3.1.7 \(\int \text {csch}^3(c+d x) (a+b \text {sech}^2(c+d x)) \, dx\) [7]

Optimal. Leaf size=54 \[ \frac {(a+3 b) \tanh ^{-1}(\cosh (c+d x))}{2 d}-\frac {(a+b) \coth (c+d x) \text {csch}(c+d x)}{2 d}-\frac {b \text {sech}(c+d x)}{d} \]

[Out]

1/2*(a+3*b)*arctanh(cosh(d*x+c))/d-1/2*(a+b)*coth(d*x+c)*csch(d*x+c)/d-b*sech(d*x+c)/d

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Rubi [A]
time = 0.05, antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {4218, 467, 464, 212} \begin {gather*} \frac {(a+3 b) \tanh ^{-1}(\cosh (c+d x))}{2 d}-\frac {(a+b) \coth (c+d x) \text {csch}(c+d x)}{2 d}-\frac {b \text {sech}(c+d x)}{d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Csch[c + d*x]^3*(a + b*Sech[c + d*x]^2),x]

[Out]

((a + 3*b)*ArcTanh[Cosh[c + d*x]])/(2*d) - ((a + b)*Coth[c + d*x]*Csch[c + d*x])/(2*d) - (b*Sech[c + d*x])/d

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 464

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[c*(e*x)^(m +
1)*((a + b*x^n)^(p + 1)/(a*e*(m + 1))), x] + Dist[(a*d*(m + 1) - b*c*(m + n*(p + 1) + 1))/(a*e^n*(m + 1)), In
t[(e*x)^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b*c - a*d, 0] && (IntegerQ[n] ||
GtQ[e, 0]) && ((GtQ[n, 0] && LtQ[m, -1]) || (LtQ[n, 0] && GtQ[m + n, -1])) &&  !ILtQ[p, -1]

Rule 467

Int[(x_)^(m_)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[(-a)^(m/2 - 1)*(b*c - a*d)*x
*((a + b*x^2)^(p + 1)/(2*b^(m/2 + 1)*(p + 1))), x] + Dist[1/(2*b^(m/2 + 1)*(p + 1)), Int[x^m*(a + b*x^2)^(p +
1)*ExpandToSum[2*b*(p + 1)*Together[(b^(m/2)*(c + d*x^2) - (-a)^(m/2 - 1)*(b*c - a*d)*x^(-m + 2))/(a + b*x^2)]
- ((-a)^(m/2 - 1)*(b*c - a*d))/x^m, x], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] &
& ILtQ[m/2, 0] && (IntegerQ[p] || EqQ[m + 2*p + 1, 0])

Rule 4218

Int[((a_) + (b_.)*sec[(e_.) + (f_.)*(x_)]^(n_))^(p_.)*sin[(e_.) + (f_.)*(x_)]^(m_.), x_Symbol] :> With[{ff = F
reeFactors[Cos[e + f*x], x]}, Dist[-ff/f, Subst[Int[(1 - ff^2*x^2)^((m - 1)/2)*((b + a*(ff*x)^n)^p/(ff*x)^(n*p
)), x], x, Cos[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f}, x] && IntegerQ[(m - 1)/2] && IntegerQ[n] && IntegerQ[p
]

Rubi steps

\begin {align*} \int \text {csch}^3(c+d x) \left (a+b \text {sech}^2(c+d x)\right ) \, dx &=\frac {\text {Subst}\left (\int \frac {b+a x^2}{x^2 \left (1-x^2\right )^2} \, dx,x,\cosh (c+d x)\right )}{d}\\ &=-\frac {(a+b) \coth (c+d x) \text {csch}(c+d x)}{2 d}-\frac {\text {Subst}\left (\int \frac {-2 b-(a+b) x^2}{x^2 \left (1-x^2\right )} \, dx,x,\cosh (c+d x)\right )}{2 d}\\ &=-\frac {(a+b) \coth (c+d x) \text {csch}(c+d x)}{2 d}-\frac {b \text {sech}(c+d x)}{d}+\frac {(a+3 b) \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\cosh (c+d x)\right )}{2 d}\\ &=\frac {(a+3 b) \tanh ^{-1}(\cosh (c+d x))}{2 d}-\frac {(a+b) \coth (c+d x) \text {csch}(c+d x)}{2 d}-\frac {b \text {sech}(c+d x)}{d}\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(131\) vs. \(2(54)=108\).
time = 0.04, size = 131, normalized size = 2.43 \begin {gather*} -\frac {a \text {csch}^2\left (\frac {1}{2} (c+d x)\right )}{8 d}-\frac {b \text {csch}^2\left (\frac {1}{2} (c+d x)\right )}{8 d}-\frac {a \log \left (\tanh \left (\frac {1}{2} (c+d x)\right )\right )}{2 d}-\frac {3 b \log \left (\tanh \left (\frac {1}{2} (c+d x)\right )\right )}{2 d}-\frac {a \text {sech}^2\left (\frac {1}{2} (c+d x)\right )}{8 d}-\frac {b \text {sech}^2\left (\frac {1}{2} (c+d x)\right )}{8 d}-\frac {b \text {sech}(c+d x)}{d} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Csch[c + d*x]^3*(a + b*Sech[c + d*x]^2),x]

