3.3.16 \(\int \text {csch}^{13}(c+d x) (a+b \sinh ^4(c+d x))^3 \, dx\) [216]

Optimal. Leaf size=220 \[ -\frac {\left (231 a^3+840 a^2 b+1152 a b^2+1024 b^3\right ) \tanh ^{-1}(\cosh (c+d x))}{1024 d}+\frac {3 a \left (77 a^2+280 a b+384 b^2\right ) \coth (c+d x) \text {csch}(c+d x)}{1024 d}-\frac {a \left (77 a^2+280 a b+384 b^2\right ) \coth (c+d x) \text {csch}^3(c+d x)}{512 d}+\frac {7 a^2 (11 a+40 b) \coth (c+d x) \text {csch}^5(c+d x)}{640 d}-\frac {3 a^2 (11 a+40 b) \coth (c+d x) \text {csch}^7(c+d x)}{320 d}+\frac {11 a^3 \coth (c+d x) \text {csch}^9(c+d x)}{120 d}-\frac {a^3 \coth (c+d x) \text {csch}^{11}(c+d x)}{12 d} \]

[Out]

-1/1024*(231*a^3+840*a^2*b+1152*a*b^2+1024*b^3)*arctanh(cosh(d*x+c))/d+3/1024*a*(77*a^2+280*a*b+384*b^2)*coth(
d*x+c)*csch(d*x+c)/d-1/512*a*(77*a^2+280*a*b+384*b^2)*coth(d*x+c)*csch(d*x+c)^3/d+7/640*a^2*(11*a+40*b)*coth(d
*x+c)*csch(d*x+c)^5/d-3/320*a^2*(11*a+40*b)*coth(d*x+c)*csch(d*x+c)^7/d+11/120*a^3*coth(d*x+c)*csch(d*x+c)^9/d
-1/12*a^3*coth(d*x+c)*csch(d*x+c)^11/d

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Rubi [A]
time = 0.28, antiderivative size = 220, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {3294, 1171, 1828, 393, 212} \begin {gather*} -\frac {a^3 \coth (c+d x) \text {csch}^{11}(c+d x)}{12 d}+\frac {11 a^3 \coth (c+d x) \text {csch}^9(c+d x)}{120 d}-\frac {a \left (77 a^2+280 a b+384 b^2\right ) \coth (c+d x) \text {csch}^3(c+d x)}{512 d}+\frac {3 a \left (77 a^2+280 a b+384 b^2\right ) \coth (c+d x) \text {csch}(c+d x)}{1024 d}-\frac {3 a^2 (11 a+40 b) \coth (c+d x) \text {csch}^7(c+d x)}{320 d}+\frac {7 a^2 (11 a+40 b) \coth (c+d x) \text {csch}^5(c+d x)}{640 d}-\frac {\left (231 a^3+840 a^2 b+1152 a b^2+1024 b^3\right ) \tanh ^{-1}(\cosh (c+d x))}{1024 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Csch[c + d*x]^13*(a + b*Sinh[c + d*x]^4)^3,x]

[Out]

-1/1024*((231*a^3 + 840*a^2*b + 1152*a*b^2 + 1024*b^3)*ArcTanh[Cosh[c + d*x]])/d + (3*a*(77*a^2 + 280*a*b + 38
4*b^2)*Coth[c + d*x]*Csch[c + d*x])/(1024*d) - (a*(77*a^2 + 280*a*b + 384*b^2)*Coth[c + d*x]*Csch[c + d*x]^3)/
(512*d) + (7*a^2*(11*a + 40*b)*Coth[c + d*x]*Csch[c + d*x]^5)/(640*d) - (3*a^2*(11*a + 40*b)*Coth[c + d*x]*Csc
h[c + d*x]^7)/(320*d) + (11*a^3*Coth[c + d*x]*Csch[c + d*x]^9)/(120*d) - (a^3*Coth[c + d*x]*Csch[c + d*x]^11)/
(12*d)

Rule 212

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

Rule 393

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[(-(b*c - a*d))*x*((a + b*x^n)^(p
+ 1)/(a*b*n*(p + 1))), x] - Dist[(a*d - b*c*(n*(p + 1) + 1))/(a*b*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x]
/; FreeQ[{a, b, c, d, n, p}, x] && NeQ[b*c - a*d, 0] && (LtQ[p, -1] || ILtQ[1/n + p, 0])

Rule 1171

Int[((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> With[{Qx = PolynomialQ
uotient[(a + b*x^2 + c*x^4)^p, d + e*x^2, x], R = Coeff[PolynomialRemainder[(a + b*x^2 + c*x^4)^p, d + e*x^2,
x], x, 0]}, Simp[(-R)*x*((d + e*x^2)^(q + 1)/(2*d*(q + 1))), x] + Dist[1/(2*d*(q + 1)), Int[(d + e*x^2)^(q + 1
)*ExpandToSum[2*d*(q + 1)*Qx + R*(2*q + 3), x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] &&
NeQ[c*d^2 - b*d*e + a*e^2, 0] && IGtQ[p, 0] && LtQ[q, -1]

Rule 1828

Int[(Pq_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[Pq, a + b*x^2, x], f = Coeff[P
olynomialRemainder[Pq, a + b*x^2, x], x, 0], g = Coeff[PolynomialRemainder[Pq, a + b*x^2, x], x, 1]}, Simp[(a*
g - b*f*x)*((a + b*x^2)^(p + 1)/(2*a*b*(p + 1))), x] + Dist[1/(2*a*(p + 1)), Int[(a + b*x^2)^(p + 1)*ExpandToS
um[2*a*(p + 1)*Q + f*(2*p + 3), x], x], x]] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] && LtQ[p, -1]

