3.1.71 \(\int \text {csch}^3(c+d x) (a+b \tanh ^3(c+d x))^3 \, dx\) [71]

Optimal. Leaf size=232 \[ \frac {3 a^2 b \text {ArcTan}(\sinh (c+d x))}{2 d}+\frac {5 b^3 \text {ArcTan}(\sinh (c+d x))}{128 d}+\frac {a^3 \tanh ^{-1}(\cosh (c+d x))}{2 d}-\frac {a^3 \coth (c+d x) \text {csch}(c+d x)}{2 d}-\frac {a b^2 \text {sech}^3(c+d x)}{d}+\frac {3 a b^2 \text {sech}^5(c+d x)}{5 d}+\frac {3 a^2 b \text {sech}(c+d x) \tanh (c+d x)}{2 d}+\frac {5 b^3 \text {sech}(c+d x) \tanh (c+d x)}{128 d}-\frac {5 b^3 \text {sech}^3(c+d x) \tanh (c+d x)}{64 d}-\frac {5 b^3 \text {sech}^3(c+d x) \tanh ^3(c+d x)}{48 d}-\frac {b^3 \text {sech}^3(c+d x) \tanh ^5(c+d x)}{8 d} \]

[Out]

3/2*a^2*b*arctan(sinh(d*x+c))/d+5/128*b^3*arctan(sinh(d*x+c))/d+1/2*a^3*arctanh(cosh(d*x+c))/d-1/2*a^3*coth(d*
x+c)*csch(d*x+c)/d-a*b^2*sech(d*x+c)^3/d+3/5*a*b^2*sech(d*x+c)^5/d+3/2*a^2*b*sech(d*x+c)*tanh(d*x+c)/d+5/128*b
^3*sech(d*x+c)*tanh(d*x+c)/d-5/64*b^3*sech(d*x+c)^3*tanh(d*x+c)/d-5/48*b^3*sech(d*x+c)^3*tanh(d*x+c)^3/d-1/8*b
^3*sech(d*x+c)^3*tanh(d*x+c)^5/d

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Rubi [A]
time = 0.22, antiderivative size = 232, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 6, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {3747, 3853, 3855, 2686, 14, 2691} \begin {gather*} \frac {a^3 \tanh ^{-1}(\cosh (c+d x))}{2 d}-\frac {a^3 \coth (c+d x) \text {csch}(c+d x)}{2 d}+\frac {3 a^2 b \text {ArcTan}(\sinh (c+d x))}{2 d}+\frac {3 a^2 b \tanh (c+d x) \text {sech}(c+d x)}{2 d}+\frac {3 a b^2 \text {sech}^5(c+d x)}{5 d}-\frac {a b^2 \text {sech}^3(c+d x)}{d}+\frac {5 b^3 \text {ArcTan}(\sinh (c+d x))}{128 d}-\frac {b^3 \tanh ^5(c+d x) \text {sech}^3(c+d x)}{8 d}-\frac {5 b^3 \tanh ^3(c+d x) \text {sech}^3(c+d x)}{48 d}-\frac {5 b^3 \tanh (c+d x) \text {sech}^3(c+d x)}{64 d}+\frac {5 b^3 \tanh (c+d x) \text {sech}(c+d x)}{128 d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(3*a^2*b*ArcTan[Sinh[c + d*x]])/(2*d) + (5*b^3*ArcTan[Sinh[c + d*x]])/(128*d) + (a^3*ArcTanh[Cosh[c + d*x]])/(
2*d) - (a^3*Coth[c + d*x]*Csch[c + d*x])/(2*d) - (a*b^2*Sech[c + d*x]^3)/d + (3*a*b^2*Sech[c + d*x]^5)/(5*d) +
(3*a^2*b*Sech[c + d*x]*Tanh[c + d*x])/(2*d) + (5*b^3*Sech[c + d*x]*Tanh[c + d*x])/(128*d) - (5*b^3*Sech[c + d
*x]^3*Tanh[c + d*x])/(64*d) - (5*b^3*Sech[c + d*x]^3*Tanh[c + d*x]^3)/(48*d) - (b^3*Sech[c + d*x]^3*Tanh[c + d
*x]^5)/(8*d)

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 2686

Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[a/f, Subst[
Int[(a*x)^(m - 1)*(-1 + x^2)^((n - 1)/2), x], x, Sec[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n -
1)/2] &&  !(IntegerQ[m/2] && LtQ[0, m, n + 1])

Rule 2691

Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[b*(a*Sec[e +
f*x])^m*((b*Tan[e + f*x])^(n - 1)/(f*(m + n - 1))), x] - Dist[b^2*((n - 1)/(m + n - 1)), Int[(a*Sec[e + f*x])
^m*(b*Tan[e + f*x])^(n - 2), x], x] /; FreeQ[{a, b, e, f, m}, x] && GtQ[n, 1] && NeQ[m + n - 1, 0] && Integers
Q[2*m, 2*n]

Rule 3747

Int[((d_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((a_) + (b_.)*((c_.)*tan[(e_.) + (f_.)*(x_)])^(n_))^(p_.), x_Symbol]
:> Int[ExpandTrig[(d*sin[e + f*x])^m*(a + b*(c*tan[e + f*x])^n)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, n},
x] && IGtQ[p, 0]

