\(\int \frac {\text {csch}^4(x)}{a+a \cosh (x)} \, dx\) [163]

Optimal result

Integrand size = 13, antiderivative size = 37 \[ \int \frac {\text {csch}^4(x)}{a+a \cosh (x)} \, dx=\frac {4 \coth (x)}{5 a}-\frac {4 \coth ^3(x)}{15 a}+\frac {\text {csch}^3(x)}{5 (a+a \cosh (x))} \]

[Out]

4/5*coth(x)/a-4/15*coth(x)^3/a+1/5*csch(x)^3/(a+a*cosh(x))

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2751, 3852} \[ \int \frac {\text {csch}^4(x)}{a+a \cosh (x)} \, dx=-\frac {4 \coth ^3(x)}{15 a}+\frac {4 \coth (x)}{5 a}+\frac {\text {csch}^3(x)}{5 (a \cosh (x)+a)} \]

[In]

Int[Csch[x]^4/(a + a*Cosh[x]),x]

[Out]

(4*Coth[x])/(5*a) - (4*Coth[x]^3)/(15*a) + Csch[x]^3/(5*(a + a*Cosh[x]))

Rule 2751

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[b*(g*C
os[e + f*x])^(p + 1)*((a + b*Sin[e + f*x])^m/(a*f*g*Simplify[2*m + p + 1])), x] + Dist[Simplify[m + p + 1]/(a*
Simplify[2*m + p + 1]), Int[(g*Cos[e + f*x])^p*(a + b*Sin[e + f*x])^(m + 1), x], x] /; FreeQ[{a, b, e, f, g, m
, p}, x] && EqQ[a^2 - b^2, 0] && ILtQ[Simplify[m + p + 1], 0] && NeQ[2*m + p + 1, 0] &&  !IGtQ[m, 0]

Rule 3852

Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Dist[-d^(-1), Subst[Int[ExpandIntegrand[(1 + x^2)^(n/2 - 1), x]
, x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {\text {csch}^3(x)}{5 (a+a \cosh (x))}+\frac {4 \int \text {csch}^4(x) \, dx}{5 a} \\ & = \frac {\text {csch}^3(x)}{5 (a+a \cosh (x))}+\frac {(4 i) \text {Subst}\left (\int \left (1+x^2\right ) \, dx,x,-i \coth (x)\right )}{5 a} \\ & = \frac {4 \coth (x)}{5 a}-\frac {4 \coth ^3(x)}{15 a}+\frac {\text {csch}^3(x)}{5 (a+a \cosh (x))} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.14 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.03 \[ \int \frac {\text {csch}^4(x)}{a+a \cosh (x)} \, dx=\frac {(-6 \cosh (x)-2 \cosh (2 x)+2 \cosh (3 x)+\cosh (4 x)) \text {csch}^3(x)}{15 a (1+\cosh (x))} \]

[In]

Integrate[Csch[x]^4/(a + a*Cosh[x]),x]

[Out]

((-6*Cosh[x] - 2*Cosh[2*x] + 2*Cosh[3*x] + Cosh[4*x])*Csch[x]^3)/(15*a*(1 + Cosh[x]))

Maple [A] (verified)

Time = 2.64 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.97

method	result	size
risch	\(-\frac {16 \left (6 \,{\mathrm e}^{3 x}+2 \,{\mathrm e}^{2 x}-2 \,{\mathrm e}^{x}-1\right )}{15 a \left ({\mathrm e}^{x}+1\right )^{5} \left ({\mathrm e}^{x}-1\right )^{3}}\)	\(36\)
default	\(\frac {\frac {\tanh \left (\frac {x}{2}\right )^{5}}{5}-\frac {4 \tanh \left (\frac {x}{2}\right )^{3}}{3}+6 \tanh \left (\frac {x}{2}\right )+\frac {4}{\tanh \left (\frac {x}{2}\right )}-\frac {1}{3 \tanh \left (\frac {x}{2}\right )^{3}}}{16 a}\)	\(45\)

[In]

int(csch(x)^4/(a+a*cosh(x)),x,method=_RETURNVERBOSE)

[Out]

-16/15*(6*exp(3*x)+2*exp(2*x)-2*exp(x)-1)/a/(exp(x)+1)^5/(exp(x)-1)^3

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 250 vs. \(2 (31) = 62\).

