3.3.28 \(\int \text {csch}^{20}(c+d x) (a+b \sinh ^4(c+d x))^3 \, dx\) [228]

Optimal. Leaf size=248 \[ \frac {(a+b)^3 \coth (c+d x)}{d}-\frac {(a+b)^2 (3 a+b) \coth ^3(c+d x)}{d}+\frac {3 (a+b) \left (12 a^2+9 a b+b^2\right ) \coth ^5(c+d x)}{5 d}-\frac {\left (84 a^3+105 a^2 b+30 a b^2+b^3\right ) \coth ^7(c+d x)}{7 d}+\frac {a \left (42 a^2+35 a b+5 b^2\right ) \coth ^9(c+d x)}{3 d}-\frac {3 a \left (42 a^2+21 a b+b^2\right ) \coth ^{11}(c+d x)}{11 d}+\frac {21 a^2 (4 a+b) \coth ^{13}(c+d x)}{13 d}-\frac {a^2 (12 a+b) \coth ^{15}(c+d x)}{5 d}+\frac {9 a^3 \coth ^{17}(c+d x)}{17 d}-\frac {a^3 \coth ^{19}(c+d x)}{19 d} \]

[Out]

(a+b)^3*coth(d*x+c)/d-(a+b)^2*(3*a+b)*coth(d*x+c)^3/d+3/5*(a+b)*(12*a^2+9*a*b+b^2)*coth(d*x+c)^5/d-1/7*(84*a^3
+105*a^2*b+30*a*b^2+b^3)*coth(d*x+c)^7/d+1/3*a*(42*a^2+35*a*b+5*b^2)*coth(d*x+c)^9/d-3/11*a*(42*a^2+21*a*b+b^2
)*coth(d*x+c)^11/d+21/13*a^2*(4*a+b)*coth(d*x+c)^13/d-1/5*a^2*(12*a+b)*coth(d*x+c)^15/d+9/17*a^3*coth(d*x+c)^1
7/d-1/19*a^3*coth(d*x+c)^19/d

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Rubi [A]
time = 0.16, antiderivative size = 248, 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 ^{19}(c+d x)}{19 d}+\frac {9 a^3 \coth ^{17}(c+d x)}{17 d}-\frac {3 a \left (42 a^2+21 a b+b^2\right ) \coth ^{11}(c+d x)}{11 d}+\frac {a \left (42 a^2+35 a b+5 b^2\right ) \coth ^9(c+d x)}{3 d}+\frac {3 (a+b) \left (12 a^2+9 a b+b^2\right ) \coth ^5(c+d x)}{5 d}-\frac {a^2 (12 a+b) \coth ^{15}(c+d x)}{5 d}+\frac {21 a^2 (4 a+b) \coth ^{13}(c+d x)}{13 d}-\frac {\left (84 a^3+105 a^2 b+30 a b^2+b^3\right ) \coth ^7(c+d x)}{7 d}-\frac {(a+b)^2 (3 a+b) \coth ^3(c+d x)}{d}+\frac {(a+b)^3 \coth (c+d x)}{d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Csch[c + d*x]^20*(a + b*Sinh[c + d*x]^4)^3,x]

[Out]

((a + b)^3*Coth[c + d*x])/d - ((a + b)^2*(3*a + b)*Coth[c + d*x]^3)/d + (3*(a + b)*(12*a^2 + 9*a*b + b^2)*Coth
[c + d*x]^5)/(5*d) - ((84*a^3 + 105*a^2*b + 30*a*b^2 + b^3)*Coth[c + d*x]^7)/(7*d) + (a*(42*a^2 + 35*a*b + 5*b
^2)*Coth[c + d*x]^9)/(3*d) - (3*a*(42*a^2 + 21*a*b + b^2)*Coth[c + d*x]^11)/(11*d) + (21*a^2*(4*a + b)*Coth[c
+ d*x]^13)/(13*d) - (a^2*(12*a + b)*Coth[c + d*x]^15)/(5*d) + (9*a^3*Coth[c + d*x]^17)/(17*d) - (a^3*Coth[c +
d*x]^19)/(19*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}^{20}(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 )^3 \left (a-2 a x^2+(a+b) x^4\right )^3}{x^{20}} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac {\text {Subst}\left (\int \left (\frac {a^3}{x^{20}}-\frac {9 a^3}{x^{18}}+\frac {3 a^2 (12 a+b)}{x^{16}}-\frac {21 a^2 (4 a+b)}{x^{14}}+\frac {3 a \left (42 a^2+21 a b+b^2\right )}{x^{12}}-\frac {3 a \left (42 a^2+35 a b+5 b^2\right )}{x^{10}}+\frac {84 a^3+105 a^2 b+30 a b^2+b^3}{x^8}+\frac {3 (a+b) \left (-12 a^2-9 a b-b^2\right )}{x^6}+\frac {3 (a+b)^2 (3 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 {(a+b)^2 (3 a+b) \coth ^3(c+d x)}{d}+\frac {3 (a+b) \left (12 a^2+9 a b+b^2\right ) \coth ^5(c+d x)}{5 d}-\frac {\left (84 a^3+105 a^2 b+30 a b^2+b^3\right ) \coth ^7(c+d x)}{7 d}+\frac {a \left (42 a^2+35 a b+5 b^2\right ) \coth ^9(c+d x)}{3 d}-\frac {3 a \left (42 a^2+21 a b+b^2\right ) \coth ^{11}(c+d x)}{11 d}+\frac {21 a^2 (4 a+b) \coth ^{13}(c+d x)}{13 d}-\frac {a^2 (12 a+b) \coth ^{15}(c+d x)}{5 d}+\frac {9 a^3 \coth ^{17}(c+d x)}{17 d}-\frac {a^3 \coth ^{19}(c+d x)}{19 d}\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(512\) vs. \(2(248)=496\).
time = 5.99, size = 512, normalized size = 2.06 \begin {gather*} -\frac {\left (7759752 \left (1024 a^3+1152 a^2 b+840 a b^2+231 b^3\right ) \cosh (c+d x)-2116296 \left (3072 a^3+8576 a^2 b+7000 a b^2+2013 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))-2333786112 a^3 \cosh (7 (c+d x))-7852217856 a^2 b \cosh (7 (c+d x))-8803791360 a b^2 \cosh (7 (c+d x))-3047642598 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))-333398016 a^3 \cosh (11 (c+d x))-1121745408 a^2 b \cosh (11 (c+d x))-1302025920 a b^2 \cosh (11 (c+d x))-527386002 b^3 \cosh (11 (c+d x))[phone] 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))[phone] b^3 \cosh (13 (c+d x))-14708736 a^3 \cosh (15 (c+d x))-49488768 a^2 b \cosh (15 (c+d x))-57442320 a b^2 \cosh (15 (c+d x))-23694957 b^3 \cosh (15 (c+d x))+1634304 a^3 \cosh (17 (c+d x))+5498752 a^2 b \cosh (17 (c+d x))+6382480 a b^2 \cosh (17 (c+d x))+2632773 b^3 \cosh (17 (c+d x))-86016 a^3 \cosh (19 (c+d x))-289408 a^2 b \cosh (19 (c+d x))-335920 a b^2 \cosh (19 (c+d x))-138567 b^3 \cosh (19 (c+d x))\right ) \text {csch}^{19}(c+d x)}{79459860480 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Csch[c + d*x]^20*(a + b*Sinh[c + d*x]^4)^3,x]

