3.158 \(\int \tanh ^2(c+d x) (a+b \tanh ^2(c+d x))^3 \, dx\)

Optimal. Leaf size=94 \[ -\frac{b \left (3 a^2+3 a b+b^2\right ) \tanh ^3(c+d x)}{3 d}-\frac{b^2 (3 a+b) \tanh ^5(c+d x)}{5 d}-\frac{(a+b)^3 \tanh (c+d x)}{d}+x (a+b)^3-\frac{b^3 \tanh ^7(c+d x)}{7 d} \]

[Out]

(a + b)^3*x - ((a + b)^3*Tanh[c + d*x])/d - (b*(3*a^2 + 3*a*b + b^2)*Tanh[c + d*x]^3)/(3*d) - (b^2*(3*a + b)*T
anh[c + d*x]^5)/(5*d) - (b^3*Tanh[c + d*x]^7)/(7*d)

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Rubi [A]  time = 0.0927085, antiderivative size = 94, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, Rules used = {3670, 461, 206} \[ -\frac{b \left (3 a^2+3 a b+b^2\right ) \tanh ^3(c+d x)}{3 d}-\frac{b^2 (3 a+b) \tanh ^5(c+d x)}{5 d}-\frac{(a+b)^3 \tanh (c+d x)}{d}+x (a+b)^3-\frac{b^3 \tanh ^7(c+d x)}{7 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a + b)^3*x - ((a + b)^3*Tanh[c + d*x])/d - (b*(3*a^2 + 3*a*b + b^2)*Tanh[c + d*x]^3)/(3*d) - (b^2*(3*a + b)*T
anh[c + d*x]^5)/(5*d) - (b^3*Tanh[c + d*x]^7)/(7*d)

Rule 3670

Int[((d_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((a_) + (b_.)*((c_.)*tan[(e_.) + (f_.)*(x_)])^(n_))^(p_.), x_Symbol]
:> With[{ff = FreeFactors[Tan[e + f*x], x]}, Dist[(c*ff)/f, Subst[Int[(((d*ff*x)/c)^m*(a + b*(ff*x)^n)^p)/(c^
2 + ff^2*x^2), x], x, (c*Tan[e + f*x])/ff], x]] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && (IGtQ[p, 0] || EqQ
[n, 2] || EqQ[n, 4] || (IntegerQ[p] && RationalQ[n]))

Rule 461

Int[(((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_))/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Int[ExpandIntegr
and[((e*x)^m*(a + b*x^n)^p)/(c + d*x^n), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b*c - a*d, 0] && IGtQ[n
, 0] && IGtQ[p, 0] && (IntegerQ[m] || IGtQ[2*(m + 1), 0] ||  !RationalQ[m])

Rule 206

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

Rubi steps

\begin{align*} \int \tanh ^2(c+d x) \left (a+b \tanh ^2(c+d x)\right )^3 \, dx &=\frac{\operatorname{Subst}\left (\int \frac{x^2 \left (a+b x^2\right )^3}{1-x^2} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int \left (-(a+b)^3-b \left (3 a^2+3 a b+b^2\right ) x^2-b^2 (3 a+b) x^4-b^3 x^6+\frac{a^3+3 a^2 b+3 a b^2+b^3}{1-x^2}\right ) \, dx,x,\tanh (c+d x)\right )}{d}\\ &=-\frac{(a+b)^3 \tanh (c+d x)}{d}-\frac{b \left (3 a^2+3 a b+b^2\right ) \tanh ^3(c+d x)}{3 d}-\frac{b^2 (3 a+b) \tanh ^5(c+d x)}{5 d}-\frac{b^3 \tanh ^7(c+d x)}{7 d}+\frac{(a+b)^3 \operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=(a+b)^3 x-\frac{(a+b)^3 \tanh (c+d x)}{d}-\frac{b \left (3 a^2+3 a b+b^2\right ) \tanh ^3(c+d x)}{3 d}-\frac{b^2 (3 a+b) \tanh ^5(c+d x)}{5 d}-\frac{b^3 \tanh ^7(c+d x)}{7 d}\\ \end{align*}

