3.2.24 \(\int (a+b \text {sech}^2(c+d x))^3 \tanh ^4(c+d x) \, dx\) [124]

Optimal. Leaf size=110 \[ a^3 x-\frac {a^3 \tanh (c+d x)}{d}-\frac {a^3 \tanh ^3(c+d x)}{3 d}+\frac {b \left (3 a^2+3 a b+b^2\right ) \tanh ^5(c+d x)}{5 d}-\frac {b^2 (3 a+2 b) \tanh ^7(c+d x)}{7 d}+\frac {b^3 \tanh ^9(c+d x)}{9 d} \]

[Out]

a^3*x-a^3*tanh(d*x+c)/d-1/3*a^3*tanh(d*x+c)^3/d+1/5*b*(3*a^2+3*a*b+b^2)*tanh(d*x+c)^5/d-1/7*b^2*(3*a+2*b)*tanh
(d*x+c)^7/d+1/9*b^3*tanh(d*x+c)^9/d

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Rubi [A]
time = 0.09, antiderivative size = 110, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {4226, 1816, 212} \begin {gather*} -\frac {a^3 \tanh ^3(c+d x)}{3 d}-\frac {a^3 \tanh (c+d x)}{d}+a^3 x+\frac {b \left (3 a^2+3 a b+b^2\right ) \tanh ^5(c+d x)}{5 d}-\frac {b^2 (3 a+2 b) \tanh ^7(c+d x)}{7 d}+\frac {b^3 \tanh ^9(c+d x)}{9 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*Sech[c + d*x]^2)^3*Tanh[c + d*x]^4,x]

[Out]

a^3*x - (a^3*Tanh[c + d*x])/d - (a^3*Tanh[c + d*x]^3)/(3*d) + (b*(3*a^2 + 3*a*b + b^2)*Tanh[c + d*x]^5)/(5*d)
- (b^2*(3*a + 2*b)*Tanh[c + d*x]^7)/(7*d) + (b^3*Tanh[c + d*x]^9)/(9*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 1816

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*Pq*(a + b*x
^2)^p, x], x] /; FreeQ[{a, b, c, m}, x] && PolyQ[Pq, x] && IGtQ[p, -2]

Rule 4226

Int[((a_) + (b_.)*sec[(e_.) + (f_.)*(x_)]^(n_))^(p_.)*((d_.)*tan[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> With[
{ff = FreeFactors[Tan[e + f*x], x]}, Dist[ff/f, Subst[Int[(d*ff*x)^m*((a + b*(1 + ff^2*x^2)^(n/2))^p/(1 + ff^2
*x^2)), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, d, e, f, m, p}, x] && IntegerQ[n/2] && (IntegerQ[m/2] ||
EqQ[n, 2])

Rubi steps

\begin {align*} \int \left (a+b \text {sech}^2(c+d x)\right )^3 \tanh ^4(c+d x) \, dx &=\frac {\text {Subst}\left (\int \frac {x^4 \left (a+b \left (1-x^2\right )\right )^3}{1-x^2} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac {\text {Subst}\left (\int \left (-a^3-a^3 x^2+b \left (3 a^2+3 a b+b^2\right ) x^4-b^2 (3 a+2 b) x^6+b^3 x^8+\frac {a^3}{1-x^2}\right ) \, dx,x,\tanh (c+d x)\right )}{d}\\ &=-\frac {a^3 \tanh (c+d x)}{d}-\frac {a^3 \tanh ^3(c+d x)}{3 d}+\frac {b \left (3 a^2+3 a b+b^2\right ) \tanh ^5(c+d x)}{5 d}-\frac {b^2 (3 a+2 b) \tanh ^7(c+d x)}{7 d}+\frac {b^3 \tanh ^9(c+d x)}{9 d}+\frac {a^3 \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=a^3 x-\frac {a^3 \tanh (c+d x)}{d}-\frac {a^3 \tanh ^3(c+d x)}{3 d}+\frac {b \left (3 a^2+3 a b+b^2\right ) \tanh ^5(c+d x)}{5 d}-\frac {b^2 (3 a+2 b) \tanh ^7(c+d x)}{7 d}+\frac {b^3 \tanh ^9(c+d x)}{9 d}\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(301\) vs. \(2(110)=220\).
time = 3.84, size = 301, normalized size = 2.74 \begin {gather*} \frac {8 \left (b+a \cosh ^2(c+d x)\right )^3 \text {sech}^9(c+d x) \left (315 a^3 d x \cosh ^9(c+d x)+35 b^3 \text {sech}(c) \sinh (d x)+5 (27 a-10 b) b^2 \cosh ^2(c+d x) \text {sech}(c) \sinh (d x)+3 b \left (63 a^2-72 a b+b^2\right ) \cosh ^4(c+d x) \text {sech}(c) \sinh (d x)+\left (105 a^3-378 a^2 b+27 a b^2+4 b^3\right ) \cosh ^6(c+d x) \text {sech}(c) \sinh (d x)-\left (420 a^3-189 a^2 b-54 a b^2-8 b^3\right ) \cosh ^8(c+d x) \text {sech}(c) \sinh (d x)+35 b^3 \cosh (c+d x) \tanh (c)+5 (27 a-10 b) b^2 \cosh ^3(c+d x) \tanh (c)+3 b \left (63 a^2-72 a b+b^2\right ) \cosh ^5(c+d x) \tanh (c)+\left (105 a^3-378 a^2 b+27 a b^2+4 b^3\right ) \cosh ^7(c+d x) \tanh (c)\right )}{315 d (a+2 b+a \cosh (2 (c+d x)))^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Sech[c + d*x]^2)^3*Tanh[c + d*x]^4,x]

[Out]