[Out]

-1/79459860480*((7759752*(1024*a^3 + 1152*a^2*b + 840*a*b^2 + 231*b^3)*Cosh[c + d*x] - 2116296*(3072*a^3 + 857
6*a^2*b + 7000*a*b^2 + 2013*b^3)*Cosh[3*(c + d*x)] + 4334174208*a^3*Cosh[5*(c + d*x)] + 14582690304*a^2*b*Cosh
[5*(c + d*x)] + 14221509120*a*b^2*Cosh[5*(c + d*x)] + 4440518082*b^3*Cosh[5*(c + d*x)] - 2333786112*a^3*Cosh[7
*(c + d*x)] - 7852217856*a^2*b*Cosh[7*(c + d*x)] - 8803791360*a*b^2*Cosh[7*(c + d*x)] - 3047642598*b^3*Cosh[7*
(c + d*x)] + 1000194048*a^3*Cosh[9*(c + d*x)] + 3365236224*a^2*b*Cosh[9*(c + d*x)] + 3906077760*a*b^2*Cosh[9*(
c + d*x)] + 1489040982*b^3*Cosh[9*(c + d*x)] - 333398016*a^3*Cosh[11*(c + d*x)] - 1121745408*a^2*b*Cosh[11*(c
+ d*x)] - 1302025920*a*b^2*Cosh[11*(c + d*x)] - 527386002*b^3*Cosh[11*(c + d*x)] + 83349504*a^3*Cosh[13*(c + d
*x)] + 280436352*a^2*b*Cosh[13*(c + d*x)] + 325506480*a*b^2*Cosh[13*(c + d*x)] + 134271423*b^3*Cosh[13*(c + d*
x)] - 14708736*a^3*Cosh[15*(c + d*x)] - 49488768*a^2*b*Cosh[15*(c + d*x)] - 57442320*a*b^2*Cosh[15*(c + d*x)]
- 23694957*b^3*Cosh[15*(c + d*x)] + 1634304*a^3*Cosh[17*(c + d*x)] + 5498752*a^2*b*Cosh[17*(c + d*x)] + 638248
0*a*b^2*Cosh[17*(c + d*x)] + 2632773*b^3*Cosh[17*(c + d*x)] - 86016*a^3*Cosh[19*(c + d*x)] - 289408*a^2*b*Cosh
[19*(c + d*x)] - 335920*a*b^2*Cosh[19*(c + d*x)] - 138567*b^3*Cosh[19*(c + d*x)])*Csch[c + d*x]^19)/d

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(737\) vs. \(2(232)=464\).
time = 1.68, size = 738, normalized size = 2.98

method	result	size
risch	\(-\frac {32 \left (-6501261312 a^{3} {\mathrm e}^{16 d x +16 c}-335920 a \,b^{2}[phone] a \,b^{2} {\mathrm e}^{14 d x +14 c}-7852217856 a^{2} b \,{\mathrm e}^{12 d x +12 c}-8958986400 a \,b^{2} {\mathrm e}^{12 d x +12 c}[phone] a^{2} b \,{\mathrm e}^{10 d x +10 c}[phone] a \,b^{2} {\mathrm e}^{10 d x +10 c}[phone] a \,b^{2} {\mathrm e}^{18 d x +18 c}-20011694976 a^{2} b \,{\mathrm e}^{16 d x +16 c}-20101788720 a \,b^{2} {\mathrm e}^{16 d x +16 c}[phone] a^{2} b \,{\mathrm e}^{14 d x +14 c}-49488768 a^{2} b \,{\mathrm e}^{4 d x +4 c}+5498752 a^{2} b \,{\mathrm e}^{2 d x +2 c}-289408 a^{2} b -86016 a^{3}[phone] a^{2} b \,{\mathrm e}^{22 d x +22 c}-8897848960 a^{2} b \,{\mathrm e}^{20 d x +20 c}-138567 b^{3}-1302025920 a \,b^{2} {\mathrm e}^{8 d x +8 c}[phone] a \,b^{2} {\mathrm e}^{6 d x +6 c}-57442320 a \,b^{2} {\mathrm e}^{4 d x +4 c}-12256713040 a \,b^{2} {\mathrm e}^{20 d x +20 c}-1121745408 a^{2} b \,{\mathrm e}^{8 d x +8 c}+6382480 a \,b^{2} {\mathrm e}^{2 d x +2 c}[phone] a^{2} b \,{\mathrm e}^{6 d x +6 c}[phone] a \,b^{2} {\mathrm e}^{26 d x +26 c}-1270797957 b^{3} {\mathrm e}^{24 d x +24 c}[phone] a \,b^{2} {\mathrm e}^{22 d x +22 c}[phone] a^{2} b \,{\mathrm e}^{18 d x +18 c}[phone] b^{3} {\mathrm e}^{18 d x +18 c}-7366637421 b^{3} {\mathrm e}^{16 d x +16 c}[phone] b^{3} {\mathrm e}^{14 d x +14 c}+1634304 a^{3} {\mathrm e}^{2 d x +2 c}-2333786112 a^{3} {\mathrm e}^{12 d x +12 c}-3403621221 b^{3} {\mathrm e}^{12 d x +12 c}[phone] a^{3} {\mathrm e}^{10 d x +10 c}[phone] b^{3} {\mathrm e}^{22 d x +22 c}-5504019807 b^{3} {\mathrm e}^{20 d x +20 c}-1352413920 a \,b^{2} {\mathrm e}^{24 d x +24 c}[phone] b^{3} {\mathrm e}^{26 d x +26 c}[phone] a^{3} {\mathrm e}^{14 d x +14 c}+2632773 b^{3} {\mathrm e}^{2 d x +2 c}-532235847 b^{3} {\mathrm e}^{8 d x +8 c}-14708736 a^{3} {\mathrm e}^{4 d x +4 c}-23694957 b^{3} {\mathrm e}^{4 d x +4 c}+4849845 b^{3} {\mathrm e}^{30 d x +30 c}-61108047 b^{3} {\mathrm e}^{28 d x +28 c}[phone] a^{3} {\mathrm e}^{18 d x +18 c}[phone] b^{3} {\mathrm e}^{10 d x +10 c}-333398016 a^{3} {\mathrm e}^{8 d x +8 c}[phone] a^{3} {\mathrm e}^{6 d x +6 c}[phone] b^{3} {\mathrm e}^{6 d x +6 c}\right )}{4849845 d \left ({\mathrm e}^{2 d x +2 c}-1\right )^{19}}\)	\(738\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csch(d*x+c)^20*(a+b*sinh(d*x+c)^4)^3,x,method=_RETURNVERBOSE)

