3.3.23 \(\int \text {csch}^{10}(c+d x) (a+b \sinh ^4(c+d x))^3 \, dx\) [223]

Optimal. Leaf size=140 \[ -\frac {b^3 x}{2}-\frac {a \left (a^2+3 a b+3 b^2\right ) \coth (c+d x)}{d}+\frac {2 a^2 (2 a+3 b) \coth ^3(c+d x)}{3 d}-\frac {3 a^2 (2 a+b) \coth ^5(c+d x)}{5 d}+\frac {4 a^3 \coth ^7(c+d x)}{7 d}-\frac {a^3 \coth ^9(c+d x)}{9 d}+\frac {b^3 \cosh (c+d x) \sinh (c+d x)}{2 d} \]

[Out]

-1/2*b^3*x-a*(a^2+3*a*b+3*b^2)*coth(d*x+c)/d+2/3*a^2*(2*a+3*b)*coth(d*x+c)^3/d-3/5*a^2*(2*a+b)*coth(d*x+c)^5/d
+4/7*a^3*coth(d*x+c)^7/d-1/9*a^3*coth(d*x+c)^9/d+1/2*b^3*cosh(d*x+c)*sinh(d*x+c)/d

________________________________________________________________________________________

Rubi [A]
time = 0.15, antiderivative size = 140, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {3296, 1273, 1816, 213} \begin {gather*} -\frac {a^3 \coth ^9(c+d x)}{9 d}+\frac {4 a^3 \coth ^7(c+d x)}{7 d}-\frac {a \left (a^2+3 a b+3 b^2\right ) \coth (c+d x)}{d}-\frac {3 a^2 (2 a+b) \coth ^5(c+d x)}{5 d}+\frac {2 a^2 (2 a+3 b) \coth ^3(c+d x)}{3 d}+\frac {b^3 \sinh (c+d x) \cosh (c+d x)}{2 d}-\frac {b^3 x}{2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Csch[c + d*x]^10*(a + b*Sinh[c + d*x]^4)^3,x]

[Out]

-1/2*(b^3*x) - (a*(a^2 + 3*a*b + 3*b^2)*Coth[c + d*x])/d + (2*a^2*(2*a + 3*b)*Coth[c + d*x]^3)/(3*d) - (3*a^2*
(2*a + b)*Coth[c + d*x]^5)/(5*d) + (4*a^3*Coth[c + d*x]^7)/(7*d) - (a^3*Coth[c + d*x]^9)/(9*d) + (b^3*Cosh[c +
d*x]*Sinh[c + d*x])/(2*d)

Rule 213

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

Rule 1273

Int[(x_)^(m_)*((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Simp[(-d)^(m
/2 - 1)*(c*d^2 - b*d*e + a*e^2)^p*x*((d + e*x^2)^(q + 1)/(2*e^(2*p + m/2)*(q + 1))), x] + Dist[(-d)^(m/2 - 1)/
(2*e^(2*p)*(q + 1)), Int[x^m*(d + e*x^2)^(q + 1)*ExpandToSum[Together[(1/(d + e*x^2))*(2*(-d)^(-m/2 + 1)*e^(2*
p)*(q + 1)*(a + b*x^2 + c*x^4)^p - ((c*d^2 - b*d*e + a*e^2)^p/(e^(m/2)*x^m))*(d + e*(2*q + 3)*x^2))], x], x],
x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && IGtQ[p, 0] && ILtQ[q, -1] && ILtQ[m/2, 0]

Rule 1816

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

Rule 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}^{10}(c+d x) \left (a+b \sinh ^4(c+d x)\right )^3 \, dx &=\frac {\text {Subst}\left (\int \frac {\left (a-2 a x^2+(a+b) x^4\right )^3}{x^{10} \left (1-x^2\right )^2} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac {b^3 \cosh (c+d x) \sinh (c+d x)}{2 d}-\frac {\text {Subst}\left (\int \frac {-2 a^3+10 a^3 x^2-2 a^2 (10 a+3 b) x^4+2 a^2 (10 a+9 b) x^6-2 a \left (5 a^2+9 a b+3 b^2\right ) x^8+\left (2 a^3+6 a^2 b+6 a b^2+b^3\right ) x^{10}}{x^{10} \left (1-x^2\right )} \, dx,x,\tanh (c+d x)\right )}{2 d}\\ &=\frac {b^3 \cosh (c+d x) \sinh (c+d x)}{2 d}-\frac {\text {Subst}\left (\int \left (-\frac {2 a^3}{x^{10}}+\frac {8 a^3}{x^8}-\frac {6 a^2 (2 a+b)}{x^6}+\frac {4 a^2 (2 a+3 b)}{x^4}-\frac {2 a \left (a^2+3 a b+3 b^2\right )}{x^2}-\frac {b^3}{-1+x^2}\right ) \, dx,x,\tanh (c+d x)\right )}{2 d}\\ &=-\frac {a \left (a^2+3 a b+3 b^2\right ) \coth (c+d x)}{d}+\frac {2 a^2 (2 a+3 b) \coth ^3(c+d x)}{3 d}-\frac {3 a^2 (2 a+b) \coth ^5(c+d x)}{5 d}+\frac {4 a^3 \coth ^7(c+d x)}{7 d}-\frac {a^3 \coth ^9(c+d x)}{9 d}+\frac {b^3 \cosh (c+d x) \sinh (c+d x)}{2 d}+\frac {b^3 \text {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\tanh (c+d x)\right )}{2 d}\\ &=-\frac {b^3 x}{2}-\frac {a \left (a^2+3 a b+3 b^2\right ) \coth (c+d x)}{d}+\frac {2 a^2 (2 a+3 b) \coth ^3(c+d x)}{3 d}-\frac {3 a^2 (2 a+b) \coth ^5(c+d x)}{5 d}+\frac {4 a^3 \coth ^7(c+d x)}{7 d}-\frac {a^3 \coth ^9(c+d x)}{9 d}+\frac {b^3 \cosh (c+d x) \sinh (c+d x)}{2 d}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.45, size = 115, normalized size = 0.82 \begin {gather*} \frac {-4 a \coth (c+d x) \left (128 a^2+504 a b+945 b^2-4 a (16 a+63 b) \text {csch}^2(c+d x)+3 a (16 a+63 b) \text {csch}^4(c+d x)-40 a^2 \text {csch}^6(c+d x)+35 a^2 \text {csch}^8(c+d x)\right )+315 b^3 (-2 (c+d x)+\sinh (2 (c+d x)))}{1260 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Csch[c + d*x]^10*(a + b*Sinh[c + d*x]^4)^3,x]

