\(\int \frac {1}{(3+5 \cosh (c+d x))^3} \, dx\) [73]

Optimal result

Integrand size = 12, antiderivative size = 73 \[ \int \frac {1}{(3+5 \cosh (c+d x))^3} \, dx=\frac {43 \arctan \left (\frac {1}{2} \tanh \left (\frac {1}{2} (c+d x)\right )\right )}{1024 d}+\frac {5 \sinh (c+d x)}{32 d (3+5 \cosh (c+d x))^2}-\frac {45 \sinh (c+d x)}{512 d (3+5 \cosh (c+d x))} \] Output:

43/1024*arctan(1/2*tanh(1/2*d*x+1/2*c))/d+5/32*sinh(d*x+c)/d/(3+5*cosh(d*x 
+c))^2-45/512*sinh(d*x+c)/d/(3+5*cosh(d*x+c))
 

Mathematica [A] (verified)

Time = 0.13 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.73 \[ \int \frac {1}{(3+5 \cosh (c+d x))^3} \, dx=-\frac {43 \arctan \left (2 \coth \left (\frac {1}{2} (c+d x)\right )\right )+\frac {10 (11+45 \cosh (c+d x)) \sinh (c+d x)}{(3+5 \cosh (c+d x))^2}}{1024 d} \] Input:

Integrate[(3 + 5*Cosh[c + d*x])^(-3),x]
 

Output:

-1/1024*(43*ArcTan[2*Coth[(c + d*x)/2]] + (10*(11 + 45*Cosh[c + d*x])*Sinh 
[c + d*x])/(3 + 5*Cosh[c + d*x])^2)/d
 

Rubi [A] (verified)

Time = 0.39 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.07, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {3042, 3143, 25, 3042, 3233, 27, 3042, 3138, 219}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {1}{(5 \cosh (c+d x)+3)^3} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {1}{\left (3+5 \sin \left (i c+i d x+\frac {\pi }{2}\right )\right )^3}dx\)

\(\Big \downarrow \) 3143

\(\displaystyle \frac {1}{32} \int -\frac {6-5 \cosh (c+d x)}{(5 \cosh (c+d x)+3)^2}dx+\frac {5 \sinh (c+d x)}{32 d (5 \cosh (c+d x)+3)^2}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {5 \sinh (c+d x)}{32 d (5 \cosh (c+d x)+3)^2}-\frac {1}{32} \int \frac {6-5 \cosh (c+d x)}{(5 \cosh (c+d x)+3)^2}dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {5 \sinh (c+d x)}{32 d (5 \cosh (c+d x)+3)^2}-\frac {1}{32} \int \frac {6-5 \sin \left (i c+i d x+\frac {\pi }{2}\right )}{\left (5 \sin \left (i c+i d x+\frac {\pi }{2}\right )+3\right )^2}dx\)

\(\Big \downarrow \) 3233

\(\displaystyle \frac {1}{32} \left (-\frac {1}{16} \int -\frac {43}{5 \cosh (c+d x)+3}dx-\frac {45 \sinh (c+d x)}{16 d (5 \cosh (c+d x)+3)}\right )+\frac {5 \sinh (c+d x)}{32 d (5 \cosh (c+d x)+3)^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{32} \left (\frac {43}{16} \int \frac {1}{5 \cosh (c+d x)+3}dx-\frac {45 \sinh (c+d x)}{16 d (5 \cosh (c+d x)+3)}\right )+\frac {5 \sinh (c+d x)}{32 d (5 \cosh (c+d x)+3)^2}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {5 \sinh (c+d x)}{32 d (5 \cosh (c+d x)+3)^2}+\frac {1}{32} \left (-\frac {45 \sinh (c+d x)}{16 d (5 \cosh (c+d x)+3)}+\frac {43}{16} \int \frac {1}{5 \sin \left (i c+i d x+\frac {\pi }{2}\right )+3}dx\right )\)

\(\Big \downarrow \) 3138

\(\displaystyle \frac {5 \sinh (c+d x)}{32 d (5 \cosh (c+d x)+3)^2}+\frac {1}{32} \left (-\frac {45 \sinh (c+d x)}{16 d (5 \cosh (c+d x)+3)}-\frac {43 i \int \frac {1}{2 \tanh ^2\left (\frac {1}{2} (c+d x)\right )+8}d\left (i \tanh \left (\frac {1}{2} (c+d x)\right )\right )}{8 d}\right )\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {1}{32} \left (\frac {43 \arctan \left (\frac {1}{2} \tanh \left (\frac {1}{2} (c+d x)\right )\right )}{32 d}-\frac {45 \sinh (c+d x)}{16 d (5 \cosh (c+d x)+3)}\right )+\frac {5 \sinh (c+d x)}{32 d (5 \cosh (c+d x)+3)^2}\)

