3.3.27 \(\int \text {csch}^{18}(c+d x) (a+b \sinh ^4(c+d x))^3 \, dx\) [227]

Optimal. Leaf size=221 \[ -\frac {(a+b)^3 \coth (c+d x)}{d}+\frac {2 (a+b)^2 (4 a+b) \coth ^3(c+d x)}{3 d}-\frac {(a+b) \left (28 a^2+17 a b+b^2\right ) \coth ^5(c+d x)}{5 d}+\frac {4 a \left (14 a^2+15 a b+3 b^2\right ) \coth ^7(c+d x)}{7 d}-\frac {a \left (70 a^2+45 a b+3 b^2\right ) \coth ^9(c+d x)}{9 d}+\frac {2 a^2 (28 a+9 b) \coth ^{11}(c+d x)}{11 d}-\frac {a^2 (28 a+3 b) \coth ^{13}(c+d x)}{13 d}+\frac {8 a^3 \coth ^{15}(c+d x)}{15 d}-\frac {a^3 \coth ^{17}(c+d x)}{17 d} \]

[Out]

-(a+b)^3*coth(d*x+c)/d+2/3*(a+b)^2*(4*a+b)*coth(d*x+c)^3/d-1/5*(a+b)*(28*a^2+17*a*b+b^2)*coth(d*x+c)^5/d+4/7*a
*(14*a^2+15*a*b+3*b^2)*coth(d*x+c)^7/d-1/9*a*(70*a^2+45*a*b+3*b^2)*coth(d*x+c)^9/d+2/11*a^2*(28*a+9*b)*coth(d*
x+c)^11/d-1/13*a^2*(28*a+3*b)*coth(d*x+c)^13/d+8/15*a^3*coth(d*x+c)^15/d-1/17*a^3*coth(d*x+c)^17/d

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Rubi [A]
time = 0.14, antiderivative size = 221, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {3296, 1275} \begin {gather*} -\frac {a^3 \coth ^{17}(c+d x)}{17 d}+\frac {8 a^3 \coth ^{15}(c+d x)}{15 d}-\frac {a \left (70 a^2+45 a b+3 b^2\right ) \coth ^9(c+d x)}{9 d}+\frac {4 a \left (14 a^2+15 a b+3 b^2\right ) \coth ^7(c+d x)}{7 d}-\frac {(a+b) \left (28 a^2+17 a b+b^2\right ) \coth ^5(c+d x)}{5 d}-\frac {a^2 (28 a+3 b) \coth ^{13}(c+d x)}{13 d}+\frac {2 a^2 (28 a+9 b) \coth ^{11}(c+d x)}{11 d}+\frac {2 (a+b)^2 (4 a+b) \coth ^3(c+d x)}{3 d}-\frac {(a+b)^3 \coth (c+d x)}{d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Csch[c + d*x]^18*(a + b*Sinh[c + d*x]^4)^3,x]

[Out]

-(((a + b)^3*Coth[c + d*x])/d) + (2*(a + b)^2*(4*a + b)*Coth[c + d*x]^3)/(3*d) - ((a + b)*(28*a^2 + 17*a*b + b
^2)*Coth[c + d*x]^5)/(5*d) + (4*a*(14*a^2 + 15*a*b + 3*b^2)*Coth[c + d*x]^7)/(7*d) - (a*(70*a^2 + 45*a*b + 3*b
^2)*Coth[c + d*x]^9)/(9*d) + (2*a^2*(28*a + 9*b)*Coth[c + d*x]^11)/(11*d) - (a^2*(28*a + 3*b)*Coth[c + d*x]^13
)/(13*d) + (8*a^3*Coth[c + d*x]^15)/(15*d) - (a^3*Coth[c + d*x]^17)/(17*d)

Rule 1275

Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> In
t[ExpandIntegrand[(f*x)^m*(d + e*x^2)^q*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, q}, x] &&
NeQ[b^2 - 4*a*c, 0] && IGtQ[p, 0] && IGtQ[q, -2]

Rule 3296

Int[sin[(e_.) + (f_.)*(x_)]^(m_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^4)^(p_.), x_Symbol] :> With[{ff = FreeF
actors[Tan[e + f*x], x]}, Dist[ff^(m + 1)/f, Subst[Int[x^m*((a + 2*a*ff^2*x^2 + (a + b)*ff^4*x^4)^p/(1 + ff^2*
x^2)^(m/2 + 2*p + 1)), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f}, x] && IntegerQ[m/2] && IntegerQ[p]

Rubi steps

\begin {align*} \int \text {csch}^{18}(c+d x) \left (a+b \sinh ^4(c+d x)\right )^3 \, dx &=\frac {\text {Subst}\left (\int \frac {\left (1-x^2\right )^2 \left (a-2 a x^2+(a+b) x^4\right )^3}{x^{18}} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac {\text {Subst}\left (\int \left (\frac {a^3}{x^{18}}-\frac {8 a^3}{x^{16}}+\frac {a^2 (28 a+3 b)}{x^{14}}-\frac {2 a^2 (28 a+9 b)}{x^{12}}+\frac {a \left (70 a^2+45 a b+3 b^2\right )}{x^{10}}-\frac {4 a \left (14 a^2+15 a b+3 b^2\right )}{x^8}+\frac {(a+b) \left (28 a^2+17 a b+b^2\right )}{x^6}-\frac {2 (a+b)^2 (4 a+b)}{x^4}+\frac {(a+b)^3}{x^2}\right ) \, dx,x,\tanh (c+d x)\right )}{d}\\ &=-\frac {(a+b)^3 \coth (c+d x)}{d}+\frac {2 (a+b)^2 (4 a+b) \coth ^3(c+d x)}{3 d}-\frac {(a+b) \left (28 a^2+17 a b+b^2\right ) \coth ^5(c+d x)}{5 d}+\frac {4 a \left (14 a^2+15 a b+3 b^2\right ) \coth ^7(c+d x)}{7 d}-\frac {a \left (70 a^2+45 a b+3 b^2\right ) \coth ^9(c+d x)}{9 d}+\frac {2 a^2 (28 a+9 b) \coth ^{11}(c+d x)}{11 d}-\frac {a^2 (28 a+3 b) \coth ^{13}(c+d x)}{13 d}+\frac {8 a^3 \coth ^{15}(c+d x)}{15 d}-\frac {a^3 \coth ^{17}(c+d x)}{17 d}\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(458\) vs. \(2(221)=442\).
time = 4.63, size = 458, normalized size = 2.07 \begin {gather*} -\frac {\left (680680 \left (1024 a^3+1152 a^2 b+840 a b^2+231 b^3\right ) \cosh (c+d x)-272272 \left (2048 a^3+5760 a^2 b+4704 a b^2+1353 b^3\right ) \cosh (3 (c+d x))[phone] a^3 \cosh (5 (c+d x))[phone] a^2 b \cosh (5 (c+d x))[phone] a b^2 \cosh (5 (c+d x))[phone] b^3 \cosh (5 (c+d x))-177422336 a^3 \cosh (7 (c+d x))-605928960 a^2 b \cosh (7 (c+d x))-692659968 a b^2 \cosh (7 (c+d x))-242288046 b^3 \cosh (7 (c+d x))[phone] a^3 \cosh (9 (c+d x))[phone] a^2 b \cosh (9 (c+d x))[phone] a b^2 \cosh (9 (c+d x))[phone] b^3 \cosh (9 (c+d x))-19496960 a^3 \cosh (11 (c+d x))-66585600 a^2 b \cosh (11 (c+d x))-79347840 a b^2 \cosh (11 (c+d x))-33693660 b^3 \cosh (11 (c+d x))+3899392 a^3 \cosh (13 (c+d x))[phone] a^2 b \cosh (13 (c+d x))[phone] a b^2 \cosh (13 (c+d x))+6942936 b^3 \cosh (13 (c+d x))-487424 a^3 \cosh (15 (c+d x))-1664640 a^2 b \cosh (15 (c+d x))-1983696 a b^2 \cosh (15 (c+d x))-867867 b^3 \cosh (15 (c+d x))+28672 a^3 \cosh (17 (c+d x))+97920 a^2 b \cosh (17 (c+d x))+116688 a b^2 \cosh (17 (c+d x))+51051 b^3 \cosh (17 (c+d x))\right ) \text {csch}^{17}(c+d x)}{6273146880 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Csch[c + d*x]^18*(a + b*Sinh[c + d*x]^4)^3,x]

