\(\int \frac {\coth ^3(x)}{a+a \cosh (x)} \, dx\) [195]

Optimal result

Integrand size = 13, antiderivative size = 46 \[ \int \frac {\coth ^3(x)}{a+a \cosh (x)} \, dx=-\frac {3 \text {arctanh}(\cosh (x))}{8 a}+\frac {\coth ^4(x)}{4 a}-\frac {3 \coth (x) \text {csch}(x)}{8 a}-\frac {\coth ^3(x) \text {csch}(x)}{4 a} \]

[Out]

-3/8*arctanh(cosh(x))/a+1/4*coth(x)^4/a-3/8*coth(x)*csch(x)/a-1/4*coth(x)^3*csch(x)/a

Rubi [A] (verified)

Time = 0.10 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {2785, 2687, 30, 2691, 3855} \[ \int \frac {\coth ^3(x)}{a+a \cosh (x)} \, dx=-\frac {3 \text {arctanh}(\cosh (x))}{8 a}+\frac {\coth ^4(x)}{4 a}-\frac {\coth ^3(x) \text {csch}(x)}{4 a}-\frac {3 \coth (x) \text {csch}(x)}{8 a} \]

[In]

Int[Coth[x]^3/(a + a*Cosh[x]),x]

[Out]

(-3*ArcTanh[Cosh[x]])/(8*a) + Coth[x]^4/(4*a) - (3*Coth[x]*Csch[x])/(8*a) - (Coth[x]^3*Csch[x])/(4*a)

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 2687

Int[sec[(e_.) + (f_.)*(x_)]^(m_)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[1/f, Subst[Int[(b*x)
^n*(1 + x^2)^(m/2 - 1), x], x, Tan[e + f*x]], x] /; FreeQ[{b, e, f, n}, x] && IntegerQ[m/2] &&  !(IntegerQ[(n
- 1)/2] && LtQ[0, n, m - 1])

Rule 2691

Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[b*(a*Sec[e +
f*x])^m*((b*Tan[e + f*x])^(n - 1)/(f*(m + n - 1))), x] - Dist[b^2*((n - 1)/(m + n - 1)), Int[(a*Sec[e + f*x])
^m*(b*Tan[e + f*x])^(n - 2), x], x] /; FreeQ[{a, b, e, f, m}, x] && GtQ[n, 1] && NeQ[m + n - 1, 0] && Integers
Q[2*m, 2*n]

Rule 2785

Int[((g_.)*tan[(e_.) + (f_.)*(x_)])^(p_.)/((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[1/a, Int[S
ec[e + f*x]^2*(g*Tan[e + f*x])^p, x], x] - Dist[1/(b*g), Int[Sec[e + f*x]*(g*Tan[e + f*x])^(p + 1), x], x] /;
FreeQ[{a, b, e, f, g, p}, x] && EqQ[a^2 - b^2, 0] && NeQ[p, -1]

Rule 3855

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

Rubi steps \begin{align*} \text {integral}& = \frac {\int \coth ^4(x) \text {csch}(x) \, dx}{a}-\frac {\int \coth ^3(x) \text {csch}^2(x) \, dx}{a} \\ & = -\frac {\coth ^3(x) \text {csch}(x)}{4 a}+\frac {3 \int \coth ^2(x) \text {csch}(x) \, dx}{4 a}+\frac {\text {Subst}\left (\int x^3 \, dx,x,i \coth (x)\right )}{a} \\ & = \frac {\coth ^4(x)}{4 a}-\frac {3 \coth (x) \text {csch}(x)}{8 a}-\frac {\coth ^3(x) \text {csch}(x)}{4 a}+\frac {3 \int \text {csch}(x) \, dx}{8 a} \\ & = -\frac {3 \text {arctanh}(\cosh (x))}{8 a}+\frac {\coth ^4(x)}{4 a}-\frac {3 \coth (x) \text {csch}(x)}{8 a}-\frac {\coth ^3(x) \text {csch}(x)}{4 a} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.30 \[ \int \frac {\coth ^3(x)}{a+a \cosh (x)} \, dx=\frac {-8-2 \coth ^2\left (\frac {x}{2}\right )-12 \cosh ^2\left (\frac {x}{2}\right ) \left (\log \left (\cosh \left (\frac {x}{2}\right )\right )-\log \left (\sinh \left (\frac {x}{2}\right )\right )\right )+\text {sech}^2\left (\frac {x}{2}\right )}{16 a (1+\cosh (x))} \]

[In]

Integrate[Coth[x]^3/(a + a*Cosh[x]),x]

[Out]

(-8 - 2*Coth[x/2]^2 - 12*Cosh[x/2]^2*(Log[Cosh[x/2]] - Log[Sinh[x/2]]) + Sech[x/2]^2)/(16*a*(1 + Cosh[x]))

Maple [A] (verified)