[Out]

(-4*a*Coth[c + d*x]*(128*a^2 + 504*a*b + 945*b^2 - 4*a*(16*a + 63*b)*Csch[c + d*x]^2 + 3*a*(16*a + 63*b)*Csch[
c + d*x]^4 - 40*a^2*Csch[c + d*x]^6 + 35*a^2*Csch[c + d*x]^8) + 315*b^3*(-2*(c + d*x) + Sinh[2*(c + d*x)]))/(1
260*d)

________________________________________________________________________________________

Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(321\) vs. \(2(128)=256\).
time = 1.57, size = 322, normalized size = 2.30

method	result	size
risch	\(-\frac {b^{3} x}{2}+\frac {b^{3} {\mathrm e}^{2 d x +2 c}}{8 d}-\frac {b^{3} {\mathrm e}^{-2 d x -2 c}}{8 d}-\frac {2 a \left (945 b^{2} {\mathrm e}^{16 d x +16 c}-7560 b^{2} {\mathrm e}^{14 d x +14 c}+5040 a b \,{\mathrm e}^{12 d x +12 c}+26460 b^{2} {\mathrm e}^{12 d x +12 c}-22680 a b \,{\mathrm e}^{10 d x +10 c}-52920 b^{2} {\mathrm e}^{10 d x +10 c}+16128 a^{2} {\mathrm e}^{8 d x +8 c}+40824 a b \,{\mathrm e}^{8 d x +8 c}+66150 b^{2} {\mathrm e}^{8 d x +8 c}-10752 a^{2} {\mathrm e}^{6 d x +6 c}-37296 a b \,{\mathrm e}^{6 d x +6 c}-52920 b^{2} {\mathrm e}^{6 d x +6 c}+4608 a^{2} {\mathrm e}^{4 d x +4 c}+18144 a b \,{\mathrm e}^{4 d x +4 c}+26460 b^{2} {\mathrm e}^{4 d x +4 c}-1152 a^{2} {\mathrm e}^{2 d x +2 c}-4536 a b \,{\mathrm e}^{2 d x +2 c}-7560 b^{2} {\mathrm e}^{2 d x +2 c}+128 a^{2}+504 a b +945 b^{2}\right )}{315 d \left ({\mathrm e}^{2 d x +2 c}-1\right )^{9}}\)	\(322\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csch(d*x+c)^10*(a+b*sinh(d*x+c)^4)^3,x,method=_RETURNVERBOSE)

[Out]

-1/2*b^3*x+1/8*b^3/d*exp(2*d*x+2*c)-1/8*b^3/d*exp(-2*d*x-2*c)-2/315*a*(945*b^2*exp(16*d*x+16*c)-7560*b^2*exp(1
4*d*x+14*c)+5040*a*b*exp(12*d*x+12*c)+26460*b^2*exp(12*d*x+12*c)-22680*a*b*exp(10*d*x+10*c)-52920*b^2*exp(10*d
*x+10*c)+16128*a^2*exp(8*d*x+8*c)+40824*a*b*exp(8*d*x+8*c)+66150*b^2*exp(8*d*x+8*c)-10752*a^2*exp(6*d*x+6*c)-3
7296*a*b*exp(6*d*x+6*c)-52920*b^2*exp(6*d*x+6*c)+4608*a^2*exp(4*d*x+4*c)+18144*a*b*exp(4*d*x+4*c)+26460*b^2*ex
p(4*d*x+4*c)-1152*a^2*exp(2*d*x+2*c)-4536*a*b*exp(2*d*x+2*c)-7560*b^2*exp(2*d*x+2*c)+128*a^2+504*a*b+945*b^2)/
d/(exp(2*d*x+2*c)-1)^9

