3.5 \(\int \frac{\text{csch}^4(x)}{a-a \cosh ^2(x)} \, dx\)

Optimal. Leaf size=29 \[ \frac{\coth ^5(x)}{5 a}-\frac{2 \coth ^3(x)}{3 a}+\frac{\coth (x)}{a} \]

[Out]

Coth[x]/a - (2*Coth[x]^3)/(3*a) + Coth[x]^5/(5*a)

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Rubi [A]  time = 0.0508984, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {3175, 3767} \[ \frac{\coth ^5(x)}{5 a}-\frac{2 \coth ^3(x)}{3 a}+\frac{\coth (x)}{a} \]

Antiderivative was successfully verified.

[In]

Int[Csch[x]^4/(a - a*Cosh[x]^2),x]

[Out]

Coth[x]/a - (2*Coth[x]^3)/(3*a) + Coth[x]^5/(5*a)

Rule 3175

Int[(u_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> Dist[a^p, Int[ActivateTrig[u*cos[e + f*x
]^(2*p)], x], x] /; FreeQ[{a, b, e, f, p}, x] && EqQ[a + b, 0] && IntegerQ[p]

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 ^2(x)} \, dx &=-\frac{\int \text{csch}^6(x) \, dx}{a}\\ &=\frac{i \operatorname{Subst}\left (\int \left (1+2 x^2+x^4\right ) \, dx,x,-i \coth (x)\right )}{a}\\ &=\frac{\coth (x)}{a}-\frac{2 \coth ^3(x)}{3 a}+\frac{\coth ^5(x)}{5 a}\\ \end{align*}

Mathematica [A]  time = 0.0037742, size = 32, normalized size = 1.1 \[ -\frac{-\frac{8 \coth (x)}{15}-\frac{1}{5} \coth (x) \text{csch}^4(x)+\frac{4}{15} \coth (x) \text{csch}^2(x)}{a} \]

Antiderivative was successfully verified.

[In]

Integrate[Csch[x]^4/(a - a*Cosh[x]^2),x]

[Out]

-(((-8*Coth[x])/15 + (4*Coth[x]*Csch[x]^2)/15 - (Coth[x]*Csch[x]^4)/5)/a)

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Maple [B]  time = 0.028, size = 53, normalized size = 1.8 \begin{align*}{\frac{1}{32\,a} \left ({\frac{1}{5} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{5}}-{\frac{5}{3} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{3}}+10\,\tanh \left ( x/2 \right ) +10\, \left ( \tanh \left ( x/2 \right ) \right ) ^{-1}-{\frac{5}{3} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{-3}}+{\frac{1}{5} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{-5}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csch(x)^4/(a-a*cosh(x)^2),x)

[Out]

1/32/a*(1/5*tanh(1/2*x)^5-5/3*tanh(1/2*x)^3+10*tanh(1/2*x)+10/tanh(1/2*x)-5/3/tanh(1/2*x)^3+1/5/tanh(1/2*x)^5)

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Maxima [B]  time = 1.08527, size = 182, normalized size = 6.28 \begin{align*} \frac{16 \, e^{\left (-2 \, x\right )}}{3 \,{\left (5 \, a e^{\left (-2 \, x\right )} - 10 \, a e^{\left (-4 \, x\right )} + 10 \, a e^{\left (-6 \, x\right )} - 5 \, a e^{\left (-8 \, x\right )} + a e^{\left (-10 \, x\right )} - a\right )}} - \frac{32 \, e^{\left (-4 \, x\right )}}{3 \,{\left (5 \, a e^{\left (-2 \, x\right )} - 10 \, a e^{\left (-4 \, x\right )} + 10 \, a e^{\left (-6 \, x\right )} - 5 \, a e^{\left (-8 \, x\right )} + a e^{\left (-10 \, x\right )} - a\right )}} - \frac{16}{15 \,{\left (5 \, a e^{\left (-2 \, x\right )} - 10 \, a e^{\left (-4 \, x\right )} + 10 \, a e^{\left (-6 \, x\right )} - 5 \, a e^{\left (-8 \, x\right )} + a e^{\left (-10 \, 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)^2),x, algorithm="maxima")

[Out]