Rule 3294

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

Rubi steps

\begin {align*} \int \text {csch}^{13}(c+d x) \left (a+b \sinh ^4(c+d x)\right )^3 \, dx &=-\frac {\text {Subst}\left (\int \frac {\left (a+b-2 b x^2+b x^4\right )^3}{\left (1-x^2\right )^7} \, dx,x,\cosh (c+d x)\right )}{d}\\ &=-\frac {a^3 \coth (c+d x) \text {csch}^{11}(c+d x)}{12 d}+\frac {\text {Subst}\left (\int \frac {-11 a^3-36 a^2 b-36 a b^2-12 b^3+12 b \left (3 a^2+9 a b+5 b^2\right ) x^2-12 b^2 (9 a+10 b) x^4+12 b^2 (3 a+10 b) x^6-60 b^3 x^8+12 b^3 x^{10}}{\left (1-x^2\right )^6} \, dx,x,\cosh (c+d x)\right )}{12 d}\\ &=\frac {11 a^3 \coth (c+d x) \text {csch}^9(c+d x)}{120 d}-\frac {a^3 \coth (c+d x) \text {csch}^{11}(c+d x)}{12 d}-\frac {\text {Subst}\left (\int \frac {3 \left (33 a^3+120 a^2 b+120 a b^2+40 b^3\right )-240 b^2 (3 a+2 b) x^2+360 b^2 (a+2 b) x^4-480 b^3 x^6+120 b^3 x^8}{\left (1-x^2\right )^5} \, dx,x,\cosh (c+d x)\right )}{120 d}\\ &=-\frac {3 a^2 (11 a+40 b) \coth (c+d x) \text {csch}^7(c+d x)}{320 d}+\frac {11 a^3 \coth (c+d x) \text {csch}^9(c+d x)}{120 d}-\frac {a^3 \coth (c+d x) \text {csch}^{11}(c+d x)}{12 d}+\frac {\text {Subst}\left (\int \frac {-3 \left (231 a^3+840 a^2 b+960 a b^2+320 b^3\right )+2880 b^2 (a+b) x^2-2880 b^3 x^4+960 b^3 x^6}{\left (1-x^2\right )^4} \, dx,x,\cosh (c+d x)\right )}{960 d}\\ &=\frac {7 a^2 (11 a+40 b) \coth (c+d x) \text {csch}^5(c+d x)}{640 d}-\frac {3 a^2 (11 a+40 b) \coth (c+d x) \text {csch}^7(c+d x)}{320 d}+\frac {11 a^3 \coth (c+d x) \text {csch}^9(c+d x)}{120 d}-\frac {a^3 \coth (c+d x) \text {csch}^{11}(c+d x)}{12 d}-\frac {\text {Subst}\left (\int \frac {45 \left (77 a^3+280 a^2 b+384 a b^2+128 b^3\right )-11520 b^3 x^2+5760 b^3 x^4}{\left (1-x^2\right )^3} \, dx,x,\cosh (c+d x)\right )}{5760 d}\\ &=-\frac {a \left (77 a^2+280 a b+384 b^2\right ) \coth (c+d x) \text {csch}^3(c+d x)}{512 d}+\frac {7 a^2 (11 a+40 b) \coth (c+d x) \text {csch}^5(c+d x)}{640 d}-\frac {3 a^2 (11 a+40 b) \coth (c+d x) \text {csch}^7(c+d x)}{320 d}+\frac {11 a^3 \coth (c+d x) \text {csch}^9(c+d x)}{120 d}-\frac {a^3 \coth (c+d x) \text {csch}^{11}(c+d x)}{12 d}+\frac {\text {Subst}\left (\int \frac {-45 \left (231 a^3+840 a^2 b+1152 a b^2+512 b^3\right )+23040 b^3 x^2}{\left (1-x^2\right )^2} \, dx,x,\cosh (c+d x)\right )}{23040 d}\\ &=\frac {3 a \left (77 a^2+280 a b+384 b^2\right ) \coth (c+d x) \text {csch}(c+d x)}{1024 d}-\frac {a \left (77 a^2+280 a b+384 b^2\right ) \coth (c+d x) \text {csch}^3(c+d x)}{512 d}+\frac {7 a^2 (11 a+40 b) \coth (c+d x) \text {csch}^5(c+d x)}{640 d}-\frac {3 a^2 (11 a+40 b) \coth (c+d x) \text {csch}^7(c+d x)}{320 d}+\frac {11 a^3 \coth (c+d x) \text {csch}^9(c+d x)}{120 d}-\frac {a^3 \coth (c+d x) \text {csch}^{11}(c+d x)}{12 d}-\frac {\left (231 a^3+840 a^2 b+1152 a b^2+1024 b^3\right ) \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\cosh (c+d x)\right )}{1024 d}\\ &=-\frac {\left (231 a^3+840 a^2 b+1152 a b^2+1024 b^3\right ) \tanh ^{-1}(\cosh (c+d x))}{1024 d}+\frac {3 a \left (77 a^2+280 a b+384 b^2\right ) \coth (c+d x) \text {csch}(c+d x)}{1024 d}-\frac {a \left (77 a^2+280 a b+384 b^2\right ) \coth (c+d x) \text {csch}^3(c+d x)}{512 d}+\frac {7 a^2 (11 a+40 b) \coth (c+d x) \text {csch}^5(c+d x)}{640 d}-\frac {3 a^2 (11 a+40 b) \coth (c+d x) \text {csch}^7(c+d x)}{320 d}+\frac {11 a^3 \coth (c+d x) \text {csch}^9(c+d x)}{120 d}-\frac {a^3 \coth (c+d x) \text {csch}^{11}(c+d x)}{12 d}\\ \end {align*}