[Out]

-32/4849845*(-6501261312*a^3*exp(16*d*x+16*c)-335920*a*b^2+15573923040*a*b^2*exp(14*d*x+14*c)-7852217856*a^2*b
*exp(12*d*x+12*c)-8958986400*a*b^2*exp(12*d*x+12*c)[phone]*a^2*b*exp(10*d*x+10*c)[phone]*a*b^2*exp(10*
d*x+10*c)[phone]*a*b^2*exp(18*d*x+18*c)-20011694976*a^2*b*exp(16*d*x+16*c)-20101788720*a*b^2*exp(16*d*x+1
6*c)[phone]*a^2*b*exp(14*d*x+14*c)-49488768*a^2*b*exp(4*d*x+4*c)+5498752*a^2*b*exp(2*d*x+2*c)-289408*a^2*
b-86016*a^3+1862340480*a^2*b*exp(22*d*x+22*c)-8897848960*a^2*b*exp(20*d*x+20*c)-138567*b^3-1302025920*a*b^2*ex
p(8*d*x+8*c)[phone]*a*b^2*exp(6*d*x+6*c)-57442320*a*b^2*exp(4*d*x+4*c)-12256713040*a*b^2*exp(20*d*x+20*c)-1
121745408*a^2*b*exp(8*d*x+8*c)+6382480*a*b^2*exp(2*d*x+2*c)[phone]*a^2*b*exp(6*d*x+6*c)[phone]*a*b^2*exp
(26*d*x+26*c)-1270797957*b^3*exp(24*d*x+24*c)[phone]*a*b^2*exp(22*d*x+22*c)[phone]*a^2*b*exp(18*d*x+1
8*c)[phone]*b^3*exp(18*d*x+18*c)-7366637421*b^3*exp(16*d*x+16*c)[phone]*b^3*exp(14*d*x+14*c)+1634304*a
^3*exp(2*d*x+2*c)-2333786112*a^3*exp(12*d*x+12*c)-3403621221*b^3*exp(12*d*x+12*c)[phone]*a^3*exp(10*d*x+10
*c)[phone]*b^3*exp(22*d*x+22*c)-5504019807*b^3*exp(20*d*x+20*c)-1352413920*a*b^2*exp(24*d*x+24*c)[phone]
3*b^3*exp(26*d*x+26*c)[phone]*a^3*exp(14*d*x+14*c)+2632773*b^3*exp(2*d*x+2*c)-532235847*b^3*exp(8*d*x+8*c)
-14708736*a^3*exp(4*d*x+4*c)-23694957*b^3*exp(4*d*x+4*c)+4849845*b^3*exp(30*d*x+30*c)-61108047*b^3*exp(28*d*x+
28*c)[phone]*a^3*exp(18*d*x+18*c)[phone]*b^3*exp(10*d*x+10*c)-333398016*a^3*exp(8*d*x+8*c)[phone]*a^
3*exp(6*d*x+6*c)[phone]*b^3*exp(6*d*x+6*c))/d/(exp(2*d*x+2*c)-1)^19

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 4883 vs. \(2 (232) = 464\).
time = 0.33, size = 4883, normalized size = 19.69 \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)^20*(a+b*sinh(d*x+c)^4)^3,x, algorithm="maxima")

[Out]

