\(\int \frac {\cosh ^4(x)}{1+\coth (x)} \, dx\) [105]

Optimal result

Integrand size = 11, antiderivative size = 60 \[ \int \frac {\cosh ^4(x)}{1+\coth (x)} \, dx=\frac {x}{16}+\frac {1}{32 (1-\coth (x))^2}-\frac {1}{8 (1-\coth (x))}-\frac {1}{24 (1+\coth (x))^3}+\frac {5}{32 (1+\coth (x))^2}-\frac {3}{16 (1+\coth (x))} \]

[Out]

1/16*x+1/32/(1-coth(x))^2-1/8/(1-coth(x))-1/24/(1+coth(x))^3+5/32/(1+coth(x))^2-3/16/(1+coth(x))

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {3597, 862, 90, 213} \[ \int \frac {\cosh ^4(x)}{1+\coth (x)} \, dx=\frac {x}{16}-\frac {1}{8 (1-\coth (x))}-\frac {3}{16 (\coth (x)+1)}+\frac {1}{32 (1-\coth (x))^2}+\frac {5}{32 (\coth (x)+1)^2}-\frac {1}{24 (\coth (x)+1)^3} \]

[In]

Int[Cosh[x]^4/(1 + Coth[x]),x]

[Out]

x/16 + 1/(32*(1 - Coth[x])^2) - 1/(8*(1 - Coth[x])) - 1/(24*(1 + Coth[x])^3) + 5/(32*(1 + Coth[x])^2) - 3/(16*
(1 + Coth[x]))

Rule 90

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI
ntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] &&
(IntegerQ[p] || (GtQ[m, 0] && GeQ[n, -1]))

Rule 213

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[b, 2])^(-1))*ArcTanh[Rt[b, 2]*(x/Rt[-a, 2])]
, x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[a, 0] || GtQ[b, 0])

Rule 862

Int[((d_) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[(d + e*x)
^(m + p)*(f + g*x)^n*(a/d + (c/e)*x)^p, x] /; FreeQ[{a, c, d, e, f, g, m, n}, x] && NeQ[e*f - d*g, 0] && EqQ[c
*d^2 + a*e^2, 0] && (IntegerQ[p] || (GtQ[a, 0] && GtQ[d, 0] && EqQ[m + p, 0]))

Rule 3597

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

Rubi steps \begin{align*} \text {integral}& = -\text {Subst}\left (\int \frac {x^4}{(1+x) \left (-1+x^2\right )^3} \, dx,x,\coth (x)\right ) \\ & = -\text {Subst}\left (\int \frac {x^4}{(-1+x)^3 (1+x)^4} \, dx,x,\coth (x)\right ) \\ & = -\text {Subst}\left (\int \left (\frac {1}{16 (-1+x)^3}+\frac {1}{8 (-1+x)^2}-\frac {1}{8 (1+x)^4}+\frac {5}{16 (1+x)^3}-\frac {3}{16 (1+x)^2}+\frac {1}{16 \left (-1+x^2\right )}\right ) \, dx,x,\coth (x)\right ) \\ & = \frac {1}{32 (1-\coth (x))^2}-\frac {1}{8 (1-\coth (x))}-\frac {1}{24 (1+\coth (x))^3}+\frac {5}{32 (1+\coth (x))^2}-\frac {3}{16 (1+\coth (x))}-\frac {1}{16} \text {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\coth (x)\right ) \\ & = \frac {x}{16}+\frac {1}{32 (1-\coth (x))^2}-\frac {1}{8 (1-\coth (x))}-\frac {1}{24 (1+\coth (x))^3}+\frac {5}{32 (1+\coth (x))^2}-\frac {3}{16 (1+\coth (x))} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.23 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.70 \[ \int \frac {\cosh ^4(x)}{1+\coth (x)} \, dx=\frac {1}{192} (12 x+15 \cosh (2 x)+6 \cosh (4 x)+\cosh (6 x)+3 \sinh (2 x)-3 \sinh (4 x)-\sinh (6 x)) \]

[In]

Integrate[Cosh[x]^4/(1 + Coth[x]),x]

[Out]

(12*x + 15*Cosh[2*x] + 6*Cosh[4*x] + Cosh[6*x] + 3*Sinh[2*x] - 3*Sinh[4*x] - Sinh[6*x])/192

Maple [A] (verified)