________________________________________________________________________________________

Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 842 vs. \(2 (128) = 256\).
time = 0.27, size = 842, normalized size = 6.01 \begin {gather*} -\frac {1}{8} \, b^{3} {\left (4 \, x - \frac {e^{\left (2 \, d x + 2 \, c\right )}}{d} + \frac {e^{\left (-2 \, d x - 2 \, c\right )}}{d}\right )} - \frac {256}{315} \, a^{3} {\left (\frac {9 \, e^{\left (-2 \, d x - 2 \, c\right )}}{d {\left (9 \, e^{\left (-2 \, d x - 2 \, c\right )} - 36 \, e^{\left (-4 \, d x - 4 \, c\right )} + 84 \, e^{\left (-6 \, d x - 6 \, c\right )} - 126 \, e^{\left (-8 \, d x - 8 \, c\right )} + 126 \, e^{\left (-10 \, d x - 10 \, c\right )} - 84 \, e^{\left (-12 \, d x - 12 \, c\right )} + 36 \, e^{\left (-14 \, d x - 14 \, c\right )} - 9 \, e^{\left (-16 \, d x - 16 \, c\right )} + e^{\left (-18 \, d x - 18 \, c\right )} - 1\right )}} - \frac {36 \, e^{\left (-4 \, d x - 4 \, c\right )}}{d {\left (9 \, e^{\left (-2 \, d x - 2 \, c\right )} - 36 \, e^{\left (-4 \, d x - 4 \, c\right )} + 84 \, e^{\left (-6 \, d x - 6 \, c\right )} - 126 \, e^{\left (-8 \, d x - 8 \, c\right )} + 126 \, e^{\left (-10 \, d x - 10 \, c\right )} - 84 \, e^{\left (-12 \, d x - 12 \, c\right )} + 36 \, e^{\left (-14 \, d x - 14 \, c\right )} - 9 \, e^{\left (-16 \, d x - 16 \, c\right )} + e^{\left (-18 \, d x - 18 \, c\right )} - 1\right )}} + \frac {84 \, e^{\left (-6 \, d x - 6 \, c\right )}}{d {\left (9 \, e^{\left (-2 \, d x - 2 \, c\right )} - 36 \, e^{\left (-4 \, d x - 4 \, c\right )} + 84 \, e^{\left (-6 \, d x - 6 \, c\right )} - 126 \, e^{\left (-8 \, d x - 8 \, c\right )} + 126 \, e^{\left (-10 \, d x - 10 \, c\right )} - 84 \, e^{\left (-12 \, d x - 12 \, c\right )} + 36 \, e^{\left (-14 \, d x - 14 \, c\right )} - 9 \, e^{\left (-16 \, d x - 16 \, c\right )} + e^{\left (-18 \, d x - 18 \, c\right )} - 1\right )}} - \frac {126 \, e^{\left (-8 \, d x - 8 \, c\right )}}{d {\left (9 \, e^{\left (-2 \, d x - 2 \, c\right )} - 36 \, e^{\left (-4 \, d x - 4 \, c\right )} + 84 \, e^{\left (-6 \, d x - 6 \, c\right )} - 126 \, e^{\left (-8 \, d x - 8 \, c\right )} + 126 \, e^{\left (-10 \, d x - 10 \, c\right )} - 84 \, e^{\left (-12 \, d x - 12 \, c\right )} + 36 \, e^{\left (-14 \, d x - 14 \, c\right )} - 9 \, e^{\left (-16 \, d x - 16 \, c\right )} + e^{\left (-18 \, d x - 18 \, c\right )} - 1\right )}} - \frac {1}{d {\left (9 \, e^{\left (-2 \, d x - 2 \, c\right )} - 36 \, e^{\left (-4 \, d x - 4 \, c\right )} + 84 \, e^{\left (-6 \, d x - 6 \, c\right )} - 126 \, e^{\left (-8 \, d x - 8 \, c\right )} + 126 \, e^{\left (-10 \, d x - 10 \, c\right )} - 84 \, e^{\left (-12 \, d x - 12 \, c\right )} + 36 \, e^{\left (-14 \, d x - 14 \, c\right )} - 9 \, e^{\left (-16 \, d x - 16 \, c\right )} + e^{\left (-18 \, d x - 18 \, c\right )} - 1\right )}}\right )} - \frac {16}{5} \, a^{2} b {\left (\frac {5 \, e^{\left (-2 \, d x - 2 \, c\right )}}{d {\left (5 \, e^{\left (-2 \, d x - 2 \, c\right )} - 10 \, e^{\left (-4 \, d x - 4 \, c\right )} + 10 \, e^{\left (-6 \, d x - 6 \, c\right )} - 5 \, e^{\left (-8 \, d x - 8 \, c\right )} + e^{\left (-10 \, d x - 10 \, c\right )} - 1\right )}} - \frac {10 \, e^{\left (-4 \, d x - 4 \, c\right )}}{d {\left (5 \, e^{\left (-2 \, d x - 2 \, c\right )} - 10 \, e^{\left (-4 \, d x - 4 \, c\right )} + 10 \, e^{\left (-6 \, d x - 6 \, c\right )} - 5 \, e^{\left (-8 \, d x - 8 \, c\right )} + e^{\left (-10 \, d x - 10 \, c\right )} - 1\right )}} - \frac {1}{d {\left (5 \, e^{\left (-2 \, d x - 2 \, c\right )} - 10 \, e^{\left (-4 \, d x - 4 \, c\right )} + 10 \, e^{\left (-6 \, d x - 6 \, c\right )} - 5 \, e^{\left (-8 \, d x - 8 \, c\right )} + e^{\left (-10 \, d x - 10 \, c\right )} - 1\right )}}\right )} + \frac {6 \, a b^{2}}{d {\left (e^{\left (-2 \, d x - 2 \, c\right )} - 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(d*x+c)^10*(a+b*sinh(d*x+c)^4)^3,x, algorithm="maxima")

[Out]

-1/8*b^3*(4*x - e^(2*d*x + 2*c)/d + e^(-2*d*x - 2*c)/d) - 256/315*a^3*(9*e^(-2*d*x - 2*c)/(d*(9*e^(-2*d*x - 2*
c) - 36*e^(-4*d*x - 4*c) + 84*e^(-6*d*x - 6*c) - 126*e^(-8*d*x - 8*c) + 126*e^(-10*d*x - 10*c) - 84*e^(-12*d*x
- 12*c) + 36*e^(-14*d*x - 14*c) - 9*e^(-16*d*x - 16*c) + e^(-18*d*x - 18*c) - 1)) - 36*e^(-4*d*x - 4*c)/(d*(9
*e^(-2*d*x - 2*c) - 36*e^(-4*d*x - 4*c) + 84*e^(-6*d*x - 6*c) - 126*e^(-8*d*x - 8*c) + 126*e^(-10*d*x - 10*c)
- 84*e^(-12*d*x - 12*c) + 36*e^(-14*d*x - 14*c) - 9*e^(-16*d*x - 16*c) + e^(-18*d*x - 18*c) - 1)) + 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))) - 16/5*a^2*b*(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))) + 6*a*b^2/(d*(e^(-
2*d*x - 2*c) - 1))