Rule 3853

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d*x]*((b*Csc[c + d*x])^(n - 1)/(d*(n
- 1))), x] + Dist[b^2*((n - 2)/(n - 1)), Int[(b*Csc[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n,
1] && IntegerQ[2*n]

Rule 3855

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]

Rubi steps

\begin {align*} \int \text {csch}^3(c+d x) \left (a+b \tanh ^3(c+d x)\right )^3 \, dx &=-\left (i \int \left (i a^3 \text {csch}^3(c+d x)+3 i a^2 b \text {sech}^3(c+d x)+3 i a b^2 \text {sech}^3(c+d x) \tanh ^3(c+d x)+i b^3 \text {sech}^3(c+d x) \tanh ^6(c+d x)\right ) \, dx\right )\\ &=a^3 \int \text {csch}^3(c+d x) \, dx+\left (3 a^2 b\right ) \int \text {sech}^3(c+d x) \, dx+\left (3 a b^2\right ) \int \text {sech}^3(c+d x) \tanh ^3(c+d x) \, dx+b^3 \int \text {sech}^3(c+d x) \tanh ^6(c+d x) \, dx\\ &=-\frac {a^3 \coth (c+d x) \text {csch}(c+d x)}{2 d}+\frac {3 a^2 b \text {sech}(c+d x) \tanh (c+d x)}{2 d}-\frac {b^3 \text {sech}^3(c+d x) \tanh ^5(c+d x)}{8 d}-\frac {1}{2} a^3 \int \text {csch}(c+d x) \, dx+\frac {1}{2} \left (3 a^2 b\right ) \int \text {sech}(c+d x) \, dx+\frac {1}{8} \left (5 b^3\right ) \int \text {sech}^3(c+d x) \tanh ^4(c+d x) \, dx+\frac {\left (3 a b^2\right ) \text {Subst}\left (\int x^2 \left (-1+x^2\right ) \, dx,x,\text {sech}(c+d x)\right )}{d}\\ &=\frac {3 a^2 b \tan ^{-1}(\sinh (c+d x))}{2 d}+\frac {a^3 \tanh ^{-1}(\cosh (c+d x))}{2 d}-\frac {a^3 \coth (c+d x) \text {csch}(c+d x)}{2 d}+\frac {3 a^2 b \text {sech}(c+d x) \tanh (c+d x)}{2 d}-\frac {5 b^3 \text {sech}^3(c+d x) \tanh ^3(c+d x)}{48 d}-\frac {b^3 \text {sech}^3(c+d x) \tanh ^5(c+d x)}{8 d}+\frac {1}{16} \left (5 b^3\right ) \int \text {sech}^3(c+d x) \tanh ^2(c+d x) \, dx+\frac {\left (3 a b^2\right ) \text {Subst}\left (\int \left (-x^2+x^4\right ) \, dx,x,\text {sech}(c+d x)\right )}{d}\\ &=\frac {3 a^2 b \tan ^{-1}(\sinh (c+d x))}{2 d}+\frac {a^3 \tanh ^{-1}(\cosh (c+d x))}{2 d}-\frac {a^3 \coth (c+d x) \text {csch}(c+d x)}{2 d}-\frac {a b^2 \text {sech}^3(c+d x)}{d}+\frac {3 a b^2 \text {sech}^5(c+d x)}{5 d}+\frac {3 a^2 b \text {sech}(c+d x) \tanh (c+d x)}{2 d}-\frac {5 b^3 \text {sech}^3(c+d x) \tanh (c+d x)}{64 d}-\frac {5 b^3 \text {sech}^3(c+d x) \tanh ^3(c+d x)}{48 d}-\frac {b^3 \text {sech}^3(c+d x) \tanh ^5(c+d x)}{8 d}+\frac {1}{64} \left (5 b^3\right ) \int \text {sech}^3(c+d x) \, dx\\ &=\frac {3 a^2 b \tan ^{-1}(\sinh (c+d x))}{2 d}+\frac {a^3 \tanh ^{-1}(\cosh (c+d x))}{2 d}-\frac {a^3 \coth (c+d x) \text {csch}(c+d x)}{2 d}-\frac {a b^2 \text {sech}^3(c+d x)}{d}+\frac {3 a b^2 \text {sech}^5(c+d x)}{5 d}+\frac {3 a^2 b \text {sech}(c+d x) \tanh (c+d x)}{2 d}+\frac {5 b^3 \text {sech}(c+d x) \tanh (c+d x)}{128 d}-\frac {5 b^3 \text {sech}^3(c+d x) \tanh (c+d x)}{64 d}-\frac {5 b^3 \text {sech}^3(c+d x) \tanh ^3(c+d x)}{48 d}-\frac {b^3 \text {sech}^3(c+d x) \tanh ^5(c+d x)}{8 d}+\frac {1}{128} \left (5 b^3\right ) \int \text {sech}(c+d x) \, dx\\ &=\frac {3 a^2 b \tan ^{-1}(\sinh (c+d x))}{2 d}+\frac {5 b^3 \tan ^{-1}(\sinh (c+d x))}{128 d}+\frac {a^3 \tanh ^{-1}(\cosh (c+d x))}{2 d}-\frac {a^3 \coth (c+d x) \text {csch}(c+d x)}{2 d}-\frac {a b^2 \text {sech}^3(c+d x)}{d}+\frac {3 a b^2 \text {sech}^5(c+d x)}{5 d}+\frac {3 a^2 b \text {sech}(c+d x) \tanh (c+d x)}{2 d}+\frac {5 b^3 \text {sech}(c+d x) \tanh (c+d x)}{128 d}-\frac {5 b^3 \text {sech}^3(c+d x) \tanh (c+d x)}{64 d}-\frac {5 b^3 \text {sech}^3(c+d x) \tanh ^3(c+d x)}{48 d}-\frac {b^3 \text {sech}^3(c+d x) \tanh ^5(c+d x)}{8 d}\\ \end {align*}

