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]

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

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Rubi [A]  time = 0.06, antiderivative size = 101, normalized size of antiderivative = 1.00, 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 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]

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]

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*}

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Mathematica [A]  time = 0.08, size = 51, normalized size = 0.50 \[ \frac {\sinh (c+d x) (29 \cosh (c+d x)-8 \cosh (2 (c+d x))+\cosh (3 (c+d x))-32)}{70 d (\cosh (c+d x)-1)^4} \]

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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fricas [B]  time = 0.54, size = 347, normalized size = 3.44 \[ -\frac {4 \, {\left (35 \, \cosh \left (d x + c\right )^{2} + 10 \, {\left (7 \, \cosh \left (d x + c\right ) - 2\right )} \sinh \left (d x + c\right ) + 35 \, \sinh \left (d x + c\right )^{2} - 22 \, \cosh \left (d x + c\right ) + 7\right )}}{35 \, {\left (d \cosh \left (d x + c\right )^{6} + d \sinh \left (d x + c\right )^{6} - 7 \, d \cosh \left (d x + c\right )^{5} + {\left (6 \, d \cosh \left (d x + c\right ) - 7 \, d\right )} \sinh \left (d x + c\right )^{5} + 21 \, d \cosh \left (d x + c\right )^{4} + {\left (15 \, d \cosh \left (d x + c\right )^{2} - 35 \, d \cosh \left (d x + c\right ) + 21 \, d\right )} \sinh \left (d x + c\right )^{4} - 35 \, d \cosh \left (d x + c\right )^{3} + {\left (20 \, d \cosh \left (d x + c\right )^{3} - 70 \, d \cosh \left (d x + c\right )^{2} + 84 \, d \cosh \left (d x + c\right ) - 35 \, d\right )} \sinh \left (d x + c\right )^{3} + 35 \, d \cosh \left (d x + c\right )^{2} + {\left (15 \, d \cosh \left (d x + c\right )^{4} - 70 \, d \cosh \left (d x + c\right )^{3} + 126 \, d \cosh \left (d x + c\right )^{2} - 105 \, d \cosh \left (d x + c\right ) + 35 \, d\right )} \sinh \left (d x + c\right )^{2} - 22 \, d \cosh \left (d x + c\right ) + {\left (6 \, d \cosh \left (d x + c\right )^{5} - 35 \, d \cosh \left (d x + c\right )^{4} + 84 \, d \cosh \left (d x + c\right )^{3} - 105 \, d \cosh \left (d x + c\right )^{2} + 70 \, d \cosh \left (d x + c\right ) - 20 \, d\right )} \sinh \left (d x + c\right ) + 7 \, d\right )}} \]

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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giac [A]  time = 0.13, size = 47, normalized size = 0.47 \[ -\frac {4 \, {\left (35 \, e^{\left (3 \, d x + 3 \, c\right )} - 21 \, e^{\left (2 \, d x + 2 \, c\right )} + 7 \, e^{\left (d x + c\right )} - 1\right )}}{35 \, d {\left (e^{\left (d x + c\right )} - 1\right )}^{7}} \]

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)

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maple [A]  time = 0.08, size = 58, normalized size = 0.57 \[ \frac {-\frac {1}{56 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}-\frac {1}{8 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}+\frac {1}{8 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}+\frac {3}{40 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [B]  time = 0.33, size = 364, normalized size = 3.60 \[ \frac {4 \, e^{\left (-d x - c\right )}}{5 \, d {\left (7 \, e^{\left (-d x - c\right )} - 21 \, e^{\left (-2 \, d x - 2 \, c\right )} + 35 \, e^{\left (-3 \, d x - 3 \, c\right )} - 35 \, e^{\left (-4 \, d x - 4 \, c\right )} + 21 \, e^{\left (-5 \, d x - 5 \, c\right )} - 7 \, e^{\left (-6 \, d x - 6 \, c\right )} + e^{\left (-7 \, d x - 7 \, c\right )} - 1\right )}} - \frac {12 \, e^{\left (-2 \, d x - 2 \, c\right )}}{5 \, d {\left (7 \, e^{\left (-d x - c\right )} - 21 \, e^{\left (-2 \, d x - 2 \, c\right )} + 35 \, e^{\left (-3 \, d x - 3 \, c\right )} - 35 \, e^{\left (-4 \, d x - 4 \, c\right )} + 21 \, e^{\left (-5 \, d x - 5 \, c\right )} - 7 \, e^{\left (-6 \, d x - 6 \, c\right )} + e^{\left (-7 \, d x - 7 \, c\right )} - 1\right )}} + \frac {4 \, e^{\left (-3 \, d x - 3 \, c\right )}}{d {\left (7 \, e^{\left (-d x - c\right )} - 21 \, e^{\left (-2 \, d x - 2 \, c\right )} + 35 \, e^{\left (-3 \, d x - 3 \, c\right )} - 35 \, e^{\left (-4 \, d x - 4 \, c\right )} + 21 \, e^{\left (-5 \, d x - 5 \, c\right )} - 7 \, e^{\left (-6 \, d x - 6 \, c\right )} + e^{\left (-7 \, d x - 7 \, c\right )} - 1\right )}} - \frac {4}{35 \, d {\left (7 \, e^{\left (-d x - c\right )} - 21 \, e^{\left (-2 \, d x - 2 \, c\right )} + 35 \, e^{\left (-3 \, d x - 3 \, c\right )} - 35 \, e^{\left (-4 \, d x - 4 \, c\right )} + 21 \, e^{\left (-5 \, d x - 5 \, c\right )} - 7 \, e^{\left (-6 \, d x - 6 \, c\right )} + e^{\left (-7 \, d x - 7 \, c\right )} - 1\right )}} \]

