\(\int \frac {\tanh ^2(x)}{(a+b \coth ^2(x))^{5/2}} \, dx\) [47]

Optimal result

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

[Out]

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

Rubi [A] (verified)

Time = 0.17 (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} \[ \int \frac {\tanh ^2(x)}{\left (a+b \coth ^2(x)\right )^{5/2}} \, dx=-\frac {(3 a+2 b) (a+4 b) \tanh (x) \sqrt {a+b \coth ^2(x)}}{3 a^3 (a+b)^2}+\frac {b (7 a+4 b) \tanh (x)}{3 a^2 (a+b)^2 \sqrt {a+b \coth ^2(x)}}+\frac {\text {arctanh}\left (\frac {\sqrt {a+b} \coth (x)}{\sqrt {a+b \coth ^2(x)}}\right )}{(a+b)^{5/2}}+\frac {b \tanh (x)}{3 a (a+b) \left (a+b \coth ^2(x)\right )^{3/2}} \]

[In]

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

[Out]

ArcTanh[(Sqrt[a + b]*Coth[x])/Sqrt[a + b*Coth[x]^2]]/(a + b)^(5/2) + (b*Tanh[x])/(3*a*(a + b)*(a + b*Coth[x]^2
)^(3/2)) + (b*(7*a + 4*b)*Tanh[x])/(3*a^2*(a + b)^2*Sqrt[a + b*Coth[x]^2]) - ((3*a + 2*b)*(a + 4*b)*Sqrt[a + b
*Coth[x]^2]*Tanh[x])/(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,\coth (x)\right ) \\ & = \frac {b \tanh (x)}{3 a (a+b) \left (a+b \coth ^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,\coth (x)\right )}{3 a (a+b)} \\ & = \frac {b \tanh (x)}{3 a (a+b) \left (a+b \coth ^2(x)\right )^{3/2}}+\frac {b (7 a+4 b) \tanh (x)}{3 a^2 (a+b)^2 \sqrt {a+b \coth ^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,\coth (x)\right )}{3 a^2 (a+b)^2} \\ & = \frac {b \tanh (x)}{3 a (a+b) \left (a+b \coth ^2(x)\right )^{3/2}}+\frac {b (7 a+4 b) \tanh (x)}{3 a^2 (a+b)^2 \sqrt {a+b \coth ^2(x)}}-\frac {(3 a+2 b) (a+4 b) \sqrt {a+b \coth ^2(x)} \tanh (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,\coth (x)\right )}{3 a^3 (a+b)^2} \\ & = \frac {b \tanh (x)}{3 a (a+b) \left (a+b \coth ^2(x)\right )^{3/2}}+\frac {b (7 a+4 b) \tanh (x)}{3 a^2 (a+b)^2 \sqrt {a+b \coth ^2(x)}}-\frac {(3 a+2 b) (a+4 b) \sqrt {a+b \coth ^2(x)} \tanh (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,\coth (x)\right )}{(a+b)^2} \\ & = \frac {b \tanh (x)}{3 a (a+b) \left (a+b \coth ^2(x)\right )^{3/2}}+\frac {b (7 a+4 b) \tanh (x)}{3 a^2 (a+b)^2 \sqrt {a+b \coth ^2(x)}}-\frac {(3 a+2 b) (a+4 b) \sqrt {a+b \coth ^2(x)} \tanh (x)}{3 a^3 (a+b)^2}+\frac {\text {Subst}\left (\int \frac {1}{1-(a+b) x^2} \, dx,x,\frac {\coth (x)}{\sqrt {a+b \coth ^2(x)}}\right )}{(a+b)^2} \\ & = \frac {\text {arctanh}\left (\frac {\sqrt {a+b} \coth (x)}{\sqrt {a+b \coth ^2(x)}}\right )}{(a+b)^{5/2}}+\frac {b \tanh (x)}{3 a (a+b) \left (a+b \coth ^2(x)\right )^{3/2}}+\frac {b (7 a+4 b) \tanh (x)}{3 a^2 (a+b)^2 \sqrt {a+b \coth ^2(x)}}-\frac {(3 a+2 b) (a+4 b) \sqrt {a+b \coth ^2(x)} \tanh (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 = 8.54 (sec) , antiderivative size = 1350, normalized size of antiderivative = 10.31 \[ \int \frac {\tanh ^2(x)}{\left (a+b \coth ^2(x)\right )^{5/2}} \, dx=\frac {\sinh ^2(x) \left (\frac {16 b^3 \left (-i \coth (x)+i \coth ^3(x)\right )^2}{a (a+b)^2}+\frac {40 b \text {csch}^2(x)}{a+b}+\frac {160 b^2 \coth ^2(x) \text {csch}^2(x)}{3 a (a+b)}+\frac {64 b^3 \coth ^4(x) \text {csch}^2(x)}{3 a^2 (a+b)}-\frac {40 b^2 \text {csch}^4(x)}{(a+b)^2}+\frac {92 (a+b) \cosh ^2(x) \operatorname {Hypergeometric2F1}\left (2,2,\frac {9}{2},\frac {(a+b) \cosh ^2(x)}{a}\right )}{105 a}+\frac {124 b (a+b) \cosh ^2(x) \coth ^2(x) \operatorname {Hypergeometric2F1}\left (2,2,\frac {9}{2},\frac {(a+b) \cosh ^2(x)}{a}\right )}{35 a^2}+\frac {152 b^2 (a+b) \cosh ^2(x) \coth ^4(x) \operatorname {Hypergeometric2F1}\left (2,2,\frac {9}{2},\frac {(a+b) \cosh ^2(x)}{a}\right )}{35 a^3}+\frac {176 b^3 (a+b) \cosh ^2(x) \coth ^6(x) \operatorname {Hypergeometric2F1}\left (2,2,\frac {9}{2},\frac {(a+b) \cosh ^2(x)}{a}\right )}{105 a^4}+\frac {24 (a+b) \cosh ^2(x) \, _3F_2\left (2,2,2;1,\frac {9}{2};\frac {(a+b) \cosh ^2(x)}{a}\right )}{35 a}+\frac {16 b (a+b) \cosh ^2(x) \coth ^2(x) \, _3F_2\left (2,2,2;1,\frac {9}{2};\frac {(a+b) \cosh ^2(x)}{a}\right )}{7 a^2}+\frac {88 b^2 (a+b) \cosh ^2(x) \coth ^4(x) \, _3F_2\left (2,2,2;1,\frac {9}{2};\frac {(a+b) \cosh ^2(x)}{a}\right )}{35 a^3}+\frac {32 b^3 (a+b) \cosh ^2(x) \coth ^6(x) \, _3F_2\left (2,2,2;1,\frac {9}{2};\frac {(a+b) \cosh ^2(x)}{a}\right )}{35 a^4}+\frac {16 (a+b) \cosh ^2(x) \, _4F_3\left (2,2,2,2;1,1,\frac {9}{2};\frac {(a+b) \cosh ^2(x)}{a}\right )}{105 a}+\frac {16 b (a+b) \cosh ^2(x) \coth ^2(x) \, _4F_3\left (2,2,2,2;1,1,\frac {9}{2};\frac {(a+b) \cosh ^2(x)}{a}\right )}{35 a^2}+\frac {16 b^2 (a+b) \cosh ^2(x) \coth ^4(x) \, _4F_3\left (2,2,2,2;1,1,\frac {9}{2};\frac {(a+b) \cosh ^2(x)}{a}\right )}{35 a^3}+\frac {16 b^3 (a+b) \cosh ^2(x) \coth ^6(x) \, _4F_3\left (2,2,2,2;1,1,\frac {9}{2};\frac {(a+b) \cosh ^2(x)}{a}\right )}{105 a^4}+\frac {20 a \text {sech}^2(x)}{3 (a+b)}-\frac {30 a b \text {csch}^2(x) \text {sech}^2(x)}{(a+b)^2}-\frac {5 a^2 \text {sech}^4(x)}{(a+b)^2}+\frac {5 \arcsin \left (\sqrt {\frac {(a+b) \cosh ^2(x)}{a}}\right )}{\left (\frac {(a+b) \cosh ^2(x)}{a}\right )^{5/2} \sqrt {-\frac {\left (a+b \coth ^2(x)\right ) \sinh ^2(x)}{a}}}+\frac {30 b \arcsin \left (\sqrt {\frac {(a+b) \cosh ^2(x)}{a}}\right ) \coth ^2(x)}{a \left (\frac {(a+b) \cosh ^2(x)}{a}\right )^{5/2} \sqrt {-\frac {\left (a+b \coth ^2(x)\right ) \sinh ^2(x)}{a}}}+\frac {40 b^2 \arcsin \left (\sqrt {\frac {(a+b) \cosh ^2(x)}{a}}\right ) \coth ^4(x)}{a^2 \left (\frac {(a+b) \cosh ^2(x)}{a}\right )^{5/2} \sqrt {-\frac {\left (a+b \coth ^2(x)\right ) \sinh ^2(x)}{a}}}+\frac {16 b^3 \arcsin \left (\sqrt {\frac {(a+b) \cosh ^2(x)}{a}}\right ) \coth ^6(x)}{a^3 \left (\frac {(a+b) \cosh ^2(x)}{a}\right )^{5/2} \sqrt {-\frac {\left (a+b \coth ^2(x)\right ) \sinh ^2(x)}{a}}}+\frac {5 \arcsin \left (\sqrt {\frac {(a+b) \cosh ^2(x)}{a}}\right )}{\sqrt {-\frac {(a+b) \cosh ^2(x) \left (a+b \coth ^2(x)\right ) \sinh ^2(x)}{a^2}}}+\frac {30 b \arcsin \left (\sqrt {\frac {(a+b) \cosh ^2(x)}{a}}\right ) \coth ^2(x)}{a \sqrt {-\frac {(a+b) \cosh ^2(x) \left (a+b \coth ^2(x)\right ) \sinh ^2(x)}{a^2}}}+\frac {40 b^2 \arcsin \left (\sqrt {\frac {(a+b) \cosh ^2(x)}{a}}\right ) \coth ^4(x)}{a^2 \sqrt {-\frac {(a+b) \cosh ^2(x) \left (a+b \coth ^2(x)\right ) \sinh ^2(x)}{a^2}}}+\frac {16 b^3 \arcsin \left (\sqrt {\frac {(a+b) \cosh ^2(x)}{a}}\right ) \coth ^6(x)}{a^3 \sqrt {-\frac {(a+b) \cosh ^2(x) \left (a+b \coth ^2(x)\right ) \sinh ^2(x)}{a^2}}}-\frac {60 b \arcsin \left (\sqrt {\frac {(a+b) \cosh ^2(x)}{a}}\right ) \text {csch}^2(x)}{(a+b) \sqrt {-\frac {(a+b) \cosh ^2(x) \left (a+b \coth ^2(x)\right ) \sinh ^2(x)}{a^2}}}-\frac {80 b^2 \arcsin \left (\sqrt {\frac {(a+b) \cosh ^2(x)}{a}}\right ) \coth ^2(x) \text {csch}^2(x)}{a (a+b) \sqrt {-\frac {(a+b) \cosh ^2(x) \left (a+b \coth ^2(x)\right ) \sinh ^2(x)}{a^2}}}-\frac {32 b^3 \arcsin \left (\sqrt {\frac {(a+b) \cosh ^2(x)}{a}}\right ) \coth ^4(x) \text {csch}^2(x)}{a^2 (a+b) \sqrt {-\frac {(a+b) \cosh ^2(x) \left (a+b \coth ^2(x)\right ) \sinh ^2(x)}{a^2}}}-\frac {10 a \arcsin \left (\sqrt {\frac {(a+b) \cosh ^2(x)}{a}}\right ) \text {sech}^2(x)}{(a+b) \sqrt {-\frac {(a+b) \cosh ^2(x) \left (a+b \coth ^2(x)\right ) \sinh ^2(x)}{a^2}}}\right ) \tanh (x)}{a^2 \sqrt {a+b \coth ^2(x)} \left (1+\frac {b \coth ^2(x)}{a}\right )} \]