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Mathematica [A]
time = 6.27, size = 243, normalized size = 1.05 \begin {gather*} \frac {b \left (192 a^2+5 b^2\right ) \text {ArcTan}\left (\tanh \left (\frac {1}{2} (c+d x)\right )\right )}{64 d}-\frac {a^3 \text {csch}^2\left (\frac {1}{2} (c+d x)\right )}{8 d}-\frac {a^3 \log \left (\tanh \left (\frac {1}{2} (c+d x)\right )\right )}{2 d}-\frac {a^3 \text {sech}^2\left (\frac {1}{2} (c+d x)\right )}{8 d}-\frac {a b^2 \text {sech}^3(c+d x)}{d}+\frac {3 a b^2 \text {sech}^5(c+d x)}{5 d}+\frac {\text {sech}^2(c+d x) \left (192 a^2 b \sinh (c+d x)+5 b^3 \sinh (c+d x)\right )}{128 d}-\frac {59 b^3 \text {sech}^3(c+d x) \tanh (c+d x)}{192 d}+\frac {17 b^3 \text {sech}^5(c+d x) \tanh (c+d x)}{48 d}-\frac {b^3 \text {sech}^7(c+d x) \tanh (c+d x)}{8 d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(b*(192*a^2 + 5*b^2)*ArcTan[Tanh[(c + d*x)/2]])/(64*d) - (a^3*Csch[(c + d*x)/2]^2)/(8*d) - (a^3*Log[Tanh[(c +
d*x)/2]])/(2*d) - (a^3*Sech[(c + d*x)/2]^2)/(8*d) - (a*b^2*Sech[c + d*x]^3)/d + (3*a*b^2*Sech[c + d*x]^5)/(5*d
) + (Sech[c + d*x]^2*(192*a^2*b*Sinh[c + d*x] + 5*b^3*Sinh[c + d*x]))/(128*d) - (59*b^3*Sech[c + d*x]^3*Tanh[c
+ d*x])/(192*d) + (17*b^3*Sech[c + d*x]^5*Tanh[c + d*x])/(48*d) - (b^3*Sech[c + d*x]^7*Tanh[c + d*x])/(8*d)

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Maple [C] Result contains complex when optimal does not.
time = 5.02, size = 645, normalized size = 2.78

