Integrand size = 8, antiderivative size = 61 \[ \int \frac {1}{(1+\coth (x))^{5/2}} \, dx=\frac {\text {arctanh}\left (\frac {\sqrt {1+\coth (x)}}{\sqrt {2}}\right )}{4 \sqrt {2}}-\frac {1}{5 (1+\coth (x))^{5/2}}-\frac {1}{6 (1+\coth (x))^{3/2}}-\frac {1}{4 \sqrt {1+\coth (x)}} \] Output:
1/8*2^(1/2)*arctanh(1/2*(1+coth(x))^(1/2)*2^(1/2))-1/5/(1+coth(x))^(5/2)-1 /6/(1+coth(x))^(3/2)-1/4/(1+coth(x))^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.40 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.46 \[ \int \frac {1}{(1+\coth (x))^{5/2}} \, dx=-\frac {\operatorname {Hypergeometric2F1}\left (-\frac {5}{2},1,-\frac {3}{2},\frac {1}{2} (1+\coth (x))\right )}{5 (1+\coth (x))^{5/2}} \] Input:
Integrate[(1 + Coth[x])^(-5/2),x]
Output:
-1/5*Hypergeometric2F1[-5/2, 1, -3/2, (1 + Coth[x])/2]/(1 + Coth[x])^(5/2)
Time = 0.39 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.08, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.125, Rules used = {3042, 3960, 3042, 3960, 3042, 3960, 3042, 3961, 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.
| \(\displaystyle \int \frac {1}{(\coth (x)+1)^{5/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\left (1-i \tan \left (\frac {\pi }{2}+i x\right )\right )^{5/2}}dx\) |
| \(\Big \downarrow \) 3960 |
| \(\displaystyle \frac {1}{2} \int \frac {1}{(\coth (x)+1)^{3/2}}dx-\frac {1}{5 (\coth (x)+1)^{5/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {1}{5 (\coth (x)+1)^{5/2}}+\frac {1}{2} \int \frac {1}{\left (1-i \tan \left (i x+\frac {\pi }{2}\right )\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 3960 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{2} \int \frac {1}{\sqrt {\coth (x)+1}}dx-\frac {1}{3 (\coth (x)+1)^{3/2}}\right )-\frac {1}{5 (\coth (x)+1)^{5/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {1}{5 (\coth (x)+1)^{5/2}}+\frac {1}{2} \left (-\frac {1}{3 (\coth (x)+1)^{3/2}}+\frac {1}{2} \int \frac {1}{\sqrt {1-i \tan \left (i x+\frac {\pi }{2}\right )}}dx\right )\) |
| \(\Big \downarrow \) 3960 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{2} \left (\frac {1}{2} \int \sqrt {\coth (x)+1}dx-\frac {1}{\sqrt {\coth (x)+1}}\right )-\frac {1}{3 (\coth (x)+1)^{3/2}}\right )-\frac {1}{5 (\coth (x)+1)^{5/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {1}{5 (\coth (x)+1)^{5/2}}+\frac {1}{2} \left (-\frac {1}{3 (\coth (x)+1)^{3/2}}+\frac {1}{2} \left (-\frac {1}{\sqrt {\coth (x)+1}}+\frac {1}{2} \int \sqrt {1-i \tan \left (i x+\frac {\pi }{2}\right )}dx\right )\right )\) |
| \(\Big \downarrow \) 3961 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{2} \left (\int \frac {1}{1-\coth (x)}d\sqrt {\coth (x)+1}-\frac {1}{\sqrt {\coth (x)+1}}\right )-\frac {1}{3 (\coth (x)+1)^{3/2}}\right )-\frac {1}{5 (\coth (x)+1)^{5/2}}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{2} \left (\frac {\text {arctanh}\left (\frac {\sqrt {\coth (x)+1}}{\sqrt {2}}\right )}{\sqrt {2}}-\frac {1}{\sqrt {\coth (x)+1}}\right )-\frac {1}{3 (\coth (x)+1)^{3/2}}\right )-\frac {1}{5 (\coth (x)+1)^{5/2}}\) |