[In]

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

[Out]

(Sinh[x]^2*((16*b^3*((-I)*Coth[x] + I*Coth[x]^3)^2)/(a*(a + b)^2) + (40*b*Csch[x]^2)/(a + b) + (160*b^2*Coth[x
]^2*Csch[x]^2)/(3*a*(a + b)) + (64*b^3*Coth[x]^4*Csch[x]^2)/(3*a^2*(a + b)) - (40*b^2*Csch[x]^4)/(a + b)^2 + (
92*(a + b)*Cosh[x]^2*Hypergeometric2F1[2, 2, 9/2, ((a + b)*Cosh[x]^2)/a])/(105*a) + (124*b*(a + b)*Cosh[x]^2*C
oth[x]^2*Hypergeometric2F1[2, 2, 9/2, ((a + b)*Cosh[x]^2)/a])/(35*a^2) + (152*b^2*(a + b)*Cosh[x]^2*Coth[x]^4*
Hypergeometric2F1[2, 2, 9/2, ((a + b)*Cosh[x]^2)/a])/(35*a^3) + (176*b^3*(a + b)*Cosh[x]^2*Coth[x]^6*Hypergeom
etric2F1[2, 2, 9/2, ((a + b)*Cosh[x]^2)/a])/(105*a^4) + (24*(a + b)*Cosh[x]^2*HypergeometricPFQ[{2, 2, 2}, {1,
9/2}, ((a + b)*Cosh[x]^2)/a])/(35*a) + (16*b*(a + b)*Cosh[x]^2*Coth[x]^2*HypergeometricPFQ[{2, 2, 2}, {1, 9/2
}, ((a + b)*Cosh[x]^2)/a])/(7*a^2) + (88*b^2*(a + b)*Cosh[x]^2*Coth[x]^4*HypergeometricPFQ[{2, 2, 2}, {1, 9/2}
, ((a + b)*Cosh[x]^2)/a])/(35*a^3) + (32*b^3*(a + b)*Cosh[x]^2*Coth[x]^6*HypergeometricPFQ[{2, 2, 2}, {1, 9/2}
, ((a + b)*Cosh[x]^2)/a])/(35*a^4) + (16*(a + b)*Cosh[x]^2*HypergeometricPFQ[{2, 2, 2, 2}, {1, 1, 9/2}, ((a +
b)*Cosh[x]^2)/a])/(105*a) + (16*b*(a + b)*Cosh[x]^2*Coth[x]^2*HypergeometricPFQ[{2, 2, 2, 2}, {1, 1, 9/2}, ((a
+ b)*Cosh[x]^2)/a])/(35*a^2) + (16*b^2*(a + b)*Cosh[x]^2*Coth[x]^4*HypergeometricPFQ[{2, 2, 2, 2}, {1, 1, 9/2
}, ((a + b)*Cosh[x]^2)/a])/(35*a^3) + (16*b^3*(a + b)*Cosh[x]^2*Coth[x]^6*HypergeometricPFQ[{2, 2, 2, 2}, {1,
1, 9/2}, ((a + b)*Cosh[x]^2)/a])/(105*a^4) + (20*a*Sech[x]^2)/(3*(a + b)) - (30*a*b*Csch[x]^2*Sech[x]^2)/(a +
b)^2 - (5*a^2*Sech[x]^4)/(a + b)^2 + (5*ArcSin[Sqrt[((a + b)*Cosh[x]^2)/a]])/((((a + b)*Cosh[x]^2)/a)^(5/2)*Sq
rt[-(((a + b*Coth[x]^2)*Sinh[x]^2)/a)]) + (30*b*ArcSin[Sqrt[((a + b)*Cosh[x]^2)/a]]*Coth[x]^2)/(a*(((a + b)*Co
sh[x]^2)/a)^(5/2)*Sqrt[-(((a + b*Coth[x]^2)*Sinh[x]^2)/a)]) + (40*b^2*ArcSin[Sqrt[((a + b)*Cosh[x]^2)/a]]*Coth
[x]^4)/(a^2*(((a + b)*Cosh[x]^2)/a)^(5/2)*Sqrt[-(((a + b*Coth[x]^2)*Sinh[x]^2)/a)]) + (16*b^3*ArcSin[Sqrt[((a
+ b)*Cosh[x]^2)/a]]*Coth[x]^6)/(a^3*(((a + b)*Cosh[x]^2)/a)^(5/2)*Sqrt[-(((a + b*Coth[x]^2)*Sinh[x]^2)/a)]) +
(5*ArcSin[Sqrt[((a + b)*Cosh[x]^2)/a]])/Sqrt[-(((a + b)*Cosh[x]^2*(a + b*Coth[x]^2)*Sinh[x]^2)/a^2)] + (30*b*A
rcSin[Sqrt[((a + b)*Cosh[x]^2)/a]]*Coth[x]^2)/(a*Sqrt[-(((a + b)*Cosh[x]^2*(a + b*Coth[x]^2)*Sinh[x]^2)/a^2)])
+ (40*b^2*ArcSin[Sqrt[((a + b)*Cosh[x]^2)/a]]*Coth[x]^4)/(a^2*Sqrt[-(((a + b)*Cosh[x]^2*(a + b*Coth[x]^2)*Sin
h[x]^2)/a^2)]) + (16*b^3*ArcSin[Sqrt[((a + b)*Cosh[x]^2)/a]]*Coth[x]^6)/(a^3*Sqrt[-(((a + b)*Cosh[x]^2*(a + b*
Coth[x]^2)*Sinh[x]^2)/a^2)]) - (60*b*ArcSin[Sqrt[((a + b)*Cosh[x]^2)/a]]*Csch[x]^2)/((a + b)*Sqrt[-(((a + b)*C
osh[x]^2*(a + b*Coth[x]^2)*Sinh[x]^2)/a^2)]) - (80*b^2*ArcSin[Sqrt[((a + b)*Cosh[x]^2)/a]]*Coth[x]^2*Csch[x]^2
)/(a*(a + b)*Sqrt[-(((a + b)*Cosh[x]^2*(a + b*Coth[x]^2)*Sinh[x]^2)/a^2)]) - (32*b^3*ArcSin[Sqrt[((a + b)*Cosh
[x]^2)/a]]*Coth[x]^4*Csch[x]^2)/(a^2*(a + b)*Sqrt[-(((a + b)*Cosh[x]^2*(a + b*Coth[x]^2)*Sinh[x]^2)/a^2)]) - (
10*a*ArcSin[Sqrt[((a + b)*Cosh[x]^2)/a]]*Sech[x]^2)/((a + b)*Sqrt[-(((a + b)*Cosh[x]^2*(a + b*Coth[x]^2)*Sinh[
x]^2)/a^2)]))*Tanh[x])/(a^2*Sqrt[a + b*Coth[x]^2]*(1 + (b*Coth[x]^2)/a))