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Mathematica [A]
time = 1.49, size = 246, normalized size = 1.12 \begin {gather*} \frac {-30 a \left (76555 a^2+75816 a b+45696 b^2\right ) \coth (c+d x) \text {csch}^{11}(c+d x)+2 a \left (750629 a^2+2074200 a b+1422720 b^2\right ) \cosh (3 (c+d x)) \text {csch}^{12}(c+d x)-9 a \left (77099 a^2+280360 a b+246400 b^2\right ) \cosh (5 (c+d x)) \text {csch}^{12}(c+d x)+63 a \left (3421 a^2+12440 a b+14720 b^2\right ) \cosh (7 (c+d x)) \text {csch}^{12}(c+d x)-525 a \left (77 a^2+280 a b+384 b^2\right ) \cosh (9 (c+d x)) \text {csch}^{12}(c+d x)+45 a \left (77 a^2+280 a b+384 b^2\right ) \cosh (11 (c+d x)) \text {csch}^{12}(c+d x)+15360 \left (231 a^3+840 a^2 b+1152 a b^2+1024 b^3\right ) \log \left (\tanh \left (\frac {1}{2} (c+d x)\right )\right )}{15728640 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Csch[c + d*x]^13*(a + b*Sinh[c + d*x]^4)^3,x]

[Out]

(-30*a*(76555*a^2 + 75816*a*b + 45696*b^2)*Coth[c + d*x]*Csch[c + d*x]^11 + 2*a*(750629*a^2 + 2074200*a*b + 14
22720*b^2)*Cosh[3*(c + d*x)]*Csch[c + d*x]^12 - 9*a*(77099*a^2 + 280360*a*b + 246400*b^2)*Cosh[5*(c + d*x)]*Cs
ch[c + d*x]^12 + 63*a*(3421*a^2 + 12440*a*b + 14720*b^2)*Cosh[7*(c + d*x)]*Csch[c + d*x]^12 - 525*a*(77*a^2 +
280*a*b + 384*b^2)*Cosh[9*(c + d*x)]*Csch[c + d*x]^12 + 45*a*(77*a^2 + 280*a*b + 384*b^2)*Cosh[11*(c + d*x)]*C
sch[c + d*x]^12 + 15360*(231*a^3 + 840*a^2*b + 1152*a*b^2 + 1024*b^3)*Log[Tanh[(c + d*x)/2]])/(15728640*d)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(631\) vs. \(2(206)=412\).
time = 1.50, size = 632, normalized size = 2.87

method	result	size
risch	\(\frac {a \,{\mathrm e}^{d x +c} \left (12600 a b +3465 a^{2}-147000 a b \,{\mathrm e}^{2 d x +2 c}+17280 b^{2}-2274480 a b \,{\mathrm e}^{10 d x +10 c}-2523240 a b \,{\mathrm e}^{6 d x +6 c}+4148400 a b \,{\mathrm e}^{8 d x +8 c}+215523 a^{2} {\mathrm e}^{4 d x +4 c}+783720 a b \,{\mathrm e}^{18 d x +18 c}+12600 a b \,{\mathrm e}^{22 d x +22 c}-2523240 a b \,{\mathrm e}^{16 d x +16 c}+4148400 a b \,{\mathrm e}^{14 d x +14 c}-2274480 a b \,{\mathrm e}^{12 d x +12 c}-147000 a b \,{\mathrm e}^{20 d x +20 c}+783720 a b \,{\mathrm e}^{4 d x +4 c}-40425 a^{2} {\mathrm e}^{20 d x +20 c}-201600 b^{2} {\mathrm e}^{20 d x +20 c}+215523 a^{2} {\mathrm e}^{18 d x +18 c}+927360 b^{2} {\mathrm e}^{18 d x +18 c}-693891 a^{2} {\mathrm e}^{16 d x +16 c}-2217600 b^{2} {\mathrm e}^{16 d x +16 c}+1501258 a^{2} {\mathrm e}^{14 d x +14 c}+2845440 b^{2} {\mathrm e}^{14 d x +14 c}-2296650 a^{2} {\mathrm e}^{12 d x +12 c}+3465 a^{2} {\mathrm e}^{22 d x +22 c}+17280 b^{2} {\mathrm e}^{22 d x +22 c}-693891 a^{2} {\mathrm e}^{6 d x +6 c}+2845440 b^{2} {\mathrm e}^{8 d x +8 c}-201600 b^{2} {\mathrm e}^{2 d x +2 c}-2217600 b^{2} {\mathrm e}^{6 d x +6 c}+927360 b^{2} {\mathrm e}^{4 d x +4 c}-40425 a^{2} {\mathrm e}^{2 d x +2 c}-1370880 b^{2} {\mathrm e}^{12 d x +12 c}-2296650 a^{2} {\mathrm e}^{10 d x +10 c}-1370880 b^{2} {\mathrm e}^{10 d x +10 c}+1501258 a^{2} {\mathrm e}^{8 d x +8 c}\right )}{7680 d \left ({\mathrm e}^{2 d x +2 c}-1\right )^{12}}-\frac {231 a^{3} \ln \left ({\mathrm e}^{d x +c}+1\right )}{1024 d}-\frac {105 a^{2} b \ln \left ({\mathrm e}^{d x +c}+1\right )}{128 d}-\frac {9 a \ln \left ({\mathrm e}^{d x +c}+1\right ) b^{2}}{8 d}-\frac {\ln \left ({\mathrm e}^{d x +c}+1\right ) b^{3}}{d}+\frac {231 a^{3} \ln \left ({\mathrm e}^{d x +c}-1\right )}{1024 d}+\frac {105 a^{2} b \ln \left ({\mathrm e}^{d x +c}-1\right )}{128 d}+\frac {9 a \ln \left ({\mathrm e}^{d x +c}-1\right ) b^{2}}{8 d}+\frac {\ln \left ({\mathrm e}^{d x +c}-1\right ) b^{3}}{d}\)	\(632\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csch(d*x+c)^13*(a+b*sinh(d*x+c)^4)^3,x,method=_RETURNVERBOSE)

