Integrand size = 15, antiderivative size = 108 \[ \int \frac {\coth (x)}{\left (a+b \tanh ^2(x)\right )^{5/2}} \, dx=-\frac {\text {arctanh}\left (\frac {\sqrt {a+b \tanh ^2(x)}}{\sqrt {a}}\right )}{a^{5/2}}+\frac {\text {arctanh}\left (\frac {\sqrt {a+b \tanh ^2(x)}}{\sqrt {a+b}}\right )}{(a+b)^{5/2}}+\frac {b}{3 a (a+b) \left (a+b \tanh ^2(x)\right )^{3/2}}+\frac {b (2 a+b)}{a^2 (a+b)^2 \sqrt {a+b \tanh ^2(x)}} \]
-arctanh((a+b*tanh(x)^2)^(1/2)/a^(1/2))/a^(5/2)+arctanh((a+b*tanh(x)^2)^(1 /2)/(a+b)^(1/2))/(a+b)^(5/2)+b*(2*a+b)/a^2/(a+b)^2/(a+b*tanh(x)^2)^(1/2)+1 /3*b/a/(a+b)/(a+b*tanh(x)^2)^(3/2)
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.10 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.68 \[ \int \frac {\coth (x)}{\left (a+b \tanh ^2(x)\right )^{5/2}} \, dx=\frac {-a \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},1,-\frac {1}{2},\frac {a+b \tanh ^2(x)}{a+b}\right )+(a+b) \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},1,-\frac {1}{2},1+\frac {b \tanh ^2(x)}{a}\right )}{3 a (a+b) \left (a+b \tanh ^2(x)\right )^{3/2}} \]
(-(a*Hypergeometric2F1[-3/2, 1, -1/2, (a + b*Tanh[x]^2)/(a + b)]) + (a + b )*Hypergeometric2F1[-3/2, 1, -1/2, 1 + (b*Tanh[x]^2)/a])/(3*a*(a + b)*(a + b*Tanh[x]^2)^(3/2))
Time = 0.39 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.31, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.800, Rules used = {3042, 26, 4153, 26, 354, 96, 25, 169, 27, 174, 73, 221}
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.
\(\displaystyle \int \frac {\coth (x)}{\left (a+b \tanh ^2(x)\right )^{5/2}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {i}{\tan (i x) \left (a-b \tan (i x)^2\right )^{5/2}}dx\) |
\(\Big \downarrow \) 26 |
\(\displaystyle i \int \frac {1}{\tan (i x) \left (a-b \tan (i x)^2\right )^{5/2}}dx\) |
\(\Big \downarrow \) 4153 |
\(\displaystyle i \int -\frac {i \coth (x)}{\left (1-\tanh ^2(x)\right ) \left (b \tanh ^2(x)+a\right )^{5/2}}d\tanh (x)\) |
\(\Big \downarrow \) 26 |
\(\displaystyle \int \frac {\coth (x)}{\left (1-\tanh ^2(x)\right ) \left (a+b \tanh ^2(x)\right )^{5/2}}d\tanh (x)\) |
\(\Big \downarrow \) 354 |
\(\displaystyle \frac {1}{2} \int \frac {\coth (x)}{\left (1-\tanh ^2(x)\right ) \left (b \tanh ^2(x)+a\right )^{5/2}}d\tanh ^2(x)\) |
\(\Big \downarrow \) 96 |
\(\displaystyle \frac {1}{2} \left (\frac {2 b}{3 a (a+b) \left (a+b \tanh ^2(x)\right )^{3/2}}-\frac {\int -\frac {\coth (x) \left (-b \tanh ^2(x)+a+b\right )}{\left (1-\tanh ^2(x)\right ) \left (b \tanh ^2(x)+a\right )^{3/2}}d\tanh ^2(x)}{a (a+b)}\right )\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {1}{2} \left (\frac {\int \frac {\coth (x) \left (-b \tanh ^2(x)+a+b\right )}{\left (1-\tanh ^2(x)\right ) \left (b \tanh ^2(x)+a\right )^{3/2}}d\tanh ^2(x)}{a (a+b)}+\frac {2 b}{3 a (a+b) \left (a+b \tanh ^2(x)\right )^{3/2}}\right )\) |
\(\Big \downarrow \) 169 |
\(\displaystyle \frac {1}{2} \left (\frac {\frac {2 \int \frac {\coth (x) \left ((a+b)^2-b (2 a+b) \tanh ^2(x)\right )}{2 \left (1-\tanh ^2(x)\right ) \sqrt {b \tanh ^2(x)+a}}d\tanh ^2(x)}{a (a+b)}+\frac {2 b (2 a+b)}{a (a+b) \sqrt {a+b \tanh ^2(x)}}}{a (a+b)}+\frac {2 b}{3 a (a+b) \left (a+b \tanh ^2(x)\right )^{3/2}}\right )\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{2} \left (\frac {\frac {\int \frac {\coth (x) \left ((a+b)^2-b (2 a+b) \tanh ^2(x)\right )}{\left (1-\tanh ^2(x)\right ) \sqrt {b \tanh ^2(x)+a}}d\tanh ^2(x)}{a (a+b)}+\frac {2 b (2 a+b)}{a (a+b) \sqrt {a+b \tanh ^2(x)}}}{a (a+b)}+\frac {2 b}{3 a (a+b) \left (a+b \tanh ^2(x)\right )^{3/2}}\right )\) |
