Integrand size = 8, antiderivative size = 57 \[ \int (1+\coth (x))^{7/2} \, dx=8 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {1+\coth (x)}}{\sqrt {2}}\right )-8 \sqrt {1+\coth (x)}-\frac {4}{3} (1+\coth (x))^{3/2}-\frac {2}{5} (1+\coth (x))^{5/2} \]
-4/3*(1+coth(x))^(3/2)-2/5*(1+coth(x))^(5/2)+8*arctanh(1/2*(1+coth(x))^(1/ 2)*2^(1/2))*2^(1/2)-8*(1+coth(x))^(1/2)
Time = 0.85 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.82 \[ \int (1+\coth (x))^{7/2} \, dx=8 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {1+\coth (x)}}{\sqrt {2}}\right )-\frac {2}{15} \sqrt {1+\coth (x)} \left (73+16 \coth (x)+3 \coth ^2(x)\right ) \]
8*Sqrt[2]*ArcTanh[Sqrt[1 + Coth[x]]/Sqrt[2]] - (2*Sqrt[1 + Coth[x]]*(73 + 16*Coth[x] + 3*Coth[x]^2))/15
Time = 0.38 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.11, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.125, Rules used = {3042, 3959, 3042, 3959, 3042, 3959, 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.
\(\displaystyle \int (\coth (x)+1)^{7/2} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \left (1-i \tan \left (\frac {\pi }{2}+i x\right )\right )^{7/2}dx\) |
\(\Big \downarrow \) 3959 |
\(\displaystyle 2 \int (\coth (x)+1)^{5/2}dx-\frac {2}{5} (\coth (x)+1)^{5/2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {2}{5} (\coth (x)+1)^{5/2}+2 \int \left (1-i \tan \left (i x+\frac {\pi }{2}\right )\right )^{5/2}dx\) |
\(\Big \downarrow \) 3959 |
\(\displaystyle 2 \left (2 \int (\coth (x)+1)^{3/2}dx-\frac {2}{3} (\coth (x)+1)^{3/2}\right )-\frac {2}{5} (\coth (x)+1)^{5/2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {2}{5} (\coth (x)+1)^{5/2}+2 \left (-\frac {2}{3} (\coth (x)+1)^{3/2}+2 \int \left (1-i \tan \left (i x+\frac {\pi }{2}\right )\right )^{3/2}dx\right )\) |
\(\Big \downarrow \) 3959 |
\(\displaystyle 2 \left (2 \left (2 \int \sqrt {\coth (x)+1}dx-2 \sqrt {\coth (x)+1}\right )-\frac {2}{3} (\coth (x)+1)^{3/2}\right )-\frac {2}{5} (\coth (x)+1)^{5/2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {2}{5} (\coth (x)+1)^{5/2}+2 \left (-\frac {2}{3} (\coth (x)+1)^{3/2}+2 \left (-2 \sqrt {\coth (x)+1}+2 \int \sqrt {1-i \tan \left (i x+\frac {\pi }{2}\right )}dx\right )\right )\) |
\(\Big \downarrow \) 3961 |
\(\displaystyle 2 \left (2 \left (4 \int \frac {1}{1-\coth (x)}d\sqrt {\coth (x)+1}-2 \sqrt {\coth (x)+1}\right )-\frac {2}{3} (\coth (x)+1)^{3/2}\right )-\frac {2}{5} (\coth (x)+1)^{5/2}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle 2 \left (2 \left (2 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {\coth (x)+1}}{\sqrt {2}}\right )-2 \sqrt {\coth (x)+1}\right )-\frac {2}{3} (\coth (x)+1)^{3/2}\right )-\frac {2}{5} (\coth (x)+1)^{5/2}\) |
(-2*(1 + Coth[x])^(5/2))/5 + 2*((-2*(1 + Coth[x])^(3/2))/3 + 2*(2*Sqrt[2]* ArcTanh[Sqrt[1 + Coth[x]]/Sqrt[2]] - 2*Sqrt[1 + Coth[x]]))
3.1.70.3.1 Defintions of rubi rules used
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[b*((a + b*Tan[c + d*x])^(n - 1)/(d*(n - 1))), x] + Simp[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] && GtQ[n , 1]
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 = 0.12 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.75
method | result | size |
derivativedivides | \(-\frac {4 \left (1+\coth \left (x \right )\right )^{\frac {3}{2}}}{3}-\frac {2 \left (1+\coth \left (x \right )\right )^{\frac {5}{2}}}{5}+8 \,\operatorname {arctanh}\left (\frac {\sqrt {1+\coth \left (x \right )}\, \sqrt {2}}{2}\right ) \sqrt {2}-8 \sqrt {1+\coth \left (x \right )}\) | \(43\) |
default | \(-\frac {4 \left (1+\coth \left (x \right )\right )^{\frac {3}{2}}}{3}-\frac {2 \left (1+\coth \left (x \right )\right )^{\frac {5}{2}}}{5}+8 \,\operatorname {arctanh}\left (\frac {\sqrt {1+\coth \left (x \right )}\, \sqrt {2}}{2}\right ) \sqrt {2}-8 \sqrt {1+\coth \left (x \right )}\) | \(43\) |
-4/3*(1+coth(x))^(3/2)-2/5*(1+coth(x))^(5/2)+8*arctanh(1/2*(1+coth(x))^(1/ 2)*2^(1/2))*2^(1/2)-8*(1+coth(x))^(1/2)
Leaf count of result is larger than twice the leaf count of optimal. 438 vs. \(2 (42) = 84\).