131072/230945*a^3*(19*e^(-2*d*x - 2*c)/(d*(19*e^(-2*d*x - 2*c) - 171*e^(-4*d*x - 4*c) + 969*e^(-6*d*x - 6*c) -
3876*e^(-8*d*x - 8*c) + 11628*e^(-10*d*x - 10*c) - 27132*e^(-12*d*x - 12*c) + 50388*e^(-14*d*x - 14*c) - 7558
2*e^(-16*d*x - 16*c) + 92378*e^(-18*d*x - 18*c) - 92378*e^(-20*d*x - 20*c) + 75582*e^(-22*d*x - 22*c) - 50388*
e^(-24*d*x - 24*c) + 27132*e^(-26*d*x - 26*c) - 11628*e^(-28*d*x - 28*c) + 3876*e^(-30*d*x - 30*c) - 969*e^(-3
2*d*x - 32*c) + 171*e^(-34*d*x - 34*c) - 19*e^(-36*d*x - 36*c) + e^(-38*d*x - 38*c) - 1)) - 171*e^(-4*d*x - 4*
c)/(d*(19*e^(-2*d*x - 2*c) - 171*e^(-4*d*x - 4*c) + 969*e^(-6*d*x - 6*c) - 3876*e^(-8*d*x - 8*c) + 11628*e^(-1
0*d*x - 10*c) - 27132*e^(-12*d*x - 12*c) + 50388*e^(-14*d*x - 14*c) - 75582*e^(-16*d*x - 16*c) + 92378*e^(-18*
d*x - 18*c) - 92378*e^(-20*d*x - 20*c) + 75582*e^(-22*d*x - 22*c) - 50388*e^(-24*d*x - 24*c) + 27132*e^(-26*d*
x - 26*c) - 11628*e^(-28*d*x - 28*c) + 3876*e^(-30*d*x - 30*c) - 969*e^(-32*d*x - 32*c) + 171*e^(-34*d*x - 34*
c) - 19*e^(-36*d*x - 36*c) + e^(-38*d*x - 38*c) - 1)) + 969*e^(-6*d*x - 6*c)/(d*(19*e^(-2*d*x - 2*c) - 171*e^(
-4*d*x - 4*c) + 969*e^(-6*d*x - 6*c) - 3876*e^(-8*d*x - 8*c) + 11628*e^(-10*d*x - 10*c) - 27132*e^(-12*d*x - 1
2*c) + 50388*e^(-14*d*x - 14*c) - 75582*e^(-16*d*x - 16*c) + 92378*e^(-18*d*x - 18*c) - 92378*e^(-20*d*x - 20*
c) + 75582*e^(-22*d*x - 22*c) - 50388*e^(-24*d*x - 24*c) + 27132*e^(-26*d*x - 26*c) - 11628*e^(-28*d*x - 28*c)
+ 3876*e^(-30*d*x - 30*c) - 969*e^(-32*d*x - 32*c) + 171*e^(-34*d*x - 34*c) - 19*e^(-36*d*x - 36*c) + e^(-38*
d*x - 38*c) - 1)) - 3876*e^(-8*d*x - 8*c)/(d*(19*e^(-2*d*x - 2*c) - 171*e^(-4*d*x - 4*c) + 969*e^(-6*d*x - 6*c
) - 3876*e^(-8*d*x - 8*c) + 11628*e^(-10*d*x - 10*c) - 27132*e^(-12*d*x - 12*c) + 50388*e^(-14*d*x - 14*c) - 7
5582*e^(-16*d*x - 16*c) + 92378*e^(-18*d*x - 18*c) - 92378*e^(-20*d*x - 20*c) + 75582*e^(-22*d*x - 22*c) - 503
88*e^(-24*d*x - 24*c) + 27132*e^(-26*d*x - 26*c) - 11628*e^(-28*d*x - 28*c) + 3876*e^(-30*d*x - 30*c) - 969*e^
(-32*d*x - 32*c) + 171*e^(-34*d*x - 34*c) - 19*e^(-36*d*x - 36*c) + e^(-38*d*x - 38*c) - 1)) + 11628*e^(-10*d*
x - 10*c)/(d*(19*e^(-2*d*x - 2*c) - 171*e^(-4*d*x - 4*c) + 969*e^(-6*d*x - 6*c) - 3876*e^(-8*d*x - 8*c) + 1162
8*e^(-10*d*x - 10*c) - 27132*e^(-12*d*x - 12*c) + 50388*e^(-14*d*x - 14*c) - 75582*e^(-16*d*x - 16*c) + 92378*
e^(-18*d*x - 18*c) - 92378*e^(-20*d*x - 20*c) + 75582*e^(-22*d*x - 22*c) - 50388*e^(-24*d*x - 24*c) + 27132*e^
(-26*d*x - 26*c) - 11628*e^(-28*d*x - 28*c) + 3876*e^(-30*d*x - 30*c) - 969*e^(-32*d*x - 32*c) + 171*e^(-34*d*
x - 34*c) - 19*e^(-36*d*x - 36*c) + e^(-38*d*x - 38*c) - 1)) - 27132*e^(-12*d*x - 12*c)/(d*(19*e^(-2*d*x - 2*c
) - 171*e^(-4*d*x - 4*c) + 969*e^(-6*d*x - 6*c) - 3876*e^(-8*d*x - 8*c) + 11628*e^(-10*d*x - 10*c) - 27132*e^(
-12*d*x - 12*c) + 50388*e^(-14*d*x - 14*c) - 75582*e^(-16*d*x - 16*c) + 92378*e^(-18*d*x - 18*c) - 92378*e^(-2
0*d*x - 20*c) + 75582*e^(-22*d*x - 22*c) - 50388*e^(-24*d*x - 24*c) + 27132*e^(-26*d*x - 26*c) - 11628*e^(-28*
d*x - 28*c) + 3876*e^(-30*d*x - 30*c) - 969*e^(-32*d*x - 32*c) + 171*e^(-34*d*x - 34*c) - 19*e^(-36*d*x - 36*c
) + e^(-38*d*x - 38*c) - 1)) + 50388*e^(-14*d*x - 14*c)/(d*(19*e^(-2*d*x - 2*c) - 171*e^(-4*d*x - 4*c) + 969*e
^(-6*d*x - 6*c) - 3876*e^(-8*d*x - 8*c) + 11628*e^(-10*d*x - 10*c) - 27132*e^(-12*d*x - 12*c) + 50388*e^(-14*d
*x - 14*c) - 75582*e^(-16*d*x - 16*c) + 92378*e^(-18*d*x - 18*c) - 92378*e^(-20*d*x - 20*c) + 75582*e^(-22*d*x
- 22*c) - 50388*e^(-24*d*x - 24*c) + 27132*e^(-26*d*x - 26*c) - 11628*e^(-28*d*x - 28*c) + 3876*e^(-30*d*x -
30*c) - 969*e^(-32*d*x - 32*c) + 171*e^(-34*d*x - 34*c) - 19*e^(-36*d*x - 36*c) + e^(-38*d*x - 38*c) - 1)) - 7
5582*e^(-16*d*x - 16*c)/(d*(19*e^(-2*d*x - 2*c) - 171*e^(-4*d*x - 4*c) + 969*e^(-6*d*x - 6*c) - 3876*e^(-8*d*x
- 8*c) + 11628*e^(-10*d*x - 10*c) - 27132*e^(-12*d*x - 12*c) + 50388*e^(-14*d*x - 14*c) - 75582*e^(-16*d*x -
16*c) + 92378*e^(-18*d*x - 18*c) - 92378*e^(-20*d*x - 20*c) + 75582*e^(-22*d*x - 22*c) - 50388*e^(-24*d*x - 24
*c) + 27132*e^(-26*d*x - 26*c) - 11628*e^(-28*d*x - 28*c) + 3876*e^(-30*d*x - 30*c) - 969*e^(-32*d*x - 32*c) +
171*e^(-34*d*x - 34*c) - 19*e^(-36*d*x - 36*c) + e^(-38*d*x - 38*c) - 1)) + 92378*e^(-18*d*x - 18*c)/(d*(19*e
^(-2*d*x - 2*c) - 171*e^(-4*d*x - 4*c) + 969*e^(-6*d*x - 6*c) - 3876*e^(-8*d*x - 8*c) + 11628*e^(-10*d*x - 10*
c) - 27132*e^(-12*d*x - 12*c) + 50388*e^(-14*d*x - 14*c) - 75582*e^(-16*d*x - 16*c) + 92378*e^(-18*d*x - 18*c)
- 92378*e^(-20*d*x - 20*c) + 75582*e^(-22*d*x - 22*c) - 50388*e^(-24*d*x - 24*c) + 27132*e^(-26*d*x - 26*c) -
11628*e^(-28*d*x - 28*c) + 3876*e^(-30*d*x - 30*c) - 969*e^(-32*d*x - 32*c) + 171*e^(-34*d*x - 34*c) - 19*e^(
-36*d*x - 36*c) + e^(-38*d*x - 38*c) - 1)) - 1/(d*(19*e^(-2*d*x - 2*c) - 171*e^(-4*d*x - 4*c) + 969*e^(-6*d*x
- 6*c) - 3876*e^(-8*d*x - 8*c) + 11628*e^(-10*d*x - 10*c) - 27132*e^(-12*d*x - 12*c) + 50388*e^(-14*d*x - 14*c
) - 75582*e^(-16*d*x - 16*c) + 92378*e^(-18*d*x - 18*c) - 92378*e^(-20*d*x - 20*c) + 75582*e^(-22*d*x - 22*c)
- 50388*e^(-24*d*x - 24*c) + 27132*e^(-26*d*x -...