[Out]

-1/8*(a*Csch[(c + d*x)/2]^2)/d - (b*Csch[(c + d*x)/2]^2)/(8*d) - (a*Log[Tanh[(c + d*x)/2]])/(2*d) - (3*b*Log[T
anh[(c + d*x)/2]])/(2*d) - (a*Sech[(c + d*x)/2]^2)/(8*d) - (b*Sech[(c + d*x)/2]^2)/(8*d) - (b*Sech[c + d*x])/d

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(150\) vs. \(2(50)=100\).
time = 1.99, size = 151, normalized size = 2.80

method	result	size
risch	\(-\frac {{\mathrm e}^{d x +c} \left (a \,{\mathrm e}^{4 d x +4 c}+3 b \,{\mathrm e}^{4 d x +4 c}+2 a \,{\mathrm e}^{2 d x +2 c}-2 b \,{\mathrm e}^{2 d x +2 c}+a +3 b \right )}{d \left ({\mathrm e}^{2 d x +2 c}-1\right )^{2} \left (1+{\mathrm e}^{2 d x +2 c}\right )}-\frac {a \ln \left ({\mathrm e}^{d x +c}-1\right )}{2 d}-\frac {3 \ln \left ({\mathrm e}^{d x +c}-1\right ) b}{2 d}+\frac {a \ln \left ({\mathrm e}^{d x +c}+1\right )}{2 d}+\frac {3 \ln \left ({\mathrm e}^{d x +c}+1\right ) b}{2 d}\)	\(151\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csch(d*x+c)^3*(a+b*sech(d*x+c)^2),x,method=_RETURNVERBOSE)

[Out]

-exp(d*x+c)*(a*exp(4*d*x+4*c)+3*b*exp(4*d*x+4*c)+2*a*exp(2*d*x+2*c)-2*b*exp(2*d*x+2*c)+a+3*b)/d/(exp(2*d*x+2*c
)-1)^2/(1+exp(2*d*x+2*c))-1/2*a/d*ln(exp(d*x+c)-1)-3/2/d*ln(exp(d*x+c)-1)*b+1/2*a/d*ln(exp(d*x+c)+1)+3/2/d*ln(
exp(d*x+c)+1)*b

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 198 vs. \(2 (50) = 100\).
time = 0.28, size = 198, normalized size = 3.67 \begin {gather*} \frac {1}{2} \, b {\left (\frac {3 \, \log \left (e^{\left (-d x - c\right )} + 1\right )}{d} - \frac {3 \, \log \left (e^{\left (-d x - c\right )} - 1\right )}{d} + \frac {2 \, {\left (3 \, e^{\left (-d x - c\right )} - 2 \, e^{\left (-3 \, d x - 3 \, c\right )} + 3 \, e^{\left (-5 \, d x - 5 \, c\right )}\right )}}{d {\left (e^{\left (-2 \, d x - 2 \, c\right )} + e^{\left (-4 \, d x - 4 \, c\right )} - e^{\left (-6 \, d x - 6 \, c\right )} - 1\right )}}\right )} + \frac {1}{2} \, a {\left (\frac {\log \left (e^{\left (-d x - c\right )} + 1\right )}{d} - \frac {\log \left (e^{\left (-d x - c\right )} - 1\right )}{d} + \frac {2 \, {\left (e^{\left (-d x - c\right )} + e^{\left (-3 \, d x - 3 \, c\right )}\right )}}{d {\left (2 \, e^{\left (-2 \, d x - 2 \, c\right )} - e^{\left (-4 \, d x - 4 \, c\right )} - 1\right )}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(d*x+c)^3*(a+b*sech(d*x+c)^2),x, algorithm="maxima")