method	result	size
risch	\(-\frac {{\mathrm e}^{d x +c} \left (4608 a \,b^{2} {\mathrm e}^{14 d x +14 c}+23040 a^{2} b \,{\mathrm e}^{12 d x +12 c}-10752 a \,b^{2} {\mathrm e}^{12 d x +12 c}+17280 a^{2} b \,{\mathrm e}^{10 d x +10 c}-1536 a \,b^{2} {\mathrm e}^{10 d x +10 c}-8640 a^{2} b \,{\mathrm e}^{16 d x +16 c}+7680 a \,b^{2} {\mathrm e}^{16 d x +16 c}+8640 a^{2} b \,{\mathrm e}^{2 d x +2 c}+2880 a^{2} b +960 a^{3}+75 b^{3}-1536 a \,b^{2} {\mathrm e}^{8 d x +8 c}-10752 a \,b^{2} {\mathrm e}^{6 d x +6 c}+4608 a \,b^{2} {\mathrm e}^{4 d x +4 c}-17280 a^{2} b \,{\mathrm e}^{8 d x +8 c}+7680 a \,b^{2} {\mathrm e}^{2 d x +2 c}-23040 a^{2} b \,{\mathrm e}^{6 d x +6 c}-2880 a^{2} b \,{\mathrm e}^{18 d x +18 c}-75 b^{3} {\mathrm e}^{18 d x +18 c}+2135 b^{3} {\mathrm e}^{16 d x +16 c}-8520 b^{3} {\mathrm e}^{14 d x +14 c}+8640 a^{3} {\mathrm e}^{2 d x +2 c}+80640 a^{3} {\mathrm e}^{12 d x +12 c}+19760 b^{3} {\mathrm e}^{12 d x +12 c}+120960 a^{3} {\mathrm e}^{10 d x +10 c}+34560 a^{3} {\mathrm e}^{14 d x +14 c}-2135 b^{3} {\mathrm e}^{2 d x +2 c}+8640 a^{3} {\mathrm e}^{16 d x +16 c}+30950 b^{3} {\mathrm e}^{8 d x +8 c}+34560 a^{3} {\mathrm e}^{4 d x +4 c}+8520 b^{3} {\mathrm e}^{4 d x +4 c}+960 a^{3} {\mathrm e}^{18 d x +18 c}-30950 b^{3} {\mathrm e}^{10 d x +10 c}+120960 a^{3} {\mathrm e}^{8 d x +8 c}+80640 a^{3} {\mathrm e}^{6 d x +6 c}-19760 b^{3} {\mathrm e}^{6 d x +6 c}\right )}{960 d \left (1+{\mathrm e}^{2 d x +2 c}\right )^{8} \left ({\mathrm e}^{2 d x +2 c}-1\right )^{2}}+\frac {a^{3} \ln \left ({\mathrm e}^{d x +c}+1\right )}{2 d}-\frac {a^{3} \ln \left ({\mathrm e}^{d x +c}-1\right )}{2 d}+\frac {3 i b \ln \left ({\mathrm e}^{d x +c}+i\right ) a^{2}}{2 d}+\frac {5 i b^{3} \ln \left ({\mathrm e}^{d x +c}+i\right )}{128 d}-\frac {3 i b \ln \left ({\mathrm e}^{d x +c}-i\right ) a^{2}}{2 d}-\frac {5 i b^{3} \ln \left ({\mathrm e}^{d x +c}-i\right )}{128 d}\)	\(645\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/960*exp(d*x+c)*(4608*a*b^2*exp(14*d*x+14*c)+23040*a^2*b*exp(12*d*x+12*c)-10752*a*b^2*exp(12*d*x+12*c)+17280
*a^2*b*exp(10*d*x+10*c)-1536*a*b^2*exp(10*d*x+10*c)-8640*a^2*b*exp(16*d*x+16*c)+7680*a*b^2*exp(16*d*x+16*c)+86
40*a^2*b*exp(2*d*x+2*c)+2880*a^2*b+960*a^3+75*b^3-1536*a*b^2*exp(8*d*x+8*c)-10752*a*b^2*exp(6*d*x+6*c)+4608*a*
b^2*exp(4*d*x+4*c)-17280*a^2*b*exp(8*d*x+8*c)+7680*a*b^2*exp(2*d*x+2*c)-23040*a^2*b*exp(6*d*x+6*c)-2880*a^2*b*
exp(18*d*x+18*c)-75*b^3*exp(18*d*x+18*c)+2135*b^3*exp(16*d*x+16*c)-8520*b^3*exp(14*d*x+14*c)+8640*a^3*exp(2*d*
x+2*c)+80640*a^3*exp(12*d*x+12*c)+19760*b^3*exp(12*d*x+12*c)+120960*a^3*exp(10*d*x+10*c)+34560*a^3*exp(14*d*x+
14*c)-2135*b^3*exp(2*d*x+2*c)+8640*a^3*exp(16*d*x+16*c)+30950*b^3*exp(8*d*x+8*c)+34560*a^3*exp(4*d*x+4*c)+8520
*b^3*exp(4*d*x+4*c)+960*a^3*exp(18*d*x+18*c)-30950*b^3*exp(10*d*x+10*c)+120960*a^3*exp(8*d*x+8*c)+80640*a^3*ex
p(6*d*x+6*c)-19760*b^3*exp(6*d*x+6*c))/d/(1+exp(2*d*x+2*c))^8/(exp(2*d*x+2*c)-1)^2+1/2*a^3/d*ln(exp(d*x+c)+1)-
1/2*a^3/d*ln(exp(d*x+c)-1)+3/2*I*b/d*ln(exp(d*x+c)+I)*a^2+5/128*I*b^3/d*ln(exp(d*x+c)+I)-3/2*I*b/d*ln(exp(d*x+
c)-I)*a^2-5/128*I*b^3/d*ln(exp(d*x+c)-I)

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 586 vs. \(2 (212) = 424\).
time = 0.49, size = 586, normalized size = 2.53 \begin {gather*} -\frac {1}{192} \, b^{3} {\left (\frac {15 \, \arctan \left (e^{\left (-d x - c\right )}\right )}{d} - \frac {15 \, e^{\left (-d x - c\right )} - 397 \, e^{\left (-3 \, d x - 3 \, c\right )} + 895 \, e^{\left (-5 \, d x - 5 \, c\right )} - 1765 \, e^{\left (-7 \, d x - 7 \, c\right )} + 1765 \, e^{\left (-9 \, d x - 9 \, c\right )} - 895 \, e^{\left (-11 \, d x - 11 \, c\right )} + 397 \, e^{\left (-13 \, d x - 13 \, c\right )} - 15 \, e^{\left (-15 \, d x - 15 \, c\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 )} - 3 \, a^{2} b {\left (\frac {\arctan \left (e^{\left (-d x - c\right )}\right )}{d} - \frac {e^{\left (-d x - c\right )} - e^{\left (-3 \, d x - 3 \, c\right )}}{d {\left (2 \, e^{\left (-2 \, d x - 2 \, c\right )} + e^{\left (-4 \, d x - 4 \, c\right )} + 1\right )}}\right )} + \frac {1}{2} \, a^{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} + \frac {2 \, {\left (e^{\left (-d x - c\right )} + e^{\left (-3 \, d x - 3 \, c\right )}\right )}}{d {\left (2 \, e^{\left (-2 \, d x - 2 \, c\right )} - e^{\left (-4 \, d x - 4 \, c\right )} - 1\right )}}\right )} - \frac {8}{5} \, a b^{2} {\left (\frac {5 \, e^{\left (-3 \, d x - 3 \, 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 {2 \, e^{\left (-5 \, d x - 5 \, 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 {5 \, e^{\left (-7 \, d x - 7 \, 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 )}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/192*b^3*(15*arctan(e^(-d*x - c))/d - (15*e^(-d*x - c) - 397*e^(-3*d*x - 3*c) + 895*e^(-5*d*x - 5*c) - 1765*
e^(-7*d*x - 7*c) + 1765*e^(-9*d*x - 9*c) - 895*e^(-11*d*x - 11*c) + 397*e^(-13*d*x - 13*c) - 15*e^(-15*d*x - 1
5*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*a^2*b*(arctan(e^(-d*x
- c))/d - (e^(-d*x - c) - e^(-3*d*x - 3*c))/(d*(2*e^(-2*d*x - 2*c) + e^(-4*d*x - 4*c) + 1))) + 1/2*a^3*(log(e^
(-d*x - c) + 1)/d - log(e^(-d*x - c) - 1)/d + 2*(e^(-d*x - c) + e^(-3*d*x - 3*c))/(d*(2*e^(-2*d*x - 2*c) - e^(
-4*d*x - 4*c) - 1))) - 8/5*a*b^2*(5*e^(-3*d*x - 3*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)) - 2*e^(-5*d*x - 5*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)) + 5*e^(-7*d*x - 7*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)))

