Integrand size = 17, antiderivative size = 74 \[ \int \frac {\coth ^3(x)}{\left (a+b \coth ^2(x)\right )^{5/2}} \, dx=\frac {\text {arctanh}\left (\frac {\sqrt {a+b \coth ^2(x)}}{\sqrt {a+b}}\right )}{(a+b)^{5/2}}+\frac {a}{3 b (a+b) \left (a+b \coth ^2(x)\right )^{3/2}}-\frac {1}{(a+b)^2 \sqrt {a+b \coth ^2(x)}} \]
arctanh((a+b*coth(x)^2)^(1/2)/(a+b)^(1/2))/(a+b)^(5/2)+1/3*a/b/(a+b)/(a+b* coth(x)^2)^(3/2)-1/(a+b)^2/(a+b*coth(x)^2)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.12 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.85 \[ \int \frac {\coth ^3(x)}{\left (a+b \coth ^2(x)\right )^{5/2}} \, dx=\frac {a (a+b)-3 b \left (a+b \coth ^2(x)\right ) \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},1,\frac {1}{2},\frac {a+b \coth ^2(x)}{a+b}\right )}{3 b (a+b)^2 \left (a+b \coth ^2(x)\right )^{3/2}} \]
(a*(a + b) - 3*b*(a + b*Coth[x]^2)*Hypergeometric2F1[-1/2, 1, 1/2, (a + b* Coth[x]^2)/(a + b)])/(3*b*(a + b)^2*(a + b*Coth[x]^2)^(3/2))
Time = 0.32 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.16, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.529, Rules used = {3042, 26, 4153, 26, 354, 87, 61, 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.
| \(\displaystyle \int \frac {\coth ^3(x)}{\left (a+b \coth ^2(x)\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {i \tan \left (\frac {\pi }{2}+i x\right )^3}{\left (a-b \tan \left (\frac {\pi }{2}+i x\right )^2\right )^{5/2}}dx\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle i \int \frac {\tan \left (i x+\frac {\pi }{2}\right )^3}{\left (a-b \tan \left (i x+\frac {\pi }{2}\right )^2\right )^{5/2}}dx\) |
| \(\Big \downarrow \) 4153 |
| \(\displaystyle i \int -\frac {i \coth ^3(x)}{\left (1-\coth ^2(x)\right ) \left (b \coth ^2(x)+a\right )^{5/2}}d\coth (x)\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \int \frac {\coth ^3(x)}{\left (1-\coth ^2(x)\right ) \left (a+b \coth ^2(x)\right )^{5/2}}d\coth (x)\) |
| \(\Big \downarrow \) 354 |
| \(\displaystyle \frac {1}{2} \int \frac {\coth ^2(x)}{\left (1-\coth ^2(x)\right ) \left (b \coth ^2(x)+a\right )^{5/2}}d\coth ^2(x)\) |
| \(\Big \downarrow \) 87 |
| \(\displaystyle \frac {1}{2} \left (\frac {\int \frac {1}{\left (1-\coth ^2(x)\right ) \left (b \coth ^2(x)+a\right )^{3/2}}d\coth ^2(x)}{a+b}+\frac {2 a}{3 b (a+b) \left (a+b \coth ^2(x)\right )^{3/2}}\right )\) |
| \(\Big \downarrow \) 61 |
| \(\displaystyle \frac {1}{2} \left (\frac {\frac {\int \frac {1}{\left (1-\coth ^2(x)\right ) \sqrt {b \coth ^2(x)+a}}d\coth ^2(x)}{a+b}-\frac {2}{(a+b) \sqrt {a+b \coth ^2(x)}}}{a+b}+\frac {2 a}{3 b (a+b) \left (a+b \coth ^2(x)\right )^{3/2}}\right )\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {1}{2} \left (\frac {\frac {2 \int \frac {1}{\frac {a+b}{b}-\frac {\coth ^4(x)}{b}}d\sqrt {b \coth ^2(x)+a}}{b (a+b)}-\frac {2}{(a+b) \sqrt {a+b \coth ^2(x)}}}{a+b}+\frac {2 a}{3 b (a+b) \left (a+b \coth ^2(x)\right )^{3/2}}\right )\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {1}{2} \left (\frac {\frac {2 \text {arctanh}\left (\frac {\sqrt {a+b \coth ^2(x)}}{\sqrt {a+b}}\right )}{(a+b)^{3/2}}-\frac {2}{(a+b) \sqrt {a+b \coth ^2(x)}}}{a+b}+\frac {2 a}{3 b (a+b) \left (a+b \coth ^2(x)\right )^{3/2}}\right )\) |
((2*a)/(3*b*(a + b)*(a + b*Coth[x]^2)^(3/2)) + ((2*ArcTanh[Sqrt[a + b*Coth [x]^2]/Sqrt[a + b]])/(a + b)^(3/2) - 2/((a + b)*Sqrt[a + b*Coth[x]^2]))/(a + b))/2
3.1.43.3.1 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_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(( m + n + 2)/((b*c - a*d)*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && LtQ[m, -1] && !(LtQ[n, -1] && (EqQ[a, 0 ] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d , m, n, 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[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p _.), x_] :> Simp[(-(b*e - a*f))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p + 1)*(c*f - d*e))), x] - Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)) Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || Intege rQ[p] || !(IntegerQ[n] || !(EqQ[e, 0] || !(EqQ[c, 0] || LtQ[p, n]))))
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]))
Leaf count of result is larger than twice the leaf count of optimal. \(434\) vs. \(2(62)=124\).
