3.164 \(\int \text{csch}(c+d x) (a+b \sinh ^3(c+d x))^3 \, dx\)

Optimal. Leaf size=201 \[ \frac{3 a^2 b \sinh (c+d x) \cosh (c+d x)}{2 d}-\frac{3}{2} a^2 b x-\frac{a^3 \tanh ^{-1}(\cosh (c+d x))}{d}+\frac{3 a b^2 \cosh ^5(c+d x)}{5 d}-\frac{2 a b^2 \cosh ^3(c+d x)}{d}+\frac{3 a b^2 \cosh (c+d x)}{d}+\frac{b^3 \sinh ^7(c+d x) \cosh (c+d x)}{8 d}-\frac{7 b^3 \sinh ^5(c+d x) \cosh (c+d x)}{48 d}+\frac{35 b^3 \sinh ^3(c+d x) \cosh (c+d x)}{192 d}-\frac{35 b^3 \sinh (c+d x) \cosh (c+d x)}{128 d}+\frac{35 b^3 x}{128} \]

[Out]

(-3*a^2*b*x)/2 + (35*b^3*x)/128 - (a^3*ArcTanh[Cosh[c + d*x]])/d + (3*a*b^2*Cosh[c + d*x])/d - (2*a*b^2*Cosh[c
+ d*x]^3)/d + (3*a*b^2*Cosh[c + d*x]^5)/(5*d) + (3*a^2*b*Cosh[c + d*x]*Sinh[c + d*x])/(2*d) - (35*b^3*Cosh[c
+ d*x]*Sinh[c + d*x])/(128*d) + (35*b^3*Cosh[c + d*x]*Sinh[c + d*x]^3)/(192*d) - (7*b^3*Cosh[c + d*x]*Sinh[c +
d*x]^5)/(48*d) + (b^3*Cosh[c + d*x]*Sinh[c + d*x]^7)/(8*d)

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Rubi [A]  time = 0.188718, antiderivative size = 201, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238, Rules used = {3220, 3770, 2635, 8, 2633} \[ \frac{3 a^2 b \sinh (c+d x) \cosh (c+d x)}{2 d}-\frac{3}{2} a^2 b x-\frac{a^3 \tanh ^{-1}(\cosh (c+d x))}{d}+\frac{3 a b^2 \cosh ^5(c+d x)}{5 d}-\frac{2 a b^2 \cosh ^3(c+d x)}{d}+\frac{3 a b^2 \cosh (c+d x)}{d}+\frac{b^3 \sinh ^7(c+d x) \cosh (c+d x)}{8 d}-\frac{7 b^3 \sinh ^5(c+d x) \cosh (c+d x)}{48 d}+\frac{35 b^3 \sinh ^3(c+d x) \cosh (c+d x)}{192 d}-\frac{35 b^3 \sinh (c+d x) \cosh (c+d x)}{128 d}+\frac{35 b^3 x}{128} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*a^2*b*x)/2 + (35*b^3*x)/128 - (a^3*ArcTanh[Cosh[c + d*x]])/d + (3*a*b^2*Cosh[c + d*x])/d - (2*a*b^2*Cosh[c
+ d*x]^3)/d + (3*a*b^2*Cosh[c + d*x]^5)/(5*d) + (3*a^2*b*Cosh[c + d*x]*Sinh[c + d*x])/(2*d) - (35*b^3*Cosh[c
+ d*x]*Sinh[c + d*x])/(128*d) + (35*b^3*Cosh[c + d*x]*Sinh[c + d*x]^3)/(192*d) - (7*b^3*Cosh[c + d*x]*Sinh[c +
d*x]^5)/(48*d) + (b^3*Cosh[c + d*x]*Sinh[c + d*x]^7)/(8*d)

Rule 3220

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]
|| GtQ[p, 0] || (EqQ[p, -1] && IntegerQ[n]))

Rule 3770

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

Rule 2635

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

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 2633

Int[sin[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> -Dist[d^(-1), Subst[Int[Expand[(1 - x^2)^((n - 1)/2), x], x], x
, Cos[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[(n - 1)/2, 0]

