2(x)}}\right )}{(a+b)^{5/2}}+\frac {b \coth (x)}{3 a (a+b) \left (a+b \tanh ^2(x)\right )^{3/2}}+\frac {b (7 a+4 b) \coth (x)}{3 a^2 (a+b)^2 \sqrt {a+b \tanh ^2(x)}}-\frac {(3 a+2 b) (a+4 b) \coth (x) \sqrt {a+b \tanh ^2(x)}}{3 a^3 (a+b)^2} \]

[Out]

arctanh((a+b)^(1/2)*tanh(x)/(a+b*tanh(x)^2)^(1/2))/(a+b)^(5/2)+1/3*b*(7*a+4*b)*coth(x)/a^2/(a+b)^2/(a+b*tanh(x
)^2)^(1/2)-1/3*(3*a+2*b)*(a+4*b)*coth(x)*(a+b*tanh(x)^2)^(1/2)/a^3/(a+b)^2+1/3*b*coth(x)/a/(a+b)/(a+b*tanh(x)^
2)^(3/2)

Rubi [A] (verified)

Time = 0.18 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.412, Rules used = {3751, 483, 593, 597, 12, 385, 212} \[ \int \frac {\coth ^2(x)}{\left (a+b \tanh ^2(x)\right )^{5/2}} \, dx=-\frac {(3 a+2 b) (a+4 b) \coth (x) \sqrt {a+b \tanh ^2(x)}}{3 a^3 (a+b)^2}+\frac {b (7 a+4 b) \coth (x)}{3 a^2 (a+b)^2 \sqrt {a+b \tanh ^2(x)}}+\frac {\text {arctanh}\left (\frac {\sqrt {a+b} \tanh (x)}{\sqrt {a+b \tanh ^2(x)}}\right )}{(a+b)^{5/2}}+\frac {b \coth (x)}{3 a (a+b) \left (a+b \tanh ^2(x)\right )^{3/2}} \]

[In]

Int[Coth[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) + (b*Coth[x])/(3*a*(a + b)*(a + b*Tanh[x]^2
)^(3/2)) + (b*(7*a + 4*b)*Coth[x])/(3*a^2*(a + b)^2*Sqrt[a + b*Tanh[x]^2]) - ((3*a + 2*b)*(a + 4*b)*Coth[x]*Sq
rt[a + b*Tanh[x]^2])/(3*a^3*(a + b)^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 483

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

Rule 593

Int[((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_)*((e_) + (f_.)*(x_)^(n_)), x
_Symbol] :> Simp[(-(b*e - a*f))*(g*x)^(m + 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(a*g*n*(b*c - a*d)*(p +
1))), x] + Dist[1/(a*n*(b*c - a*d)*(p + 1)), Int[(g*x)^m*(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[c*(b*e - a*f)
*(m + 1) + e*n*(b*c - a*d)*(p + 1) + d*(b*e - a*f)*(m + n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c,
d, e, f, g, m, q}, x] && IGtQ[n, 0] && LtQ[p, -1]