Input:

Int[(3 + 5*Cosh[c + d*x])^(-3),x]
 

Output:

(5*Sinh[c + d*x])/(32*d*(3 + 5*Cosh[c + d*x])^2) + ((43*ArcTan[Tanh[(c + d 
*x)/2]/2])/(32*d) - (45*Sinh[c + d*x])/(16*d*(3 + 5*Cosh[c + d*x])))/32
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 219

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* 
ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3138

Int[((a_) + (b_.)*sin[Pi/2 + (c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{ 
e = FreeFactors[Tan[(c + d*x)/2], x]}, Simp[2*(e/d)   Subst[Int[1/(a + b + 
(a - b)*e^2*x^2), x], x, Tan[(c + d*x)/2]/e], x]] /; FreeQ[{a, b, c, d}, x] 
 && NeQ[a^2 - b^2, 0]
 

rule 3143

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

rule 3233

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + 
(f_.)*(x_)]), x_Symbol] :> Simp[(-(b*c - a*d))*Cos[e + f*x]*((a + b*Sin[e + 
 f*x])^(m + 1)/(f*(m + 1)*(a^2 - b^2))), x] + Simp[1/((m + 1)*(a^2 - b^2)) 
  Int[(a + b*Sin[e + f*x])^(m + 1)*Simp[(a*c - b*d)*(m + 1) - (b*c - a*d)*( 
m + 2)*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c 
- a*d, 0] && NeQ[a^2 - b^2, 0] && LtQ[m, -1] && IntegerQ[2*m]
 

Maple [A] (verified)

Time = 0.76 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.85

method	result	size
derivativedivides	\(\frac {\frac {-\frac {85 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{128}-\frac {35 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{32}}{4 \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+4\right )^{2}}+\frac {43 \arctan \left (\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{2}\right )}{1024}}{d}\)	\(62\)
default	\(\frac {\frac {-\frac {85 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{128}-\frac {35 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{32}}{4 \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+4\right )^{2}}+\frac {43 \arctan \left (\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{2}\right )}{1024}}{d}\)	\(62\)
risch	\(\frac {215 \,{\mathrm e}^{3 d x +3 c}+387 \,{\mathrm e}^{2 d x +2 c}+325 \,{\mathrm e}^{d x +c}+225}{256 d \left (5 \,{\mathrm e}^{2 d x +2 c}+6 \,{\mathrm e}^{d x +c}+5\right )^{2}}+\frac {43 i \ln \left ({\mathrm e}^{d x +c}+\frac {3}{5}+\frac {4 i}{5}\right )}{2048 d}-\frac {43 i \ln \left ({\mathrm e}^{d x +c}+\frac {3}{5}-\frac {4 i}{5}\right )}{2048 d}\)	\(96\)
parallelrisch	\(\frac {43 i \left (-25 \cosh \left (2 d x +2 c \right )-43-60 \cosh \left (d x +c \right )\right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-2 i\right )+43 i \left (25 \cosh \left (2 d x +2 c \right )+43+60 \cosh \left (d x +c \right )\right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+2 i\right )-440 \sinh \left (d x +c \right )-900 \sinh \left (2 d x +2 c \right )}{2048 d \left (25 \cosh \left (2 d x +2 c \right )+43+60 \cosh \left (d x +c \right )\right )}\)	\(123\)

Input:

int(1/(3+5*cosh(d*x+c))^3,x,method=_RETURNVERBOSE)
 

Output:

1/d*(1/4*(-85/128*tanh(1/2*d*x+1/2*c)^3-35/32*tanh(1/2*d*x+1/2*c))/(tanh(1 
/2*d*x+1/2*c)^2+4)^2+43/1024*arctan(1/2*tanh(1/2*d*x+1/2*c)))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 408 vs. \(2 (64) = 128\).