Rubi steps

\begin{align*} \int \text{csch}(c+d x) \left (a+b \sinh ^3(c+d x)\right )^3 \, dx &=i \int \left (-i a^3 \text{csch}(c+d x)-3 i a^2 b \sinh ^2(c+d x)-3 i a b^2 \sinh ^5(c+d x)-i b^3 \sinh ^8(c+d x)\right ) \, dx\\ &=a^3 \int \text{csch}(c+d x) \, dx+\left (3 a^2 b\right ) \int \sinh ^2(c+d x) \, dx+\left (3 a b^2\right ) \int \sinh ^5(c+d x) \, dx+b^3 \int \sinh ^8(c+d x) \, dx\\ &=-\frac{a^3 \tanh ^{-1}(\cosh (c+d x))}{d}+\frac{3 a^2 b \cosh (c+d x) \sinh (c+d x)}{2 d}+\frac{b^3 \cosh (c+d x) \sinh ^7(c+d x)}{8 d}-\frac{1}{2} \left (3 a^2 b\right ) \int 1 \, dx-\frac{1}{8} \left (7 b^3\right ) \int \sinh ^6(c+d x) \, dx+\frac{\left (3 a b^2\right ) \operatorname{Subst}\left (\int \left (1-2 x^2+x^4\right ) \, dx,x,\cosh (c+d x)\right )}{d}\\ &=-\frac{3}{2} a^2 b x-\frac{a^3 \tanh ^{-1}(\cosh (c+d x))}{d}+\frac{3 a b^2 \cosh (c+d x)}{d}-\frac{2 a b^2 \cosh ^3(c+d x)}{d}+\frac{3 a b^2 \cosh ^5(c+d x)}{5 d}+\frac{3 a^2 b \cosh (c+d x) \sinh (c+d x)}{2 d}-\frac{7 b^3 \cosh (c+d x) \sinh ^5(c+d x)}{48 d}+\frac{b^3 \cosh (c+d x) \sinh ^7(c+d x)}{8 d}+\frac{1}{48} \left (35 b^3\right ) \int \sinh ^4(c+d x) \, dx\\ &=-\frac{3}{2} a^2 b x-\frac{a^3 \tanh ^{-1}(\cosh (c+d x))}{d}+\frac{3 a b^2 \cosh (c+d x)}{d}-\frac{2 a b^2 \cosh ^3(c+d x)}{d}+\frac{3 a b^2 \cosh ^5(c+d x)}{5 d}+\frac{3 a^2 b \cosh (c+d x) \sinh (c+d x)}{2 d}+\frac{35 b^3 \cosh (c+d x) \sinh ^3(c+d x)}{192 d}-\frac{7 b^3 \cosh (c+d x) \sinh ^5(c+d x)}{48 d}+\frac{b^3 \cosh (c+d x) \sinh ^7(c+d x)}{8 d}-\frac{1}{64} \left (35 b^3\right ) \int \sinh ^2(c+d x) \, dx\\ &=-\frac{3}{2} a^2 b x-\frac{a^3 \tanh ^{-1}(\cosh (c+d x))}{d}+\frac{3 a b^2 \cosh (c+d x)}{d}-\frac{2 a b^2 \cosh ^3(c+d x)}{d}+\frac{3 a b^2 \cosh ^5(c+d x)}{5 d}+\frac{3 a^2 b \cosh (c+d x) \sinh (c+d x)}{2 d}-\frac{35 b^3 \cosh (c+d x) \sinh (c+d x)}{128 d}+\frac{35 b^3 \cosh (c+d x) \sinh ^3(c+d x)}{192 d}-\frac{7 b^3 \cosh (c+d x) \sinh ^5(c+d x)}{48 d}+\frac{b^3 \cosh (c+d x) \sinh ^7(c+d x)}{8 d}+\frac{1}{128} \left (35 b^3\right ) \int 1 \, dx\\ &=-\frac{3}{2} a^2 b x+\frac{35 b^3 x}{128}-\frac{a^3 \tanh ^{-1}(\cosh (c+d x))}{d}+\frac{3 a b^2 \cosh (c+d x)}{d}-\frac{2 a b^2 \cosh ^3(c+d x)}{d}+\frac{3 a b^2 \cosh ^5(c+d x)}{5 d}+\frac{3 a^2 b \cosh (c+d x) \sinh (c+d x)}{2 d}-\frac{35 b^3 \cosh (c+d x) \sinh (c+d x)}{128 d}+\frac{35 b^3 \cosh (c+d x) \sinh ^3(c+d x)}{192 d}-\frac{7 b^3 \cosh (c+d x) \sinh ^5(c+d x)}{48 d}+\frac{b^3 \cosh (c+d x) \sinh ^7(c+d x)}{8 d}\\ \end{align*}