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 4259 vs. \(2 (232) = 464\).
time = 0.56, size = 4259, normalized size = 17.17 \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)^20*(a+b*sinh(d*x+c)^4)^3,x, algorithm="fricas")

[Out]

64/4849845*((43008*a^3 + 144704*a^2*b + 167960*a*b^2 - 2355639*b^3)*cosh(d*x + c)^15 + 15*(43008*a^3 + 144704*
a^2*b + 167960*a*b^2 - 2355639*b^3)*cosh(d*x + c)*sinh(d*x + c)^14 - 2*(21504*a^3 + 72352*a^2*b + 83980*a*b^2
+ 1247103*b^3)*sinh(d*x + c)^15 - 19*(43008*a^3 + 144704*a^2*b + 167960*a*b^2 - 1538823*b^3)*cosh(d*x + c)^13
+ 2*(408576*a^3 + 1374688*a^2*b + 1595620*a*b^2 + 15935205*b^3 - 105*(21504*a^3 + 72352*a^2*b + 83980*a*b^2 +
1247103*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^13 + 13*(35*(43008*a^3 + 144704*a^2*b + 167960*a*b^2 - 2355639*b^3
)*cosh(d*x + c)^3 - 19*(43008*a^3 + 144704*a^2*b + 167960*a*b^2 - 1538823*b^3)*cosh(d*x + c))*sinh(d*x + c)^12
+ 57*(129024*a^3 + 434112*a^2*b - 857480*a*b^2 - 2914769*b^3)*cosh(d*x + c)^11 - 6*(455*(21504*a^3 + 72352*a^
2*b + 83980*a*b^2 + 1247103*b^3)*cosh(d*x + c)^4 + 1225728*a^3 + 4124064*a^2*b + 17719780*a*b^2 + 31639465*b^3
- 494*(21504*a^3 + 72352*a^2*b + 83980*a*b^2 + 838695*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^11 + 11*(273*(43008
*a^3 + 144704*a^2*b + 167960*a*b^2 - 2355639*b^3)*cosh(d*x + c)^5 - 494*(43008*a^3 + 144704*a^2*b + 167960*a*b
^2 - 1538823*b^3)*cosh(d*x + c)^3 + 57*(129024*a^3 + 434112*a^2*b - 857480*a*b^2 - 2914769*b^3)*cosh(d*x + c))
*sinh(d*x + c)^10 - 969*(43008*a^3 + 144704*a^2*b - 529880*a*b^2 - 586443*b^3)*cosh(d*x + c)^9 - 2*(5005*(2150
4*a^3 + 72352*a^2*b + 83980*a*b^2 + 1247103*b^3)*cosh(d*x + c)^6 - 13585*(21504*a^3 + 72352*a^2*b + 83980*a*b^
2 + 838695*b^3)*cosh(d*x + c)^4 - 20837376*a^3 - 70109088*a^2*b - 419480100*a*b^2 - 351267345*b^3 + 3135*(6451
2*a^3 + 217056*a^2*b + 932620*a*b^2 + 1665235*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^9 + 9*(715*(43008*a^3 + 1447
04*a^2*b + 167960*a*b^2 - 2355639*b^3)*cosh(d*x + c)^7 - 2717*(43008*a^3 + 144704*a^2*b + 167960*a*b^2 - 15388
23*b^3)*cosh(d*x + c)^5 + 1045*(129024*a^3 + 434112*a^2*b - 857480*a*b^2 - 2914769*b^3)*cosh(d*x + c)^3 - 969*
(43008*a^3 + 144704*a^2*b - 529880*a*b^2 - 586443*b^3)*cosh(d*x + c))*sinh(d*x + c)^8 + 6783*(24576*a^3 - 5459
2*a^2*b - 293800*a*b^2 - 189761*b^3)*cosh(d*x + c)^7 - 6*(2145*(21504*a^3 + 72352*a^2*b + 83980*a*b^2 + 124710
3*b^3)*cosh(d*x + c)^8 - 10868*(21504*a^3 + 72352*a^2*b + 83980*a*b^2 + 838695*b^3)*cosh(d*x + c)^6 + 6270*(64
512*a^3 + 217056*a^2*b + 932620*a*b^2 + 1665235*b^3)*cosh(d*x + c)^4 + 27783168*a^3 + 248673824*a^2*b + 549145
220*a*b^2 + 303230785*b^3 - 11628*(21504*a^3 + 72352*a^2*b + 432900*a*b^2 + 362505*b^3)*cosh(d*x + c)^2)*sinh(
d*x + c)^7 + (5005*(43008*a^3 + 144704*a^2*b + 167960*a*b^2 - 2355639*b^3)*cosh(d*x + c)^9 - 32604*(43008*a^3
+ 144704*a^2*b + 167960*a*b^2 - 1538823*b^3)*cosh(d*x + c)^7 + 26334*(129024*a^3 + 434112*a^2*b - 857480*a*b^2
- 2914769*b^3)*cosh(d*x + c)^5 - 81396*(43008*a^3 + 144704*a^2*b - 529880*a*b^2 - 586443*b^3)*cosh(d*x + c)^3
+ 47481*(24576*a^3 - 54592*a^2*b - 293800*a*b^2 - 189761*b^3)*cosh(d*x + c))*sinh(d*x + c)^6 - 323*(1548288*a
^3 - 8564416*a^2*b - 12926680*a*b^2 - 6120543*b^3)*cosh(d*x + c)^5 - 2*(3003*(21504*a^3 + 72352*a^2*b + 83980*
a*b^2 + 1247103*b^3)*cosh(d*x + c)^10 - 24453*(21504*a^3 + 72352*a^2*b + 83980*a*b^2 + 838695*b^3)*cosh(d*x +
c)^8 + 26334*(64512*a^3 + 217056*a^2*b + 932620*a*b^2 + 1665235*b^3)*cosh(d*x + c)^6 - 122094*(21504*a^3 + 723
52*a^2*b + 432900*a*b^2 + 362505*b^3)*cosh(d*x + c)^4 - 250048512*a^3 - 3065771296*a^2*b - 4040697700*a*b^2 -
1763542209*b^3 + 20349*(86016*a^3 + 769888*a^2*b + 1700140*a*b^2 + 938795*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^
5 + (1365*(43008*a^3 + 144704*a^2*b + 167960*a*b^2 - 2355639*b^3)*cosh(d*x + c)^11 - 13585*(43008*a^3 + 144704
*a^2*b + 167960*a*b^2 - 1538823*b^3)*cosh(d*x + c)^9 + 18810*(129024*a^3 + 434112*a^2*b - 857480*a*b^2 - 29147
69*b^3)*cosh(d*x + c)^7 - 122094*(43008*a^3 + 144704*a^2*b - 529880*a*b^2 - 586443*b^3)*cosh(d*x + c)^5 + 2374
05*(24576*a^3 - 54592*a^2*b - 293800*a*b^2 - 189761*b^3)*cosh(d*x + c)^3 - 1615*(1548288*a^3 - 8564416*a^2*b -
12926680*a*b^2 - 6120543*b^3)*cosh(d*x + c))*sinh(d*x + c)^4 - 323*(8687616*a^3 + 15456448*a^2*b + 15194920*a
*b^2 + 6026163*b^3)*cosh(d*x + c)^3 - 2*(455*(21504*a^3 + 72352*a^2*b + 83980*a*b^2 + 1247103*b^3)*cosh(d*x +
c)^12 - 5434*(21504*a^3 + 72352*a^2*b + 83980*a*b^2 + 838695*b^3)*cosh(d*x + c)^10 + 9405*(64512*a^3 + 217056*
a^2*b + 932620*a*b^2 + 1665235*b^3)*cosh(d*x + c)^8 - 81396*(21504*a^3 + 72352*a^2*b + 432900*a*b^2 + 362505*b
^3)*cosh(d*x + c)^6 + 33915*(86016*a^3 + 769888*a^2*b + 1700140*a*b^2 + 938795*b^3)*cosh(d*x + c)^4 + 25699430
40*a^3 + 6422325280*a^2*b + 6933472780*a*b^2 + 2675035935*b^3 - 3230*(774144*a^3 + 9491552*a^2*b + 12509900*a*
b^2 + 5459883*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^3 + (105*(43008*a^3 + 144704*a^2*b + 167960*a*b^2 - 2355639*
b^3)*cosh(d*x + c)^13 - 1482*(43008*a^3 + 144704*a^2*b + 167960*a*b^2 - 1538823*b^3)*cosh(d*x + c)^11 + 3135*(
129024*a^3 + 434112*a^2*b - 857480*a*b^2 - 2914769*b^3)*cosh(d*x + c)^9 - 34884*(43008*a^3 + 144704*a^2*b - 52
9880*a*b^2 - 586443*b^3)*cosh(d*x + c)^7 + 142443*(24576*a^3 - 54592*a^2*b - 293800*a*b^2 - 189761*b^3)*cosh(d
*x + c)^5 - 3230*(1548288*a^3 - 8564416*a^2*b -...

