Integrand size = 12, antiderivative size = 81 \[ \int \frac {1}{(5+3 \cosh (c+d x))^3} \, dx=\frac {59 x}{2048}-\frac {59 \text {arctanh}\left (\frac {\sinh (c+d x)}{3+\cosh (c+d x)}\right )}{1024 d}-\frac {3 \sinh (c+d x)}{32 d (5+3 \cosh (c+d x))^2}-\frac {45 \sinh (c+d x)}{512 d (5+3 \cosh (c+d x))} \] Output:
59/2048*x-59/1024*arctanh(sinh(d*x+c)/(3+cosh(d*x+c)))/d-3/32*sinh(d*x+c)/ d/(5+3*cosh(d*x+c))^2-45/512*sinh(d*x+c)/d/(5+3*cosh(d*x+c))
Leaf count is larger than twice the leaf count of optimal. \(217\) vs. \(2(81)=162\).
Time = 0.16 (sec) , antiderivative size = 217, normalized size of antiderivative = 2.68 \[ \int \frac {1}{(5+3 \cosh (c+d x))^3} \, dx=-\frac {59 \log \left (2 \cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )\right )}{2048 d}+\frac {59 \log \left (2 \cosh \left (\frac {1}{2} (c+d x)\right )+\sinh \left (\frac {1}{2} (c+d x)\right )\right )}{2048 d}-\frac {3}{512 d \left (2 \cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )\right )^2}-\frac {45 \sinh \left (\frac {1}{2} (c+d x)\right )}{2048 d \left (2 \cosh \left (\frac {1}{2} (c+d x)\right )-\sinh \left (\frac {1}{2} (c+d x)\right )\right )}+\frac {3}{512 d \left (2 \cosh \left (\frac {1}{2} (c+d x)\right )+\sinh \left (\frac {1}{2} (c+d x)\right )\right )^2}-\frac {45 \sinh \left (\frac {1}{2} (c+d x)\right )}{2048 d \left (2 \cosh \left (\frac {1}{2} (c+d x)\right )+\sinh \left (\frac {1}{2} (c+d x)\right )\right )} \] Input:
Integrate[(5 + 3*Cosh[c + d*x])^(-3),x]
Output:
(-59*Log[2*Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2]])/(2048*d) + (59*Log[2*Co sh[(c + d*x)/2] + Sinh[(c + d*x)/2]])/(2048*d) - 3/(512*d*(2*Cosh[(c + d*x )/2] - Sinh[(c + d*x)/2])^2) - (45*Sinh[(c + d*x)/2])/(2048*d*(2*Cosh[(c + d*x)/2] - Sinh[(c + d*x)/2])) + 3/(512*d*(2*Cosh[(c + d*x)/2] + Sinh[(c + d*x)/2])^2) - (45*Sinh[(c + d*x)/2])/(2048*d*(2*Cosh[(c + d*x)/2] + Sinh[ (c + d*x)/2]))
Time = 0.39 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.12, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {3042, 3143, 25, 3042, 3233, 27, 3042, 3136}
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.