Mathematica [A]  time = 0.21892, size = 158, normalized size = 0.79 \[ \frac{11520 a^2 b \sinh (2 (c+d x))-23040 a^2 b c-23040 a^2 b d x+15360 a^3 \log \left (\tanh \left (\frac{1}{2} (c+d x)\right )\right )+28800 a b^2 \cosh (c+d x)-4800 a b^2 \cosh (3 (c+d x))+576 a b^2 \cosh (5 (c+d x))-3360 b^3 \sinh (2 (c+d x))+840 b^3 \sinh (4 (c+d x))-160 b^3 \sinh (6 (c+d x))+15 b^3 \sinh (8 (c+d x))+4200 b^3 c+4200 b^3 d x}{15360 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-23040*a^2*b*c + 4200*b^3*c - 23040*a^2*b*d*x + 4200*b^3*d*x + 28800*a*b^2*Cosh[c + d*x] - 4800*a*b^2*Cosh[3*
(c + d*x)] + 576*a*b^2*Cosh[5*(c + d*x)] + 15360*a^3*Log[Tanh[(c + d*x)/2]] + 11520*a^2*b*Sinh[2*(c + d*x)] -
3360*b^3*Sinh[2*(c + d*x)] + 840*b^3*Sinh[4*(c + d*x)] - 160*b^3*Sinh[6*(c + d*x)] + 15*b^3*Sinh[8*(c + d*x)])
/(15360*d)

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Maple [A]  time = 0.072, size = 138, normalized size = 0.7 \begin{align*}{\frac{1}{d} \left ( -2\,{a}^{3}{\it Artanh} \left ({{\rm e}^{dx+c}} \right ) +3\,{a}^{2}b \left ( 1/2\,\cosh \left ( dx+c \right ) \sinh \left ( dx+c \right ) -1/2\,dx-c/2 \right ) +3\,a{b}^{2} \left ({\frac{8}{15}}+1/5\, \left ( \sinh \left ( dx+c \right ) \right ) ^{4}-{\frac{4\, \left ( \sinh \left ( dx+c \right ) \right ) ^{2}}{15}} \right ) \cosh \left ( dx+c \right ) +{b}^{3} \left ( \left ({\frac{ \left ( \sinh \left ( dx+c \right ) \right ) ^{7}}{8}}-{\frac{7\, \left ( \sinh \left ( dx+c \right ) \right ) ^{5}}{48}}+{\frac{35\, \left ( \sinh \left ( dx+c \right ) \right ) ^{3}}{192}}-{\frac{35\,\sinh \left ( dx+c \right ) }{128}} \right ) \cosh \left ( dx+c \right ) +{\frac{35\,dx}{128}}+{\frac{35\,c}{128}} \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/d*(-2*a^3*arctanh(exp(d*x+c))+3*a^2*b*(1/2*cosh(d*x+c)*sinh(d*x+c)-1/2*d*x-1/2*c)+3*a*b^2*(8/15+1/5*sinh(d*x
+c)^4-4/15*sinh(d*x+c)^2)*cosh(d*x+c)+b^3*((1/8*sinh(d*x+c)^7-7/48*sinh(d*x+c)^5+35/192*sinh(d*x+c)^3-35/128*s
inh(d*x+c))*cosh(d*x+c)+35/128*d*x+35/128*c))