[Out]

-1/6273146880*((680680*(1024*a^3 + 1152*a^2*b + 840*a*b^2 + 231*b^3)*Cosh[c + d*x] - 272272*(2048*a^3 + 5760*a
^2*b + 4704*a*b^2 + 1353*b^3)*Cosh[3*(c + d*x)] + 354844672*a^3*Cosh[5*(c + d*x)] + 1211857920*a^2*b*Cosh[5*(c
+ d*x)] + 1189284096*a*b^2*Cosh[5*(c + d*x)] + 372263892*b^3*Cosh[5*(c + d*x)] - 177422336*a^3*Cosh[7*(c + d*
x)] - 605928960*a^2*b*Cosh[7*(c + d*x)] - 692659968*a*b^2*Cosh[7*(c + d*x)] - 242288046*b^3*Cosh[7*(c + d*x)]
+ 68239360*a^3*Cosh[9*(c + d*x)] + 233049600*a^2*b*Cosh[9*(c + d*x)] + 277717440*a*b^2*Cosh[9*(c + d*x)] + 108
738630*b^3*Cosh[9*(c + d*x)] - 19496960*a^3*Cosh[11*(c + d*x)] - 66585600*a^2*b*Cosh[11*(c + d*x)] - 79347840*
a*b^2*Cosh[11*(c + d*x)] - 33693660*b^3*Cosh[11*(c + d*x)] + 3899392*a^3*Cosh[13*(c + d*x)] + 13317120*a^2*b*C
osh[13*(c + d*x)] + 15869568*a*b^2*Cosh[13*(c + d*x)] + 6942936*b^3*Cosh[13*(c + d*x)] - 487424*a^3*Cosh[15*(c
+ d*x)] - 1664640*a^2*b*Cosh[15*(c + d*x)] - 1983696*a*b^2*Cosh[15*(c + d*x)] - 867867*b^3*Cosh[15*(c + d*x)]
+ 28672*a^3*Cosh[17*(c + d*x)] + 97920*a^2*b*Cosh[17*(c + d*x)] + 116688*a*b^2*Cosh[17*(c + d*x)] + 51051*b^3
*Cosh[17*(c + d*x)])*Csch[c + d*x]^17)/d

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(679\) vs. \(2(205)=410\).
time = 1.64, size = 680, normalized size = 3.08

method	result	size
risch	\(-\frac {16 \left (697016320 a^{3} {\mathrm e}^{16 d x +16 c}+116688 a \,b^{2}-1775057856 a \,b^{2} {\mathrm e}^{14 d x +14 c}[phone] a^{2} b \,{\mathrm e}^{12 d x +12 c}[phone] a \,b^{2} {\mathrm e}^{12 d x +12 c}-605928960 a^{2} b \,{\mathrm e}^{10 d x +10 c}-707362656 a \,b^{2} {\mathrm e}^{10 d x +10 c}-1132457040 a \,b^{2} {\mathrm e}^{18 d x +18 c}[phone] a^{2} b \,{\mathrm e}^{16 d x +16 c}[phone] a \,b^{2} {\mathrm e}^{16 d x +16 c}-1736317440 a^{2} b \,{\mathrm e}^{14 d x +14 c}[phone] a^{2} b \,{\mathrm e}^{4 d x +4 c}-1664640 a^{2} b \,{\mathrm e}^{2 d x +2 c}+97920 a^{2} b +28672 a^{3}[phone] a^{2} b \,{\mathrm e}^{20 d x +20 c}+51051 b^{3}[phone] a \,b^{2} {\mathrm e}^{8 d x +8 c}-79347840 a \,b^{2} {\mathrm e}^{6 d x +6 c}[phone] a \,b^{2} {\mathrm e}^{4 d x +4 c}[phone] a \,b^{2} {\mathrm e}^{20 d x +20 c}[phone] a^{2} b \,{\mathrm e}^{8 d x +8 c}-1983696 a \,b^{2} {\mathrm e}^{2 d x +2 c}-66585600 a^{2} b \,{\mathrm e}^{6 d x +6 c}[phone] b^{3} {\mathrm e}^{24 d x +24 c}-127423296 a \,b^{2} {\mathrm e}^{22 d x +22 c}-798145920 a^{2} b \,{\mathrm e}^{18 d x +18 c}-541906365 b^{3} {\mathrm e}^{18 d x +18 c}[phone] b^{3} {\mathrm e}^{16 d x +16 c}-680611932 b^{3} {\mathrm e}^{14 d x +14 c}-487424 a^{3} {\mathrm e}^{2 d x +2 c}[phone] a^{3} {\mathrm e}^{12 d x +12 c}[phone] b^{3} {\mathrm e}^{12 d x +12 c}-177422336 a^{3} {\mathrm e}^{10 d x +10 c}-129771642 b^{3} {\mathrm e}^{22 d x +22 c}[phone] b^{3} {\mathrm e}^{20 d x +20 c}[phone] a \,b^{2} {\mathrm e}^{24 d x +24 c}-6381375 b^{3} {\mathrm e}^{26 d x +26 c}-557613056 a^{3} {\mathrm e}^{14 d x +14 c}-867867 b^{3} {\mathrm e}^{2 d x +2 c}[phone] b^{3} {\mathrm e}^{8 d x +8 c}+3899392 a^{3} {\mathrm e}^{4 d x +4 c}+6942936 b^{3} {\mathrm e}^{4 d x +4 c}+510510 b^{3} {\mathrm e}^{28 d x +28 c}-279095817 b^{3} {\mathrm e}^{10 d x +10 c}[phone] a^{3} {\mathrm e}^{8 d x +8 c}-19496960 a^{3} {\mathrm e}^{6 d x +6 c}-34204170 b^{3} {\mathrm e}^{6 d x +6 c}\right )}{765765 d \left ({\mathrm e}^{2 d x +2 c}-1\right )^{17}}\)	\(680\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csch(d*x+c)^18*(a+b*sinh(d*x+c)^4)^3,x,method=_RETURNVERBOSE)