| \(\displaystyle \int \frac {1}{(3 \cosh (c+d x)+5)^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\left (5+3 \sin \left (i c+i d x+\frac {\pi }{2}\right )\right )^3}dx\) |
| \(\Big \downarrow \) 3143 |
| \(\displaystyle -\frac {1}{32} \int -\frac {10-3 \cosh (c+d x)}{(3 \cosh (c+d x)+5)^2}dx-\frac {3 \sinh (c+d x)}{32 d (3 \cosh (c+d x)+5)^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{32} \int \frac {10-3 \cosh (c+d x)}{(3 \cosh (c+d x)+5)^2}dx-\frac {3 \sinh (c+d x)}{32 d (3 \cosh (c+d x)+5)^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {3 \sinh (c+d x)}{32 d (3 \cosh (c+d x)+5)^2}+\frac {1}{32} \int \frac {10-3 \sin \left (i c+i d x+\frac {\pi }{2}\right )}{\left (3 \sin \left (i c+i d x+\frac {\pi }{2}\right )+5\right )^2}dx\) |
| \(\Big \downarrow \) 3233 |
| \(\displaystyle \frac {1}{32} \left (-\frac {1}{16} \int -\frac {59}{3 \cosh (c+d x)+5}dx-\frac {45 \sinh (c+d x)}{16 d (3 \cosh (c+d x)+5)}\right )-\frac {3 \sinh (c+d x)}{32 d (3 \cosh (c+d x)+5)^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{32} \left (\frac {59}{16} \int \frac {1}{3 \cosh (c+d x)+5}dx-\frac {45 \sinh (c+d x)}{16 d (3 \cosh (c+d x)+5)}\right )-\frac {3 \sinh (c+d x)}{32 d (3 \cosh (c+d x)+5)^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {3 \sinh (c+d x)}{32 d (3 \cosh (c+d x)+5)^2}+\frac {1}{32} \left (-\frac {45 \sinh (c+d x)}{16 d (3 \cosh (c+d x)+5)}+\frac {59}{16} \int \frac {1}{3 \sin \left (i c+i d x+\frac {\pi }{2}\right )+5}dx\right )\) |
| \(\Big \downarrow \) 3136 |
| \(\displaystyle \frac {1}{32} \left (\frac {59}{16} \left (\frac {x}{4}-\frac {\text {arctanh}\left (\frac {\sinh (c+d x)}{\cosh (c+d x)+3}\right )}{2 d}\right )-\frac {45 \sinh (c+d x)}{16 d (3 \cosh (c+d x)+5)}\right )-\frac {3 \sinh (c+d x)}{32 d (3 \cosh (c+d x)+5)^2}\) |
Input:
Int[(5 + 3*Cosh[c + d*x])^(-3),x]
Output:
(-3*Sinh[c + d*x])/(32*d*(5 + 3*Cosh[c + d*x])^2) + ((59*(x/4 - ArcTanh[Si nh[c + d*x]/(3 + Cosh[c + d*x])]/(2*d)))/16 - (45*Sinh[c + d*x])/(16*d*(5 + 3*Cosh[c + d*x])))/32
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{q = Rt[ a^2 - b^2, 2]}, Simp[x/q, x] + Simp[(2/(d*q))*ArcTan[b*(Cos[c + d*x]/(a + q + b*Sin[c + d*x]))], x]] /; FreeQ[{a, b, c, d}, x] && GtQ[a^2 - b^2, 0] && PosQ[a]
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]
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]
Time = 0.62 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.11
| method | result | size |
| risch | \(\frac {\frac {177 \,{\mathrm e}^{3 d x +3 c}}{256}+\frac {885 \,{\mathrm e}^{2 d x +2 c}}{256}+\frac {723 \,{\mathrm e}^{d x +c}}{256}+\frac {135}{256}}{d \left (3 \,{\mathrm e}^{2 d x +2 c}+10 \,{\mathrm e}^{d x +c}+3\right )^{2}}-\frac {59 \ln \left (3+{\mathrm e}^{d x +c}\right )}{2048 d}+\frac {59 \ln \left ({\mathrm e}^{d x +c}+\frac {1}{3}\right )}{2048 d}\) | \(90\) |
| derivativedivides | \(\frac {-\frac {9}{512 \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+2\right )^{2}}+\frac {69}{1024 \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+2\right )}+\frac {59 \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+2\right )}{2048}+\frac {9}{512 \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-2\right )^{2}}+\frac {69}{1024 \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-2\right )}-\frac {59 \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-2\right )}{2048}}{d}\) | \(94\) |
| default | \(\frac {-\frac {9}{512 \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+2\right )^{2}}+\frac {69}{1024 \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+2\right )}+\frac {59 \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+2\right )}{2048}+\frac {9}{512 \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-2\right )^{2}}+\frac {69}{1024 \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-2\right )}-\frac {59 \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-2\right )}{2048}}{d}\) | \(94\) |
| parallelrisch | \(\frac {\left (-3540 \cosh \left (d x +c \right )-531 \cosh \left (2 d x +2 c \right )-3481\right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-2\right )+\left (3540 \cosh \left (d x +c \right )+531 \cosh \left (2 d x +2 c \right )+3481\right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+2\right )-2184 \sinh \left (d x +c \right )-540 \sinh \left (2 d x +2 c \right )}{2048 d \left (59+9 \cosh \left (2 d x +2 c \right )+60 \cosh \left (d x +c \right )\right )}\) | \(117\) |
Input:
int(1/(5+3*cosh(d*x+c))^3,x,method=_RETURNVERBOSE)
Output:
3/256*(59*exp(3*d*x+3*c)+295*exp(2*d*x+2*c)+241*exp(d*x+c)+45)/d/(3*exp(2* d*x+2*c)+10*exp(d*x+c)+3)^2-59/2048/d*ln(3+exp(d*x+c))+59/2048/d*ln(exp(d* x+c)+1/3)
Leaf count of result is larger than twice the leaf count of optimal. 563 vs. \(2 (73) = 146\).
