3.2.30 \(\int \frac {\coth ^3(x)}{1+\coth (x)} \, dx\) [130]

Optimal. Leaf size=31 \[ \frac {3 x}{2}-\frac {3 \coth (x)}{2}+\frac {\coth ^2(x)}{2 (1+\coth (x))}-\log (\sinh (x)) \]

[Out]

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

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Rubi [A]
time = 0.04, antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {3631, 3606, 3556} \begin {gather*} \frac {3 x}{2}+\frac {\coth ^2(x)}{2 (\coth (x)+1)}-\frac {3 \coth (x)}{2}-\log (\sinh (x)) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Coth[x]^3/(1 + Coth[x]),x]

[Out]

(3*x)/2 - (3*Coth[x])/2 + Coth[x]^2/(2*(1 + Coth[x])) - Log[Sinh[x]]

Rule 3556

Int[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Log[RemoveContent[Cos[c + d*x], x]]/d, x] /; FreeQ[{c, d}, x]

Rule 3606

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

Rule 3631

Int[((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_)/((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(b*
c - a*d)*((c + d*Tan[e + f*x])^(n - 1)/(2*a*f*(a + b*Tan[e + f*x]))), x] + Dist[1/(2*a^2), Int[(c + d*Tan[e +
f*x])^(n - 2)*Simp[a*c^2 + a*d^2*(n - 1) - b*c*d*n - d*(a*c*(n - 2) + b*d*n)*Tan[e + f*x], x], x], x] /; FreeQ
[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && GtQ[n, 1]

Rubi steps

\begin {align*} \int \frac {\coth ^3(x)}{1+\coth (x)} \, dx &=\frac {\coth ^2(x)}{2 (1+\coth (x))}-\frac {1}{2} \int (2-3 \coth (x)) \coth (x) \, dx\\ &=\frac {3 x}{2}-\frac {3 \coth (x)}{2}+\frac {\coth ^2(x)}{2 (1+\coth (x))}-\int \coth (x) \, dx\\ &=\frac {3 x}{2}-\frac {3 \coth (x)}{2}+\frac {\coth ^2(x)}{2 (1+\coth (x))}-\log (\sinh (x))\\ \end {align*}

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Mathematica [A]
time = 0.03, size = 27, normalized size = 0.87 \begin {gather*} \frac {1}{4} (6 x-\cosh (2 x)-4 \coth (x)-4 \log (\sinh (x))+\sinh (2 x)) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Coth[x]^3/(1 + Coth[x]),x]

[Out]

(6*x - Cosh[2*x] - 4*Coth[x] - 4*Log[Sinh[x]] + Sinh[2*x])/4

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Maple [A]
time = 0.35, size = 28, normalized size = 0.90

method	result	size
derivativedivides	\(-\coth \left (x \right )-\frac {\ln \left (\coth \left (x \right )-1\right )}{4}+\frac {1}{2+2 \coth \left (x \right )}+\frac {5 \ln \left (1+\coth \left (x \right )\right )}{4}\)	\(28\)
default	\(-\coth \left (x \right )-\frac {\ln \left (\coth \left (x \right )-1\right )}{4}+\frac {1}{2+2 \coth \left (x \right )}+\frac {5 \ln \left (1+\coth \left (x \right )\right )}{4}\)	\(28\)
risch	\(\frac {5 x}{2}-\frac {{\mathrm e}^{-2 x}}{4}-\frac {2}{{\mathrm e}^{2 x}-1}-\ln \left ({\mathrm e}^{2 x}-1\right )\)	\(30\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-coth(x)-1/4*ln(coth(x)-1)+1/2/(1+coth(x))+5/4*ln(1+coth(x))

