3.136 \(\int \coth ^2(x) (1+\coth (x))^{3/2} \, dx\)

Optimal. Leaf size=45 \[ -\frac {2}{5} (\coth (x)+1)^{5/2}-2 \sqrt {\coth (x)+1}+2 \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {\coth (x)+1}}{\sqrt {2}}\right ) \]

[Out]

-2/5*(1+coth(x))^(5/2)+2*arctanh(1/2*(1+coth(x))^(1/2)*2^(1/2))*2^(1/2)-2*(1+coth(x))^(1/2)

________________________________________________________________________________________

Rubi [A]  time = 0.06, antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {3543, 3478, 3480, 206} \[ -\frac {2}{5} (\coth (x)+1)^{5/2}-2 \sqrt {\coth (x)+1}+2 \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {\coth (x)+1}}{\sqrt {2}}\right ) \]

Antiderivative was successfully verified.

[In]

Int[Coth[x]^2*(1 + Coth[x])^(3/2),x]

[Out]

2*Sqrt[2]*ArcTanh[Sqrt[1 + Coth[x]]/Sqrt[2]] - 2*Sqrt[1 + Coth[x]] - (2*(1 + Coth[x])^(5/2))/5

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 3478

Int[((a_) + (b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*(a + b*Tan[c + d*x])^(n - 1))/(d*(n - 1)
), x] + Dist[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] && G
tQ[n, 1]

Rule 3480

Int[Sqrt[(a_) + (b_.)*tan[(c_.) + (d_.)*(x_)]], x_Symbol] :> Dist[(-2*b)/d, Subst[Int[1/(2*a - x^2), x], x, Sq
rt[a + b*Tan[c + d*x]]], x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 + b^2, 0]

Rule 3543

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])

Rubi steps

\begin {align*} \int \coth ^2(x) (1+\coth (x))^{3/2} \, dx &=-\frac {2}{5} (1+\coth (x))^{5/2}+\int (1+\coth (x))^{3/2} \, dx\\ &=-2 \sqrt {1+\coth (x)}-\frac {2}{5} (1+\coth (x))^{5/2}+2 \int \sqrt {1+\coth (x)} \, dx\\ &=-2 \sqrt {1+\coth (x)}-\frac {2}{5} (1+\coth (x))^{5/2}+4 \operatorname {Subst}\left (\int \frac {1}{2-x^2} \, dx,x,\sqrt {1+\coth (x)}\right )\\ &=2 \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {1+\coth (x)}}{\sqrt {2}}\right )-2 \sqrt {1+\coth (x)}-\frac {2}{5} (1+\coth (x))^{5/2}\\ \end {align*}

________________________________________________________________________________________

Mathematica [C]  time = 0.26, size = 70, normalized size = 1.56 \[ -\frac {2 \left (2 \coth ^2(x)+\text {csch}^2(x)+(5+5 i) \sqrt {i (\coth (x)+1)} \tan ^{-1}\left (\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {i (\coth (x)+1)}\right )+\coth (x) \left (\text {csch}^2(x)+9\right )+7\right )}{5 \sqrt {\coth (x)+1}} \]

Antiderivative was successfully verified.

[In]

Integrate[Coth[x]^2*(1 + Coth[x])^(3/2),x]

[Out]

(-2*(7 + 2*Coth[x]^2 + (5 + 5*I)*ArcTan[(1/2 + I/2)*Sqrt[I*(1 + Coth[x])]]*Sqrt[I*(1 + Coth[x])] + Csch[x]^2 +
Coth[x]*(9 + Csch[x]^2)))/(5*Sqrt[1 + Coth[x]])

