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

Optimal. Leaf size=156 \[ \frac {a^3 \tanh ^{-1}(\cosh (c+d x))}{2 d}-\frac {a^3 \coth (c+d x) \text {csch}(c+d x)}{2 d}+3 a^2 b x+\frac {a b^2 \cosh ^3(c+d x)}{d}-\frac {3 a b^2 \cosh (c+d x)}{d}+\frac {b^3 \sinh ^5(c+d x) \cosh (c+d x)}{6 d}-\frac {5 b^3 \sinh ^3(c+d x) \cosh (c+d x)}{24 d}+\frac {5 b^3 \sinh (c+d x) \cosh (c+d x)}{16 d}-\frac {5 b^3 x}{16} \]

[Out]

3*a^2*b*x-5/16*b^3*x+1/2*a^3*arctanh(cosh(d*x+c))/d-3*a*b^2*cosh(d*x+c)/d+a*b^2*cosh(d*x+c)^3/d-1/2*a^3*coth(d
*x+c)*csch(d*x+c)/d+5/16*b^3*cosh(d*x+c)*sinh(d*x+c)/d-5/24*b^3*cosh(d*x+c)*sinh(d*x+c)^3/d+1/6*b^3*cosh(d*x+c
)*sinh(d*x+c)^5/d

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Rubi [A]  time = 0.17, antiderivative size = 156, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 6, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {3220, 3768, 3770, 2633, 2635, 8} \[ 3 a^2 b x+\frac {a^3 \tanh ^{-1}(\cosh (c+d x))}{2 d}-\frac {a^3 \coth (c+d x) \text {csch}(c+d x)}{2 d}+\frac {a b^2 \cosh ^3(c+d x)}{d}-\frac {3 a b^2 \cosh (c+d x)}{d}+\frac {b^3 \sinh ^5(c+d x) \cosh (c+d x)}{6 d}-\frac {5 b^3 \sinh ^3(c+d x) \cosh (c+d x)}{24 d}+\frac {5 b^3 \sinh (c+d x) \cosh (c+d x)}{16 d}-\frac {5 b^3 x}{16} \]

Antiderivative was successfully verified.

[In]

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

[Out]

3*a^2*b*x - (5*b^3*x)/16 + (a^3*ArcTanh[Cosh[c + d*x]])/(2*d) - (3*a*b^2*Cosh[c + d*x])/d + (a*b^2*Cosh[c + d*
x]^3)/d - (a^3*Coth[c + d*x]*Csch[c + d*x])/(2*d) + (5*b^3*Cosh[c + d*x]*Sinh[c + d*x])/(16*d) - (5*b^3*Cosh[c
+ d*x]*Sinh[c + d*x]^3)/(24*d) + (b^3*Cosh[c + d*x]*Sinh[c + d*x]^5)/(6*d)

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]

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 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 3768

