3.227 \(\int \text{csch}^{18}(c+d x) (a+b \sinh ^4(c+d x))^3 \, dx\)
Optimal. Leaf size=221 \[ -\frac{a \left (70 a^2+45 a b+3 b^2\right ) \coth ^9(c+d x)}{9 d}+\frac{4 a \left (14 a^2+15 a b+3 b^2\right ) \coth ^7(c+d x)}{7 d}-\frac{(a+b) \left (28 a^2+17 a b+b^2\right ) \coth ^5(c+d x)}{5 d}-\frac{a^2 (28 a+3 b) \coth ^{13}(c+d x)}{13 d}+\frac{2 a^2 (28 a+9 b) \coth ^{11}(c+d x)}{11 d}-\frac{a^3 \coth ^{17}(c+d x)}{17 d}+\frac{8 a^3 \coth ^{15}(c+d x)}{15 d}+\frac{2 (a+b)^2 (4 a+b) \coth ^3(c+d x)}{3 d}-\frac{(a+b)^3 \coth (c+d x)}{d} \]
[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)
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Rubi [A] time = 0.191817, antiderivative size = 221, normalized size of antiderivative = 1.,
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 =
{3217, 1261} \[ -\frac{a \left (70 a^2+45 a b+3 b^2\right ) \coth ^9(c+d x)}{9 d}+\frac{4 a \left (14 a^2+15 a b+3 b^2\right ) \coth ^7(c+d x)}{7 d}-\frac{(a+b) \left (28 a^2+17 a b+b^2\right ) \coth ^5(c+d x)}{5 d}-\frac{a^2 (28 a+3 b) \coth ^{13}(c+d x)}{13 d}+\frac{2 a^2 (28 a+9 b) \coth ^{11}(c+d x)}{11 d}-\frac{a^3 \coth ^{17}(c+d x)}{17 d}+\frac{8 a^3 \coth ^{15}(c+d x)}{15 d}+\frac{2 (a+b)^2 (4 a+b) \coth ^3(c+d x)}{3 d}-\frac{(a+b)^3 \coth (c+d x)}{d} \]
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 3217
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]
Rule 1261
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]
Rubi steps
\begin{align*} \int \text{csch}^{18}(c+d x) \left (a+b \sinh ^4(c+d x)\right )^3 \, dx &=\frac{\operatorname{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{\operatorname{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*}
Mathematica [B] time = 6.18111, size = 494, normalized size = 2.24 \[ \frac{\text{csch}^{17}(c+d x) \left (-784143360 a^2 b \cosh (c+d x)+1568286720 a^2 b \cosh (3 (c+d x))-1211857920 a^2 b \cosh (5 (c+d x))+605928960 a^2 b \cosh (7 (c+d x))-233049600 a^2 b \cosh (9 (c+d x))+66585600 a^2 b \cosh (11 (c+d x))-13317120 a^2 b \cosh (13 (c+d x))+1664640 a^2 b \cosh (15 (c+d x))-97920 a^2 b \cosh (17 (c+d x))-697016320 a^3 \cosh (c+d x)+557613056 a^3 \cosh (3 (c+d x))-354844672 a^3 \cosh (5 (c+d x))+177422336 a^3 \cosh (7 (c+d x))-68239360 a^3 \cosh (9 (c+d x))+19496960 a^3 \cosh (11 (c+d x))-3899392 a^3 \cosh (13 (c+d x))+487424 a^3 \cosh (15 (c+d x))-28672 a^3 \cosh (17 (c+d x))-571771200 a b^2 \cosh (c+d x)+1280767488 a b^2 \cosh (3 (c+d x))-1189284096 a b^2 \cosh (5 (c+d x))+692659968 a b^2 \cosh (7 (c+d x))-277717440 a b^2 \cosh (9 (c+d x))+79347840 a b^2 \cosh (11 (c+d x))-15869568 a b^2 \cosh (13 (c+d x))+1983696 a b^2 \cosh (15 (c+d x))-116688 a b^2 \cosh (17 (c+d x))-157237080 b^3 \cosh (c+d x)+368384016 b^3 \cosh (3 (c+d x))-372263892 b^3 \cosh (5 (c+d x))+242288046 b^3 \cosh (7 (c+d x))-108738630 b^3 \cosh (9 (c+d x))+33693660 b^3 \cosh (11 (c+d x))-6942936 b^3 \cosh (13 (c+d x))+867867 b^3 \cosh (15 (c+d x))-51051 b^3 \cosh (17 (c+d x))\right )}{6273146880 d} \]
Antiderivative was successfully verified.