Mathematica [A]  time = 1.61293, size = 108, normalized size = 1.15 \[ \frac{\tanh (c+d x) \left (-35 b \left (3 a^2+3 a b+b^2\right ) \tanh ^2(c+d x)-21 b^2 (3 a+b) \tanh ^4(c+d x)+\frac{105 (a+b)^3 \tanh ^{-1}\left (\sqrt{\tanh ^2(c+d x)}\right )}{\sqrt{\tanh ^2(c+d x)}}-105 (a+b)^3-15 b^3 \tanh ^6(c+d x)\right )}{105 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Tanh[c + d*x]*(-105*(a + b)^3 - 35*b*(3*a^2 + 3*a*b + b^2)*Tanh[c + d*x]^2 - 21*b^2*(3*a + b)*Tanh[c + d*x]^4
- 15*b^3*Tanh[c + d*x]^6 + (105*(a + b)^3*ArcTanh[Sqrt[Tanh[c + d*x]^2]])/Sqrt[Tanh[c + d*x]^2]))/(105*d)

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Maple [B]  time = 0.005, size = 299, normalized size = 3.2 \begin{align*} -{\frac{{a}^{3}\ln \left ( \tanh \left ( dx+c \right ) -1 \right ) }{2\,d}}-{\frac{3\,\ln \left ( \tanh \left ( dx+c \right ) -1 \right ){a}^{2}b}{2\,d}}-{\frac{3\,\ln \left ( \tanh \left ( dx+c \right ) -1 \right ) a{b}^{2}}{2\,d}}-{\frac{\ln \left ( \tanh \left ( dx+c \right ) -1 \right ){b}^{3}}{2\,d}}-{\frac{{b}^{3} \left ( \tanh \left ( dx+c \right ) \right ) ^{7}}{7\,d}}-{\frac{{b}^{3} \left ( \tanh \left ( dx+c \right ) \right ) ^{5}}{5\,d}}-{\frac{{b}^{3} \left ( \tanh \left ( dx+c \right ) \right ) ^{3}}{3\,d}}-{\frac{{b}^{3}\tanh \left ( dx+c \right ) }{d}}-{\frac{{a}^{2}b \left ( \tanh \left ( dx+c \right ) \right ) ^{3}}{d}}-{\frac{a{b}^{2} \left ( \tanh \left ( dx+c \right ) \right ) ^{3}}{d}}-{\frac{3\,a{b}^{2} \left ( \tanh \left ( dx+c \right ) \right ) ^{5}}{5\,d}}-3\,{\frac{{a}^{2}b\tanh \left ( dx+c \right ) }{d}}+{\frac{\ln \left ( \tanh \left ( dx+c \right ) +1 \right ){a}^{3}}{2\,d}}+{\frac{3\,\ln \left ( \tanh \left ( dx+c \right ) +1 \right ){a}^{2}b}{2\,d}}+{\frac{3\,\ln \left ( \tanh \left ( dx+c \right ) +1 \right ) a{b}^{2}}{2\,d}}+{\frac{\ln \left ( \tanh \left ( dx+c \right ) +1 \right ){b}^{3}}{2\,d}}-{\frac{{a}^{3}\tanh \left ( dx+c \right ) }{d}}-3\,{\frac{a{b}^{2}\tanh \left ( dx+c \right ) }{d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tanh(d*x+c)^2*(a+b*tanh(d*x+c)^2)^3,x)

[Out]

-1/2/d*a^3*ln(tanh(d*x+c)-1)-3/2/d*ln(tanh(d*x+c)-1)*a^2*b-3/2/d*ln(tanh(d*x+c)-1)*a*b^2-1/2/d*ln(tanh(d*x+c)-
1)*b^3-1/7*b^3*tanh(d*x+c)^7/d-1/5*b^3*tanh(d*x+c)^5/d-1/3*b^3*tanh(d*x+c)^3/d-b^3*tanh(d*x+c)/d-a^2*b*tanh(d*
x+c)^3/d-a*b^2*tanh(d*x+c)^3/d-3/5*a*b^2*tanh(d*x+c)^5/d-3*a^2*b*tanh(d*x+c)/d+1/2/d*ln(tanh(d*x+c)+1)*a^3+3/2
/d*ln(tanh(d*x+c)+1)*a^2*b+3/2/d*ln(tanh(d*x+c)+1)*a*b^2+1/2/d*ln(tanh(d*x+c)+1)*b^3-a^3*tanh(d*x+c)/d-3*a*b^2
*tanh(d*x+c)/d