Time = 0.26 (sec) , antiderivative size = 438, normalized size of antiderivative = 7.68 \[ \int (1+\coth (x))^{7/2} \, dx=-\frac {4 \, {\left (2 \, \sqrt {2} {\left (23 \, \sqrt {2} \cosh \left (x\right )^{5} + 115 \, \sqrt {2} \cosh \left (x\right ) \sinh \left (x\right )^{4} + 23 \, \sqrt {2} \sinh \left (x\right )^{5} + 5 \, {\left (46 \, \sqrt {2} \cosh \left (x\right )^{2} - 7 \, \sqrt {2}\right )} \sinh \left (x\right )^{3} - 35 \, \sqrt {2} \cosh \left (x\right )^{3} + 5 \, {\left (46 \, \sqrt {2} \cosh \left (x\right )^{3} - 21 \, \sqrt {2} \cosh \left (x\right )\right )} \sinh \left (x\right )^{2} + 5 \, {\left (23 \, \sqrt {2} \cosh \left (x\right )^{4} - 21 \, \sqrt {2} \cosh \left (x\right )^{2} + 3 \, \sqrt {2}\right )} \sinh \left (x\right ) + 15 \, \sqrt {2} \cosh \left (x\right )\right )} \sqrt {\frac {\sinh \left (x\right )}{\cosh \left (x\right ) - \sinh \left (x\right )}} - 15 \, {\left (\sqrt {2} \cosh \left (x\right )^{6} + 6 \, \sqrt {2} \cosh \left (x\right ) \sinh \left (x\right )^{5} + \sqrt {2} \sinh \left (x\right )^{6} + 3 \, {\left (5 \, \sqrt {2} \cosh \left (x\right )^{2} - \sqrt {2}\right )} \sinh \left (x\right )^{4} - 3 \, \sqrt {2} \cosh \left (x\right )^{4} + 4 \, {\left (5 \, \sqrt {2} \cosh \left (x\right )^{3} - 3 \, \sqrt {2} \cosh \left (x\right )\right )} \sinh \left (x\right )^{3} + 3 \, {\left (5 \, \sqrt {2} \cosh \left (x\right )^{4} - 6 \, \sqrt {2} \cosh \left (x\right )^{2} + \sqrt {2}\right )} \sinh \left (x\right )^{2} + 3 \, \sqrt {2} \cosh \left (x\right )^{2} + 6 \, {\left (\sqrt {2} \cosh \left (x\right )^{5} - 2 \, \sqrt {2} \cosh \left (x\right )^{3} + \sqrt {2} \cosh \left (x\right )\right )} \sinh \left (x\right ) - \sqrt {2}\right )} \log \left (2 \, \sqrt {2} \sqrt {\frac {\sinh \left (x\right )}{\cosh \left (x\right ) - \sinh \left (x\right )}} {\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )} + 2 \, \cosh \left (x\right )^{2} + 4 \, \cosh \left (x\right ) \sinh \left (x\right ) + 2 \, \sinh \left (x\right )^{2} - 1\right )\right )}}{15 \, {\left (\cosh \left (x\right )^{6} + 6 \, \cosh \left (x\right ) \sinh \left (x\right )^{5} + \sinh \left (x\right )^{6} + 3 \, {\left (5 \, \cosh \left (x\right )^{2} - 1\right )} \sinh \left (x\right )^{4} - 3 \, \cosh \left (x\right )^{4} + 4 \, {\left (5 \, \cosh \left (x\right )^{3} - 3 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{3} + 3 \, {\left (5 \, \cosh \left (x\right )^{4} - 6 \, \cosh \left (x\right )^{2} + 1\right )} \sinh \left (x\right )^{2} + 3 \, \cosh \left (x\right )^{2} + 6 \, {\left (\cosh \left (x\right )^{5} - 2 \, \cosh \left (x\right )^{3} + \cosh \left (x\right )\right )} \sinh \left (x\right ) - 1\right )}} \]