[Out]

-16/765765*(697016320*a^3*exp(16*d*x+16*c)+116688*a*b^2-1775057856*a*b^2*exp(14*d*x+14*c)[phone]*a^2*b*exp
(12*d*x+12*c)[phone]*a*b^2*exp(12*d*x+12*c)-605928960*a^2*b*exp(10*d*x+10*c)-707362656*a*b^2*exp(10*d*x+10
*c)-1132457040*a*b^2*exp(18*d*x+18*c)[phone]*a^2*b*exp(16*d*x+16*c)[phone]*a*b^2*exp(16*d*x+16*c)-1736
317440*a^2*b*exp(14*d*x+14*c)[phone]*a^2*b*exp(4*d*x+4*c)-1664640*a^2*b*exp(2*d*x+2*c)+97920*a^2*b+28672*a^3
[phone]*a^2*b*exp(20*d*x+20*c)+51051*b^3+277717440*a*b^2*exp(8*d*x+8*c)-79347840*a*b^2*exp(6*d*x+6*c)+15869
568*a*b^2*exp(4*d*x+4*c)[phone]*a*b^2*exp(20*d*x+20*c)[phone]*a^2*b*exp(8*d*x+8*c)-1983696*a*b^2*exp(2*d
*x+2*c)-66585600*a^2*b*exp(6*d*x+6*c)[phone]*b^3*exp(24*d*x+24*c)-127423296*a*b^2*exp(22*d*x+22*c)-798145920
*a^2*b*exp(18*d*x+18*c)-541906365*b^3*exp(18*d*x+18*c)[phone]*b^3*exp(16*d*x+16*c)-680611932*b^3*exp(14*d*x
+14*c)-487424*a^3*exp(2*d*x+2*c)[phone]*a^3*exp(12*d*x+12*c)[phone]*b^3*exp(12*d*x+12*c)-177422336*a^3*e
xp(10*d*x+10*c)-129771642*b^3*exp(22*d*x+22*c)[phone]*b^3*exp(20*d*x+20*c)[phone]*a*b^2*exp(24*d*x+24*c)-
6381375*b^3*exp(26*d*x+26*c)-557613056*a^3*exp(14*d*x+14*c)-867867*b^3*exp(2*d*x+2*c)[phone]*b^3*exp(8*d*x+
8*c)+3899392*a^3*exp(4*d*x+4*c)+6942936*b^3*exp(4*d*x+4*c)+510510*b^3*exp(28*d*x+28*c)-279095817*b^3*exp(10*d*
x+10*c)[phone]*a^3*exp(8*d*x+8*c)-19496960*a^3*exp(6*d*x+6*c)-34204170*b^3*exp(6*d*x+6*c))/d/(exp(2*d*x+2*c)
-1)^17

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 3719 vs. \(2 (205) = 410\).
time = 0.30, size = 3719, normalized size = 16.83 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(d*x+c)^18*(a+b*sinh(d*x+c)^4)^3,x, algorithm="maxima")

[Out]