Time = 0.13 (sec) , antiderivative size = 563, normalized size of antiderivative = 6.95 \[ \int \frac {1}{(5+3 \cosh (c+d x))^3} \, dx =\text {Too large to display} \] Input:
integrate(1/(5+3*cosh(d*x+c))^3,x, algorithm="fricas")
Output:
1/2048*(1416*cosh(d*x + c)^3 + 1416*(3*cosh(d*x + c) + 5)*sinh(d*x + c)^2 + 1416*sinh(d*x + c)^3 + 7080*cosh(d*x + c)^2 + 59*(9*cosh(d*x + c)^4 + 12 *(3*cosh(d*x + c) + 5)*sinh(d*x + c)^3 + 9*sinh(d*x + c)^4 + 60*cosh(d*x + c)^3 + 2*(27*cosh(d*x + c)^2 + 90*cosh(d*x + c) + 59)*sinh(d*x + c)^2 + 1 18*cosh(d*x + c)^2 + 4*(9*cosh(d*x + c)^3 + 45*cosh(d*x + c)^2 + 59*cosh(d *x + c) + 15)*sinh(d*x + c) + 60*cosh(d*x + c) + 9)*log(3*cosh(d*x + c) + 3*sinh(d*x + c) + 1) - 59*(9*cosh(d*x + c)^4 + 12*(3*cosh(d*x + c) + 5)*si nh(d*x + c)^3 + 9*sinh(d*x + c)^4 + 60*cosh(d*x + c)^3 + 2*(27*cosh(d*x + c)^2 + 90*cosh(d*x + c) + 59)*sinh(d*x + c)^2 + 118*cosh(d*x + c)^2 + 4*(9 *cosh(d*x + c)^3 + 45*cosh(d*x + c)^2 + 59*cosh(d*x + c) + 15)*sinh(d*x + c) + 60*cosh(d*x + c) + 9)*log(cosh(d*x + c) + sinh(d*x + c) + 3) + 24*(17 7*cosh(d*x + c)^2 + 590*cosh(d*x + c) + 241)*sinh(d*x + c) + 5784*cosh(d*x + c) + 1080)/(9*d*cosh(d*x + c)^4 + 9*d*sinh(d*x + c)^4 + 60*d*cosh(d*x + c)^3 + 12*(3*d*cosh(d*x + c) + 5*d)*sinh(d*x + c)^3 + 118*d*cosh(d*x + c) ^2 + 2*(27*d*cosh(d*x + c)^2 + 90*d*cosh(d*x + c) + 59*d)*sinh(d*x + c)^2 + 60*d*cosh(d*x + c) + 4*(9*d*cosh(d*x + c)^3 + 45*d*cosh(d*x + c)^2 + 59* d*cosh(d*x + c) + 15*d)*sinh(d*x + c) + 9*d)
Leaf count of result is larger than twice the leaf count of optimal. 445 vs. \(2 (71) = 142\).
