∫ csch6(c + dx)(a + b sinh3(c + dx))2 dx [159]

Optimal. Leaf size=88 \[ b^2 x+\frac {a b \tanh ^{-1}(\cosh (c+d x))}{d}-\frac {a^2 \coth (c+d x)}{d}+\frac {2 a^2 \coth ^3(c+d x)}{3 d}-\frac {a^2 \coth ^5(c+d x)}{5 d}-\frac {a b \coth (c+d x) \text {csch}(c+d x)}{d} \]

b^2*x+a*b*arctanh(cosh(d*x+c))/d-a^2*coth(d*x+c)/d+2/3*a^2*coth(d*x+c)^3/d-1/5*a^2*coth(d*x+c)^5/d-a*b*coth(d*

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Rubi [A]
time = 0.07, antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {3299, 3853, 3855, 3852} \begin {gather*} -\frac {a^2 \coth ^5(c+d x)}{5 d}+\frac {2 a^2 \coth ^3(c+d x)}{3 d}-\frac {a^2 \coth (c+d x)}{d}+\frac {a b \tanh ^{-1}(\cosh (c+d x))}{d}-\frac {a b \coth (c+d x) \text {csch}(c+d x)}{d}+b^2 x \end {gather*}

b^2*x + (a*b*ArcTanh[Cosh[c + d*x]])/d - (a^2*Coth[c + d*x])/d + (2*a^2*Coth[c + d*x]^3)/(3*d) - (a^2*Coth[c +

Int[sin[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^(n_))^(p_.), x_Symbol] :> Int[ExpandTr

ig[sin[e + f*x]^m*(a + b*sin[e + f*x]^n)^p, x], x] /; FreeQ[{a, b, e, f}, x] && IntegersQ[m, p] && (EqQ[n, 4]

Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Dist[-d^(-1), Subst[Int[ExpandIntegrand[(1 + x^2)^(n/2 - 1), x]

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,

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

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

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(197\) vs. \(2(88)=176\).
time = 0.67, size = 197, normalized size = 2.24 \begin {gather*} \frac {-256 a^2 \coth \left (\frac {1}{2} (c+d x)\right )-240 a b \text {csch}^2\left (\frac {1}{2} (c+d x)\right )+19 a^2 \text {csch}^4\left (\frac {1}{2} (c+d x)\right ) \sinh (c+d x)-3 a^2 \text {csch}^6\left (\frac {1}{2} (c+d x)\right ) \sinh (c+d x)+16 \left (60 b^2 c+60 b^2 d x-60 a b \log \left (\tanh \left (\frac {1}{2} (c+d x)\right )\right )-15 a b \text {sech}^2\left (\frac {1}{2} (c+d x)\right )-19 a^2 \text {csch}^3(c+d x) \sinh ^4\left (\frac {1}{2} (c+d x)\right )-12 a^2 \text {csch}^5(c+d x) \sinh ^6\left (\frac {1}{2} (c+d x)\right )-16 a^2 \tanh \left (\frac {1}{2} (c+d x)\right )\right )}{960 d} \end {gather*}

(-256*a^2*Coth[(c + d*x)/2] - 240*a*b*Csch[(c + d*x)/2]^2 + 19*a^2*Csch[(c + d*x)/2]^4*Sinh[c + d*x] - 3*a^2*C

sch[(c + d*x)/2]^6*Sinh[c + d*x] + 16*(60*b^2*c + 60*b^2*d*x - 60*a*b*Log[Tanh[(c + d*x)/2]] - 15*a*b*Sech[(c

+ d*x)/2]^2 - 19*a^2*Csch[c + d*x]^3*Sinh[(c + d*x)/2]^4 - 12*a^2*Csch[c + d*x]^5*Sinh[(c + d*x)/2]^6 - 16*a^2

