\(\int \frac {1}{(1+\cosh (c+d x))^4} \, dx\) [35]

Optimal result

Integrand size = 10, antiderivative size = 93 \[ \int \frac {1}{(1+\cosh (c+d x))^4} \, dx=\frac {\sinh (c+d x)}{7 d (1+\cosh (c+d x))^4}+\frac {3 \sinh (c+d x)}{35 d (1+\cosh (c+d x))^3}+\frac {2 \sinh (c+d x)}{35 d (1+\cosh (c+d x))^2}+\frac {2 \sinh (c+d x)}{35 d (1+\cosh (c+d x))} \]

[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))

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2729, 2727} \[ \int \frac {1}{(1+\cosh (c+d x))^4} \, dx=\frac {2 \sinh (c+d x)}{35 d (\cosh (c+d x)+1)}+\frac {2 \sinh (c+d x)}{35 d (\cosh (c+d x)+1)^2}+\frac {3 \sinh (c+d x)}{35 d (\cosh (c+d x)+1)^3}+\frac {\sinh (c+d x)}{7 d (\cosh (c+d x)+1)^4} \]

[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 2727

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 2729

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*} \text {integral}& = \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] (verified)

Time = 0.07 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.58 \[ \int \frac {1}{(1+\cosh (c+d x))^4} \, dx=\frac {56 \sinh (c+d x)+28 \sinh (2 (c+d x))+8 \sinh (3 (c+d x))+\sinh (4 (c+d x))}{140 d (1+\cosh (c+d x))^4} \]

[In]

Integrate[(1 + Cosh[c + d*x])^(-4),x]

[Out]

(56*Sinh[c + d*x] + 28*Sinh[2*(c + d*x)] + 8*Sinh[3*(c + d*x)] + Sinh[4*(c + d*x)])/(140*d*(1 + Cosh[c + d*x])
^4)

Maple [A] (verified)

Time = 0.08 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.52

method	result	size
risch	\(-\frac {4 \left (35 \,{\mathrm e}^{3 d x +3 c}+21 \,{\mathrm e}^{2 d x +2 c}+7 \,{\mathrm e}^{d x +c}+1\right )}{35 d \left ({\mathrm e}^{d x +c}+1\right )^{7}}\)	\(48\)
parallelrisch	\(-\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}-\frac {21 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{5}+7 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-7\right )}{56 d}\)	\(54\)
derivativedivides	\(\frac {-\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{56}+\frac {3 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{40}-\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{8}+\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{8}}{d}\)	\(56\)
default	\(\frac {-\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{56}+\frac {3 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{40}-\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{8}+\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{8}}{d}\)	\(56\)

[In]

int(1/(cosh(d*x+c)+1)^4,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 347 vs. \(2 (85) = 170\).

Time = 0.25 (sec) , antiderivative size = 347, normalized size of antiderivative = 3.73 \[ \int \frac {1}{(1+\cosh (c+d x))^4} \, dx=-\frac {4 \, {\left (35 \, \cosh \left (d x + c\right )^{2} + 10 \, {\left (7 \, \cosh \left (d x + c\right ) + 2\right )} \sinh \left (d x + c\right ) + 35 \, \sinh \left (d x + c\right )^{2} + 22 \, \cosh \left (d x + c\right ) + 7\right )}}{35 \, {\left (d \cosh \left (d x + c\right )^{6} + d \sinh \left (d x + c\right )^{6} + 7 \, d \cosh \left (d x + c\right )^{5} + {\left (6 \, d \cosh \left (d x + c\right ) + 7 \, d\right )} \sinh \left (d x + c\right )^{5} + 21 \, d \cosh \left (d x + c\right )^{4} + {\left (15 \, d \cosh \left (d x + c\right )^{2} + 35 \, d \cosh \left (d x + c\right ) + 21 \, d\right )} \sinh \left (d x + c\right )^{4} + 35 \, d \cosh \left (d x + c\right )^{3} + {\left (20 \, d \cosh \left (d x + c\right )^{3} + 70 \, d \cosh \left (d x + c\right )^{2} + 84 \, d \cosh \left (d x + c\right ) + 35 \, d\right )} \sinh \left (d x + c\right )^{3} + 35 \, d \cosh \left (d x + c\right )^{2} + {\left (15 \, d \cosh \left (d x + c\right )^{4} + 70 \, d \cosh \left (d x + c\right )^{3} + 126 \, d \cosh \left (d x + c\right )^{2} + 105 \, d \cosh \left (d x + c\right ) + 35 \, d\right )} \sinh \left (d x + c\right )^{2} + 22 \, d \cosh \left (d x + c\right ) + {\left (6 \, d \cosh \left (d x + c\right )^{5} + 35 \, d \cosh \left (d x + c\right )^{4} + 84 \, d \cosh \left (d x + c\right )^{3} + 105 \, d \cosh \left (d x + c\right )^{2} + 70 \, d \cosh \left (d x + c\right ) + 20 \, d\right )} \sinh \left (d x + c\right ) + 7 \, d\right )}} \]

[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)

Sympy [A] (verification not implemented)

Time = 1.75 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.73 \[ \int \frac {1}{(1+\cosh (c+d x))^4} \, dx=\begin {cases} - \frac {\tanh ^{7}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{56 d} + \frac {3 \tanh ^{5}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{40 d} - \frac {\tanh ^{3}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{8 d} + \frac {\tanh {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{8 d} & \text {for}\: d \neq 0 \\\frac {x}{\left (\cosh {\left (c \right )} + 1\right )^{4}} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise((-tanh(c/2 + d*x/2)**7/(56*d) + 3*tanh(c/2 + d*x/2)**5/(40*d) - tanh(c/2 + d*x/2)**3/(8*d) + tanh(c/
2 + d*x/2)/(8*d), Ne(d, 0)), (x/(cosh(c) + 1)**4, True))

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 364 vs. \(2 (85) = 170\).