Time = 0.10 (sec) , antiderivative size = 408, normalized size of antiderivative = 5.59 \[ \int \frac {1}{(3+5 \cosh (c+d x))^3} \, dx=\frac {860 \, \cosh \left (d x + c\right )^{3} + 516 \, {\left (5 \, \cosh \left (d x + c\right ) + 3\right )} \sinh \left (d x + c\right )^{2} + 860 \, \sinh \left (d x + c\right )^{3} + 43 \, {\left (25 \, \cosh \left (d x + c\right )^{4} + 20 \, {\left (5 \, \cosh \left (d x + c\right ) + 3\right )} \sinh \left (d x + c\right )^{3} + 25 \, \sinh \left (d x + c\right )^{4} + 60 \, \cosh \left (d x + c\right )^{3} + 2 \, {\left (75 \, \cosh \left (d x + c\right )^{2} + 90 \, \cosh \left (d x + c\right ) + 43\right )} \sinh \left (d x + c\right )^{2} + 86 \, \cosh \left (d x + c\right )^{2} + 4 \, {\left (25 \, \cosh \left (d x + c\right )^{3} + 45 \, \cosh \left (d x + c\right )^{2} + 43 \, \cosh \left (d x + c\right ) + 15\right )} \sinh \left (d x + c\right ) + 60 \, \cosh \left (d x + c\right ) + 25\right )} \arctan \left (\frac {5}{4} \, \cosh \left (d x + c\right ) + \frac {5}{4} \, \sinh \left (d x + c\right ) + \frac {3}{4}\right ) + 1548 \, \cosh \left (d x + c\right )^{2} + 4 \, {\left (645 \, \cosh \left (d x + c\right )^{2} + 774 \, \cosh \left (d x + c\right ) + 325\right )} \sinh \left (d x + c\right ) + 1300 \, \cosh \left (d x + c\right ) + 900}{1024 \, {\left (25 \, d \cosh \left (d x + c\right )^{4} + 25 \, d \sinh \left (d x + c\right )^{4} + 60 \, d \cosh \left (d x + c\right )^{3} + 20 \, {\left (5 \, d \cosh \left (d x + c\right ) + 3 \, d\right )} \sinh \left (d x + c\right )^{3} + 86 \, d \cosh \left (d x + c\right )^{2} + 2 \, {\left (75 \, d \cosh \left (d x + c\right )^{2} + 90 \, d \cosh \left (d x + c\right ) + 43 \, d\right )} \sinh \left (d x + c\right )^{2} + 60 \, d \cosh \left (d x + c\right ) + 4 \, {\left (25 \, d \cosh \left (d x + c\right )^{3} + 45 \, d \cosh \left (d x + c\right )^{2} + 43 \, d \cosh \left (d x + c\right ) + 15 \, d\right )} \sinh \left (d x + c\right ) + 25 \, d\right )}} \] Input:

integrate(1/(3+5*cosh(d*x+c))^3,x, algorithm="fricas")
 

Output:

1/1024*(860*cosh(d*x + c)^3 + 516*(5*cosh(d*x + c) + 3)*sinh(d*x + c)^2 + 
860*sinh(d*x + c)^3 + 43*(25*cosh(d*x + c)^4 + 20*(5*cosh(d*x + c) + 3)*si 
nh(d*x + c)^3 + 25*sinh(d*x + c)^4 + 60*cosh(d*x + c)^3 + 2*(75*cosh(d*x + 
 c)^2 + 90*cosh(d*x + c) + 43)*sinh(d*x + c)^2 + 86*cosh(d*x + c)^2 + 4*(2 
5*cosh(d*x + c)^3 + 45*cosh(d*x + c)^2 + 43*cosh(d*x + c) + 15)*sinh(d*x + 
 c) + 60*cosh(d*x + c) + 25)*arctan(5/4*cosh(d*x + c) + 5/4*sinh(d*x + c) 
+ 3/4) + 1548*cosh(d*x + c)^2 + 4*(645*cosh(d*x + c)^2 + 774*cosh(d*x + c) 
 + 325)*sinh(d*x + c) + 1300*cosh(d*x + c) + 900)/(25*d*cosh(d*x + c)^4 + 
25*d*sinh(d*x + c)^4 + 60*d*cosh(d*x + c)^3 + 20*(5*d*cosh(d*x + c) + 3*d) 
*sinh(d*x + c)^3 + 86*d*cosh(d*x + c)^2 + 2*(75*d*cosh(d*x + c)^2 + 90*d*c 
osh(d*x + c) + 43*d)*sinh(d*x + c)^2 + 60*d*cosh(d*x + c) + 4*(25*d*cosh(d 
*x + c)^3 + 45*d*cosh(d*x + c)^2 + 43*d*cosh(d*x + c) + 15*d)*sinh(d*x + c 
) + 25*d)
 