Time = 0.10 (sec) , antiderivative size = 435, normalized size of antiderivative = 5.88
| method | result | size |
| derivativedivides | \(\frac {1}{3 b \left (a +b \coth \left (x \right )^{2}\right )^{\frac {3}{2}}}-\frac {1}{6 \left (a +b \right ) \left (b \left (\coth \left (x \right )-1\right )^{2}+2 b \left (\coth \left (x \right )-1\right )+a +b \right )^{\frac {3}{2}}}+\frac {b \coth \left (x \right )}{6 \left (a +b \right ) a \left (b \left (\coth \left (x \right )-1\right )^{2}+2 b \left (\coth \left (x \right )-1\right )+a +b \right )^{\frac {3}{2}}}+\frac {b \coth \left (x \right )}{3 \left (a +b \right ) a^{2} \sqrt {b \left (\coth \left (x \right )-1\right )^{2}+2 b \left (\coth \left (x \right )-1\right )+a +b}}-\frac {1}{2 \left (a +b \right )^{2} \sqrt {b \left (\coth \left (x \right )-1\right )^{2}+2 b \left (\coth \left (x \right )-1\right )+a +b}}+\frac {b \coth \left (x \right )}{2 \left (a +b \right )^{2} a \sqrt {b \left (\coth \left (x \right )-1\right )^{2}+2 b \left (\coth \left (x \right )-1\right )+a +b}}+\frac {\ln \left (\frac {2 a +2 b +2 b \left (\coth \left (x \right )-1\right )+2 \sqrt {a +b}\, \sqrt {b \left (\coth \left (x \right )-1\right )^{2}+2 b \left (\coth \left (x \right )-1\right )+a +b}}{\coth \left (x \right )-1}\right )}{2 \left (a +b \right )^{\frac {5}{2}}}-\frac {1}{6 \left (a +b \right ) \left (b \left (1+\coth \left (x \right )\right )^{2}-2 b \left (1+\coth \left (x \right )\right )+a +b \right )^{\frac {3}{2}}}-\frac {b \coth \left (x \right )}{6 \left (a +b \right ) a \left (b \left (1+\coth \left (x \right )\right )^{2}-2 b \left (1+\coth \left (x \right )\right )+a +b \right )^{\frac {3}{2}}}-\frac {b \coth \left (x \right )}{3 \left (a +b \right ) a^{2} \sqrt {b \left (1+\coth \left (x \right )\right )^{2}-2 b \left (1+\coth \left (x \right )\right )+a +b}}-\frac {1}{2 \left (a +b \right )^{2} \sqrt {b \left (1+\coth \left (x \right )\right )^{2}-2 b \left (1+\coth \left (x \right )\right )+a +b}}-\frac {b \coth \left (x \right )}{2 \left (a +b \right )^{2} a \sqrt {b \left (1+\coth \left (x \right )\right )^{2}-2 b \left (1+\coth \left (x \right )\right )+a +b}}+\frac {\ln \left (\frac {2 a +2 b -2 b \left (1+\coth \left (x \right )\right )+2 \sqrt {a +b}\, \sqrt {b \left (1+\coth \left (x \right )\right )^{2}-2 b \left (1+\coth \left (x \right )\right )+a +b}}{1+\coth \left (x \right )}\right )}{2 \left (a +b \right )^{\frac {5}{2}}}\) | \(435\) |
| default | \(\frac {1}{3 b \left (a +b \coth \left (x \right )^{2}\right )^{\frac {3}{2}}}-\frac {1}{6 \left (a +b \right ) \left (b \left (\coth \left (x \right )-1\right )^{2}+2 b \left (\coth \left (x \right )-1\right )+a +b \right )^{\frac {3}{2}}}+\frac {b \coth \left (x \right )}{6 \left (a +b \right ) a \left (b \left (\coth \left (x \right )-1\right )^{2}+2 b \left (\coth \left (x \right )-1\right )+a +b \right )^{\frac {3}{2}}}+\frac {b \coth \left (x \right )}{3 \left (a +b \right ) a^{2} \sqrt {b \left (\coth \left (x \right )-1\right )^{2}+2 b \left (\coth \left (x \right )-1\right )+a +b}}-\frac {1}{2 \left (a +b \right )^{2} \sqrt {b \left (\coth \left (x \right )-1\right )^{2}+2 b \left (\coth \left (x \right )-1\right )+a +b}}+\frac {b \coth \left (x \right )}{2 \left (a +b \right )^{2} a \sqrt {b \left (\coth \left (x \right )-1\right )^{2}+2 b \left (\coth \left (x \right )-1\right )+a +b}}+\frac {\ln \left (\frac {2 a +2 b +2 b \left (\coth \left (x \right )-1\right )+2 \sqrt {a +b}\, \sqrt {b \left (\coth \left (x \right )-1\right )^{2}+2 b \left (\coth \left (x \right )-1\right )+a +b}}{\coth \left (x \right )-1}\right )}{2 \left (a +b \right )^{\frac {5}{2}}}-\frac {1}{6 \left (a +b \right ) \left (b \left (1+\coth \left (x \right )\right )^{2}-2 b \left (1+\coth \left (x \right )\right )+a +b \right )^{\frac {3}{2}}}-\frac {b \coth \left (x \right )}{6 \left (a +b \right ) a \left (b \left (1+\coth \left (x \right )\right )^{2}-2 b \left (1+\coth \left (x \right )\right )+a +b \right )^{\frac {3}{2}}}-\frac {b \coth \left (x \right )}{3 \left (a +b \right ) a^{2} \sqrt {b \left (1+\coth \left (x \right )\right )^{2}-2 b \left (1+\coth \left (x \right )\right )+a +b}}-\frac {1}{2 \left (a +b \right )^{2} \sqrt {b \left (1+\coth \left (x \right )\right )^{2}-2 b \left (1+\coth \left (x \right )\right )+a +b}}-\frac {b \coth \left (x \right )}{2 \left (a +b \right )^{2} a \sqrt {b \left (1+\coth \left (x \right )\right )^{2}-2 b \left (1+\coth \left (x \right )\right )+a +b}}+\frac {\ln \left (\frac {2 a +2 b -2 b \left (1+\coth \left (x \right )\right )+2 \sqrt {a +b}\, \sqrt {b \left (1+\coth \left (x \right )\right )^{2}-2 b \left (1+\coth \left (x \right )\right )+a +b}}{1+\coth \left (x \right )}\right )}{2 \left (a +b \right )^{\frac {5}{2}}}\) | \(435\) |
1/3/b/(a+b*coth(x)^2)^(3/2)-1/6/(a+b)/(b*(coth(x)-1)^2+2*b*(coth(x)-1)+a+b )^(3/2)+1/6*b/(a+b)/a/(b*(coth(x)-1)^2+2*b*(coth(x)-1)+a+b)^(3/2)*coth(x)+ 1/3*b/(a+b)/a^2/(b*(coth(x)-1)^2+2*b*(coth(x)-1)+a+b)^(1/2)*coth(x)-1/2/(a +b)^2/(b*(coth(x)-1)^2+2*b*(coth(x)-1)+a+b)^(1/2)+1/2/(a+b)^2/a/(b*(coth(x )-1)^2+2*b*(coth(x)-1)+a+b)^(1/2)*b*coth(x)+1/2/(a+b)^(5/2)*ln((2*a+2*b+2* b*(coth(x)-1)+2*(a+b)^(1/2)*(b*(coth(x)-1)^2+2*b*(coth(x)-1)+a+b)^(1/2))/( coth(x)-1))-1/6/(a+b)/(b*(1+coth(x))^2-2*b*(1+coth(x))+a+b)^(3/2)-1/6*b/(a +b)/a/(b*(1+coth(x))^2-2*b*(1+coth(x))+a+b)^(3/2)*coth(x)-1/3*b/(a+b)/a^2/ (b*(1+coth(x))^2-2*b*(1+coth(x))+a+b)^(1/2)*coth(x)-1/2/(a+b)^2/(b*(1+coth (x))^2-2*b*(1+coth(x))+a+b)^(1/2)-1/2/(a+b)^2/a/(b*(1+coth(x))^2-2*b*(1+co th(x))+a+b)^(1/2)*b*coth(x)+1/2/(a+b)^(5/2)*ln((2*a+2*b-2*b*(1+coth(x))+2* (a+b)^(1/2)*(b*(1+coth(x))^2-2*b*(1+coth(x))+a+b)^(1/2))/(1+coth(x)))
Leaf count of result is larger than twice the leaf count of optimal. 2964 vs. \(2 (62) = 124\).