[Out]

1/7680*a*exp(d*x+c)*(12600*a*b+3465*a^2-147000*a*b*exp(2*d*x+2*c)+17280*b^2-2274480*a*b*exp(10*d*x+10*c)-25232
40*a*b*exp(6*d*x+6*c)+4148400*a*b*exp(8*d*x+8*c)+215523*a^2*exp(4*d*x+4*c)+783720*a*b*exp(18*d*x+18*c)+12600*a
*b*exp(22*d*x+22*c)-2523240*a*b*exp(16*d*x+16*c)+4148400*a*b*exp(14*d*x+14*c)-2274480*a*b*exp(12*d*x+12*c)-147
000*a*b*exp(20*d*x+20*c)+783720*a*b*exp(4*d*x+4*c)-40425*a^2*exp(20*d*x+20*c)-201600*b^2*exp(20*d*x+20*c)+2155
23*a^2*exp(18*d*x+18*c)+927360*b^2*exp(18*d*x+18*c)-693891*a^2*exp(16*d*x+16*c)-2217600*b^2*exp(16*d*x+16*c)+1
501258*a^2*exp(14*d*x+14*c)+2845440*b^2*exp(14*d*x+14*c)-2296650*a^2*exp(12*d*x+12*c)+3465*a^2*exp(22*d*x+22*c
)+17280*b^2*exp(22*d*x+22*c)-693891*a^2*exp(6*d*x+6*c)+2845440*b^2*exp(8*d*x+8*c)-201600*b^2*exp(2*d*x+2*c)-22
17600*b^2*exp(6*d*x+6*c)+927360*b^2*exp(4*d*x+4*c)-40425*a^2*exp(2*d*x+2*c)-1370880*b^2*exp(12*d*x+12*c)-22966
50*a^2*exp(10*d*x+10*c)-1370880*b^2*exp(10*d*x+10*c)+1501258*a^2*exp(8*d*x+8*c))/d/(exp(2*d*x+2*c)-1)^12-231/1
024*a^3/d*ln(exp(d*x+c)+1)-105/128*a^2*b/d*ln(exp(d*x+c)+1)-9/8*a/d*ln(exp(d*x+c)+1)*b^2-1/d*ln(exp(d*x+c)+1)*
b^3+231/1024*a^3/d*ln(exp(d*x+c)-1)+105/128*a^2*b/d*ln(exp(d*x+c)-1)+9/8*a/d*ln(exp(d*x+c)-1)*b^2+1/d*ln(exp(d
*x+c)-1)*b^3

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 720 vs. \(2 (206) = 412\).
time = 0.30, size = 720, normalized size = 3.27 \begin {gather*} -\frac {1}{15360} \, a^{3} {\left (\frac {3465 \, \log \left (e^{\left (-d x - c\right )} + 1\right )}{d} - \frac {3465 \, \log \left (e^{\left (-d x - c\right )} - 1\right )}{d} + \frac {2 \, {\left (3465 \, e^{\left (-d x - c\right )} - 40425 \, e^{\left (-3 \, d x - 3 \, c\right )} + 215523 \, e^{\left (-5 \, d x - 5 \, c\right )} - 693891 \, e^{\left (-7 \, d x - 7 \, c\right )} + 1501258 \, e^{\left (-9 \, d x - 9 \, c\right )} - 2296650 \, e^{\left (-11 \, d x - 11 \, c\right )} - 2296650 \, e^{\left (-13 \, d x - 13 \, c\right )} + 1501258 \, e^{\left (-15 \, d x - 15 \, c\right )} - 693891 \, e^{\left (-17 \, d x - 17 \, c\right )} + 215523 \, e^{\left (-19 \, d x - 19 \, c\right )} - 40425 \, e^{\left (-21 \, d x - 21 \, c\right )} + 3465 \, e^{\left (-23 \, d x - 23 \, c\right )}\right )}}{d {\left (12 \, e^{\left (-2 \, d x - 2 \, c\right )} - 66 \, e^{\left (-4 \, d x - 4 \, c\right )} + 220 \, e^{\left (-6 \, d x - 6 \, c\right )} - 495 \, e^{\left (-8 \, d x - 8 \, c\right )} + 792 \, e^{\left (-10 \, d x - 10 \, c\right )} - 924 \, e^{\left (-12 \, d x - 12 \, c\right )} + 792 \, e^{\left (-14 \, d x - 14 \, c\right )} - 495 \, e^{\left (-16 \, d x - 16 \, c\right )} + 220 \, e^{\left (-18 \, d x - 18 \, c\right )} - 66 \, e^{\left (-20 \, d x - 20 \, c\right )} + 12 \, e^{\left (-22 \, d x - 22 \, c\right )} - e^{\left (-24 \, d x - 24 \, c\right )} - 1\right )}}\right )} - \frac {1}{128} \, a^{2} b {\left (\frac {105 \, \log \left (e^{\left (-d x - c\right )} + 1\right )}{d} - \frac {105 \, \log \left (e^{\left (-d x - c\right )} - 1\right )}{d} + \frac {2 \, {\left (105 \, e^{\left (-d x - c\right )} - 805 \, e^{\left (-3 \, d x - 3 \, c\right )} + 2681 \, e^{\left (-5 \, d x - 5 \, c\right )} - 5053 \, e^{\left (-7 \, d x - 7 \, c\right )} - 5053 \, e^{\left (-9 \, d x - 9 \, c\right )} + 2681 \, e^{\left (-11 \, d x - 11 \, c\right )} - 805 \, e^{\left (-13 \, d x - 13 \, c\right )} + 105 \, e^{\left (-15 \, d x - 15 \, c\right )}\right )}}{d {\left (8 \, e^{\left (-2 \, d x - 2 \, c\right )} - 28 \, e^{\left (-4 \, d x - 4 \, c\right )} + 56 \, e^{\left (-6 \, d x - 6 \, c\right )} - 70 \, e^{\left (-8 \, d x - 8 \, c\right )} + 56 \, e^{\left (-10 \, d x - 10 \, c\right )} - 28 \, e^{\left (-12 \, d x - 12 \, c\right )} + 8 \, e^{\left (-14 \, d x - 14 \, c\right )} - e^{\left (-16 \, d x - 16 \, c\right )} - 1\right )}}\right )} - \frac {3}{8} \, a b^{2} {\left (\frac {3 \, \log \left (e^{\left (-d x - c\right )} + 1\right )}{d} - \frac {3 \, \log \left (e^{\left (-d x - c\right )} - 1\right )}{d} + \frac {2 \, {\left (3 \, e^{\left (-d x - c\right )} - 11 \, e^{\left (-3 \, d x - 3 \, c\right )} - 11 \, e^{\left (-5 \, d x - 5 \, c\right )} + 3 \, e^{\left (-7 \, d x - 7 \, c\right )}\right )}}{d {\left (4 \, e^{\left (-2 \, d x - 2 \, c\right )} - 6 \, e^{\left (-4 \, d x - 4 \, c\right )} + 4 \, e^{\left (-6 \, d x - 6 \, c\right )} - e^{\left (-8 \, d x - 8 \, c\right )} - 1\right )}}\right )} - b^{3} {\left (\frac {\log \left (e^{\left (-d x - c\right )} + 1\right )}{d} - \frac {\log \left (e^{\left (-d x - c\right )} - 1\right )}{d}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(d*x+c)^13*(a+b*sinh(d*x+c)^4)^3,x, algorithm="maxima")