Rule 597

Int[((g_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)),
x_Symbol] :> Simp[e*(g*x)^(m + 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(a*c*g*(m + 1))), x] + Dist[1/(a*c*
g^n*(m + 1)), Int[(g*x)^(m + n)*(a + b*x^n)^p*(c + d*x^n)^q*Simp[a*f*c*(m + 1) - e*(b*c + a*d)*(m + n + 1) - e
*n*(b*c*p + a*d*q) - b*e*d*(m + n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p, q}, x] &&
IGtQ[n, 0] && LtQ[m, -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 {1}{x^2 \left (1-x^2\right ) \left (a+b x^2\right )^{5/2}} \, dx,x,\tanh (x)\right ) \\ & = \frac {b \coth (x)}{3 a (a+b) \left (a+b \tanh ^2(x)\right )^{3/2}}-\frac {\text {Subst}\left (\int \frac {-3 a-4 b+4 b x^2}{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 \coth (x)}{3 a (a+b) \left (a+b \tanh ^2(x)\right )^{3/2}}+\frac {b (7 a+4 b) \coth (x)}{3 a^2 (a+b)^2 \sqrt {a+b \tanh ^2(x)}}+\frac {\text {Subst}\left (\int \frac {(3 a+2 b) (a+4 b)-2 b (7 a+4 b) x^2}{x^2 \left (1-x^2\right ) \sqrt {a+b x^2}} \, dx,x,\tanh (x)\right )}{3 a^2 (a+b)^2} \\ & = \frac {b \coth (x)}{3 a (a+b) \left (a+b \tanh ^2(x)\right )^{3/2}}+\frac {b (7 a+4 b) \coth (x)}{3 a^2 (a+b)^2 \sqrt {a+b \tanh ^2(x)}}-\frac {(3 a+2 b) (a+4 b) \coth (x) \sqrt {a+b \tanh ^2(x)}}{3 a^3 (a+b)^2}-\frac {\text {Subst}\left (\int -\frac {3 a^3}{\left (1-x^2\right ) \sqrt {a+b x^2}} \, dx,x,\tanh (x)\right )}{3 a^3 (a+b)^2} \\ & = \frac {b \coth (x)}{3 a (a+b) \left (a+b \tanh ^2(x)\right )^{3/2}}+\frac {b (7 a+4 b) \coth (x)}{3 a^2 (a+b)^2 \sqrt {a+b \tanh ^2(x)}}-\frac {(3 a+2 b) (a+4 b) \coth (x) \sqrt {a+b \tanh ^2(x)}}{3 a^3 (a+b)^2}+\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 \coth (x)}{3 a (a+b) \left (a+b \tanh ^2(x)\right )^{3/2}}+\frac {b (7 a+4 b) \coth (x)}{3 a^2 (a+b)^2 \sqrt {a+b \tanh ^2(x)}}-\frac {(3 a+2 b) (a+4 b) \coth (x) \sqrt {a+b \tanh ^2(x)}}{3 a^3 (a+b)^2}+\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 \coth (x)}{3 a (a+b) \left (a+b \tanh ^2(x)\right )^{3/2}}+\frac {b (7 a+4 b) \coth (x)}{3 a^2 (a+b)^2 \sqrt {a+b \tanh ^2(x)}}-\frac {(3 a+2 b) (a+4 b) \coth (x) \sqrt {a+b \tanh ^2(x)}}{3 a^3 (a+b)^2} \\ \end{align*}

Mathematica [C] (warning: unable to verify)