-65536/109395*a^3*(17*e^(-2*d*x - 2*c)/(d*(17*e^(-2*d*x - 2*c) - 136*e^(-4*d*x - 4*c) + 680*e^(-6*d*x - 6*c) -
2380*e^(-8*d*x - 8*c) + 6188*e^(-10*d*x - 10*c) - 12376*e^(-12*d*x - 12*c) + 19448*e^(-14*d*x - 14*c) - 24310
*e^(-16*d*x - 16*c) + 24310*e^(-18*d*x - 18*c) - 19448*e^(-20*d*x - 20*c) + 12376*e^(-22*d*x - 22*c) - 6188*e^
(-24*d*x - 24*c) + 2380*e^(-26*d*x - 26*c) - 680*e^(-28*d*x - 28*c) + 136*e^(-30*d*x - 30*c) - 17*e^(-32*d*x -
32*c) + e^(-34*d*x - 34*c) - 1)) - 136*e^(-4*d*x - 4*c)/(d*(17*e^(-2*d*x - 2*c) - 136*e^(-4*d*x - 4*c) + 680*
e^(-6*d*x - 6*c) - 2380*e^(-8*d*x - 8*c) + 6188*e^(-10*d*x - 10*c) - 12376*e^(-12*d*x - 12*c) + 19448*e^(-14*d
*x - 14*c) - 24310*e^(-16*d*x - 16*c) + 24310*e^(-18*d*x - 18*c) - 19448*e^(-20*d*x - 20*c) + 12376*e^(-22*d*x
- 22*c) - 6188*e^(-24*d*x - 24*c) + 2380*e^(-26*d*x - 26*c) - 680*e^(-28*d*x - 28*c) + 136*e^(-30*d*x - 30*c)
- 17*e^(-32*d*x - 32*c) + e^(-34*d*x - 34*c) - 1)) + 680*e^(-6*d*x - 6*c)/(d*(17*e^(-2*d*x - 2*c) - 136*e^(-4
*d*x - 4*c) + 680*e^(-6*d*x - 6*c) - 2380*e^(-8*d*x - 8*c) + 6188*e^(-10*d*x - 10*c) - 12376*e^(-12*d*x - 12*c
) + 19448*e^(-14*d*x - 14*c) - 24310*e^(-16*d*x - 16*c) + 24310*e^(-18*d*x - 18*c) - 19448*e^(-20*d*x - 20*c)
+ 12376*e^(-22*d*x - 22*c) - 6188*e^(-24*d*x - 24*c) + 2380*e^(-26*d*x - 26*c) - 680*e^(-28*d*x - 28*c) + 136*
e^(-30*d*x - 30*c) - 17*e^(-32*d*x - 32*c) + e^(-34*d*x - 34*c) - 1)) - 2380*e^(-8*d*x - 8*c)/(d*(17*e^(-2*d*x
- 2*c) - 136*e^(-4*d*x - 4*c) + 680*e^(-6*d*x - 6*c) - 2380*e^(-8*d*x - 8*c) + 6188*e^(-10*d*x - 10*c) - 1237
6*e^(-12*d*x - 12*c) + 19448*e^(-14*d*x - 14*c) - 24310*e^(-16*d*x - 16*c) + 24310*e^(-18*d*x - 18*c) - 19448*
e^(-20*d*x - 20*c) + 12376*e^(-22*d*x - 22*c) - 6188*e^(-24*d*x - 24*c) + 2380*e^(-26*d*x - 26*c) - 680*e^(-28
*d*x - 28*c) + 136*e^(-30*d*x - 30*c) - 17*e^(-32*d*x - 32*c) + e^(-34*d*x - 34*c) - 1)) + 6188*e^(-10*d*x - 1
0*c)/(d*(17*e^(-2*d*x - 2*c) - 136*e^(-4*d*x - 4*c) + 680*e^(-6*d*x - 6*c) - 2380*e^(-8*d*x - 8*c) + 6188*e^(-
10*d*x - 10*c) - 12376*e^(-12*d*x - 12*c) + 19448*e^(-14*d*x - 14*c) - 24310*e^(-16*d*x - 16*c) + 24310*e^(-18
*d*x - 18*c) - 19448*e^(-20*d*x - 20*c) + 12376*e^(-22*d*x - 22*c) - 6188*e^(-24*d*x - 24*c) + 2380*e^(-26*d*x
- 26*c) - 680*e^(-28*d*x - 28*c) + 136*e^(-30*d*x - 30*c) - 17*e^(-32*d*x - 32*c) + e^(-34*d*x - 34*c) - 1))
- 12376*e^(-12*d*x - 12*c)/(d*(17*e^(-2*d*x - 2*c) - 136*e^(-4*d*x - 4*c) + 680*e^(-6*d*x - 6*c) - 2380*e^(-8*
d*x - 8*c) + 6188*e^(-10*d*x - 10*c) - 12376*e^(-12*d*x - 12*c) + 19448*e^(-14*d*x - 14*c) - 24310*e^(-16*d*x
- 16*c) + 24310*e^(-18*d*x - 18*c) - 19448*e^(-20*d*x - 20*c) + 12376*e^(-22*d*x - 22*c) - 6188*e^(-24*d*x - 2
4*c) + 2380*e^(-26*d*x - 26*c) - 680*e^(-28*d*x - 28*c) + 136*e^(-30*d*x - 30*c) - 17*e^(-32*d*x - 32*c) + e^(
-34*d*x - 34*c) - 1)) + 19448*e^(-14*d*x - 14*c)/(d*(17*e^(-2*d*x - 2*c) - 136*e^(-4*d*x - 4*c) + 680*e^(-6*d*
x - 6*c) - 2380*e^(-8*d*x - 8*c) + 6188*e^(-10*d*x - 10*c) - 12376*e^(-12*d*x - 12*c) + 19448*e^(-14*d*x - 14*
c) - 24310*e^(-16*d*x - 16*c) + 24310*e^(-18*d*x - 18*c) - 19448*e^(-20*d*x - 20*c) + 12376*e^(-22*d*x - 22*c)
- 6188*e^(-24*d*x - 24*c) + 2380*e^(-26*d*x - 26*c) - 680*e^(-28*d*x - 28*c) + 136*e^(-30*d*x - 30*c) - 17*e^
(-32*d*x - 32*c) + e^(-34*d*x - 34*c) - 1)) - 24310*e^(-16*d*x - 16*c)/(d*(17*e^(-2*d*x - 2*c) - 136*e^(-4*d*x
- 4*c) + 680*e^(-6*d*x - 6*c) - 2380*e^(-8*d*x - 8*c) + 6188*e^(-10*d*x - 10*c) - 12376*e^(-12*d*x - 12*c) +
19448*e^(-14*d*x - 14*c) - 24310*e^(-16*d*x - 16*c) + 24310*e^(-18*d*x - 18*c) - 19448*e^(-20*d*x - 20*c) + 12
376*e^(-22*d*x - 22*c) - 6188*e^(-24*d*x - 24*c) + 2380*e^(-26*d*x - 26*c) - 680*e^(-28*d*x - 28*c) + 136*e^(-
30*d*x - 30*c) - 17*e^(-32*d*x - 32*c) + e^(-34*d*x - 34*c) - 1)) - 1/(d*(17*e^(-2*d*x - 2*c) - 136*e^(-4*d*x
- 4*c) + 680*e^(-6*d*x - 6*c) - 2380*e^(-8*d*x - 8*c) + 6188*e^(-10*d*x - 10*c) - 12376*e^(-12*d*x - 12*c) + 1
9448*e^(-14*d*x - 14*c) - 24310*e^(-16*d*x - 16*c) + 24310*e^(-18*d*x - 18*c) - 19448*e^(-20*d*x - 20*c) + 123
76*e^(-22*d*x - 22*c) - 6188*e^(-24*d*x - 24*c) + 2380*e^(-26*d*x - 26*c) - 680*e^(-28*d*x - 28*c) + 136*e^(-3
0*d*x - 30*c) - 17*e^(-32*d*x - 32*c) + e^(-34*d*x - 34*c) - 1))) - 2048/1001*a^2*b*(13*e^(-2*d*x - 2*c)/(d*(1
3*e^(-2*d*x - 2*c) - 78*e^(-4*d*x - 4*c) + 286*e^(-6*d*x - 6*c) - 715*e^(-8*d*x - 8*c) + 1287*e^(-10*d*x - 10*
c) - 1716*e^(-12*d*x - 12*c) + 1716*e^(-14*d*x - 14*c) - 1287*e^(-16*d*x - 16*c) + 715*e^(-18*d*x - 18*c) - 28
6*e^(-20*d*x - 20*c) + 78*e^(-22*d*x - 22*c) - 13*e^(-24*d*x - 24*c) + e^(-26*d*x - 26*c) - 1)) - 78*e^(-4*d*x
- 4*c)/(d*(13*e^(-2*d*x - 2*c) - 78*e^(-4*d*x - 4*c) + 286*e^(-6*d*x - 6*c) - 715*e^(-8*d*x - 8*c) + 1287*e^(
-10*d*x - 10*c) - 1716*e^(-12*d*x - 12*c) + 1716*e^(-14*d*x - 14*c) - 1287*e^(-16*d*x - 16*c) + 715*e^(-18*d*x
- 18*c) - 286*e^(-20*d*x - 20*c) + 78*e^(-22*d*x - 22*c) - 13*e^(-24*d*x - 24*c) + e^(-26*d*x - 26*c) - 1)) +
286*e^(-6*d*x - 6*c)/(d*(13*e^(-2*d*x - 2*c) - 78*e^(-4*d*x - 4*c) + 286*e^(-6*d*x - 6*c) - 715*e^(-8*d*x - 8
*c) + 1287*e^(-10*d*x - 10*c) - 1716*e^(-12*d*x...