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Maxima [A]  time = 1.18444, size = 347, normalized size = 1.73 \begin{align*} -\frac{3}{8} \, a^{2} b{\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{1}{6144} \, b^{3}{\left (\frac{{\left (32 \, e^{\left (-2 \, d x - 2 \, c\right )} - 168 \, e^{\left (-4 \, d x - 4 \, c\right )} + 672 \, e^{\left (-6 \, d x - 6 \, c\right )} - 3\right )} e^{\left (8 \, d x + 8 \, c\right )}}{d} - \frac{1680 \,{\left (d x + c\right )}}{d} - \frac{672 \, e^{\left (-2 \, d x - 2 \, c\right )} - 168 \, e^{\left (-4 \, d x - 4 \, c\right )} + 32 \, e^{\left (-6 \, d x - 6 \, c\right )} - 3 \, e^{\left (-8 \, d x - 8 \, c\right )}}{d}\right )} + \frac{1}{160} \, a b^{2}{\left (\frac{3 \, e^{\left (5 \, d x + 5 \, c\right )}}{d} - \frac{25 \, e^{\left (3 \, d x + 3 \, c\right )}}{d} + \frac{150 \, e^{\left (d x + c\right )}}{d} + \frac{150 \, e^{\left (-d x - c\right )}}{d} - \frac{25 \, e^{\left (-3 \, d x - 3 \, c\right )}}{d} + \frac{3 \, e^{\left (-5 \, d x - 5 \, c\right )}}{d}\right )} + \frac{a^{3} \log \left (\tanh \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )\right )}{d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3/8*a^2*b*(4*x - e^(2*d*x + 2*c)/d + e^(-2*d*x - 2*c)/d) - 1/6144*b^3*((32*e^(-2*d*x - 2*c) - 168*e^(-4*d*x -
4*c) + 672*e^(-6*d*x - 6*c) - 3)*e^(8*d*x + 8*c)/d - 1680*(d*x + c)/d - (672*e^(-2*d*x - 2*c) - 168*e^(-4*d*x
- 4*c) + 32*e^(-6*d*x - 6*c) - 3*e^(-8*d*x - 8*c))/d) + 1/160*a*b^2*(3*e^(5*d*x + 5*c)/d - 25*e^(3*d*x + 3*c)
/d + 150*e^(d*x + c)/d + 150*e^(-d*x - c)/d - 25*e^(-3*d*x - 3*c)/d + 3*e^(-5*d*x - 5*c)/d) + a^3*log(tanh(1/2
*d*x + 1/2*c))/d