(8*(b + a*Cosh[c + d*x]^2)^3*Sech[c + d*x]^9*(315*a^3*d*x*Cosh[c + d*x]^9 + 35*b^3*Sech[c]*Sinh[d*x] + 5*(27*a
- 10*b)*b^2*Cosh[c + d*x]^2*Sech[c]*Sinh[d*x] + 3*b*(63*a^2 - 72*a*b + b^2)*Cosh[c + d*x]^4*Sech[c]*Sinh[d*x]
+ (105*a^3 - 378*a^2*b + 27*a*b^2 + 4*b^3)*Cosh[c + d*x]^6*Sech[c]*Sinh[d*x] - (420*a^3 - 189*a^2*b - 54*a*b^
2 - 8*b^3)*Cosh[c + d*x]^8*Sech[c]*Sinh[d*x] + 35*b^3*Cosh[c + d*x]*Tanh[c] + 5*(27*a - 10*b)*b^2*Cosh[c + d*x
]^3*Tanh[c] + 3*b*(63*a^2 - 72*a*b + b^2)*Cosh[c + d*x]^5*Tanh[c] + (105*a^3 - 378*a^2*b + 27*a*b^2 + 4*b^3)*C
osh[c + d*x]^7*Tanh[c]))/(315*d*(a + 2*b + a*Cosh[2*(c + d*x)])^3)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(468\) vs. \(2(102)=204\).
time = 2.72, size = 469, normalized size = 4.26

method	result	size
risch	\(a^{3} x +\frac {-\frac {12 a \,b^{2}}{35}-12 a \,b^{2} {\mathrm e}^{14 d x +14 c}-48 a^{2} b \,{\mathrm e}^{12 d x +12 c}-12 a \,b^{2} {\mathrm e}^{12 d x +12 c}-72 a^{2} b \,{\mathrm e}^{10 d x +10 c}-12 a \,b^{2} {\mathrm e}^{10 d x +10 c}-6 a^{2} b \,{\mathrm e}^{16 d x +16 c}-24 a^{2} b \,{\mathrm e}^{14 d x +14 c}-\frac {96 a^{2} b \,{\mathrm e}^{4 d x +4 c}}{5}-\frac {24 a^{2} b \,{\mathrm e}^{2 d x +2 c}}{5}-\frac {6 a^{2} b}{5}+\frac {8 a^{3}}{3}-\frac {16 b^{3}}{315}-\frac {156 a \,b^{2} {\mathrm e}^{8 d x +8 c}}{5}-\frac {84 a \,b^{2} {\mathrm e}^{6 d x +6 c}}{5}-\frac {12 a \,b^{2} {\mathrm e}^{4 d x +4 c}}{35}-\frac {396 a^{2} b \,{\mathrm e}^{8 d x +8 c}}{5}-\frac {108 a \,b^{2} {\mathrm e}^{2 d x +2 c}}{35}-\frac {264 a^{2} b \,{\mathrm e}^{6 d x +6 c}}{5}+28 a^{3} {\mathrm e}^{14 d x +14 c}+20 a^{3} {\mathrm e}^{2 d x +2 c}+\frac {260 a^{3} {\mathrm e}^{12 d x +12 c}}{3}-\frac {32 b^{3} {\mathrm e}^{12 d x +12 c}}{3}+156 a^{3} {\mathrm e}^{10 d x +10 c}-\frac {16 b^{3} {\mathrm e}^{2 d x +2 c}}{35}+4 a^{3} {\mathrm e}^{16 d x +16 c}-\frac {112 b^{3} {\mathrm e}^{8 d x +8 c}}{5}+68 a^{3} {\mathrm e}^{4 d x +4 c}-\frac {64 b^{3} {\mathrm e}^{4 d x +4 c}}{35}+16 b^{3} {\mathrm e}^{10 d x +10 c}+180 a^{3} {\mathrm e}^{8 d x +8 c}+\frac {412 a^{3} {\mathrm e}^{6 d x +6 c}}{3}+\frac {32 b^{3} {\mathrm e}^{6 d x +6 c}}{5}}{d \left (1+{\mathrm e}^{2 d x +2 c}\right )^{9}}\)	\(469\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*sech(d*x+c)^2)^3*tanh(d*x+c)^4,x,method=_RETURNVERBOSE)

[Out]

a^3*x+2/315*(-54*a*b^2-1890*a*b^2*exp(14*d*x+14*c)-7560*a^2*b*exp(12*d*x+12*c)-1890*a*b^2*exp(12*d*x+12*c)-113
40*a^2*b*exp(10*d*x+10*c)-1890*a*b^2*exp(10*d*x+10*c)-945*a^2*b*exp(16*d*x+16*c)-3780*a^2*b*exp(14*d*x+14*c)-3
024*a^2*b*exp(4*d*x+4*c)-756*a^2*b*exp(2*d*x+2*c)-189*a^2*b+420*a^3-8*b^3-4914*a*b^2*exp(8*d*x+8*c)-2646*a*b^2
*exp(6*d*x+6*c)-54*a*b^2*exp(4*d*x+4*c)-12474*a^2*b*exp(8*d*x+8*c)-486*a*b^2*exp(2*d*x+2*c)-8316*a^2*b*exp(6*d
*x+6*c)+4410*a^3*exp(14*d*x+14*c)+3150*a^3*exp(2*d*x+2*c)+13650*a^3*exp(12*d*x+12*c)-1680*b^3*exp(12*d*x+12*c)
+24570*a^3*exp(10*d*x+10*c)-72*b^3*exp(2*d*x+2*c)+630*a^3*exp(16*d*x+16*c)-3528*b^3*exp(8*d*x+8*c)+10710*a^3*e
xp(4*d*x+4*c)-288*b^3*exp(4*d*x+4*c)+2520*b^3*exp(10*d*x+10*c)+28350*a^3*exp(8*d*x+8*c)+21630*a^3*exp(6*d*x+6*
c)+1008*b^3*exp(6*d*x+6*c))/d/(1+exp(2*d*x+2*c))^9

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 1453 vs. \(2 (102) = 204\).
time = 0.28, size = 1453, normalized size = 13.21 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*sech(d*x+c)^2)^3*tanh(d*x+c)^4,x, algorithm="maxima")

[Out]