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

Time = 8.66 (sec) , antiderivative size = 1375, normalized size of antiderivative = 10.50 \[ \int \frac {\coth ^2(x)}{\left (a+b \tanh ^2(x)\right )^{5/2}} \, dx=-\frac {\cosh ^2(x) \coth (x) \left (-\frac {20 a \text {csch}^2(x)}{3 (a+b)}-\frac {5 a^2 \text {csch}^4(x)}{(a+b)^2}-\frac {40 b \text {sech}^2(x)}{a+b}-\frac {30 a b \text {csch}^2(x) \text {sech}^2(x)}{(a+b)^2}-\frac {40 b^2 \text {sech}^4(x)}{(a+b)^2}-\frac {92 (a+b) \operatorname {Hypergeometric2F1}\left (2,2,\frac {9}{2},-\frac {(a+b) \sinh ^2(x)}{a}\right ) \sinh ^2(x)}{105 a}-\frac {24 (a+b) \, _3F_2\left (2,2,2;1,\frac {9}{2};-\frac {(a+b) \sinh ^2(x)}{a}\right ) \sinh ^2(x)}{35 a}-\frac {16 (a+b) \, _4F_3\left (2,2,2,2;1,1,\frac {9}{2};-\frac {(a+b) \sinh ^2(x)}{a}\right ) \sinh ^2(x)}{105 a}-\frac {160 b^2 \text {sech}^2(x) \tanh ^2(x)}{3 a (a+b)}-\frac {124 b (a+b) \operatorname {Hypergeometric2F1}\left (2,2,\frac {9}{2},-\frac {(a+b) \sinh ^2(x)}{a}\right ) \sinh ^2(x) \tanh ^2(x)}{35 a^2}-\frac {16 b (a+b) \, _3F_2\left (2,2,2;1,\frac {9}{2};-\frac {(a+b) \sinh ^2(x)}{a}\right ) \sinh ^2(x) \tanh ^2(x)}{7 a^2}-\frac {16 b (a+b) \, _4F_3\left (2,2,2,2;1,1,\frac {9}{2};-\frac {(a+b) \sinh ^2(x)}{a}\right ) \sinh ^2(x) \tanh ^2(x)}{35 a^2}-\frac {64 b^3 \text {sech}^2(x) \tanh ^4(x)}{3 a^2 (a+b)}-\frac {152 b^2 (a+b) \operatorname {Hypergeometric2F1}\left (2,2,\frac {9}{2},-\frac {(a+b) \sinh ^2(x)}{a}\right ) \sinh ^2(x) \tanh ^4(x)}{35 a^3}-\frac {88 b^2 (a+b) \, _3F_2\left (2,2,2;1,\frac {9}{2};-\frac {(a+b) \sinh ^2(x)}{a}\right ) \sinh ^2(x) \tanh ^4(x)}{35 a^3}-\frac {16 b^2 (a+b) \, _4F_3\left (2,2,2,2;1,1,\frac {9}{2};-\frac {(a+b) \sinh ^2(x)}{a}\right ) \sinh ^2(x) \tanh ^4(x)}{35 a^3}-\frac {176 b^3 (a+b) \operatorname {Hypergeometric2F1}\left (2,2,\frac {9}{2},-\frac {(a+b) \sinh ^2(x)}{a}\right ) \sinh ^2(x) \tanh ^6(x)}{105 a^4}-\frac {32 b^3 (a+b) \, _3F_2\left (2,2,2;1,\frac {9}{2};-\frac {(a+b) \sinh ^2(x)}{a}\right ) \sinh ^2(x) \tanh ^6(x)}{35 a^4}-\frac {16 b^3 (a+b) \, _4F_3\left (2,2,2,2;1,1,\frac {9}{2};-\frac {(a+b) \sinh ^2(x)}{a}\right ) \sinh ^2(x) \tanh ^6(x)}{105 a^4}+\frac {5 \arcsin \left (\sqrt {-\frac {(a+b) \sinh ^2(x)}{a}}\right )}{\left (-\frac {(a+b) \sinh ^2(x)}{a}\right )^{5/2} \sqrt {\frac {\cosh ^2(x) \left (a+b \tanh ^2(x)\right )}{a}}}+\frac {30 b \arcsin \left (\sqrt {-\frac {(a+b) \sinh ^2(x)}{a}}\right ) \tanh ^2(x)}{a \left (-\frac {(a+b) \sinh ^2(x)}{a}\right )^{5/2} \sqrt {\frac {\cosh ^2(x) \left (a+b \tanh ^2(x)\right )}{a}}}+\frac {40 b^2 \arcsin \left (\sqrt {-\frac {(a+b) \sinh ^2(x)}{a}}\right ) \tanh ^4(x)}{a^2 \left (-\frac {(a+b) \sinh ^2(x)}{a}\right )^{5/2} \sqrt {\frac {\cosh ^2(x) \left (a+b \tanh ^2(x)\right )}{a}}}+\frac {16 b^3 \arcsin \left (\sqrt {-\frac {(a+b) \sinh ^2(x)}{a}}\right ) \tanh ^6(x)}{a^3 \left (-\frac {(a+b) \sinh ^2(x)}{a}\right )^{5/2} \sqrt {\frac {\cosh ^2(x) \left (a+b \tanh ^2(x)\right )}{a}}}+\frac {5 \arcsin \left (\sqrt {-\frac {(a+b) \sinh ^2(x)}{a}}\right )}{\sqrt {-\frac {(a+b) \cosh ^2(x) \sinh ^2(x) \left (a+b \tanh ^2(x)\right )}{a^2}}}+\frac {10 a \arcsin \left (\sqrt {-\frac {(a+b) \sinh ^2(x)}{a}}\right ) \text {csch}^2(x)}{(a+b) \sqrt {-\frac {(a+b) \cosh ^2(x) \sinh ^2(x) \left (a+b \tanh ^2(x)\right )}{a^2}}}+\frac {60 b \arcsin \left (\sqrt {-\frac {(a+b) \sinh ^2(x)}{a}}\right ) \text {sech}^2(x)}{(a+b) \sqrt {-\frac {(a+b) \cosh ^2(x) \sinh ^2(x) \left (a+b \tanh ^2(x)\right )}{a^2}}}+\frac {30 b \arcsin \left (\sqrt {-\frac {(a+b) \sinh ^2(x)}{a}}\right ) \tanh ^2(x)}{a \sqrt {-\frac {(a+b) \cosh ^2(x) \sinh ^2(x) \left (a+b \tanh ^2(x)\right )}{a^2}}}+\frac {80 b^2 \arcsin \left (\sqrt {-\frac {(a+b) \sinh ^2(x)}{a}}\right ) \text {sech}^2(x) \tanh ^2(x)}{a (a+b) \sqrt {-\frac {(a+b) \cosh ^2(x) \sinh ^2(x) \left (a+b \tanh ^2(x)\right )}{a^2}}}+\frac {40 b^2 \arcsin \left (\sqrt {-\frac {(a+b) \sinh ^2(x)}{a}}\right ) \tanh ^4(x)}{a^2 \sqrt {-\frac {(a+b) \cosh ^2(x) \sinh ^2(x) \left (a+b \tanh ^2(x)\right )}{a^2}}}+\frac {32 b^3 \arcsin \left (\sqrt {-\frac {(a+b) \sinh ^2(x)}{a}}\right ) \text {sech}^2(x) \tanh ^4(x)}{a^2 (a+b) \sqrt {-\frac {(a+b) \cosh ^2(x) \sinh ^2(x) \left (a+b \tanh ^2(x)\right )}{a^2}}}+\frac {16 b^3 \arcsin \left (\sqrt {-\frac {(a+b) \sinh ^2(x)}{a}}\right ) \tanh ^6(x)}{a^3 \sqrt {-\frac {(a+b) \cosh ^2(x) \sinh ^2(x) \left (a+b \tanh ^2(x)\right )}{a^2}}}+\frac {16 b^3 \left (i \tanh (x)-i \tanh ^3(x)\right )^2}{a (a+b)^2}\right )}{a^2 \sqrt {a+b \tanh ^2(x)} \left (1+\frac {b \tanh ^2(x)}{a}\right )} \]