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method	result	size

risch	\(b^{2} x -\frac {2 a \left (15 b \,{\mathrm e}^{9 d x +9 c}-30 b \,{\mathrm e}^{7 d x +7 c}+80 a \,{\mathrm e}^{4 d x +4 c}+30 b \,{\mathrm e}^{3 d x +3 c}-40 a \,{\mathrm e}^{2 d x +2 c}-15 b \,{\mathrm e}^{d x +c}+8 a \right )}{15 d \left ({\mathrm e}^{2 d x +2 c}-1\right )^{5}}+\frac {a b \ln \left ({\mathrm e}^{d x +c}+1\right )}{d}-\frac {a b \ln \left ({\mathrm e}^{d x +c}-1\right )}{d}\)	\(130\)

b^2*x-2/15*a*(15*b*exp(9*d*x+9*c)-30*b*exp(7*d*x+7*c)+80*a*exp(4*d*x+4*c)+30*b*exp(3*d*x+3*c)-40*a*exp(2*d*x+2

*c)-15*b*exp(d*x+c)+8*a)/d/(exp(2*d*x+2*c)-1)^5+a*b/d*ln(exp(d*x+c)+1)-a*b/d*ln(exp(d*x+c)-1)

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 303 vs. \(2 (84) = 168\).
time = 0.28, size = 303, normalized size = 3.44 \begin {gather*} b^{2} x + a b {\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 {16}{15} \, a^{2} {\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 )} \end {gather*}

b^2*x + a*b*(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))) - 16/15*a^2*(5*e^(-2*d*x - 2*c)/(d*(5*e^(-2*d*x - 2*c) - 10*e^(-4*d*x

- 4*c) + 10*e^(-6*d*x - 6*c) - 5*e^(-8*d*x - 8*c) + e^(-10*d*x - 10*c) - 1)) - 10*e^(-4*d*x - 4*c)/(d*(5*e^(-2

*d*x - 2*c) - 10*e^(-4*d*x - 4*c) + 10*e^(-6*d*x - 6*c) - 5*e^(-8*d*x - 8*c) + e^(-10*d*x - 10*c) - 1)) - 1/(d

*(5*e^(-2*d*x - 2*c) - 10*e^(-4*d*x - 4*c) + 10*e^(-6*d*x - 6*c) - 5*e^(-8*d*x - 8*c) + e^(-10*d*x - 10*c) - 1

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 2310 vs. \(2 (84) = 168\).
time = 0.47, size = 2310, normalized size = 26.25 \begin {gather*} \text {Too large to display} \end {gather*}

1/15*(15*b^2*d*x*cosh(d*x + c)^10 + 15*b^2*d*x*sinh(d*x + c)^10 - 75*b^2*d*x*cosh(d*x + c)^8 - 30*a*b*cosh(d*x

+ c)^9 + 150*b^2*d*x*cosh(d*x + c)^6 + 30*(5*b^2*d*x*cosh(d*x + c) - a*b)*sinh(d*x + c)^9 + 60*a*b*cosh(d*x +

c)^7 + 15*(45*b^2*d*x*cosh(d*x + c)^2 - 5*b^2*d*x - 18*a*b*cosh(d*x + c))*sinh(d*x + c)^8 + 60*(30*b^2*d*x*co

sh(d*x + c)^3 - 10*b^2*d*x*cosh(d*x + c) - 18*a*b*cosh(d*x + c)^2 + a*b)*sinh(d*x + c)^7 + 30*(105*b^2*d*x*cos

h(d*x + c)^4 - 70*b^2*d*x*cosh(d*x + c)^2 - 84*a*b*cosh(d*x + c)^3 + 5*b^2*d*x + 14*a*b*cosh(d*x + c))*sinh(d*

x + c)^6 + 60*(63*b^2*d*x*cosh(d*x + c)^5 - 70*b^2*d*x*cosh(d*x + c)^3 - 63*a*b*cosh(d*x + c)^4 + 15*b^2*d*x*c

osh(d*x + c) + 21*a*b*cosh(d*x + c)^2)*sinh(d*x + c)^5 - 60*a*b*cosh(d*x + c)^3 - 10*(15*b^2*d*x + 16*a^2)*cos

h(d*x + c)^4 + 10*(315*b^2*d*x*cosh(d*x + c)^6 - 525*b^2*d*x*cosh(d*x + c)^4 - 378*a*b*cosh(d*x + c)^5 + 225*b