[In]
Integrate[Csch[c + d*x]^18*(a + b*Sinh[c + d*x]^4)^3,x]
[Out]
((-697016320*a^3*Cosh[c + d*x] - 784143360*a^2*b*Cosh[c + d*x] - 571771200*a*b^2*Cosh[c + d*x] - 157237080*b^3
*Cosh[c + d*x] + 557613056*a^3*Cosh[3*(c + d*x)] + 1568286720*a^2*b*Cosh[3*(c + d*x)] + 1280767488*a*b^2*Cosh[
3*(c + d*x)] + 368384016*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)] - 1087
38630*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*Co
sh[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)/(6273146880*d)
________________________________________________________________________________________
Maple [A] time = 0.085, size = 258, normalized size = 1.2 \begin{align*}{\frac{1}{d} \left ({a}^{3} \left ( -{\frac{32768}{109395}}-{\frac{ \left ({\rm csch} \left (dx+c\right ) \right ) ^{16}}{17}}+{\frac{16\, \left ({\rm csch} \left (dx+c\right ) \right ) ^{14}}{255}}-{\frac{224\, \left ({\rm csch} \left (dx+c\right ) \right ) ^{12}}{3315}}+{\frac{896\, \left ({\rm csch} \left (dx+c\right ) \right ) ^{10}}{12155}}-{\frac{1792\, \left ({\rm csch} \left (dx+c\right ) \right ) ^{8}}{21879}}+{\frac{2048\, \left ({\rm csch} \left (dx+c\right ) \right ) ^{6}}{21879}}-{\frac{4096\, \left ({\rm csch} \left (dx+c\right ) \right ) ^{4}}{36465}}+{\frac{16384\, \left ({\rm csch} \left (dx+c\right ) \right ) ^{2}}{109395}} \right ){\rm coth} \left (dx+c\right )+3\,{a}^{2}b \left ( -{\frac{1024}{3003}}-1/13\, \left ({\rm csch} \left (dx+c\right ) \right ) ^{12}+{\frac{12\, \left ({\rm csch} \left (dx+c\right ) \right ) ^{10}}{143}}-{\frac{40\, \left ({\rm csch} \left (dx+c\right ) \right ) ^{8}}{429}}+{\frac{320\, \left ({\rm csch} \left (dx+c\right ) \right ) ^{6}}{3003}}-{\frac{128\, \left ({\rm csch} \left (dx+c\right ) \right ) ^{4}}{1001}}+{\frac{512\, \left ({\rm csch} \left (dx+c\right ) \right ) ^{2}}{3003}} \right ){\rm coth} \left (dx+c\right )+3\,a{b}^{2} \left ( -{\frac{128}{315}}-1/9\, \left ({\rm csch} \left (dx+c\right ) \right ) ^{8}+{\frac{8\, \left ({\rm csch} \left (dx+c\right ) \right ) ^{6}}{63}}-{\frac{16\, \left ({\rm csch} \left (dx+c\right ) \right ) ^{4}}{105}}+{\frac{64\, \left ({\rm csch} \left (dx+c\right ) \right ) ^{2}}{315}} \right ){\rm coth} \left (dx+c\right )+{b}^{3} \left ( -{\frac{8}{15}}-{\frac{ \left ({\rm csch} \left (dx+c\right ) \right ) ^{4}}{5}}+{\frac{4\, \left ({\rm csch} \left (dx+c\right ) \right ) ^{2}}{15}} \right ){\rm coth} \left (dx+c\right ) \right ) } \end{align*}
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)
[Out]
1/d*(a^3*(-32768/109395-1/17*csch(d*x+c)^16+16/255*csch(d*x+c)^14-224/3315*csch(d*x+c)^12+896/12155*csch(d*x+c
)^10-1792/21879*csch(d*x+c)^8+2048/21879*csch(d*x+c)^6-4096/36465*csch(d*x+c)^4+16384/109395*csch(d*x+c)^2)*co