[Out]

-1/15360*a^3*(3465*log(e^(-d*x - c) + 1)/d - 3465*log(e^(-d*x - c) - 1)/d + 2*(3465*e^(-d*x - c) - 40425*e^(-3
*d*x - 3*c) + 215523*e^(-5*d*x - 5*c) - 693891*e^(-7*d*x - 7*c) + 1501258*e^(-9*d*x - 9*c) - 2296650*e^(-11*d*
x - 11*c) - 2296650*e^(-13*d*x - 13*c) + 1501258*e^(-15*d*x - 15*c) - 693891*e^(-17*d*x - 17*c) + 215523*e^(-1
9*d*x - 19*c) - 40425*e^(-21*d*x - 21*c) + 3465*e^(-23*d*x - 23*c))/(d*(12*e^(-2*d*x - 2*c) - 66*e^(-4*d*x - 4
*c) + 220*e^(-6*d*x - 6*c) - 495*e^(-8*d*x - 8*c) + 792*e^(-10*d*x - 10*c) - 924*e^(-12*d*x - 12*c) + 792*e^(-
14*d*x - 14*c) - 495*e^(-16*d*x - 16*c) + 220*e^(-18*d*x - 18*c) - 66*e^(-20*d*x - 20*c) + 12*e^(-22*d*x - 22*
c) - e^(-24*d*x - 24*c) - 1))) - 1/128*a^2*b*(105*log(e^(-d*x - c) + 1)/d - 105*log(e^(-d*x - c) - 1)/d + 2*(1
05*e^(-d*x - c) - 805*e^(-3*d*x - 3*c) + 2681*e^(-5*d*x - 5*c) - 5053*e^(-7*d*x - 7*c) - 5053*e^(-9*d*x - 9*c)
+ 2681*e^(-11*d*x - 11*c) - 805*e^(-13*d*x - 13*c) + 105*e^(-15*d*x - 15*c))/(d*(8*e^(-2*d*x - 2*c) - 28*e^(-
4*d*x - 4*c) + 56*e^(-6*d*x - 6*c) - 70*e^(-8*d*x - 8*c) + 56*e^(-10*d*x - 10*c) - 28*e^(-12*d*x - 12*c) + 8*e
^(-14*d*x - 14*c) - e^(-16*d*x - 16*c) - 1))) - 3/8*a*b^2*(3*log(e^(-d*x - c) + 1)/d - 3*log(e^(-d*x - c) - 1)
/d + 2*(3*e^(-d*x - c) - 11*e^(-3*d*x - 3*c) - 11*e^(-5*d*x - 5*c) + 3*e^(-7*d*x - 7*c))/(d*(4*e^(-2*d*x - 2*c
) - 6*e^(-4*d*x - 4*c) + 4*e^(-6*d*x - 6*c) - e^(-8*d*x - 8*c) - 1))) - b^3*(log(e^(-d*x - c) + 1)/d - log(e^(
-d*x - c) - 1)/d)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 17811 vs. \(2 (206) = 412\).
time = 0.53, size = 17811, normalized size = 80.96 \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)^13*(a+b*sinh(d*x+c)^4)^3,x, algorithm="fricas")

[Out]