^2*d*x*cosh(d*x + c)^2 + 210*a*b*cosh(d*x + c)^3 - 15*b^2*d*x - 16*a^2)*sinh(d*x + c)^4 - 15*b^2*d*x + 20*(90*

b^2*d*x*cosh(d*x + c)^7 - 210*b^2*d*x*cosh(d*x + c)^5 - 126*a*b*cosh(d*x + c)^6 + 150*b^2*d*x*cosh(d*x + c)^3

+ 105*a*b*cosh(d*x + c)^4 - 3*a*b - 2*(15*b^2*d*x + 16*a^2)*cosh(d*x + c))*sinh(d*x + c)^3 + 30*a*b*cosh(d*x +

c) + 5*(15*b^2*d*x + 16*a^2)*cosh(d*x + c)^2 + 5*(135*b^2*d*x*cosh(d*x + c)^8 - 420*b^2*d*x*cosh(d*x + c)^6 -

216*a*b*cosh(d*x + c)^7 + 450*b^2*d*x*cosh(d*x + c)^4 + 252*a*b*cosh(d*x + c)^5 + 15*b^2*d*x - 36*a*b*cosh(d*

x + c) - 12*(15*b^2*d*x + 16*a^2)*cosh(d*x + c)^2 + 16*a^2)*sinh(d*x + c)^2 - 16*a^2 + 15*(a*b*cosh(d*x + c)^1

0 + 10*a*b*cosh(d*x + c)*sinh(d*x + c)^9 + a*b*sinh(d*x + c)^10 - 5*a*b*cosh(d*x + c)^8 + 5*(9*a*b*cosh(d*x +

c)^2 - a*b)*sinh(d*x + c)^8 + 10*a*b*cosh(d*x + c)^6 + 40*(3*a*b*cosh(d*x + c)^3 - a*b*cosh(d*x + c))*sinh(d*x

+ c)^7 + 10*(21*a*b*cosh(d*x + c)^4 - 14*a*b*cosh(d*x + c)^2 + a*b)*sinh(d*x + c)^6 - 10*a*b*cosh(d*x + c)^4

+ 4*(63*a*b*cosh(d*x + c)^5 - 70*a*b*cosh(d*x + c)^3 + 15*a*b*cosh(d*x + c))*sinh(d*x + c)^5 + 10*(21*a*b*cosh

(d*x + c)^6 - 35*a*b*cosh(d*x + c)^4 + 15*a*b*cosh(d*x + c)^2 - a*b)*sinh(d*x + c)^4 + 5*a*b*cosh(d*x + c)^2 +

40*(3*a*b*cosh(d*x + c)^7 - 7*a*b*cosh(d*x + c)^5 + 5*a*b*cosh(d*x + c)^3 - a*b*cosh(d*x + c))*sinh(d*x + c)^

3 + 5*(9*a*b*cosh(d*x + c)^8 - 28*a*b*cosh(d*x + c)^6 + 30*a*b*cosh(d*x + c)^4 - 12*a*b*cosh(d*x + c)^2 + a*b)

*sinh(d*x + c)^2 - a*b + 10*(a*b*cosh(d*x + c)^9 - 4*a*b*cosh(d*x + c)^7 + 6*a*b*cosh(d*x + c)^5 - 4*a*b*cosh(

d*x + c)^3 + a*b*cosh(d*x + c))*sinh(d*x + c))*log(cosh(d*x + c) + sinh(d*x + c) + 1) - 15*(a*b*cosh(d*x + c)^

10 + 10*a*b*cosh(d*x + c)*sinh(d*x + c)^9 + a*b*sinh(d*x + c)^10 - 5*a*b*cosh(d*x + c)^8 + 5*(9*a*b*cosh(d*x +

c)^2 - a*b)*sinh(d*x + c)^8 + 10*a*b*cosh(d*x + c)^6 + 40*(3*a*b*cosh(d*x + c)^3 - a*b*cosh(d*x + c))*sinh(d*

x + c)^7 + 10*(21*a*b*cosh(d*x + c)^4 - 14*a*b*cosh(d*x + c)^2 + a*b)*sinh(d*x + c)^6 - 10*a*b*cosh(d*x + c)^4