th(d*x+c)+3*a^2*b*(-1024/3003-1/13*csch(d*x+c)^12+12/143*csch(d*x+c)^10-40/429*csch(d*x+c)^8+320/3003*csch(d*x
+c)^6-128/1001*csch(d*x+c)^4+512/3003*csch(d*x+c)^2)*coth(d*x+c)+3*a*b^2*(-128/315-1/9*csch(d*x+c)^8+8/63*csch
(d*x+c)^6-16/105*csch(d*x+c)^4+64/315*csch(d*x+c)^2)*coth(d*x+c)+b^3*(-8/15-1/5*csch(d*x+c)^4+4/15*csch(d*x+c)
^2)*coth(d*x+c))
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Maxima [B] time = 1.29654, size = 5021, normalized size = 22.72 \begin{align*} \text{result too large to display} \end{align*}
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 - 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)) - 715*e^(-8*d*x - 8*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)) + 1287*e^(-10*d*x - 10*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)) - 1716*e^(-12*d*x - 12*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)) - 1/(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 - 2
2*c) - 13*e^(-24*d*x - 24*c) + e^(-26*d*x - 26*c) - 1))) - 256/105*a*b^2*(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)) + 84*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)) - 126*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)) - 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))) - 16/15*b^3*(5*e^(-2*d*x - 2*c)/(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)) - 10*e^(-4*d*x - 4*c)/(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)) - 1/(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)))
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Fricas [B] time = 1.90939, size = 11491, normalized size = 52. \begin{align*} \text{result too large to display} \end{align*}
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*(200704*a^3 - 217440*a^2*b - 480876*a*b^2 - 297297*b^3)*cosh(d*x
+ c)^3 - 5967*(57344*a^3 + 62080*a^2*b + 64944*a*b^2 + 33033*b^3)*cosh(d*x + c))*sinh(d*x + c))/(d*cosh(d*x +
c)^20 + 20*d*cosh(d*x + c)*sinh(d*x + c)^19 + d*sinh(d*x + c)^20 - 17*d*cosh(d*x + c)^18 + (190*d*cosh(d*x + c
)^2 - 17*d)*sinh(d*x + c)^18 + 6*(190*d*cosh(d*x + c)^3 - 51*d*cosh(d*x + c))*sinh(d*x + c)^17 + 136*d*cosh(d*
x + c)^16 + 17*(285*d*cosh(d*x + c)^4 - 153*d*cosh(d*x + c)^2 + 8*d)*sinh(d*x + c)^16 + 272*(57*d*cosh(d*x + c
)^5 - 51*d*cosh(d*x + c)^3 + 8*d*cosh(d*x + c))*sinh(d*x + c)^15 - 681*d*cosh(d*x + c)^14 + 3*(12920*d*cosh(d*
x + c)^6 - 17340*d*cosh(d*x + c)^4 + 5440*d*cosh(d*x + c)^2 - 227*d)*sinh(d*x + c)^14 + 2*(38760*d*cosh(d*x +
c)^7 - 72828*d*cosh(d*x + c)^5 + 38080*d*cosh(d*x + c)^3 - 4753*d*cosh(d*x + c))*sinh(d*x + c)^13 + 2397*d*cos
h(d*x + c)^12 + (125970*d*cosh(d*x + c)^8 - 315588*d*cosh(d*x + c)^6 + 247520*d*cosh(d*x + c)^4 - 61971*d*cosh
(d*x + c)^2 + 2397*d)*sinh(d*x + c)^12 + 4*(41990*d*cosh(d*x + c)^9 - 135252*d*cosh(d*x + c)^7 + 148512*d*cosh
