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]

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

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Rubi [A]  time = 0.05, antiderivative size = 37, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.06, 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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fricas [B]  time = 1.09, size = 250, normalized size = 6.76 \[ -\frac {16 \, {\left (6 \, \cosh \relax (x)^{2} + 3 \, {\left (4 \, \cosh \relax (x) + 1\right )} \sinh \relax (x) + 6 \, \sinh \relax (x)^{2} + \cosh \relax (x) - 2\right )}}{15 \, {\left (a \cosh \relax (x)^{7} + a \sinh \relax (x)^{7} + 2 \, a \cosh \relax (x)^{6} + {\left (7 \, a \cosh \relax (x) + 2 \, a\right )} \sinh \relax (x)^{6} - 2 \, a \cosh \relax (x)^{5} + {\left (21 \, a \cosh \relax (x)^{2} + 12 \, a \cosh \relax (x) - 2 \, a\right )} \sinh \relax (x)^{5} - 6 \, a \cosh \relax (x)^{4} + {\left (35 \, a \cosh \relax (x)^{3} + 30 \, a \cosh \relax (x)^{2} - 10 \, a \cosh \relax (x) - 6 \, a\right )} \sinh \relax (x)^{4} + {\left (35 \, a \cosh \relax (x)^{4} + 40 \, a \cosh \relax (x)^{3} - 20 \, a \cosh \relax (x)^{2} - 24 \, a \cosh \relax (x)\right )} \sinh \relax (x)^{3} + 6 \, a \cosh \relax (x)^{2} + {\left (21 \, a \cosh \relax (x)^{5} + 30 \, a \cosh \relax (x)^{4} - 20 \, a \cosh \relax (x)^{3} - 36 \, a \cosh \relax (x)^{2} + 6 \, a\right )} \sinh \relax (x)^{2} + a \cosh \relax (x) + {\left (7 \, a \cosh \relax (x)^{6} + 12 \, a \cosh \relax (x)^{5} - 10 \, a \cosh \relax (x)^{4} - 24 \, a \cosh \relax (x)^{3} + 12 \, a \cosh \relax (x) + 3 \, a\right )} \sinh \relax (x) - 2 \, a\right )}} \]

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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giac [A]  time = 0.15, size = 59, normalized size = 1.59 \[ \frac {9 \, e^{\left (2 \, x\right )} - 24 \, e^{x} + 11}{24 \, a {\left (e^{x} - 1\right )}^{3}} - \frac {45 \, e^{\left (4 \, x\right )} + 240 \, e^{\left (3 \, x\right )} + 490 \, e^{\left (2 \, x\right )} + 320 \, e^{x} + 73}{120 \, a {\left (e^{x} + 1\right )}^{5}} \]

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)

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maple [A]  time = 0.09, size = 45, normalized size = 1.22 \[ \frac {\frac {\left (\tanh ^{5}\left (\frac {x}{2}\right )\right )}{5}-\frac {4 \left (\tanh ^{3}\left (\frac {x}{2}\right )\right )}{3}+6 \tanh \left (\frac {x}{2}\right )-\frac {1}{3 \tanh \left (\frac {x}{2}\right )^{3}}+\frac {4}{\tanh \left (\frac {x}{2}\right )}}{16 a} \]

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

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maxima [B]  time = 0.31, size = 233, normalized size = 6.30 \[ \frac {32 \, e^{\left (-x\right )}}{15 \, {\left (2 \, a e^{\left (-x\right )} - 2 \, a e^{\left (-2 \, x\right )} - 6 \, a e^{\left (-3 \, x\right )} + 6 \, a e^{\left (-5 \, x\right )} + 2 \, a e^{\left (-6 \, x\right )} - 2 \, a e^{\left (-7 \, x\right )} - a e^{\left (-8 \, x\right )} + a\right )}} - \frac {32 \, e^{\left (-2 \, x\right )}}{15 \, {\left (2 \, a e^{\left (-x\right )} - 2 \, a e^{\left (-2 \, x\right )} - 6 \, a e^{\left (-3 \, x\right )} + 6 \, a e^{\left (-5 \, x\right )} + 2 \, a e^{\left (-6 \, x\right )} - 2 \, a e^{\left (-7 \, x\right )} - a e^{\left (-8 \, x\right )} + a\right )}} - \frac {32 \, e^{\left (-3 \, x\right )}}{5 \, {\left (2 \, a e^{\left (-x\right )} - 2 \, a e^{\left (-2 \, x\right )} - 6 \, a e^{\left (-3 \, x\right )} + 6 \, a e^{\left (-5 \, x\right )} + 2 \, a e^{\left (-6 \, x\right )} - 2 \, a e^{\left (-7 \, x\right )} - a e^{\left (-8 \, x\right )} + a\right )}} + \frac {16}{15 \, {\left (2 \, a e^{\left (-x\right )} - 2 \, a e^{\left (-2 \, x\right )} - 6 \, a e^{\left (-3 \, x\right )} + 6 \, a e^{\left (-5 \, x\right )} + 2 \, a e^{\left (-6 \, x\right )} - 2 \, a e^{\left (-7 \, x\right )} - a e^{\left (-8 \, x\right )} + a\right )}} \]

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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mupad [B]  time = 0.95, size = 263, normalized size = 7.11 \[ \frac {1}{6\,a\,\left (3\,{\mathrm {e}}^{2\,x}-{\mathrm {e}}^{3\,x}-3\,{\mathrm {e}}^x+1\right )}-\frac {\frac {3\,{\mathrm {e}}^{2\,x}}{8\,a}+\frac {3\,{\mathrm {e}}^{3\,x}}{40\,a}+\frac {1}{8\,a}+\frac {5\,{\mathrm {e}}^x}{8\,a}}{6\,{\mathrm {e}}^{2\,x}+4\,{\mathrm {e}}^{3\,x}+{\mathrm {e}}^{4\,x}+4\,{\mathrm {e}}^x+1}-\frac {\frac {3\,{\mathrm {e}}^{2\,x}}{40\,a}+\frac {5}{24\,a}+\frac {{\mathrm {e}}^x}{4\,a}}{3\,{\mathrm {e}}^{2\,x}+{\mathrm {e}}^{3\,x}+3\,{\mathrm {e}}^x+1}-\frac {\frac {1}{8\,a}+\frac {3\,{\mathrm {e}}^x}{40\,a}}{{\mathrm {e}}^{2\,x}+2\,{\mathrm {e}}^x+1}-\frac {\frac {5\,{\mathrm {e}}^{2\,x}}{4\,a}+\frac {{\mathrm {e}}^{3\,x}}{2\,a}+\frac {3\,{\mathrm {e}}^{4\,x}}{40\,a}+\frac {3}{40\,a}+\frac {{\mathrm {e}}^x}{2\,a}}{10\,{\mathrm {e}}^{2\,x}+10\,{\mathrm {e}}^{3\,x}+5\,{\mathrm {e}}^{4\,x}+{\mathrm {e}}^{5\,x}+5\,{\mathrm {e}}^x+1}-\frac {1}{4\,a\,\left ({\mathrm {e}}^{2\,x}-2\,{\mathrm {e}}^x+1\right )}+\frac {3}{8\,a\,\left ({\mathrm {e}}^x-1\right )}-\frac {3}{40\,a\,\left ({\mathrm {e}}^x+1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[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))

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {\operatorname {csch}^{4}{\relax (x )}}{\cosh {\relax (x )} + 1}\, dx}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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