Integrand size = 13, antiderivative size = 34 \[ \int \coth ^2(x) \sqrt {1+\coth (x)} \, dx=\sqrt {2} \text {arctanh}\left (\frac {\sqrt {1+\coth (x)}}{\sqrt {2}}\right )-\frac {2}{3} (1+\coth (x))^{3/2} \] Output:
2^(1/2)*arctanh(1/2*(1+coth(x))^(1/2)*2^(1/2))-2/3*(1+coth(x))^(3/2)
Time = 0.54 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.09 \[ \int \coth ^2(x) \sqrt {1+\coth (x)} \, dx=-2 \left (-\frac {\text {arctanh}\left (\frac {\sqrt {1+\coth (x)}}{\sqrt {2}}\right )}{\sqrt {2}}+\frac {1}{3} (1+\coth (x))^{3/2}\right ) \] Input:
Integrate[Coth[x]^2*Sqrt[1 + Coth[x]],x]
Output:
-2*(-(ArcTanh[Sqrt[1 + Coth[x]]/Sqrt[2]]/Sqrt[2]) + (1 + Coth[x])^(3/2)/3)
Time = 0.27 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.538, Rules used = {3042, 25, 4026, 25, 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.
\(\displaystyle \int \coth ^2(x) \sqrt {\coth (x)+1} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int -\sqrt {1-i \tan \left (\frac {\pi }{2}+i x\right )} \tan \left (\frac {\pi }{2}+i x\right )^2dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\int \sqrt {1-i \tan \left (i x+\frac {\pi }{2}\right )} \tan \left (i x+\frac {\pi }{2}\right )^2dx\) |
\(\Big \downarrow \) 4026 |
\(\displaystyle -\int -\sqrt {\coth (x)+1}dx-\frac {2}{3} (\coth (x)+1)^{3/2}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \int \sqrt {\coth (x)+1}dx-\frac {2}{3} (\coth (x)+1)^{3/2}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {2}{3} (\coth (x)+1)^{3/2}+\int \sqrt {1-i \tan \left (i x+\frac {\pi }{2}\right )}dx\) |
\(\Big \downarrow \) 3961 |
\(\displaystyle 2 \int \frac {1}{1-\coth (x)}d\sqrt {\coth (x)+1}-\frac {2}{3} (\coth (x)+1)^{3/2}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \sqrt {2} \text {arctanh}\left (\frac {\sqrt {\coth (x)+1}}{\sqrt {2}}\right )-\frac {2}{3} (\coth (x)+1)^{3/2}\) |
Input:
Int[Coth[x]^2*Sqrt[1 + Coth[x]],x]
Output:
Sqrt[2]*ArcTanh[Sqrt[1 + Coth[x]]/Sqrt[2]] - (2*(1 + Coth[x])^(3/2))/3
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_)])^2, x_Symbol] :> Simp[d^2*((a + b*Tan[e + f*x])^(m + 1)/(b*f*( m + 1))), x] + Int[(a + b*Tan[e + f*x])^m*Simp[c^2 - d^2 + 2*c*d*Tan[e + f* x], x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b*c - a*d, 0] && !LeQ [m, -1] && !(EqQ[m, 2] && EqQ[a, 0])
Time = 0.05 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.76
method | result | size |
derivativedivides | \(\sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {1+\coth \left (x \right )}\, \sqrt {2}}{2}\right )-\frac {2 \left (1+\coth \left (x \right )\right )^{\frac {3}{2}}}{3}\) | \(26\) |
default | \(\sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {1+\coth \left (x \right )}\, \sqrt {2}}{2}\right )-\frac {2 \left (1+\coth \left (x \right )\right )^{\frac {3}{2}}}{3}\) | \(26\) |
Input:
int(coth(x)^2*(1+coth(x))^(1/2),x,method=_RETURNVERBOSE)
Output:
2^(1/2)*arctanh(1/2*(1+coth(x))^(1/2)*2^(1/2))-2/3*(1+coth(x))^(3/2)
Leaf count of result is larger than twice the leaf count of optimal. 198 vs. \(2 (25) = 50\).