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Maxima [A]
time = 0.26, size = 38, normalized size = 1.23 \begin {gather*} \frac {1}{2} \, x + \frac {2}{e^{\left (-2 \, x\right )} - 1} - \frac {1}{4} \, e^{\left (-2 \, x\right )} - \log \left (e^{\left (-x\right )} + 1\right ) - \log \left (e^{\left (-x\right )} - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*x + 2/(e^(-2*x) - 1) - 1/4*e^(-2*x) - log(e^(-x) + 1) - log(e^(-x) - 1)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 196 vs. \(2 (25) = 50\).
time = 0.38, size = 196, normalized size = 6.32 \begin {gather*} \frac {10 \, x \cosh \left (x\right )^{4} + 40 \, x \cosh \left (x\right ) \sinh \left (x\right )^{3} + 10 \, x \sinh \left (x\right )^{4} - {\left (10 \, x + 9\right )} \cosh \left (x\right )^{2} + {\left (60 \, x \cosh \left (x\right )^{2} - 10 \, x - 9\right )} \sinh \left (x\right )^{2} - 4 \, {\left (\cosh \left (x\right )^{4} + 4 \, \cosh \left (x\right ) \sinh \left (x\right )^{3} + \sinh \left (x\right )^{4} + {\left (6 \, \cosh \left (x\right )^{2} - 1\right )} \sinh \left (x\right )^{2} - \cosh \left (x\right )^{2} + 2 \, {\left (2 \, \cosh \left (x\right )^{3} - \cosh \left (x\right )\right )} \sinh \left (x\right )\right )} \log \left (\frac {2 \, \sinh \left (x\right )}{\cosh \left (x\right ) - \sinh \left (x\right )}\right ) + 2 \, {\left (20 \, x \cosh \left (x\right )^{3} - {\left (10 \, x + 9\right )} \cosh \left (x\right )\right )} \sinh \left (x\right ) + 1}{4 \, {\left (\cosh \left (x\right )^{4} + 4 \, \cosh \left (x\right ) \sinh \left (x\right )^{3} + \sinh \left (x\right )^{4} + {\left (6 \, \cosh \left (x\right )^{2} - 1\right )} \sinh \left (x\right )^{2} - \cosh \left (x\right )^{2} + 2 \, {\left (2 \, \cosh \left (x\right )^{3} - \cosh \left (x\right )\right )} \sinh \left (x\right )\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(10*x*cosh(x)^4 + 40*x*cosh(x)*sinh(x)^3 + 10*x*sinh(x)^4 - (10*x + 9)*cosh(x)^2 + (60*x*cosh(x)^2 - 10*x
- 9)*sinh(x)^2 - 4*(cosh(x)^4 + 4*cosh(x)*sinh(x)^3 + sinh(x)^4 + (6*cosh(x)^2 - 1)*sinh(x)^2 - cosh(x)^2 + 2*
(2*cosh(x)^3 - cosh(x))*sinh(x))*log(2*sinh(x)/(cosh(x) - sinh(x))) + 2*(20*x*cosh(x)^3 - (10*x + 9)*cosh(x))*
sinh(x) + 1)/(cosh(x)^4 + 4*cosh(x)*sinh(x)^3 + sinh(x)^4 + (6*cosh(x)^2 - 1)*sinh(x)^2 - cosh(x)^2 + 2*(2*cos
h(x)^3 - cosh(x))*sinh(x))

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 160 vs. \(2 (27) = 54\).
time = 0.56, size = 160, normalized size = 5.16 \begin {gather*} \frac {x \tanh ^{2}{\left (x \right )}}{2 \tanh ^{2}{\left (x \right )} + 2 \tanh {\left (x \right )}} + \frac {x \tanh {\left (x \right )}}{2 \tanh ^{2}{\left (x \right )} + 2 \tanh {\left (x \right )}} + \frac {2 \log {\left (\tanh {\left (x \right )} + 1 \right )} \tanh ^{2}{\left (x \right )}}{2 \tanh ^{2}{\left (x \right )} + 2 \tanh {\left (x \right )}} + \frac {2 \log {\left (\tanh {\left (x \right )} + 1 \right )} \tanh {\left (x \right )}}{2 \tanh ^{2}{\left (x \right )} + 2 \tanh {\left (x \right )}} - \frac {2 \log {\left (\tanh {\left (x \right )} \right )} \tanh ^{2}{\left (x \right )}}{2 \tanh ^{2}{\left (x \right )} + 2 \tanh {\left (x \right )}} - \frac {2 \log {\left (\tanh {\left (x \right )} \right )} \tanh {\left (x \right )}}{2 \tanh ^{2}{\left (x \right )} + 2 \tanh {\left (x \right )}} - \frac {3 \tanh {\left (x \right )}}{2 \tanh ^{2}{\left (x \right )} + 2 \tanh {\left (x \right )}} - \frac {2}{2 \tanh ^{2}{\left (x \right )} + 2 \tanh {\left (x \right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*tanh(x)**2/(2*tanh(x)**2 + 2*tanh(x)) + x*tanh(x)/(2*tanh(x)**2 + 2*tanh(x)) + 2*log(tanh(x) + 1)*tanh(x)**2
/(2*tanh(x)**2 + 2*tanh(x)) + 2*log(tanh(x) + 1)*tanh(x)/(2*tanh(x)**2 + 2*tanh(x)) - 2*log(tanh(x))*tanh(x)**
2/(2*tanh(x)**2 + 2*tanh(x)) - 2*log(tanh(x))*tanh(x)/(2*tanh(x)**2 + 2*tanh(x)) - 3*tanh(x)/(2*tanh(x)**2 + 2
*tanh(x)) - 2/(2*tanh(x)**2 + 2*tanh(x))

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Giac [A]
time = 0.42, size = 36, normalized size = 1.16 \begin {gather*} \frac {5}{2} \, x - \frac {{\left (9 \, e^{\left (2 \, x\right )} - 1\right )} e^{\left (-2 \, x\right )}}{4 \, {\left (e^{\left (2 \, x\right )} - 1\right )}} - \log \left ({\left | e^{\left (2 \, x\right )} - 1 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

5/2*x - 1/4*(9*e^(2*x) - 1)*e^(-2*x)/(e^(2*x) - 1) - log(abs(e^(2*x) - 1))

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Mupad [B]
time = 1.16, size = 21, normalized size = 0.68 \begin {gather*} \frac {x}{2}+\ln \left (\mathrm {coth}\left (x\right )+1\right )-\mathrm {coth}\left (x\right )+\frac {1}{2\,\left (\mathrm {coth}\left (x\right )+1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(coth(x)^3/(coth(x) + 1),x)

[Out]

x/2 + log(coth(x) + 1) - coth(x) + 1/(2*(coth(x) + 1))

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