3.1.76 \(\int \frac {1}{(1+\coth (x))^{5/2}} \, dx\) [76]

Optimal. Leaf size=61 \[ \frac {\tanh ^{-1}\left (\frac {\sqrt {1+\coth (x)}}{\sqrt {2}}\right )}{4 \sqrt {2}}-\frac {1}{5 (1+\coth (x))^{5/2}}-\frac {1}{6 (1+\coth (x))^{3/2}}-\frac {1}{4 \sqrt {1+\coth (x)}} \]

[Out]

-1/5/(1+coth(x))^(5/2)-1/6/(1+coth(x))^(3/2)+1/8*arctanh(1/2*(1+coth(x))^(1/2)*2^(1/2))*2^(1/2)-1/4/(1+coth(x)
)^(1/2)

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Rubi [A]
time = 0.03, antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 5, 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}{4 \sqrt {\coth (x)+1}}-\frac {1}{6 (\coth (x)+1)^{3/2}}-\frac {1}{5 (\coth (x)+1)^{5/2}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {\coth (x)+1}}{\sqrt {2}}\right )}{4 \sqrt {2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 + Coth[x])^(-5/2),x]

[Out]

ArcTanh[Sqrt[1 + Coth[x]]/Sqrt[2]]/(4*Sqrt[2]) - 1/(5*(1 + Coth[x])^(5/2)) - 1/(6*(1 + Coth[x])^(3/2)) - 1/(4*
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))^{5/2}} \, dx &=-\frac {1}{5 (1+\coth (x))^{5/2}}+\frac {1}{2} \int \frac {1}{(1+\coth (x))^{3/2}} \, dx\\ &=-\frac {1}{5 (1+\coth (x))^{5/2}}-\frac {1}{6 (1+\coth (x))^{3/2}}+\frac {1}{4} \int \frac {1}{\sqrt {1+\coth (x)}} \, dx\\ &=-\frac {1}{5 (1+\coth (x))^{5/2}}-\frac {1}{6 (1+\coth (x))^{3/2}}-\frac {1}{4 \sqrt {1+\coth (x)}}+\frac {1}{8} \int \sqrt {1+\coth (x)} \, dx\\ &=-\frac {1}{5 (1+\coth (x))^{5/2}}-\frac {1}{6 (1+\coth (x))^{3/2}}-\frac {1}{4 \sqrt {1+\coth (x)}}+\frac {1}{4} \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 )}{4 \sqrt {2}}-\frac {1}{5 (1+\coth (x))^{5/2}}-\frac {1}{6 (1+\coth (x))^{3/2}}-\frac {1}{4 \sqrt {1+\coth (x)}}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 0.53, size = 94, normalized size = 1.54 \begin {gather*} \frac {\left (\frac {1}{8}+\frac {i}{8}\right ) \text {ArcTan}\left (\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {i (1+\coth (x))}\right ) (1+\coth (x))^{3/2}}{(i (1+\coth (x)))^{3/2}}-\frac {1}{60} \sqrt {1+\coth (x)} (\cosh (3 x)-\sinh (3 x)) (-10 \cosh (x)+10 \cosh (3 x)-24 \sinh (x)+13 \sinh (3 x)) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1 + Coth[x])^(-5/2),x]

[Out]

((1/8 + I/8)*ArcTan[(1/2 + I/2)*Sqrt[I*(1 + Coth[x])]]*(1 + Coth[x])^(3/2))/(I*(1 + Coth[x]))^(3/2) - (Sqrt[1
+ Coth[x]]*(Cosh[3*x] - Sinh[3*x])*(-10*Cosh[x] + 10*Cosh[3*x] - 24*Sinh[x] + 13*Sinh[3*x]))/60

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Maple [A]
time = 0.66, size = 43, normalized size = 0.70

method	result	size
derivativedivides	\(-\frac {1}{5 \left (1+\coth \left (x \right )\right )^{\frac {5}{2}}}-\frac {1}{6 \left (1+\coth \left (x \right )\right )^{\frac {3}{2}}}+\frac {\arctanh \left (\frac {\sqrt {1+\coth \left (x \right )}\, \sqrt {2}}{2}\right ) \sqrt {2}}{8}-\frac {1}{4 \sqrt {1+\coth \left (x \right )}}\)	\(43\)
default	\(-\frac {1}{5 \left (1+\coth \left (x \right )\right )^{\frac {5}{2}}}-\frac {1}{6 \left (1+\coth \left (x \right )\right )^{\frac {3}{2}}}+\frac {\arctanh \left (\frac {\sqrt {1+\coth \left (x \right )}\, \sqrt {2}}{2}\right ) \sqrt {2}}{8}-\frac {1}{4 \sqrt {1+\coth \left (x \right )}}\)	\(43\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(1+coth(x))^(5/2),x,method=_RETURNVERBOSE)

[Out]

-1/5/(1+coth(x))^(5/2)-1/6/(1+coth(x))^(3/2)+1/8*arctanh(1/2*(1+coth(x))^(1/2)*2^(1/2))*2^(1/2)-1/4/(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(1/(1+coth(x))^(5/2),x, algorithm="maxima")

