3.2.38 \(\int \frac {\coth ^2(x)}{\sqrt {1+\coth (x)}} \, dx\) [138]

Optimal. Leaf size=42 \[ \frac {\tanh ^{-1}\left (\frac {\sqrt {1+\coth (x)}}{\sqrt {2}}\right )}{\sqrt {2}}-\frac {1}{\sqrt {1+\coth (x)}}-2 \sqrt {1+\coth (x)} \]

[Out]

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

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Rubi [A]
time = 0.04, antiderivative size = 42, 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 = {3624, 3560, 3561, 212} \begin {gather*} -2 \sqrt {\coth (x)+1}-\frac {1}{\sqrt {\coth (x)+1}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {\coth (x)+1}}{\sqrt {2}}\right )}{\sqrt {2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Coth[x]^2/Sqrt[1 + Coth[x]],x]

[Out]

ArcTanh[Sqrt[1 + Coth[x]]/Sqrt[2]]/Sqrt[2] - 1/Sqrt[1 + Coth[x]] - 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]

Rule 3624

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 \frac {\coth ^2(x)}{\sqrt {1+\coth (x)}} \, dx &=-2 \sqrt {1+\coth (x)}+\int \frac {1}{\sqrt {1+\coth (x)}} \, dx\\ &=-\frac {1}{\sqrt {1+\coth (x)}}-2 \sqrt {1+\coth (x)}+\frac {1}{2} \int \sqrt {1+\coth (x)} \, dx\\ &=-\frac {1}{\sqrt {1+\coth (x)}}-2 \sqrt {1+\coth (x)}+\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 )}{\sqrt {2}}-\frac {1}{\sqrt {1+\coth (x)}}-2 \sqrt {1+\coth (x)}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 0.25, size = 81, normalized size = 1.93 \begin {gather*} \frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \text {csch}(x) (\cosh (x)+\sinh (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}{2}-\frac {i}{2}\right ) (-5+\cosh (2 x)-\sinh (2 x))\right )}{\sqrt {1+\coth (x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Coth[x]^2/Sqrt[1 + Coth[x]],x]

[Out]

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

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Maple [A]
time = 0.77, size = 35, normalized size = 0.83

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

integrate(coth(x)^2/sqrt(coth(x) + 1), x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 189 vs. \(2 (34) = 68\).
time = 0.36, size = 189, normalized size = 4.50 \begin {gather*} -\frac {2 \, \sqrt {2} {\left (5 \, \sqrt {2} \cosh \left (x\right )^{2} + 10 \, \sqrt {2} \cosh \left (x\right ) \sinh \left (x\right ) + 5 \, \sqrt {2} \sinh \left (x\right )^{2} - \sqrt {2}\right )} \sqrt {\frac {\sinh \left (x\right )}{\cosh \left (x\right ) - \sinh \left (x\right )}} - {\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 )} \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 )}{4 \, {\left (\cosh \left (x\right )^{3} + 3 \, \cosh \left (x\right ) \sinh \left (x\right )^{2} + \sinh \left (x\right )^{3} + {\left (3 \, \cosh \left (x\right )^{2} - 1\right )} \sinh \left (x\right ) - \cosh \left (x\right )\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*(2*sqrt(2)*(5*sqrt(2)*cosh(x)^2 + 10*sqrt(2)*cosh(x)*sinh(x) + 5*sqrt(2)*sinh(x)^2 - sqrt(2))*sqrt(sinh(x
)/(cosh(x) - sinh(x))) - (sqrt(2)*cosh(x)^3 + 3*sqrt(2)*cosh(x)*sinh(x)^2 + sqrt(2)*sinh(x)^3 + (3*sqrt(2)*cos
h(x)^2 - sqrt(2))*sinh(x) - sqrt(2)*cosh(x))*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)*sinh(x)^2 + sinh(x)^3 + (3*cos
h(x)^2 - 1)*sinh(x) - cosh(x))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 88 vs. \(2 (34) = 68\).
time = 0.42, size = 88, normalized size = 2.10 \begin {gather*} -\frac {\frac {5 \, \sqrt {2} e^{\left (2 \, x\right )}}{\mathrm {sgn}\left (e^{\left (2 \, x\right )} - 1\right )} - \frac {\sqrt {2}}{\mathrm {sgn}\left (e^{\left (2 \, x\right )} - 1\right )}}{2 \, \sqrt {e^{\left (4 \, x\right )} - e^{\left (2 \, x\right )}}} - \frac {\sqrt {2} \log \left ({\left | 4 \, \sqrt {e^{\left (4 \, x\right )} - e^{\left (2 \, x\right )}} - 4 \, e^{\left (2 \, x\right )} + 2 \right |}\right )}{4 \, \mathrm {sgn}\left (e^{\left (2 \, x\right )} - 1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(5*sqrt(2)*e^(2*x)/sgn(e^(2*x) - 1) - sqrt(2)/sgn(e^(2*x) - 1))/sqrt(e^(4*x) - e^(2*x)) - 1/4*sqrt(2)*log
(abs(4*sqrt(e^(4*x) - e^(2*x)) - 4*e^(2*x) + 2))/sgn(e^(2*x) - 1)

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Mupad [B]
time = 1.26, size = 36, normalized size = 0.86 \begin {gather*} \frac {\sqrt {2}\,\mathrm {atanh}\left (\frac {\sqrt {2}\,\sqrt {\mathrm {coth}\left (x\right )+1}}{2}\right )}{2}-\frac {3}{\sqrt {\mathrm {coth}\left (x\right )+1}}-\frac {2\,\mathrm {coth}\left (x\right )}{\sqrt {\mathrm {coth}\left (x\right )+1}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(coth(x)^2/(coth(x) + 1)^(1/2),x)

[Out]

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

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