Integrand size = 17, antiderivative size = 85 \[ \int \frac {\tanh ^2(x)}{\left (a+b \coth ^2(x)\right )^{3/2}} \, dx=\frac {\text {arctanh}\left (\frac {\sqrt {a+b} \coth (x)}{\sqrt {a+b \coth ^2(x)}}\right )}{(a+b)^{3/2}}+\frac {b \tanh (x)}{a (a+b) \sqrt {a+b \coth ^2(x)}}-\frac {(a+2 b) \sqrt {a+b \coth ^2(x)} \tanh (x)}{a^2 (a+b)} \]
arctanh(coth(x)*(a+b)^(1/2)/(a+b*coth(x)^2)^(1/2))/(a+b)^(3/2)+b*tanh(x)/a /(a+b)/(a+b*coth(x)^2)^(1/2)-(a+2*b)*(a+b*coth(x)^2)^(1/2)*tanh(x)/a^2/(a+ b)
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 7.93 (sec) , antiderivative size = 260, normalized size of antiderivative = 3.06 \[ \int \frac {\tanh ^2(x)}{\left (a+b \coth ^2(x)\right )^{3/2}} \, dx=\frac {\sinh ^2(x) \tanh (x) \left (\frac {8 (a+b) \cosh ^2(x) \left (2 a^2+5 a b \coth ^2(x)+3 b^2 \coth ^4(x)\right ) \operatorname {Hypergeometric2F1}\left (2,2,\frac {7}{2},\frac {(a+b) \cosh ^2(x)}{a}\right )}{15 a^3}-\frac {8 (a+b) \left (-i a \coth (x)-i b \coth ^3(x)\right )^2 \, _3F_2\left (2,2,2;1,\frac {7}{2};\frac {(a+b) \cosh ^2(x)}{a}\right ) \sinh ^2(x)}{15 a^3}-\frac {\left (3 a^2+12 a b \coth ^2(x)+8 b^2 \coth ^4(x)\right ) \left (\arcsin \left (\sqrt {\frac {(a+b) \cosh ^2(x)}{a}}\right ) \left (-a-b \coth ^2(x)\right )-a \text {csch}^2(x) \sqrt {-\frac {(a+b) \cosh ^2(x) \left (a+b \coth ^2(x)\right ) \sinh ^2(x)}{a^2}}\right ) \tanh ^2(x)}{a^2 (a+b) \sqrt {-\frac {(a+b) \cosh ^2(x) \left (a+b \coth ^2(x)\right ) \sinh ^2(x)}{a^2}}}\right )}{a \sqrt {a+b \coth ^2(x)}} \]
(Sinh[x]^2*Tanh[x]*((8*(a + b)*Cosh[x]^2*(2*a^2 + 5*a*b*Coth[x]^2 + 3*b^2* Coth[x]^4)*Hypergeometric2F1[2, 2, 7/2, ((a + b)*Cosh[x]^2)/a])/(15*a^3) - (8*(a + b)*((-I)*a*Coth[x] - I*b*Coth[x]^3)^2*HypergeometricPFQ[{2, 2, 2} , {1, 7/2}, ((a + b)*Cosh[x]^2)/a]*Sinh[x]^2)/(15*a^3) - ((3*a^2 + 12*a*b* Coth[x]^2 + 8*b^2*Coth[x]^4)*(ArcSin[Sqrt[((a + b)*Cosh[x]^2)/a]]*(-a - b* Coth[x]^2) - a*Csch[x]^2*Sqrt[-(((a + b)*Cosh[x]^2*(a + b*Coth[x]^2)*Sinh[ x]^2)/a^2)])*Tanh[x]^2)/(a^2*(a + b)*Sqrt[-(((a + b)*Cosh[x]^2*(a + b*Coth [x]^2)*Sinh[x]^2)/a^2)])))/(a*Sqrt[a + b*Coth[x]^2])
Time = 0.37 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.07, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.647, Rules used = {3042, 25, 4153, 25, 374, 25, 445, 25, 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.