Time = 1.60 (sec) , antiderivative size = 445, normalized size of antiderivative = 5.49 \[ \int \frac {1}{(5+3 \cosh (c+d x))^3} \, dx =\text {Too large to display} \] Input:
integrate(1/(5+3*cosh(d*x+c))**3,x)
Output:
Piecewise((-59*log(tanh(c/2 + d*x/2) - 2)*tanh(c/2 + d*x/2)**4/(2048*d*tan h(c/2 + d*x/2)**4 - 16384*d*tanh(c/2 + d*x/2)**2 + 32768*d) + 472*log(tanh (c/2 + d*x/2) - 2)*tanh(c/2 + d*x/2)**2/(2048*d*tanh(c/2 + d*x/2)**4 - 163 84*d*tanh(c/2 + d*x/2)**2 + 32768*d) - 944*log(tanh(c/2 + d*x/2) - 2)/(204 8*d*tanh(c/2 + d*x/2)**4 - 16384*d*tanh(c/2 + d*x/2)**2 + 32768*d) + 59*lo g(tanh(c/2 + d*x/2) + 2)*tanh(c/2 + d*x/2)**4/(2048*d*tanh(c/2 + d*x/2)**4 - 16384*d*tanh(c/2 + d*x/2)**2 + 32768*d) - 472*log(tanh(c/2 + d*x/2) + 2 )*tanh(c/2 + d*x/2)**2/(2048*d*tanh(c/2 + d*x/2)**4 - 16384*d*tanh(c/2 + d *x/2)**2 + 32768*d) + 944*log(tanh(c/2 + d*x/2) + 2)/(2048*d*tanh(c/2 + d* x/2)**4 - 16384*d*tanh(c/2 + d*x/2)**2 + 32768*d) + 276*tanh(c/2 + d*x/2)* *3/(2048*d*tanh(c/2 + d*x/2)**4 - 16384*d*tanh(c/2 + d*x/2)**2 + 32768*d) - 816*tanh(c/2 + d*x/2)/(2048*d*tanh(c/2 + d*x/2)**4 - 16384*d*tanh(c/2 + d*x/2)**2 + 32768*d), Ne(d, 0)), (x/(3*cosh(c) + 5)**3, True))
Time = 0.04 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.54 \[ \int \frac {1}{(5+3 \cosh (c+d x))^3} \, dx=-\frac {59 \, \log \left (3 \, e^{\left (-d x - c\right )} + 1\right )}{2048 \, d} + \frac {59 \, \log \left (e^{\left (-d x - c\right )} + 3\right )}{2048 \, d} - \frac {3 \, {\left (241 \, e^{\left (-d x - c\right )} + 295 \, e^{\left (-2 \, d x - 2 \, c\right )} + 59 \, e^{\left (-3 \, d x - 3 \, c\right )} + 45\right )}}{256 \, d {\left (60 \, e^{\left (-d x - c\right )} + 118 \, e^{\left (-2 \, d x - 2 \, c\right )} + 60 \, e^{\left (-3 \, d x - 3 \, c\right )} + 9 \, e^{\left (-4 \, d x - 4 \, c\right )} + 9\right )}} \] Input:
integrate(1/(5+3*cosh(d*x+c))^3,x, algorithm="maxima")
Output:
-59/2048*log(3*e^(-d*x - c) + 1)/d + 59/2048*log(e^(-d*x - c) + 3)/d - 3/2 56*(241*e^(-d*x - c) + 295*e^(-2*d*x - 2*c) + 59*e^(-3*d*x - 3*c) + 45)/(d *(60*e^(-d*x - c) + 118*e^(-2*d*x - 2*c) + 60*e^(-3*d*x - 3*c) + 9*e^(-4*d *x - 4*c) + 9))
Time = 0.11 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.07 \[ \int \frac {1}{(5+3 \cosh (c+d x))^3} \, dx=\frac {\frac {24 \, {\left (59 \, e^{\left (3 \, d x + 3 \, c\right )} + 295 \, e^{\left (2 \, d x + 2 \, c\right )} + 241 \, e^{\left (d x + c\right )} + 45\right )}}{{\left (3 \, e^{\left (2 \, d x + 2 \, c\right )} + 10 \, e^{\left (d x + c\right )} + 3\right )}^{2}} + 59 \, \log \left (3 \, e^{\left (d x + c\right )} + 1\right ) - 59 \, \log \left (e^{\left (d x + c\right )} + 3\right )}{2048 \, d} \] Input:
integrate(1/(5+3*cosh(d*x+c))^3,x, algorithm="giac")
