3.39 \(\int \frac{1}{(1-\cosh (c+d x))^4} \, dx\)

Optimal. Leaf size=101 \[ -\frac{2 \sinh (c+d x)}{35 d (1-\cosh (c+d x))}-\frac{2 \sinh (c+d x)}{35 d (1-\cosh (c+d x))^2}-\frac{3 \sinh (c+d x)}{35 d (1-\cosh (c+d x))^3}-\frac{\sinh (c+d x)}{7 d (1-\cosh (c+d x))^4} \]

[Out]

-Sinh[c + d*x]/(7*d*(1 - Cosh[c + d*x])^4) - (3*Sinh[c + d*x])/(35*d*(1 - Cosh[c + d*x])^3) - (2*Sinh[c + d*x]
)/(35*d*(1 - Cosh[c + d*x])^2) - (2*Sinh[c + d*x])/(35*d*(1 - Cosh[c + d*x]))

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Rubi [A]  time = 0.0571527, antiderivative size = 101, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {2650, 2648} \[ -\frac{2 \sinh (c+d x)}{35 d (1-\cosh (c+d x))}-\frac{2 \sinh (c+d x)}{35 d (1-\cosh (c+d x))^2}-\frac{3 \sinh (c+d x)}{35 d (1-\cosh (c+d x))^3}-\frac{\sinh (c+d x)}{7 d (1-\cosh (c+d x))^4} \]

Antiderivative was successfully verified.

[In]

Int[(1 - Cosh[c + d*x])^(-4),x]

[Out]

-Sinh[c + d*x]/(7*d*(1 - Cosh[c + d*x])^4) - (3*Sinh[c + d*x])/(35*d*(1 - Cosh[c + d*x])^3) - (2*Sinh[c + d*x]
)/(35*d*(1 - Cosh[c + d*x])^2) - (2*Sinh[c + d*x])/(35*d*(1 - Cosh[c + d*x]))

Rule 2650

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*Cos[c + d*x]*(a + b*Sin[c + d*x])^n)/(a*
d*(2*n + 1)), x] + Dist[(n + 1)/(a*(2*n + 1)), Int[(a + b*Sin[c + d*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d},
x] && EqQ[a^2 - b^2, 0] && LtQ[n, -1] && IntegerQ[2*n]

Rule 2648

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> -Simp[Cos[c + d*x]/(d*(b + a*Sin[c + d*x])), x]
/; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0]

Rubi steps

\begin{align*} \int \frac{1}{(1-\cosh (c+d x))^4} \, dx &=-\frac{\sinh (c+d x)}{7 d (1-\cosh (c+d x))^4}+\frac{3}{7} \int \frac{1}{(1-\cosh (c+d x))^3} \, dx\\ &=-\frac{\sinh (c+d x)}{7 d (1-\cosh (c+d x))^4}-\frac{3 \sinh (c+d x)}{35 d (1-\cosh (c+d x))^3}+\frac{6}{35} \int \frac{1}{(1-\cosh (c+d x))^2} \, dx\\ &=-\frac{\sinh (c+d x)}{7 d (1-\cosh (c+d x))^4}-\frac{3 \sinh (c+d x)}{35 d (1-\cosh (c+d x))^3}-\frac{2 \sinh (c+d x)}{35 d (1-\cosh (c+d x))^2}+\frac{2}{35} \int \frac{1}{1-\cosh (c+d x)} \, dx\\ &=-\frac{\sinh (c+d x)}{7 d (1-\cosh (c+d x))^4}-\frac{3 \sinh (c+d x)}{35 d (1-\cosh (c+d x))^3}-\frac{2 \sinh (c+d x)}{35 d (1-\cosh (c+d x))^2}-\frac{2 \sinh (c+d x)}{35 d (1-\cosh (c+d x))}\\ \end{align*}

Mathematica [A]  time = 0.0711323, size = 51, normalized size = 0.5 \[ \frac{\sinh (c+d x) (29 \cosh (c+d x)-8 \cosh (2 (c+d x))+\cosh (3 (c+d x))-32)}{70 d (\cosh (c+d x)-1)^4} \]

Antiderivative was successfully verified.

[In]

Integrate[(1 - Cosh[c + d*x])^(-4),x]

[Out]

((-32 + 29*Cosh[c + d*x] - 8*Cosh[2*(c + d*x)] + Cosh[3*(c + d*x)])*Sinh[c + d*x])/(70*d*(-1 + Cosh[c + d*x])^
4)

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Maple [A]  time = 0.014, size = 58, normalized size = 0.6 \begin{align*}{\frac{1}{d} \left ({\frac{1}{8} \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-1}}-{\frac{1}{8} \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-3}}-{\frac{1}{56} \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-7}}+{\frac{3}{40} \left ( \tanh \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-5}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(1-cosh(d*x+c))^4,x)

[Out]

1/d*(1/8/tanh(1/2*d*x+1/2*c)-1/8/tanh(1/2*d*x+1/2*c)^3-1/56/tanh(1/2*d*x+1/2*c)^7+3/40/tanh(1/2*d*x+1/2*c)^5)