________________________________________________________________________________________

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)**20*(a+b*sinh(d*x+c)**4)**3,x)

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 737 vs. \(2 (232) = 464\).
time = 0.65, size = 737, normalized size = 2.97 \begin {gather*} -\frac {32 \, {\left (4849845 \, b^{3} e^{\left (30 \, d x + 30 \, c\right )} - 61108047 \, b^{3} e^{\left (28 \, d x + 28 \, c\right )} + 155195040 \, a b^{2} e^{\left (26 \, d x + 26 \, c\right )} + 355978623 \, b^{3} e^{\left (26 \, d x + 26 \, c\right )} - 1352413920 \, a b^{2} e^{\left (24 \, d x + 24 \, c\right )} - 1270797957 \, b^{3} e^{\left (24 \, d x + 24 \, c\right )} + 1862340480 \, a^{2} b e^{\left (22 \, d x + 22 \, c\right )} + 5287716720 \, a b^{2} e^{\left (22 \, d x + 22 \, c\right )} + 3106533573 \, b^{3} e^{\left (22 \, d x + 22 \, c\right )} - 8897848960 \, a^{2} b e^{\left (20 \, d x + 20 \, c\right )} - 12256713040 \, a b^{2} e^{\left (20 \, d x + 20 \, c\right )} - 5504019807 \, b^{3} e^{\left (20 \, d x + 20 \, c\right )} + 7945986048 \, a^{3} e^{\left (18 \, d x + 18 \, c\right )} + 17837083264 \, a^{2} b e^{\left (18 \, d x + 18 \, c\right )} + 18774904720 \, a b^{2} e^{\left (18 \, d x + 18 \, c\right )} + 7296522519 \, b^{3} e^{\left (18 \, d x + 18 \, c\right )} - 6501261312 \, a^{3} e^{\left (16 \, d x + 16 \, c\right )} - 20011694976 \, a^{2} b e^{\left (16 \, d x + 16 \, c\right )} - 20101788720 \, a b^{2} e^{\left (16 \, d x + 16 \, c\right )} - 7366637421 \, b^{3} e^{\left (16 \, d x + 16 \, c\right )} + 4334174208 \, a^{3} e^{\left (14 \, d x + 14 \, c\right )} + 14582690304 \, a^{2} b e^{\left (14 \, d x + 14 \, c\right )} + 15573923040 \, a b^{2} e^{\left (14 \, d x + 14 \, c\right )} + 5711316039 \, b^{3} e^{\left (14 \, d x + 14 \, c\right )} - 2333786112 \, a^{3} e^{\left (12 \, d x + 12 \, c\right )} - 7852217856 \, a^{2} b e^{\left (12 \, d x + 12 \, c\right )} - 8958986400 \, a b^{2} e^{\left (12 \, d x + 12 \, c\right )} - 3403621221 \, b^{3} e^{\left (12 \, d x + 12 \, c\right )} + 1000194048 \, a^{3} e^{\left (10 \, d x + 10 \, c\right )} + 3365236224 \, a^{2} b e^{\left (10 \, d x + 10 \, c\right )} + 3906077760 \, a b^{2} e^{\left (10 \, d x + 10 \, c\right )} + 1550149029 \, b^{3} e^{\left (10 \, d x + 10 \, c\right )} - 333398016 \, a^{3} e^{\left (8 \, d x + 8 \, c\right )} - 1121745408 \, a^{2} b e^{\left (8 \, d x + 8 \, c\right )} - 1302025920 \, a b^{2} e^{\left (8 \, d x + 8 \, c\right )} - 532235847 \, b^{3} e^{\left (8 \, d x + 8 \, c\right )} + 83349504 \, a^{3} e^{\left (6 \, d x + 6 \, c\right )} + 280436352 \, a^{2} b e^{\left (6 \, d x + 6 \, c\right )} + 325506480 \, a b^{2} e^{\left (6 \, d x + 6 \, c\right )} + 134271423 \, b^{3} e^{\left (6 \, d x + 6 \, c\right )} - 14708736 \, a^{3} e^{\left (4 \, d x + 4 \, c\right )} - 49488768 \, a^{2} b e^{\left (4 \, d x + 4 \, c\right )} - 57442320 \, a b^{2} e^{\left (4 \, d x + 4 \, c\right )} - 23694957 \, b^{3} e^{\left (4 \, d x + 4 \, c\right )} + 1634304 \, a^{3} e^{\left (2 \, d x + 2 \, c\right )} + 5498752 \, a^{2} b e^{\left (2 \, d x + 2 \, c\right )} + 6382480 \, a b^{2} e^{\left (2 \, d x + 2 \, c\right )} + 2632773 \, b^{3} e^{\left (2 \, d x + 2 \, c\right )} - 86016 \, a^{3} - 289408 \, a^{2} b - 335920 \, a b^{2} - 138567 \, b^{3}\right )}}{4849845 \, d {\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )}^{19}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(d*x+c)^20*(a+b*sinh(d*x+c)^4)^3,x, algorithm="giac")