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1314 vs. \(2 (128) = 256\).
time = 0.40, size = 1314, normalized size = 9.39 \begin {gather*} \frac {315 \, b^{3} \cosh \left (d x + c\right )^{11} + 3465 \, b^{3} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{10} - {\left (1024 \, a^{3} + 4032 \, a^{2} b + 7560 \, a b^{2} + 2835 \, b^{3}\right )} \cosh \left (d x + c\right )^{9} - 4 \, {\left (315 \, b^{3} d x - 256 \, a^{3} - 1008 \, a^{2} b - 1890 \, a b^{2}\right )} \sinh \left (d x + c\right )^{9} + 9 \, {\left (5775 \, b^{3} \cosh \left (d x + c\right )^{3} - {\left (1024 \, a^{3} + 4032 \, a^{2} b + 7560 \, a b^{2} + 2835 \, b^{3}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{8} + 9 \, {\left (1024 \, a^{3} + 4032 \, a^{2} b + 5880 \, a b^{2} + 1225 \, b^{3}\right )} \cosh \left (d x + c\right )^{7} + 36 \, {\left (315 \, b^{3} d x - 256 \, a^{3} - 1008 \, a^{2} b - 1890 \, a b^{2} - 4 \, {\left (315 \, b^{3} d x - 256 \, a^{3} - 1008 \, a^{2} b - 1890 \, a b^{2}\right )} \cosh \left (d x + c\right )^{2}\right )} \sinh \left (d x + c\right )^{7} + 21 \, {\left (6930 \, b^{3} \cosh \left (d x + c\right )^{5} - 4 \, {\left (1024 \, a^{3} + 4032 \, a^{2} b + 7560 \, a b^{2} + 2835 \, b^{3}\right )} \cosh \left (d x + c\right )^{3} + 3 \, {\left (1024 \, a^{3} + 4032 \, a^{2} b + 5880 \, a b^{2} + 1225 \, b^{3}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{6} - 9 \, {\left (4096 \, a^{3} + 16128 \, a^{2} b + 16800 \, a b^{2} + 2625 \, b^{3}\right )} \cosh \left (d x + c\right )^{5} - 36 \, {\left (1260 \, b^{3} d x + 14 \, {\left (315 \, b^{3} d x - 256 \, a^{3} - 1008 \, a^{2} b - 1890 \, a b^{2}\right )} \cosh \left (d x + c\right )^{4} - 1024 \, a^{3} - 4032 \, a^{2} b - 7560 \, a b^{2} - 21 \, {\left (315 \, b^{3} d x - 256 \, a^{3} - 1008 \, a^{2} b - 1890 \, a b^{2}\right )} \cosh \left (d x + c\right )^{2}\right )} \sinh \left (d x + c\right )^{5} + 9 \, {\left (11550 \, b^{3} \cosh \left (d x + c\right )^{7} - 14 \, {\left (1024 \, a^{3} + 4032 \, a^{2} b + 7560 \, a b^{2} + 2835 \, b^{3}\right )} \cosh \left (d x + c\right )^{5} + 35 \, {\left (1024 \, a^{3} + 4032 \, a^{2} b + 5880 \, a b^{2} + 1225 \, b^{3}\right )} \cosh \left (d x + c\right )^{3} - 5 \, {\left (4096 \, a^{3} + 16128 \, a^{2} b + 16800 \, a b^{2} + 2625 \, b^{3}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{4} + 42 \, {\left (2048 \, a^{3} + 6144 \, a^{2} b + 5040 \, a b^{2} + 675 \, b^{3}\right )} \cosh \left (d x + c\right )^{3} - 12 \, {\left (28 \, {\left (315 \, b^{3} d x - 256 \, a^{3} - 1008 \, a^{2} b - 1890 \, a b^{2}\right )} \cosh \left (d x + c\right )^{6} - 8820 \, b^{3} d x - 105 \, {\left (315 \, b^{3} d x - 256 \, a^{3} - 1008 \, a^{2} b - 1890 \, a b^{2}\right )} \cosh \left (d x + c\right )^{4} + 7168 \, a^{3} + 28224 \, a^{2} b + 52920 \, a b^{2} + 120 \, {\left (315 \, b^{3} d x - 256 \, a^{3} - 1008 \, a^{2} b - 1890 \, a b^{2}\right )} \cosh \left (d x + c\right )^{2}\right )} \sinh \left (d x + c\right )^{3} + 9 \, {\left (1925 \, b^{3} \cosh \left (d x + c\right )^{9} - 4 \, {\left (1024 \, a^{3} + 4032 \, a^{2} b + 7560 \, a b^{2} + 2835 \, b^{3}\right )} \cosh \left (d x + c\right )^{7} + 21 \, {\left (1024 \, a^{3} + 4032 \, a^{2} b + 5880 \, a b^{2} + 1225 \, b^{3}\right )} \cosh \left (d x + c\right )^{5} - 10 \, {\left (4096 \, a^{3} + 16128 \, a^{2} b + 16800 \, a b^{2} + 2625 \, b^{3}\right )} \cosh \left (d x + c\right )^{3} + 14 \, {\left (2048 \, a^{3} + 6144 \, a^{2} b + 5040 \, a b^{2} + 675 \, b^{3}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{2} - 126 \, {\left (1024 \, a^{3} + 1152 \, a^{2} b + 840 \, a b^{2} + 105 \, b^{3}\right )} \cosh \left (d x + c\right ) - 36 \, {\left ({\left (315 \, b^{3} d x - 256 \, a^{3} - 1008 \, a^{2} b - 1890 \, a b^{2}\right )} \cosh \left (d x + c\right )^{8} - 7 \, {\left (315 \, b^{3} d x - 256 \, a^{3} - 1008 \, a^{2} b - 1890 \, a b^{2}\right )} \cosh \left (d x + c\right )^{6} + 4410 \, b^{3} d x + 20 \, {\left (315 \, b^{3} d x - 256 \, a^{3} - 1008 \, a^{2} b - 1890 \, a b^{2}\right )} \cosh \left (d x + c\right )^{4} - 3584 \, a^{3} - 14112 \, a^{2} b - 26460 \, a b^{2} - 28 \, {\left (315 \, b^{3} d x - 256 \, a^{3} - 1008 \, a^{2} b - 1890 \, a b^{2}\right )} \cosh \left (d x + c\right )^{2}\right )} \sinh \left (d x + c\right )}{2520 \, {\left (d \sinh \left (d x + c\right )^{9} + 9 \, {\left (4 \, d \cosh \left (d x + c\right )^{2} - d\right )} \sinh \left (d x + c\right )^{7} + 9 \, {\left (14 \, d \cosh \left (d x + c\right )^{4} - 21 \, d \cosh \left (d x + c\right )^{2} + 4 \, d\right )} \sinh \left (d x + c\right )^{5} + 3 \, {\left (28 \, d \cosh \left (d x + c\right )^{6} - 105 \, d \cosh \left (d x + c\right )^{4} + 120 \, d \cosh \left (d x + c\right )^{2} - 28 \, d\right )} \sinh \left (d x + c\right )^{3} + 9 \, {\left (d \cosh \left (d x + c\right )^{8} - 7 \, d \cosh \left (d x + c\right )^{6} + 20 \, d \cosh \left (d x + c\right )^{4} - 28 \, d \cosh \left (d x + c\right )^{2} + 14 \, d\right )} \sinh \left (d x + c\right )\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(d*x+c)^10*(a+b*sinh(d*x+c)^4)^3,x, algorithm="fricas")

