3.1.75 \(\int \frac {1}{(1+\coth (x))^{3/2}} \, dx\) [75]

Optimal. Leaf size=49 \[ \frac {\tanh ^{-1}\left (\frac {\sqrt {1+\coth (x)}}{\sqrt {2}}\right )}{2 \sqrt {2}}-\frac {1}{3 (1+\coth (x))^{3/2}}-\frac {1}{2 \sqrt {1+\coth (x)}} \]

[Out]

-1/3/(1+coth(x))^(3/2)+1/4*arctanh(1/2*(1+coth(x))^(1/2)*2^(1/2))*2^(1/2)-1/2/(1+coth(x))^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 0.02, antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {3560, 3561, 212} \begin {gather*} -\frac {1}{2 \sqrt {\coth (x)+1}}-\frac {1}{3 (\coth (x)+1)^{3/2}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {\coth (x)+1}}{\sqrt {2}}\right )}{2 \sqrt {2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 + Coth[x])^(-3/2),x]

[Out]

ArcTanh[Sqrt[1 + Coth[x]]/Sqrt[2]]/(2*Sqrt[2]) - 1/(3*(1 + Coth[x])^(3/2)) - 1/(2*Sqrt[1 + Coth[x]])

Rule 212

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] && (GtQ[a, 0] || LtQ[b, 0])

Rule 3560

Int[((a_) + (b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[a*((a + b*Tan[c + d*x])^n/(2*b*d*n)), x] +
Dist[1/(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] && LtQ[n
, 0]

Rule 3561

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]

Rubi steps

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

________________________________________________________________________________________

Mathematica [C] Result contains complex when optimal does not.
time = 0.22, size = 86, normalized size = 1.76 \begin {gather*} \left (\frac {1}{4}+\frac {i}{4}\right ) \sqrt {1+\coth (x)} \left (-\frac {i \text {ArcTan}\left (\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {i (1+\coth (x))}\right )}{\sqrt {i (1+\coth (x))}}+\left (\frac {1}{6}-\frac {i}{6}\right ) (-4+5 \cosh (2 x)-\cosh (4 x)-5 \sinh (2 x)+\sinh (4 x))\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1 + Coth[x])^(-3/2),x]

[Out]

(1/4 + I/4)*Sqrt[1 + Coth[x]]*(((-I)*ArcTan[(1/2 + I/2)*Sqrt[I*(1 + Coth[x])]])/Sqrt[I*(1 + Coth[x])] + (1/6 -
I/6)*(-4 + 5*Cosh[2*x] - Cosh[4*x] - 5*Sinh[2*x] + Sinh[4*x]))

________________________________________________________________________________________

Maple [A]
time = 0.67, size = 35, normalized size = 0.71

method	result	size
derivativedivides	\(-\frac {1}{3 \left (1+\coth \left (x \right )\right )^{\frac {3}{2}}}+\frac {\arctanh \left (\frac {\sqrt {1+\coth \left (x \right )}\, \sqrt {2}}{2}\right ) \sqrt {2}}{4}-\frac {1}{2 \sqrt {1+\coth \left (x \right )}}\)	\(35\)
default	\(-\frac {1}{3 \left (1+\coth \left (x \right )\right )^{\frac {3}{2}}}+\frac {\arctanh \left (\frac {\sqrt {1+\coth \left (x \right )}\, \sqrt {2}}{2}\right ) \sqrt {2}}{4}-\frac {1}{2 \sqrt {1+\coth \left (x \right )}}\)	\(35\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(1+coth(x))^(3/2),x,method=_RETURNVERBOSE)

[Out]

-1/3/(1+coth(x))^(3/2)+1/4*arctanh(1/2*(1+coth(x))^(1/2)*2^(1/2))*2^(1/2)-1/2/(1+coth(x))^(1/2)

________________________________________________________________________________________

Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((coth(x) + 1)^(-3/2), x)

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 168 vs. \(2 (34) = 68\).
time = 0.39, size = 168, normalized size = 3.43 \begin {gather*} -\frac {2 \, \sqrt {2} {\left (4 \, \sqrt {2} \cosh \left (x\right )^{2} + 8 \, \sqrt {2} \cosh \left (x\right ) \sinh \left (x\right ) + 4 \, \sqrt {2} \sinh \left (x\right )^{2} - \sqrt {2}\right )} \sqrt {\frac {\sinh \left (x\right )}{\cosh \left (x\right ) - \sinh \left (x\right )}} - 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 \, \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 )}{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 )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/24*(2*sqrt(2)*(4*sqrt(2)*cosh(x)^2 + 8*sqrt(2)*cosh(x)*sinh(x) + 4*sqrt(2)*sinh(x)^2 - sqrt(2))*sqrt(sinh(x
)/(cosh(x) - sinh(x))) - 3*(sqrt(2)*cosh(x)^3 + 3*sqrt(2)*cosh(x)^2*sinh(x) + 3*sqrt(2)*cosh(x)*sinh(x)^2 + sq
rt(2)*sinh(x)^3)*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)^3 + 3*cosh(x)^2*sinh(x) + 3*cosh(x)*sinh(x)^2 + sinh(x)^3)

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (\coth {\left (x \right )} + 1\right )^{\frac {3}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((coth(x) + 1)**(-3/2), x)

________________________________________________________________________________________

Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 113 vs. \(2 (34) = 68\).
time = 0.43, size = 113, normalized size = 2.31 \begin {gather*} \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 )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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)

________________________________________________________________________________________

Mupad [B]
time = 1.20, size = 32, normalized size = 0.65 \begin {gather*} \frac {\sqrt {2}\,\mathrm {atanh}\left (\frac {\sqrt {2}\,\sqrt {\mathrm {coth}\left (x\right )+1}}{2}\right )}{4}-\frac {\frac {\mathrm {coth}\left (x\right )}{2}+\frac {5}{6}}{{\left (\mathrm {coth}\left (x\right )+1\right )}^{3/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(coth(x) + 1)^(3/2),x)

[Out]

(2^(1/2)*atanh((2^(1/2)*(coth(x) + 1)^(1/2))/2))/4 - (coth(x)/2 + 5/6)/(coth(x) + 1)^(3/2)

________________________________________________________________________________________