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Maxima [B]  time = 1.06923, size = 540, normalized size = 5.74 \begin{align*} \frac{1}{105} \, b^{3}{\left (105 \, x + \frac{105 \, c}{d} - \frac{8 \,{\left (203 \, e^{\left (-2 \, d x - 2 \, c\right )} + 609 \, e^{\left (-4 \, d x - 4 \, c\right )} + 770 \, e^{\left (-6 \, d x - 6 \, c\right )} + 770 \, e^{\left (-8 \, d x - 8 \, c\right )} + 315 \, e^{\left (-10 \, d x - 10 \, c\right )} + 105 \, e^{\left (-12 \, d x - 12 \, c\right )} + 44\right )}}{d{\left (7 \, e^{\left (-2 \, d x - 2 \, c\right )} + 21 \, e^{\left (-4 \, d x - 4 \, c\right )} + 35 \, e^{\left (-6 \, d x - 6 \, c\right )} + 35 \, e^{\left (-8 \, d x - 8 \, c\right )} + 21 \, e^{\left (-10 \, d x - 10 \, c\right )} + 7 \, e^{\left (-12 \, d x - 12 \, c\right )} + e^{\left (-14 \, d x - 14 \, c\right )} + 1\right )}}\right )} + \frac{1}{5} \, a b^{2}{\left (15 \, x + \frac{15 \, c}{d} - \frac{2 \,{\left (70 \, e^{\left (-2 \, d x - 2 \, c\right )} + 140 \, e^{\left (-4 \, d x - 4 \, c\right )} + 90 \, e^{\left (-6 \, d x - 6 \, c\right )} + 45 \, e^{\left (-8 \, d x - 8 \, c\right )} + 23\right )}}{d{\left (5 \, e^{\left (-2 \, d x - 2 \, c\right )} + 10 \, e^{\left (-4 \, d x - 4 \, c\right )} + 10 \, e^{\left (-6 \, d x - 6 \, c\right )} + 5 \, e^{\left (-8 \, d x - 8 \, c\right )} + e^{\left (-10 \, d x - 10 \, c\right )} + 1\right )}}\right )} + a^{2} b{\left (3 \, x + \frac{3 \, c}{d} - \frac{4 \,{\left (3 \, e^{\left (-2 \, d x - 2 \, c\right )} + 3 \, e^{\left (-4 \, d x - 4 \, c\right )} + 2\right )}}{d{\left (3 \, e^{\left (-2 \, d x - 2 \, c\right )} + 3 \, e^{\left (-4 \, d x - 4 \, c\right )} + e^{\left (-6 \, d x - 6 \, c\right )} + 1\right )}}\right )} + a^{3}{\left (x + \frac{c}{d} - \frac{2}{d{\left (e^{\left (-2 \, d x - 2 \, c\right )} + 1\right )}}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/105*b^3*(105*x + 105*c/d - 8*(203*e^(-2*d*x - 2*c) + 609*e^(-4*d*x - 4*c) + 770*e^(-6*d*x - 6*c) + 770*e^(-8
*d*x - 8*c) + 315*e^(-10*d*x - 10*c) + 105*e^(-12*d*x - 12*c) + 44)/(d*(7*e^(-2*d*x - 2*c) + 21*e^(-4*d*x - 4*
c) + 35*e^(-6*d*x - 6*c) + 35*e^(-8*d*x - 8*c) + 21*e^(-10*d*x - 10*c) + 7*e^(-12*d*x - 12*c) + e^(-14*d*x - 1
4*c) + 1))) + 1/5*a*b^2*(15*x + 15*c/d - 2*(70*e^(-2*d*x - 2*c) + 140*e^(-4*d*x - 4*c) + 90*e^(-6*d*x - 6*c) +
45*e^(-8*d*x - 8*c) + 23)/(d*(5*e^(-2*d*x - 2*c) + 10*e^(-4*d*x - 4*c) + 10*e^(-6*d*x - 6*c) + 5*e^(-8*d*x -
8*c) + e^(-10*d*x - 10*c) + 1))) + a^2*b*(3*x + 3*c/d - 4*(3*e^(-2*d*x - 2*c) + 3*e^(-4*d*x - 4*c) + 2)/(d*(3*
e^(-2*d*x - 2*c) + 3*e^(-4*d*x - 4*c) + e^(-6*d*x - 6*c) + 1))) + a^3*(x + c/d - 2/(d*(e^(-2*d*x - 2*c) + 1)))