[Out]

-32/4849845*(4849845*b^3*e^(30*d*x + 30*c) - 61108047*b^3*e^(28*d*x + 28*c) + 155195040*a*b^2*e^(26*d*x + 26*c
) + 355978623*b^3*e^(26*d*x + 26*c) - 1352413920*a*b^2*e^(24*d*x + 24*c) - 1270797957*b^3*e^(24*d*x + 24*c) +
1862340480*a^2*b*e^(22*d*x + 22*c) + 5287716720*a*b^2*e^(22*d*x + 22*c) + 3106533573*b^3*e^(22*d*x + 22*c) - 8
897848960*a^2*b*e^(20*d*x + 20*c) - 12256713040*a*b^2*e^(20*d*x + 20*c) - 5504019807*b^3*e^(20*d*x + 20*c) + 7
945986048*a^3*e^(18*d*x + 18*c) + 17837083264*a^2*b*e^(18*d*x + 18*c) + 18774904720*a*b^2*e^(18*d*x + 18*c) +
7296522519*b^3*e^(18*d*x + 18*c) - 6501261312*a^3*e^(16*d*x + 16*c) - 20011694976*a^2*b*e^(16*d*x + 16*c) - 20
101788720*a*b^2*e^(16*d*x + 16*c) - 7366637421*b^3*e^(16*d*x + 16*c) + 4334174208*a^3*e^(14*d*x + 14*c) + 1458
2690304*a^2*b*e^(14*d*x + 14*c) + 15573923040*a*b^2*e^(14*d*x + 14*c) + 5711316039*b^3*e^(14*d*x + 14*c) - 233
3786112*a^3*e^(12*d*x + 12*c) - 7852217856*a^2*b*e^(12*d*x + 12*c) - 8958986400*a*b^2*e^(12*d*x + 12*c) - 3403
621221*b^3*e^(12*d*x + 12*c) + 1000194048*a^3*e^(10*d*x + 10*c) + 3365236224*a^2*b*e^(10*d*x + 10*c) + 3906077
760*a*b^2*e^(10*d*x + 10*c) + 1550149029*b^3*e^(10*d*x + 10*c) - 333398016*a^3*e^(8*d*x + 8*c) - 1121745408*a^
2*b*e^(8*d*x + 8*c) - 1302025920*a*b^2*e^(8*d*x + 8*c) - 532235847*b^3*e^(8*d*x + 8*c) + 83349504*a^3*e^(6*d*x
+ 6*c) + 280436352*a^2*b*e^(6*d*x + 6*c) + 325506480*a*b^2*e^(6*d*x + 6*c) + 134271423*b^3*e^(6*d*x + 6*c) -
14708736*a^3*e^(4*d*x + 4*c) - 49488768*a^2*b*e^(4*d*x + 4*c) - 57442320*a*b^2*e^(4*d*x + 4*c) - 23694957*b^3*
e^(4*d*x + 4*c) + 1634304*a^3*e^(2*d*x + 2*c) + 5498752*a^2*b*e^(2*d*x + 2*c) + 6382480*a*b^2*e^(2*d*x + 2*c)
+ 2632773*b^3*e^(2*d*x + 2*c) - 86016*a^3 - 289408*a^2*b - 335920*a*b^2 - 138567*b^3)/(d*(e^(2*d*x + 2*c) - 1)
^19)

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Mupad [B]
time = 1.42, size = 2500, normalized size = 10.08 \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)^20,x)

[Out]

