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

Optimal. Leaf size=37 \[ -\frac{4 \coth ^3(x)}{15 a}+\frac{4 \coth (x)}{5 a}+\frac{\text{csch}^3(x)}{5 (a \cosh (x)+a)} \]

[Out]

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

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Rubi [A]  time = 0.0507936, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {2672, 3767} \[ -\frac{4 \coth ^3(x)}{15 a}+\frac{4 \coth (x)}{5 a}+\frac{\text{csch}^3(x)}{5 (a \cosh (x)+a)} \]

Antiderivative was successfully verified.

[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 2672

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[(b*(g*
Cos[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 3767

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*} \int \frac{\text{csch}^4(x)}{a+a \cosh (x)} \, dx &=\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) \operatorname{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]  time = 0.0533196, size = 38, normalized size = 1.03 \[ \frac{(-6 \cosh (x)-2 \cosh (2 x)+2 \cosh (3 x)+\cosh (4 x)) \text{csch}^3(x)}{15 a (\cosh (x)+1)} \]

Antiderivative was successfully verified.

[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]))

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Maple [A]  time = 0.02, size = 45, normalized size = 1.2 \begin{align*}{\frac{1}{16\,a} \left ({\frac{1}{5} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{5}}-{\frac{4}{3} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{3}}+6\,\tanh \left ( x/2 \right ) +4\, \left ( \tanh \left ( x/2 \right ) \right ) ^{-1}-{\frac{1}{3} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{-3}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/16/a*(1/5*tanh(1/2*x)^5-4/3*tanh(1/2*x)^3+6*tanh(1/2*x)+4/tanh(1/2*x)-1/3/tanh(1/2*x)^3)

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Maxima [B]  time = 1.03411, size = 315, normalized size = 8.51 \begin{align*} \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 )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [B]  time = 1.75972, size = 809, normalized size = 21.86 \begin{align*} -\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 )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [A]  time = 1.1316, size = 80, normalized size = 2.16 \begin{align*} \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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)