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Fricas [B]  time = 2.07054, size = 7191, normalized size = 35.78 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/30720*(15*b^3*cosh(d*x + c)^16 + 240*b^3*cosh(d*x + c)*sinh(d*x + c)^15 + 15*b^3*sinh(d*x + c)^16 - 160*b^3*
cosh(d*x + c)^14 + 576*a*b^2*cosh(d*x + c)^13 + 840*b^3*cosh(d*x + c)^12 + 40*(45*b^3*cosh(d*x + c)^2 - 4*b^3)
*sinh(d*x + c)^14 - 4800*a*b^2*cosh(d*x + c)^11 + 16*(525*b^3*cosh(d*x + c)^3 - 140*b^3*cosh(d*x + c) + 36*a*b
^2)*sinh(d*x + c)^13 + 4*(6825*b^3*cosh(d*x + c)^4 - 3640*b^3*cosh(d*x + c)^2 + 1872*a*b^2*cosh(d*x + c) + 210
*b^3)*sinh(d*x + c)^12 + 28800*a*b^2*cosh(d*x + c)^9 + 16*(4095*b^3*cosh(d*x + c)^5 - 3640*b^3*cosh(d*x + c)^3
+ 2808*a*b^2*cosh(d*x + c)^2 + 630*b^3*cosh(d*x + c) - 300*a*b^2)*sinh(d*x + c)^11 - 240*(192*a^2*b - 35*b^3)
*d*x*cosh(d*x + c)^8 + 480*(24*a^2*b - 7*b^3)*cosh(d*x + c)^10 + 8*(15015*b^3*cosh(d*x + c)^6 - 20020*b^3*cosh
(d*x + c)^4 + 20592*a*b^2*cosh(d*x + c)^3 + 6930*b^3*cosh(d*x + c)^2 - 6600*a*b^2*cosh(d*x + c) + 1440*a^2*b -
420*b^3)*sinh(d*x + c)^10 + 28800*a*b^2*cosh(d*x + c)^7 + 80*(2145*b^3*cosh(d*x + c)^7 - 4004*b^3*cosh(d*x +
c)^5 + 5148*a*b^2*cosh(d*x + c)^4 + 2310*b^3*cosh(d*x + c)^3 - 3300*a*b^2*cosh(d*x + c)^2 + 360*a*b^2 + 60*(24
*a^2*b - 7*b^3)*cosh(d*x + c))*sinh(d*x + c)^9 + 6*(32175*b^3*cosh(d*x + c)^8 - 80080*b^3*cosh(d*x + c)^6 + 12
3552*a*b^2*cosh(d*x + c)^5 + 69300*b^3*cosh(d*x + c)^4 - 132000*a*b^2*cosh(d*x + c)^3 + 43200*a*b^2*cosh(d*x +
c) - 40*(192*a^2*b - 35*b^3)*d*x + 3600*(24*a^2*b - 7*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^8 - 4800*a*b^2*cosh
(d*x + c)^5 + 48*(3575*b^3*cosh(d*x + c)^9 - 11440*b^3*cosh(d*x + c)^7 + 20592*a*b^2*cosh(d*x + c)^6 + 13860*b
^3*cosh(d*x + c)^5 - 33000*a*b^2*cosh(d*x + c)^4 + 21600*a*b^2*cosh(d*x + c)^2 - 40*(192*a^2*b - 35*b^3)*d*x*c
osh(d*x + c) + 1200*(24*a^2*b - 7*b^3)*cosh(d*x + c)^3 + 600*a*b^2)*sinh(d*x + c)^7 - 840*b^3*cosh(d*x + c)^4
- 480*(24*a^2*b - 7*b^3)*cosh(d*x + c)^6 + 24*(5005*b^3*cosh(d*x + c)^10 - 20020*b^3*cosh(d*x + c)^8 + 41184*a
*b^2*cosh(d*x + c)^7 + 32340*b^3*cosh(d*x + c)^6 - 92400*a*b^2*cosh(d*x + c)^5 + 100800*a*b^2*cosh(d*x + c)^3
- 280*(192*a^2*b - 35*b^3)*d*x*cosh(d*x + c)^2 + 4200*(24*a^2*b - 7*b^3)*cosh(d*x + c)^4 + 8400*a*b^2*cosh(d*x
+ c) - 480*a^2*b + 140*b^3)*sinh(d*x + c)^6 + 576*a*b^2*cosh(d*x + c)^3 + 16*(4095*b^3*cosh(d*x + c)^11 - 200
20*b^3*cosh(d*x + c)^9 + 46332*a*b^2*cosh(d*x + c)^8 + 41580*b^3*cosh(d*x + c)^7 - 138600*a*b^2*cosh(d*x + c)^