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 3585 vs. \(2 (205) = 410\).
time = 0.38, size = 3585, normalized size = 16.22 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(d*x+c)^18*(a+b*sinh(d*x+c)^4)^3,x, algorithm="fricas")

[Out]

-16/765765*((28672*a^3 + 97920*a^2*b + 116688*a*b^2 + 561561*b^3)*cosh(d*x + c)^14 - 14*(28672*a^3 + 97920*a^2
*b + 116688*a*b^2 - 459459*b^3)*cosh(d*x + c)*sinh(d*x + c)^13 + (28672*a^3 + 97920*a^2*b + 116688*a*b^2 + 561
561*b^3)*sinh(d*x + c)^14 - 34*(14336*a^3 + 48960*a^2*b + 58344*a*b^2 + 213213*b^3)*cosh(d*x + c)^12 - (487424
*a^3 + 1664640*a^2*b + 1983696*a*b^2 + 7249242*b^3 - 91*(28672*a^3 + 97920*a^2*b + 116688*a*b^2 + 561561*b^3)*
cosh(d*x + c)^2)*sinh(d*x + c)^12 - 4*(91*(28672*a^3 + 97920*a^2*b + 116688*a*b^2 - 459459*b^3)*cosh(d*x + c)^
3 - 204*(7168*a^3 + 24480*a^2*b + 29172*a*b^2 - 81081*b^3)*cosh(d*x + c))*sinh(d*x + c)^11 + 17*(229376*a^3 +
783360*a^2*b + 1798368*a*b^2 + 2573571*b^3)*cosh(d*x + c)^10 + (1001*(28672*a^3 + 97920*a^2*b + 116688*a*b^2 +
561561*b^3)*cosh(d*x + c)^4 + 3899392*a^3 + 13317120*a^2*b + 30572256*a*b^2 + 43750707*b^3 - 2244*(14336*a^3
+ 48960*a^2*b + 58344*a*b^2 + 213213*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^10 - 2*(1001*(28672*a^3 + 97920*a^2*b
+ 116688*a*b^2 - 459459*b^3)*cosh(d*x + c)^5 - 7480*(7168*a^3 + 24480*a^2*b + 29172*a*b^2 - 81081*b^3)*cosh(d
*x + c)^3 + 85*(229376*a^3 + 783360*a^2*b + 68640*a*b^2 - 1756755*b^3)*cosh(d*x + c))*sinh(d*x + c)^9 - 68*(28
6720*a^3 + 979200*a^2*b + 3040752*a*b^2 + 2411409*b^3)*cosh(d*x + c)^8 + (3003*(28672*a^3 + 97920*a^2*b + 1166
88*a*b^2 + 561561*b^3)*cosh(d*x + c)^6 - 16830*(14336*a^3 + 48960*a^2*b + 58344*a*b^2 + 213213*b^3)*cosh(d*x +
c)^4 - 19496960*a^3 - 66585600*a^2*b - 206771136*a*b^2 - 163975812*b^3 + 765*(229376*a^3 + 783360*a^2*b + 179
8368*a*b^2 + 2573571*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^8 - 8*(429*(28672*a^3 + 97920*a^2*b + 116688*a*b^2 -
459459*b^3)*cosh(d*x + c)^7 - 6732*(7168*a^3 + 24480*a^2*b + 29172*a*b^2 - 81081*b^3)*cosh(d*x + c)^5 + 255*(2
29376*a^3 + 783360*a^2*b + 68640*a*b^2 - 1756755*b^3)*cosh(d*x + c)^3 - 272*(71680*a^3 + 244800*a^2*b - 176748
*a*b^2 - 351351*b^3)*cosh(d*x + c))*sinh(d*x + c)^7 + 17*(4014080*a^3 + 23592960*a^2*b + 45412224*a*b^2 + 2513
8113*b^3)*cosh(d*x + c)^6 + (3003*(28672*a^3 + 97920*a^2*b + 116688*a*b^2 + 561561*b^3)*cosh(d*x + c)^8 - 3141
6*(14336*a^3 + 48960*a^2*b + 58344*a*b^2 + 213213*b^3)*cosh(d*x + c)^6 + 3570*(229376*a^3 + 783360*a^2*b + 179
8368*a*b^2 + 2573571*b^3)*cosh(d*x + c)^4 + 68239360*a^3 + 401080320*a^2*b + 772007808*a*b^2 + 427347921*b^3 -
1904*(286720*a^3 + 979200*a^2*b + 3040752*a*b^2 + 2411409*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^6 - 2*(1001*(28
672*a^3 + 97920*a^2*b + 116688*a*b^2 - 459459*b^3)*cosh(d*x + c)^9 - 26928*(7168*a^3 + 24480*a^2*b + 29172*a*b
^2 - 81081*b^3)*cosh(d*x + c)^7 + 2142*(229376*a^3 + 783360*a^2*b + 68640*a*b^2 - 1756755*b^3)*cosh(d*x + c)^5
- 7616*(71680*a^3 + 244800*a^2*b - 176748*a*b^2 - 351351*b^3)*cosh(d*x + c)^3 + 51*(4014080*a^3 + 3824640*a^2
*b - 12739584*a*b^2 - 11594583*b^3)*cosh(d*x + c))*sinh(d*x + c)^5 - 442*(401408*a^3 + 3176640*a^2*b + 4162488
*a*b^2 + 1857471*b^3)*cosh(d*x + c)^4 + (1001*(28672*a^3 + 97920*a^2*b + 116688*a*b^2 + 561561*b^3)*cosh(d*x +
c)^10 - 16830*(14336*a^3 + 48960*a^2*b + 58344*a*b^2 + 213213*b^3)*cosh(d*x + c)^8 + 3570*(229376*a^3 + 78336
0*a^2*b + 1798368*a*b^2 + 2573571*b^3)*cosh(d*x + c)^6 - 4760*(286720*a^3 + 979200*a^2*b + 3040752*a*b^2 + 241
1409*b^3)*cosh(d*x + c)^4 - 177422336*a^3 - 1404074880*a^2*b - 1839819696*a*b^2 - 821002182*b^3 + 255*(4014080
*a^3 + 23592960*a^2*b + 45412224*a*b^2 + 25138113*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^4 - 4*(91*(28672*a^3 + 9
7920*a^2*b + 116688*a*b^2 - 459459*b^3)*cosh(d*x + c)^11 - 3740*(7168*a^3 + 24480*a^2*b + 29172*a*b^2 - 81081*
b^3)*cosh(d*x + c)^9 + 510*(229376*a^3 + 783360*a^2*b + 68640*a*b^2 - 1756755*b^3)*cosh(d*x + c)^7 - 3808*(716
80*a^3 + 244800*a^2*b - 176748*a*b^2 - 351351*b^3)*cosh(d*x + c)^5 + 85*(4014080*a^3 + 3824640*a^2*b - 1273958
4*a*b^2 - 11594583*b^3)*cosh(d*x + c)^3 - 884*(200704*a^3 - 217440*a^2*b - 480876*a*b^2 - 297297*b^3)*cosh(d*x
+ c))*sinh(d*x + c)^3 - 557613056*a^3 - 1736317440*a^2*b - 1775057856*a*b^2 - 680611932*b^3 + 221*(4759552*a^
3 + 12643200*a^2*b + 13669392*a*b^2 + 5435199*b^3)*cosh(d*x + c)^2 + (91*(28672*a^3 + 97920*a^2*b + 116688*a*b
^2 + 561561*b^3)*cosh(d*x + c)^12 - 2244*(14336*a^3 + 48960*a^2*b + 58344*a*b^2 + 213213*b^3)*cosh(d*x + c)^10
+ 765*(229376*a^3 + 783360*a^2*b + 1798368*a*b^2 + 2573571*b^3)*cosh(d*x + c)^8 - 1904*(286720*a^3 + 979200*a
^2*b + 3040752*a*b^2 + 2411409*b^3)*cosh(d*x + c)^6 + 255*(4014080*a^3 + 23592960*a^2*b + 45412224*a*b^2 + 251
38113*b^3)*cosh(d*x + c)^4 + 1051860992*a^3 + 2794147200*a^2*b + 3020935632*a*b^2 + 1201178979*b^3 - 2652*(401
408*a^3 + 3176640*a^2*b + 4162488*a*b^2 + 1857471*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^2 - 2*(7*(28672*a^3 + 97
920*a^2*b + 116688*a*b^2 - 459459*b^3)*cosh(d*x + c)^13 - 408*(7168*a^3 + 24480*a^2*b + 29172*a*b^2 - 81081*b^
3)*cosh(d*x + c)^11 + 85*(229376*a^3 + 783360*a^2*b + 68640*a*b^2 - 1756755*b^3)*cosh(d*x + c)^9 - 1088*(71680
*a^3 + 244800*a^2*b - 176748*a*b^2 - 351351*b^3)*cosh(d*x + c)^7 + 51*(4014080*a^3 + 3824640*a^2*b - 12739584*
a*b^2 - 11594583*b^3)*cosh(d*x + c)^5 - 1768*(2...

