3.62 \(\int (1+\coth (x))^4 \, dx\)

Optimal. Leaf size=31 \[ 8 x-\frac{1}{3} (\coth (x)+1)^3-(\coth (x)+1)^2-4 \coth (x)+8 \log (\sinh (x)) \]

[Out]

8*x - 4*Coth[x] - (1 + Coth[x])^2 - (1 + Coth[x])^3/3 + 8*Log[Sinh[x]]

________________________________________________________________________________________

Rubi [A]  time = 0.0309066, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 6, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {3478, 3477, 3475} \[ 8 x-\frac{1}{3} (\coth (x)+1)^3-(\coth (x)+1)^2-4 \coth (x)+8 \log (\sinh (x)) \]

Antiderivative was successfully verified.

[In]

Int[(1 + Coth[x])^4,x]

[Out]

8*x - 4*Coth[x] - (1 + Coth[x])^2 - (1 + Coth[x])^3/3 + 8*Log[Sinh[x]]

Rule 3478

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 3477

Int[((a_) + (b_.)*tan[(c_.) + (d_.)*(x_)])^2, x_Symbol] :> Simp[(a^2 - b^2)*x, x] + (Dist[2*a*b, Int[Tan[c + d
*x], x], x] + Simp[(b^2*Tan[c + d*x])/d, x]) /; FreeQ[{a, b, c, d}, x]

Rule 3475

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

Rubi steps

\begin{align*} \int (1+\coth (x))^4 \, dx &=-\frac{1}{3} (1+\coth (x))^3+2 \int (1+\coth (x))^3 \, dx\\ &=-(1+\coth (x))^2-\frac{1}{3} (1+\coth (x))^3+4 \int (1+\coth (x))^2 \, dx\\ &=8 x-4 \coth (x)-(1+\coth (x))^2-\frac{1}{3} (1+\coth (x))^3+8 \int \coth (x) \, dx\\ &=8 x-4 \coth (x)-(1+\coth (x))^2-\frac{1}{3} (1+\coth (x))^3+8 \log (\sinh (x))\\ \end{align*}

Mathematica [C]  time = 0.179641, size = 84, normalized size = 2.71 \[ \frac{\sinh (x) (\coth (x)+1)^4 \left (3 \sinh (x) \left (-6 \sinh (x) \cosh (x) \, _2F_1\left (-\frac{1}{2},1;\frac{1}{2};\tanh ^2(x)\right )-2 \cosh ^2(x)+\sinh ^2(x) (x+8 \log (\tanh (x))+8 \log (\cosh (x)))\right )-\cosh ^3(x) \, _2F_1\left (-\frac{3}{2},1;-\frac{1}{2};\tanh ^2(x)\right )\right )}{3 (\sinh (x)+\cosh (x))^4} \]

Antiderivative was successfully verified.

[In]

Integrate[(1 + Coth[x])^4,x]

[Out]

((1 + Coth[x])^4*Sinh[x]*(-(Cosh[x]^3*Hypergeometric2F1[-3/2, 1, -1/2, Tanh[x]^2]) + 3*Sinh[x]*(-2*Cosh[x]^2 -
6*Cosh[x]*Hypergeometric2F1[-1/2, 1, 1/2, Tanh[x]^2]*Sinh[x] + (x + 8*Log[Cosh[x]] + 8*Log[Tanh[x]])*Sinh[x]^
2)))/(3*(Cosh[x] + Sinh[x])^4)

________________________________________________________________________________________

Maple [A]  time = 0.004, size = 25, normalized size = 0.8 \begin{align*} -{\frac{ \left ({\rm coth} \left (x\right ) \right ) ^{3}}{3}}-2\, \left ({\rm coth} \left (x\right ) \right ) ^{2}-7\,{\rm coth} \left (x\right )-8\,\ln \left ({\rm coth} \left (x\right )-1 \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1+coth(x))^4,x)

[Out]

-1/3*coth(x)^3-2*coth(x)^2-7*coth(x)-8*ln(coth(x)-1)

________________________________________________________________________________________

Maxima [B]  time = 1.10124, size = 128, normalized size = 4.13 \begin{align*} 12 \, x - \frac{4 \,{\left (3 \, e^{\left (-2 \, x\right )} - 3 \, e^{\left (-4 \, x\right )} - 2\right )}}{3 \,{\left (3 \, e^{\left (-2 \, x\right )} - 3 \, e^{\left (-4 \, x\right )} + e^{\left (-6 \, x\right )} - 1\right )}} + \frac{8 \, e^{\left (-2 \, x\right )}}{2 \, e^{\left (-2 \, x\right )} - e^{\left (-4 \, x\right )} - 1} + \frac{12}{e^{\left (-2 \, x\right )} - 1} + 4 \, \log \left (e^{\left (-x\right )} + 1\right ) + 4 \, \log \left (e^{\left (-x\right )} - 1\right ) + 4 \, \log \left (\sinh \left (x\right )\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+coth(x))^4,x, algorithm="maxima")

[Out]

12*x - 4/3*(3*e^(-2*x) - 3*e^(-4*x) - 2)/(3*e^(-2*x) - 3*e^(-4*x) + e^(-6*x) - 1) + 8*e^(-2*x)/(2*e^(-2*x) - e
^(-4*x) - 1) + 12/(e^(-2*x) - 1) + 4*log(e^(-x) + 1) + 4*log(e^(-x) - 1) + 4*log(sinh(x))

