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

Optimal. Leaf size=73 \[ \frac {43 \text {ArcTan}\left (\frac {1}{2} \tanh \left (\frac {1}{2} (c+d x)\right )\right )}{1024 d}+\frac {5 \sinh (c+d x)}{32 d (3+5 \cosh (c+d x))^2}-\frac {45 \sinh (c+d x)}{512 d (3+5 \cosh (c+d x))} \]

[Out]

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*cos
h(d*x+c))

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Rubi [A]
time = 0.04, antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {2743, 2833, 12, 2738, 212} \begin {gather*} \frac {43 \text {ArcTan}\left (\frac {1}{2} \tanh \left (\frac {1}{2} (c+d x)\right )\right )}{1024 d}-\frac {45 \sinh (c+d x)}{512 d (5 \cosh (c+d x)+3)}+\frac {5 \sinh (c+d x)}{32 d (5 \cosh (c+d x)+3)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(3 + 5*Cosh[c + d*x])^(-3),x]

[Out]

(43*ArcTan[Tanh[(c + d*x)/2]/2])/(1024*d) + (5*Sinh[c + d*x])/(32*d*(3 + 5*Cosh[c + d*x])^2) - (45*Sinh[c + d*
x])/(512*d*(3 + 5*Cosh[c + d*x]))

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 212

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] && (GtQ[a, 0] || LtQ[b, 0])

Rule 2738

Int[((a_) + (b_.)*sin[Pi/2 + (c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{e = FreeFactors[Tan[(c + d*x)/2], x
]}, Dist[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 2743

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] + Dist[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] && Integ
erQ[2*n]

Rule 2833

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] + Dist[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]

Rubi steps

\begin {align*} \int \frac {1}{(3+5 \cosh (c+d x))^3} \, dx &=\frac {5 \sinh (c+d x)}{32 d (3+5 \cosh (c+d x))^2}+\frac {1}{32} \int \frac {-6+5 \cosh (c+d x)}{(3+5 \cosh (c+d x))^2} \, dx\\ &=\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))}+\frac {1}{512} \int \frac {43}{3+5 \cosh (c+d x)} \, dx\\ &=\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))}+\frac {43}{512} \int \frac {1}{3+5 \cosh (c+d x)} \, dx\\ &=\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))}-\frac {(43 i) \text {Subst}\left (\int \frac {1}{8-2 x^2} \, dx,x,\tan \left (\frac {1}{2} (i c+i d x)\right )\right )}{256 d}\\ &=\frac {43 \tan ^{-1}\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))}\\ \end {align*}

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Mathematica [A]
time = 0.12, size = 55, normalized size = 0.75 \begin {gather*} \frac {43 \text {ArcTan}\left (\frac {1}{2} \tanh \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} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 0.88, size = 62, normalized size = 0.85

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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)))

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Maxima [A]
time = 0.47, size = 108, normalized size = 1.48 \begin {gather*} -\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 )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-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))

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 408 vs. \(2 (64) = 128\).
time = 0.45, size = 408, normalized size = 5.59 \begin {gather*} \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 )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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)*sinh(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*(25*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*cos
h(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*cosh(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)

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Sympy [C] Result contains complex when optimal does not.
time = 2.36, size = 600, normalized size = 8.22 \begin {gather*} \begin {cases} \frac {x}{125 \cosh ^{3}{\left (d x + \log {\left (- 3 e^{- d x} - 4 i e^{- d x} \right )} - \log {\left (5 \right )} \right )} + 225 \cosh ^{2}{\left (d x + \log {\left (- 3 e^{- d x} - 4 i e^{- d x} \right )} - \log {\left (5 \right )} \right )} + 135 \cosh {\left (d x + \log {\left (- 3 e^{- d x} - 4 i e^{- d x} \right )} - \log {\left (5 \right )} \right )} + 27} & \text {for}\: c = \log {\left (\left (- \frac {3}{5} - \frac {4 i}{5}\right ) e^{- d x} \right )} \\- \frac {\log {\left (- 3 e^{- d x} + 4 i e^{- d x} \right )}}{125 d \cosh ^{3}{\left (d x + \log {\left (- 3 e^{- d x} + 4 i e^{- d x} \right )} - \log {\left (5 \right )} \right )} + 225 d \cosh ^{2}{\left (d x + \log {\left (- 3 e^{- d x} + 4 i e^{- d x} \right )} - \log {\left (5 \right )} \right )} + 135 d \cosh {\left (d x + \log {\left (- 3 e^{- d x} + 4 i e^{- d x} \right )} - \log {\left (5 \right )} \right )} + 27 d} + \frac {\log {\left (5 \right )}}{125 d \cosh ^{3}{\left (d x + \log {\left (- 3 e^{- d x} + 4 i e^{- d x} \right )} - \log {\left (5 \right )} \right )} + 225 d \cosh ^{2}{\left (d x + \log {\left (- 3 e^{- d x} + 4 i e^{- d x} \right )} - \log {\left (5 \right )} \right )} + 135 d \cosh {\left (d x + \log {\left (- 3 e^{- d x} + 4 i e^{- d x} \right )} - \log {\left (5 \right )} \right )} + 27 d} & \text {for}\: c = \log {\left (\left (- \frac {3}{5} + \frac {4 i}{5}\right ) e^{- d x} \right )} \\\frac {x}{\left (5 \cosh {\left (c \right )} + 3\right )^{3}} & \text {for}\: d = 0 \\\frac {43 \tanh ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} \operatorname {atan}{\left (\frac {\tanh {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{2} \right )}}{1024 d \tanh ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 8192 d \tanh ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 16384 d} - \frac {170 \tanh ^{3}{\left (\frac {c}{2} + \frac {d x}{2} \right )}}{1024 d \tanh ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 8192 d \tanh ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 16384 d} + \frac {344 \tanh ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} \operatorname {atan}{\left (\frac {\tanh {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{2} \right )}}{1024 d \tanh ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 8192 d \tanh ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 16384 d} - \frac {280 \tanh {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{1024 d \tanh ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 8192 d \tanh ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 16384 d} + \frac {688 \operatorname {atan}{\left (\frac {\tanh {\left (\frac {c}{2} + \frac {d x}{2} \right )}}{2} \right )}}{1024 d \tanh ^{4}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 8192 d \tanh ^{2}{\left (\frac {c}{2} + \frac {d x}{2} \right )} + 16384 d} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((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((-
3/5 - 4*I/5)*exp(-d*x)))), (-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) + log(5)/(125*d*cosh(d*x + log(-3*exp(-d*x) + 4*I*exp(-d*x)) - lo
g(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((-3/5 + 4*I/5)*exp(-d*x)))), (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), True))

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Giac [A]
time = 0.41, size = 76, normalized size = 1.04 \begin {gather*} \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/1024*(4*(215*e^(3*d*x + 3*c) + 387*e^(2*d*x + 2*c) + 325*e^(d*x + c) + 225)/(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

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Mupad [B]
time = 0.96, size = 137, normalized size = 1.88 \begin {gather*} \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}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(5*cosh(c + d*x) + 3)^3,x)

[Out]

((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*at
an((3/(4*d) + (5*exp(d*x)*exp(c))/(4*d))*(d^2)^(1/2)))/(1024*(d^2)^(1/2))

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