Integrand size = 13, antiderivative size = 49 \[ \int \frac {\coth ^2(x)}{(1+\coth (x))^{3/2}} \, dx=\frac {\text {arctanh}\left (\frac {\sqrt {1+\coth (x)}}{\sqrt {2}}\right )}{2 \sqrt {2}}-\frac {1}{3 (1+\coth (x))^{3/2}}+\frac {3}{2 \sqrt {1+\coth (x)}} \] Output:
1/4*2^(1/2)*arctanh(1/2*(1+coth(x))^(1/2)*2^(1/2))-1/3/(1+coth(x))^(3/2)+3 /2/(1+coth(x))^(1/2)
Time = 0.56 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.98 \[ \int \frac {\coth ^2(x)}{(1+\coth (x))^{3/2}} \, dx=\frac {14+18 \coth (x)+3 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {1+\coth (x)}}{\sqrt {2}}\right ) (1+\coth (x))^{3/2}}{12 (1+\coth (x))^{3/2}} \] Input:
Integrate[Coth[x]^2/(1 + Coth[x])^(3/2),x]
Output:
(14 + 18*Coth[x] + 3*Sqrt[2]*ArcTanh[Sqrt[1 + Coth[x]]/Sqrt[2]]*(1 + Coth[ x])^(3/2))/(12*(1 + Coth[x])^(3/2))
Time = 0.38 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.615, Rules used = {3042, 25, 4023, 3042, 4009, 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.
\(\displaystyle \int \frac {\coth ^2(x)}{(\coth (x)+1)^{3/2}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int -\frac {\tan \left (\frac {\pi }{2}+i x\right )^2}{\left (1-i \tan \left (\frac {\pi }{2}+i x\right )\right )^{3/2}}dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\int \frac {\tan \left (i x+\frac {\pi }{2}\right )^2}{\left (1-i \tan \left (i x+\frac {\pi }{2}\right )\right )^{3/2}}dx\) |
\(\Big \downarrow \) 4023 |
\(\displaystyle -\frac {1}{2} \int \frac {1-2 \coth (x)}{\sqrt {\coth (x)+1}}dx-\frac {1}{3 (\coth (x)+1)^{3/2}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {1}{3 (\coth (x)+1)^{3/2}}-\frac {1}{2} \int \frac {2 i \tan \left (i x+\frac {\pi }{2}\right )+1}{\sqrt {1-i \tan \left (i x+\frac {\pi }{2}\right )}}dx\) |
\(\Big \downarrow \) 4009 |
\(\displaystyle \frac {1}{2} \left (\frac {1}{2} \int \sqrt {\coth (x)+1}dx+\frac {3}{\sqrt {\coth (x)+1}}\right )-\frac {1}{3 (\coth (x)+1)^{3/2}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {1}{3 (\coth (x)+1)^{3/2}}+\frac {1}{2} \left (\frac {3}{\sqrt {\coth (x)+1}}+\frac {1}{2} \int \sqrt {1-i \tan \left (i x+\frac {\pi }{2}\right )}dx\right )\) |
\(\Big \downarrow \) 3961 |
\(\displaystyle \frac {1}{2} \left (\int \frac {1}{1-\coth (x)}d\sqrt {\coth (x)+1}+\frac {3}{\sqrt {\coth (x)+1}}\right )-\frac {1}{3 (\coth (x)+1)^{3/2}}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {1}{2} \left (\frac {\text {arctanh}\left (\frac {\sqrt {\coth (x)+1}}{\sqrt {2}}\right )}{\sqrt {2}}+\frac {3}{\sqrt {\coth (x)+1}}\right )-\frac {1}{3 (\coth (x)+1)^{3/2}}\) |
Input:
Int[Coth[x]^2/(1 + Coth[x])^(3/2),x]
Output:
-1/3*1/(1 + Coth[x])^(3/2) + (ArcTanh[Sqrt[1 + Coth[x]]/Sqrt[2]]/Sqrt[2] + 3/Sqrt[1 + Coth[x]])/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[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]
Int[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(-(b*c - a*d))*((a + b*Tan[e + f*x])^m/(2*a *f*m)), x] + Simp[(b*c + a*d)/(2*a*b) Int[(a + b*Tan[e + f*x])^(m + 1), x ], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2 , 0] && LtQ[m, 0]
Int[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^2, x_Symbol] :> Simp[(-b)*(a*c + b*d)^2*((a + b*Tan[e + f*x])^ m/(2*a^3*f*m)), x] + Simp[1/(2*a^2) Int[(a + b*Tan[e + f*x])^(m + 1)*Simp [a*c^2 - 2*b*c*d + a*d^2 - 2*b*d^2*Tan[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && LeQ[m, -1] && EqQ[a^2 + b^2, 0]
Time = 0.05 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.71
method | result | size |