________________________________________________________________________________________

fricas [B]  time = 0.40, size = 436, normalized size = 9.69 \[ -\frac {2 \, \sqrt {2} {\left (9 \, \sqrt {2} \cosh \relax (x)^{5} + 45 \, \sqrt {2} \cosh \relax (x) \sinh \relax (x)^{4} + 9 \, \sqrt {2} \sinh \relax (x)^{5} + 10 \, {\left (9 \, \sqrt {2} \cosh \relax (x)^{2} - \sqrt {2}\right )} \sinh \relax (x)^{3} - 10 \, \sqrt {2} \cosh \relax (x)^{3} + 30 \, {\left (3 \, \sqrt {2} \cosh \relax (x)^{3} - \sqrt {2} \cosh \relax (x)\right )} \sinh \relax (x)^{2} + 5 \, {\left (9 \, \sqrt {2} \cosh \relax (x)^{4} - 6 \, \sqrt {2} \cosh \relax (x)^{2} + \sqrt {2}\right )} \sinh \relax (x) + 5 \, \sqrt {2} \cosh \relax (x)\right )} \sqrt {\frac {\sinh \relax (x)}{\cosh \relax (x) - \sinh \relax (x)}} - 5 \, {\left (\sqrt {2} \cosh \relax (x)^{6} + 6 \, \sqrt {2} \cosh \relax (x) \sinh \relax (x)^{5} + \sqrt {2} \sinh \relax (x)^{6} + 3 \, {\left (5 \, \sqrt {2} \cosh \relax (x)^{2} - \sqrt {2}\right )} \sinh \relax (x)^{4} - 3 \, \sqrt {2} \cosh \relax (x)^{4} + 4 \, {\left (5 \, \sqrt {2} \cosh \relax (x)^{3} - 3 \, \sqrt {2} \cosh \relax (x)\right )} \sinh \relax (x)^{3} + 3 \, {\left (5 \, \sqrt {2} \cosh \relax (x)^{4} - 6 \, \sqrt {2} \cosh \relax (x)^{2} + \sqrt {2}\right )} \sinh \relax (x)^{2} + 3 \, \sqrt {2} \cosh \relax (x)^{2} + 6 \, {\left (\sqrt {2} \cosh \relax (x)^{5} - 2 \, \sqrt {2} \cosh \relax (x)^{3} + \sqrt {2} \cosh \relax (x)\right )} \sinh \relax (x) - \sqrt {2}\right )} \log \left (2 \, \sqrt {2} \sqrt {\frac {\sinh \relax (x)}{\cosh \relax (x) - \sinh \relax (x)}} {\left (\cosh \relax (x) + \sinh \relax (x)\right )} + 2 \, \cosh \relax (x)^{2} + 4 \, \cosh \relax (x) \sinh \relax (x) + 2 \, \sinh \relax (x)^{2} - 1\right )}{5 \, {\left (\cosh \relax (x)^{6} + 6 \, \cosh \relax (x) \sinh \relax (x)^{5} + \sinh \relax (x)^{6} + 3 \, {\left (5 \, \cosh \relax (x)^{2} - 1\right )} \sinh \relax (x)^{4} - 3 \, \cosh \relax (x)^{4} + 4 \, {\left (5 \, \cosh \relax (x)^{3} - 3 \, \cosh \relax (x)\right )} \sinh \relax (x)^{3} + 3 \, {\left (5 \, \cosh \relax (x)^{4} - 6 \, \cosh \relax (x)^{2} + 1\right )} \sinh \relax (x)^{2} + 3 \, \cosh \relax (x)^{2} + 6 \, {\left (\cosh \relax (x)^{5} - 2 \, \cosh \relax (x)^{3} + \cosh \relax (x)\right )} \sinh \relax (x) - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(x)^2*(1+coth(x))^(3/2),x, algorithm="fricas")

[Out]

-1/5*(2*sqrt(2)*(9*sqrt(2)*cosh(x)^5 + 45*sqrt(2)*cosh(x)*sinh(x)^4 + 9*sqrt(2)*sinh(x)^5 + 10*(9*sqrt(2)*cosh
(x)^2 - sqrt(2))*sinh(x)^3 - 10*sqrt(2)*cosh(x)^3 + 30*(3*sqrt(2)*cosh(x)^3 - sqrt(2)*cosh(x))*sinh(x)^2 + 5*(
9*sqrt(2)*cosh(x)^4 - 6*sqrt(2)*cosh(x)^2 + sqrt(2))*sinh(x) + 5*sqrt(2)*cosh(x))*sqrt(sinh(x)/(cosh(x) - sinh
(x))) - 5*(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)*co
sh(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*sqrt(2)*co
sh(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)

