Integrand size = 11, antiderivative size = 60 \[ \int \frac {\cosh ^4(x)}{1+\coth (x)} \, dx=\frac {x}{16}+\frac {1}{32 (1-\coth (x))^2}-\frac {1}{8 (1-\coth (x))}-\frac {1}{24 (1+\coth (x))^3}+\frac {5}{32 (1+\coth (x))^2}-\frac {3}{16 (1+\coth (x))} \] Output:
1/16*x+1/32/(1-coth(x))^2-1/(8-8*coth(x))-1/24/(1+coth(x))^3+5/32/(1+coth( x))^2-3/(16+16*coth(x))
Time = 0.14 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.70 \[ \int \frac {\cosh ^4(x)}{1+\coth (x)} \, dx=\frac {1}{192} (12 x+15 \cosh (2 x)+6 \cosh (4 x)+\cosh (6 x)+3 \sinh (2 x)-3 \sinh (4 x)-\sinh (6 x)) \] Input:
Integrate[Cosh[x]^4/(1 + Coth[x]),x]
Output:
(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
Time = 0.48 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.03, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.545, Rules used = {3042, 3999, 25, 516, 99, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {\cosh ^4(x)}{\coth (x)+1} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\sin \left (\frac {\pi }{2}+i x\right )^4}{1-i \tan \left (\frac {\pi }{2}+i x\right )}dx\) |
\(\Big \downarrow \) 3999 |
\(\displaystyle -\int -\frac {\coth ^4(x)}{(\coth (x)+1) \left (1-\coth ^2(x)\right )^3}d\coth (x)\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \int \frac {\coth ^4(x)}{(\coth (x)+1) \left (1-\coth ^2(x)\right )^3}d\coth (x)\) |
\(\Big \downarrow \) 516 |
\(\displaystyle \int \frac {\coth ^4(x)}{(1-\coth (x))^3 (\coth (x)+1)^4}d\coth (x)\) |
\(\Big \downarrow \) 99 |
\(\displaystyle \int \left (-\frac {1}{16 \left (\coth ^2(x)-1\right )}-\frac {1}{8 (\coth (x)-1)^2}+\frac {3}{16 (\coth (x)+1)^2}-\frac {1}{16 (\coth (x)-1)^3}-\frac {5}{16 (\coth (x)+1)^3}+\frac {1}{8 (\coth (x)+1)^4}\right )d\coth (x)\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{16} \text {arctanh}(\coth (x))-\frac {1}{8 (1-\coth (x))}-\frac {3}{16 (\coth (x)+1)}+\frac {1}{32 (1-\coth (x))^2}+\frac {5}{32 (\coth (x)+1)^2}-\frac {1}{24 (\coth (x)+1)^3}\) |
Input:
Int[Cosh[x]^4/(1 + Coth[x]),x]
Output:
ArcTanh[Coth[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]))
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Int[ExpandIntegrand[(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]))
Int[((e_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2)^(p_. ), x_Symbol] :> Int[(e*x)^m*(c + d*x)^(n + p)*(a/c + (b/d)*x)^p, x] /; Free Q[{a, b, c, d, e, m, n, p}, x] && EqQ[b*c^2 + a*d^2, 0] && (IntegerQ[p] || (GtQ[a, 0] && GtQ[c, 0] && !IntegerQ[n]))
Int[sin[(e_.) + (f_.)*(x_)]^(m_)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(n_ ), x_Symbol] :> Simp[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]
Time = 1.53 (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 | \(\frac {\left (-12 \cosh \left (x \right )-12 \sinh \left (x \right )\right ) \ln \left (1-\tanh \left (x \right )\right )+\left (12 \cosh \left (x \right )+12 \sinh \left (x \right )\right ) \ln \left (\tanh \left (x \right )+1\right )+64 \cosh \left (x \right )+40 \sinh \left (x \right )+27 \cosh \left (3 x \right )+5 \cosh \left (5 x \right )+9 \sinh \left (3 x \right )+\sinh \left (5 x \right )}{384 \sinh \left (x \right )+384 \cosh \left (x \right )}\) | \(76\) |
default | \(\frac {1}{3 \left (\tanh \left (\frac {x}{2}\right )+1\right )^{6}}-\frac {1}{\left (\tanh \left (\frac {x}{2}\right )+1\right )^{5}}+\frac {13}{8 \left (\tanh \left (\frac {x}{2}\right )+1\right )^{4}}-\frac {19}{12 \left (\tanh \left (\frac {x}{2}\right )+1\right )^{3}}+\frac {1}{\left (\tanh \left (\frac {x}{2}\right )+1\right )^{2}}-\frac {3}{8 \left (\tanh \left (\frac {x}{2}\right )+1\right )}+\frac {\ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{16}+\frac {1}{8 \left (\tanh \left (\frac {x}{2}\right )-1\right )^{4}}+\frac {1}{4 \left (\tanh \left (\frac {x}{2}\right )-1\right )^{3}}+\frac {3}{8 \left (\tanh \left (\frac {x}{2}\right )-1\right )^{2}}+\frac {1}{4 \tanh \left (\frac {x}{2}\right )-4}-\frac {\ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{16}\) | \(118\) |
Input:
int(cosh(x)^4/(1+coth(x)),x,method=_RETURNVERBOSE)
Output:
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*e xp(-6*x)
Leaf count of result is larger than twice the leaf count of optimal. 92 vs. \(2 (44) = 88\).