6 + 226800*a*b^2*cosh(d*x + c)^4 - 840*(192*a^2*b - 35*b^3)*d*x*cosh(d*x + c)^3 + 7560*(24*a^2*b - 7*b^3)*cosh
(d*x + c)^5 + 37800*a*b^2*cosh(d*x + c)^2 - 300*a*b^2 - 180*(24*a^2*b - 7*b^3)*cosh(d*x + c))*sinh(d*x + c)^5
+ 160*b^3*cosh(d*x + c)^2 + 20*(1365*b^3*cosh(d*x + c)^12 - 8008*b^3*cosh(d*x + c)^10 + 20592*a*b^2*cosh(d*x +
c)^9 + 20790*b^3*cosh(d*x + c)^8 - 79200*a*b^2*cosh(d*x + c)^7 + 181440*a*b^2*cosh(d*x + c)^5 - 840*(192*a^2*
b - 35*b^3)*d*x*cosh(d*x + c)^4 + 5040*(24*a^2*b - 7*b^3)*cosh(d*x + c)^6 + 50400*a*b^2*cosh(d*x + c)^3 - 1200
*a*b^2*cosh(d*x + c) - 42*b^3 - 360*(24*a^2*b - 7*b^3)*cosh(d*x + c)^2)*sinh(d*x + c)^4 + 16*(525*b^3*cosh(d*x
+ c)^13 - 3640*b^3*cosh(d*x + c)^11 + 10296*a*b^2*cosh(d*x + c)^10 + 11550*b^3*cosh(d*x + c)^9 - 49500*a*b^2*
cosh(d*x + c)^8 + 151200*a*b^2*cosh(d*x + c)^6 - 840*(192*a^2*b - 35*b^3)*d*x*cosh(d*x + c)^5 + 3600*(24*a^2*b
- 7*b^3)*cosh(d*x + c)^7 + 63000*a*b^2*cosh(d*x + c)^4 - 3000*a*b^2*cosh(d*x + c)^2 - 210*b^3*cosh(d*x + c) -
600*(24*a^2*b - 7*b^3)*cosh(d*x + c)^3 + 36*a*b^2)*sinh(d*x + c)^3 - 15*b^3 + 8*(225*b^3*cosh(d*x + c)^14 - 1
820*b^3*cosh(d*x + c)^12 + 5616*a*b^2*cosh(d*x + c)^11 + 6930*b^3*cosh(d*x + c)^10 - 33000*a*b^2*cosh(d*x + c)
^9 + 129600*a*b^2*cosh(d*x + c)^7 - 840*(192*a^2*b - 35*b^3)*d*x*cosh(d*x + c)^6 + 2700*(24*a^2*b - 7*b^3)*cos
h(d*x + c)^8 + 75600*a*b^2*cosh(d*x + c)^5 - 6000*a*b^2*cosh(d*x + c)^3 - 630*b^3*cosh(d*x + c)^2 - 900*(24*a^
2*b - 7*b^3)*cosh(d*x + c)^4 + 216*a*b^2*cosh(d*x + c) + 20*b^3)*sinh(d*x + c)^2 - 30720*(a^3*cosh(d*x + c)^8
+ 8*a^3*cosh(d*x + c)^7*sinh(d*x + c) + 28*a^3*cosh(d*x + c)^6*sinh(d*x + c)^2 + 56*a^3*cosh(d*x + c)^5*sinh(d
*x + c)^3 + 70*a^3*cosh(d*x + c)^4*sinh(d*x + c)^4 + 56*a^3*cosh(d*x + c)^3*sinh(d*x + c)^5 + 28*a^3*cosh(d*x
+ c)^2*sinh(d*x + c)^6 + 8*a^3*cosh(d*x + c)*sinh(d*x + c)^7 + a^3*sinh(d*x + c)^8)*log(cosh(d*x + c) + sinh(d
*x + c) + 1) + 30720*(a^3*cosh(d*x + c)^8 + 8*a^3*cosh(d*x + c)^7*sinh(d*x + c) + 28*a^3*cosh(d*x + c)^6*sinh(
d*x + c)^2 + 56*a^3*cosh(d*x + c)^5*sinh(d*x + c)^3 + 70*a^3*cosh(d*x + c)^4*sinh(d*x + c)^4 + 56*a^3*cosh(d*x
+ c)^3*sinh(d*x + c)^5 + 28*a^3*cosh(d*x + c)^2*sinh(d*x + c)^6 + 8*a^3*cosh(d*x + c)*sinh(d*x + c)^7 + a^3*s
inh(d*x + c)^8)*log(cosh(d*x + c) + sinh(d*x + c) - 1) + 16*(15*b^3*cosh(d*x + c)^15 - 140*b^3*cosh(d*x + c)^1
3 + 468*a*b^2*cosh(d*x + c)^12 + 630*b^3*cosh(d*x + c)^11 - 3300*a*b^2*cosh(d*x + c)^10 + 16200*a*b^2*cosh(d*x