\(\Big \downarrow \) 174 |
\(\displaystyle \frac {1}{2} \left (\frac {\frac {a^2 \int \frac {1}{\left (1-\tanh ^2(x)\right ) \sqrt {b \tanh ^2(x)+a}}d\tanh ^2(x)+(a+b)^2 \int \frac {\coth (x)}{\sqrt {b \tanh ^2(x)+a}}d\tanh ^2(x)}{a (a+b)}+\frac {2 b (2 a+b)}{a (a+b) \sqrt {a+b \tanh ^2(x)}}}{a (a+b)}+\frac {2 b}{3 a (a+b) \left (a+b \tanh ^2(x)\right )^{3/2}}\right )\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {1}{2} \left (\frac {\frac {\frac {2 a^2 \int \frac {1}{\frac {a+b}{b}-\frac {\tanh ^4(x)}{b}}d\sqrt {b \tanh ^2(x)+a}}{b}+\frac {2 (a+b)^2 \int \frac {1}{\frac {\tanh ^4(x)}{b}-\frac {a}{b}}d\sqrt {b \tanh ^2(x)+a}}{b}}{a (a+b)}+\frac {2 b (2 a+b)}{a (a+b) \sqrt {a+b \tanh ^2(x)}}}{a (a+b)}+\frac {2 b}{3 a (a+b) \left (a+b \tanh ^2(x)\right )^{3/2}}\right )\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {1}{2} \left (\frac {\frac {\frac {2 a^2 \text {arctanh}\left (\frac {\sqrt {a+b \tanh ^2(x)}}{\sqrt {a+b}}\right )}{\sqrt {a+b}}-\frac {2 (a+b)^2 \text {arctanh}\left (\frac {\sqrt {a+b \tanh ^2(x)}}{\sqrt {a}}\right )}{\sqrt {a}}}{a (a+b)}+\frac {2 b (2 a+b)}{a (a+b) \sqrt {a+b \tanh ^2(x)}}}{a (a+b)}+\frac {2 b}{3 a (a+b) \left (a+b \tanh ^2(x)\right )^{3/2}}\right )\) |
((2*b)/(3*a*(a + b)*(a + b*Tanh[x]^2)^(3/2)) + (((-2*(a + b)^2*ArcTanh[Sqr t[a + b*Tanh[x]^2]/Sqrt[a]])/Sqrt[a] + (2*a^2*ArcTanh[Sqrt[a + b*Tanh[x]^2 ]/Sqrt[a + b]])/Sqrt[a + b])/(a*(a + b)) + (2*b*(2*a + b))/(a*(a + b)*Sqrt [a + b*Tanh[x]^2]))/(a*(a + b)))/2
3.3.53.3.1 Defintions of rubi rules used
Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a]) I nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
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_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
Int[((e_.) + (f_.)*(x_))^(p_)/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_] :> Simp[f*((e + f*x)^(p + 1)/((p + 1)*(b*e - a*f)*(d*e - c*f))), x] + S imp[1/((b*e - a*f)*(d*e - c*f)) Int[(b*d*e - b*c*f - a*d*f - b*d*f*x)*((e + f*x)^(p + 1)/((a + b*x)*(c + d*x))), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && LtQ[p, -1]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[(b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + S imp[1/((m + 1)*(b*c - a*d)*(b*e - a*f)) Int[(a + b*x)^(m + 1)*(c + d*x)^n *(e + f*x)^p*Simp[(a*d*f*g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a* h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && LtQ[m, -1] && IntegersQ[ 2*m, 2*n, 2*p]
Int[(((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/(((a_.) + (b_.)*(x_))* ((c_.) + (d_.)*(x_))), x_] :> Simp[(b*g - a*h)/(b*c - a*d) Int[(e + f*x)^ p/(a + b*x), x], x] - Simp[(d*g - c*h)/(b*c - a*d) Int[(e + f*x)^p/(c + d *x), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.), x_S ymbol] :> Simp[1/2 Subst[Int[x^((m - 1)/2)*(a + b*x)^p*(c + d*x)^q, x], x , x^2], x] /; FreeQ[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && IntegerQ [(m - 1)/2]
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]))
\[\int \frac {\coth \left (x \right )}{\left (a +b \tanh \left (x \right )^{2}\right )^{\frac {5}{2}}}d x\]
Leaf count of result is larger than twice the leaf count of optimal. 4510 vs. \(2 (90) = 180\).