[Out]

1/2520*(315*b^3*cosh(d*x + c)^11 + 3465*b^3*cosh(d*x + c)*sinh(d*x + c)^10 - (1024*a^3 + 4032*a^2*b + 7560*a*b
^2 + 2835*b^3)*cosh(d*x + c)^9 - 4*(315*b^3*d*x - 256*a^3 - 1008*a^2*b - 1890*a*b^2)*sinh(d*x + c)^9 + 9*(5775
*b^3*cosh(d*x + c)^3 - (1024*a^3 + 4032*a^2*b + 7560*a*b^2 + 2835*b^3)*cosh(d*x + c))*sinh(d*x + c)^8 + 9*(102
4*a^3 + 4032*a^2*b + 5880*a*b^2 + 1225*b^3)*cosh(d*x + c)^7 + 36*(315*b^3*d*x - 256*a^3 - 1008*a^2*b - 1890*a*
b^2 - 4*(315*b^3*d*x - 256*a^3 - 1008*a^2*b - 1890*a*b^2)*cosh(d*x + c)^2)*sinh(d*x + c)^7 + 21*(6930*b^3*cosh
(d*x + c)^5 - 4*(1024*a^3 + 4032*a^2*b + 7560*a*b^2 + 2835*b^3)*cosh(d*x + c)^3 + 3*(1024*a^3 + 4032*a^2*b + 5
880*a*b^2 + 1225*b^3)*cosh(d*x + c))*sinh(d*x + c)^6 - 9*(4096*a^3 + 16128*a^2*b + 16800*a*b^2 + 2625*b^3)*cos
h(d*x + c)^5 - 36*(1260*b^3*d*x + 14*(315*b^3*d*x - 256*a^3 - 1008*a^2*b - 1890*a*b^2)*cosh(d*x + c)^4 - 1024*
a^3 - 4032*a^2*b - 7560*a*b^2 - 21*(315*b^3*d*x - 256*a^3 - 1008*a^2*b - 1890*a*b^2)*cosh(d*x + c)^2)*sinh(d*x
+ c)^5 + 9*(11550*b^3*cosh(d*x + c)^7 - 14*(1024*a^3 + 4032*a^2*b + 7560*a*b^2 + 2835*b^3)*cosh(d*x + c)^5 +
35*(1024*a^3 + 4032*a^2*b + 5880*a*b^2 + 1225*b^3)*cosh(d*x + c)^3 - 5*(4096*a^3 + 16128*a^2*b + 16800*a*b^2 +
2625*b^3)*cosh(d*x + c))*sinh(d*x + c)^4 + 42*(2048*a^3 + 6144*a^2*b + 5040*a*b^2 + 675*b^3)*cosh(d*x + c)^3
- 12*(28*(315*b^3*d*x - 256*a^3 - 1008*a^2*b - 1890*a*b^2)*cosh(d*x + c)^6 - 8820*b^3*d*x - 105*(315*b^3*d*x -
256*a^3 - 1008*a^2*b - 1890*a*b^2)*cosh(d*x + c)^4 + 7168*a^3 + 28224*a^2*b + 52920*a*b^2 + 120*(315*b^3*d*x
- 256*a^3 - 1008*a^2*b - 1890*a*b^2)*cosh(d*x + c)^2)*sinh(d*x + c)^3 + 9*(1925*b^3*cosh(d*x + c)^9 - 4*(1024*
a^3 + 4032*a^2*b + 7560*a*b^2 + 2835*b^3)*cosh(d*x + c)^7 + 21*(1024*a^3 + 4032*a^2*b + 5880*a*b^2 + 1225*b^3)
*cosh(d*x + c)^5 - 10*(4096*a^3 + 16128*a^2*b + 16800*a*b^2 + 2625*b^3)*cosh(d*x + c)^3 + 14*(2048*a^3 + 6144*
a^2*b + 5040*a*b^2 + 675*b^3)*cosh(d*x + c))*sinh(d*x + c)^2 - 126*(1024*a^3 + 1152*a^2*b + 840*a*b^2 + 105*b^
3)*cosh(d*x + c) - 36*((315*b^3*d*x - 256*a^3 - 1008*a^2*b - 1890*a*b^2)*cosh(d*x + c)^8 - 7*(315*b^3*d*x - 25
6*a^3 - 1008*a^2*b - 1890*a*b^2)*cosh(d*x + c)^6 + 4410*b^3*d*x + 20*(315*b^3*d*x - 256*a^3 - 1008*a^2*b - 189
0*a*b^2)*cosh(d*x + c)^4 - 3584*a^3 - 14112*a^2*b - 26460*a*b^2 - 28*(315*b^3*d*x - 256*a^3 - 1008*a^2*b - 189
0*a*b^2)*cosh(d*x + c)^2)*sinh(d*x + c))/(d*sinh(d*x + c)^9 + 9*(4*d*cosh(d*x + c)^2 - d)*sinh(d*x + c)^7 + 9*
(14*d*cosh(d*x + c)^4 - 21*d*cosh(d*x + c)^2 + 4*d)*sinh(d*x + c)^5 + 3*(28*d*cosh(d*x + c)^6 - 105*d*cosh(d*x
+ c)^4 + 120*d*cosh(d*x + c)^2 - 28*d)*sinh(d*x + c)^3 + 9*(d*cosh(d*x + c)^8 - 7*d*cosh(d*x + c)^6 + 20*d*co
sh(d*x + c)^4 - 28*d*cosh(d*x + c)^2 + 14*d)*sinh(d*x + c))