[Out]

1/2*b*(3*log(e^(-d*x - c) + 1)/d - 3*log(e^(-d*x - c) - 1)/d + 2*(3*e^(-d*x - c) - 2*e^(-3*d*x - 3*c) + 3*e^(-
5*d*x - 5*c))/(d*(e^(-2*d*x - 2*c) + e^(-4*d*x - 4*c) - e^(-6*d*x - 6*c) - 1))) + 1/2*a*(log(e^(-d*x - c) + 1)
/d - log(e^(-d*x - c) - 1)/d + 2*(e^(-d*x - c) + e^(-3*d*x - 3*c))/(d*(2*e^(-2*d*x - 2*c) - e^(-4*d*x - 4*c) -
1)))

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 924 vs. \(2 (50) = 100\).
time = 0.57, size = 924, normalized size = 17.11 \begin {gather*} -\frac {2 \, {\left (a + 3 \, b\right )} \cosh \left (d x + c\right )^{5} + 10 \, {\left (a + 3 \, b\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{4} + 2 \, {\left (a + 3 \, b\right )} \sinh \left (d x + c\right )^{5} + 4 \, {\left (a - b\right )} \cosh \left (d x + c\right )^{3} + 4 \, {\left (5 \, {\left (a + 3 \, b\right )} \cosh \left (d x + c\right )^{2} + a - b\right )} \sinh \left (d x + c\right )^{3} + 4 \, {\left (5 \, {\left (a + 3 \, b\right )} \cosh \left (d x + c\right )^{3} + 3 \, {\left (a - b\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{2} + 2 \, {\left (a + 3 \, b\right )} \cosh \left (d x + c\right ) - {\left ({\left (a + 3 \, b\right )} \cosh \left (d x + c\right )^{6} + 6 \, {\left (a + 3 \, b\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{5} + {\left (a + 3 \, b\right )} \sinh \left (d x + c\right )^{6} - {\left (a + 3 \, b\right )} \cosh \left (d x + c\right )^{4} + {\left (15 \, {\left (a + 3 \, b\right )} \cosh \left (d x + c\right )^{2} - a - 3 \, b\right )} \sinh \left (d x + c\right )^{4} + 4 \, {\left (5 \, {\left (a + 3 \, b\right )} \cosh \left (d x + c\right )^{3} - {\left (a + 3 \, b\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{3} - {\left (a + 3 \, b\right )} \cosh \left (d x + c\right )^{2} + {\left (15 \, {\left (a + 3 \, b\right )} \cosh \left (d x + c\right )^{4} - 6 \, {\left (a + 3 \, b\right )} \cosh \left (d x + c\right )^{2} - a - 3 \, b\right )} \sinh \left (d x + c\right )^{2} + 2 \, {\left (3 \, {\left (a + 3 \, b\right )} \cosh \left (d x + c\right )^{5} - 2 \, {\left (a + 3 \, b\right )} \cosh \left (d x + c\right )^{3} - {\left (a + 3 \, b\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right ) + a + 3 \, b\right )} \log \left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right ) + 1\right ) + {\left ({\left (a + 3 \, b\right )} \cosh \left (d x + c\right )^{6} + 6 \, {\left (a + 3 \, b\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{5} + {\left (a + 3 \, b\right )} \sinh \left (d x + c\right )^{6} - {\left (a + 3 \, b\right )} \cosh \left (d x + c\right )^{4} + {\left (15 \, {\left (a + 3 \, b\right )} \cosh \left (d x + c\right )^{2} - a - 3 \, b\right )} \sinh \left (d x + c\right )^{4} + 4 \, {\left (5 \, {\left (a + 3 \, b\right )} \cosh \left (d x + c\right )^{3} - {\left (a + 3 \, b\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{3} - {\left (a + 3 \, b\right )} \cosh \left (d x + c\right )^{2} + {\left (15 \, {\left (a + 3 \, b\right )} \cosh \left (d x + c\right )^{4} - 6 \, {\left (a + 3 \, b\right )} \cosh \left (d x + c\right )^{2} - a - 3 \, b\right )} \sinh \left (d x + c\right )^{2} + 2 \, {\left (3 \, {\left (a + 3 \, b\right )} \cosh \left (d x + c\right )^{5} - 2 \, {\left (a + 3 \, b\right )} \cosh \left (d x + c\right )^{3} - {\left (a + 3 \, b\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right ) + a + 3 \, b\right )} \log \left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right ) - 1\right ) + 2 \, {\left (5 \, {\left (a + 3 \, b\right )} \cosh \left (d x + c\right )^{4} + 6 \, {\left (a - b\right )} \cosh \left (d x + c\right )^{2} + a + 3 \, b\right )} \sinh \left (d x + c\right )}{2 \, {\left (d \cosh \left (d x + c\right )^{6} + 6 \, d \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{5} + d \sinh \left (d x + c\right )^{6} - d \cosh \left (d x + c\right )^{4} + {\left (15 \, d \cosh \left (d x + c\right )^{2} - d\right )} \sinh \left (d x + c\right )^{4} + 4 \, {\left (5 \, d \cosh \left (d x + c\right )^{3} - d \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{3} - d \cosh \left (d x + c\right )^{2} + {\left (15 \, d \cosh \left (d x + c\right )^{4} - 6 \, d \cosh \left (d x + c\right )^{2} - d\right )} \sinh \left (d x + c\right )^{2} + 2 \, {\left (3 \, d \cosh \left (d x + c\right )^{5} - 2 \, d \cosh \left (d x + c\right )^{3} - d \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right ) + d\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(d*x+c)^3*(a+b*sech(d*x+c)^2),x, algorithm="fricas")