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 10985 vs. \(2 (212) = 424\).
time = 0.44, size = 10985, normalized size = 47.35 \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)^3*(a+b*tanh(d*x+c)^3)^3,x, algorithm="fricas")

[Out]

-1/960*(15*(64*a^3 - 192*a^2*b - 5*b^3)*cosh(d*x + c)^19 + 285*(64*a^3 - 192*a^2*b - 5*b^3)*cosh(d*x + c)*sinh
(d*x + c)^18 + 15*(64*a^3 - 192*a^2*b - 5*b^3)*sinh(d*x + c)^19 + 5*(1728*a^3 - 1728*a^2*b + 1536*a*b^2 + 427*
b^3)*cosh(d*x + c)^17 + 5*(1728*a^3 - 1728*a^2*b + 1536*a*b^2 + 427*b^3 + 513*(64*a^3 - 192*a^2*b - 5*b^3)*cos
h(d*x + c)^2)*sinh(d*x + c)^17 + 85*(171*(64*a^3 - 192*a^2*b - 5*b^3)*cosh(d*x + c)^3 + (1728*a^3 - 1728*a^2*b
+ 1536*a*b^2 + 427*b^3)*cosh(d*x + c))*sinh(d*x + c)^16 + 24*(1440*a^3 + 192*a*b^2 - 355*b^3)*cosh(d*x + c)^1
5 + 4*(14535*(64*a^3 - 192*a^2*b - 5*b^3)*cosh(d*x + c)^4 + 8640*a^3 + 1152*a*b^2 - 2130*b^3 + 170*(1728*a^3 -
1728*a^2*b + 1536*a*b^2 + 427*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^15 + 20*(8721*(64*a^3 - 192*a^2*b - 5*b^3)*
cosh(d*x + c)^5 + 170*(1728*a^3 - 1728*a^2*b + 1536*a*b^2 + 427*b^3)*cosh(d*x + c)^3 + 18*(1440*a^3 + 192*a*b^
2 - 355*b^3)*cosh(d*x + c))*sinh(d*x + c)^14 + 16*(5040*a^3 + 1440*a^2*b - 672*a*b^2 + 1235*b^3)*cosh(d*x + c)
^13 + 4*(101745*(64*a^3 - 192*a^2*b - 5*b^3)*cosh(d*x + c)^6 + 2975*(1728*a^3 - 1728*a^2*b + 1536*a*b^2 + 427*
b^3)*cosh(d*x + c)^4 + 20160*a^3 + 5760*a^2*b - 2688*a*b^2 + 4940*b^3 + 630*(1440*a^3 + 192*a*b^2 - 355*b^3)*c
osh(d*x + c)^2)*sinh(d*x + c)^13 + 52*(14535*(64*a^3 - 192*a^2*b - 5*b^3)*cosh(d*x + c)^7 + 595*(1728*a^3 - 17
28*a^2*b + 1536*a*b^2 + 427*b^3)*cosh(d*x + c)^5 + 210*(1440*a^3 + 192*a*b^2 - 355*b^3)*cosh(d*x + c)^3 + 4*(5
040*a^3 + 1440*a^2*b - 672*a*b^2 + 1235*b^3)*cosh(d*x + c))*sinh(d*x + c)^12 + 2*(60480*a^3 + 8640*a^2*b - 768
*a*b^2 - 15475*b^3)*cosh(d*x + c)^11 + 2*(566865*(64*a^3 - 192*a^2*b - 5*b^3)*cosh(d*x + c)^8 + 30940*(1728*a^
3 - 1728*a^2*b + 1536*a*b^2 + 427*b^3)*cosh(d*x + c)^6 + 16380*(1440*a^3 + 192*a*b^2 - 355*b^3)*cosh(d*x + c)^
4 + 60480*a^3 + 8640*a^2*b - 768*a*b^2 - 15475*b^3 + 624*(5040*a^3 + 1440*a^2*b - 672*a*b^2 + 1235*b^3)*cosh(d
*x + c)^2)*sinh(d*x + c)^11 + 22*(62985*(64*a^3 - 192*a^2*b - 5*b^3)*cosh(d*x + c)^9 + 4420*(1728*a^3 - 1728*a
^2*b + 1536*a*b^2 + 427*b^3)*cosh(d*x + c)^7 + 3276*(1440*a^3 + 192*a*b^2 - 355*b^3)*cosh(d*x + c)^5 + 208*(50
40*a^3 + 1440*a^2*b - 672*a*b^2 + 1235*b^3)*cosh(d*x + c)^3 + (60480*a^3 + 8640*a^2*b - 768*a*b^2 - 15475*b^3)
*cosh(d*x + c))*sinh(d*x + c)^10 + 2*(60480*a^3 - 8640*a^2*b - 768*a*b^2 + 15475*b^3)*cosh(d*x + c)^9 + 2*(692
835*(64*a^3 - 192*a^2*b - 5*b^3)*cosh(d*x + c)^10 + 60775*(1728*a^3 - 1728*a^2*b + 1536*a*b^2 + 427*b^3)*cosh(