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 2.46 (sec) , antiderivative size = 507, normalized size of antiderivative = 6.95 \[ \int \frac {1}{(3+5 \cosh (c+d x))^3} \, dx =\text {Too large to display} \] Input:

integrate(1/(3+5*cosh(d*x+c))**3,x)
 

Output:

Piecewise((-log(-3*exp(-d*x) - 4*I*exp(-d*x))/(125*d*cosh(d*x + log(-3*exp 
(-d*x) - 4*I*exp(-d*x)) - log(5))**3 + 225*d*cosh(d*x + log(-3*exp(-d*x) - 
 4*I*exp(-d*x)) - log(5))**2 + 135*d*cosh(d*x + log(-3*exp(-d*x) - 4*I*exp 
(-d*x)) - log(5)) + 27*d), Eq(c, log((-3 - 4*I)*exp(-d*x)) - log(5))), (x/ 
(125*cosh(d*x + log(-3*exp(-d*x) + 4*I*exp(-d*x)) - log(5))**3 + 225*cosh( 
d*x + log(-3*exp(-d*x) + 4*I*exp(-d*x)) - log(5))**2 + 135*cosh(d*x + log( 
-3*exp(-d*x) + 4*I*exp(-d*x)) - log(5)) + 27), Eq(c, log((-3 + 4*I)*exp(-d 
*x)) - log(5))), (x/(5*cosh(c) + 3)**3, Eq(d, 0)), (43*tanh(c/2 + d*x/2)** 
4*atan(tanh(c/2 + d*x/2)/2)/(1024*d*tanh(c/2 + d*x/2)**4 + 8192*d*tanh(c/2 
 + d*x/2)**2 + 16384*d) - 170*tanh(c/2 + d*x/2)**3/(1024*d*tanh(c/2 + d*x/ 
2)**4 + 8192*d*tanh(c/2 + d*x/2)**2 + 16384*d) + 344*tanh(c/2 + d*x/2)**2* 
atan(tanh(c/2 + d*x/2)/2)/(1024*d*tanh(c/2 + d*x/2)**4 + 8192*d*tanh(c/2 + 
 d*x/2)**2 + 16384*d) - 280*tanh(c/2 + d*x/2)/(1024*d*tanh(c/2 + d*x/2)**4 
 + 8192*d*tanh(c/2 + d*x/2)**2 + 16384*d) + 688*atan(tanh(c/2 + d*x/2)/2)/ 
(1024*d*tanh(c/2 + d*x/2)**4 + 8192*d*tanh(c/2 + d*x/2)**2 + 16384*d), Tru 
e))
 

Maxima [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.48 \[ \int \frac {1}{(3+5 \cosh (c+d x))^3} \, dx=-\frac {43 \, \arctan \left (\frac {5}{4} \, e^{\left (-d x - c\right )} + \frac {3}{4}\right )}{1024 \, d} - \frac {325 \, e^{\left (-d x - c\right )} + 387 \, e^{\left (-2 \, d x - 2 \, c\right )} + 215 \, e^{\left (-3 \, d x - 3 \, c\right )} + 225}{256 \, d {\left (60 \, e^{\left (-d x - c\right )} + 86 \, e^{\left (-2 \, d x - 2 \, c\right )} + 60 \, e^{\left (-3 \, d x - 3 \, c\right )} + 25 \, e^{\left (-4 \, d x - 4 \, c\right )} + 25\right )}} \] Input:

integrate(1/(3+5*cosh(d*x+c))^3,x, algorithm="maxima")
 

Output:

-43/1024*arctan(5/4*e^(-d*x - c) + 3/4)/d - 1/256*(325*e^(-d*x - c) + 387* 
e^(-2*d*x - 2*c) + 215*e^(-3*d*x - 3*c) + 225)/(d*(60*e^(-d*x - c) + 86*e^ 
(-2*d*x - 2*c) + 60*e^(-3*d*x - 3*c) + 25*e^(-4*d*x - 4*c) + 25))
 

Giac [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.04 \[ \int \frac {1}{(3+5 \cosh (c+d x))^3} \, dx=\frac {\frac {4 \, {\left (215 \, e^{\left (3 \, d x + 3 \, c\right )} + 387 \, e^{\left (2 \, d x + 2 \, c\right )} + 325 \, e^{\left (d x + c\right )} + 225\right )}}{{\left (5 \, e^{\left (2 \, d x + 2 \, c\right )} + 6 \, e^{\left (d x + c\right )} + 5\right )}^{2}} + 43 \, \arctan \left (\frac {5}{4} \, e^{\left (d x + c\right )} + \frac {3}{4}\right )}{1024 \, d} \] Input:

integrate(1/(3+5*cosh(d*x+c))^3,x, algorithm="giac")
 