[Out]

-1/2*(2*(a + 3*b)*cosh(d*x + c)^5 + 10*(a + 3*b)*cosh(d*x + c)*sinh(d*x + c)^4 + 2*(a + 3*b)*sinh(d*x + c)^5 +
4*(a - b)*cosh(d*x + c)^3 + 4*(5*(a + 3*b)*cosh(d*x + c)^2 + a - b)*sinh(d*x + c)^3 + 4*(5*(a + 3*b)*cosh(d*x
+ c)^3 + 3*(a - b)*cosh(d*x + c))*sinh(d*x + c)^2 + 2*(a + 3*b)*cosh(d*x + c) - ((a + 3*b)*cosh(d*x + c)^6 +
6*(a + 3*b)*cosh(d*x + c)*sinh(d*x + c)^5 + (a + 3*b)*sinh(d*x + c)^6 - (a + 3*b)*cosh(d*x + c)^4 + (15*(a + 3
*b)*cosh(d*x + c)^2 - a - 3*b)*sinh(d*x + c)^4 + 4*(5*(a + 3*b)*cosh(d*x + c)^3 - (a + 3*b)*cosh(d*x + c))*sin
h(d*x + c)^3 - (a + 3*b)*cosh(d*x + c)^2 + (15*(a + 3*b)*cosh(d*x + c)^4 - 6*(a + 3*b)*cosh(d*x + c)^2 - a - 3
*b)*sinh(d*x + c)^2 + 2*(3*(a + 3*b)*cosh(d*x + c)^5 - 2*(a + 3*b)*cosh(d*x + c)^3 - (a + 3*b)*cosh(d*x + c))*
sinh(d*x + c) + a + 3*b)*log(cosh(d*x + c) + sinh(d*x + c) + 1) + ((a + 3*b)*cosh(d*x + c)^6 + 6*(a + 3*b)*cos
h(d*x + c)*sinh(d*x + c)^5 + (a + 3*b)*sinh(d*x + c)^6 - (a + 3*b)*cosh(d*x + c)^4 + (15*(a + 3*b)*cosh(d*x +
c)^2 - a - 3*b)*sinh(d*x + c)^4 + 4*(5*(a + 3*b)*cosh(d*x + c)^3 - (a + 3*b)*cosh(d*x + c))*sinh(d*x + c)^3 -
(a + 3*b)*cosh(d*x + c)^2 + (15*(a + 3*b)*cosh(d*x + c)^4 - 6*(a + 3*b)*cosh(d*x + c)^2 - a - 3*b)*sinh(d*x +
c)^2 + 2*(3*(a + 3*b)*cosh(d*x + c)^5 - 2*(a + 3*b)*cosh(d*x + c)^3 - (a + 3*b)*cosh(d*x + c))*sinh(d*x + c) +
a + 3*b)*log(cosh(d*x + c) + sinh(d*x + c) - 1) + 2*(5*(a + 3*b)*cosh(d*x + c)^4 + 6*(a - b)*cosh(d*x + c)^2
+ a + 3*b)*sinh(d*x + c))/(d*cosh(d*x + c)^6 + 6*d*cosh(d*x + c)*sinh(d*x + c)^5 + d*sinh(d*x + c)^6 - d*cosh(
d*x + c)^4 + (15*d*cosh(d*x + c)^2 - d)*sinh(d*x + c)^4 + 4*(5*d*cosh(d*x + c)^3 - d*cosh(d*x + c))*sinh(d*x +
c)^3 - d*cosh(d*x + c)^2 + (15*d*cosh(d*x + c)^4 - 6*d*cosh(d*x + c)^2 - d)*sinh(d*x + c)^2 + 2*(3*d*cosh(d*x
+ c)^5 - 2*d*cosh(d*x + c)^3 - d*cosh(d*x + c))*sinh(d*x + c) + d)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (a + b \operatorname {sech}^{2}{\left (c + d x \right )}\right ) \operatorname {csch}^{3}{\left (c + d x \right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(d*x+c)**3*(a+b*sech(d*x+c)**2),x)