Maple [F]

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

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 5085 vs. \(2 (113) = 226\).

Time = 1.30 (sec) , antiderivative size = 10729, normalized size of antiderivative = 81.90 \[ \int \frac {\tanh ^2(x)}{\left (a+b \coth ^2(x)\right )^{5/2}} \, dx=\text {Too large to display} \]

[In]

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

[Out]

Too large to include

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1133 vs. \(2 (113) = 226\).

Time = 0.77 (sec) , antiderivative size = 1133, normalized size of antiderivative = 8.65 \[ \int \frac {\tanh ^2(x)}{\left (a+b \coth ^2(x)\right )^{5/2}} \, dx=\text {Too large to display} \]

[In]

integrate(tanh(x)^2/(a+b*coth(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*sgn(e^(2*x) - 1) + 6*a^15*b^3*sgn(e^(2*x) - 1) + 15*a^14*b^4*sgn(e^(2*x) - 1) + 20*a^13*b^5*sgn(e^
(2*x) - 1) + 15*a^12*b^6*sgn(e^(2*x) - 1) + 6*a^11*b^7*sgn(e^(2*x) - 1) + a^10*b^8*sgn(e^(2*x) - 1)) - 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*sgn(e^(2*x)
- 1) + 6*a^15*b^3*sgn(e^(2*x) - 1) + 15*a^14*b^4*sgn(e^(2*x) - 1) + 20*a^13*b^5*sgn(e^(2*x) - 1) + 15*a^12*b^6
*sgn(e^(2*x) - 1) + 6*a^11*b^7*sgn(e^(2*x) - 1) + a^10*b^8*sgn(e^(2*x) - 1)))*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*sgn(e^(2*x) - 1) + 6*a^15*b
^3*sgn(e^(2*x) - 1) + 15*a^14*b^4*sgn(e^(2*x) - 1) + 20*a^13*b^5*sgn(e^(2*x) - 1) + 15*a^12*b^6*sgn(e^(2*x) -
1) + 6*a^11*b^7*sgn(e^(2*x) - 1) + a^10*b^8*sgn(e^(2*x) - 1)))*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*sgn(e^(2*x) - 1) + 6*a^15*b^3*sgn(e^(2*x)
- 1) + 15*a^14*b^4*sgn(e^(2*x) - 1) + 20*a^13*b^5*sgn(e^(2*x) - 1) + 15*a^12*b^6*sgn(e^(2*x) - 1) + 6*a^11*b^
7*sgn(e^(2*x) - 1) + a^10*b^8*sgn(e^(2*x) - 1)))/(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))*sqr
t(a + b) - a + b))/((a^2 + 2*a*b + b^2)*sqrt(a + b)*sgn(e^(2*x) - 1)) + 1/2*log(abs((sqrt(a + b)*e^(2*x) - sqr
t(a*e^(4*x) + b*e^(4*x) - 2*a*e^(2*x) + 2*b*e^(2*x) + a + b))*sqrt(a + b) - a - b))/((a^2 + 2*a*b + b^2)*sqrt(
a + b)*sgn(e^(2*x) - 1)) - 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)*sgn(e^(2*x) - 1)) - 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))/(((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))^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*sgn(e^(2*x) - 1))

Mupad [F(-1)]

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

[In]

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

[Out]

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