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________

Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 360 vs. \(2 (128) = 256\).
time = 0.59, size = 360, normalized size = 2.57 \begin {gather*} -\frac {1260 \, {\left (d x + c\right )} b^{3} - 315 \, b^{3} e^{\left (2 \, d x + 2 \, c\right )} - 315 \, {\left (2 \, b^{3} e^{\left (2 \, d x + 2 \, c\right )} - b^{3}\right )} e^{\left (-2 \, d x - 2 \, c\right )} + \frac {16 \, {\left (945 \, a b^{2} e^{\left (16 \, d x + 16 \, c\right )} - 7560 \, a b^{2} e^{\left (14 \, d x + 14 \, c\right )} + 5040 \, a^{2} b e^{\left (12 \, d x + 12 \, c\right )} + 26460 \, a b^{2} e^{\left (12 \, d x + 12 \, c\right )} - 22680 \, a^{2} b e^{\left (10 \, d x + 10 \, c\right )} - 52920 \, a b^{2} e^{\left (10 \, d x + 10 \, c\right )} + 16128 \, a^{3} e^{\left (8 \, d x + 8 \, c\right )} + 40824 \, a^{2} b e^{\left (8 \, d x + 8 \, c\right )} + 66150 \, a b^{2} e^{\left (8 \, d x + 8 \, c\right )} - 10752 \, a^{3} e^{\left (6 \, d x + 6 \, c\right )} - 37296 \, a^{2} b e^{\left (6 \, d x + 6 \, c\right )} - 52920 \, a b^{2} e^{\left (6 \, d x + 6 \, c\right )} + 4608 \, a^{3} e^{\left (4 \, d x + 4 \, c\right )} + 18144 \, a^{2} b e^{\left (4 \, d x + 4 \, c\right )} + 26460 \, a b^{2} e^{\left (4 \, d x + 4 \, c\right )} - 1152 \, a^{3} e^{\left (2 \, d x + 2 \, c\right )} - 4536 \, a^{2} b e^{\left (2 \, d x + 2 \, c\right )} - 7560 \, a b^{2} e^{\left (2 \, d x + 2 \, c\right )} + 128 \, a^{3} + 504 \, a^{2} b + 945 \, a b^{2}\right )}}{{\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )}^{9}}}{2520 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(d*x+c)^10*(a+b*sinh(d*x+c)^4)^3,x, algorithm="giac")

[Out]

-1/2520*(1260*(d*x + c)*b^3 - 315*b^3*e^(2*d*x + 2*c) - 315*(2*b^3*e^(2*d*x + 2*c) - b^3)*e^(-2*d*x - 2*c) + 1
6*(945*a*b^2*e^(16*d*x + 16*c) - 7560*a*b^2*e^(14*d*x + 14*c) + 5040*a^2*b*e^(12*d*x + 12*c) + 26460*a*b^2*e^(
12*d*x + 12*c) - 22680*a^2*b*e^(10*d*x + 10*c) - 52920*a*b^2*e^(10*d*x + 10*c) + 16128*a^3*e^(8*d*x + 8*c) + 4
0824*a^2*b*e^(8*d*x + 8*c) + 66150*a*b^2*e^(8*d*x + 8*c) - 10752*a^3*e^(6*d*x + 6*c) - 37296*a^2*b*e^(6*d*x +
6*c) - 52920*a*b^2*e^(6*d*x + 6*c) + 4608*a^3*e^(4*d*x + 4*c) + 18144*a^2*b*e^(4*d*x + 4*c) + 26460*a*b^2*e^(4
*d*x + 4*c) - 1152*a^3*e^(2*d*x + 2*c) - 4536*a^2*b*e^(2*d*x + 2*c) - 7560*a*b^2*e^(2*d*x + 2*c) + 128*a^3 + 5
04*a^2*b + 945*a*b^2)/(e^(2*d*x + 2*c) - 1)^9)/d