3/5*a^2*b*tanh(d*x + c)^5/d + 1/3*a^3*(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))) + 16/315*b^3*(9*e^(-2*d*x - 2*c)/(d*(9*e^(-2*d*x
- 2*c) + 36*e^(-4*d*x - 4*c) + 84*e^(-6*d*x - 6*c) + 126*e^(-8*d*x - 8*c) + 126*e^(-10*d*x - 10*c) + 84*e^(-12
*d*x - 12*c) + 36*e^(-14*d*x - 14*c) + 9*e^(-16*d*x - 16*c) + e^(-18*d*x - 18*c) + 1)) + 36*e^(-4*d*x - 4*c)/(
d*(9*e^(-2*d*x - 2*c) + 36*e^(-4*d*x - 4*c) + 84*e^(-6*d*x - 6*c) + 126*e^(-8*d*x - 8*c) + 126*e^(-10*d*x - 10
*c) + 84*e^(-12*d*x - 12*c) + 36*e^(-14*d*x - 14*c) + 9*e^(-16*d*x - 16*c) + e^(-18*d*x - 18*c) + 1)) - 126*e^
(-6*d*x - 6*c)/(d*(9*e^(-2*d*x - 2*c) + 36*e^(-4*d*x - 4*c) + 84*e^(-6*d*x - 6*c) + 126*e^(-8*d*x - 8*c) + 126
*e^(-10*d*x - 10*c) + 84*e^(-12*d*x - 12*c) + 36*e^(-14*d*x - 14*c) + 9*e^(-16*d*x - 16*c) + e^(-18*d*x - 18*c
) + 1)) + 441*e^(-8*d*x - 8*c)/(d*(9*e^(-2*d*x - 2*c) + 36*e^(-4*d*x - 4*c) + 84*e^(-6*d*x - 6*c) + 126*e^(-8*
d*x - 8*c) + 126*e^(-10*d*x - 10*c) + 84*e^(-12*d*x - 12*c) + 36*e^(-14*d*x - 14*c) + 9*e^(-16*d*x - 16*c) + e
^(-18*d*x - 18*c) + 1)) - 315*e^(-10*d*x - 10*c)/(d*(9*e^(-2*d*x - 2*c) + 36*e^(-4*d*x - 4*c) + 84*e^(-6*d*x -
6*c) + 126*e^(-8*d*x - 8*c) + 126*e^(-10*d*x - 10*c) + 84*e^(-12*d*x - 12*c) + 36*e^(-14*d*x - 14*c) + 9*e^(-
16*d*x - 16*c) + e^(-18*d*x - 18*c) + 1)) + 210*e^(-12*d*x - 12*c)/(d*(9*e^(-2*d*x - 2*c) + 36*e^(-4*d*x - 4*c
) + 84*e^(-6*d*x - 6*c) + 126*e^(-8*d*x - 8*c) + 126*e^(-10*d*x - 10*c) + 84*e^(-12*d*x - 12*c) + 36*e^(-14*d*
x - 14*c) + 9*e^(-16*d*x - 16*c) + e^(-18*d*x - 18*c) + 1)) + 1/(d*(9*e^(-2*d*x - 2*c) + 36*e^(-4*d*x - 4*c) +
84*e^(-6*d*x - 6*c) + 126*e^(-8*d*x - 8*c) + 126*e^(-10*d*x - 10*c) + 84*e^(-12*d*x - 12*c) + 36*e^(-14*d*x -
14*c) + 9*e^(-16*d*x - 16*c) + e^(-18*d*x - 18*c) + 1))) + 12/35*a*b^2*(7*e^(-2*d*x - 2*c)/(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 - 14*c) + 1)) - 14*e^(-4*d*x - 4*c)/(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 - 14*c) + 1))
+ 70*e^(-6*d*x - 6*c)/(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 - 14*c) + 1)) - 35*e^(-8*d*x - 8*c)/(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 - 14*c) + 1)) + 35*e^(-10*d*x - 10*c)/(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 - 14*c)
+ 1)) + 1/(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^(-1
0*d*x - 10*c) + 7*e^(-12*d*x - 12*c) + e^(-14*d*x - 14*c) + 1)))