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Fricas [B]  time = 1.97666, size = 2693, normalized size = 28.65 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/105*((105*a^3 + 420*a^2*b + 483*a*b^2 + 176*b^3 + 105*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*d*x)*cosh(d*x + c)^7 +
7*(105*a^3 + 420*a^2*b + 483*a*b^2 + 176*b^3 + 105*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*d*x)*cosh(d*x + c)*sinh(d*
x + c)^6 - (105*a^3 + 420*a^2*b + 483*a*b^2 + 176*b^3)*sinh(d*x + c)^7 + 7*(105*a^3 + 420*a^2*b + 483*a*b^2 +
176*b^3 + 105*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*d*x)*cosh(d*x + c)^5 - 7*(75*a^3 + 240*a^2*b + 213*a*b^2 + 56*b^
3 + 3*(105*a^3 + 420*a^2*b + 483*a*b^2 + 176*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^5 + 35*((105*a^3 + 420*a^2*b
+ 483*a*b^2 + 176*b^3 + 105*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*d*x)*cosh(d*x + c)^3 + (105*a^3 + 420*a^2*b + 483*
a*b^2 + 176*b^3 + 105*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*d*x)*cosh(d*x + c))*sinh(d*x + c)^4 + 21*(105*a^3 + 420*
a^2*b + 483*a*b^2 + 176*b^3 + 105*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*d*x)*cosh(d*x + c)^3 - 7*(5*(105*a^3 + 420*a
^2*b + 483*a*b^2 + 176*b^3)*cosh(d*x + c)^4 + 135*a^3 + 360*a^2*b + 369*a*b^2 + 168*b^3 + 10*(75*a^3 + 240*a^2
*b + 213*a*b^2 + 56*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^3 + 7*(3*(105*a^3 + 420*a^2*b + 483*a*b^2 + 176*b^3 +
105*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*d*x)*cosh(d*x + c)^5 + 10*(105*a^3 + 420*a^2*b + 483*a*b^2 + 176*b^3 + 105
*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*d*x)*cosh(d*x + c)^3 + 9*(105*a^3 + 420*a^2*b + 483*a*b^2 + 176*b^3 + 105*(a^
3 + 3*a^2*b + 3*a*b^2 + b^3)*d*x)*cosh(d*x + c))*sinh(d*x + c)^2 + 35*(105*a^3 + 420*a^2*b + 483*a*b^2 + 176*b
^3 + 105*(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*d*x)*cosh(d*x + c) - 7*((105*a^3 + 420*a^2*b + 483*a*b^2 + 176*b^3)*c
osh(d*x + c)^6 + 5*(75*a^3 + 240*a^2*b + 213*a*b^2 + 56*b^3)*cosh(d*x + c)^4 + 75*a^3 + 180*a^2*b + 225*a*b^2
+ 9*(45*a^3 + 120*a^2*b + 123*a*b^2 + 56*b^3)*cosh(d*x + c)^2)*sinh(d*x + c))/(d*cosh(d*x + c)^7 + 7*d*cosh(d*
x + c)*sinh(d*x + c)^6 + 7*d*cosh(d*x + c)^5 + 35*(d*cosh(d*x + c)^3 + d*cosh(d*x + c))*sinh(d*x + c)^4 + 21*d
*cosh(d*x + c)^3 + 7*(3*d*cosh(d*x + c)^5 + 10*d*cosh(d*x + c)^3 + 9*d*cosh(d*x + c))*sinh(d*x + c)^2 + 35*d*c
osh(d*x + c))

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Sympy [A]  time = 1.69827, size = 192, normalized size = 2.04 \begin{align*} \begin{cases} a^{3} x - \frac{a^{3} \tanh{\left (c + d x \right )}}{d} + 3 a^{2} b x - \frac{a^{2} b \tanh ^{3}{\left (c + d x \right )}}{d} - \frac{3 a^{2} b \tanh{\left (c + d x \right )}}{d} + 3 a b^{2} x - \frac{3 a b^{2} \tanh ^{5}{\left (c + d x \right )}}{5 d} - \frac{a b^{2} \tanh ^{3}{\left (c + d x \right )}}{d} - \frac{3 a b^{2} \tanh{\left (c + d x \right )}}{d} + b^{3} x - \frac{b^{3} \tanh ^{7}{\left (c + d x \right )}}{7 d} - \frac{b^{3} \tanh ^{5}{\left (c + d x \right )}}{5 d} - \frac{b^{3} \tanh ^{3}{\left (c + d x \right )}}{3 d} - \frac{b^{3} \tanh{\left (c + d x \right )}}{d} & \text{for}\: d \neq 0 \\x \left (a + b \tanh ^{2}{\left (c \right )}\right )^{3} \tanh ^{2}{\left (c \right )} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((a**3*x - a**3*tanh(c + d*x)/d + 3*a**2*b*x - a**2*b*tanh(c + d*x)**3/d - 3*a**2*b*tanh(c + d*x)/d +
3*a*b**2*x - 3*a*b**2*tanh(c + d*x)**5/(5*d) - a*b**2*tanh(c + d*x)**3/d - 3*a*b**2*tanh(c + d*x)/d + b**3*x
- b**3*tanh(c + d*x)**7/(7*d) - b**3*tanh(c + d*x)**5/(5*d) - b**3*tanh(c + d*x)**3/(3*d) - b**3*tanh(c + d*x)
/d, Ne(d, 0)), (x*(a + b*tanh(c)**2)**3*tanh(c)**2, True))