Output:

1/1024*(4*(215*e^(3*d*x + 3*c) + 387*e^(2*d*x + 2*c) + 325*e^(d*x + c) + 2 
25)/(5*e^(2*d*x + 2*c) + 6*e^(d*x + c) + 5)^2 + 43*arctan(5/4*e^(d*x + c) 
+ 3/4))/d
 

Mupad [B] (verification not implemented)

Time = 2.07 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.88 \[ \int \frac {1}{(3+5 \cosh (c+d x))^3} \, dx=\frac {\frac {43\,{\mathrm {e}}^{c+d\,x}}{256\,d}+\frac {129}{1280\,d}}{6\,{\mathrm {e}}^{c+d\,x}+5\,{\mathrm {e}}^{2\,c+2\,d\,x}+5}-\frac {\frac {7\,{\mathrm {e}}^{c+d\,x}}{40\,d}-\frac {3}{8\,d}}{60\,{\mathrm {e}}^{c+d\,x}+86\,{\mathrm {e}}^{2\,c+2\,d\,x}+60\,{\mathrm {e}}^{3\,c+3\,d\,x}+25\,{\mathrm {e}}^{4\,c+4\,d\,x}+25}+\frac {43\,\mathrm {atan}\left (\left (\frac {3}{4\,d}+\frac {5\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c}{4\,d}\right )\,\sqrt {d^2}\right )}{1024\,\sqrt {d^2}} \] Input:

int(1/(5*cosh(c + d*x) + 3)^3,x)
 

Output:

((43*exp(c + d*x))/(256*d) + 129/(1280*d))/(6*exp(c + d*x) + 5*exp(2*c + 2 
*d*x) + 5) - ((7*exp(c + d*x))/(40*d) - 3/(8*d))/(60*exp(c + d*x) + 86*exp 
(2*c + 2*d*x) + 60*exp(3*c + 3*d*x) + 25*exp(4*c + 4*d*x) + 25) + (43*atan 
((3/(4*d) + (5*exp(d*x)*exp(c))/(4*d))*(d^2)^(1/2)))/(1024*(d^2)^(1/2))
 

Reduce [B] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 196, normalized size of antiderivative = 2.68 \[ \int \frac {1}{(3+5 \cosh (c+d x))^3} \, dx=\frac {3225 e^{4 d x +4 c} \mathit {atan} \left (\frac {5 e^{d x +c}}{4}+\frac {3}{4}\right )+7740 e^{3 d x +3 c} \mathit {atan} \left (\frac {5 e^{d x +c}}{4}+\frac {3}{4}\right )+11094 e^{2 d x +2 c} \mathit {atan} \left (\frac {5 e^{d x +c}}{4}+\frac {3}{4}\right )+7740 e^{d x +c} \mathit {atan} \left (\frac {5 e^{d x +c}}{4}+\frac {3}{4}\right )+3225 \mathit {atan} \left (\frac {5 e^{d x +c}}{4}+\frac {3}{4}\right )-1075 e^{4 d x +4 c}+946 e^{2 d x +2 c}+1320 e^{d x +c}+1625}{3072 d \left (25 e^{4 d x +4 c}+60 e^{3 d x +3 c}+86 e^{2 d x +2 c}+60 e^{d x +c}+25\right )} \] Input:

int(1/(3+5*cosh(d*x+c))^3,x)
 

Output:

(3225*e**(4*c + 4*d*x)*atan((5*e**(c + d*x) + 3)/4) + 7740*e**(3*c + 3*d*x 
)*atan((5*e**(c + d*x) + 3)/4) + 11094*e**(2*c + 2*d*x)*atan((5*e**(c + d* 
x) + 3)/4) + 7740*e**(c + d*x)*atan((5*e**(c + d*x) + 3)/4) + 3225*atan((5 
*e**(c + d*x) + 3)/4) - 1075*e**(4*c + 4*d*x) + 946*e**(2*c + 2*d*x) + 132 
0*e**(c + d*x) + 1625)/(3072*d*(25*e**(4*c + 4*d*x) + 60*e**(3*c + 3*d*x) 
+ 86*e**(2*c + 2*d*x) + 60*e**(c + d*x) + 25))