| \(\displaystyle \int \frac {\tanh ^2(x)}{\left (a+b \coth ^2(x)\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int -\frac {1}{\tan \left (\frac {\pi }{2}+i x\right )^2 \left (a-b \tan \left (\frac {\pi }{2}+i x\right )^2\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\int \frac {1}{\tan \left (i x+\frac {\pi }{2}\right )^2 \left (a-b \tan \left (i x+\frac {\pi }{2}\right )^2\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 4153 |
| \(\displaystyle -\int -\frac {\tanh ^2(x)}{\left (1-\coth ^2(x)\right ) \left (b \coth ^2(x)+a\right )^{3/2}}d\coth (x)\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \int \frac {\tanh ^2(x)}{\left (1-\coth ^2(x)\right ) \left (a+b \coth ^2(x)\right )^{3/2}}d\coth (x)\) |
| \(\Big \downarrow \) 374 |
| \(\displaystyle \frac {b \tanh (x)}{a (a+b) \sqrt {a+b \coth ^2(x)}}-\frac {\int -\frac {\left (-2 b \coth ^2(x)+a+2 b\right ) \tanh ^2(x)}{\left (1-\coth ^2(x)\right ) \sqrt {b \coth ^2(x)+a}}d\coth (x)}{a (a+b)}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {\left (-2 b \coth ^2(x)+a+2 b\right ) \tanh ^2(x)}{\left (1-\coth ^2(x)\right ) \sqrt {b \coth ^2(x)+a}}d\coth (x)}{a (a+b)}+\frac {b \tanh (x)}{a (a+b) \sqrt {a+b \coth ^2(x)}}\) |
| \(\Big \downarrow \) 445 |
| \(\displaystyle \frac {-\frac {\int -\frac {a^2}{\left (1-\coth ^2(x)\right ) \sqrt {b \coth ^2(x)+a}}d\coth (x)}{a}-\frac {(a+2 b) \tanh (x) \sqrt {a+b \coth ^2(x)}}{a}}{a (a+b)}+\frac {b \tanh (x)}{a (a+b) \sqrt {a+b \coth ^2(x)}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\int \frac {a^2}{\left (1-\coth ^2(x)\right ) \sqrt {b \coth ^2(x)+a}}d\coth (x)}{a}-\frac {(a+2 b) \tanh (x) \sqrt {a+b \coth ^2(x)}}{a}}{a (a+b)}+\frac {b \tanh (x)}{a (a+b) \sqrt {a+b \coth ^2(x)}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {a \int \frac {1}{\left (1-\coth ^2(x)\right ) \sqrt {b \coth ^2(x)+a}}d\coth (x)-\frac {(a+2 b) \tanh (x) \sqrt {a+b \coth ^2(x)}}{a}}{a (a+b)}+\frac {b \tanh (x)}{a (a+b) \sqrt {a+b \coth ^2(x)}}\) |
| \(\Big \downarrow \) 291 |
| \(\displaystyle \frac {a \int \frac {1}{1-\frac {(a+b) \coth ^2(x)}{b \coth ^2(x)+a}}d\frac {\coth (x)}{\sqrt {b \coth ^2(x)+a}}-\frac {(a+2 b) \tanh (x) \sqrt {a+b \coth ^2(x)}}{a}}{a (a+b)}+\frac {b \tanh (x)}{a (a+b) \sqrt {a+b \coth ^2(x)}}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\frac {a \text {arctanh}\left (\frac {\sqrt {a+b} \coth (x)}{\sqrt {a+b \coth ^2(x)}}\right )}{\sqrt {a+b}}-\frac {(a+2 b) \tanh (x) \sqrt {a+b \coth ^2(x)}}{a}}{a (a+b)}+\frac {b \tanh (x)}{a (a+b) \sqrt {a+b \coth ^2(x)}}\) |
(b*Tanh[x])/(a*(a + b)*Sqrt[a + b*Coth[x]^2]) + ((a*ArcTanh[(Sqrt[a + b]*C oth[x])/Sqrt[a + b*Coth[x]^2]])/Sqrt[a + b] - ((a + 2*b)*Sqrt[a + b*Coth[x ]^2]*Tanh[x])/a)/(a*(a + b))
3.1.42.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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])
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]
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_ ), x_Symbol] :> Simp[(-b)*(e*x)^(m + 1)*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q + 1)/(a*e*2*(b*c - a*d)*(p + 1))), x] + Simp[1/(a*2*(b*c - a*d)*(p + 1)) Int[(e*x)^m*(a + b*x^2)^(p + 1)*(c + d*x^2)^q*Simp[b*c*(m + 1) + 2*(b*c - a*d)*(p + 1) + d*b*(m + 2*(p + q + 2) + 1)*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, m, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && IntBinomialQ[a, b, c, d, e, m, 2, p, q, x]
Int[((g_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_ .)*((e_) + (f_.)*(x_)^2), x_Symbol] :> Simp[e*(g*x)^(m + 1)*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q + 1)/(a*c*g*(m + 1))), x] + Simp[1/(a*c*g^2*(m + 1)) Int[(g*x)^(m + 2)*(a + b*x^2)^p*(c + d*x^2)^q*Simp[a*f*c*(m + 1) - e*(b*c + a*d)*(m + 2 + 1) - e*2*(b*c*p + a*d*q) - b*e*d*(m + 2*(p + q + 2) + 1)*x^ 2, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p, q}, x] && LtQ[m, -1]
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]))
\[\int \frac {\tanh \left (x \right )^{2}}{\left (a +b \coth \left (x \right )^{2}\right )^{\frac {3}{2}}}d x\]
Leaf count of result is larger than twice the leaf count of optimal. 1686 vs. \(2 (75) = 150\).