+ 4*(63*a*b*cosh(d*x + c)^5 - 70*a*b*cosh(d*x + c)^3 + 15*a*b*cosh(d*x + c))*sinh(d*x + c)^5 + 10*(21*a*b*cos

h(d*x + c)^6 - 35*a*b*cosh(d*x + c)^4 + 15*a*b*cosh(d*x + c)^2 - a*b)*sinh(d*x + c)^4 + 5*a*b*cosh(d*x + c)^2

+ 40*(3*a*b*cosh(d*x + c)^7 - 7*a*b*cosh(d*x + c)^5 + 5*a*b*cosh(d*x + c)^3 - a*b*cosh(d*x + c))*sinh(d*x + c)

^3 + 5*(9*a*b*cosh(d*x + c)^8 - 28*a*b*cosh(d*x + c)^6 + 30*a*b*cosh(d*x + c)^4 - 12*a*b*cosh(d*x + c)^2 + a*b

)*sinh(d*x + c)^2 - a*b + 10*(a*b*cosh(d*x + c)^9 - 4*a*b*cosh(d*x + c)^7 + 6*a*b*cosh(d*x + c)^5 - 4*a*b*cosh

(d*x + c)^3 + a*b*cosh(d*x + c))*sinh(d*x + c))*log(cosh(d*x + c) + sinh(d*x + c) - 1) + 10*(15*b^2*d*x*cosh(d

*x + c)^9 - 60*b^2*d*x*cosh(d*x + c)^7 - 27*a*b*cosh(d*x + c)^8 + 90*b^2*d*x*cosh(d*x + c)^5 + 42*a*b*cosh(d*x

+ c)^6 - 18*a*b*cosh(d*x + c)^2 - 4*(15*b^2*d*x + 16*a^2)*cosh(d*x + c)^3 + 3*a*b + (15*b^2*d*x + 16*a^2)*cos

h(d*x + c))*sinh(d*x + c))/(d*cosh(d*x + c)^10 + 10*d*cosh(d*x + c)*sinh(d*x + c)^9 + d*sinh(d*x + c)^10 - 5*d

*cosh(d*x + c)^8 + 5*(9*d*cosh(d*x + c)^2 - d)*sinh(d*x + c)^8 + 40*(3*d*cosh(d*x + c)^3 - d*cosh(d*x + c))*si

nh(d*x + c)^7 + 10*d*cosh(d*x + c)^6 + 10*(21*d*cosh(d*x + c)^4 - 14*d*cosh(d*x + c)^2 + d)*sinh(d*x + c)^6 +

4*(63*d*cosh(d*x + c)^5 - 70*d*cosh(d*x + c)^3 + 15*d*cosh(d*x + c))*sinh(d*x + c)^5 - 10*d*cosh(d*x + c)^4 +

10*(21*d*cosh(d*x + c)^6 - 35*d*cosh(d*x + c)^4 + 15*d*cosh(d*x + c)^2 - d)*sinh(d*x + c)^4 + 40*(3*d*cosh(d*x

+ c)^7 - 7*d*cosh(d*x + c)^5 + 5*d*cosh(d*x + c)^3 - d*cosh(d*x + c))*sinh(d*x + c)^3 + 5*d*cosh(d*x + c)^2 +

5*(9*d*cosh(d*x + c)^8 - 28*d*cosh(d*x + c)^6 + 30*d*cosh(d*x + c)^4 - 12*d*cosh(d*x + c)^2 + d)*sinh(d*x + c

)^2 + 10*(d*cosh(d*x + c)^9 - 4*d*cosh(d*x + c)^7 + 6*d*cosh(d*x + c)^5 - 4*d*cosh(d*x + c)^3 + d*cosh(d*x + c

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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*}