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Giac [B]  time = 1.46132, size = 564, normalized size = 6. \begin{align*} \frac{{\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )}{\left (d x + c\right )}}{d} + \frac{2 \,{\left (105 \, a^{3} e^{\left (12 \, d x + 12 \, c\right )} + 630 \, a^{2} b e^{\left (12 \, d x + 12 \, c\right )} + 945 \, a b^{2} e^{\left (12 \, d x + 12 \, c\right )} + 420 \, b^{3} e^{\left (12 \, d x + 12 \, c\right )} + 630 \, a^{3} e^{\left (10 \, d x + 10 \, c\right )} + 3150 \, a^{2} b e^{\left (10 \, d x + 10 \, c\right )} + 3780 \, a b^{2} e^{\left (10 \, d x + 10 \, c\right )} + 1260 \, b^{3} e^{\left (10 \, d x + 10 \, c\right )} + 1575 \, a^{3} e^{\left (8 \, d x + 8 \, c\right )} + 6720 \, a^{2} b e^{\left (8 \, d x + 8 \, c\right )} + 7665 \, a b^{2} e^{\left (8 \, d x + 8 \, c\right )} + 3080 \, b^{3} e^{\left (8 \, d x + 8 \, c\right )} + 2100 \, a^{3} e^{\left (6 \, d x + 6 \, c\right )} + 7980 \, a^{2} b e^{\left (6 \, d x + 6 \, c\right )} + 9240 \, a b^{2} e^{\left (6 \, d x + 6 \, c\right )} + 3080 \, b^{3} e^{\left (6 \, d x + 6 \, c\right )} + 1575 \, a^{3} e^{\left (4 \, d x + 4 \, c\right )} + 5670 \, a^{2} b e^{\left (4 \, d x + 4 \, c\right )} + 6363 \, a b^{2} e^{\left (4 \, d x + 4 \, c\right )} + 2436 \, b^{3} e^{\left (4 \, d x + 4 \, c\right )} + 630 \, a^{3} e^{\left (2 \, d x + 2 \, c\right )} + 2310 \, a^{2} b e^{\left (2 \, d x + 2 \, c\right )} + 2436 \, a b^{2} e^{\left (2 \, d x + 2 \, c\right )} + 812 \, b^{3} e^{\left (2 \, d x + 2 \, c\right )} + 105 \, a^{3} + 420 \, a^{2} b + 483 \, a b^{2} + 176 \, b^{3}\right )}}{105 \, d{\left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right )}^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(a^3 + 3*a^2*b + 3*a*b^2 + b^3)*(d*x + c)/d + 2/105*(105*a^3*e^(12*d*x + 12*c) + 630*a^2*b*e^(12*d*x + 12*c) +
945*a*b^2*e^(12*d*x + 12*c) + 420*b^3*e^(12*d*x + 12*c) + 630*a^3*e^(10*d*x + 10*c) + 3150*a^2*b*e^(10*d*x +
10*c) + 3780*a*b^2*e^(10*d*x + 10*c) + 1260*b^3*e^(10*d*x + 10*c) + 1575*a^3*e^(8*d*x + 8*c) + 6720*a^2*b*e^(8
*d*x + 8*c) + 7665*a*b^2*e^(8*d*x + 8*c) + 3080*b^3*e^(8*d*x + 8*c) + 2100*a^3*e^(6*d*x + 6*c) + 7980*a^2*b*e^
(6*d*x + 6*c) + 9240*a*b^2*e^(6*d*x + 6*c) + 3080*b^3*e^(6*d*x + 6*c) + 1575*a^3*e^(4*d*x + 4*c) + 5670*a^2*b*
e^(4*d*x + 4*c) + 6363*a*b^2*e^(4*d*x + 4*c) + 2436*b^3*e^(4*d*x + 4*c) + 630*a^3*e^(2*d*x + 2*c) + 2310*a^2*b
*e^(2*d*x + 2*c) + 2436*a*b^2*e^(2*d*x + 2*c) + 812*b^3*e^(2*d*x + 2*c) + 105*a^3 + 420*a^2*b + 483*a*b^2 + 17
6*b^3)/(d*(e^(2*d*x + 2*c) + 1)^7)