d*x + c)^8 + 60060*(1440*a^3 + 192*a*b^2 - 355*b^3)*cosh(d*x + c)^6 + 5720*(5040*a^3 + 1440*a^2*b - 672*a*b^2
+ 1235*b^3)*cosh(d*x + c)^4 + 60480*a^3 - 8640*a^2*b - 768*a*b^2 + 15475*b^3 + 55*(60480*a^3 + 8640*a^2*b - 76
8*a*b^2 - 15475*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^9 + 2*(566865*(64*a^3 - 192*a^2*b - 5*b^3)*cosh(d*x + c)^1
1 + 60775*(1728*a^3 - 1728*a^2*b + 1536*a*b^2 + 427*b^3)*cosh(d*x + c)^9 + 77220*(1440*a^3 + 192*a*b^2 - 355*b
^3)*cosh(d*x + c)^7 + 10296*(5040*a^3 + 1440*a^2*b - 672*a*b^2 + 1235*b^3)*cosh(d*x + c)^5 + 165*(60480*a^3 +
8640*a^2*b - 768*a*b^2 - 15475*b^3)*cosh(d*x + c)^3 + 9*(60480*a^3 - 8640*a^2*b - 768*a*b^2 + 15475*b^3)*cosh(
d*x + c))*sinh(d*x + c)^8 + 16*(5040*a^3 - 1440*a^2*b - 672*a*b^2 - 1235*b^3)*cosh(d*x + c)^7 + 4*(188955*(64*
a^3 - 192*a^2*b - 5*b^3)*cosh(d*x + c)^12 + 24310*(1728*a^3 - 1728*a^2*b + 1536*a*b^2 + 427*b^3)*cosh(d*x + c)
^10 + 38610*(1440*a^3 + 192*a*b^2 - 355*b^3)*cosh(d*x + c)^8 + 6864*(5040*a^3 + 1440*a^2*b - 672*a*b^2 + 1235*
b^3)*cosh(d*x + c)^6 + 165*(60480*a^3 + 8640*a^2*b - 768*a*b^2 - 15475*b^3)*cosh(d*x + c)^4 + 20160*a^3 - 5760
*a^2*b - 2688*a*b^2 - 4940*b^3 + 18*(60480*a^3 - 8640*a^2*b - 768*a*b^2 + 15475*b^3)*cosh(d*x + c)^2)*sinh(d*x
+ c)^7 + 4*(101745*(64*a^3 - 192*a^2*b - 5*b^3)*cosh(d*x + c)^13 + 15470*(1728*a^3 - 1728*a^2*b + 1536*a*b^2
+ 427*b^3)*cosh(d*x + c)^11 + 30030*(1440*a^3 + 192*a*b^2 - 355*b^3)*cosh(d*x + c)^9 + 6864*(5040*a^3 + 1440*a
^2*b - 672*a*b^2 + 1235*b^3)*cosh(d*x + c)^7 + 231*(60480*a^3 + 8640*a^2*b - 768*a*b^2 - 15475*b^3)*cosh(d*x +
c)^5 + 42*(60480*a^3 - 8640*a^2*b - 768*a*b^2 + 15475*b^3)*cosh(d*x + c)^3 + 28*(5040*a^3 - 1440*a^2*b - 672*
a*b^2 - 1235*b^3)*cosh(d*x + c))*sinh(d*x + c)^6 + 24*(1440*a^3 + 192*a*b^2 + 355*b^3)*cosh(d*x + c)^5 + 4*(43
605*(64*a^3 - 192*a^2*b - 5*b^3)*cosh(d*x + c)^14 + 7735*(1728*a^3 - 1728*a^2*b + 1536*a*b^2 + 427*b^3)*cosh(d
*x + c)^12 + 18018*(1440*a^3 + 192*a*b^2 - 355*b^3)*cosh(d*x + c)^10 + 5148*(5040*a^3 + 1440*a^2*b - 672*a*b^2
+ 1235*b^3)*cosh(d*x + c)^8 + 231*(60480*a^3 + 8640*a^2*b - 768*a*b^2 - 15475*b^3)*cosh(d*x + c)^6 + 63*(6048
0*a^3 - 8640*a^2*b - 768*a*b^2 + 15475*b^3)*cosh(d*x + c)^4 + 8640*a^3 + 1152*a*b^2 + 2130*b^3 + 84*(5040*a^3
- 1440*a^2*b - 672*a*b^2 - 1235*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^5 + 4*(14535*(64*a^3 - 192*a^2*b - 5*b^3)*
cosh(d*x + c)^15 + 2975*(1728*a^3 - 1728*a^2*b + 1536*a*b^2 + 427*b^3)*cosh(d*x + c)^13 + 8190*(1440*a^3 + 192
*a*b^2 - 355*b^3)*cosh(d*x + c)^11 + 2860*(5040*a^3 + 1440*a^2*b - 672*a*b^2 + 1235*b^3)*cosh(d*x + c)^9 + 165
*(60480*a^3 + 8640*a^2*b - 768*a*b^2 - 15475*b^...