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1323 vs. \(2 (102) = 204\).
time = 0.38, size = 1323, normalized size = 12.03 \begin {gather*} \frac {{\left (315 \, a^{3} d x + 420 \, a^{3} - 189 \, a^{2} b - 54 \, a b^{2} - 8 \, b^{3}\right )} \cosh \left (d x + c\right )^{9} + 9 \, {\left (315 \, a^{3} d x + 420 \, a^{3} - 189 \, a^{2} b - 54 \, a b^{2} - 8 \, b^{3}\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{8} - {\left (420 \, a^{3} - 189 \, a^{2} b - 54 \, a b^{2} - 8 \, b^{3}\right )} \sinh \left (d x + c\right )^{9} + 9 \, {\left (315 \, a^{3} d x + 420 \, a^{3} - 189 \, a^{2} b - 54 \, a b^{2} - 8 \, b^{3}\right )} \cosh \left (d x + c\right )^{7} - 9 \, {\left (280 \, a^{3} + 21 \, a^{2} b - 54 \, a b^{2} - 8 \, b^{3} + 4 \, {\left (420 \, a^{3} - 189 \, a^{2} b - 54 \, a b^{2} - 8 \, b^{3}\right )} \cosh \left (d x + c\right )^{2}\right )} \sinh \left (d x + c\right )^{7} + 21 \, {\left (4 \, {\left (315 \, a^{3} d x + 420 \, a^{3} - 189 \, a^{2} b - 54 \, a b^{2} - 8 \, b^{3}\right )} \cosh \left (d x + c\right )^{3} + 3 \, {\left (315 \, a^{3} d x + 420 \, a^{3} - 189 \, a^{2} b - 54 \, a b^{2} - 8 \, b^{3}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{6} + 36 \, {\left (315 \, a^{3} d x + 420 \, a^{3} - 189 \, a^{2} b - 54 \, a b^{2} - 8 \, b^{3}\right )} \cosh \left (d x + c\right )^{5} - 9 \, {\left (14 \, {\left (420 \, a^{3} - 189 \, a^{2} b - 54 \, a b^{2} - 8 \, b^{3}\right )} \cosh \left (d x + c\right )^{4} + 700 \, a^{3} + 84 \, a^{2} b + 204 \, a b^{2} - 32 \, b^{3} + 21 \, {\left (280 \, a^{3} + 21 \, a^{2} b - 54 \, a b^{2} - 8 \, b^{3}\right )} \cosh \left (d x + c\right )^{2}\right )} \sinh \left (d x + c\right )^{5} + 9 \, {\left (14 \, {\left (315 \, a^{3} d x + 420 \, a^{3} - 189 \, a^{2} b - 54 \, a b^{2} - 8 \, b^{3}\right )} \cosh \left (d x + c\right )^{5} + 35 \, {\left (315 \, a^{3} d x + 420 \, a^{3} - 189 \, a^{2} b - 54 \, a b^{2} - 8 \, b^{3}\right )} \cosh \left (d x + c\right )^{3} + 20 \, {\left (315 \, a^{3} d x + 420 \, a^{3} - 189 \, a^{2} b - 54 \, a b^{2} - 8 \, b^{3}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{4} + 84 \, {\left (315 \, a^{3} d x + 420 \, a^{3} - 189 \, a^{2} b - 54 \, a b^{2} - 8 \, b^{3}\right )} \cosh \left (d x + c\right )^{3} - 3 \, {\left (28 \, {\left (420 \, a^{3} - 189 \, a^{2} b - 54 \, a b^{2} - 8 \, b^{3}\right )} \cosh \left (d x + c\right )^{6} + 105 \, {\left (280 \, a^{3} + 21 \, a^{2} b - 54 \, a b^{2} - 8 \, b^{3}\right )} \cosh \left (d x + c\right )^{4} + 2660 \, a^{3} - 252 \, a^{2} b - 252 \, a b^{2} + 896 \, b^{3} + 120 \, {\left (175 \, a^{3} + 21 \, a^{2} b + 51 \, a b^{2} - 8 \, b^{3}\right )} \cosh \left (d x + c\right )^{2}\right )} \sinh \left (d x + c\right )^{3} + 9 \, {\left (4 \, {\left (315 \, a^{3} d x + 420 \, a^{3} - 189 \, a^{2} b - 54 \, a b^{2} - 8 \, b^{3}\right )} \cosh \left (d x + c\right )^{7} + 21 \, {\left (315 \, a^{3} d x + 420 \, a^{3} - 189 \, a^{2} b - 54 \, a b^{2} - 8 \, b^{3}\right )} \cosh \left (d x + c\right )^{5} + 40 \, {\left (315 \, a^{3} d x + 420 \, a^{3} - 189 \, a^{2} b - 54 \, a b^{2} - 8 \, b^{3}\right )} \cosh \left (d x + c\right )^{3} + 28 \, {\left (315 \, a^{3} d x + 420 \, a^{3} - 189 \, a^{2} b - 54 \, a b^{2} - 8 \, b^{3}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{2} + 126 \, {\left (315 \, a^{3} d x + 420 \, a^{3} - 189 \, a^{2} b - 54 \, a b^{2} - 8 \, b^{3}\right )} \cosh \left (d x + c\right ) - 9 \, {\left ({\left (420 \, a^{3} - 189 \, a^{2} b - 54 \, a b^{2} - 8 \, b^{3}\right )} \cosh \left (d x + c\right )^{8} + 7 \, {\left (280 \, a^{3} + 21 \, a^{2} b - 54 \, a b^{2} - 8 \, b^{3}\right )} \cosh \left (d x + c\right )^{6} + 20 \, {\left (175 \, a^{3} + 21 \, a^{2} b + 51 \, a b^{2} - 8 \, b^{3}\right )} \cosh \left (d x + c\right )^{4} + 420 \, a^{3} - 126 \, a^{2} b - 336 \, a b^{2} - 672 \, b^{3} + 28 \, {\left (95 \, a^{3} - 9 \, a^{2} b - 9 \, a b^{2} + 32 \, b^{3}\right )} \cosh \left (d x + c\right )^{2}\right )} \sinh \left (d x + c\right )}{315 \, {\left (d \cosh \left (d x + c\right )^{9} + 9 \, d \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{8} + 9 \, d \cosh \left (d x + c\right )^{7} + 21 \, {\left (4 \, d \cosh \left (d x + c\right )^{3} + 3 \, d \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{6} + 36 \, d \cosh \left (d x + c\right )^{5} + 9 \, {\left (14 \, d \cosh \left (d x + c\right )^{5} + 35 \, d \cosh \left (d x + c\right )^{3} + 20 \, d \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{4} + 84 \, d \cosh \left (d x + c\right )^{3} + 9 \, {\left (4 \, d \cosh \left (d x + c\right )^{7} + 21 \, d \cosh \left (d x + c\right )^{5} + 40 \, d \cosh \left (d x + c\right )^{3} + 28 \, d \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{2} + 126 \, d \cosh \left (d x + c\right )\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*sech(d*x+c)^2)^3*tanh(d*x+c)^4,x, algorithm="fricas")

[Out]