[Out]

integrate((coth(x) + 1)^(-5/2), x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 266 vs. \(2 (42) = 84\).
time = 0.38, size = 266, normalized size = 4.36 \begin {gather*} -\frac {2 \, \sqrt {2} {\left (23 \, \sqrt {2} \cosh \left (x\right )^{4} + 92 \, \sqrt {2} \cosh \left (x\right ) \sinh \left (x\right )^{3} + 23 \, \sqrt {2} \sinh \left (x\right )^{4} + {\left (138 \, \sqrt {2} \cosh \left (x\right )^{2} - 11 \, \sqrt {2}\right )} \sinh \left (x\right )^{2} - 11 \, \sqrt {2} \cosh \left (x\right )^{2} + 2 \, {\left (46 \, \sqrt {2} \cosh \left (x\right )^{3} - 11 \, \sqrt {2} \cosh \left (x\right )\right )} \sinh \left (x\right ) + 3 \, \sqrt {2}\right )} \sqrt {\frac {\sinh \left (x\right )}{\cosh \left (x\right ) - \sinh \left (x\right )}} - 15 \, {\left (\sqrt {2} \cosh \left (x\right )^{5} + 5 \, \sqrt {2} \cosh \left (x\right )^{4} \sinh \left (x\right ) + 10 \, \sqrt {2} \cosh \left (x\right )^{3} \sinh \left (x\right )^{2} + 10 \, \sqrt {2} \cosh \left (x\right )^{2} \sinh \left (x\right )^{3} + 5 \, \sqrt {2} \cosh \left (x\right ) \sinh \left (x\right )^{4} + \sqrt {2} \sinh \left (x\right )^{5}\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 )}{240 \, {\left (\cosh \left (x\right )^{5} + 5 \, \cosh \left (x\right )^{4} \sinh \left (x\right ) + 10 \, \cosh \left (x\right )^{3} \sinh \left (x\right )^{2} + 10 \, \cosh \left (x\right )^{2} \sinh \left (x\right )^{3} + 5 \, \cosh \left (x\right ) \sinh \left (x\right )^{4} + \sinh \left (x\right )^{5}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1+coth(x))^(5/2),x, algorithm="fricas")

[Out]

-1/240*(2*sqrt(2)*(23*sqrt(2)*cosh(x)^4 + 92*sqrt(2)*cosh(x)*sinh(x)^3 + 23*sqrt(2)*sinh(x)^4 + (138*sqrt(2)*c
osh(x)^2 - 11*sqrt(2))*sinh(x)^2 - 11*sqrt(2)*cosh(x)^2 + 2*(46*sqrt(2)*cosh(x)^3 - 11*sqrt(2)*cosh(x))*sinh(x
) + 3*sqrt(2))*sqrt(sinh(x)/(cosh(x) - sinh(x))) - 15*(sqrt(2)*cosh(x)^5 + 5*sqrt(2)*cosh(x)^4*sinh(x) + 10*sq
rt(2)*cosh(x)^3*sinh(x)^2 + 10*sqrt(2)*cosh(x)^2*sinh(x)^3 + 5*sqrt(2)*cosh(x)*sinh(x)^4 + sqrt(2)*sinh(x)^5)*
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)^5 + 5*cosh(x)^4*sinh(x) + 10*cosh(x)^3*sinh(x)^2 + 10*cosh(x)^2*sinh(x)^3 + 5*cosh(x)*sin
h(x)^4 + sinh(x)^5)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (\coth {\left (x \right )} + 1\right )^{\frac {5}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1+coth(x))**(5/2),x)

[Out]

Integral((coth(x) + 1)**(-5/2), x)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 161 vs. \(2 (42) = 84\).
time = 0.44, size = 161, normalized size = 2.64 \begin {gather*} \frac {\sqrt {2} {\left (\frac {2 \, {\left (45 \, {\left (\sqrt {e^{\left (4 \, x\right )} - e^{\left (2 \, x\right )}} - e^{\left (2 \, x\right )}\right )}^{4} + 45 \, {\left (\sqrt {e^{\left (4 \, x\right )} - e^{\left (2 \, x\right )}} - e^{\left (2 \, x\right )}\right )}^{3} + 35 \, {\left (\sqrt {e^{\left (4 \, x\right )} - e^{\left (2 \, x\right )}} - e^{\left (2 \, x\right )}\right )}^{2} + 15 \, \sqrt {e^{\left (4 \, x\right )} - e^{\left (2 \, x\right )}} - 15 \, e^{\left (2 \, x\right )} + 3\right )}}{{\left (\sqrt {e^{\left (4 \, x\right )} - e^{\left (2 \, x\right )}} - e^{\left (2 \, x\right )}\right )}^{5}} - 15 \, \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 )}}{240 \, \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))^(5/2),x, algorithm="giac")

[Out]

1/240*sqrt(2)*(2*(45*(sqrt(e^(4*x) - e^(2*x)) - e^(2*x))^4 + 45*(sqrt(e^(4*x) - e^(2*x)) - e^(2*x))^3 + 35*(sq
rt(e^(4*x) - e^(2*x)) - e^(2*x))^2 + 15*sqrt(e^(4*x) - e^(2*x)) - 15*e^(2*x) + 3)/(sqrt(e^(4*x) - e^(2*x)) - e
^(2*x))^5 - 15*log(abs(2*sqrt(e^(4*x) - e^(2*x)) - 2*e^(2*x) + 1)))/sgn(e^(2*x) - 1)

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Mupad [B]
time = 1.20, size = 40, normalized size = 0.66 \begin {gather*} \frac {\sqrt {2}\,\mathrm {atanh}\left (\frac {\sqrt {2}\,\sqrt {\mathrm {coth}\left (x\right )+1}}{2}\right )}{8}-\frac {\frac {\mathrm {coth}\left (x\right )}{6}+\frac {{\left (\mathrm {coth}\left (x\right )+1\right )}^2}{4}+\frac {11}{30}}{{\left (\mathrm {coth}\left (x\right )+1\right )}^{5/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(coth(x) + 1)^(5/2),x)

[Out]

(2^(1/2)*atanh((2^(1/2)*(coth(x) + 1)^(1/2))/2))/8 - (coth(x)/6 + (coth(x) + 1)^2/4 + 11/30)/(coth(x) + 1)^(5/
2)

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