(d*x + c)^5 - 61789*d*cosh(d*x + c)^3 + 7089*d*cosh(d*x + c))*sinh(d*x + c)^11 - 6324*d*cosh(d*x + c)^10 + (18
4756*d*cosh(d*x + c)^10 - 743886*d*cosh(d*x + c)^8 + 1089088*d*cosh(d*x + c)^6 - 681681*d*cosh(d*x + c)^4 + 15
8202*d*cosh(d*x + c)^2 - 6324*d)*sinh(d*x + c)^10 + 2*(83980*d*cosh(d*x + c)^11 - 413270*d*cosh(d*x + c)^9 + 7
77920*d*cosh(d*x + c)^7 - 679679*d*cosh(d*x + c)^5 + 259930*d*cosh(d*x + c)^3 - 30260*d*cosh(d*x + c))*sinh(d*
x + c)^9 + 13056*d*cosh(d*x + c)^8 + 3*(41990*d*cosh(d*x + c)^12 - 247962*d*cosh(d*x + c)^10 + 583440*d*cosh(d
*x + c)^8 - 681681*d*cosh(d*x + c)^6 + 395505*d*cosh(d*x + c)^4 - 94860*d*cosh(d*x + c)^2 + 4352*d)*sinh(d*x +
c)^8 + 8*(9690*d*cosh(d*x + c)^13 - 67626*d*cosh(d*x + c)^11 + 194480*d*cosh(d*x + c)^9 - 291291*d*cosh(d*x +
c)^7 + 233937*d*cosh(d*x + c)^5 - 90780*d*cosh(d*x + c)^3 + 11696*d*cosh(d*x + c))*sinh(d*x + c)^7 - 21828*d*
cosh(d*x + c)^6 + (38760*d*cosh(d*x + c)^14 - 315588*d*cosh(d*x + c)^12 + 1089088*d*cosh(d*x + c)^10 - 2045043
*d*cosh(d*x + c)^8 + 2214828*d*cosh(d*x + c)^6 - 1328040*d*cosh(d*x + c)^4 + 365568*d*cosh(d*x + c)^2 - 21828*
d)*sinh(d*x + c)^6 + 2*(7752*d*cosh(d*x + c)^15 - 72828*d*cosh(d*x + c)^13 + 297024*d*cosh(d*x + c)^11 - 67967
9*d*cosh(d*x + c)^9 + 935748*d*cosh(d*x + c)^7 - 762552*d*cosh(d*x + c)^5 + 327488*d*cosh(d*x + c)^3 - 51204*d
*cosh(d*x + c))*sinh(d*x + c)^5 + 30498*d*cosh(d*x + c)^4 + (4845*d*cosh(d*x + c)^16 - 52020*d*cosh(d*x + c)^1
4 + 247520*d*cosh(d*x + c)^12 - 681681*d*cosh(d*x + c)^10 + 1186515*d*cosh(d*x + c)^8 - 1328040*d*cosh(d*x + c
)^6 + 913920*d*cosh(d*x + c)^4 - 327420*d*cosh(d*x + c)^2 + 30498*d)*sinh(d*x + c)^4 + 4*(285*d*cosh(d*x + c)^
17 - 3468*d*cosh(d*x + c)^15 + 19040*d*cosh(d*x + c)^13 - 61789*d*cosh(d*x + c)^11 + 129965*d*cosh(d*x + c)^9
- 181560*d*cosh(d*x + c)^7 + 163744*d*cosh(d*x + c)^5 - 85340*d*cosh(d*x + c)^3 + 18122*d*cosh(d*x + c))*sinh(
d*x + c)^3 - 36686*d*cosh(d*x + c)^2 + (190*d*cosh(d*x + c)^18 - 2601*d*cosh(d*x + c)^16 + 16320*d*cosh(d*x +
c)^14 - 61971*d*cosh(d*x + c)^12 + 158202*d*cosh(d*x + c)^10 - 284580*d*cosh(d*x + c)^8 + 365568*d*cosh(d*x +
c)^6 - 327420*d*cosh(d*x + c)^4 + 182988*d*cosh(d*x + c)^2 - 36686*d)*sinh(d*x + c)^2 + 2*(10*d*cosh(d*x + c)^
19 - 153*d*cosh(d*x + c)^17 + 1088*d*cosh(d*x + c)^15 - 4753*d*cosh(d*x + c)^13 + 14178*d*cosh(d*x + c)^11 - 3
0260*d*cosh(d*x + c)^9 + 46784*d*cosh(d*x + c)^7 - 51204*d*cosh(d*x + c)^5 + 36244*d*cosh(d*x + c)^3 - 11934*d
*cosh(d*x + c))*sinh(d*x + c) + 19448*d)
________________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
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] time = 1.6542, size = 917, normalized size = 4.15 \begin{align*} \text{result too large to display} \end{align*}
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)