1/315*((315*a^3*d*x + 420*a^3 - 189*a^2*b - 54*a*b^2 - 8*b^3)*cosh(d*x + c)^9 + 9*(315*a^3*d*x + 420*a^3 - 189
*a^2*b - 54*a*b^2 - 8*b^3)*cosh(d*x + c)*sinh(d*x + c)^8 - (420*a^3 - 189*a^2*b - 54*a*b^2 - 8*b^3)*sinh(d*x +
c)^9 + 9*(315*a^3*d*x + 420*a^3 - 189*a^2*b - 54*a*b^2 - 8*b^3)*cosh(d*x + c)^7 - 9*(280*a^3 + 21*a^2*b - 54*
a*b^2 - 8*b^3 + 4*(420*a^3 - 189*a^2*b - 54*a*b^2 - 8*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^7 + 21*(4*(315*a^3*d
*x + 420*a^3 - 189*a^2*b - 54*a*b^2 - 8*b^3)*cosh(d*x + c)^3 + 3*(315*a^3*d*x + 420*a^3 - 189*a^2*b - 54*a*b^2
- 8*b^3)*cosh(d*x + c))*sinh(d*x + c)^6 + 36*(315*a^3*d*x + 420*a^3 - 189*a^2*b - 54*a*b^2 - 8*b^3)*cosh(d*x
+ c)^5 - 9*(14*(420*a^3 - 189*a^2*b - 54*a*b^2 - 8*b^3)*cosh(d*x + c)^4 + 700*a^3 + 84*a^2*b + 204*a*b^2 - 32*
b^3 + 21*(280*a^3 + 21*a^2*b - 54*a*b^2 - 8*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^5 + 9*(14*(315*a^3*d*x + 420*a
^3 - 189*a^2*b - 54*a*b^2 - 8*b^3)*cosh(d*x + c)^5 + 35*(315*a^3*d*x + 420*a^3 - 189*a^2*b - 54*a*b^2 - 8*b^3)
*cosh(d*x + c)^3 + 20*(315*a^3*d*x + 420*a^3 - 189*a^2*b - 54*a*b^2 - 8*b^3)*cosh(d*x + c))*sinh(d*x + c)^4 +
84*(315*a^3*d*x + 420*a^3 - 189*a^2*b - 54*a*b^2 - 8*b^3)*cosh(d*x + c)^3 - 3*(28*(420*a^3 - 189*a^2*b - 54*a*
b^2 - 8*b^3)*cosh(d*x + c)^6 + 105*(280*a^3 + 21*a^2*b - 54*a*b^2 - 8*b^3)*cosh(d*x + c)^4 + 2660*a^3 - 252*a^
2*b - 252*a*b^2 + 896*b^3 + 120*(175*a^3 + 21*a^2*b + 51*a*b^2 - 8*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^3 + 9*(
4*(315*a^3*d*x + 420*a^3 - 189*a^2*b - 54*a*b^2 - 8*b^3)*cosh(d*x + c)^7 + 21*(315*a^3*d*x + 420*a^3 - 189*a^2
*b - 54*a*b^2 - 8*b^3)*cosh(d*x + c)^5 + 40*(315*a^3*d*x + 420*a^3 - 189*a^2*b - 54*a*b^2 - 8*b^3)*cosh(d*x +
c)^3 + 28*(315*a^3*d*x + 420*a^3 - 189*a^2*b - 54*a*b^2 - 8*b^3)*cosh(d*x + c))*sinh(d*x + c)^2 + 126*(315*a^3
*d*x + 420*a^3 - 189*a^2*b - 54*a*b^2 - 8*b^3)*cosh(d*x + c) - 9*((420*a^3 - 189*a^2*b - 54*a*b^2 - 8*b^3)*cos
h(d*x + c)^8 + 7*(280*a^3 + 21*a^2*b - 54*a*b^2 - 8*b^3)*cosh(d*x + c)^6 + 20*(175*a^3 + 21*a^2*b + 51*a*b^2 -
8*b^3)*cosh(d*x + c)^4 + 420*a^3 - 126*a^2*b - 336*a*b^2 - 672*b^3 + 28*(95*a^3 - 9*a^2*b - 9*a*b^2 + 32*b^3)
*cosh(d*x + c)^2)*sinh(d*x + c))/(d*cosh(d*x + c)^9 + 9*d*cosh(d*x + c)*sinh(d*x + c)^8 + 9*d*cosh(d*x + c)^7
+ 21*(4*d*cosh(d*x + c)^3 + 3*d*cosh(d*x + c))*sinh(d*x + c)^6 + 36*d*cosh(d*x + c)^5 + 9*(14*d*cosh(d*x + c)^
5 + 35*d*cosh(d*x + c)^3 + 20*d*cosh(d*x + c))*sinh(d*x + c)^4 + 84*d*cosh(d*x + c)^3 + 9*(4*d*cosh(d*x + c)^7
+ 21*d*cosh(d*x + c)^5 + 40*d*cosh(d*x + c)^3 + 28*d*cosh(d*x + c))*sinh(d*x + c)^2 + 126*d*cosh(d*x + c))

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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 )^{3} \tanh ^{4}{\left (c + d x \right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*sech(d*x+c)**2)**3*tanh(d*x+c)**4,x)

[Out]

Integral((a + b*sech(c + d*x)**2)**3*tanh(c + d*x)**4, x)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 475 vs. \(2 (102) = 204\).
time = 0.45, size = 475, normalized size = 4.32 \begin {gather*} \frac {315 \, {\left (d x + c\right )} a^{3} + \frac {2 \, {\left (630 \, a^{3} e^{\left (16 \, d x + 16 \, c\right )} - 945 \, a^{2} b e^{\left (16 \, d x + 16 \, c\right )} + 4410 \, a^{3} e^{\left (14 \, d x + 14 \, c\right )} - 3780 \, a^{2} b e^{\left (14 \, d x + 14 \, c\right )} - 1890 \, a b^{2} e^{\left (14 \, d x + 14 \, c\right )} + 13650 \, a^{3} e^{\left (12 \, d x + 12 \, c\right )} - 7560 \, a^{2} b e^{\left (12 \, d x + 12 \, c\right )} - 1890 \, a b^{2} e^{\left (12 \, d x + 12 \, c\right )} - 1680 \, b^{3} e^{\left (12 \, d x + 12 \, c\right )} + 24570 \, a^{3} e^{\left (10 \, d x + 10 \, c\right )} - 11340 \, a^{2} b e^{\left (10 \, d x + 10 \, c\right )} - 1890 \, a b^{2} e^{\left (10 \, d x + 10 \, c\right )} + 2520 \, b^{3} e^{\left (10 \, d x + 10 \, c\right )} + 28350 \, a^{3} e^{\left (8 \, d x + 8 \, c\right )} - 12474 \, a^{2} b e^{\left (8 \, d x + 8 \, c\right )} - 4914 \, a b^{2} e^{\left (8 \, d x + 8 \, c\right )} - 3528 \, b^{3} e^{\left (8 \, d x + 8 \, c\right )} + 21630 \, a^{3} e^{\left (6 \, d x + 6 \, c\right )} - 8316 \, a^{2} b e^{\left (6 \, d x + 6 \, c\right )} - 2646 \, a b^{2} e^{\left (6 \, d x + 6 \, c\right )} + 1008 \, b^{3} e^{\left (6 \, d x + 6 \, c\right )} + 10710 \, a^{3} e^{\left (4 \, d x + 4 \, c\right )} - 3024 \, a^{2} b e^{\left (4 \, d x + 4 \, c\right )} - 54 \, a b^{2} e^{\left (4 \, d x + 4 \, c\right )} - 288 \, b^{3} e^{\left (4 \, d x + 4 \, c\right )} + 3150 \, a^{3} e^{\left (2 \, d x + 2 \, c\right )} - 756 \, a^{2} b e^{\left (2 \, d x + 2 \, c\right )} - 486 \, a b^{2} e^{\left (2 \, d x + 2 \, c\right )} - 72 \, b^{3} e^{\left (2 \, d x + 2 \, c\right )} + 420 \, a^{3} - 189 \, a^{2} b - 54 \, a b^{2} - 8 \, b^{3}\right )}}{{\left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right )}^{9}}}{315 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*sech(d*x+c)^2)^3*tanh(d*x+c)^4,x, algorithm="giac")

[Out]