Time = 0.14 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.83

method	result	size
default	\(\frac {\frac {\tanh \left (\frac {x}{2}\right )^{4}}{4}+\frac {3 \tanh \left (\frac {x}{2}\right )^{2}}{2}+3 \ln \left (\tanh \left (\frac {x}{2}\right )\right )-\frac {1}{2 \tanh \left (\frac {x}{2}\right )^{2}}}{8 a}\)	\(38\)
risch	\(-\frac {{\mathrm e}^{x} \left (5 \,{\mathrm e}^{4 x}+2 \,{\mathrm e}^{3 x}+2 \,{\mathrm e}^{2 x}+2 \,{\mathrm e}^{x}+5\right )}{4 \left ({\mathrm e}^{x}+1\right )^{4} a \left ({\mathrm e}^{x}-1\right )^{2}}-\frac {3 \ln \left ({\mathrm e}^{x}+1\right )}{8 a}+\frac {3 \ln \left ({\mathrm e}^{x}-1\right )}{8 a}\)	\(65\)

[In]

int(coth(x)^3/(a+a*cosh(x)),x,method=_RETURNVERBOSE)

[Out]

1/8/a*(1/4*tanh(1/2*x)^4+3/2*tanh(1/2*x)^2+3*ln(tanh(1/2*x))-1/2/tanh(1/2*x)^2)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 631 vs. \(2 (38) = 76\).

Time = 0.26 (sec) , antiderivative size = 631, normalized size of antiderivative = 13.72 \[ \int \frac {\coth ^3(x)}{a+a \cosh (x)} \, dx=\text {Too large to display} \]

[In]

integrate(coth(x)^3/(a+a*cosh(x)),x, algorithm="fricas")

[Out]

-1/8*(10*cosh(x)^5 + 2*(25*cosh(x) + 2)*sinh(x)^4 + 10*sinh(x)^5 + 4*cosh(x)^4 + 4*(25*cosh(x)^2 + 4*cosh(x) +
1)*sinh(x)^3 + 4*cosh(x)^3 + 4*(25*cosh(x)^3 + 6*cosh(x)^2 + 3*cosh(x) + 1)*sinh(x)^2 + 4*cosh(x)^2 + 3*(cosh
(x)^6 + 2*(3*cosh(x) + 1)*sinh(x)^5 + sinh(x)^6 + 2*cosh(x)^5 + (15*cosh(x)^2 + 10*cosh(x) - 1)*sinh(x)^4 - co
sh(x)^4 + 4*(5*cosh(x)^3 + 5*cosh(x)^2 - cosh(x) - 1)*sinh(x)^3 - 4*cosh(x)^3 + (15*cosh(x)^4 + 20*cosh(x)^3 -
6*cosh(x)^2 - 12*cosh(x) - 1)*sinh(x)^2 - cosh(x)^2 + 2*(3*cosh(x)^5 + 5*cosh(x)^4 - 2*cosh(x)^3 - 6*cosh(x)^
2 - cosh(x) + 1)*sinh(x) + 2*cosh(x) + 1)*log(cosh(x) + sinh(x) + 1) - 3*(cosh(x)^6 + 2*(3*cosh(x) + 1)*sinh(x
)^5 + sinh(x)^6 + 2*cosh(x)^5 + (15*cosh(x)^2 + 10*cosh(x) - 1)*sinh(x)^4 - cosh(x)^4 + 4*(5*cosh(x)^3 + 5*cos
h(x)^2 - cosh(x) - 1)*sinh(x)^3 - 4*cosh(x)^3 + (15*cosh(x)^4 + 20*cosh(x)^3 - 6*cosh(x)^2 - 12*cosh(x) - 1)*s
inh(x)^2 - cosh(x)^2 + 2*(3*cosh(x)^5 + 5*cosh(x)^4 - 2*cosh(x)^3 - 6*cosh(x)^2 - cosh(x) + 1)*sinh(x) + 2*cos
h(x) + 1)*log(cosh(x) + sinh(x) - 1) + 2*(25*cosh(x)^4 + 8*cosh(x)^3 + 6*cosh(x)^2 + 4*cosh(x) + 5)*sinh(x) +
10*cosh(x))/(a*cosh(x)^6 + a*sinh(x)^6 + 2*a*cosh(x)^5 + 2*(3*a*cosh(x) + a)*sinh(x)^5 - a*cosh(x)^4 + (15*a*c
osh(x)^2 + 10*a*cosh(x) - a)*sinh(x)^4 - 4*a*cosh(x)^3 + 4*(5*a*cosh(x)^3 + 5*a*cosh(x)^2 - a*cosh(x) - a)*sin
h(x)^3 - a*cosh(x)^2 + (15*a*cosh(x)^4 + 20*a*cosh(x)^3 - 6*a*cosh(x)^2 - 12*a*cosh(x) - a)*sinh(x)^2 + 2*a*co
sh(x) + 2*(3*a*cosh(x)^5 + 5*a*cosh(x)^4 - 2*a*cosh(x)^3 - 6*a*cosh(x)^2 - a*cosh(x) + a)*sinh(x) + a)

Sympy [F]

\[ \int \frac {\coth ^3(x)}{a+a \cosh (x)} \, dx=\frac {\int \frac {\coth ^{3}{\left (x \right )}}{\cosh {\left (x \right )} + 1}\, dx}{a} \]

[In]

integrate(coth(x)**3/(a+a*cosh(x)),x)

[Out]

Integral(coth(x)**3/(cosh(x) + 1), x)/a

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 103 vs. \(2 (38) = 76\).