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Maxima [B]  time = 1.08714, size = 491, normalized size = 4.86 \begin{align*} \frac{4 \, e^{\left (-d x - c\right )}}{5 \, d{\left (7 \, e^{\left (-d x - c\right )} - 21 \, e^{\left (-2 \, d x - 2 \, c\right )} + 35 \, e^{\left (-3 \, d x - 3 \, c\right )} - 35 \, e^{\left (-4 \, d x - 4 \, c\right )} + 21 \, e^{\left (-5 \, d x - 5 \, c\right )} - 7 \, e^{\left (-6 \, d x - 6 \, c\right )} + e^{\left (-7 \, d x - 7 \, c\right )} - 1\right )}} - \frac{12 \, e^{\left (-2 \, d x - 2 \, c\right )}}{5 \, d{\left (7 \, e^{\left (-d x - c\right )} - 21 \, e^{\left (-2 \, d x - 2 \, c\right )} + 35 \, e^{\left (-3 \, d x - 3 \, c\right )} - 35 \, e^{\left (-4 \, d x - 4 \, c\right )} + 21 \, e^{\left (-5 \, d x - 5 \, c\right )} - 7 \, e^{\left (-6 \, d x - 6 \, c\right )} + e^{\left (-7 \, d x - 7 \, c\right )} - 1\right )}} + \frac{4 \, e^{\left (-3 \, d x - 3 \, c\right )}}{d{\left (7 \, e^{\left (-d x - c\right )} - 21 \, e^{\left (-2 \, d x - 2 \, c\right )} + 35 \, e^{\left (-3 \, d x - 3 \, c\right )} - 35 \, e^{\left (-4 \, d x - 4 \, c\right )} + 21 \, e^{\left (-5 \, d x - 5 \, c\right )} - 7 \, e^{\left (-6 \, d x - 6 \, c\right )} + e^{\left (-7 \, d x - 7 \, c\right )} - 1\right )}} - \frac{4}{35 \, d{\left (7 \, e^{\left (-d x - c\right )} - 21 \, e^{\left (-2 \, d x - 2 \, c\right )} + 35 \, e^{\left (-3 \, d x - 3 \, c\right )} - 35 \, e^{\left (-4 \, d x - 4 \, c\right )} + 21 \, e^{\left (-5 \, d x - 5 \, c\right )} - 7 \, e^{\left (-6 \, d x - 6 \, c\right )} + e^{\left (-7 \, d x - 7 \, c\right )} - 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1-cosh(d*x+c))^4,x, algorithm="maxima")

[Out]

4/5*e^(-d*x - c)/(d*(7*e^(-d*x - c) - 21*e^(-2*d*x - 2*c) + 35*e^(-3*d*x - 3*c) - 35*e^(-4*d*x - 4*c) + 21*e^(
-5*d*x - 5*c) - 7*e^(-6*d*x - 6*c) + e^(-7*d*x - 7*c) - 1)) - 12/5*e^(-2*d*x - 2*c)/(d*(7*e^(-d*x - c) - 21*e^
(-2*d*x - 2*c) + 35*e^(-3*d*x - 3*c) - 35*e^(-4*d*x - 4*c) + 21*e^(-5*d*x - 5*c) - 7*e^(-6*d*x - 6*c) + e^(-7*
d*x - 7*c) - 1)) + 4*e^(-3*d*x - 3*c)/(d*(7*e^(-d*x - c) - 21*e^(-2*d*x - 2*c) + 35*e^(-3*d*x - 3*c) - 35*e^(-
4*d*x - 4*c) + 21*e^(-5*d*x - 5*c) - 7*e^(-6*d*x - 6*c) + e^(-7*d*x - 7*c) - 1)) - 4/35/(d*(7*e^(-d*x - c) - 2
1*e^(-2*d*x - 2*c) + 35*e^(-3*d*x - 3*c) - 35*e^(-4*d*x - 4*c) + 21*e^(-5*d*x - 5*c) - 7*e^(-6*d*x - 6*c) + e^
(-7*d*x - 7*c) - 1))