Time = 0.09 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.53 \[ \int \frac {\cosh ^4(x)}{1+\coth (x)} \, dx=\frac {5 \, \cosh \left (x\right )^{5} + 25 \, \cosh \left (x\right ) \sinh \left (x\right )^{4} + \sinh \left (x\right )^{5} + {\left (10 \, \cosh \left (x\right )^{2} + 9\right )} \sinh \left (x\right )^{3} + 27 \, \cosh \left (x\right )^{3} + {\left (50 \, \cosh \left (x\right )^{3} + 81 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{2} + 12 \, {\left (2 \, x + 1\right )} \cosh \left (x\right ) + {\left (5 \, \cosh \left (x\right )^{4} + 27 \, \cosh \left (x\right )^{2} + 24 \, x - 12\right )} \sinh \left (x\right )}{384 \, {\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )}} \] Input:
integrate(cosh(x)^4/(1+coth(x)),x, algorithm="fricas")
Output:
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))
\[ \int \frac {\cosh ^4(x)}{1+\coth (x)} \, dx=\int \frac {\cosh ^{4}{\left (x \right )}}{\coth {\left (x \right )} + 1}\, dx \] Input:
integrate(cosh(x)**4/(1+coth(x)),x)
Output:
Integral(cosh(x)**4/(coth(x) + 1), x)
Time = 0.03 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.60 \[ \int \frac {\cosh ^4(x)}{1+\coth (x)} \, dx=\frac {1}{128} \, {\left (6 \, e^{\left (-2 \, x\right )} + 1\right )} e^{\left (4 \, x\right )} + \frac {1}{16} \, x + \frac {1}{32} \, e^{\left (-2 \, x\right )} + \frac {3}{128} \, e^{\left (-4 \, x\right )} + \frac {1}{192} \, e^{\left (-6 \, x\right )} \] Input:
integrate(cosh(x)^4/(1+coth(x)),x, algorithm="maxima")
Output:
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)
Time = 0.12 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.70 \[ \int \frac {\cosh ^4(x)}{1+\coth (x)} \, dx=-\frac {1}{384} \, {\left (22 \, e^{\left (6 \, x\right )} - 12 \, e^{\left (4 \, x\right )} - 9 \, e^{\left (2 \, x\right )} - 2\right )} e^{\left (-6 \, x\right )} + \frac {1}{16} \, x + \frac {1}{128} \, e^{\left (4 \, x\right )} + \frac {3}{64} \, e^{\left (2 \, x\right )} \] Input:
integrate(cosh(x)^4/(1+coth(x)),x, algorithm="giac")
Output:
-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)
Time = 2.59 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.57 \[ \int \frac {\cosh ^4(x)}{1+\coth (x)} \, dx=\frac {x}{16}+\frac {{\mathrm {e}}^{-2\,x}}{32}+\frac {3\,{\mathrm {e}}^{2\,x}}{64}+\frac {3\,{\mathrm {e}}^{-4\,x}}{128}+\frac {{\mathrm {e}}^{4\,x}}{128}+\frac {{\mathrm {e}}^{-6\,x}}{192} \] Input:
int(cosh(x)^4/(coth(x) + 1),x)
Output:
x/16 + exp(-2*x)/32 + (3*exp(2*x))/64 + (3*exp(-4*x))/128 + exp(4*x)/128 + exp(-6*x)/192
Time = 0.25 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.78 \[ \int \frac {\cosh ^4(x)}{1+\coth (x)} \, dx=\frac {3 e^{10 x}+18 e^{8 x}+24 e^{6 x} x +12 e^{4 x}+9 e^{2 x}+2}{384 e^{6 x}} \] Input:
int(cosh(x)^4/(1+coth(x)),x)
Output:
(3*e**(10*x) + 18*e**(8*x) + 24*e**(6*x)*x + 12*e**(4*x) + 9*e**(2*x) + 2) /(384*e**(6*x))