________________________________________________________________________________________

Fricas [B]  time = 2.06893, size = 915, normalized size = 29.52 \begin{align*} -\frac{4 \,{\left (18 \, \cosh \left (x\right )^{4} + 72 \, \cosh \left (x\right ) \sinh \left (x\right )^{3} + 18 \, \sinh \left (x\right )^{4} + 27 \,{\left (4 \, \cosh \left (x\right )^{2} - 1\right )} \sinh \left (x\right )^{2} - 27 \, \cosh \left (x\right )^{2} - 6 \,{\left (\cosh \left (x\right )^{6} + 6 \, \cosh \left (x\right ) \sinh \left (x\right )^{5} + \sinh \left (x\right )^{6} + 3 \,{\left (5 \, \cosh \left (x\right )^{2} - 1\right )} \sinh \left (x\right )^{4} - 3 \, \cosh \left (x\right )^{4} + 4 \,{\left (5 \, \cosh \left (x\right )^{3} - 3 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{3} + 3 \,{\left (5 \, \cosh \left (x\right )^{4} - 6 \, \cosh \left (x\right )^{2} + 1\right )} \sinh \left (x\right )^{2} + 3 \, \cosh \left (x\right )^{2} + 6 \,{\left (\cosh \left (x\right )^{5} - 2 \, \cosh \left (x\right )^{3} + \cosh \left (x\right )\right )} \sinh \left (x\right ) - 1\right )} \log \left (\frac{2 \, \sinh \left (x\right )}{\cosh \left (x\right ) - \sinh \left (x\right )}\right ) + 18 \,{\left (4 \, \cosh \left (x\right )^{3} - 3 \, \cosh \left (x\right )\right )} \sinh \left (x\right ) + 11\right )}}{3 \,{\left (\cosh \left (x\right )^{6} + 6 \, \cosh \left (x\right ) \sinh \left (x\right )^{5} + \sinh \left (x\right )^{6} + 3 \,{\left (5 \, \cosh \left (x\right )^{2} - 1\right )} \sinh \left (x\right )^{4} - 3 \, \cosh \left (x\right )^{4} + 4 \,{\left (5 \, \cosh \left (x\right )^{3} - 3 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{3} + 3 \,{\left (5 \, \cosh \left (x\right )^{4} - 6 \, \cosh \left (x\right )^{2} + 1\right )} \sinh \left (x\right )^{2} + 3 \, \cosh \left (x\right )^{2} + 6 \,{\left (\cosh \left (x\right )^{5} - 2 \, \cosh \left (x\right )^{3} + \cosh \left (x\right )\right )} \sinh \left (x\right ) - 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+coth(x))^4,x, algorithm="fricas")

[Out]

-4/3*(18*cosh(x)^4 + 72*cosh(x)*sinh(x)^3 + 18*sinh(x)^4 + 27*(4*cosh(x)^2 - 1)*sinh(x)^2 - 27*cosh(x)^2 - 6*(
cosh(x)^6 + 6*cosh(x)*sinh(x)^5 + sinh(x)^6 + 3*(5*cosh(x)^2 - 1)*sinh(x)^4 - 3*cosh(x)^4 + 4*(5*cosh(x)^3 - 3
*cosh(x))*sinh(x)^3 + 3*(5*cosh(x)^4 - 6*cosh(x)^2 + 1)*sinh(x)^2 + 3*cosh(x)^2 + 6*(cosh(x)^5 - 2*cosh(x)^3 +
cosh(x))*sinh(x) - 1)*log(2*sinh(x)/(cosh(x) - sinh(x))) + 18*(4*cosh(x)^3 - 3*cosh(x))*sinh(x) + 11)/(cosh(x
)^6 + 6*cosh(x)*sinh(x)^5 + sinh(x)^6 + 3*(5*cosh(x)^2 - 1)*sinh(x)^4 - 3*cosh(x)^4 + 4*(5*cosh(x)^3 - 3*cosh(
x))*sinh(x)^3 + 3*(5*cosh(x)^4 - 6*cosh(x)^2 + 1)*sinh(x)^2 + 3*cosh(x)^2 + 6*(cosh(x)^5 - 2*cosh(x)^3 + cosh(
x))*sinh(x) - 1)

________________________________________________________________________________________

Sympy [A]  time = 1.88352, size = 37, normalized size = 1.19 \begin{align*} 16 x - 8 \log{\left (\tanh{\left (x \right )} + 1 \right )} + 8 \log{\left (\tanh{\left (x \right )} \right )} - \frac{7}{\tanh{\left (x \right )}} - \frac{2}{\tanh ^{2}{\left (x \right )}} - \frac{1}{3 \tanh ^{3}{\left (x \right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+coth(x))**4,x)

[Out]

16*x - 8*log(tanh(x) + 1) + 8*log(tanh(x)) - 7/tanh(x) - 2/tanh(x)**2 - 1/(3*tanh(x)**3)

________________________________________________________________________________________

Giac [A]  time = 1.1945, size = 47, normalized size = 1.52 \begin{align*} -\frac{4 \,{\left (18 \, e^{\left (4 \, x\right )} - 27 \, e^{\left (2 \, x\right )} + 11\right )}}{3 \,{\left (e^{\left (2 \, x\right )} - 1\right )}^{3}} + 8 \, \log \left ({\left | e^{\left (2 \, x\right )} - 1 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+coth(x))^4,x, algorithm="giac")

[Out]

-4/3*(18*e^(4*x) - 27*e^(2*x) + 11)/(e^(2*x) - 1)^3 + 8*log(abs(e^(2*x) - 1))