Time = 0.20 (sec) , antiderivative size = 103, normalized size of antiderivative = 2.24 \[ \int \frac {\coth ^3(x)}{a+a \cosh (x)} \, dx=-\frac {5 \, e^{\left (-x\right )} + 2 \, e^{\left (-2 \, x\right )} + 2 \, e^{\left (-3 \, x\right )} + 2 \, e^{\left (-4 \, x\right )} + 5 \, e^{\left (-5 \, x\right )}}{4 \, {\left (2 \, a e^{\left (-x\right )} - a e^{\left (-2 \, x\right )} - 4 \, a e^{\left (-3 \, x\right )} - a e^{\left (-4 \, x\right )} + 2 \, a e^{\left (-5 \, x\right )} + a e^{\left (-6 \, x\right )} + a\right )}} - \frac {3 \, \log \left (e^{\left (-x\right )} + 1\right )}{8 \, a} + \frac {3 \, \log \left (e^{\left (-x\right )} - 1\right )}{8 \, a} \]

[In]

integrate(coth(x)^3/(a+a*cosh(x)),x, algorithm="maxima")

[Out]

-1/4*(5*e^(-x) + 2*e^(-2*x) + 2*e^(-3*x) + 2*e^(-4*x) + 5*e^(-5*x))/(2*a*e^(-x) - a*e^(-2*x) - 4*a*e^(-3*x) -
a*e^(-4*x) + 2*a*e^(-5*x) + a*e^(-6*x) + a) - 3/8*log(e^(-x) + 1)/a + 3/8*log(e^(-x) - 1)/a

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 94 vs. \(2 (38) = 76\).

Time = 0.26 (sec) , antiderivative size = 94, normalized size of antiderivative = 2.04 \[ \int \frac {\coth ^3(x)}{a+a \cosh (x)} \, dx=-\frac {3 \, \log \left (e^{\left (-x\right )} + e^{x} + 2\right )}{16 \, a} + \frac {3 \, \log \left (e^{\left (-x\right )} + e^{x} - 2\right )}{16 \, a} - \frac {3 \, e^{\left (-x\right )} + 3 \, e^{x} - 2}{16 \, a {\left (e^{\left (-x\right )} + e^{x} - 2\right )}} + \frac {9 \, {\left (e^{\left (-x\right )} + e^{x}\right )}^{2} + 4 \, e^{\left (-x\right )} + 4 \, e^{x} - 12}{32 \, a {\left (e^{\left (-x\right )} + e^{x} + 2\right )}^{2}} \]

[In]

integrate(coth(x)^3/(a+a*cosh(x)),x, algorithm="giac")

[Out]

-3/16*log(e^(-x) + e^x + 2)/a + 3/16*log(e^(-x) + e^x - 2)/a - 1/16*(3*e^(-x) + 3*e^x - 2)/(a*(e^(-x) + e^x -
2)) + 1/32*(9*(e^(-x) + e^x)^2 + 4*e^(-x) + 4*e^x - 12)/(a*(e^(-x) + e^x + 2)^2)

Mupad [B] (verification not implemented)

Time = 1.68 (sec) , antiderivative size = 132, normalized size of antiderivative = 2.87 \[ \int \frac {\coth ^3(x)}{a+a \cosh (x)} \, dx=\frac {3}{2\,a\,\left ({\mathrm {e}}^{2\,x}+2\,{\mathrm {e}}^x+1\right )}-\frac {1}{4\,a\,\left ({\mathrm {e}}^{2\,x}-2\,{\mathrm {e}}^x+1\right )}+\frac {1}{2\,a\,\left (6\,{\mathrm {e}}^{2\,x}+4\,{\mathrm {e}}^{3\,x}+{\mathrm {e}}^{4\,x}+4\,{\mathrm {e}}^x+1\right )}-\frac {1}{4\,a\,\left ({\mathrm {e}}^x-1\right )}-\frac {1}{a\,\left ({\mathrm {e}}^x+1\right )}-\frac {3\,\mathrm {atan}\left (\frac {{\mathrm {e}}^x\,\sqrt {-a^2}}{a}\right )}{4\,\sqrt {-a^2}}-\frac {1}{a\,\left (3\,{\mathrm {e}}^{2\,x}+{\mathrm {e}}^{3\,x}+3\,{\mathrm {e}}^x+1\right )} \]

[In]

int(coth(x)^3/(a + a*cosh(x)),x)

[Out]

3/(2*a*(exp(2*x) + 2*exp(x) + 1)) - 1/(4*a*(exp(2*x) - 2*exp(x) + 1)) + 1/(2*a*(6*exp(2*x) + 4*exp(3*x) + exp(
4*x) + 4*exp(x) + 1)) - 1/(4*a*(exp(x) - 1)) - 1/(a*(exp(x) + 1)) - (3*atan((exp(x)*(-a^2)^(1/2))/a))/(4*(-a^2
)^(1/2)) - 1/(a*(3*exp(2*x) + exp(3*x) + 3*exp(x) + 1))