________________________________________________________________________________________

Mupad [B]
time = 1.08, size = 1500, normalized size = 10.71 \begin {gather*} \frac {\frac {2\,a\,b^2}{3\,d}-\frac {2\,{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (4\,a^2\,b+7\,a\,b^2\right )}{3\,d}+\frac {2\,{\mathrm {e}}^{4\,c+4\,d\,x}\,\left (8\,a^2\,b+7\,a\,b^2\right )}{d}+\frac {10\,{\mathrm {e}}^{8\,c+8\,d\,x}\,\left (8\,a^2\,b+7\,a\,b^2\right )}{3\,d}-\frac {2\,{\mathrm {e}}^{10\,c+10\,d\,x}\,\left (4\,a^2\,b+7\,a\,b^2\right )}{d}-\frac {2\,{\mathrm {e}}^{6\,c+6\,d\,x}\,\left (128\,a^3+144\,a^2\,b+105\,a\,b^2\right )}{9\,d}+\frac {14\,a\,b^2\,{\mathrm {e}}^{12\,c+12\,d\,x}}{3\,d}-\frac {2\,a\,b^2\,{\mathrm {e}}^{14\,c+14\,d\,x}}{3\,d}}{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}-\frac {\frac {2\,\left (128\,a^3+144\,a^2\,b+105\,a\,b^2\right )}{315\,d}-\frac {8\,{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (8\,a^2\,b+7\,a\,b^2\right )}{21\,d}+\frac {4\,{\mathrm {e}}^{4\,c+4\,d\,x}\,\left (4\,a^2\,b+7\,a\,b^2\right )}{7\,d}-\frac {8\,a\,b^2\,{\mathrm {e}}^{6\,c+6\,d\,x}}{3\,d}+\frac {2\,a\,b^2\,{\mathrm {e}}^{8\,c+8\,d\,x}}{3\,d}}{5\,{\mathrm {e}}^{2\,c+2\,d\,x}-10\,{\mathrm {e}}^{4\,c+4\,d\,x}+10\,{\mathrm {e}}^{6\,c+6\,d\,x}-5\,{\mathrm {e}}^{8\,c+8\,d\,x}+{\mathrm {e}}^{10\,c+10\,d\,x}-1}-\frac {\frac {2\,\left (4\,a^2\,b+7\,a\,b^2\right )}{21\,d}-\frac {4\,a\,b^2\,{\mathrm {e}}^{2\,c+2\,d\,x}}{3\,d}+\frac {2\,a\,b^2\,{\mathrm {e}}^{4\,c+4\,d\,x}}{3\,d}}{3\,{\mathrm {e}}^{2\,c+2\,d\,x}-3\,{\mathrm {e}}^{4\,c+4\,d\,x}+{\mathrm {e}}^{6\,c+6\,d\,x}-1}-\frac {\frac {2\,a\,b^2}{3\,d}+\frac {8\,{\mathrm {e}}^{4\,c+4\,d\,x}\,\left (4\,a^2\,b+7\,a\,b^2\right )}{3\,d}-\frac {16\,{\mathrm {e}}^{6\,c+6\,d\,x}\,\left (8\,a^2\,b+7\,a\,b^2\right )}{3\,d}-\frac {16\,{\mathrm {e}}^{10\,c+10\,d\,x}\,\left (8\,a^2\,b+7\,a\,b^2\right )}{3\,d}+\frac {8\,{\mathrm {e}}^{12\,c+12\,d\,x}\,\left (4\,a^2\,b+7\,a\,b^2\right )}{3\,d}+\frac {4\,{\mathrm {e}}^{8\,c+8\,d\,x}\,\left (128\,a^3+144\,a^2\,b+105\,a\,b^2\right )}{9\,d}-\frac {16\,a\,b^2\,{\mathrm {e}}^{2\,c+2\,d\,x}}{3\,d}-\frac {16\,a\,b^2\,{\mathrm {e}}^{14\,c+14\,d\,x}}{3\,d}+\frac {2\,a\,b^2\,{\mathrm {e}}^{16\,c+16\,d\,x}}{3\,d}}{9\,{\mathrm {e}}^{2\,c+2\,d\,x}-36\,{\mathrm {e}}^{4\,c+4\,d\,x}+84\,{\mathrm {e}}^{6\,c+6\,d\,x}-126\,{\mathrm {e}}^{8\,c+8\,d\,x}+126\,{\mathrm {e}}^{10\,c+10\,d\,x}-84\,{\mathrm {e}}^{12\,c+12\,d\,x}+36\,{\mathrm {e}}^{14\,c+14\,d\,x}-9\,{\mathrm {e}}^{16\,c+16\,d\,x}+{\mathrm {e}}^{18\,c+18\,d\,x}-1}+\frac {\frac {2\,\left (8\,a^2\,b+7\,a\,b^2\right )}{21\,d}+\frac {20\,{\mathrm {e}}^{4\,c+4\,d\,x}\,\left (8\,a^2\,b+7\,a\,b^2\right )}{21\,d}-\frac {20\,{\mathrm {e}}^{6\,c+6\,d\,x}\,\left (4\,a^2\,b+7\,a\,b^2\right )}{21\,d}-\frac {2\,{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (128\,a^3+144\,a^2\,b+105\,a\,b^2\right )}{63\,d}+\frac {10\,a\,b^2\,{\mathrm {e}}^{8\,c+8\,d\,x}}{3\,d}-\frac {2\,a\,b^2\,{\mathrm {e}}^{10\,c+10\,d\,x}}{3\,d}}{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}+\frac {\frac {2\,\left (8\,a^2\,b+7\,a\,b^2\right )}{21\,d}-\frac {2\,{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (4\,a^2\,b+7\,a\,b^2\right )}{7\,d}+\frac {2\,a\,b^2\,{\mathrm {e}}^{4\,c+4\,d\,x}}{d}-\frac {2\,a\,b^2\,{\mathrm {e}}^{6\,c+6\,d\,x}}{3\,d}}{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}-\frac {b^3\,x}{2}-\frac {\frac {2\,\left (4\,a^2\,b+7\,a\,b^2\right )}{21\,d}-\frac {4\,{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (8\,a^2\,b+7\,a\,b^2\right )}{7\,d}-\frac {40\,{\mathrm {e}}^{6\,c+6\,d\,x}\,\left (8\,a^2\,b+7\,a\,b^2\right )}{21\,d}+\frac {10\,{\mathrm {e}}^{8\,c+8\,d\,x}\,\left (4\,a^2\,b+7\,a\,b^2\right )}{7\,d}+\frac {2\,{\mathrm {e}}^{4\,c+4\,d\,x}\,\left (128\,a^3+144\,a^2\,b+105\,a\,b^2\right )}{21\,d}-\frac {4\,a\,b^2\,{\mathrm {e}}^{10\,c+10\,d\,x}}{d}+\frac {2\,a\,b^2\,{\mathrm {e}}^{12\,c+12\,d\,x}}{3\,d}}{7\,{\mathrm {e}}^{2\,c+2\,d\,x}-21\,{\mathrm {e}}^{4\,c+4\,d\,x}+35\,{\mathrm {e}}^{6\,c+6\,d\,x}-35\,{\mathrm {e}}^{8\,c+8\,d\,x}+21\,{\mathrm {e}}^{10\,c+10\,d\,x}-7\,{\mathrm {e}}^{12\,c+12\,d\,x}+{\mathrm {e}}^{14\,c+14\,d\,x}-1}-\frac {b^3\,{\mathrm {e}}^{-2\,c-2\,d\,x}}{8\,d}+\frac {b^3\,{\mathrm {e}}^{2\,c+2\,d\,x}}{8\,d}-\frac {4\,a\,b^2}{3\,d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}-1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*sinh(c + d*x)^4)^3/sinh(c + d*x)^10,x)