Time = 0.25 (sec) , antiderivative size = 250, normalized size of antiderivative = 6.76 \[ \int \frac {\text {csch}^4(x)}{a+a \cosh (x)} \, dx=-\frac {16 \, {\left (6 \, \cosh \left (x\right )^{2} + 3 \, {\left (4 \, \cosh \left (x\right ) + 1\right )} \sinh \left (x\right ) + 6 \, \sinh \left (x\right )^{2} + \cosh \left (x\right ) - 2\right )}}{15 \, {\left (a \cosh \left (x\right )^{7} + a \sinh \left (x\right )^{7} + 2 \, a \cosh \left (x\right )^{6} + {\left (7 \, a \cosh \left (x\right ) + 2 \, a\right )} \sinh \left (x\right )^{6} - 2 \, a \cosh \left (x\right )^{5} + {\left (21 \, a \cosh \left (x\right )^{2} + 12 \, a \cosh \left (x\right ) - 2 \, a\right )} \sinh \left (x\right )^{5} - 6 \, a \cosh \left (x\right )^{4} + {\left (35 \, a \cosh \left (x\right )^{3} + 30 \, a \cosh \left (x\right )^{2} - 10 \, a \cosh \left (x\right ) - 6 \, a\right )} \sinh \left (x\right )^{4} + {\left (35 \, a \cosh \left (x\right )^{4} + 40 \, a \cosh \left (x\right )^{3} - 20 \, a \cosh \left (x\right )^{2} - 24 \, a \cosh \left (x\right )\right )} \sinh \left (x\right )^{3} + 6 \, a \cosh \left (x\right )^{2} + {\left (21 \, a \cosh \left (x\right )^{5} + 30 \, a \cosh \left (x\right )^{4} - 20 \, a \cosh \left (x\right )^{3} - 36 \, a \cosh \left (x\right )^{2} + 6 \, a\right )} \sinh \left (x\right )^{2} + a \cosh \left (x\right ) + {\left (7 \, a \cosh \left (x\right )^{6} + 12 \, a \cosh \left (x\right )^{5} - 10 \, a \cosh \left (x\right )^{4} - 24 \, a \cosh \left (x\right )^{3} + 12 \, a \cosh \left (x\right ) + 3 \, a\right )} \sinh \left (x\right ) - 2 \, a\right )}} \]

[In]

integrate(csch(x)^4/(a+a*cosh(x)),x, algorithm="fricas")

[Out]

-16/15*(6*cosh(x)^2 + 3*(4*cosh(x) + 1)*sinh(x) + 6*sinh(x)^2 + cosh(x) - 2)/(a*cosh(x)^7 + a*sinh(x)^7 + 2*a*
cosh(x)^6 + (7*a*cosh(x) + 2*a)*sinh(x)^6 - 2*a*cosh(x)^5 + (21*a*cosh(x)^2 + 12*a*cosh(x) - 2*a)*sinh(x)^5 -
6*a*cosh(x)^4 + (35*a*cosh(x)^3 + 30*a*cosh(x)^2 - 10*a*cosh(x) - 6*a)*sinh(x)^4 + (35*a*cosh(x)^4 + 40*a*cosh
(x)^3 - 20*a*cosh(x)^2 - 24*a*cosh(x))*sinh(x)^3 + 6*a*cosh(x)^2 + (21*a*cosh(x)^5 + 30*a*cosh(x)^4 - 20*a*cos
h(x)^3 - 36*a*cosh(x)^2 + 6*a)*sinh(x)^2 + a*cosh(x) + (7*a*cosh(x)^6 + 12*a*cosh(x)^5 - 10*a*cosh(x)^4 - 24*a
*cosh(x)^3 + 12*a*cosh(x) + 3*a)*sinh(x) - 2*a)

Sympy [F]

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

[In]

integrate(csch(x)**4/(a+a*cosh(x)),x)

[Out]

Integral(csch(x)**4/(cosh(x) + 1), x)/a

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 233 vs. \(2 (31) = 62\).