Time = 1.60 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.58

method	result	size
risch	\(\frac {x}{16}+\frac {{\mathrm e}^{4 x}}{128}+\frac {3 \,{\mathrm e}^{2 x}}{64}+\frac {{\mathrm e}^{-2 x}}{32}+\frac {3 \,{\mathrm e}^{-4 x}}{128}+\frac {{\mathrm e}^{-6 x}}{192}\)	\(35\)
parallelrisch	\(\frac {13}{96}+\frac {\cosh \left (4 x \right )}{32}+\frac {\cosh \left (6 x \right )}{192}+\frac {5 \cosh \left (2 x \right )}{64}-\frac {\sinh \left (4 x \right )}{64}+\frac {\sinh \left (2 x \right )}{64}-\frac {\sinh \left (6 x \right )}{192}-\frac {\ln \left (1-\tanh \left (x \right )\right )}{32}+\frac {\ln \left (1+\tanh \left (x \right )\right )}{32}\)	\(55\)
default	\(\frac {1}{8 \left (\tanh \left (\frac {x}{2}\right )-1\right )^{4}}+\frac {1}{4 \left (\tanh \left (\frac {x}{2}\right )-1\right )^{3}}+\frac {3}{8 \left (\tanh \left (\frac {x}{2}\right )-1\right )^{2}}+\frac {1}{4 \tanh \left (\frac {x}{2}\right )-4}-\frac {\ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{16}+\frac {1}{3 \left (\tanh \left (\frac {x}{2}\right )+1\right )^{6}}-\frac {1}{\left (\tanh \left (\frac {x}{2}\right )+1\right )^{5}}+\frac {13}{8 \left (\tanh \left (\frac {x}{2}\right )+1\right )^{4}}-\frac {19}{12 \left (\tanh \left (\frac {x}{2}\right )+1\right )^{3}}+\frac {1}{\left (\tanh \left (\frac {x}{2}\right )+1\right )^{2}}-\frac {3}{8 \left (\tanh \left (\frac {x}{2}\right )+1\right )}+\frac {\ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{16}\)	\(118\)

[In]

int(cosh(x)^4/(1+coth(x)),x,method=_RETURNVERBOSE)

[Out]

1/16*x+1/128*exp(4*x)+3/64*exp(2*x)+1/32*exp(-2*x)+3/128*exp(-4*x)+1/192*exp(-6*x)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 92 vs. \(2 (44) = 88\).

Time = 0.26 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.53 \[ \int \frac {\cosh ^4(x)}{1+\coth (x)} \, dx=\frac {5 \, \cosh \left (x\right )^{5} + 25 \, \cosh \left (x\right ) \sinh \left (x\right )^{4} + \sinh \left (x\right )^{5} + {\left (10 \, \cosh \left (x\right )^{2} + 9\right )} \sinh \left (x\right )^{3} + 27 \, \cosh \left (x\right )^{3} + {\left (50 \, \cosh \left (x\right )^{3} + 81 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{2} + 12 \, {\left (2 \, x + 1\right )} \cosh \left (x\right ) + {\left (5 \, \cosh \left (x\right )^{4} + 27 \, \cosh \left (x\right )^{2} + 24 \, x - 12\right )} \sinh \left (x\right )}{384 \, {\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )}} \]

[In]

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

[Out]

1/384*(5*cosh(x)^5 + 25*cosh(x)*sinh(x)^4 + sinh(x)^5 + (10*cosh(x)^2 + 9)*sinh(x)^3 + 27*cosh(x)^3 + (50*cosh
(x)^3 + 81*cosh(x))*sinh(x)^2 + 12*(2*x + 1)*cosh(x) + (5*cosh(x)^4 + 27*cosh(x)^2 + 24*x - 12)*sinh(x))/(cosh
(x) + sinh(x))

Sympy [F]

\[ \int \frac {\cosh ^4(x)}{1+\coth (x)} \, dx=\int \frac {\cosh ^{4}{\left (x \right )}}{\coth {\left (x \right )} + 1}\, dx \]

[In]

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

[Out]

Integral(cosh(x)**4/(coth(x) + 1), x)

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.60 \[ \int \frac {\cosh ^4(x)}{1+\coth (x)} \, dx=\frac {1}{128} \, {\left (6 \, e^{\left (-2 \, x\right )} + 1\right )} e^{\left (4 \, x\right )} + \frac {1}{16} \, x + \frac {1}{32} \, e^{\left (-2 \, x\right )} + \frac {3}{128} \, e^{\left (-4 \, x\right )} + \frac {1}{192} \, e^{\left (-6 \, x\right )} \]

[In]

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

[Out]

1/128*(6*e^(-2*x) + 1)*e^(4*x) + 1/16*x + 1/32*e^(-2*x) + 3/128*e^(-4*x) + 1/192*e^(-6*x)

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.70 \[ \int \frac {\cosh ^4(x)}{1+\coth (x)} \, dx=-\frac {1}{384} \, {\left (22 \, e^{\left (6 \, x\right )} - 12 \, e^{\left (4 \, x\right )} - 9 \, e^{\left (2 \, x\right )} - 2\right )} e^{\left (-6 \, x\right )} + \frac {1}{16} \, x + \frac {1}{128} \, e^{\left (4 \, x\right )} + \frac {3}{64} \, e^{\left (2 \, x\right )} \]

[In]

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

[Out]

-1/384*(22*e^(6*x) - 12*e^(4*x) - 9*e^(2*x) - 2)*e^(-6*x) + 1/16*x + 1/128*e^(4*x) + 3/64*e^(2*x)

Mupad [B] (verification not implemented)

Time = 2.11 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.57 \[ \int \frac {\cosh ^4(x)}{1+\coth (x)} \, dx=\frac {x}{16}+\frac {{\mathrm {e}}^{-2\,x}}{32}+\frac {3\,{\mathrm {e}}^{2\,x}}{64}+\frac {3\,{\mathrm {e}}^{-4\,x}}{128}+\frac {{\mathrm {e}}^{4\,x}}{128}+\frac {{\mathrm {e}}^{-6\,x}}{192} \]

[In]

int(cosh(x)^4/(coth(x) + 1),x)

[Out]

x/16 + exp(-2*x)/32 + (3*exp(2*x))/64 + (3*exp(-4*x))/128 + exp(4*x)/128 + exp(-6*x)/192