Input:
Int[(1 + Coth[x])^(-5/2),x]
Output:
-1/5*1/(1 + Coth[x])^(5/2) + (-1/3*1/(1 + Coth[x])^(3/2) + (ArcTanh[Sqrt[1 + Coth[x]]/Sqrt[2]]/Sqrt[2] - 1/Sqrt[1 + Coth[x]])/2)/2
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[((a_) + (b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[a*((a + b*Tan[c + d*x])^n/(2*b*d*n)), x] + Simp[1/(2*a) Int[(a + b*Tan[c + d*x])^ (n + 1), x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 + b^2, 0] && LtQ[n, 0]
Int[Sqrt[(a_) + (b_.)*tan[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[-2*(b/d) Subst[Int[1/(2*a - x^2), x], x, Sqrt[a + b*Tan[c + d*x]]], x] /; FreeQ[{a , b, c, d}, x] && EqQ[a^2 + b^2, 0]
Time = 0.05 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.70
| method | result | size |
| derivativedivides | \(\frac {\sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {1+\coth \left (x \right )}\, \sqrt {2}}{2}\right )}{8}-\frac {1}{5 \left (1+\coth \left (x \right )\right )^{\frac {5}{2}}}-\frac {1}{6 \left (1+\coth \left (x \right )\right )^{\frac {3}{2}}}-\frac {1}{4 \sqrt {1+\coth \left (x \right )}}\) | \(43\) |
| default | \(\frac {\sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {1+\coth \left (x \right )}\, \sqrt {2}}{2}\right )}{8}-\frac {1}{5 \left (1+\coth \left (x \right )\right )^{\frac {5}{2}}}-\frac {1}{6 \left (1+\coth \left (x \right )\right )^{\frac {3}{2}}}-\frac {1}{4 \sqrt {1+\coth \left (x \right )}}\) | \(43\) |
Input:
int(1/(1+coth(x))^(5/2),x,method=_RETURNVERBOSE)
Output:
1/8*2^(1/2)*arctanh(1/2*(1+coth(x))^(1/2)*2^(1/2))-1/5/(1+coth(x))^(5/2)-1 /6/(1+coth(x))^(3/2)-1/4/(1+coth(x))^(1/2)
Leaf count of result is larger than twice the leaf count of optimal. 340 vs. \(2 (42) = 84\).
Time = 0.10 (sec) , antiderivative size = 340, normalized size of antiderivative = 5.57 \[ \int \frac {1}{(1+\coth (x))^{5/2}} \, dx=\frac {15 \, {\left (\sqrt {2} \cosh \left (x\right )^{5} + 5 \, \sqrt {2} \cosh \left (x\right )^{4} \sinh \left (x\right ) + 10 \, \sqrt {2} \cosh \left (x\right )^{3} \sinh \left (x\right )^{2} + 10 \, \sqrt {2} \cosh \left (x\right )^{2} \sinh \left (x\right )^{3} + 5 \, \sqrt {2} \cosh \left (x\right ) \sinh \left (x\right )^{4} + \sqrt {2} \sinh \left (x\right )^{5}\right )} \log \left (2 \, \cosh \left (x\right )^{2} + 4 \, \cosh \left (x\right ) \sinh \left (x\right ) + 2 \, \sinh \left (x\right )^{2} + \frac {\sqrt {2} {\left (\sqrt {2} \cosh \left (x\right )^{3} + 3 \, \sqrt {2} \cosh \left (x\right ) \sinh \left (x\right )^{2} + \sqrt {2} \sinh \left (x\right )^{3} + {\left (3 \, \sqrt {2} \cosh \left (x\right )^{2} - \sqrt {2}\right )} \sinh \left (x\right ) - \sqrt {2} \cosh \left (x\right )\right )}}{\sqrt {\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2} - 1}} - 1\right ) - \frac {2 \, \sqrt {2} {\left (23 \, \cosh \left (x\right )^{6} + 138 \, \cosh \left (x\right ) \sinh \left (x\right )^{5} + 23 \, \sinh \left (x\right )^{6} + {\left (345 \, \cosh \left (x\right )^{2} - 34\right )} \sinh \left (x\right )^{4} - 34 \, \cosh \left (x\right )^{4} + 4 \, {\left (115 \, \cosh \left (x\right )^{3} - 34 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{3} + {\left (345 \, \cosh \left (x\right )^{4} - 204 \, \cosh \left (x\right )^{2} + 14\right )} \sinh \left (x\right )^{2} + 14 \, \cosh \left (x\right )^{2} + 2 \, {\left (69 \, \cosh \left (x\right )^{5} - 68 \, \cosh \left (x\right )^{3} + 14 \, \cosh \left (x\right )\right )} \sinh \left (x\right ) - 3\right )}}{\sqrt {\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2} - 1}}}{240 \, {\left (\cosh \left (x\right )^{5} + 5 \, \cosh \left (x\right )^{4} \sinh \left (x\right ) + 10 \, \cosh \left (x\right )^{3} \sinh \left (x\right )^{2} + 10 \, \cosh \left (x\right )^{2} \sinh \left (x\right )^{3} + 5 \, \cosh \left (x\right ) \sinh \left (x\right )^{4} + \sinh \left (x\right )^{5}\right )}} \] Input:
integrate(1/(1+coth(x))^(5/2),x, algorithm="fricas")
Output:
1/240*(15*(sqrt(2)*cosh(x)^5 + 5*sqrt(2)*cosh(x)^4*sinh(x) + 10*sqrt(2)*co sh(x)^3*sinh(x)^2 + 10*sqrt(2)*cosh(x)^2*sinh(x)^3 + 5*sqrt(2)*cosh(x)*sin h(x)^4 + sqrt(2)*sinh(x)^5)*log(2*cosh(x)^2 + 4*cosh(x)*sinh(x) + 2*sinh(x )^2 + sqrt(2)*(sqrt(2)*cosh(x)^3 + 3*sqrt(2)*cosh(x)*sinh(x)^2 + sqrt(2)*s inh(x)^3 + (3*sqrt(2)*cosh(x)^2 - sqrt(2))*sinh(x) - sqrt(2)*cosh(x))/sqrt (cosh(x)^2 + 2*cosh(x)*sinh(x) + sinh(x)^2 - 1) - 1) - 2*sqrt(2)*(23*cosh( x)^6 + 138*cosh(x)*sinh(x)^5 + 23*sinh(x)^6 + (345*cosh(x)^2 - 34)*sinh(x) ^4 - 34*cosh(x)^4 + 4*(115*cosh(x)^3 - 34*cosh(x))*sinh(x)^3 + (345*cosh(x )^4 - 204*cosh(x)^2 + 14)*sinh(x)^2 + 14*cosh(x)^2 + 2*(69*cosh(x)^5 - 68* cosh(x)^3 + 14*cosh(x))*sinh(x) - 3)/sqrt(cosh(x)^2 + 2*cosh(x)*sinh(x) + sinh(x)^2 - 1))/(cosh(x)^5 + 5*cosh(x)^4*sinh(x) + 10*cosh(x)^3*sinh(x)^2 + 10*cosh(x)^2*sinh(x)^3 + 5*cosh(x)*sinh(x)^4 + sinh(x)^5)
\[ \int \frac {1}{(1+\coth (x))^{5/2}} \, dx=\int \frac {1}{\left (\coth {\left (x \right )} + 1\right )^{\frac {5}{2}}}\, dx \] Input:
integrate(1/(1+coth(x))**(5/2),x)
Output:
Integral((coth(x) + 1)**(-5/2), x)
\[ \int \frac {1}{(1+\coth (x))^{5/2}} \, dx=\int { \frac {1}{{\left (\coth \left (x\right ) + 1\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate(1/(1+coth(x))^(5/2),x, algorithm="maxima")
Output:
integrate((coth(x) + 1)^(-5/2), x)
Leaf count of result is larger than twice the leaf count of optimal. 161 vs. \(2 (42) = 84\).