Time = 0.69 (sec) , antiderivative size = 6560, normalized size of antiderivative = 88.65 \[ \int \frac {\coth ^3(x)}{\left (a+b \coth ^2(x)\right )^{5/2}} \, dx=\text {Too large to display} \]
\[ \int \frac {\coth ^3(x)}{\left (a+b \coth ^2(x)\right )^{5/2}} \, dx=\int \frac {\coth ^{3}{\left (x \right )}}{\left (a + b \coth ^{2}{\left (x \right )}\right )^{\frac {5}{2}}}\, dx \]
\[ \int \frac {\coth ^3(x)}{\left (a+b \coth ^2(x)\right )^{5/2}} \, dx=\int { \frac {\coth \left (x\right )^{3}}{{\left (b \coth \left (x\right )^{2} + a\right )}^{\frac {5}{2}}} \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 951 vs. \(2 (62) = 124\).
Time = 0.60 (sec) , antiderivative size = 951, normalized size of antiderivative = 12.85 \[ \int \frac {\coth ^3(x)}{\left (a+b \coth ^2(x)\right )^{5/2}} \, dx=\text {Too large to display} \]
1/3*((((a^8*b*sgn(e^(2*x) - 1) + 2*a^7*b^2*sgn(e^(2*x) - 1) - 5*a^6*b^3*sg n(e^(2*x) - 1) - 20*a^5*b^4*sgn(e^(2*x) - 1) - 25*a^4*b^5*sgn(e^(2*x) - 1) - 14*a^3*b^6*sgn(e^(2*x) - 1) - 3*a^2*b^7*sgn(e^(2*x) - 1))*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^8*b*sgn(e^(2*x) - 1) + 2*a^7*b^2*sgn(e^(2*x) - 1) - a^6*b^3*sgn (e^(2*x) - 1) - 4*a^5*b^4*sgn(e^(2*x) - 1) - a^4*b^5*sgn(e^(2*x) - 1) + 2* a^3*b^6*sgn(e^(2*x) - 1) + a^2*b^7*sgn(e^(2*x) - 1))/(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^8*b*sgn(e^(2*x) - 1) + 2*a^7*b^2*sgn(e^(2*x) - 1) - a^6*b^3*sgn(e^(2*x ) - 1) - 4*a^5*b^4*sgn(e^(2*x) - 1) - a^4*b^5*sgn(e^(2*x) - 1) + 2*a^3*b^6 *sgn(e^(2*x) - 1) + a^2*b^7*sgn(e^(2*x) - 1))/(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) - (a^8*b*s gn(e^(2*x) - 1) + 2*a^7*b^2*sgn(e^(2*x) - 1) - 5*a^6*b^3*sgn(e^(2*x) - 1) - 20*a^5*b^4*sgn(e^(2*x) - 1) - 25*a^4*b^5*sgn(e^(2*x) - 1) - 14*a^3*b^6*s gn(e^(2*x) - 1) - 3*a^2*b^7*sgn(e^(2*x) - 1))/(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) )*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*...
Time = 4.25 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.11 \[ \int \frac {\coth ^3(x)}{\left (a+b \coth ^2(x)\right )^{5/2}} \, dx=\frac {\mathrm {atanh}\left (\frac {\sqrt {b\,{\mathrm {coth}\left (x\right )}^2+a}\,\left (2\,a^2+4\,a\,b+2\,b^2\right )}{2\,{\left (a+b\right )}^{5/2}}\right )}{{\left (a+b\right )}^{5/2}}+\frac {\frac {a}{3\,\left (a+b\right )}-\frac {b\,\left (b\,{\mathrm {coth}\left (x\right )}^2+a\right )}{{\left (a+b\right )}^2}}{b\,{\left (b\,{\mathrm {coth}\left (x\right )}^2+a\right )}^{3/2}} \]