Time = 1.28 (sec) , antiderivative size = 19305, normalized size of antiderivative = 178.75 \[ \int \frac {\coth (x)}{\left (a+b \tanh ^2(x)\right )^{5/2}} \, dx=\text {Too large to display} \]
\[ \int \frac {\coth (x)}{\left (a+b \tanh ^2(x)\right )^{5/2}} \, dx=\int \frac {\coth {\left (x \right )}}{\left (a + b \tanh ^{2}{\left (x \right )}\right )^{\frac {5}{2}}}\, dx \]
\[ \int \frac {\coth (x)}{\left (a+b \tanh ^2(x)\right )^{5/2}} \, dx=\int { \frac {\coth \left (x\right )}{{\left (b \tanh \left (x\right )^{2} + a\right )}^{\frac {5}{2}}} \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 808 vs. \(2 (90) = 180\).
Time = 0.65 (sec) , antiderivative size = 808, normalized size of antiderivative = 7.48 \[ \int \frac {\coth (x)}{\left (a+b \tanh ^2(x)\right )^{5/2}} \, dx=\frac {{\left ({\left (\frac {{\left (7 \, a^{14} b^{3} + 38 \, a^{13} b^{4} + 85 \, a^{12} b^{5} + 100 \, a^{11} b^{6} + 65 \, a^{10} b^{7} + 22 \, a^{9} b^{8} + 3 \, a^{8} b^{9}\right )} e^{\left (2 \, x\right )}}{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}} + \frac {3 \, {\left (7 \, a^{14} b^{3} + 30 \, a^{13} b^{4} + 49 \, a^{12} b^{5} + 36 \, a^{11} b^{6} + 9 \, a^{10} b^{7} - 2 \, a^{9} b^{8} - a^{8} b^{9}\right )}}{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}}\right )} e^{\left (2 \, x\right )} + \frac {3 \, {\left (7 \, a^{14} b^{3} + 30 \, a^{13} b^{4} + 49 \, a^{12} b^{5} + 36 \, a^{11} b^{6} + 9 \, a^{10} b^{7} - 2 \, a^{9} b^{8} - a^{8} b^{9}\right )}}{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}}\right )} e^{\left (2 \, x\right )} + \frac {7 \, a^{14} b^{3} + 38 \, a^{13} b^{4} + 85 \, a^{12} b^{5} + 100 \, a^{11} b^{6} + 65 \, a^{10} b^{7} + 22 \, a^{9} b^{8} + 3 \, a^{8} b^{9}}{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 \, {\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}} + \frac {2 \, \arctan \left (-\frac {\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}}{2 \, \sqrt {-a}}\right )}{\sqrt {-a} a^{2}} \]
1/3*((((7*a^14*b^3 + 38*a^13*b^4 + 85*a^12*b^5 + 100*a^11*b^6 + 65*a^10*b^ 7 + 22*a^9*b^8 + 3*a^8*b^9)*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*(7*a^14*b^3 + 30*a ^13*b^4 + 49*a^12*b^5 + 36*a^11*b^6 + 9*a^10*b^7 - 2*a^9*b^8 - a^8*b^9)/(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) + 3*(7*a^14*b^3 + 30*a^13*b^4 + 49*a^12*b^5 + 36*a^ 11*b^6 + 9*a^10*b^7 - 2*a^9*b^8 - a^8*b^9)/(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) + (7*a ^14*b^3 + 38*a^13*b^4 + 85*a^12*b^5 + 100*a^11*b^6 + 65*a^10*b^7 + 22*a^9* b^8 + 3*a^8*b^9)/(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)) + 2*arctan(-1/2*(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))/sqrt(-a))/(sqrt(-a)*a^2)
Timed out. \[ \int \frac {\coth (x)}{\left (a+b \tanh ^2(x)\right )^{5/2}} \, dx=\int \frac {\mathrm {coth}\left (x\right )}{{\left (b\,{\mathrm {tanh}\left (x\right )}^2+a\right )}^{5/2}} \,d x \]