Integrand size = 8, antiderivative size = 45 \[ \int (1+\coth (x))^{5/2} \, dx=4 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {1+\coth (x)}}{\sqrt {2}}\right )-4 \sqrt {1+\coth (x)}-\frac {2}{3} (1+\coth (x))^{3/2} \] Output:
4*2^(1/2)*arctanh(1/2*(1+coth(x))^(1/2)*2^(1/2))-4*(1+coth(x))^(1/2)-2/3*( 1+coth(x))^(3/2)
Time = 0.46 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.87 \[ \int (1+\coth (x))^{5/2} \, dx=4 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {1+\coth (x)}}{\sqrt {2}}\right )-\frac {2}{3} \sqrt {1+\coth (x)} (7+\coth (x)) \] Input:
Integrate[(1 + Coth[x])^(5/2),x]
Output:
4*Sqrt[2]*ArcTanh[Sqrt[1 + Coth[x]]/Sqrt[2]] - (2*Sqrt[1 + Coth[x]]*(7 + C oth[x]))/3
Time = 0.31 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.07, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.875, Rules used = {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.
\(\displaystyle \int (\coth (x)+1)^{5/2} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \left (1-i \tan \left (\frac {\pi }{2}+i x\right )\right )^{5/2}dx\) |
\(\Big \downarrow \) 3959 |
\(\displaystyle 2 \int (\coth (x)+1)^{3/2}dx-\frac {2}{3} (\coth (x)+1)^{3/2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {2}{3} (\coth (x)+1)^{3/2}+2 \int \left (1-i \tan \left (i x+\frac {\pi }{2}\right )\right )^{3/2}dx\) |
\(\Big \downarrow \) 3959 |
\(\displaystyle 2 \left (2 \int \sqrt {\coth (x)+1}dx-2 \sqrt {\coth (x)+1}\right )-\frac {2}{3} (\coth (x)+1)^{3/2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {2}{3} (\coth (x)+1)^{3/2}+2 \left (-2 \sqrt {\coth (x)+1}+2 \int \sqrt {1-i \tan \left (i x+\frac {\pi }{2}\right )}dx\right )\) |
\(\Big \downarrow \) 3961 |
\(\displaystyle 2 \left (4 \int \frac {1}{1-\coth (x)}d\sqrt {\coth (x)+1}-2 \sqrt {\coth (x)+1}\right )-\frac {2}{3} (\coth (x)+1)^{3/2}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle 2 \left (2 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {\coth (x)+1}}{\sqrt {2}}\right )-2 \sqrt {\coth (x)+1}\right )-\frac {2}{3} (\coth (x)+1)^{3/2}\) |
Input:
Int[(1 + Coth[x])^(5/2),x]
Output:
(-2*(1 + Coth[x])^(3/2))/3 + 2*(2*Sqrt[2]*ArcTanh[Sqrt[1 + Coth[x]]/Sqrt[2 ]] - 2*Sqrt[1 + Coth[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[((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 = 0.05 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.78
method | result | size |
derivativedivides | \(4 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {1+\coth \left (x \right )}\, \sqrt {2}}{2}\right )-4 \sqrt {1+\coth \left (x \right )}-\frac {2 \left (1+\coth \left (x \right )\right )^{\frac {3}{2}}}{3}\) | \(35\) |
default | \(4 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {1+\coth \left (x \right )}\, \sqrt {2}}{2}\right )-4 \sqrt {1+\coth \left (x \right )}-\frac {2 \left (1+\coth \left (x \right )\right )^{\frac {3}{2}}}{3}\) | \(35\) |
Input:
int((1+coth(x))^(5/2),x,method=_RETURNVERBOSE)
Output:
4*2^(1/2)*arctanh(1/2*(1+coth(x))^(1/2)*2^(1/2))-4*(1+coth(x))^(1/2)-2/3*( 1+coth(x))^(3/2)
Leaf count of result is larger than twice the leaf count of optimal. 210 vs. \(2 (34) = 68\).