Time = 0.19 (sec) , antiderivative size = 233, normalized size of antiderivative = 6.30 \[ \int \frac {\text {csch}^4(x)}{a+a \cosh (x)} \, dx=\frac {32 \, e^{\left (-x\right )}}{15 \, {\left (2 \, a e^{\left (-x\right )} - 2 \, a e^{\left (-2 \, x\right )} - 6 \, a e^{\left (-3 \, x\right )} + 6 \, a e^{\left (-5 \, x\right )} + 2 \, a e^{\left (-6 \, x\right )} - 2 \, a e^{\left (-7 \, x\right )} - a e^{\left (-8 \, x\right )} + a\right )}} - \frac {32 \, e^{\left (-2 \, x\right )}}{15 \, {\left (2 \, a e^{\left (-x\right )} - 2 \, a e^{\left (-2 \, x\right )} - 6 \, a e^{\left (-3 \, x\right )} + 6 \, a e^{\left (-5 \, x\right )} + 2 \, a e^{\left (-6 \, x\right )} - 2 \, a e^{\left (-7 \, x\right )} - a e^{\left (-8 \, x\right )} + a\right )}} - \frac {32 \, e^{\left (-3 \, x\right )}}{5 \, {\left (2 \, a e^{\left (-x\right )} - 2 \, a e^{\left (-2 \, x\right )} - 6 \, a e^{\left (-3 \, x\right )} + 6 \, a e^{\left (-5 \, x\right )} + 2 \, a e^{\left (-6 \, x\right )} - 2 \, a e^{\left (-7 \, x\right )} - a e^{\left (-8 \, x\right )} + a\right )}} + \frac {16}{15 \, {\left (2 \, a e^{\left (-x\right )} - 2 \, a e^{\left (-2 \, x\right )} - 6 \, a e^{\left (-3 \, x\right )} + 6 \, a e^{\left (-5 \, x\right )} + 2 \, a e^{\left (-6 \, x\right )} - 2 \, a e^{\left (-7 \, x\right )} - a e^{\left (-8 \, x\right )} + a\right )}} \]

[In]

integrate(csch(x)^4/(a+a*cosh(x)),x, algorithm="maxima")

[Out]

32/15*e^(-x)/(2*a*e^(-x) - 2*a*e^(-2*x) - 6*a*e^(-3*x) + 6*a*e^(-5*x) + 2*a*e^(-6*x) - 2*a*e^(-7*x) - a*e^(-8*
x) + a) - 32/15*e^(-2*x)/(2*a*e^(-x) - 2*a*e^(-2*x) - 6*a*e^(-3*x) + 6*a*e^(-5*x) + 2*a*e^(-6*x) - 2*a*e^(-7*x
) - a*e^(-8*x) + a) - 32/5*e^(-3*x)/(2*a*e^(-x) - 2*a*e^(-2*x) - 6*a*e^(-3*x) + 6*a*e^(-5*x) + 2*a*e^(-6*x) -
2*a*e^(-7*x) - a*e^(-8*x) + a) + 16/15/(2*a*e^(-x) - 2*a*e^(-2*x) - 6*a*e^(-3*x) + 6*a*e^(-5*x) + 2*a*e^(-6*x)
- 2*a*e^(-7*x) - a*e^(-8*x) + a)

Giac [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.59 \[ \int \frac {\text {csch}^4(x)}{a+a \cosh (x)} \, dx=\frac {9 \, e^{\left (2 \, x\right )} - 24 \, e^{x} + 11}{24 \, a {\left (e^{x} - 1\right )}^{3}} - \frac {45 \, e^{\left (4 \, x\right )} + 240 \, e^{\left (3 \, x\right )} + 490 \, e^{\left (2 \, x\right )} + 320 \, e^{x} + 73}{120 \, a {\left (e^{x} + 1\right )}^{5}} \]