1/15360*(90*(77*a^3 + 280*a^2*b + 384*a*b^2)*cosh(d*x + c)^23 + 2070*(77*a^3 + 280*a^2*b + 384*a*b^2)*cosh(d*x
+ c)*sinh(d*x + c)^22 + 90*(77*a^3 + 280*a^2*b + 384*a*b^2)*sinh(d*x + c)^23 - 1050*(77*a^3 + 280*a^2*b + 384
*a*b^2)*cosh(d*x + c)^21 - 30*(2695*a^3 + 9800*a^2*b + 13440*a*b^2 - 759*(77*a^3 + 280*a^2*b + 384*a*b^2)*cosh
(d*x + c)^2)*sinh(d*x + c)^21 + 630*(253*(77*a^3 + 280*a^2*b + 384*a*b^2)*cosh(d*x + c)^3 - 35*(77*a^3 + 280*a
^2*b + 384*a*b^2)*cosh(d*x + c))*sinh(d*x + c)^20 + 126*(3421*a^3 + 12440*a^2*b + 14720*a*b^2)*cosh(d*x + c)^1
9 + 126*(6325*(77*a^3 + 280*a^2*b + 384*a*b^2)*cosh(d*x + c)^4 + 3421*a^3 + 12440*a^2*b + 14720*a*b^2 - 1750*(
77*a^3 + 280*a^2*b + 384*a*b^2)*cosh(d*x + c)^2)*sinh(d*x + c)^19 + 798*(3795*(77*a^3 + 280*a^2*b + 384*a*b^2)
*cosh(d*x + c)^5 - 1750*(77*a^3 + 280*a^2*b + 384*a*b^2)*cosh(d*x + c)^3 + 3*(3421*a^3 + 12440*a^2*b + 14720*a
*b^2)*cosh(d*x + c))*sinh(d*x + c)^18 - 18*(77099*a^3 + 280360*a^2*b + 246400*a*b^2)*cosh(d*x + c)^17 + 18*(50
4735*(77*a ...

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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)**13*(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. 537 vs. \(2 (206) = 412\).
time = 0.62, size = 537, normalized size = 2.44 \begin {gather*} -\frac {15 \, {\left (231 \, a^{3} + 840 \, a^{2} b + 1152 \, a b^{2} + 1024 \, b^{3}\right )} \log \left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )} + 2\right ) - 15 \, {\left (231 \, a^{3} + 840 \, a^{2} b + 1152 \, a b^{2} + 1024 \, b^{3}\right )} \log \left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )} - 2\right ) - \frac {4 \, {\left (3465 \, a^{3} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{11} + 12600 \, a^{2} b {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{11} + 17280 \, a b^{2} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{11} - 78540 \, a^{3} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{9} - 285600 \, a^{2} b {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{9} - 391680 \, a b^{2} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{9} + 731808 \, a^{3} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{7} + 2661120 \, a^{2} b {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{7} + 3502080 \, a b^{2} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{7} - 3560832 \, a^{3} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{5} - 12948480 \, a^{2} b {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{5} - 15482880 \, a b^{2} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{5} + 9391360 \, a^{3} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{3} + 32839680 \, a^{2} b {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{3} + 33914880 \, a b^{2} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{3} - 12180480 \, a^{3} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )} - 34283520 \, a^{2} b {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )} - 29491200 \, a b^{2} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}\right )}}{{\left ({\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{2} - 4\right )}^{6}}}{30720 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(d*x+c)^13*(a+b*sinh(d*x+c)^4)^3,x, algorithm="giac")

[Out]

-1/30720*(15*(231*a^3 + 840*a^2*b + 1152*a*b^2 + 1024*b^3)*log(e^(d*x + c) + e^(-d*x - c) + 2) - 15*(231*a^3 +
840*a^2*b + 1152*a*b^2 + 1024*b^3)*log(e^(d*x + c) + e^(-d*x - c) - 2) - 4*(3465*a^3*(e^(d*x + c) + e^(-d*x -
c))^11 + 12600*a^2*b*(e^(d*x + c) + e^(-d*x - c))^11 + 17280*a*b^2*(e^(d*x + c) + e^(-d*x - c))^11 - 78540*a^
3*(e^(d*x + c) + e^(-d*x - c))^9 - 285600*a^2*b*(e^(d*x + c) + e^(-d*x - c))^9 - 391680*a*b^2*(e^(d*x + c) + e
^(-d*x - c))^9 + 731808*a^3*(e^(d*x + c) + e^(-d*x - c))^7 + 2661120*a^2*b*(e^(d*x + c) + e^(-d*x - c))^7 + 35
02080*a*b^2*(e^(d*x + c) + e^(-d*x - c))^7 - 3560832*a^3*(e^(d*x + c) + e^(-d*x - c))^5 - 12948480*a^2*b*(e^(d
*x + c) + e^(-d*x - c))^5 - 15482880*a*b^2*(e^(d*x + c) + e^(-d*x - c))^5 + 9391360*a^3*(e^(d*x + c) + e^(-d*x
- c))^3 + 32839680*a^2*b*(e^(d*x + c) + e^(-d*x - c))^3 + 33914880*a*b^2*(e^(d*x + c) + e^(-d*x - c))^3 - 121
80480*a^3*(e^(d*x + c) + e^(-d*x - c)) - 34283520*a^2*b*(e^(d*x + c) + e^(-d*x - c)) - 29491200*a*b^2*(e^(d*x
+ c) + e^(-d*x - c)))/((e^(d*x + c) + e^(-d*x - c))^2 - 4)^6)/d