[Out]

((2*a*b^2)/(3*d) - (2*exp(2*c + 2*d*x)*(7*a*b^2 + 4*a^2*b))/(3*d) + (2*exp(4*c + 4*d*x)*(7*a*b^2 + 8*a^2*b))/d
+ (10*exp(8*c + 8*d*x)*(7*a*b^2 + 8*a^2*b))/(3*d) - (2*exp(10*c + 10*d*x)*(7*a*b^2 + 4*a^2*b))/d - (2*exp(6*c
+ 6*d*x)*(105*a*b^2 + 144*a^2*b + 128*a^3))/(9*d) + (14*a*b^2*exp(12*c + 12*d*x))/(3*d) - (2*a*b^2*exp(14*c +
14*d*x))/(3*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*ex
p(10*c + 10*d*x) + 28*exp(12*c + 12*d*x) - 8*exp(14*c + 14*d*x) + exp(16*c + 16*d*x) + 1) - ((2*(105*a*b^2 + 1
44*a^2*b + 128*a^3))/(315*d) - (8*exp(2*c + 2*d*x)*(7*a*b^2 + 8*a^2*b))/(21*d) + (4*exp(4*c + 4*d*x)*(7*a*b^2
+ 4*a^2*b))/(7*d) - (8*a*b^2*exp(6*c + 6*d*x))/(3*d) + (2*a*b^2*exp(8*c + 8*d*x))/(3*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) - ((2*(7*a*b^2 + 4*a
^2*b))/(21*d) - (4*a*b^2*exp(2*c + 2*d*x))/(3*d) + (2*a*b^2*exp(4*c + 4*d*x))/(3*d))/(3*exp(2*c + 2*d*x) - 3*e
xp(4*c + 4*d*x) + exp(6*c + 6*d*x) - 1) - ((2*a*b^2)/(3*d) + (8*exp(4*c + 4*d*x)*(7*a*b^2 + 4*a^2*b))/(3*d) -
(16*exp(6*c + 6*d*x)*(7*a*b^2 + 8*a^2*b))/(3*d) - (16*exp(10*c + 10*d*x)*(7*a*b^2 + 8*a^2*b))/(3*d) + (8*exp(1
2*c + 12*d*x)*(7*a*b^2 + 4*a^2*b))/(3*d) + (4*exp(8*c + 8*d*x)*(105*a*b^2 + 144*a^2*b + 128*a^3))/(9*d) - (16*
a*b^2*exp(2*c + 2*d*x))/(3*d) - (16*a*b^2*exp(14*c + 14*d*x))/(3*d) + (2*a*b^2*exp(16*c + 16*d*x))/(3*d))/(9*e
xp(2*c + 2*d*x) - 36*exp(4*c + 4*d*x) + 84*exp(6*c + 6*d*x) - 126*exp(8*c + 8*d*x) + 126*exp(10*c + 10*d*x) -
84*exp(12*c + 12*d*x) + 36*exp(14*c + 14*d*x) - 9*exp(16*c + 16*d*x) + exp(18*c + 18*d*x) - 1) + ((2*(7*a*b^2
+ 8*a^2*b))/(21*d) + (20*exp(4*c + 4*d*x)*(7*a*b^2 + 8*a^2*b))/(21*d) - (20*exp(6*c + 6*d*x)*(7*a*b^2 + 4*a^2*
b))/(21*d) - (2*exp(2*c + 2*d*x)*(105*a*b^2 + 144*a^2*b + 128*a^3))/(63*d) + (10*a*b^2*exp(8*c + 8*d*x))/(3*d)
- (2*a*b^2*exp(10*c + 10*d*x))/(3*d))/(15*exp(4*c + 4*d*x) - 6*exp(2*c + 2*d*x) - 20*exp(6*c + 6*d*x) + 15*ex
p(8*c + 8*d*x) - 6*exp(10*c + 10*d*x) + exp(12*c + 12*d*x) + 1) + ((2*(7*a*b^2 + 8*a^2*b))/(21*d) - (2*exp(2*c
+ 2*d*x)*(7*a*b^2 + 4*a^2*b))/(7*d) + (2*a*b^2*exp(4*c + 4*d*x))/d - (2*a*b^2*exp(6*c + 6*d*x))/(3*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) - (b^3*x)/2 - ((2*(7*a*b^2 + 4
*a^2*b))/(21*d) - (4*exp(2*c + 2*d*x)*(7*a*b^2 + 8*a^2*b))/(7*d) - (40*exp(6*c + 6*d*x)*(7*a*b^2 + 8*a^2*b))/(
21*d) + (10*exp(8*c + 8*d*x)*(7*a*b^2 + 4*a^2*b))/(7*d) + (2*exp(4*c + 4*d*x)*(105*a*b^2 + 144*a^2*b + 128*a^3
))/(21*d) - (4*a*b^2*exp(10*c + 10*d*x))/d + (2*a*b^2*exp(12*c + 12*d*x))/(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) - (b^3*exp(- 2*c - 2*d*x))/(8*d) + (b^3*exp(2*c + 2*d*x))/(8*d) - (4*a*b^2)/(3*d*(exp(2*c
+ 2*d*x) - 1))

________________________________________________________________________________________