derivativedivides | \(\frac {\sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {1+\coth \left (x \right )}\, \sqrt {2}}{2}\right )}{4}-\frac {1}{3 \left (1+\coth \left (x \right )\right )^{\frac {3}{2}}}+\frac {3}{2 \sqrt {1+\coth \left (x \right )}}\) | \(35\) |
default | \(\frac {\sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {1+\coth \left (x \right )}\, \sqrt {2}}{2}\right )}{4}-\frac {1}{3 \left (1+\coth \left (x \right )\right )^{\frac {3}{2}}}+\frac {3}{2 \sqrt {1+\coth \left (x \right )}}\) | \(35\) |
Input:
int(coth(x)^2/(1+coth(x))^(3/2),x,method=_RETURNVERBOSE)
Output:
1/4*2^(1/2)*arctanh(1/2*(1+coth(x))^(1/2)*2^(1/2))-1/3/(1+coth(x))^(3/2)+3 /2/(1+coth(x))^(1/2)
Leaf count of result is larger than twice the leaf count of optimal. 246 vs. \(2 (34) = 68\).
Time = 0.09 (sec) , antiderivative size = 246, normalized size of antiderivative = 5.02 \[ \int \frac {\coth ^2(x)}{(1+\coth (x))^{3/2}} \, dx=\frac {3 \, {\left (\sqrt {2} \cosh \left (x\right )^{3} + 3 \, \sqrt {2} \cosh \left (x\right )^{2} \sinh \left (x\right ) + 3 \, \sqrt {2} \cosh \left (x\right ) \sinh \left (x\right )^{2} + \sqrt {2} \sinh \left (x\right )^{3}\right )} \log \left (2 \, \cosh \left (x\right )^{2} + 4 \, \cosh \left (x\right ) \sinh \left (x\right ) + 2 \, \sinh \left (x\right )^{2} + \frac {\sqrt {2} {\left (\sqrt {2} \cosh \left (x\right )^{3} + 3 \, \sqrt {2} \cosh \left (x\right ) \sinh \left (x\right )^{2} + \sqrt {2} \sinh \left (x\right )^{3} + {\left (3 \, \sqrt {2} \cosh \left (x\right )^{2} - \sqrt {2}\right )} \sinh \left (x\right ) - \sqrt {2} \cosh \left (x\right )\right )}}{\sqrt {\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2} - 1}} - 1\right ) + \frac {2 \, \sqrt {2} {\left (8 \, \cosh \left (x\right )^{4} + 32 \, \cosh \left (x\right ) \sinh \left (x\right )^{3} + 8 \, \sinh \left (x\right )^{4} + {\left (48 \, \cosh \left (x\right )^{2} - 7\right )} \sinh \left (x\right )^{2} - 7 \, \cosh \left (x\right )^{2} + 2 \, {\left (16 \, \cosh \left (x\right )^{3} - 7 \, \cosh \left (x\right )\right )} \sinh \left (x\right ) - 1\right )}}{\sqrt {\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2} - 1}}}{24 \, {\left (\cosh \left (x\right )^{3} + 3 \, \cosh \left (x\right )^{2} \sinh \left (x\right ) + 3 \, \cosh \left (x\right ) \sinh \left (x\right )^{2} + \sinh \left (x\right )^{3}\right )}} \] Input:
integrate(coth(x)^2/(1+coth(x))^(3/2),x, algorithm="fricas")
Output:
1/24*(3*(sqrt(2)*cosh(x)^3 + 3*sqrt(2)*cosh(x)^2*sinh(x) + 3*sqrt(2)*cosh( x)*sinh(x)^2 + sqrt(2)*sinh(x)^3)*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 + sqr t(2)*sinh(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)*(8* cosh(x)^4 + 32*cosh(x)*sinh(x)^3 + 8*sinh(x)^4 + (48*cosh(x)^2 - 7)*sinh(x )^2 - 7*cosh(x)^2 + 2*(16*cosh(x)^3 - 7*cosh(x))*sinh(x) - 1)/sqrt(cosh(x) ^2 + 2*cosh(x)*sinh(x) + sinh(x)^2 - 1))/(cosh(x)^3 + 3*cosh(x)^2*sinh(x) + 3*cosh(x)*sinh(x)^2 + sinh(x)^3)
\[ \int \frac {\coth ^2(x)}{(1+\coth (x))^{3/2}} \, dx=\int \frac {\coth ^{2}{\left (x \right )}}{\left (\coth {\left (x \right )} + 1\right )^{\frac {3}{2}}}\, dx \] Input:
integrate(coth(x)**2/(1+coth(x))**(3/2),x)
Output:
Integral(coth(x)**2/(coth(x) + 1)**(3/2), x)
\[ \int \frac {\coth ^2(x)}{(1+\coth (x))^{3/2}} \, dx=\int { \frac {\coth \left (x\right )^{2}}{{\left (\coth \left (x\right ) + 1\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate(coth(x)^2/(1+coth(x))^(3/2),x, algorithm="maxima")
Output:
integrate(coth(x)^2/(coth(x) + 1)^(3/2), x)
Leaf count of result is larger than twice the leaf count of optimal. 113 vs. \(2 (34) = 68\).