Output:
1/2048*(24*(59*e^(3*d*x + 3*c) + 295*e^(2*d*x + 2*c) + 241*e^(d*x + c) + 4 5)/(3*e^(2*d*x + 2*c) + 10*e^(d*x + c) + 3)^2 + 59*log(3*e^(d*x + c) + 1) - 59*log(e^(d*x + c) + 3))/d
Time = 2.02 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.74 \[ \int \frac {1}{(5+3 \cosh (c+d x))^3} \, dx=\frac {\frac {59\,{\mathrm {e}}^{c+d\,x}}{256\,d}+\frac {295}{768\,d}}{10\,{\mathrm {e}}^{c+d\,x}+3\,{\mathrm {e}}^{2\,c+2\,d\,x}+3}-\frac {59\,\mathrm {atan}\left (\left (\frac {5}{4\,d}+\frac {3\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c}{4\,d}\right )\,\sqrt {-d^2}\right )}{1024\,\sqrt {-d^2}}-\frac {\frac {41\,{\mathrm {e}}^{c+d\,x}}{24\,d}+\frac {5}{8\,d}}{60\,{\mathrm {e}}^{c+d\,x}+118\,{\mathrm {e}}^{2\,c+2\,d\,x}+60\,{\mathrm {e}}^{3\,c+3\,d\,x}+9\,{\mathrm {e}}^{4\,c+4\,d\,x}+9} \] Input:
int(1/(3*cosh(c + d*x) + 5)^3,x)
Output:
((59*exp(c + d*x))/(256*d) + 295/(768*d))/(10*exp(c + d*x) + 3*exp(2*c + 2 *d*x) + 3) - (59*atan((5/(4*d) + (3*exp(d*x)*exp(c))/(4*d))*(-d^2)^(1/2))) /(1024*(-d^2)^(1/2)) - ((41*exp(c + d*x))/(24*d) + 5/(8*d))/(60*exp(c + d* x) + 118*exp(2*c + 2*d*x) + 60*exp(3*c + 3*d*x) + 9*exp(4*c + 4*d*x) + 9)
Time = 0.22 (sec) , antiderivative size = 293, normalized size of antiderivative = 3.62 \[ \int \frac {1}{(5+3 \cosh (c+d x))^3} \, dx=\frac {-2655 e^{4 d x +4 c} \mathrm {log}\left (e^{d x +c}+3\right )+2655 e^{4 d x +4 c} \mathrm {log}\left (3 e^{d x +c}+1\right )-1062 e^{4 d x +4 c}-17700 e^{3 d x +3 c} \mathrm {log}\left (e^{d x +c}+3\right )+17700 e^{3 d x +3 c} \mathrm {log}\left (3 e^{d x +c}+1\right )-34810 e^{2 d x +2 c} \mathrm {log}\left (e^{d x +c}+3\right )+34810 e^{2 d x +2 c} \mathrm {log}\left (3 e^{d x +c}+1\right )+21476 e^{2 d x +2 c}-17700 e^{d x +c} \mathrm {log}\left (e^{d x +c}+3\right )+17700 e^{d x +c} \mathrm {log}\left (3 e^{d x +c}+1\right )+21840 e^{d x +c}-2655 \,\mathrm {log}\left (e^{d x +c}+3\right )+2655 \,\mathrm {log}\left (3 e^{d x +c}+1\right )+4338}{10240 d \left (9 e^{4 d x +4 c}+60 e^{3 d x +3 c}+118 e^{2 d x +2 c}+60 e^{d x +c}+9\right )} \] Input:
int(1/(5+3*cosh(d*x+c))^3,x)
Output:
( - 2655*e**(4*c + 4*d*x)*log(e**(c + d*x) + 3) + 2655*e**(4*c + 4*d*x)*lo g(3*e**(c + d*x) + 1) - 1062*e**(4*c + 4*d*x) - 17700*e**(3*c + 3*d*x)*log (e**(c + d*x) + 3) + 17700*e**(3*c + 3*d*x)*log(3*e**(c + d*x) + 1) - 3481 0*e**(2*c + 2*d*x)*log(e**(c + d*x) + 3) + 34810*e**(2*c + 2*d*x)*log(3*e* *(c + d*x) + 1) + 21476*e**(2*c + 2*d*x) - 17700*e**(c + d*x)*log(e**(c + d*x) + 3) + 17700*e**(c + d*x)*log(3*e**(c + d*x) + 1) + 21840*e**(c + d*x ) - 2655*log(e**(c + d*x) + 3) + 2655*log(3*e**(c + d*x) + 1) + 4338)/(102 40*d*(9*e**(4*c + 4*d*x) + 60*e**(3*c + 3*d*x) + 118*e**(2*c + 2*d*x) + 60 *e**(c + d*x) + 9))