________________________________________________________________________________________

Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(d*x+c)**18*(a+b*sinh(d*x+c)**4)**3,x)

[Out]

Timed out

________________________________________________________________________________________

Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 679 vs. \(2 (205) = 410\).
time = 0.64, size = 679, normalized size = 3.07 \begin {gather*} -\frac {16 \, {\left (510510 \, b^{3} e^{\left (28 \, d x + 28 \, c\right )} - 6381375 \, b^{3} e^{\left (26 \, d x + 26 \, c\right )} + 14702688 \, a b^{2} e^{\left (24 \, d x + 24 \, c\right )} + 36807771 \, b^{3} e^{\left (24 \, d x + 24 \, c\right )} - 127423296 \, a b^{2} e^{\left (22 \, d x + 22 \, c\right )} - 129771642 \, b^{3} e^{\left (22 \, d x + 22 \, c\right )} + 168030720 \, a^{2} b e^{\left (20 \, d x + 20 \, c\right )} + 494290368 \, a b^{2} e^{\left (20 \, d x + 20 \, c\right )} + 312227916 \, b^{3} e^{\left (20 \, d x + 20 \, c\right )} - 798145920 \, a^{2} b e^{\left (18 \, d x + 18 \, c\right )} - 1132457040 \, a b^{2} e^{\left (18 \, d x + 18 \, c\right )} - 541906365 \, b^{3} e^{\left (18 \, d x + 18 \, c\right )} + 697016320 \, a^{3} e^{\left (16 \, d x + 16 \, c\right )} + 1582289280 \, a^{2} b e^{\left (16 \, d x + 16 \, c\right )} + 1704228240 \, a b^{2} e^{\left (16 \, d x + 16 \, c\right )} + 699143445 \, b^{3} e^{\left (16 \, d x + 16 \, c\right )} - 557613056 \, a^{3} e^{\left (14 \, d x + 14 \, c\right )} - 1736317440 \, a^{2} b e^{\left (14 \, d x + 14 \, c\right )} - 1775057856 \, a b^{2} e^{\left (14 \, d x + 14 \, c\right )} - 680611932 \, b^{3} e^{\left (14 \, d x + 14 \, c\right )} + 354844672 \, a^{3} e^{\left (12 \, d x + 12 \, c\right )} + 1211857920 \, a^{2} b e^{\left (12 \, d x + 12 \, c\right )} + 1316707392 \, a b^{2} e^{\left (12 \, d x + 12 \, c\right )} + 502035534 \, b^{3} e^{\left (12 \, d x + 12 \, c\right )} - 177422336 \, a^{3} e^{\left (10 \, d x + 10 \, c\right )} - 605928960 \, a^{2} b e^{\left (10 \, d x + 10 \, c\right )} - 707362656 \, a b^{2} e^{\left (10 \, d x + 10 \, c\right )} - 279095817 \, b^{3} e^{\left (10 \, d x + 10 \, c\right )} + 68239360 \, a^{3} e^{\left (8 \, d x + 8 \, c\right )} + 233049600 \, a^{2} b e^{\left (8 \, d x + 8 \, c\right )} + 277717440 \, a b^{2} e^{\left (8 \, d x + 8 \, c\right )} + 115120005 \, b^{3} e^{\left (8 \, d x + 8 \, c\right )} - 19496960 \, a^{3} e^{\left (6 \, d x + 6 \, c\right )} - 66585600 \, a^{2} b e^{\left (6 \, d x + 6 \, c\right )} - 79347840 \, a b^{2} e^{\left (6 \, d x + 6 \, c\right )} - 34204170 \, b^{3} e^{\left (6 \, d x + 6 \, c\right )} + 3899392 \, a^{3} e^{\left (4 \, d x + 4 \, c\right )} + 13317120 \, a^{2} b e^{\left (4 \, d x + 4 \, c\right )} + 15869568 \, a b^{2} e^{\left (4 \, d x + 4 \, c\right )} + 6942936 \, b^{3} e^{\left (4 \, d x + 4 \, c\right )} - 487424 \, a^{3} e^{\left (2 \, d x + 2 \, c\right )} - 1664640 \, a^{2} b e^{\left (2 \, d x + 2 \, c\right )} - 1983696 \, a b^{2} e^{\left (2 \, d x + 2 \, c\right )} - 867867 \, b^{3} e^{\left (2 \, d x + 2 \, c\right )} + 28672 \, a^{3} + 97920 \, a^{2} b + 116688 \, a b^{2} + 51051 \, b^{3}\right )}}{765765 \, d {\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )}^{17}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(d*x+c)^18*(a+b*sinh(d*x+c)^4)^3,x, algorithm="giac")