Time = 0.11 (sec) , antiderivative size = 198, normalized size of antiderivative = 5.82 \[ \int \coth ^2(x) \sqrt {1+\coth (x)} \, dx=\frac {3 \, {\left (\sqrt {2} \cosh \left (x\right )^{2} + 2 \, \sqrt {2} \cosh \left (x\right ) \sinh \left (x\right ) + \sqrt {2} \sinh \left (x\right )^{2} - \sqrt {2}\right )} \log \left (2 \, \cosh \left (x\right )^{2} + 4 \, \cosh \left (x\right ) \sinh \left (x\right ) + 2 \, \sinh \left (x\right )^{2} + \frac {\sqrt {2} {\left (\sqrt {2} \cosh \left (x\right )^{3} + 3 \, \sqrt {2} \cosh \left (x\right ) \sinh \left (x\right )^{2} + \sqrt {2} \sinh \left (x\right )^{3} + {\left (3 \, \sqrt {2} \cosh \left (x\right )^{2} - \sqrt {2}\right )} \sinh \left (x\right ) - \sqrt {2} \cosh \left (x\right )\right )}}{\sqrt {\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2} - 1}} - 1\right ) - \frac {8 \, \sqrt {2} {\left (\cosh \left (x\right )^{3} + 3 \, \cosh \left (x\right )^{2} \sinh \left (x\right ) + 3 \, \cosh \left (x\right ) \sinh \left (x\right )^{2} + \sinh \left (x\right )^{3}\right )}}{\sqrt {\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2} - 1}}}{6 \, {\left (\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2} - 1\right )}} \] Input:
integrate(coth(x)^2*(1+coth(x))^(1/2),x, algorithm="fricas")
Output:
1/6*(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) - 8*sqrt(2)*(cosh(x)^3 + 3*cosh(x)^2*s inh(x) + 3*cosh(x)*sinh(x)^2 + sinh(x)^3)/sqrt(cosh(x)^2 + 2*cosh(x)*sinh( x) + sinh(x)^2 - 1))/(cosh(x)^2 + 2*cosh(x)*sinh(x) + sinh(x)^2 - 1)
\[ \int \coth ^2(x) \sqrt {1+\coth (x)} \, dx=\int \sqrt {\coth {\left (x \right )} + 1} \coth ^{2}{\left (x \right )}\, dx \] Input:
integrate(coth(x)**2*(1+coth(x))**(1/2),x)
Output:
Integral(sqrt(coth(x) + 1)*coth(x)**2, x)
\[ \int \coth ^2(x) \sqrt {1+\coth (x)} \, dx=\int { \sqrt {\coth \left (x\right ) + 1} \coth \left (x\right )^{2} \,d x } \] Input:
integrate(coth(x)^2*(1+coth(x))^(1/2),x, algorithm="maxima")
Output:
integrate(sqrt(coth(x) + 1)*coth(x)^2, x)
Leaf count of result is larger than twice the leaf count of optimal. 133 vs. \(2 (25) = 50\).
Time = 0.16 (sec) , antiderivative size = 133, normalized size of antiderivative = 3.91 \[ \int \coth ^2(x) \sqrt {1+\coth (x)} \, dx=-\frac {1}{6} \, \sqrt {2} {\left (3 \, \log \left ({\left | 2 \, \sqrt {e^{\left (4 \, x\right )} - e^{\left (2 \, x\right )}} - 2 \, e^{\left (2 \, x\right )} + 1 \right |}\right ) \mathrm {sgn}\left (e^{\left (2 \, x\right )} - 1\right ) + \frac {8 \, {\left (3 \, {\left (\sqrt {e^{\left (4 \, x\right )} - e^{\left (2 \, x\right )}} - e^{\left (2 \, x\right )}\right )}^{2} \mathrm {sgn}\left (e^{\left (2 \, x\right )} - 1\right ) + 3 \, {\left (\sqrt {e^{\left (4 \, x\right )} - e^{\left (2 \, x\right )}} - e^{\left (2 \, x\right )}\right )} \mathrm {sgn}\left (e^{\left (2 \, x\right )} - 1\right ) + \mathrm {sgn}\left (e^{\left (2 \, x\right )} - 1\right )\right )}}{{\left (\sqrt {e^{\left (4 \, x\right )} - e^{\left (2 \, x\right )}} - e^{\left (2 \, x\right )} + 1\right )}^{3}}\right )} \] Input:
integrate(coth(x)^2*(1+coth(x))^(1/2),x, algorithm="giac")
Output:
-1/6*sqrt(2)*(3*log(abs(2*sqrt(e^(4*x) - e^(2*x)) - 2*e^(2*x) + 1))*sgn(e^ (2*x) - 1) + 8*(3*(sqrt(e^(4*x) - e^(2*x)) - e^(2*x))^2*sgn(e^(2*x) - 1) + 3*(sqrt(e^(4*x) - e^(2*x)) - e^(2*x))*sgn(e^(2*x) - 1) + sgn(e^(2*x) - 1) )/(sqrt(e^(4*x) - e^(2*x)) - e^(2*x) + 1)^3)
Time = 2.40 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.74 \[ \int \coth ^2(x) \sqrt {1+\coth (x)} \, dx=\sqrt {2}\,\mathrm {atanh}\left (\frac {\sqrt {2}\,\sqrt {\mathrm {coth}\left (x\right )+1}}{2}\right )-\frac {2\,{\left (\mathrm {coth}\left (x\right )+1\right )}^{3/2}}{3} \] Input:
int(coth(x)^2*(coth(x) + 1)^(1/2),x)
Output:
2^(1/2)*atanh((2^(1/2)*(coth(x) + 1)^(1/2))/2) - (2*(coth(x) + 1)^(3/2))/3
\[ \int \coth ^2(x) \sqrt {1+\coth (x)} \, dx=\int \sqrt {\coth \left (x \right )+1}\, \coth \left (x \right )^{2}d x \] Input:
int(coth(x)^2*(1+coth(x))^(1/2),x)
Output:
int(sqrt(coth(x) + 1)*coth(x)**2,x)