Time = 0.09 (sec) , antiderivative size = 210, normalized size of antiderivative = 4.67 \[ \int (1+\coth (x))^{5/2} \, dx=\frac {2 \, {\left (3 \, {\left (\sqrt {2} \cosh \left (x\right )^{2} + 2 \, \sqrt {2} \cosh \left (x\right ) \sinh \left (x\right ) + \sqrt {2} \sinh \left (x\right )^{2} - \sqrt {2}\right )} \log \left (2 \, \cosh \left (x\right )^{2} + 4 \, \cosh \left (x\right ) \sinh \left (x\right ) + 2 \, \sinh \left (x\right )^{2} + \frac {\sqrt {2} {\left (\sqrt {2} \cosh \left (x\right )^{3} + 3 \, \sqrt {2} \cosh \left (x\right ) \sinh \left (x\right )^{2} + \sqrt {2} \sinh \left (x\right )^{3} + {\left (3 \, \sqrt {2} \cosh \left (x\right )^{2} - \sqrt {2}\right )} \sinh \left (x\right ) - \sqrt {2} \cosh \left (x\right )\right )}}{\sqrt {\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2} - 1}} - 1\right ) - \frac {2 \, \sqrt {2} {\left (4 \, \cosh \left (x\right )^{3} + 12 \, \cosh \left (x\right ) \sinh \left (x\right )^{2} + 4 \, \sinh \left (x\right )^{3} + 3 \, {\left (4 \, \cosh \left (x\right )^{2} - 1\right )} \sinh \left (x\right ) - 3 \, \cosh \left (x\right )\right )}}{\sqrt {\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2} - 1}}\right )}}{3 \, {\left (\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2} - 1\right )}} \] Input:
integrate((1+coth(x))^(5/2),x, algorithm="fricas")
Output:
2/3*(3*(sqrt(2)*cosh(x)^2 + 2*sqrt(2)*cosh(x)*sinh(x) + sqrt(2)*sinh(x)^2 - sqrt(2))*log(2*cosh(x)^2 + 4*cosh(x)*sinh(x) + 2*sinh(x)^2 + sqrt(2)*(sq rt(2)*cosh(x)^3 + 3*sqrt(2)*cosh(x)*sinh(x)^2 + sqrt(2)*sinh(x)^3 + (3*sqr t(2)*cosh(x)^2 - sqrt(2))*sinh(x) - sqrt(2)*cosh(x))/sqrt(cosh(x)^2 + 2*co sh(x)*sinh(x) + sinh(x)^2 - 1) - 1) - 2*sqrt(2)*(4*cosh(x)^3 + 12*cosh(x)* sinh(x)^2 + 4*sinh(x)^3 + 3*(4*cosh(x)^2 - 1)*sinh(x) - 3*cosh(x))/sqrt(co sh(x)^2 + 2*cosh(x)*sinh(x) + sinh(x)^2 - 1))/(cosh(x)^2 + 2*cosh(x)*sinh( x) + sinh(x)^2 - 1)
\[ \int (1+\coth (x))^{5/2} \, dx=\int \left (\coth {\left (x \right )} + 1\right )^{\frac {5}{2}}\, dx \] Input:
integrate((1+coth(x))**(5/2),x)
Output:
Integral((coth(x) + 1)**(5/2), x)
\[ \int (1+\coth (x))^{5/2} \, dx=\int { {\left (\coth \left (x\right ) + 1\right )}^{\frac {5}{2}} \,d x } \] Input:
integrate((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. 112 vs. \(2 (34) = 68\).
Time = 0.12 (sec) , antiderivative size = 112, normalized size of antiderivative = 2.49 \[ \int (1+\coth (x))^{5/2} \, dx=-\frac {2}{3} \, \sqrt {2} {\left (\frac {2 \, {\left (6 \, {\left (\sqrt {e^{\left (4 \, x\right )} - e^{\left (2 \, x\right )}} - e^{\left (2 \, x\right )}\right )}^{2} + 9 \, \sqrt {e^{\left (4 \, x\right )} - e^{\left (2 \, x\right )}} - 9 \, e^{\left (2 \, x\right )} + 4\right )}}{{\left (\sqrt {e^{\left (4 \, x\right )} - e^{\left (2 \, x\right )}} - e^{\left (2 \, x\right )} + 1\right )}^{3}} + 3 \, \log \left ({\left | 2 \, \sqrt {e^{\left (4 \, x\right )} - e^{\left (2 \, x\right )}} - 2 \, e^{\left (2 \, x\right )} + 1 \right |}\right )\right )} \mathrm {sgn}\left (e^{\left (2 \, x\right )} - 1\right ) \] Input:
integrate((1+coth(x))^(5/2),x, algorithm="giac")
Output:
-2/3*sqrt(2)*(2*(6*(sqrt(e^(4*x) - e^(2*x)) - e^(2*x))^2 + 9*sqrt(e^(4*x) - e^(2*x)) - 9*e^(2*x) + 4)/(sqrt(e^(4*x) - e^(2*x)) - e^(2*x) + 1)^3 + 3* log(abs(2*sqrt(e^(4*x) - e^(2*x)) - 2*e^(2*x) + 1)))*sgn(e^(2*x) - 1)
Time = 2.36 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.20 \[ \int (1+\coth (x))^{5/2} \, dx=\sqrt {8}\,\ln \left (-2\,\sqrt {8}\,\sqrt {\mathrm {coth}\left (x\right )+1}-8\right )-\frac {2\,{\left (\mathrm {coth}\left (x\right )+1\right )}^{3/2}}{3}-2\,\sqrt {2}\,\ln \left (4\,\sqrt {2}\,\sqrt {\mathrm {coth}\left (x\right )+1}-8\right )-4\,\sqrt {\mathrm {coth}\left (x\right )+1} \] Input:
int((coth(x) + 1)^(5/2),x)
Output:
8^(1/2)*log(- 2*8^(1/2)*(coth(x) + 1)^(1/2) - 8) - (2*(coth(x) + 1)^(3/2)) /3 - 2*2^(1/2)*log(4*2^(1/2)*(coth(x) + 1)^(1/2) - 8) - 4*(coth(x) + 1)^(1 /2)
\[ \int (1+\coth (x))^{5/2} \, dx=\int \sqrt {\coth \left (x \right )+1}d x +2 \left (\int \sqrt {\coth \left (x \right )+1}\, \coth \left (x \right )d x \right )+\int \sqrt {\coth \left (x \right )+1}\, \coth \left (x \right )^{2}d x \] Input:
int((1+coth(x))^(5/2),x)
Output:
int(sqrt(coth(x) + 1),x) + 2*int(sqrt(coth(x) + 1)*coth(x),x) + int(sqrt(c oth(x) + 1)*coth(x)**2,x)