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Giac [A]
time = 0.45, size = 141, normalized size = 1.60 \begin {gather*} \frac {15 \, {\left (d x + c\right )} b^{2} + 15 \, a b \log \left (e^{\left (d x + c\right )} + 1\right ) - 15 \, a b \log \left ({\left | e^{\left (d x + c\right )} - 1 \right |}\right ) - \frac {2 \, {\left (15 \, a b e^{\left (9 \, d x + 9 \, c\right )} - 30 \, a b e^{\left (7 \, d x + 7 \, c\right )} + 80 \, a^{2} e^{\left (4 \, d x + 4 \, c\right )} + 30 \, a b e^{\left (3 \, d x + 3 \, c\right )} - 40 \, a^{2} e^{\left (2 \, d x + 2 \, c\right )} - 15 \, a b e^{\left (d x + c\right )} + 8 \, a^{2}\right )}}{{\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )}^{5}}}{15 \, d} \end {gather*}

1/15*(15*(d*x + c)*b^2 + 15*a*b*log(e^(d*x + c) + 1) - 15*a*b*log(abs(e^(d*x + c) - 1)) - 2*(15*a*b*e^(9*d*x +

9*c) - 30*a*b*e^(7*d*x + 7*c) + 80*a^2*e^(4*d*x + 4*c) + 30*a*b*e^(3*d*x + 3*c) - 40*a^2*e^(2*d*x + 2*c) - 15

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Mupad [B]
time = 0.65, size = 351, normalized size = 3.99 \begin {gather*} b^2\,x-\frac {\frac {32\,a^2\,{\mathrm {e}}^{4\,c+4\,d\,x}}{5\,d}-\frac {8\,a\,b\,{\mathrm {e}}^{c+d\,x}}{5\,d}+\frac {24\,a\,b\,{\mathrm {e}}^{3\,c+3\,d\,x}}{5\,d}-\frac {24\,a\,b\,{\mathrm {e}}^{5\,c+5\,d\,x}}{5\,d}+\frac {8\,a\,b\,{\mathrm {e}}^{7\,c+7\,d\,x}}{5\,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 {2\,\mathrm {atan}\left (\frac {a\,b\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\sqrt {-d^2}}{d\,\sqrt {a^2\,b^2}}\right )\,\sqrt {a^2\,b^2}}{\sqrt {-d^2}}-\frac {64\,a^2}{15\,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 {16\,a^2}{5\,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 {2\,a\,b\,{\mathrm {e}}^{c+d\,x}}{d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}-1\right )}-\frac {12\,a\,b\,{\mathrm {e}}^{c+d\,x}}{5\,d\,\left ({\mathrm {e}}^{4\,c+4\,d\,x}-2\,{\mathrm {e}}^{2\,c+2\,d\,x}+1\right )} \end {gather*}

b^2*x - ((32*a^2*exp(4*c + 4*d*x))/(5*d) - (8*a*b*exp(c + d*x))/(5*d) + (24*a*b*exp(3*c + 3*d*x))/(5*d) - (24*

a*b*exp(5*c + 5*d*x))/(5*d) + (8*a*b*exp(7*c + 7*d*x))/(5*d))/(5*exp(2*c + 2*d*x) - 10*exp(4*c + 4*d*x) + 10*e

xp(6*c + 6*d*x) - 5*exp(8*c + 8*d*x) + exp(10*c + 10*d*x) - 1) + (2*atan((a*b*exp(d*x)*exp(c)*(-d^2)^(1/2))/(d

*(a^2*b^2)^(1/2)))*(a^2*b^2)^(1/2))/(-d^2)^(1/2) - (64*a^2)/(15*d*(3*exp(2*c + 2*d*x) - 3*exp(4*c + 4*d*x) + e

xp(6*c + 6*d*x) - 1)) - (16*a^2)/(5*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)) - (2*a*b*exp(c + d*x))/(d*(exp(2*c + 2*d*x) - 1)) - (12*a*b*exp(c + d*x))/(5*d*(exp(4*c + 4*d*x

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3.2.59 \(\int \text {csch}^6(c+d x) (a+b \sinh ^3(c+d x))^2 \, dx\) [159]