1/315*(315*(d*x + c)*a^3 + 2*(630*a^3*e^(16*d*x + 16*c) - 945*a^2*b*e^(16*d*x + 16*c) + 4410*a^3*e^(14*d*x + 1
4*c) - 3780*a^2*b*e^(14*d*x + 14*c) - 1890*a*b^2*e^(14*d*x + 14*c) + 13650*a^3*e^(12*d*x + 12*c) - 7560*a^2*b*
e^(12*d*x + 12*c) - 1890*a*b^2*e^(12*d*x + 12*c) - 1680*b^3*e^(12*d*x + 12*c) + 24570*a^3*e^(10*d*x + 10*c) -
11340*a^2*b*e^(10*d*x + 10*c) - 1890*a*b^2*e^(10*d*x + 10*c) + 2520*b^3*e^(10*d*x + 10*c) + 28350*a^3*e^(8*d*x
+ 8*c) - 12474*a^2*b*e^(8*d*x + 8*c) - 4914*a*b^2*e^(8*d*x + 8*c) - 3528*b^3*e^(8*d*x + 8*c) + 21630*a^3*e^(6
*d*x + 6*c) - 8316*a^2*b*e^(6*d*x + 6*c) - 2646*a*b^2*e^(6*d*x + 6*c) + 1008*b^3*e^(6*d*x + 6*c) + 10710*a^3*e
^(4*d*x + 4*c) - 3024*a^2*b*e^(4*d*x + 4*c) - 54*a*b^2*e^(4*d*x + 4*c) - 288*b^3*e^(4*d*x + 4*c) + 3150*a^3*e^
(2*d*x + 2*c) - 756*a^2*b*e^(2*d*x + 2*c) - 486*a*b^2*e^(2*d*x + 2*c) - 72*b^3*e^(2*d*x + 2*c) + 420*a^3 - 189
*a^2*b - 54*a*b^2 - 8*b^3)/(e^(2*d*x + 2*c) + 1)^9)/d