[In]

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

[Out]

-((Cosh[x]^2*Coth[x]*((-20*a*Csch[x]^2)/(3*(a + b)) - (5*a^2*Csch[x]^4)/(a + b)^2 - (40*b*Sech[x]^2)/(a + b) -
(30*a*b*Csch[x]^2*Sech[x]^2)/(a + b)^2 - (40*b^2*Sech[x]^4)/(a + b)^2 - (92*(a + b)*Hypergeometric2F1[2, 2, 9
/2, -(((a + b)*Sinh[x]^2)/a)]*Sinh[x]^2)/(105*a) - (24*(a + b)*HypergeometricPFQ[{2, 2, 2}, {1, 9/2}, -(((a +
b)*Sinh[x]^2)/a)]*Sinh[x]^2)/(35*a) - (16*(a + b)*HypergeometricPFQ[{2, 2, 2, 2}, {1, 1, 9/2}, -(((a + b)*Sinh
[x]^2)/a)]*Sinh[x]^2)/(105*a) - (160*b^2*Sech[x]^2*Tanh[x]^2)/(3*a*(a + b)) - (124*b*(a + b)*Hypergeometric2F1
[2, 2, 9/2, -(((a + b)*Sinh[x]^2)/a)]*Sinh[x]^2*Tanh[x]^2)/(35*a^2) - (16*b*(a + b)*HypergeometricPFQ[{2, 2, 2
}, {1, 9/2}, -(((a + b)*Sinh[x]^2)/a)]*Sinh[x]^2*Tanh[x]^2)/(7*a^2) - (16*b*(a + b)*HypergeometricPFQ[{2, 2, 2
, 2}, {1, 1, 9/2}, -(((a + b)*Sinh[x]^2)/a)]*Sinh[x]^2*Tanh[x]^2)/(35*a^2) - (64*b^3*Sech[x]^2*Tanh[x]^4)/(3*a
^2*(a + b)) - (152*b^2*(a + b)*Hypergeometric2F1[2, 2, 9/2, -(((a + b)*Sinh[x]^2)/a)]*Sinh[x]^2*Tanh[x]^4)/(35
*a^3) - (88*b^2*(a + b)*HypergeometricPFQ[{2, 2, 2}, {1, 9/2}, -(((a + b)*Sinh[x]^2)/a)]*Sinh[x]^2*Tanh[x]^4)/
(35*a^3) - (16*b^2*(a + b)*HypergeometricPFQ[{2, 2, 2, 2}, {1, 1, 9/2}, -(((a + b)*Sinh[x]^2)/a)]*Sinh[x]^2*Ta
nh[x]^4)/(35*a^3) - (176*b^3*(a + b)*Hypergeometric2F1[2, 2, 9/2, -(((a + b)*Sinh[x]^2)/a)]*Sinh[x]^2*Tanh[x]^
6)/(105*a^4) - (32*b^3*(a + b)*HypergeometricPFQ[{2, 2, 2}, {1, 9/2}, -(((a + b)*Sinh[x]^2)/a)]*Sinh[x]^2*Tanh
[x]^6)/(35*a^4) - (16*b^3*(a + b)*HypergeometricPFQ[{2, 2, 2, 2}, {1, 1, 9/2}, -(((a + b)*Sinh[x]^2)/a)]*Sinh[
x]^2*Tanh[x]^6)/(105*a^4) + (5*ArcSin[Sqrt[-(((a + b)*Sinh[x]^2)/a)]])/((-(((a + b)*Sinh[x]^2)/a))^(5/2)*Sqrt[
(Cosh[x]^2*(a + b*Tanh[x]^2))/a]) + (30*b*ArcSin[Sqrt[-(((a + b)*Sinh[x]^2)/a)]]*Tanh[x]^2)/(a*(-(((a + b)*Sin
h[x]^2)/a))^(5/2)*Sqrt[(Cosh[x]^2*(a + b*Tanh[x]^2))/a]) + (40*b^2*ArcSin[Sqrt[-(((a + b)*Sinh[x]^2)/a)]]*Tanh
[x]^4)/(a^2*(-(((a + b)*Sinh[x]^2)/a))^(5/2)*Sqrt[(Cosh[x]^2*(a + b*Tanh[x]^2))/a]) + (16*b^3*ArcSin[Sqrt[-(((
a + b)*Sinh[x]^2)/a)]]*Tanh[x]^6)/(a^3*(-(((a + b)*Sinh[x]^2)/a))^(5/2)*Sqrt[(Cosh[x]^2*(a + b*Tanh[x]^2))/a])
+ (5*ArcSin[Sqrt[-(((a + b)*Sinh[x]^2)/a)]])/Sqrt[-(((a + b)*Cosh[x]^2*Sinh[x]^2*(a + b*Tanh[x]^2))/a^2)] + (
10*a*ArcSin[Sqrt[-(((a + b)*Sinh[x]^2)/a)]]*Csch[x]^2)/((a + b)*Sqrt[-(((a + b)*Cosh[x]^2*Sinh[x]^2*(a + b*Tan
h[x]^2))/a^2)]) + (60*b*ArcSin[Sqrt[-(((a + b)*Sinh[x]^2)/a)]]*Sech[x]^2)/((a + b)*Sqrt[-(((a + b)*Cosh[x]^2*S
inh[x]^2*(a + b*Tanh[x]^2))/a^2)]) + (30*b*ArcSin[Sqrt[-(((a + b)*Sinh[x]^2)/a)]]*Tanh[x]^2)/(a*Sqrt[-(((a + b
)*Cosh[x]^2*Sinh[x]^2*(a + b*Tanh[x]^2))/a^2)]) + (80*b^2*ArcSin[Sqrt[-(((a + b)*Sinh[x]^2)/a)]]*Sech[x]^2*Tan
h[x]^2)/(a*(a + b)*Sqrt[-(((a + b)*Cosh[x]^2*Sinh[x]^2*(a + b*Tanh[x]^2))/a^2)]) + (40*b^2*ArcSin[Sqrt[-(((a +
b)*Sinh[x]^2)/a)]]*Tanh[x]^4)/(a^2*Sqrt[-(((a + b)*Cosh[x]^2*Sinh[x]^2*(a + b*Tanh[x]^2))/a^2)]) + (32*b^3*Ar
cSin[Sqrt[-(((a + b)*Sinh[x]^2)/a)]]*Sech[x]^2*Tanh[x]^4)/(a^2*(a + b)*Sqrt[-(((a + b)*Cosh[x]^2*Sinh[x]^2*(a
+ b*Tanh[x]^2))/a^2)]) + (16*b^3*ArcSin[Sqrt[-(((a + b)*Sinh[x]^2)/a)]]*Tanh[x]^6)/(a^3*Sqrt[-(((a + b)*Cosh[x
]^2*Sinh[x]^2*(a + b*Tanh[x]^2))/a^2)]) + (16*b^3*(I*Tanh[x] - I*Tanh[x]^3)^2)/(a*(a + b)^2)))/(a^2*Sqrt[a + b
*Tanh[x]^2]*(1 + (b*Tanh[x]^2)/a)))