Time = 0.51 (sec) , antiderivative size = 3931, normalized size of antiderivative = 46.25 \[ \int \frac {\tanh ^2(x)}{\left (a+b \coth ^2(x)\right )^{3/2}} \, dx=\text {Too large to display} \]
[1/4*(((a^3 + a^2*b)*cosh(x)^6 + 6*(a^3 + a^2*b)*cosh(x)*sinh(x)^5 + (a^3 + a^2*b)*sinh(x)^6 - (a^3 - 3*a^2*b)*cosh(x)^4 - (a^3 - 3*a^2*b - 15*(a^3 + a^2*b)*cosh(x)^2)*sinh(x)^4 + 4*(5*(a^3 + a^2*b)*cosh(x)^3 - (a^3 - 3*a^ 2*b)*cosh(x))*sinh(x)^3 + a^3 + a^2*b - (a^3 - 3*a^2*b)*cosh(x)^2 + (15*(a ^3 + a^2*b)*cosh(x)^4 - a^3 + 3*a^2*b - 6*(a^3 - 3*a^2*b)*cosh(x)^2)*sinh( x)^2 + 2*(3*(a^3 + a^2*b)*cosh(x)^5 - 2*(a^3 - 3*a^2*b)*cosh(x)^3 - (a^3 - 3*a^2*b)*cosh(x))*sinh(x))*sqrt(a + b)*log(((a*b^2 + b^3)*cosh(x)^8 + 8*( a*b^2 + b^3)*cosh(x)*sinh(x)^7 + (a*b^2 + b^3)*sinh(x)^8 + 2*(a*b^2 + 2*b^ 3)*cosh(x)^6 + 2*(a*b^2 + 2*b^3 + 14*(a*b^2 + b^3)*cosh(x)^2)*sinh(x)^6 + 4*(14*(a*b^2 + b^3)*cosh(x)^3 + 3*(a*b^2 + 2*b^3)*cosh(x))*sinh(x)^5 + (a^ 3 - a^2*b + 4*a*b^2 + 6*b^3)*cosh(x)^4 + (70*(a*b^2 + b^3)*cosh(x)^4 + a^3 - a^2*b + 4*a*b^2 + 6*b^3 + 30*(a*b^2 + 2*b^3)*cosh(x)^2)*sinh(x)^4 + 4*( 14*(a*b^2 + b^3)*cosh(x)^5 + 10*(a*b^2 + 2*b^3)*cosh(x)^3 + (a^3 - a^2*b + 4*a*b^2 + 6*b^3)*cosh(x))*sinh(x)^3 + a^3 + 3*a^2*b + 3*a*b^2 + b^3 - 2*( a^3 - 3*a*b^2 - 2*b^3)*cosh(x)^2 + 2*(14*(a*b^2 + b^3)*cosh(x)^6 + 15*(a*b ^2 + 2*b^3)*cosh(x)^4 - a^3 + 3*a*b^2 + 2*b^3 + 3*(a^3 - a^2*b + 4*a*b^2 + 6*b^3)*cosh(x)^2)*sinh(x)^2 + sqrt(2)*(b^2*cosh(x)^6 + 6*b^2*cosh(x)*sinh (x)^5 + b^2*sinh(x)^6 + 3*b^2*cosh(x)^4 + 3*(5*b^2*cosh(x)^2 + b^2)*sinh(x )^4 + 4*(5*b^2*cosh(x)^3 + 3*b^2*cosh(x))*sinh(x)^3 - (a^2 - 2*a*b - 3*b^2 )*cosh(x)^2 + (15*b^2*cosh(x)^4 + 18*b^2*cosh(x)^2 - a^2 + 2*a*b + 3*b^...