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Mupad [B]
time = 1.62, size = 1834, normalized size = 16.67 \begin {gather*} \frac {\frac {13\,a^3+3\,a\,b^2+16\,b^3}{63\,d}+\frac {10\,{\mathrm {e}}^{4\,c+4\,d\,x}\,\left (13\,a^3+3\,a\,b^2+16\,b^3\right )}{63\,d}+\frac {20\,{\mathrm {e}}^{6\,c+6\,d\,x}\,\left (8\,a^3+3\,a^2\,b+6\,a\,b^2-4\,b^3\right )}{63\,d}-\frac {2\,{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (-10\,a^3+3\,a^2\,b+8\,a\,b^2+16\,b^3\right )}{21\,d}-\frac {5\,{\mathrm {e}}^{8\,c+8\,d\,x}\,\left (a\,b^2-a^3\right )}{3\,d}-\frac {2\,{\mathrm {e}}^{10\,c+10\,d\,x}\,\left (3\,a^2\,b-2\,a^3\right )}{9\,d}}{6\,{\mathrm {e}}^{2\,c+2\,d\,x}+15\,{\mathrm {e}}^{4\,c+4\,d\,x}+20\,{\mathrm {e}}^{6\,c+6\,d\,x}+15\,{\mathrm {e}}^{8\,c+8\,d\,x}+6\,{\mathrm {e}}^{10\,c+10\,d\,x}+{\mathrm {e}}^{12\,c+12\,d\,x}+1}-\frac {\frac {2\,{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (a\,b^2-a^3\right )}{3\,d}-\frac {2\,\left (8\,a^3+3\,a^2\,b+6\,a\,b^2-4\,b^3\right )}{63\,d}+\frac {2\,{\mathrm {e}}^{4\,c+4\,d\,x}\,\left (3\,a^2\,b-2\,a^3\right )}{9\,d}}{3\,{\mathrm {e}}^{2\,c+2\,d\,x}+3\,{\mathrm {e}}^{4\,c+4\,d\,x}+{\mathrm {e}}^{6\,c+6\,d\,x}+1}+\frac {\frac {13\,a^3+3\,a\,b^2+16\,b^3}{63\,d}+\frac {2\,{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (8\,a^3+3\,a^2\,b+6\,a\,b^2-4\,b^3\right )}{21\,d}-\frac {{\mathrm {e}}^{4\,c+4\,d\,x}\,\left (a\,b^2-a^3\right )}{d}-\frac {2\,{\mathrm {e}}^{6\,c+6\,d\,x}\,\left (3\,a^2\,b-2\,a^3\right )}{9\,d}}{4\,{\mathrm {e}}^{2\,c+2\,d\,x}+6\,{\mathrm {e}}^{4\,c+4\,d\,x}+4\,{\mathrm {e}}^{6\,c+6\,d\,x}+{\mathrm {e}}^{8\,c+8\,d\,x}+1}-\frac {\frac {a\,b^2-a^3}{3\,d}+\frac {2\,{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (3\,a^2\,b-2\,a^3\right )}{9\,d}}{2\,{\mathrm {e}}^{2\,c+2\,d\,x}+{\mathrm {e}}^{4\,c+4\,d\,x}+1}+a^3\,x+\frac {\frac {2\,\left (8\,a^3+3\,a^2\,b+6\,a\,b^2-4\,b^3\right )}{63\,d}+\frac {2\,{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (13\,a^3+3\,a\,b^2+16\,b^3\right )}{21\,d}+\frac {20\,{\mathrm {e}}^{6\,c+6\,d\,x}\,\left (13\,a^3+3\,a\,b^2+16\,b^3\right )}{63\,d}+\frac {10\,{\mathrm {e}}^{8\,c+8\,d\,x}\,\left (8\,a^3+3\,a^2\,b+6\,a\,b^2-4\,b^3\right )}{21\,d}-\frac {2\,{\mathrm {e}}^{4\,c+4\,d\,x}\,\left (-10\,a^3+3\,a^2\,b+8\,a\,b^2+16\,b^3\right )}{7\,d}-\frac {2\,{\mathrm {e}}^{10\,c+10\,d\,x}\,\left (a\,b^2-a^3\right )}{d}-\frac {2\,{\mathrm {e}}^{12\,c+12\,d\,x}\,\left (3\,a^2\,b-2\,a^3\right )}{9\,d}}{7\,{\mathrm {e}}^{2\,c+2\,d\,x}+21\,{\mathrm {e}}^{4\,c+4\,d\,x}+35\,{\mathrm {e}}^{6\,c+6\,d\,x}+35\,{\mathrm {e}}^{8\,c+8\,d\,x}+21\,{\mathrm {e}}^{10\,c+10\,d\,x}+7\,{\mathrm {e}}^{12\,c+12\,d\,x}+{\mathrm {e}}^{14\,c+14\,d\,x}+1}-\frac {\frac {a\,b^2-a^3}{3\,d}-\frac {{\mathrm {e}}^{4\,c+4\,d\,x}\,\left (13\,a^3+3\,a\,b^2+16\,b^3\right )}{3\,d}-\frac {5\,{\mathrm {e}}^{8\,c+8\,d\,x}\,\left (13\,a^3+3\,a\,b^2+16\,b^3\right )}{9\,d}-\frac {2\,{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (8\,a^3+3\,a^2\,b+6\,a\,b^2-4\,b^3\right )}{9\,d}-\frac {2\,{\mathrm {e}}^{10\,c+10\,d\,x}\,\left (8\,a^3+3\,a^2\,b+6\,a\,b^2-4\,b^3\right )}{3\,d}+\frac {2\,{\mathrm {e}}^{6\,c+6\,d\,x}\,\left (-10\,a^3+3\,a^2\,b+8\,a\,b^2+16\,b^3\right )}{3\,d}+\frac {7\,{\mathrm {e}}^{12\,c+12\,d\,x}\,\left (a\,b^2-a^3\right )}{3\,d}+\frac {2\,{\mathrm {e}}^{14\,c+14\,d\,x}\,\left (3\,a^2\,b-2\,a^3\right )}{9\,d}}{8\,{\mathrm {e}}^{2\,c+2\,d\,x}+28\,{\mathrm {e}}^{4\,c+4\,d\,x}+56\,{\mathrm {e}}^{6\,c+6\,d\,x}+70\,{\mathrm {e}}^{8\,c+8\,d\,x}+56\,{\mathrm {e}}^{10\,c+10\,d\,x}+28\,{\mathrm {e}}^{12\,c+12\,d\,x}+8\,{\mathrm {e}}^{14\,c+14\,d\,x}+{\mathrm {e}}^{16\,c+16\,d\,x}+1}-\frac {\frac {2\,\left (3\,a^2\,b-2\,a^3\right )}{9\,d}-\frac {8\,{\mathrm {e}}^{6\,c+6\,d\,x}\,\left (13\,a^3+3\,a\,b^2+16\,b^3\right )}{9\,d}-\frac {8\,{\mathrm {e}}^{10\,c+10\,d\,x}\,\left (13\,a^3+3\,a\,b^2+16\,b^3\right )}{9\,d}-\frac {8\,{\mathrm {e}}^{4\,c+4\,d\,x}\,\left (8\,a^3+3\,a^2\,b+6\,a\,b^2-4\,b^3\right )}{9\,d}-\frac {8\,{\mathrm {e}}^{12\,c+12\,d\,x}\,\left (8\,a^3+3\,a^2\,b+6\,a\,b^2-4\,b^3\right )}{9\,d}+\frac {4\,{\mathrm {e}}^{8\,c+8\,d\,x}\,\left (-10\,a^3+3\,a^2\,b+8\,a\,b^2+16\,b^3\right )}{3\,d}+\frac {8\,{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (a\,b^2-a^3\right )}{3\,d}+\frac {8\,{\mathrm {e}}^{14\,c+14\,d\,x}\,\left (a\,b^2-a^3\right )}{3\,d}+\frac {2\,{\mathrm {e}}^{16\,c+16\,d\,x}\,\left (3\,a^2\,b-2\,a^3\right )}{9\,d}}{9\,{\mathrm {e}}^{2\,c+2\,d\,x}+36\,{\mathrm {e}}^{4\,c+4\,d\,x}+84\,{\mathrm {e}}^{6\,c+6\,d\,x}+126\,{\mathrm {e}}^{8\,c+8\,d\,x}+126\,{\mathrm {e}}^{10\,c+10\,d\,x}+84\,{\mathrm {e}}^{12\,c+12\,d\,x}+36\,{\mathrm {e}}^{14\,c+14\,d\,x}+9\,{\mathrm {e}}^{16\,c+16\,d\,x}+{\mathrm {e}}^{18\,c+18\,d\,x}+1}-\frac {\frac {2\,\left (-10\,a^3+3\,a^2\,b+8\,a\,b^2+16\,b^3\right )}{105\,d}-\frac {4\,{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (13\,a^3+3\,a\,b^2+16\,b^3\right )}{63\,d}-\frac {4\,{\mathrm {e}}^{4\,c+4\,d\,x}\,\left (8\,a^3+3\,a^2\,b+6\,a\,b^2-4\,b^3\right )}{21\,d}+\frac {4\,{\mathrm {e}}^{6\,c+6\,d\,x}\,\left (a\,b^2-a^3\right )}{3\,d}+\frac {2\,{\mathrm {e}}^{8\,c+8\,d\,x}\,\left (3\,a^2\,b-2\,a^3\right )}{9\,d}}{5\,{\mathrm {e}}^{2\,c+2\,d\,x}+10\,{\mathrm {e}}^{4\,c+4\,d\,x}+10\,{\mathrm {e}}^{6\,c+6\,d\,x}+5\,{\mathrm {e}}^{8\,c+8\,d\,x}+{\mathrm {e}}^{10\,c+10\,d\,x}+1}-\frac {2\,\left (3\,a^2\,b-2\,a^3\right )}{9\,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(tanh(c + d*x)^4*(a + b/cosh(c + d*x)^2)^3,x)

[Out]