[Out]

-16/765765*(510510*b^3*e^(28*d*x + 28*c) - 6381375*b^3*e^(26*d*x + 26*c) + 14702688*a*b^2*e^(24*d*x + 24*c) +
36807771*b^3*e^(24*d*x + 24*c) - 127423296*a*b^2*e^(22*d*x + 22*c) - 129771642*b^3*e^(22*d*x + 22*c) + 1680307
20*a^2*b*e^(20*d*x + 20*c) + 494290368*a*b^2*e^(20*d*x + 20*c) + 312227916*b^3*e^(20*d*x + 20*c) - 798145920*a
^2*b*e^(18*d*x + 18*c) - 1132457040*a*b^2*e^(18*d*x + 18*c) - 541906365*b^3*e^(18*d*x + 18*c) + 697016320*a^3*
e^(16*d*x + 16*c) + 1582289280*a^2*b*e^(16*d*x + 16*c) + 1704228240*a*b^2*e^(16*d*x + 16*c) + 699143445*b^3*e^
(16*d*x + 16*c) - 557613056*a^3*e^(14*d*x + 14*c) - 1736317440*a^2*b*e^(14*d*x + 14*c) - 1775057856*a*b^2*e^(1
4*d*x + 14*c) - 680611932*b^3*e^(14*d*x + 14*c) + 354844672*a^3*e^(12*d*x + 12*c) + 1211857920*a^2*b*e^(12*d*x
+ 12*c) + 1316707392*a*b^2*e^(12*d*x + 12*c) + 502035534*b^3*e^(12*d*x + 12*c) - 177422336*a^3*e^(10*d*x + 10
*c) - 605928960*a^2*b*e^(10*d*x + 10*c) - 707362656*a*b^2*e^(10*d*x + 10*c) - 279095817*b^3*e^(10*d*x + 10*c)
+ 68239360*a^3*e^(8*d*x + 8*c) + 233049600*a^2*b*e^(8*d*x + 8*c) + 277717440*a*b^2*e^(8*d*x + 8*c) + 115120005
*b^3*e^(8*d*x + 8*c) - 19496960*a^3*e^(6*d*x + 6*c) - 66585600*a^2*b*e^(6*d*x + 6*c) - 79347840*a*b^2*e^(6*d*x
+ 6*c) - 34204170*b^3*e^(6*d*x + 6*c) + 3899392*a^3*e^(4*d*x + 4*c) + 13317120*a^2*b*e^(4*d*x + 4*c) + 158695
68*a*b^2*e^(4*d*x + 4*c) + 6942936*b^3*e^(4*d*x + 4*c) - 487424*a^3*e^(2*d*x + 2*c) - 1664640*a^2*b*e^(2*d*x +
2*c) - 1983696*a*b^2*e^(2*d*x + 2*c) - 867867*b^3*e^(2*d*x + 2*c) + 28672*a^3 + 97920*a^2*b + 116688*a*b^2 +
51051*b^3)/(d*(e^(2*d*x + 2*c) - 1)^17)

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Mupad [B]
time = 1.31, size = 2500, normalized size = 11.31 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*sinh(c + d*x)^4)^3/sinh(c + d*x)^18,x)

[Out]