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Mupad [B]
time = 1.10, size = 1314, normalized size = 5.97 \begin {gather*} \frac {3\,{\mathrm {e}}^{c+d\,x}\,\left (77\,a^3+280\,a^2\,b+384\,a\,b^2\right )}{512\,d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}-1\right )}-\frac {5632\,a^3\,{\mathrm {e}}^{c+d\,x}}{3\,d\,\left (11\,{\mathrm {e}}^{2\,c+2\,d\,x}-55\,{\mathrm {e}}^{4\,c+4\,d\,x}+165\,{\mathrm {e}}^{6\,c+6\,d\,x}-330\,{\mathrm {e}}^{8\,c+8\,d\,x}+462\,{\mathrm {e}}^{10\,c+10\,d\,x}-462\,{\mathrm {e}}^{12\,c+12\,d\,x}+330\,{\mathrm {e}}^{14\,c+14\,d\,x}-165\,{\mathrm {e}}^{16\,c+16\,d\,x}+55\,{\mathrm {e}}^{18\,c+18\,d\,x}-11\,{\mathrm {e}}^{20\,c+20\,d\,x}+{\mathrm {e}}^{22\,c+22\,d\,x}-1\right )}-\frac {1024\,a^3\,{\mathrm {e}}^{c+d\,x}}{3\,d\,\left (66\,{\mathrm {e}}^{4\,c+4\,d\,x}-12\,{\mathrm {e}}^{2\,c+2\,d\,x}-220\,{\mathrm {e}}^{6\,c+6\,d\,x}+495\,{\mathrm {e}}^{8\,c+8\,d\,x}-792\,{\mathrm {e}}^{10\,c+10\,d\,x}+924\,{\mathrm {e}}^{12\,c+12\,d\,x}-792\,{\mathrm {e}}^{14\,c+14\,d\,x}+495\,{\mathrm {e}}^{16\,c+16\,d\,x}-220\,{\mathrm {e}}^{18\,c+18\,d\,x}+66\,{\mathrm {e}}^{20\,c+20\,d\,x}-12\,{\mathrm {e}}^{22\,c+22\,d\,x}+{\mathrm {e}}^{24\,c+24\,d\,x}+1\right )}-\frac {{\mathrm {e}}^{c+d\,x}\,\left (a^3+2424\,b\,a^2\right )}{6\,d\,\left (15\,{\mathrm {e}}^{4\,c+4\,d\,x}-6\,{\mathrm {e}}^{2\,c+2\,d\,x}-20\,{\mathrm {e}}^{6\,c+6\,d\,x}+15\,{\mathrm {e}}^{8\,c+8\,d\,x}-6\,{\mathrm {e}}^{10\,c+10\,d\,x}+{\mathrm {e}}^{12\,c+12\,d\,x}+1\right )}-\frac {\mathrm {atan}\left (\frac {{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\left (231\,a^3\,\sqrt {-d^2}+1024\,b^3\,\sqrt {-d^2}+1152\,a\,b^2\,\sqrt {-d^2}+840\,a^2\,b\,\sqrt {-d^2}\right )}{d\,\sqrt {53361\,a^6+388080\,a^5\,b+1237824\,a^4\,b^2+2408448\,a^3\,b^3+3047424\,a^2\,b^4+2359296\,a\,b^5+1048576\,b^6}}\right )\,\sqrt {53361\,a^6+388080\,a^5\,b+1237824\,a^4\,b^2+2408448\,a^3\,b^3+3047424\,a^2\,b^4+2359296\,a\,b^5+1048576\,b^6}}{512\,\sqrt {-d^2}}-\frac {{\mathrm {e}}^{c+d\,x}\,\left (10200\,a^2\,b-11\,a^3\right )}{60\,d\,\left (5\,{\mathrm {e}}^{2\,c+2\,d\,x}-10\,{\mathrm {e}}^{4\,c+4\,d\,x}+10\,{\mathrm {e}}^{6\,c+6\,d\,x}-5\,{\mathrm {e}}^{8\,c+8\,d\,x}+{\mathrm {e}}^{10\,c+10\,d\,x}-1\right )}-\frac {{\mathrm {e}}^{c+d\,x}\,\left (77\,a^3+280\,a^2\,b+384\,a\,b^2\right )}{256\,d\,\left ({\mathrm {e}}^{4\,c+4\,d\,x}-2\,{\mathrm {e}}^{2\,c+2\,d\,x}+1\right )}+\frac {{\mathrm {e}}^{c+d\,x}\,\left (77\,a^3+280\,a^2\,b-5760\,a\,b^2\right )}{320\,d\,\left (3\,{\mathrm {e}}^{2\,c+2\,d\,x}-3\,{\mathrm {e}}^{4\,c+4\,d\,x}+{\mathrm {e}}^{6\,c+6\,d\,x}-1\right )}-\frac {42\,{\mathrm {e}}^{c+d\,x}\,\left (15\,a^3+8\,b\,a^2\right )}{d\,\left (7\,{\mathrm {e}}^{2\,c+2\,d\,x}-21\,{\mathrm {e}}^{4\,c+4\,d\,x}+35\,{\mathrm {e}}^{6\,c+6\,d\,x}-35\,{\mathrm {e}}^{8\,c+8\,d\,x}+21\,{\mathrm {e}}^{10\,c+10\,d\,x}-7\,{\mathrm {e}}^{12\,c+12\,d\,x}+{\mathrm {e}}^{14\,c+14\,d\,x}-1\right )}-\frac {23488\,a^3\,{\mathrm {e}}^{c+d\,x}}{5\,d\,\left (9\,{\mathrm {e}}^{2\,c+2\,d\,x}-36\,{\mathrm {e}}^{4\,c+4\,d\,x}+84\,{\mathrm {e}}^{6\,c+6\,d\,x}-126\,{\mathrm {e}}^{8\,c+8\,d\,x}+126\,{\mathrm {e}}^{10\,c+10\,d\,x}-84\,{\mathrm {e}}^{12\,c+12\,d\,x}+36\,{\mathrm {e}}^{14\,c+14\,d\,x}-9\,{\mathrm {e}}^{16\,c+16\,d\,x}+{\mathrm {e}}^{18\,c+18\,d\,x}-1\right )}-\frac {3\,{\mathrm {e}}^{c+d\,x}\,\left (11\,a^3+40\,a^2\,b+640\,a\,b^2\right )}{160\,d\,\left (6\,{\mathrm {e}}^{4\,c+4\,d\,x}-4\,{\mathrm {e}}^{2\,c+2\,d\,x}-4\,{\mathrm {e}}^{6\,c+6\,d\,x}+{\mathrm {e}}^{8\,c+8\,d\,x}+1\right )}-\frac {4\,{\mathrm {e}}^{c+d\,x}\,\left (3361\,a^3+120\,b\,a^2\right )}{5\,d\,\left (28\,{\mathrm {e}}^{4\,c+4\,d\,x}-8\,{\mathrm {e}}^{2\,c+2\,d\,x}-56\,{\mathrm {e}}^{6\,c+6\,d\,x}+70\,{\mathrm {e}}^{8\,c+8\,d\,x}-56\,{\mathrm {e}}^{10\,c+10\,d\,x}+28\,{\mathrm {e}}^{12\,c+12\,d\,x}-8\,{\mathrm {e}}^{14\,c+14\,d\,x}+{\mathrm {e}}^{16\,c+16\,d\,x}+1\right )}-\frac {20864\,a^3\,{\mathrm {e}}^{c+d\,x}}{5\,d\,\left (45\,{\mathrm {e}}^{4\,c+4\,d\,x}-10\,{\mathrm {e}}^{2\,c+2\,d\,x}-120\,{\mathrm {e}}^{6\,c+6\,d\,x}+210\,{\mathrm {e}}^{8\,c+8\,d\,x}-252\,{\mathrm {e}}^{10\,c+10\,d\,x}+210\,{\mathrm {e}}^{12\,c+12\,d\,x}-120\,{\mathrm {e}}^{14\,c+14\,d\,x}+45\,{\mathrm {e}}^{16\,c+16\,d\,x}-10\,{\mathrm {e}}^{18\,c+18\,d\,x}+{\mathrm {e}}^{20\,c+20\,d\,x}+1\right )} \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)^13,x)