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (a + b \tanh ^{3}{\left (c + d x \right )}\right )^{3} \operatorname {csch}^{3}{\left (c + d x \right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(d*x+c)**3*(a+b*tanh(d*x+c)**3)**3,x)

[Out]

Integral((a + b*tanh(c + d*x)**3)**3*csch(c + d*x)**3, x)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 426 vs. \(2 (212) = 424\).
time = 0.67, size = 426, normalized size = 1.84 \begin {gather*} \frac {480 \, a^{3} \log \left (e^{\left (d x + c\right )} + 1\right ) - 480 \, a^{3} \log \left ({\left | e^{\left (d x + c\right )} - 1 \right |}\right ) + 15 \, {\left (192 \, a^{2} b + 5 \, b^{3}\right )} \arctan \left (e^{\left (d x + c\right )}\right ) - \frac {960 \, {\left (a^{3} e^{\left (3 \, d x + 3 \, c\right )} + a^{3} e^{\left (d x + c\right )}\right )}}{{\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )}^{2}} + \frac {2880 \, a^{2} b e^{\left (15 \, d x + 15 \, c\right )} + 75 \, b^{3} e^{\left (15 \, d x + 15 \, c\right )} + 14400 \, a^{2} b e^{\left (13 \, d x + 13 \, c\right )} - 7680 \, a b^{2} e^{\left (13 \, d x + 13 \, c\right )} - 1985 \, b^{3} e^{\left (13 \, d x + 13 \, c\right )} + 25920 \, a^{2} b e^{\left (11 \, d x + 11 \, c\right )} - 19968 \, a b^{2} e^{\left (11 \, d x + 11 \, c\right )} + 4475 \, b^{3} e^{\left (11 \, d x + 11 \, c\right )} + 14400 \, a^{2} b e^{\left (9 \, d x + 9 \, c\right )} - 21504 \, a b^{2} e^{\left (9 \, d x + 9 \, c\right )} - 8825 \, b^{3} e^{\left (9 \, d x + 9 \, c\right )} - 14400 \, a^{2} b e^{\left (7 \, d x + 7 \, c\right )} - 21504 \, a b^{2} e^{\left (7 \, d x + 7 \, c\right )} + 8825 \, b^{3} e^{\left (7 \, d x + 7 \, c\right )} - 25920 \, a^{2} b e^{\left (5 \, d x + 5 \, c\right )} - 19968 \, a b^{2} e^{\left (5 \, d x + 5 \, c\right )} - 4475 \, b^{3} e^{\left (5 \, d x + 5 \, c\right )} - 14400 \, a^{2} b e^{\left (3 \, d x + 3 \, c\right )} - 7680 \, a b^{2} e^{\left (3 \, d x + 3 \, c\right )} + 1985 \, b^{3} e^{\left (3 \, d x + 3 \, c\right )} - 2880 \, a^{2} b e^{\left (d x + c\right )} - 75 \, b^{3} e^{\left (d x + c\right )}}{{\left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right )}^{8}}}{960 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/960*(480*a^3*log(e^(d*x + c) + 1) - 480*a^3*log(abs(e^(d*x + c) - 1)) + 15*(192*a^2*b + 5*b^3)*arctan(e^(d*x
+ c)) - 960*(a^3*e^(3*d*x + 3*c) + a^3*e^(d*x + c))/(e^(2*d*x + 2*c) - 1)^2 + (2880*a^2*b*e^(15*d*x + 15*c) +
75*b^3*e^(15*d*x + 15*c) + 14400*a^2*b*e^(13*d*x + 13*c) - 7680*a*b^2*e^(13*d*x + 13*c) - 1985*b^3*e^(13*d*x
+ 13*c) + 25920*a^2*b*e^(11*d*x + 11*c) - 19968*a*b^2*e^(11*d*x + 11*c) + 4475*b^3*e^(11*d*x + 11*c) + 14400*a
^2*b*e^(9*d*x + 9*c) - 21504*a*b^2*e^(9*d*x + 9*c) - 8825*b^3*e^(9*d*x + 9*c) - 14400*a^2*b*e^(7*d*x + 7*c) -
21504*a*b^2*e^(7*d*x + 7*c) + 8825*b^3*e^(7*d*x + 7*c) - 25920*a^2*b*e^(5*d*x + 5*c) - 19968*a*b^2*e^(5*d*x +
5*c) - 4475*b^3*e^(5*d*x + 5*c) - 14400*a^2*b*e^(3*d*x + 3*c) - 7680*a*b^2*e^(3*d*x + 3*c) + 1985*b^3*e^(3*d*x
+ 3*c) - 2880*a^2*b*e^(d*x + c) - 75*b^3*e^(d*x + c))/(e^(2*d*x + 2*c) + 1)^8)/d