Time = 0.12 (sec) , antiderivative size = 161, normalized size of antiderivative = 2.64 \[ \int \frac {1}{(1+\coth (x))^{5/2}} \, dx=\frac {\sqrt {2} {\left (\frac {2 \, {\left (45 \, {\left (\sqrt {e^{\left (4 \, x\right )} - e^{\left (2 \, x\right )}} - e^{\left (2 \, x\right )}\right )}^{4} + 45 \, {\left (\sqrt {e^{\left (4 \, x\right )} - e^{\left (2 \, x\right )}} - e^{\left (2 \, x\right )}\right )}^{3} + 35 \, {\left (\sqrt {e^{\left (4 \, x\right )} - e^{\left (2 \, x\right )}} - e^{\left (2 \, x\right )}\right )}^{2} + 15 \, \sqrt {e^{\left (4 \, x\right )} - e^{\left (2 \, x\right )}} - 15 \, e^{\left (2 \, x\right )} + 3\right )}}{{\left (\sqrt {e^{\left (4 \, x\right )} - e^{\left (2 \, x\right )}} - e^{\left (2 \, x\right )}\right )}^{5}} - 15 \, \log \left ({\left | 2 \, \sqrt {e^{\left (4 \, x\right )} - e^{\left (2 \, x\right )}} - 2 \, e^{\left (2 \, x\right )} + 1 \right |}\right )\right )}}{240 \, \mathrm {sgn}\left (e^{\left (2 \, x\right )} - 1\right )} \] Input:
integrate(1/(1+coth(x))^(5/2),x, algorithm="giac")
Output:
1/240*sqrt(2)*(2*(45*(sqrt(e^(4*x) - e^(2*x)) - e^(2*x))^4 + 45*(sqrt(e^(4 *x) - e^(2*x)) - e^(2*x))^3 + 35*(sqrt(e^(4*x) - e^(2*x)) - e^(2*x))^2 + 1 5*sqrt(e^(4*x) - e^(2*x)) - 15*e^(2*x) + 3)/(sqrt(e^(4*x) - e^(2*x)) - e^( 2*x))^5 - 15*log(abs(2*sqrt(e^(4*x) - e^(2*x)) - 2*e^(2*x) + 1)))/sgn(e^(2 *x) - 1)
Time = 2.34 (sec) , antiderivative size = 40, normalized size of antiderivative = 0.66 \[ \int \frac {1}{(1+\coth (x))^{5/2}} \, dx=\frac {\sqrt {2}\,\mathrm {atanh}\left (\frac {\sqrt {2}\,\sqrt {\mathrm {coth}\left (x\right )+1}}{2}\right )}{8}-\frac {\frac {\mathrm {coth}\left (x\right )}{6}+\frac {{\left (\mathrm {coth}\left (x\right )+1\right )}^2}{4}+\frac {11}{30}}{{\left (\mathrm {coth}\left (x\right )+1\right )}^{5/2}} \] Input:
int(1/(coth(x) + 1)^(5/2),x)
Output:
(2^(1/2)*atanh((2^(1/2)*(coth(x) + 1)^(1/2))/2))/8 - (coth(x)/6 + (coth(x) + 1)^2/4 + 11/30)/(coth(x) + 1)^(5/2)
\[ \int \frac {1}{(1+\coth (x))^{5/2}} \, dx=\int \frac {\sqrt {\coth \left (x \right )+1}}{\coth \left (x \right )^{3}+3 \coth \left (x \right )^{2}+3 \coth \left (x \right )+1}d x \] Input:
int(1/(1+coth(x))^(5/2),x)
Output:
int(sqrt(coth(x) + 1)/(coth(x)**3 + 3*coth(x)**2 + 3*coth(x) + 1),x)