-4/15*(2*sqrt(2)*(23*sqrt(2)*cosh(x)^5 + 115*sqrt(2)*cosh(x)*sinh(x)^4 + 2 3*sqrt(2)*sinh(x)^5 + 5*(46*sqrt(2)*cosh(x)^2 - 7*sqrt(2))*sinh(x)^3 - 35* sqrt(2)*cosh(x)^3 + 5*(46*sqrt(2)*cosh(x)^3 - 21*sqrt(2)*cosh(x))*sinh(x)^ 2 + 5*(23*sqrt(2)*cosh(x)^4 - 21*sqrt(2)*cosh(x)^2 + 3*sqrt(2))*sinh(x) + 15*sqrt(2)*cosh(x))*sqrt(sinh(x)/(cosh(x) - sinh(x))) - 15*(sqrt(2)*cosh(x )^6 + 6*sqrt(2)*cosh(x)*sinh(x)^5 + sqrt(2)*sinh(x)^6 + 3*(5*sqrt(2)*cosh( x)^2 - sqrt(2))*sinh(x)^4 - 3*sqrt(2)*cosh(x)^4 + 4*(5*sqrt(2)*cosh(x)^3 - 3*sqrt(2)*cosh(x))*sinh(x)^3 + 3*(5*sqrt(2)*cosh(x)^4 - 6*sqrt(2)*cosh(x) ^2 + sqrt(2))*sinh(x)^2 + 3*sqrt(2)*cosh(x)^2 + 6*(sqrt(2)*cosh(x)^5 - 2*s qrt(2)*cosh(x)^3 + sqrt(2)*cosh(x))*sinh(x) - sqrt(2))*log(2*sqrt(2)*sqrt( sinh(x)/(cosh(x) - sinh(x)))*(cosh(x) + sinh(x)) + 2*cosh(x)^2 + 4*cosh(x) *sinh(x) + 2*sinh(x)^2 - 1))/(cosh(x)^6 + 6*cosh(x)*sinh(x)^5 + sinh(x)^6 + 3*(5*cosh(x)^2 - 1)*sinh(x)^4 - 3*cosh(x)^4 + 4*(5*cosh(x)^3 - 3*cosh(x) )*sinh(x)^3 + 3*(5*cosh(x)^4 - 6*cosh(x)^2 + 1)*sinh(x)^2 + 3*cosh(x)^2 + 6*(cosh(x)^5 - 2*cosh(x)^3 + cosh(x))*sinh(x) - 1)
Timed out. \[ \int (1+\coth (x))^{7/2} \, dx=\text {Timed out} \]
\[ \int (1+\coth (x))^{7/2} \, dx=\int { {\left (\coth \left (x\right ) + 1\right )}^{\frac {7}{2}} \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 160 vs. \(2 (42) = 84\).
Time = 0.29 (sec) , antiderivative size = 160, normalized size of antiderivative = 2.81 \[ \int (1+\coth (x))^{7/2} \, dx=-\frac {4}{15} \, \sqrt {2} {\left (\frac {2 \, {\left (45 \, {\left (\sqrt {e^{\left (4 \, x\right )} - e^{\left (2 \, x\right )}} - e^{\left (2 \, x\right )}\right )}^{4} + 135 \, {\left (\sqrt {e^{\left (4 \, x\right )} - e^{\left (2 \, x\right )}} - e^{\left (2 \, x\right )}\right )}^{3} + 170 \, {\left (\sqrt {e^{\left (4 \, x\right )} - e^{\left (2 \, x\right )}} - e^{\left (2 \, x\right )}\right )}^{2} + 100 \, \sqrt {e^{\left (4 \, x\right )} - e^{\left (2 \, x\right )}} - 100 \, e^{\left (2 \, x\right )} + 23\right )}}{{\left (\sqrt {e^{\left (4 \, x\right )} - e^{\left (2 \, x\right )}} - e^{\left (2 \, x\right )} + 1\right )}^{5}} + 15 \, \log \left ({\left | 2 \, \sqrt {e^{\left (4 \, x\right )} - e^{\left (2 \, x\right )}} - 2 \, e^{\left (2 \, x\right )} + 1 \right |}\right )\right )} \mathrm {sgn}\left (e^{\left (2 \, x\right )} - 1\right ) \]
-4/15*sqrt(2)*(2*(45*(sqrt(e^(4*x) - e^(2*x)) - e^(2*x))^4 + 135*(sqrt(e^( 4*x) - e^(2*x)) - e^(2*x))^3 + 170*(sqrt(e^(4*x) - e^(2*x)) - e^(2*x))^2 + 100*sqrt(e^(4*x) - e^(2*x)) - 100*e^(2*x) + 23)/(sqrt(e^(4*x) - e^(2*x)) - e^(2*x) + 1)^5 + 15*log(abs(2*sqrt(e^(4*x) - e^(2*x)) - 2*e^(2*x) + 1))) *sgn(e^(2*x) - 1)
Time = 1.94 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.77 \[ \int (1+\coth (x))^{7/2} \, dx=-8\,\sqrt {\mathrm {coth}\left (x\right )+1}-\frac {4\,{\left (\mathrm {coth}\left (x\right )+1\right )}^{3/2}}{3}-\frac {2\,{\left (\mathrm {coth}\left (x\right )+1\right )}^{5/2}}{5}-\sqrt {2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,\sqrt {\mathrm {coth}\left (x\right )+1}\,1{}\mathrm {i}}{2}\right )\,8{}\mathrm {i} \]