\[ \int \frac {\tanh ^2(x)}{\left (a+b \coth ^2(x)\right )^{3/2}} \, dx=\int \frac {\tanh ^{2}{\left (x \right )}}{\left (a + b \coth ^{2}{\left (x \right )}\right )^{\frac {3}{2}}}\, dx \]
\[ \int \frac {\tanh ^2(x)}{\left (a+b \coth ^2(x)\right )^{3/2}} \, dx=\int { \frac {\tanh \left (x\right )^{2}}{{\left (b \coth \left (x\right )^{2} + a\right )}^{\frac {3}{2}}} \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 540 vs. \(2 (75) = 150\).
Time = 0.60 (sec) , antiderivative size = 540, normalized size of antiderivative = 6.35 \[ \int \frac {\tanh ^2(x)}{\left (a+b \coth ^2(x)\right )^{3/2}} \, dx=-\frac {\frac {{\left (a^{2} b^{3} + a b^{4}\right )} e^{\left (2 \, x\right )}}{a^{5} b \mathrm {sgn}\left (e^{\left (2 \, x\right )} - 1\right ) + 2 \, a^{4} b^{2} \mathrm {sgn}\left (e^{\left (2 \, x\right )} - 1\right ) + a^{3} b^{3} \mathrm {sgn}\left (e^{\left (2 \, x\right )} - 1\right )} + \frac {a^{2} b^{3} + a b^{4}}{a^{5} b \mathrm {sgn}\left (e^{\left (2 \, x\right )} - 1\right ) + 2 \, a^{4} b^{2} \mathrm {sgn}\left (e^{\left (2 \, x\right )} - 1\right ) + a^{3} b^{3} \mathrm {sgn}\left (e^{\left (2 \, x\right )} - 1\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}} - \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 )} \sqrt {a + b} - a + b \right |}\right )}{2 \, {\left (a + b\right )}^{\frac {3}{2}} \mathrm {sgn}\left (e^{\left (2 \, x\right )} - 1\right )} + \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 )} \sqrt {a + b} - a - b \right |}\right )}{2 \, {\left (a + b\right )}^{\frac {3}{2}} \mathrm {sgn}\left (e^{\left (2 \, x\right )} - 1\right )} - \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 + b\right )}^{\frac {3}{2}} \mathrm {sgn}\left (e^{\left (2 \, x\right )} - 1\right )} - \frac {4 \, {\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 )}}{{\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 )}^{2} + 2 \, {\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 )} \sqrt {a + b} - 3 \, a + b\right )} a \mathrm {sgn}\left (e^{\left (2 \, x\right )} - 1\right )} \]
-((a^2*b^3 + a*b^4)*e^(2*x)/(a^5*b*sgn(e^(2*x) - 1) + 2*a^4*b^2*sgn(e^(2*x ) - 1) + a^3*b^3*sgn(e^(2*x) - 1)) + (a^2*b^3 + a*b^4)/(a^5*b*sgn(e^(2*x) - 1) + 2*a^4*b^2*sgn(e^(2*x) - 1) + a^3*b^3*sgn(e^(2*x) - 1)))/sqrt(a*e^(4 *x) + b*e^(4*x) - 2*a*e^(2*x) + 2*b*e^(2*x) + 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 + b))/((a + b)^(3/2)*sgn(e^(2*x) - 1)) + 1/2*log(ab s((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 - b))/((a + b)^(3/2)*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 + b)^(3/2)*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*sgn(e^(2*x) - 1))
Timed out. \[ \int \frac {\tanh ^2(x)}{\left (a+b \coth ^2(x)\right )^{3/2}} \, dx=\int \frac {{\mathrm {tanh}\left (x\right )}^2}{{\left (b\,{\mathrm {coth}\left (x\right )}^2+a\right )}^{3/2}} \,d x \]