((3*a*b^2 + 13*a^3 + 16*b^3)/(63*d) + (10*exp(4*c + 4*d*x)*(3*a*b^2 + 13*a^3 + 16*b^3))/(63*d) + (20*exp(6*c +
6*d*x)*(6*a*b^2 + 3*a^2*b + 8*a^3 - 4*b^3))/(63*d) - (2*exp(2*c + 2*d*x)*(8*a*b^2 + 3*a^2*b - 10*a^3 + 16*b^3
))/(21*d) - (5*exp(8*c + 8*d*x)*(a*b^2 - a^3))/(3*d) - (2*exp(10*c + 10*d*x)*(3*a^2*b - 2*a^3))/(9*d))/(6*exp(
2*c + 2*d*x) + 15*exp(4*c + 4*d*x) + 20*exp(6*c + 6*d*x) + 15*exp(8*c + 8*d*x) + 6*exp(10*c + 10*d*x) + exp(12
*c + 12*d*x) + 1) - ((2*exp(2*c + 2*d*x)*(a*b^2 - a^3))/(3*d) - (2*(6*a*b^2 + 3*a^2*b + 8*a^3 - 4*b^3))/(63*d)
+ (2*exp(4*c + 4*d*x)*(3*a^2*b - 2*a^3))/(9*d))/(3*exp(2*c + 2*d*x) + 3*exp(4*c + 4*d*x) + exp(6*c + 6*d*x) +
1) + ((3*a*b^2 + 13*a^3 + 16*b^3)/(63*d) + (2*exp(2*c + 2*d*x)*(6*a*b^2 + 3*a^2*b + 8*a^3 - 4*b^3))/(21*d) -
(exp(4*c + 4*d*x)*(a*b^2 - a^3))/d - (2*exp(6*c + 6*d*x)*(3*a^2*b - 2*a^3))/(9*d))/(4*exp(2*c + 2*d*x) + 6*exp
(4*c + 4*d*x) + 4*exp(6*c + 6*d*x) + exp(8*c + 8*d*x) + 1) - ((a*b^2 - a^3)/(3*d) + (2*exp(2*c + 2*d*x)*(3*a^2
*b - 2*a^3))/(9*d))/(2*exp(2*c + 2*d*x) + exp(4*c + 4*d*x) + 1) + a^3*x + ((2*(6*a*b^2 + 3*a^2*b + 8*a^3 - 4*b
^3))/(63*d) + (2*exp(2*c + 2*d*x)*(3*a*b^2 + 13*a^3 + 16*b^3))/(21*d) + (20*exp(6*c + 6*d*x)*(3*a*b^2 + 13*a^3
+ 16*b^3))/(63*d) + (10*exp(8*c + 8*d*x)*(6*a*b^2 + 3*a^2*b + 8*a^3 - 4*b^3))/(21*d) - (2*exp(4*c + 4*d*x)*(8
*a*b^2 + 3*a^2*b - 10*a^3 + 16*b^3))/(7*d) - (2*exp(10*c + 10*d*x)*(a*b^2 - a^3))/d - (2*exp(12*c + 12*d*x)*(3
*a^2*b - 2*a^3))/(9*d))/(7*exp(2*c + 2*d*x) + 21*exp(4*c + 4*d*x) + 35*exp(6*c + 6*d*x) + 35*exp(8*c + 8*d*x)
+ 21*exp(10*c + 10*d*x) + 7*exp(12*c + 12*d*x) + exp(14*c + 14*d*x) + 1) - ((a*b^2 - a^3)/(3*d) - (exp(4*c + 4
*d*x)*(3*a*b^2 + 13*a^3 + 16*b^3))/(3*d) - (5*exp(8*c + 8*d*x)*(3*a*b^2 + 13*a^3 + 16*b^3))/(9*d) - (2*exp(2*c
+ 2*d*x)*(6*a*b^2 + 3*a^2*b + 8*a^3 - 4*b^3))/(9*d) - (2*exp(10*c + 10*d*x)*(6*a*b^2 + 3*a^2*b + 8*a^3 - 4*b^
3))/(3*d) + (2*exp(6*c + 6*d*x)*(8*a*b^2 + 3*a^2*b - 10*a^3 + 16*b^3))/(3*d) + (7*exp(12*c + 12*d*x)*(a*b^2 -
a^3))/(3*d) + (2*exp(14*c + 14*d*x)*(3*a^2*b - 2*a^3))/(9*d))/(8*exp(2*c + 2*d*x) + 28*exp(4*c + 4*d*x) + 56*e
xp(6*c + 6*d*x) + 70*exp(8*c + 8*d*x) + 56*exp(10*c + 10*d*x) + 28*exp(12*c + 12*d*x) + 8*exp(14*c + 14*d*x) +
exp(16*c + 16*d*x) + 1) - ((2*(3*a^2*b - 2*a^3))/(9*d) - (8*exp(6*c + 6*d*x)*(3*a*b^2 + 13*a^3 + 16*b^3))/(9*
d) - (8*exp(10*c + 10*d*x)*(3*a*b^2 + 13*a^3 + 16*b^3))/(9*d) - (8*exp(4*c + 4*d*x)*(6*a*b^2 + 3*a^2*b + 8*a^3
- 4*b^3))/(9*d) - (8*exp(12*c + 12*d*x)*(6*a*b^2 + 3*a^2*b + 8*a^3 - 4*b^3))/(9*d) + (4*exp(8*c + 8*d*x)*(8*a
*b^2 + 3*a^2*b - 10*a^3 + 16*b^3))/(3*d) + (8*exp(2*c + 2*d*x)*(a*b^2 - a^3))/(3*d) + (8*exp(14*c + 14*d*x)*(a
*b^2 - a^3))/(3*d) + (2*exp(16*c + 16*d*x)*(3*a^2*b - 2*a^3))/(9*d))/(9*exp(2*c + 2*d*x) + 36*exp(4*c + 4*d*x)
+ 84*exp(6*c + 6*d*x) + 126*exp(8*c + 8*d*x) + 126*exp(10*c + 10*d*x) + 84*exp(12*c + 12*d*x) + 36*exp(14*c +
14*d*x) + 9*exp(16*c + 16*d*x) + exp(18*c + 18*d*x) + 1) - ((2*(8*a*b^2 + 3*a^2*b - 10*a^3 + 16*b^3))/(105*d)
- (4*exp(2*c + 2*d*x)*(3*a*b^2 + 13*a^3 + 16*b^3))/(63*d) - (4*exp(4*c + 4*d*x)*(6*a*b^2 + 3*a^2*b + 8*a^3 -
4*b^3))/(21*d) + (4*exp(6*c + 6*d*x)*(a*b^2 - a^3))/(3*d) + (2*exp(8*c + 8*d*x)*(3*a^2*b - 2*a^3))/(9*d))/(5*e
xp(2*c + 2*d*x) + 10*exp(4*c + 4*d*x) + 10*exp(6*c + 6*d*x) + 5*exp(8*c + 8*d*x) + exp(10*c + 10*d*x) + 1) - (
2*(3*a^2*b - 2*a^3))/(9*d*(exp(2*c + 2*d*x) + 1))

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