+ c)^8 - 120*(192*a^2*b - 35*b^3)*d*x*cosh(d*x + c)^7 + 300*(24*a^2*b - 7*b^3)*cosh(d*x + c)^9 + 12600*a*b^2*
cosh(d*x + c)^6 - 1500*a*b^2*cosh(d*x + c)^4 - 210*b^3*cosh(d*x + c)^3 - 180*(24*a^2*b - 7*b^3)*cosh(d*x + c)^
5 + 108*a*b^2*cosh(d*x + c)^2 + 20*b^3*cosh(d*x + c))*sinh(d*x + c))/(d*cosh(d*x + c)^8 + 8*d*cosh(d*x + c)^7*
sinh(d*x + c) + 28*d*cosh(d*x + c)^6*sinh(d*x + c)^2 + 56*d*cosh(d*x + c)^5*sinh(d*x + c)^3 + 70*d*cosh(d*x +
c)^4*sinh(d*x + c)^4 + 56*d*cosh(d*x + c)^3*sinh(d*x + c)^5 + 28*d*cosh(d*x + c)^2*sinh(d*x + c)^6 + 8*d*cosh(
d*x + c)*sinh(d*x + c)^7 + d*sinh(d*x + c)^8)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [A]  time = 1.40701, size = 427, normalized size = 2.12 \begin{align*} -\frac{a^{3} \log \left (e^{\left (d x + c\right )} + 1\right )}{d} + \frac{a^{3} \log \left ({\left | e^{\left (d x + c\right )} - 1 \right |}\right )}{d} - \frac{{\left (192 \, a^{2} b - 35 \, b^{3}\right )}{\left (d x + c\right )}}{128 \, d} + \frac{{\left (28800 \, a b^{2} e^{\left (7 \, d x + 7 \, c\right )} - 4800 \, a b^{2} e^{\left (5 \, d x + 5 \, c\right )} - 840 \, b^{3} e^{\left (4 \, d x + 4 \, c\right )} + 576 \, a b^{2} e^{\left (3 \, d x + 3 \, c\right )} + 160 \, b^{3} e^{\left (2 \, d x + 2 \, c\right )} - 15 \, b^{3} - 480 \,{\left (24 \, a^{2} b - 7 \, b^{3}\right )} e^{\left (6 \, d x + 6 \, c\right )}\right )} e^{\left (-8 \, d x - 8 \, c\right )}}{30720 \, d} + \frac{15 \, b^{3} d^{7} e^{\left (8 \, d x + 8 \, c\right )} - 160 \, b^{3} d^{7} e^{\left (6 \, d x + 6 \, c\right )} + 576 \, a b^{2} d^{7} e^{\left (5 \, d x + 5 \, c\right )} + 840 \, b^{3} d^{7} e^{\left (4 \, d x + 4 \, c\right )} - 4800 \, a b^{2} d^{7} e^{\left (3 \, d x + 3 \, c\right )} + 11520 \, a^{2} b d^{7} e^{\left (2 \, d x + 2 \, c\right )} - 3360 \, b^{3} d^{7} e^{\left (2 \, d x + 2 \, c\right )} + 28800 \, a b^{2} d^{7} e^{\left (d x + c\right )}}{30720 \, d^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-a^3*log(e^(d*x + c) + 1)/d + a^3*log(abs(e^(d*x + c) - 1))/d - 1/128*(192*a^2*b - 35*b^3)*(d*x + c)/d + 1/307
20*(28800*a*b^2*e^(7*d*x + 7*c) - 4800*a*b^2*e^(5*d*x + 5*c) - 840*b^3*e^(4*d*x + 4*c) + 576*a*b^2*e^(3*d*x +
3*c) + 160*b^3*e^(2*d*x + 2*c) - 15*b^3 - 480*(24*a^2*b - 7*b^3)*e^(6*d*x + 6*c))*e^(-8*d*x - 8*c)/d + 1/30720
*(15*b^3*d^7*e^(8*d*x + 8*c) - 160*b^3*d^7*e^(6*d*x + 6*c) + 576*a*b^2*d^7*e^(5*d*x + 5*c) + 840*b^3*d^7*e^(4*
d*x + 4*c) - 4800*a*b^2*d^7*e^(3*d*x + 3*c) + 11520*a^2*b*d^7*e^(2*d*x + 2*c) - 3360*b^3*d^7*e^(2*d*x + 2*c) +
28800*a*b^2*d^7*e^(d*x + c))/d^8