((24*b^3)/(595*d) - (64*exp(10*c + 10*d*x)*(840*a*b^2 + 1152*a^2*b + 1024*a^3 + 231*b^3))/(85*d) + (6864*b^3*e
xp(20*c + 20*d*x))/(595*d) - (104*b^3*exp(22*c + 22*d*x))/(85*d) + (48*b*exp(8*c + 8*d*x)*(112*a*b + 128*a^2 +
33*b^2))/(17*d) + (576*b*exp(12*c + 12*d*x)*(112*a*b + 128*a^2 + 33*b^2))/(85*d) - (24*b*exp(6*c + 6*d*x)*(44
8*a*b + 256*a^2 + 165*b^2))/(119*d) - (144*b*exp(14*c + 14*d*x)*(448*a*b + 256*a^2 + 165*b^2))/(119*d) - (48*b
^2*exp(2*c + 2*d*x)*(8*a + 11*b))/(595*d) - (528*b^2*exp(18*c + 18*d*x)*(8*a + 11*b))/(119*d) + (16*b^2*exp(4*
c + 4*d*x)*(96*a + 55*b))/(119*d) + (264*b^2*exp(16*c + 16*d*x)*(96*a + 55*b))/(119*d))/(91*exp(4*c + 4*d*x) -
14*exp(2*c + 2*d*x) - 364*exp(6*c + 6*d*x) + 1001*exp(8*c + 8*d*x) - 2002*exp(10*c + 10*d*x) + 3003*exp(12*c
+ 12*d*x) - 3432*exp(14*c + 14*d*x) + 3003*exp(16*c + 16*d*x) - 2002*exp(18*c + 18*d*x) + 1001*exp(20*c + 20*d
*x) - 364*exp(22*c + 22*d*x) + 91*exp(24*c + 24*d*x) - 14*exp(26*c + 26*d*x) + exp(28*c + 28*d*x) + 1) + ((24*
b^3)/(595*d) - (4*b^3*exp(2*c + 2*d*x))/(85*d))/(6*exp(4*c + 4*d*x) - 4*exp(2*c + 2*d*x) - 4*exp(6*c + 6*d*x)
+ exp(8*c + 8*d*x) + 1) - ((64*exp(8*c + 8*d*x)*(840*a*b^2 + 1152*a^2*b + 1024*a^3 + 231*b^3))/(221*d) - (1056
*b^3*exp(18*c + 18*d*x))/(119*d) + (88*b^3*exp(20*c + 20*d*x))/(85*d) + (48*b^2*(8*a + 11*b))/(7735*d) - (192*
b*exp(6*c + 6*d*x)*(112*a*b + 128*a^2 + 33*b^2))/(221*d) - (3456*b*exp(10*c + 10*d*x)*(112*a*b + 128*a^2 + 33*
b^2))/(1105*d) + (72*b*exp(4*c + 4*d*x)*(448*a*b + 256*a^2 + 165*b^2))/(1547*d) + (144*b*exp(12*c + 12*d*x)*(4
48*a*b + 256*a^2 + 165*b^2))/(221*d) + (4752*b^2*exp(16*c + 16*d*x)*(8*a + 11*b))/(1547*d) - (32*b^2*exp(2*c +
2*d*x)*(96*a + 55*b))/(1547*d) - (2112*b^2*exp(14*c + 14*d*x)*(96*a + 55*b))/(1547*d))/(13*exp(2*c + 2*d*x) -
78*exp(4*c + 4*d*x) + 286*exp(6*c + 6*d*x) - 715*exp(8*c + 8*d*x) + 1287*exp(10*c + 10*d*x) - 1716*exp(12*c +
12*d*x) + 1716*exp(14*c + 14*d*x) - 1287*exp(16*c + 16*d*x) + 715*exp(18*c + 18*d*x) - 286*exp(20*c + 20*d*x)
+ 78*exp(22*c + 22*d*x) - 13*exp(24*c + 24*d*x) + exp(26*c + 26*d*x) - 1) + ((48*b^3*exp(4*c + 4*d*x))/(119*d
) - (8*b^3*exp(6*c + 6*d*x))/(51*d) + (8*b^2*(96*a + 55*b))/(4641*d) - (48*b^2*exp(2*c + 2*d*x)*(8*a + 11*b))/
(1547*d))/(15*exp(4*c + 4*d*x) - 6*exp(2*c + 2*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) - ((64*(840*a*b^2 + 1152*a^2*b + 1024*a^3 + 231*b^3))/(109395*d) - (192*b^3
*exp(10*c + 10*d*x))/(85*d) + (112*b^3*exp(12*c + 12*d*x))/(255*d) - (384*b*exp(2*c + 2*d*x)*(112*a*b + 128*a^
2 + 33*b^2))/(12155*d) + (48*b*exp(4*c + 4*d*x)*(448*a*b + 256*a^2 + 165*b^2))/(2431*d) + (96*b^2*exp(8*c + 8*
d*x)*(8*a + 11*b))/(221*d) - (64*b^2*exp(6*c + 6*d*x)*(96*a + 55*b))/(663*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*e
xp(14*c + 14*d*x) - 9*exp(16*c + 16*d*x) + exp(18*c + 18*d*x) - 1) - ((64*exp(6*c + 6*d*x)*(840*a*b^2 + 1152*a
^2*b + 1024*a^3 + 231*b^3))/(663*d) - (792*b^3*exp(16*c + 16*d*x))/(119*d) + (44*b^3*exp(18*c + 18*d*x))/(51*d
) - (8*b^2*(96*a + 55*b))/(4641*d) - (48*b*exp(4*c + 4*d*x)*(112*a*b + 128*a^2 + 33*b^2))/(221*d) - (288*b*exp
(8*c + 8*d*x)*(112*a*b + 128*a^2 + 33*b^2))/(221*d) + (12*b*exp(2*c + 2*d*x)*(448*a*b + 256*a^2 + 165*b^2))/(1
547*d) + (72*b*exp(10*c + 10*d*x)*(448*a*b + 256*a^2 + 165*b^2))/(221*d) + (3168*b^2*exp(14*c + 14*d*x)*(8*a +
11*b))/(1547*d) - (176*b^2*exp(12*c + 12*d*x)*(96*a + 55*b))/(221*d))/(66*exp(4*c + 4*d*x) - 12*exp(2*c + 2*d
*x) - 220*exp(6*c + 6*d*x) + 495*exp(8*c + 8*d*x) - 792*exp(10*c + 10*d*x) + 924*exp(12*c + 12*d*x) - 792*exp(
14*c + 14*d*x) + 495*exp(16*c + 16*d*x) - 220*exp(18*c + 18*d*x) + 66*exp(20*c + 20*d*x) - 12*exp(22*c + 22*d*
x) + exp(24*c + 24*d*x) + 1) - ((64*exp(14*c + 14*d*x)*(840*a*b^2 + 1152*a^2*b + 1024*a^3 + 231*b^3))/(17*d) +
(4*b^3*exp(2*c + 2*d*x))/(17*d) - (72*b^3*exp(4*c + 4*d*x))/(17*d) - (312*b^3*exp(24*c + 24*d*x))/(17*d) + (2
8*b^3*exp(26*c + 26*d*x))/(17*d) - (336*b*exp(12*c + 12*d*x)*(112*a*b + 128*a^2 + 33*b^2))/(17*d) - (432*b*exp
(16*c + 16*d*x)*(112*a*b + 128*a^2 + 33*b^2))/(17*d) + (36*b*exp(10*c + 10*d*x)*(448*a*b + 256*a^2 + 165*b^2))
/(17*d) + (60*b*exp(18*c + 18*d*x)*(448*a*b + 256*a^2 + 165*b^2))/(17*d) + (48*b^2*exp(6*c + 6*d*x)*(8*a + 11*
b))/(17*d) + (144*b^2*exp(22*c + 22*d*x)*(8*a + 11*b))/(17*d) - (40*b^2*exp(8*c + 8*d*x)*(96*a + 55*b))/(17*d)
- (88*b^2*exp(20*c + 20*d*x)*(96*a + 55*b))/(17*d))/(120*exp(4*c + 4*d*x) - 16*exp(2*c + 2*d*x) - 560*exp(6*c
+ 6*d*x) + 1820*exp(8*c + 8*d*x) - 4368*exp(10*c + 10*d*x) + 8008*exp(12*c + 12*d*x) - 11440*exp(14*c + 14*d*
x) + 12870*exp(16*c + 16*d*x) - 11440*exp(18*c + 18*d*x) + 8008*exp(20*c + 20*d*x) - 4368*exp(22*c + 22*d*x) +
1820*exp(24*c + 24*d*x) - 560*exp(26*c + 26*d*x) + 120*exp(28*c + 28*d*x) - 16*exp(30*c + 30*d*x) + exp(32*c
+ 32*d*x) + 1) - ((12*b*(448*a*b + 256*a^2 + 165*b^2))/(17017*d) - (96*b^3*exp(6*c + 6*d*x))/(119*d) + (4*b^3*
exp(8*c + 8*d*x))/(17*d) + (144*b^2*exp(4*c + 4...

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