________________________________________________________________________________________

giac [B]  time = 0.17, size = 197, normalized size = 4.38 \[ -\frac {1}{5} \, \sqrt {2} {\left (5 \, \log \left ({\left | 2 \, \sqrt {e^{\left (4 \, x\right )} - e^{\left (2 \, x\right )}} - 2 \, e^{\left (2 \, x\right )} + 1 \right |}\right ) \mathrm {sgn}\left (e^{\left (2 \, x\right )} - 1\right ) + \frac {2 \, {\left (25 \, {\left (\sqrt {e^{\left (4 \, x\right )} - e^{\left (2 \, x\right )}} - e^{\left (2 \, x\right )}\right )}^{4} \mathrm {sgn}\left (e^{\left (2 \, x\right )} - 1\right ) + 60 \, {\left (\sqrt {e^{\left (4 \, x\right )} - e^{\left (2 \, x\right )}} - e^{\left (2 \, x\right )}\right )}^{3} \mathrm {sgn}\left (e^{\left (2 \, x\right )} - 1\right ) + 70 \, {\left (\sqrt {e^{\left (4 \, x\right )} - e^{\left (2 \, x\right )}} - e^{\left (2 \, x\right )}\right )}^{2} \mathrm {sgn}\left (e^{\left (2 \, x\right )} - 1\right ) + 40 \, {\left (\sqrt {e^{\left (4 \, x\right )} - e^{\left (2 \, x\right )}} - e^{\left (2 \, x\right )}\right )} \mathrm {sgn}\left (e^{\left (2 \, x\right )} - 1\right ) + 9 \, \mathrm {sgn}\left (e^{\left (2 \, x\right )} - 1\right )\right )}}{{\left (\sqrt {e^{\left (4 \, x\right )} - e^{\left (2 \, x\right )}} - e^{\left (2 \, x\right )} + 1\right )}^{5}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(x)^2*(1+coth(x))^(3/2),x, algorithm="giac")

[Out]

-1/5*sqrt(2)*(5*log(abs(2*sqrt(e^(4*x) - e^(2*x)) - 2*e^(2*x) + 1))*sgn(e^(2*x) - 1) + 2*(25*(sqrt(e^(4*x) - e
^(2*x)) - e^(2*x))^4*sgn(e^(2*x) - 1) + 60*(sqrt(e^(4*x) - e^(2*x)) - e^(2*x))^3*sgn(e^(2*x) - 1) + 70*(sqrt(e
^(4*x) - e^(2*x)) - e^(2*x))^2*sgn(e^(2*x) - 1) + 40*(sqrt(e^(4*x) - e^(2*x)) - e^(2*x))*sgn(e^(2*x) - 1) + 9*
sgn(e^(2*x) - 1))/(sqrt(e^(4*x) - e^(2*x)) - e^(2*x) + 1)^5)

________________________________________________________________________________________

maple [A]  time = 0.07, size = 35, normalized size = 0.78 \[ -\frac {2 \left (1+\coth \relax (x )\right )^{\frac {5}{2}}}{5}+2 \arctanh \left (\frac {\sqrt {1+\coth \relax (x )}\, \sqrt {2}}{2}\right ) \sqrt {2}-2 \sqrt {1+\coth \relax (x )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(coth(x)^2*(1+coth(x))^(3/2),x)

[Out]

-2/5*(1+coth(x))^(5/2)+2*arctanh(1/2*(1+coth(x))^(1/2)*2^(1/2))*2^(1/2)-2*(1+coth(x))^(1/2)

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (\coth \relax (x) + 1\right )}^{\frac {3}{2}} \coth \relax (x)^{2}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(x)^2*(1+coth(x))^(3/2),x, algorithm="maxima")

[Out]

integrate((coth(x) + 1)^(3/2)*coth(x)^2, x)

________________________________________________________________________________________

mupad [B]  time = 1.25, size = 34, normalized size = 0.76 \[ 2\,\sqrt {2}\,\mathrm {atanh}\left (\frac {\sqrt {2}\,\sqrt {\mathrm {coth}\relax (x)+1}}{2}\right )-2\,\sqrt {\mathrm {coth}\relax (x)+1}-\frac {2\,{\left (\mathrm {coth}\relax (x)+1\right )}^{5/2}}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(coth(x)^2*(coth(x) + 1)^(3/2),x)

[Out]

2*2^(1/2)*atanh((2^(1/2)*(coth(x) + 1)^(1/2))/2) - 2*(coth(x) + 1)^(1/2) - (2*(coth(x) + 1)^(5/2))/5

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (\coth {\relax (x )} + 1\right )^{\frac {3}{2}} \coth ^{2}{\relax (x )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(x)**2*(1+coth(x))**(3/2),x)

[Out]

Integral((coth(x) + 1)**(3/2)*coth(x)**2, x)

________________________________________________________________________________________