Maple [F]

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

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 5021 vs. \(2 (113) = 226\).

Time = 1.34 (sec) , antiderivative size = 10671, normalized size of antiderivative = 81.46 \[ \int \frac {\coth ^2(x)}{\left (a+b \tanh ^2(x)\right )^{5/2}} \, dx=\text {Too large to display} \]

[In]

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

[Out]

Too large to include

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

integrate(coth(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. 898 vs. \(2 (113) = 226\).

Time = 0.74 (sec) , antiderivative size = 898, normalized size of antiderivative = 6.85 \[ \int \frac {\coth ^2(x)}{\left (a+b \tanh ^2(x)\right )^{5/2}} \, dx=\text {Too large to display} \]

[In]

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

[Out]

-1/3*((((9*a^13*b^4 + 50*a^12*b^5 + 115*a^11*b^6 + 140*a^10*b^7 + 95*a^9*b^8 + 34*a^8*b^9 + 5*a^7*b^10)*e^(2*x
)/(a^16*b^2 + 6*a^15*b^3 + 15*a^14*b^4 + 20*a^13*b^5 + 15*a^12*b^6 + 6*a^11*b^7 + a^10*b^8) + 3*(3*a^13*b^4 +
6*a^12*b^5 - 11*a^11*b^6 - 44*a^10*b^7 - 51*a^9*b^8 - 26*a^8*b^9 - 5*a^7*b^10)/(a^16*b^2 + 6*a^15*b^3 + 15*a^1
4*b^4 + 20*a^13*b^5 + 15*a^12*b^6 + 6*a^11*b^7 + a^10*b^8))*e^(2*x) - 3*(3*a^13*b^4 + 6*a^12*b^5 - 11*a^11*b^6
- 44*a^10*b^7 - 51*a^9*b^8 - 26*a^8*b^9 - 5*a^7*b^10)/(a^16*b^2 + 6*a^15*b^3 + 15*a^14*b^4 + 20*a^13*b^5 + 15
*a^12*b^6 + 6*a^11*b^7 + a^10*b^8))*e^(2*x) - (9*a^13*b^4 + 50*a^12*b^5 + 115*a^11*b^6 + 140*a^10*b^7 + 95*a^9
*b^8 + 34*a^8*b^9 + 5*a^7*b^10)/(a^16*b^2 + 6*a^15*b^3 + 15*a^14*b^4 + 20*a^13*b^5 + 15*a^12*b^6 + 6*a^11*b^7
+ a^10*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))
+ 4*(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))/(((s
qrt(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)*a^2)

Mupad [F(-1)]

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

[In]

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

[Out]

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