[Out]

(3*exp(c + d*x)*(384*a*b^2 + 280*a^2*b + 77*a^3))/(512*d*(exp(2*c + 2*d*x) - 1)) - (5632*a^3*exp(c + d*x))/(3*
d*(11*exp(2*c + 2*d*x) - 55*exp(4*c + 4*d*x) + 165*exp(6*c + 6*d*x) - 330*exp(8*c + 8*d*x) + 462*exp(10*c + 10
*d*x) - 462*exp(12*c + 12*d*x) + 330*exp(14*c + 14*d*x) - 165*exp(16*c + 16*d*x) + 55*exp(18*c + 18*d*x) - 11*
exp(20*c + 20*d*x) + exp(22*c + 22*d*x) - 1)) - (1024*a^3*exp(c + d*x))/(3*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) - 79
2*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)) - (exp(c + d*x)*(2424*a^2*b + a^3))/(6*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)) - (atan
((exp(d*x)*exp(c)*(231*a^3*(-d^2)^(1/2) + 1024*b^3*(-d^2)^(1/2) + 1152*a*b^2*(-d^2)^(1/2) + 840*a^2*b*(-d^2)^(
1/2)))/(d*(2359296*a*b^5 + 388080*a^5*b + 53361*a^6 + 1048576*b^6 + 3047424*a^2*b^4 + 2408448*a^3*b^3 + 123782
4*a^4*b^2)^(1/2)))*(2359296*a*b^5 + 388080*a^5*b + 53361*a^6 + 1048576*b^6 + 3047424*a^2*b^4 + 2408448*a^3*b^3
+ 1237824*a^4*b^2)^(1/2))/(512*(-d^2)^(1/2)) - (exp(c + d*x)*(10200*a^2*b - 11*a^3))/(60*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)) - (exp(c + d*x)*
(384*a*b^2 + 280*a^2*b + 77*a^3))/(256*d*(exp(4*c + 4*d*x) - 2*exp(2*c + 2*d*x) + 1)) + (exp(c + d*x)*(280*a^2
*b - 5760*a*b^2 + 77*a^3))/(320*d*(3*exp(2*c + 2*d*x) - 3*exp(4*c + 4*d*x) + exp(6*c + 6*d*x) - 1)) - (42*exp(
c + d*x)*(8*a^2*b + 15*a^3))/(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)) - (23488*a^3*exp(c + d*x))/(
5*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)) - (3*e
xp(c + d*x)*(640*a*b^2 + 40*a^2*b + 11*a^3))/(160*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)) - (4*exp(c + d*x)*(120*a^2*b + 3361*a^3))/(5*d*(28*exp(4*c + 4*d*x) - 8*exp(2*c +
2*d*x) - 56*exp(6*c + 6*d*x) + 70*exp(8*c + 8*d*x) - 56*exp(10*c + 10*d*x) + 28*exp(12*c + 12*d*x) - 8*exp(14
*c + 14*d*x) + exp(16*c + 16*d*x) + 1)) - (20864*a^3*exp(c + d*x))/(5*d*(45*exp(4*c + 4*d*x) - 10*exp(2*c + 2*
d*x) - 120*exp(6*c + 6*d*x) + 210*exp(8*c + 8*d*x) - 252*exp(10*c + 10*d*x) + 210*exp(12*c + 12*d*x) - 120*exp
(14*c + 14*d*x) + 45*exp(16*c + 16*d*x) - 10*exp(18*c + 18*d*x) + exp(20*c + 20*d*x) + 1))

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