[In]

integrate(csch(x)^4/(a+a*cosh(x)),x, algorithm="giac")

[Out]

1/24*(9*e^(2*x) - 24*e^x + 11)/(a*(e^x - 1)^3) - 1/120*(45*e^(4*x) + 240*e^(3*x) + 490*e^(2*x) + 320*e^x + 73)
/(a*(e^x + 1)^5)

Mupad [B] (verification not implemented)

Time = 1.67 (sec) , antiderivative size = 263, normalized size of antiderivative = 7.11 \[ \int \frac {\text {csch}^4(x)}{a+a \cosh (x)} \, dx=\frac {1}{6\,a\,\left (3\,{\mathrm {e}}^{2\,x}-{\mathrm {e}}^{3\,x}-3\,{\mathrm {e}}^x+1\right )}-\frac {\frac {3\,{\mathrm {e}}^{2\,x}}{8\,a}+\frac {3\,{\mathrm {e}}^{3\,x}}{40\,a}+\frac {1}{8\,a}+\frac {5\,{\mathrm {e}}^x}{8\,a}}{6\,{\mathrm {e}}^{2\,x}+4\,{\mathrm {e}}^{3\,x}+{\mathrm {e}}^{4\,x}+4\,{\mathrm {e}}^x+1}-\frac {\frac {3\,{\mathrm {e}}^{2\,x}}{40\,a}+\frac {5}{24\,a}+\frac {{\mathrm {e}}^x}{4\,a}}{3\,{\mathrm {e}}^{2\,x}+{\mathrm {e}}^{3\,x}+3\,{\mathrm {e}}^x+1}-\frac {\frac {1}{8\,a}+\frac {3\,{\mathrm {e}}^x}{40\,a}}{{\mathrm {e}}^{2\,x}+2\,{\mathrm {e}}^x+1}-\frac {\frac {5\,{\mathrm {e}}^{2\,x}}{4\,a}+\frac {{\mathrm {e}}^{3\,x}}{2\,a}+\frac {3\,{\mathrm {e}}^{4\,x}}{40\,a}+\frac {3}{40\,a}+\frac {{\mathrm {e}}^x}{2\,a}}{10\,{\mathrm {e}}^{2\,x}+10\,{\mathrm {e}}^{3\,x}+5\,{\mathrm {e}}^{4\,x}+{\mathrm {e}}^{5\,x}+5\,{\mathrm {e}}^x+1}-\frac {1}{4\,a\,\left ({\mathrm {e}}^{2\,x}-2\,{\mathrm {e}}^x+1\right )}+\frac {3}{8\,a\,\left ({\mathrm {e}}^x-1\right )}-\frac {3}{40\,a\,\left ({\mathrm {e}}^x+1\right )} \]

[In]

int(1/(sinh(x)^4*(a + a*cosh(x))),x)

[Out]

1/(6*a*(3*exp(2*x) - exp(3*x) - 3*exp(x) + 1)) - ((3*exp(2*x))/(8*a) + (3*exp(3*x))/(40*a) + 1/(8*a) + (5*exp(
x))/(8*a))/(6*exp(2*x) + 4*exp(3*x) + exp(4*x) + 4*exp(x) + 1) - ((3*exp(2*x))/(40*a) + 5/(24*a) + exp(x)/(4*a
))/(3*exp(2*x) + exp(3*x) + 3*exp(x) + 1) - (1/(8*a) + (3*exp(x))/(40*a))/(exp(2*x) + 2*exp(x) + 1) - ((5*exp(
2*x))/(4*a) + exp(3*x)/(2*a) + (3*exp(4*x))/(40*a) + 3/(40*a) + exp(x)/(2*a))/(10*exp(2*x) + 10*exp(3*x) + 5*e
xp(4*x) + exp(5*x) + 5*exp(x) + 1) - 1/(4*a*(exp(2*x) - 2*exp(x) + 1)) + 3/(8*a*(exp(x) - 1)) - 3/(40*a*(exp(x
) + 1))