Time = 0.15 (sec) , antiderivative size = 113, normalized size of antiderivative = 2.31 \[ \int \frac {\coth ^2(x)}{(1+\coth (x))^{3/2}} \, dx=-\frac {\sqrt {2} {\left (\frac {2 \, {\left (6 \, {\left (\sqrt {e^{\left (4 \, x\right )} - e^{\left (2 \, x\right )}} - e^{\left (2 \, x\right )}\right )}^{2} - 3 \, \sqrt {e^{\left (4 \, x\right )} - e^{\left (2 \, x\right )}} + 3 \, e^{\left (2 \, x\right )} - 1\right )}}{{\left (\sqrt {e^{\left (4 \, x\right )} - e^{\left (2 \, x\right )}} - e^{\left (2 \, x\right )}\right )}^{3}} + 3 \, \log \left ({\left | 2 \, \sqrt {e^{\left (4 \, x\right )} - e^{\left (2 \, x\right )}} - 2 \, e^{\left (2 \, x\right )} + 1 \right |}\right )\right )}}{24 \, \mathrm {sgn}\left (e^{\left (2 \, x\right )} - 1\right )} \] Input:
integrate(coth(x)^2/(1+coth(x))^(3/2),x, algorithm="giac")
Output:
-1/24*sqrt(2)*(2*(6*(sqrt(e^(4*x) - e^(2*x)) - e^(2*x))^2 - 3*sqrt(e^(4*x) - e^(2*x)) + 3*e^(2*x) - 1)/(sqrt(e^(4*x) - e^(2*x)) - e^(2*x))^3 + 3*log (abs(2*sqrt(e^(4*x) - e^(2*x)) - 2*e^(2*x) + 1)))/sgn(e^(2*x) - 1)
Time = 2.43 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.63 \[ \int \frac {\coth ^2(x)}{(1+\coth (x))^{3/2}} \, dx=\frac {\sqrt {2}\,\mathrm {atanh}\left (\frac {\sqrt {2}\,\sqrt {\mathrm {coth}\left (x\right )+1}}{2}\right )}{4}+\frac {\frac {3\,\mathrm {coth}\left (x\right )}{2}+\frac {7}{6}}{{\left (\mathrm {coth}\left (x\right )+1\right )}^{3/2}} \] Input:
int(coth(x)^2/(coth(x) + 1)^(3/2),x)
Output:
(2^(1/2)*atanh((2^(1/2)*(coth(x) + 1)^(1/2))/2))/4 + ((3*coth(x))/2 + 7/6) /(coth(x) + 1)^(3/2)
\[ \int \frac {\coth ^2(x)}{(1+\coth (x))^{3/2}} \, dx=\int \frac {\sqrt {\coth \left (x \right )+1}\, \coth \left (x \right )^{2}}{\coth \left (x \right )^{2}+2 \coth \left (x \right )+1}d x \] Input:
int(coth(x)^2/(1+coth(x))^(3/2),x)
Output:
int((sqrt(coth(x) + 1)*coth(x)**2)/(coth(x)**2 + 2*coth(x) + 1),x)