[Out]

Integral((a + b*sech(c + d*x)**2)*csch(c + d*x)**3, x)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 142 vs. \(2 (50) = 100\).
time = 0.41, size = 142, normalized size = 2.63 \begin {gather*} \frac {{\left (a + 3 \, b\right )} \log \left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )} + 2\right ) - {\left (a + 3 \, b\right )} \log \left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )} - 2\right ) - \frac {4 \, {\left (a {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{2} + 3 \, b {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{2} - 8 \, b\right )}}{{\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{3} - 4 \, e^{\left (d x + c\right )} - 4 \, e^{\left (-d x - c\right )}}}{4 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(d*x+c)^3*(a+b*sech(d*x+c)^2),x, algorithm="giac")

[Out]

1/4*((a + 3*b)*log(e^(d*x + c) + e^(-d*x - c) + 2) - (a + 3*b)*log(e^(d*x + c) + e^(-d*x - c) - 2) - 4*(a*(e^(
d*x + c) + e^(-d*x - c))^2 + 3*b*(e^(d*x + c) + e^(-d*x - c))^2 - 8*b)/((e^(d*x + c) + e^(-d*x - c))^3 - 4*e^(
d*x + c) - 4*e^(-d*x - c)))/d

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Mupad [B]
time = 0.17, size = 160, normalized size = 2.96 \begin {gather*} \frac {\mathrm {atan}\left (\frac {{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\left (a\,\sqrt {-d^2}+3\,b\,\sqrt {-d^2}\right )}{d\,\sqrt {a^2+6\,a\,b+9\,b^2}}\right )\,\sqrt {a^2+6\,a\,b+9\,b^2}}{\sqrt {-d^2}}-\frac {{\mathrm {e}}^{c+d\,x}\,\left (a+b\right )}{d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}-1\right )}-\frac {2\,{\mathrm {e}}^{c+d\,x}\,\left (a+b\right )}{d\,\left ({\mathrm {e}}^{4\,c+4\,d\,x}-2\,{\mathrm {e}}^{2\,c+2\,d\,x}+1\right )}-\frac {2\,b\,{\mathrm {e}}^{c+d\,x}}{d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b/cosh(c + d*x)^2)/sinh(c + d*x)^3,x)

[Out]

(atan((exp(d*x)*exp(c)*(a*(-d^2)^(1/2) + 3*b*(-d^2)^(1/2)))/(d*(6*a*b + a^2 + 9*b^2)^(1/2)))*(6*a*b + a^2 + 9*
b^2)^(1/2))/(-d^2)^(1/2) - (exp(c + d*x)*(a + b))/(d*(exp(2*c + 2*d*x) - 1)) - (2*exp(c + d*x)*(a + b))/(d*(ex
p(4*c + 4*d*x) - 2*exp(2*c + 2*d*x) + 1)) - (2*b*exp(c + d*x))/(d*(exp(2*c + 2*d*x) + 1))

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