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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mupad [B]  time = 0.08, size = 283, normalized size = 2.80 \[ -\frac {4}{35\,d\,\left (6\,{\mathrm {e}}^{2\,c+2\,d\,x}-4\,{\mathrm {e}}^{c+d\,x}-4\,{\mathrm {e}}^{3\,c+3\,d\,x}+{\mathrm {e}}^{4\,c+4\,d\,x}+1\right )}-\frac {16\,{\mathrm {e}}^{c+d\,x}}{35\,d\,\left (5\,{\mathrm {e}}^{c+d\,x}-10\,{\mathrm {e}}^{2\,c+2\,d\,x}+10\,{\mathrm {e}}^{3\,c+3\,d\,x}-5\,{\mathrm {e}}^{4\,c+4\,d\,x}+{\mathrm {e}}^{5\,c+5\,d\,x}-1\right )}-\frac {8\,{\mathrm {e}}^{2\,c+2\,d\,x}}{7\,d\,\left (15\,{\mathrm {e}}^{2\,c+2\,d\,x}-6\,{\mathrm {e}}^{c+d\,x}-20\,{\mathrm {e}}^{3\,c+3\,d\,x}+15\,{\mathrm {e}}^{4\,c+4\,d\,x}-6\,{\mathrm {e}}^{5\,c+5\,d\,x}+{\mathrm {e}}^{6\,c+6\,d\,x}+1\right )}-\frac {16\,{\mathrm {e}}^{3\,c+3\,d\,x}}{7\,d\,\left (7\,{\mathrm {e}}^{c+d\,x}-21\,{\mathrm {e}}^{2\,c+2\,d\,x}+35\,{\mathrm {e}}^{3\,c+3\,d\,x}-35\,{\mathrm {e}}^{4\,c+4\,d\,x}+21\,{\mathrm {e}}^{5\,c+5\,d\,x}-7\,{\mathrm {e}}^{6\,c+6\,d\,x}+{\mathrm {e}}^{7\,c+7\,d\,x}-1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(cosh(c + d*x) - 1)^4,x)

[Out]

- 4/(35*d*(6*exp(2*c + 2*d*x) - 4*exp(c + d*x) - 4*exp(3*c + 3*d*x) + exp(4*c + 4*d*x) + 1)) - (16*exp(c + d*x
))/(35*d*(5*exp(c + d*x) - 10*exp(2*c + 2*d*x) + 10*exp(3*c + 3*d*x) - 5*exp(4*c + 4*d*x) + exp(5*c + 5*d*x) -
1)) - (8*exp(2*c + 2*d*x))/(7*d*(15*exp(2*c + 2*d*x) - 6*exp(c + d*x) - 20*exp(3*c + 3*d*x) + 15*exp(4*c + 4*
d*x) - 6*exp(5*c + 5*d*x) + exp(6*c + 6*d*x) + 1)) - (16*exp(3*c + 3*d*x))/(7*d*(7*exp(c + d*x) - 21*exp(2*c +
2*d*x) + 35*exp(3*c + 3*d*x) - 35*exp(4*c + 4*d*x) + 21*exp(5*c + 5*d*x) - 7*exp(6*c + 6*d*x) + exp(7*c + 7*d
*x) - 1))

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sympy [A]  time = 5.61, size = 87, normalized size = 0.86 \[ \begin {cases} \tilde {\infty } x & \text {for}\: \left (c = 0 \vee c = - d x\right ) \wedge \left (c = - d x \vee d = 0\right ) \\\frac {x}{\left (1 - \cosh {\relax (c )}\right )^{4}} & \text {for}\: d = 0 \\\frac {1}{8 d \tanh {\left (\frac {c}{2} + \frac {d x}{2} \right )}} - \frac {1}{8 d \tanh ^{3}{\left (\frac {c}{2} + \frac {d x}{2} \right )}} + \frac {3}{40 d \tanh ^{5}{\left (\frac {c}{2} + \frac {d x}{2} \right )}} - \frac {1}{56 d \tanh ^{7}{\left (\frac {c}{2} + \frac {d x}{2} \right )}} & \text {otherwise} \end {cases} \]

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(c, -d*x)) & (Eq(d, 0) | Eq(c, -d*x))), (x/(1 - cosh(c))**4, Eq(d, 0)), (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), True))

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