Time = 0.19 (sec) , antiderivative size = 364, normalized size of antiderivative = 3.91 \[ \int \frac {1}{(1+\cosh (c+d x))^4} \, dx=\frac {4 \, e^{\left (-d x - c\right )}}{5 \, d {\left (7 \, e^{\left (-d x - c\right )} + 21 \, e^{\left (-2 \, d x - 2 \, c\right )} + 35 \, e^{\left (-3 \, d x - 3 \, c\right )} + 35 \, e^{\left (-4 \, d x - 4 \, c\right )} + 21 \, e^{\left (-5 \, d x - 5 \, c\right )} + 7 \, e^{\left (-6 \, d x - 6 \, c\right )} + e^{\left (-7 \, d x - 7 \, c\right )} + 1\right )}} + \frac {12 \, e^{\left (-2 \, d x - 2 \, c\right )}}{5 \, d {\left (7 \, e^{\left (-d x - c\right )} + 21 \, e^{\left (-2 \, d x - 2 \, c\right )} + 35 \, e^{\left (-3 \, d x - 3 \, c\right )} + 35 \, e^{\left (-4 \, d x - 4 \, c\right )} + 21 \, e^{\left (-5 \, d x - 5 \, c\right )} + 7 \, e^{\left (-6 \, d x - 6 \, c\right )} + e^{\left (-7 \, d x - 7 \, c\right )} + 1\right )}} + \frac {4 \, e^{\left (-3 \, d x - 3 \, c\right )}}{d {\left (7 \, e^{\left (-d x - c\right )} + 21 \, e^{\left (-2 \, d x - 2 \, c\right )} + 35 \, e^{\left (-3 \, d x - 3 \, c\right )} + 35 \, e^{\left (-4 \, d x - 4 \, c\right )} + 21 \, e^{\left (-5 \, d x - 5 \, c\right )} + 7 \, e^{\left (-6 \, d x - 6 \, c\right )} + e^{\left (-7 \, d x - 7 \, c\right )} + 1\right )}} + \frac {4}{35 \, d {\left (7 \, e^{\left (-d x - c\right )} + 21 \, e^{\left (-2 \, d x - 2 \, c\right )} + 35 \, e^{\left (-3 \, d x - 3 \, c\right )} + 35 \, e^{\left (-4 \, d x - 4 \, c\right )} + 21 \, e^{\left (-5 \, d x - 5 \, c\right )} + 7 \, e^{\left (-6 \, d x - 6 \, c\right )} + e^{\left (-7 \, d x - 7 \, c\right )} + 1\right )}} \]

[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))

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.51 \[ \int \frac {1}{(1+\cosh (c+d x))^4} \, dx=-\frac {4 \, {\left (35 \, e^{\left (3 \, d x + 3 \, c\right )} + 21 \, e^{\left (2 \, d x + 2 \, c\right )} + 7 \, e^{\left (d x + c\right )} + 1\right )}}{35 \, d {\left (e^{\left (d x + c\right )} + 1\right )}^{7}} \]

[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)

Mupad [B] (verification not implemented)

Time = 1.72 (sec) , antiderivative size = 283, normalized size of antiderivative = 3.04 \[ \int \frac {1}{(1+\cosh (c+d x))^4} \, dx=-\frac {4}{35\,d\,\left (4\,{\mathrm {e}}^{c+d\,x}+6\,{\mathrm {e}}^{2\,c+2\,d\,x}+4\,{\mathrm {e}}^{3\,c+3\,d\,x}+{\mathrm {e}}^{4\,c+4\,d\,x}+1\right )}-\frac {16\,{\mathrm {e}}^{c+d\,x}}{35\,d\,\left (5\,{\mathrm {e}}^{c+d\,x}+10\,{\mathrm {e}}^{2\,c+2\,d\,x}+10\,{\mathrm {e}}^{3\,c+3\,d\,x}+5\,{\mathrm {e}}^{4\,c+4\,d\,x}+{\mathrm {e}}^{5\,c+5\,d\,x}+1\right )}-\frac {8\,{\mathrm {e}}^{2\,c+2\,d\,x}}{7\,d\,\left (6\,{\mathrm {e}}^{c+d\,x}+15\,{\mathrm {e}}^{2\,c+2\,d\,x}+20\,{\mathrm {e}}^{3\,c+3\,d\,x}+15\,{\mathrm {e}}^{4\,c+4\,d\,x}+6\,{\mathrm {e}}^{5\,c+5\,d\,x}+{\mathrm {e}}^{6\,c+6\,d\,x}+1\right )}-\frac {16\,{\mathrm {e}}^{3\,c+3\,d\,x}}{7\,d\,\left (7\,{\mathrm {e}}^{c+d\,x}+21\,{\mathrm {e}}^{2\,c+2\,d\,x}+35\,{\mathrm {e}}^{3\,c+3\,d\,x}+35\,{\mathrm {e}}^{4\,c+4\,d\,x}+21\,{\mathrm {e}}^{5\,c+5\,d\,x}+7\,{\mathrm {e}}^{6\,c+6\,d\,x}+{\mathrm {e}}^{7\,c+7\,d\,x}+1\right )} \]

[In]

int(1/(cosh(c + d*x) + 1)^4,x)

[Out]

- 4/(35*d*(4*exp(c + d*x) + 6*exp(2*c + 2*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*(6*exp(c + d*x) + 15*exp(2*c + 2*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))