3.2.32 \(\int \coth (x) (1+\coth (x))^{3/2} \, dx\) [132]

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

[Out]

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

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Rubi [A]
time = 0.04, antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {3608, 3559, 3561, 212} \begin {gather*} -\frac {2}{3} (\coth (x)+1)^{3/2}-2 \sqrt {\coth (x)+1}+2 \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {\coth (x)+1}}{\sqrt {2}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Coth[x]*(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])^(3/2))/3

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 3559

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 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 3608

Int[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[d*(
(a + b*Tan[e + f*x])^m/(f*m)), x] + Dist[(b*c + a*d)/b, Int[(a + b*Tan[e + f*x])^m, x], x] /; FreeQ[{a, b, c,
d, e, f, m}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] &&  !LtQ[m, 0]

Rubi steps

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

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Mathematica [C] Result contains complex when optimal does not.
time = 0.10, size = 90, normalized size = 2.00 \begin {gather*} -\frac {2 (1+\coth (x))^{3/2} \left (\cosh (x) \sqrt {i (1+\coth (x))}-(3-3 i) \text {ArcTan}\left (\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {i (1+\coth (x))}\right ) \sinh (x)+4 \sqrt {i (1+\coth (x))} \sinh (x)\right )}{3 \sqrt {i (1+\coth (x))} (\cosh (x)+\sinh (x))} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(1 + Coth[x])^(3/2)*(Cosh[x]*Sqrt[I*(1 + Coth[x])] - (3 - 3*I)*ArcTan[(1/2 + I/2)*Sqrt[I*(1 + Coth[x])]]*S
inh[x] + 4*Sqrt[I*(1 + Coth[x])]*Sinh[x]))/(3*Sqrt[I*(1 + Coth[x])]*(Cosh[x] + Sinh[x]))

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Maple [A]
time = 0.60, size = 35, normalized size = 0.78

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

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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 = 0.34, size = 259, normalized size = 5.76 \begin {gather*} -\frac {2 \, \sqrt {2} {\left (5 \, \sqrt {2} \cosh \left (x\right )^{3} + 15 \, \sqrt {2} \cosh \left (x\right ) \sinh \left (x\right )^{2} + 5 \, \sqrt {2} \sinh \left (x\right )^{3} + 3 \, {\left (5 \, \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 )}{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*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*(2*sqrt(2)*(5*sqrt(2)*cosh(x)^3 + 15*sqrt(2)*cosh(x)*sinh(x)^2 + 5*sqrt(2)*sinh(x)^3 + 3*(5*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 {3}{2}} \coth {\left (x \right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 135 vs. \(2 (34) = 68\).
time = 0.41, size = 135, normalized size = 3.00 \begin {gather*} -\frac {1}{3} \, \sqrt {2} {\left (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 ) \mathrm {sgn}\left (e^{\left (2 \, x\right )} - 1\right ) + \frac {2 \, {\left (9 \, {\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 ) + 12 \, {\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 ) + 5 \, \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 )}^{3}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*sqrt(2)*(3*log(abs(2*sqrt(e^(4*x) - e^(2*x)) - 2*e^(2*x) + 1))*sgn(e^(2*x) - 1) + 2*(9*(sqrt(e^(4*x) - e^
(2*x)) - e^(2*x))^2*sgn(e^(2*x) - 1) + 12*(sqrt(e^(4*x) - e^(2*x)) - e^(2*x))*sgn(e^(2*x) - 1) + 5*sgn(e^(2*x)
- 1))/(sqrt(e^(4*x) - e^(2*x)) - e^(2*x) + 1)^3)

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Mupad [B]
time = 1.24, size = 34, normalized size = 0.76 \begin {gather*} 2\,\sqrt {2}\,\mathrm {atanh}\left (\frac {\sqrt {2}\,\sqrt {\mathrm {coth}\left (x\right )+1}}{2}\right )-2\,\sqrt {\mathrm {coth}\left (x\right )+1}-\frac {2\,{\left (\mathrm {coth}\left (x\right )+1\right )}^{3/2}}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(coth(x)*(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)^(3/2))/3

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