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Fricas [B]  time = 1.74523, size = 984, normalized size = 9.74 \begin{align*} -\frac{4 \,{\left (35 \, \cosh \left (d x + c\right )^{2} + 10 \,{\left (7 \, \cosh \left (d x + c\right ) - 2\right )} \sinh \left (d x + c\right ) + 35 \, \sinh \left (d x + c\right )^{2} - 22 \, \cosh \left (d x + c\right ) + 7\right )}}{35 \,{\left (d \cosh \left (d x + c\right )^{6} + d \sinh \left (d x + c\right )^{6} - 7 \, d \cosh \left (d x + c\right )^{5} +{\left (6 \, d \cosh \left (d x + c\right ) - 7 \, d\right )} \sinh \left (d x + c\right )^{5} + 21 \, d \cosh \left (d x + c\right )^{4} +{\left (15 \, d \cosh \left (d x + c\right )^{2} - 35 \, d \cosh \left (d x + c\right ) + 21 \, d\right )} \sinh \left (d x + c\right )^{4} - 35 \, d \cosh \left (d x + c\right )^{3} +{\left (20 \, d \cosh \left (d x + c\right )^{3} - 70 \, d \cosh \left (d x + c\right )^{2} + 84 \, d \cosh \left (d x + c\right ) - 35 \, d\right )} \sinh \left (d x + c\right )^{3} + 35 \, d \cosh \left (d x + c\right )^{2} +{\left (15 \, d \cosh \left (d x + c\right )^{4} - 70 \, d \cosh \left (d x + c\right )^{3} + 126 \, d \cosh \left (d x + c\right )^{2} - 105 \, d \cosh \left (d x + c\right ) + 35 \, d\right )} \sinh \left (d x + c\right )^{2} - 22 \, d \cosh \left (d x + c\right ) +{\left (6 \, d \cosh \left (d x + c\right )^{5} - 35 \, d \cosh \left (d x + c\right )^{4} + 84 \, d \cosh \left (d x + c\right )^{3} - 105 \, d \cosh \left (d x + c\right )^{2} + 70 \, d \cosh \left (d x + c\right ) - 20 \, d\right )} \sinh \left (d x + c\right ) + 7 \, d\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1-cosh(d*x+c))^4,x, algorithm="fricas")

[Out]

-4/35*(35*cosh(d*x + c)^2 + 10*(7*cosh(d*x + c) - 2)*sinh(d*x + c) + 35*sinh(d*x + c)^2 - 22*cosh(d*x + c) + 7
)/(d*cosh(d*x + c)^6 + d*sinh(d*x + c)^6 - 7*d*cosh(d*x + c)^5 + (6*d*cosh(d*x + c) - 7*d)*sinh(d*x + c)^5 + 2
1*d*cosh(d*x + c)^4 + (15*d*cosh(d*x + c)^2 - 35*d*cosh(d*x + c) + 21*d)*sinh(d*x + c)^4 - 35*d*cosh(d*x + c)^
3 + (20*d*cosh(d*x + c)^3 - 70*d*cosh(d*x + c)^2 + 84*d*cosh(d*x + c) - 35*d)*sinh(d*x + c)^3 + 35*d*cosh(d*x
+ c)^2 + (15*d*cosh(d*x + c)^4 - 70*d*cosh(d*x + c)^3 + 126*d*cosh(d*x + c)^2 - 105*d*cosh(d*x + c) + 35*d)*si
nh(d*x + c)^2 - 22*d*cosh(d*x + c) + (6*d*cosh(d*x + c)^5 - 35*d*cosh(d*x + c)^4 + 84*d*cosh(d*x + c)^3 - 105*
d*cosh(d*x + c)^2 + 70*d*cosh(d*x + c) - 20*d)*sinh(d*x + c) + 7*d)

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Sympy [A]  time = 9.60131, size = 82, normalized size = 0.81 \begin{align*} \begin{cases} \tilde{\infty } x & \text{for}\: c = 0 \wedge d = 0 \\\frac{x}{\left (1 - \cosh{\left (c \right )}\right )^{4}} & \text{for}\: d = 0 \\\tilde{\infty } x & \text{for}\: c = - d x \\\frac{1}{8 d \tanh{\left (\frac{c}{2} + \frac{d x}{2} \right )}} - \frac{1}{8 d \tanh ^{3}{\left (\frac{c}{2} + \frac{d x}{2} \right )}} + \frac{3}{40 d \tanh ^{5}{\left (\frac{c}{2} + \frac{d x}{2} \right )}} - \frac{1}{56 d \tanh ^{7}{\left (\frac{c}{2} + \frac{d x}{2} \right )}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1-cosh(d*x+c))**4,x)

[Out]

Piecewise((zoo*x, Eq(c, 0) & Eq(d, 0)), (x/(1 - cosh(c))**4, Eq(d, 0)), (zoo*x, Eq(c, -d*x)), (1/(8*d*tanh(c/2
+ d*x/2)) - 1/(8*d*tanh(c/2 + d*x/2)**3) + 3/(40*d*tanh(c/2 + d*x/2)**5) - 1/(56*d*tanh(c/2 + d*x/2)**7), Tru
e))

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Giac [A]  time = 1.13852, size = 63, normalized size = 0.62 \begin{align*} -\frac{4 \,{\left (35 \, e^{\left (3 \, d x + 3 \, c\right )} - 21 \, e^{\left (2 \, d x + 2 \, c\right )} + 7 \, e^{\left (d x + c\right )} - 1\right )}}{35 \, d{\left (e^{\left (d x + c\right )} - 1\right )}^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1-cosh(d*x+c))^4,x, algorithm="giac")

[Out]

-4/35*(35*e^(3*d*x + 3*c) - 21*e^(2*d*x + 2*c) + 7*e^(d*x + c) - 1)/(d*(e^(d*x + c) - 1)^7)