∫ (1 + coth(x))5∕2 dx [71]

Optimal. Leaf size=45 \[ 4 \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {1+\coth (x)}}{\sqrt {2}}\right )-4 \sqrt {1+\coth (x)}-\frac {2}{3} (1+\coth (x))^{3/2} \]

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Rubi [A]
time = 0.02, antiderivative size = 45, 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 = {3559, 3561, 212} \begin {gather*} -\frac {2}{3} (\coth (x)+1)^{3/2}-4 \sqrt {\coth (x)+1}+4 \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {\coth (x)+1}}{\sqrt {2}}\right ) \end {gather*}

4*Sqrt[2]*ArcTanh[Sqrt[1 + Coth[x]]/Sqrt[2]] - 4*Sqrt[1 + Coth[x]] - (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]

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

Int[Sqrt[(a_) + (b_.)*tan[(c_.) + (d_.)*(x_)]], x_Symbol] :> Dist[-2*(b/d), Subst[Int[1/(2*a - x^2), x], x, Sq

\begin {align*} \int (1+\coth (x))^{5/2} \, dx &=-\frac {2}{3} (1+\coth (x))^{3/2}+2 \int (1+\coth (x))^{3/2} \, dx\\ &=-4 \sqrt {1+\coth (x)}-\frac {2}{3} (1+\coth (x))^{3/2}+4 \int \sqrt {1+\coth (x)} \, dx\\ &=-4 \sqrt {1+\coth (x)}-\frac {2}{3} (1+\coth (x))^{3/2}+8 \text {Subst}\left (\int \frac {1}{2-x^2} \, dx,x,\sqrt {1+\coth (x)}\right )\\ &=4 \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {1+\coth (x)}}{\sqrt {2}}\right )-4 \sqrt {1+\coth (x)}-\frac {2}{3} (1+\coth (x))^{3/2}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 0.11, size = 92, normalized size = 2.04 \begin {gather*} -\frac {2 (1+\coth (x))^{5/2} \sinh (x) \left (\cosh (x) \sqrt {i (1+\coth (x))}+\left ((-6+6 i) \text {ArcTan}\left (\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {i (1+\coth (x))}\right )+7 \sqrt {i (1+\coth (x))}\right ) \sinh (x)\right )}{3 \sqrt {i (1+\coth (x))} (\cosh (x)+\sinh (x))^2} \end {gather*}

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

th[x])]] + 7*Sqrt[I*(1 + Coth[x])])*Sinh[x]))/(3*Sqrt[I*(1 + Coth[x])]*(Cosh[x] + Sinh[x])^2)

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method	result	size

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

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 259 vs. \(2 (34) = 68\).
time = 0.35, size = 259, normalized size = 5.76 \begin {gather*} -\frac {2 \, {\left (2 \, \sqrt {2} {\left (4 \, \sqrt {2} \cosh \left (x\right )^{3} + 12 \, \sqrt {2} \cosh \left (x\right ) \sinh \left (x\right )^{2} + 4 \, \sqrt {2} \sinh \left (x\right )^{3} + 3 \, {\left (4 \, \sqrt {2} \cosh \left (x\right )^{2} - \sqrt {2}\right )} \sinh \left (x\right ) - 3 \, \sqrt {2} \cosh \left (x\right )\right )} \sqrt {\frac {\sinh \left (x\right )}{\cosh \left (x\right ) - \sinh \left (x\right )}} - 3 \, {\left (\sqrt {2} \cosh \left (x\right )^{4} + 4 \, \sqrt {2} \cosh \left (x\right ) \sinh \left (x\right )^{3} + \sqrt {2} \sinh \left (x\right )^{4} + 2 \, {\left (3 \, \sqrt {2} \cosh \left (x\right )^{2} - \sqrt {2}\right )} \sinh \left (x\right )^{2} - 2 \, \sqrt {2} \cosh \left (x\right )^{2} + 4 \, {\left (\sqrt {2} \cosh \left (x\right )^{3} - \sqrt {2} \cosh \left (x\right )\right )} \sinh \left (x\right ) + \sqrt {2}\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 )\right )}}{3 \, {\left (\cosh \left (x\right )^{4} + 4 \, \cosh \left (x\right ) \sinh \left (x\right )^{3} + \sinh \left (x\right )^{4} + 2 \, {\left (3 \, \cosh \left (x\right )^{2} - 1\right )} \sinh \left (x\right )^{2} - 2 \, \cosh \left (x\right )^{2} + 4 \, {\left (\cosh \left (x\right )^{3} - \cosh \left (x\right )\right )} \sinh \left (x\right ) + 1\right )}} \end {gather*}

-2/3*(2*sqrt(2)*(4*sqrt(2)*cosh(x)^3 + 12*sqrt(2)*cosh(x)*sinh(x)^2 + 4*sqrt(2)*sinh(x)^3 + 3*(4*sqrt(2)*cosh(

x)^2 - sqrt(2))*sinh(x) - 3*sqrt(2)*cosh(x))*sqrt(sinh(x)/(cosh(x) - sinh(x))) - 3*(sqrt(2)*cosh(x)^4 + 4*sqrt

(2)*cosh(x)*sinh(x)^3 + sqrt(2)*sinh(x)^4 + 2*(3*sqrt(2)*cosh(x)^2 - sqrt(2))*sinh(x)^2 - 2*sqrt(2)*cosh(x)^2

+ 4*(sqrt(2)*cosh(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)^4 + 4*cosh(x)*sinh(x)^3 + si

nh(x)^4 + 2*(3*cosh(x)^2 - 1)*sinh(x)^2 - 2*cosh(x)^2 + 4*(cosh(x)^3 - cosh(x))*sinh(x) + 1)

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

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 112 vs. \(2 (34) = 68\).
time = 0.42, size = 112, normalized size = 2.49 \begin {gather*} -\frac {2}{3} \, \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} + 9 \, \sqrt {e^{\left (4 \, x\right )} - e^{\left (2 \, x\right )}} - 9 \, e^{\left (2 \, x\right )} + 4\right )}}{{\left (\sqrt {e^{\left (4 \, x\right )} - e^{\left (2 \, x\right )}} - e^{\left (2 \, x\right )} + 1\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 )} \mathrm {sgn}\left (e^{\left (2 \, x\right )} - 1\right ) \end {gather*}

-2/3*sqrt(2)*(2*(6*(sqrt(e^(4*x) - e^(2*x)) - e^(2*x))^2 + 9*sqrt(e^(4*x) - e^(2*x)) - 9*e^(2*x) + 4)/(sqrt(e^

(4*x) - e^(2*x)) - e^(2*x) + 1)^3 + 3*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 = 1.19, size = 54, normalized size = 1.20 \begin {gather*} \sqrt {8}\,\ln \left (-2\,\sqrt {8}\,\sqrt {\mathrm {coth}\left (x\right )+1}-8\right )-\frac {2\,{\left (\mathrm {coth}\left (x\right )+1\right )}^{3/2}}{3}-2\,\sqrt {2}\,\ln \left (4\,\sqrt {2}\,\sqrt {\mathrm {coth}\left (x\right )+1}-8\right )-4\,\sqrt {\mathrm {coth}\left (x\right )+1} \end {gather*}

8^(1/2)*log(- 2*8^(1/2)*(coth(x) + 1)^(1/2) - 8) - (2*(coth(x) + 1)^(3/2))/3 - 2*2^(1/2)*log(4*2^(1/2)*(coth(x

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3.1.71 \(\int (1+\coth (x))^{5/2} \, dx\) [71]