((512*b*(112*a*b + 128*a^2 + 33*b^2))/(138567*d) + (1792*b^3*exp(8*c + 8*d*x))/(969*d) - (448*b^3*exp(10*c + 1
0*d*x))/(969*d) - (64*b*exp(2*c + 2*d*x)*(448*a*b + 256*a^2 + 165*b^2))/(12597*d) - (256*b^2*exp(6*c + 6*d*x)*
(8*a + 11*b))/(969*d) + (512*b^2*exp(4*c + 4*d*x)*(96*a + 55*b))/(12597*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) - ((8*b*(448*a*b + 256*a^2 + 165*b^2))/(12597
*d) + (128*exp(4*c + 4*d*x)*(840*a*b^2 + 1152*a^2*b + 1024*a^3 + 231*b^3))/(4199*d) - (2816*b^3*exp(14*c + 14*
d*x))/(323*d) + (440*b^3*exp(16*c + 16*d*x))/(323*d) - (512*b*exp(2*c + 2*d*x)*(112*a*b + 128*a^2 + 33*b^2))/(
12597*d) - (2560*b*exp(6*c + 6*d*x)*(112*a*b + 128*a^2 + 33*b^2))/(4199*d) + (880*b*exp(8*c + 8*d*x)*(448*a*b
+ 256*a^2 + 165*b^2))/(4199*d) + (704*b^2*exp(12*c + 12*d*x)*(8*a + 11*b))/(323*d) - (2816*b^2*exp(10*c + 10*d
*x)*(96*a + 55*b))/(4199*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) - ((512*exp(1
8*c + 18*d*x)*(840*a*b^2 + 1152*a^2*b + 1024*a^3 + 231*b^3))/(19*d) + (128*b^3*exp(6*c + 6*d*x))/(19*d) - (153
6*b^3*exp(8*c + 8*d*x))/(19*d) - (1536*b^3*exp(28*c + 28*d*x))/(19*d) + (128*b^3*exp(30*c + 30*d*x))/(19*d) -
(3072*b*exp(16*c + 16*d*x)*(112*a*b + 128*a^2 + 33*b^2))/(19*d) - (3072*b*exp(20*c + 20*d*x)*(112*a*b + 128*a^
2 + 33*b^2))/(19*d) + (384*b*exp(14*c + 14*d*x)*(448*a*b + 256*a^2 + 165*b^2))/(19*d) + (384*b*exp(22*c + 22*d
*x)*(448*a*b + 256*a^2 + 165*b^2))/(19*d) + (768*b^2*exp(10*c + 10*d*x)*(8*a + 11*b))/(19*d) + (768*b^2*exp(26
*c + 26*d*x)*(8*a + 11*b))/(19*d) - (512*b^2*exp(12*c + 12*d*x)*(96*a + 55*b))/(19*d) - (512*b^2*exp(24*c + 24
*d*x)*(96*a + 55*b))/(19*d))/(19*exp(2*c + 2*d*x) - 171*exp(4*c + 4*d*x) + 969*exp(6*c + 6*d*x) - 3876*exp(8*c
+ 8*d*x) + 11628*exp(10*c + 10*d*x) - 27132*exp(12*c + 12*d*x) + 50388*exp(14*c + 14*d*x) - 75582*exp(16*c +
16*d*x) + 92378*exp(18*c + 18*d*x) - 92378*exp(20*c + 20*d*x) + 75582*exp(22*c + 22*d*x) - 50388*exp(24*c + 24
*d*x) + 27132*exp(26*c + 26*d*x) - 11628*exp(28*c + 28*d*x) + 3876*exp(30*c + 30*d*x) - 969*exp(32*c + 32*d*x)
+ 171*exp(34*c + 34*d*x) - 19*exp(36*c + 36*d*x) + exp(38*c + 38*d*x) - 1) + ((128*b^3)/(4845*d) - (1792*exp(
10*c + 10*d*x)*(840*a*b^2 + 1152*a^2*b + 1024*a^3 + 231*b^3))/(1615*d) + (128128*b^3*exp(20*c + 20*d*x))/(4845
*d) - (2912*b^3*exp(22*c + 22*d*x))/(969*d) + (3584*b*exp(8*c + 8*d*x)*(112*a*b + 128*a^2 + 33*b^2))/(969*d) +
(3584*b*exp(12*c + 12*d*x)*(112*a*b + 128*a^2 + 33*b^2))/(323*d) - (224*b*exp(6*c + 6*d*x)*(448*a*b + 256*a^2
+ 165*b^2))/(969*d) - (704*b*exp(14*c + 14*d*x)*(448*a*b + 256*a^2 + 165*b^2))/(323*d) - (64*b^2*exp(2*c + 2*
d*x)*(8*a + 11*b))/(969*d) - (9152*b^2*exp(18*c + 18*d*x)*(8*a + 11*b))/(969*d) + (128*b^2*exp(4*c + 4*d*x)*(9
6*a + 55*b))/(969*d) + (1408*b^2*exp(16*c + 16*d*x)*(96*a + 55*b))/(323*d))/(15*exp(2*c + 2*d*x) - 105*exp(4*c
+ 4*d*x) + 455*exp(6*c + 6*d*x) - 1365*exp(8*c + 8*d*x) + 3003*exp(10*c + 10*d*x) - 5005*exp(12*c + 12*d*x) +
6435*exp(14*c + 14*d*x) - 6435*exp(16*c + 16*d*x) + 5005*exp(18*c + 18*d*x) - 3003*exp(20*c + 20*d*x) + 1365*
exp(22*c + 22*d*x) - 455*exp(24*c + 24*d*x) + 105*exp(26*c + 26*d*x) - 15*exp(28*c + 28*d*x) + exp(30*c + 30*d
*x) - 1) + ((128*b^3)/(4845*d) - (32*b^3*exp(2*c + 2*d*x))/(969*d))/(5*exp(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) - ((128*exp(8*c + 8*d*x)*(840*a*b^2 + 115
2*a^2*b + 1024*a^3 + 231*b^3))/(323*d) - (18304*b^3*exp(18*c + 18*d*x))/(969*d) + (2288*b^3*exp(20*c + 20*d*x)
)/(969*d) + (32*b^2*(8*a + 11*b))/(6783*d) - (1024*b*exp(6*c + 6*d*x)*(112*a*b + 128*a^2 + 33*b^2))/(969*d) -
(1536*b*exp(10*c + 10*d*x)*(112*a*b + 128*a^2 + 33*b^2))/(323*d) + (16*b*exp(4*c + 4*d*x)*(448*a*b + 256*a^2 +
165*b^2))/(323*d) + (352*b*exp(12*c + 12*d*x)*(448*a*b + 256*a^2 + 165*b^2))/(323*d) + (13728*b^2*exp(16*c +
16*d*x)*(8*a + 11*b))/(2261*d) - (128*b^2*exp(2*c + 2*d*x)*(96*a + 55*b))/(6783*d) - (5632*b^2*exp(14*c + 14*d
*x)*(96*a + 55*b))/(2261*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*e
xp(26*c + 26*d*x) + exp(28*c + 28*d*x) + 1) + ((128*b^3*exp(4*c + 4*d*x))/(323*d) - (160*b^3*exp(6*c + 6*d*x))
/(969*d) + (128*b^2*(96*a + 55*b))/(88179*d) - (64*b^2*exp(2*c + 2*d*x)*(8*a + 11*b))/(2261*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) - ((128*(840...

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