16/3*e^(-2*x)/(5*a*e^(-2*x) - 10*a*e^(-4*x) + 10*a*e^(-6*x) - 5*a*e^(-8*x) + a*e^(-10*x) - a) - 32/3*e^(-4*x)/
(5*a*e^(-2*x) - 10*a*e^(-4*x) + 10*a*e^(-6*x) - 5*a*e^(-8*x) + a*e^(-10*x) - a) - 16/15/(5*a*e^(-2*x) - 10*a*e
^(-4*x) + 10*a*e^(-6*x) - 5*a*e^(-8*x) + a*e^(-10*x) - a)

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Fricas [B]  time = 1.80449, size = 694, normalized size = 23.93 \begin{align*} \frac{16 \,{\left (11 \, \cosh \left (x\right )^{2} + 18 \, \cosh \left (x\right ) \sinh \left (x\right ) + 11 \, \sinh \left (x\right )^{2} - 5\right )}}{15 \,{\left (a \cosh \left (x\right )^{8} + 8 \, a \cosh \left (x\right ) \sinh \left (x\right )^{7} + a \sinh \left (x\right )^{8} - 5 \, a \cosh \left (x\right )^{6} +{\left (28 \, a \cosh \left (x\right )^{2} - 5 \, a\right )} \sinh \left (x\right )^{6} + 2 \,{\left (28 \, a \cosh \left (x\right )^{3} - 15 \, a \cosh \left (x\right )\right )} \sinh \left (x\right )^{5} + 10 \, a \cosh \left (x\right )^{4} + 5 \,{\left (14 \, a \cosh \left (x\right )^{4} - 15 \, a \cosh \left (x\right )^{2} + 2 \, a\right )} \sinh \left (x\right )^{4} + 4 \,{\left (14 \, a \cosh \left (x\right )^{5} - 25 \, a \cosh \left (x\right )^{3} + 10 \, a \cosh \left (x\right )\right )} \sinh \left (x\right )^{3} - 11 \, a \cosh \left (x\right )^{2} +{\left (28 \, a \cosh \left (x\right )^{6} - 75 \, a \cosh \left (x\right )^{4} + 60 \, a \cosh \left (x\right )^{2} - 11 \, a\right )} \sinh \left (x\right )^{2} + 2 \,{\left (4 \, a \cosh \left (x\right )^{7} - 15 \, a \cosh \left (x\right )^{5} + 20 \, a \cosh \left (x\right )^{3} - 9 \, a \cosh \left (x\right )\right )} \sinh \left (x\right ) + 5 \, a\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(x)^4/(a-a*cosh(x)^2),x, algorithm="fricas")

[Out]

16/15*(11*cosh(x)^2 + 18*cosh(x)*sinh(x) + 11*sinh(x)^2 - 5)/(a*cosh(x)^8 + 8*a*cosh(x)*sinh(x)^7 + a*sinh(x)^
8 - 5*a*cosh(x)^6 + (28*a*cosh(x)^2 - 5*a)*sinh(x)^6 + 2*(28*a*cosh(x)^3 - 15*a*cosh(x))*sinh(x)^5 + 10*a*cosh
(x)^4 + 5*(14*a*cosh(x)^4 - 15*a*cosh(x)^2 + 2*a)*sinh(x)^4 + 4*(14*a*cosh(x)^5 - 25*a*cosh(x)^3 + 10*a*cosh(x
))*sinh(x)^3 - 11*a*cosh(x)^2 + (28*a*cosh(x)^6 - 75*a*cosh(x)^4 + 60*a*cosh(x)^2 - 11*a)*sinh(x)^2 + 2*(4*a*c
osh(x)^7 - 15*a*cosh(x)^5 + 20*a*cosh(x)^3 - 9*a*cosh(x))*sinh(x) + 5*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)**2),x)

[Out]

Timed out

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Giac [A]  time = 1.28108, size = 36, normalized size = 1.24 \begin{align*} \frac{16 \,{\left (10 \, e^{\left (4 \, x\right )} - 5 \, e^{\left (2 \, x\right )} + 1\right )}}{15 \, a{\left (e^{\left (2 \, x\right )} - 1\right )}^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(x)^4/(a-a*cosh(x)^2),x, algorithm="giac")

[Out]

16/15*(10*e^(4*x) - 5*e^(2*x) + 1)/(a*(e^(2*x) - 1)^5)