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Mupad [B]
time = 8.05, size = 731, normalized size = 3.15 \begin {gather*} \frac {a^3\,\ln \left ({\mathrm {e}}^{c+d\,x}+1\right )}{2\,d}-\frac {a^3\,\ln \left ({\mathrm {e}}^{c+d\,x}-1\right )}{2\,d}+\frac {{\mathrm {e}}^{c+d\,x}\,\left (192\,a^2\,b+5\,b^3\right )}{64\,d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )}+\frac {{\mathrm {e}}^{c+d\,x}\,\left (2245\,b^3+3264\,a\,b^2\right )}{120\,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 {{\mathrm {e}}^{c+d\,x}\,\left (1325\,b^3+768\,a\,b^2\right )}{20\,d\,\left (4\,{\mathrm {e}}^{2\,c+2\,d\,x}+6\,{\mathrm {e}}^{4\,c+4\,d\,x}+4\,{\mathrm {e}}^{6\,c+6\,d\,x}+{\mathrm {e}}^{8\,c+8\,d\,x}+1\right )}-\frac {500\,b^3\,{\mathrm {e}}^{c+d\,x}}{3\,d\,\left (6\,{\mathrm {e}}^{2\,c+2\,d\,x}+15\,{\mathrm {e}}^{4\,c+4\,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 {2\,{\mathrm {e}}^{c+d\,x}\,\left (1025\,b^3+144\,a\,b^2\right )}{15\,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 {112\,b^3\,{\mathrm {e}}^{c+d\,x}}{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 {{\mathrm {e}}^{c+d\,x}\,\left (576\,a^2\,b+768\,a\,b^2+251\,b^3\right )}{96\,d\,\left (2\,{\mathrm {e}}^{2\,c+2\,d\,x}+{\mathrm {e}}^{4\,c+4\,d\,x}+1\right )}-\frac {32\,b^3\,{\mathrm {e}}^{c+d\,x}}{d\,\left (8\,{\mathrm {e}}^{2\,c+2\,d\,x}+28\,{\mathrm {e}}^{4\,c+4\,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 {a^3\,{\mathrm {e}}^{c+d\,x}}{d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}-1\right )}-\frac {2\,a^3\,{\mathrm {e}}^{c+d\,x}}{d\,\left ({\mathrm {e}}^{4\,c+4\,d\,x}-2\,{\mathrm {e}}^{2\,c+2\,d\,x}+1\right )}-\frac {b\,\ln \left ({\mathrm {e}}^{c+d\,x}-\mathrm {i}\right )\,\left (192\,a^2+5\,b^2\right )\,1{}\mathrm {i}}{128\,d}+\frac {b\,\ln \left ({\mathrm {e}}^{c+d\,x}+1{}\mathrm {i}\right )\,\left (192\,a^2+5\,b^2\right )\,1{}\mathrm {i}}{128\,d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(a^3*log(exp(c + d*x) + 1))/(2*d) - (a^3*log(exp(c + d*x) - 1))/(2*d) + (exp(c + d*x)*(192*a^2*b + 5*b^3))/(64
*d*(exp(2*c + 2*d*x) + 1)) + (exp(c + d*x)*(3264*a*b^2 + 2245*b^3))/(120*d*(3*exp(2*c + 2*d*x) + 3*exp(4*c + 4
*d*x) + exp(6*c + 6*d*x) + 1)) - (b*log(exp(c + d*x) - 1i)*(192*a^2 + 5*b^2)*1i)/(128*d) + (b*log(exp(c + d*x)
+ 1i)*(192*a^2 + 5*b^2)*1i)/(128*d) - (exp(c + d*x)*(768*a*b^2 + 1325*b^3))/(20*d*(4*exp(2*c + 2*d*x) + 6*exp
(4*c + 4*d*x) + 4*exp(6*c + 6*d*x) + exp(8*c + 8*d*x) + 1)) - (500*b^3*exp(c + d*x))/(3*d*(6*exp(2*c + 2*d*x)
+ 15*exp(4*c + 4*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)) + (2*exp(c + d*x)*(144*a*b^2 + 1025*b^3))/(15*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)) + (112*b^3*exp(c + d*x))/(d*(7*exp(2*c + 2*d*x) + 21*e
xp(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) + e
xp(14*c + 14*d*x) + 1)) - (exp(c + d*x)*(768*a*b^2 + 576*a^2*b + 251*b^3))/(96*d*(2*exp(2*c + 2*d*x) + exp(4*c
+ 4*d*x) + 1)) - (32*b^3*exp(c + d*x))/(d*(8*exp(2*c + 2*d*x) + 28*exp(4*c + 4*d*x) + 56*exp(6*c + 6*d*x) + 7
0*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)) - (a^3*exp(c + d*x))/(d*(exp(2*c + 2*d*x) - 1)) - (2*a^3*exp(c + d*x))/(d*(exp(4*c + 4*d*x) - 2*exp(2*c
+ 2*d*x) + 1))

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