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 3770

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 \sinh ^3(c+d x)\right )^3 \, dx &=-\left (i \int \left (3 i a^2 b+i a^3 \text {csch}^3(c+d x)+3 i a b^2 \sinh ^3(c+d x)+i b^3 \sinh ^6(c+d x)\right ) \, dx\right )\\ &=3 a^2 b x+a^3 \int \text {csch}^3(c+d x) \, dx+\left (3 a b^2\right ) \int \sinh ^3(c+d x) \, dx+b^3 \int \sinh ^6(c+d x) \, dx\\ &=3 a^2 b x-\frac {a^3 \coth (c+d x) \text {csch}(c+d x)}{2 d}+\frac {b^3 \cosh (c+d x) \sinh ^5(c+d x)}{6 d}-\frac {1}{2} a^3 \int \text {csch}(c+d x) \, dx-\frac {1}{6} \left (5 b^3\right ) \int \sinh ^4(c+d x) \, dx-\frac {\left (3 a b^2\right ) \operatorname {Subst}\left (\int \left (1-x^2\right ) \, dx,x,\cosh (c+d x)\right )}{d}\\ &=3 a^2 b x+\frac {a^3 \tanh ^{-1}(\cosh (c+d x))}{2 d}-\frac {3 a b^2 \cosh (c+d x)}{d}+\frac {a b^2 \cosh ^3(c+d x)}{d}-\frac {a^3 \coth (c+d x) \text {csch}(c+d x)}{2 d}-\frac {5 b^3 \cosh (c+d x) \sinh ^3(c+d x)}{24 d}+\frac {b^3 \cosh (c+d x) \sinh ^5(c+d x)}{6 d}+\frac {1}{8} \left (5 b^3\right ) \int \sinh ^2(c+d x) \, dx\\ &=3 a^2 b x+\frac {a^3 \tanh ^{-1}(\cosh (c+d x))}{2 d}-\frac {3 a b^2 \cosh (c+d x)}{d}+\frac {a b^2 \cosh ^3(c+d x)}{d}-\frac {a^3 \coth (c+d x) \text {csch}(c+d x)}{2 d}+\frac {5 b^3 \cosh (c+d x) \sinh (c+d x)}{16 d}-\frac {5 b^3 \cosh (c+d x) \sinh ^3(c+d x)}{24 d}+\frac {b^3 \cosh (c+d x) \sinh ^5(c+d x)}{6 d}-\frac {1}{16} \left (5 b^3\right ) \int 1 \, dx\\ &=3 a^2 b x-\frac {5 b^3 x}{16}+\frac {a^3 \tanh ^{-1}(\cosh (c+d x))}{2 d}-\frac {3 a b^2 \cosh (c+d x)}{d}+\frac {a b^2 \cosh ^3(c+d x)}{d}-\frac {a^3 \coth (c+d x) \text {csch}(c+d x)}{2 d}+\frac {5 b^3 \cosh (c+d x) \sinh (c+d x)}{16 d}-\frac {5 b^3 \cosh (c+d x) \sinh ^3(c+d x)}{24 d}+\frac {b^3 \cosh (c+d x) \sinh ^5(c+d x)}{6 d}\\ \end {align*}

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Mathematica [A]  time = 3.44, size = 150, normalized size = 0.96 \[ \frac {-24 a^3 \text {csch}^2\left (\frac {1}{2} (c+d x)\right )-24 a^3 \text {sech}^2\left (\frac {1}{2} (c+d x)\right )-96 a^3 \log \left (\tanh \left (\frac {1}{2} (c+d x)\right )\right )+576 a^2 b c+576 a^2 b d x-432 a b^2 \cosh (c+d x)+48 a b^2 \cosh (3 (c+d x))+45 b^3 \sinh (2 (c+d x))-9 b^3 \sinh (4 (c+d x))+b^3 \sinh (6 (c+d x))-60 b^3 c-60 b^3 d x}{192 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(576*a^2*b*c - 60*b^3*c + 576*a^2*b*d*x - 60*b^3*d*x - 432*a*b^2*Cosh[c + d*x] + 48*a*b^2*Cosh[3*(c + d*x)] -
24*a^3*Csch[(c + d*x)/2]^2 - 96*a^3*Log[Tanh[(c + d*x)/2]] - 24*a^3*Sech[(c + d*x)/2]^2 + 45*b^3*Sinh[2*(c + d
*x)] - 9*b^3*Sinh[4*(c + d*x)] + b^3*Sinh[6*(c + d*x)])/(192*d)

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fricas [B]  time = 0.63, size = 3627, normalized size = 23.25 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/384*(b^3*cosh(d*x + c)^16 + 16*b^3*cosh(d*x + c)*sinh(d*x + c)^15 + b^3*sinh(d*x + c)^16 - 11*b^3*cosh(d*x +
c)^14 + 48*a*b^2*cosh(d*x + c)^13 + 64*b^3*cosh(d*x + c)^12 + (120*b^3*cosh(d*x + c)^2 - 11*b^3)*sinh(d*x + c
)^14 - 528*a*b^2*cosh(d*x + c)^11 + 2*(280*b^3*cosh(d*x + c)^3 - 77*b^3*cosh(d*x + c) + 24*a*b^2)*sinh(d*x + c
)^13 + (1820*b^3*cosh(d*x + c)^4 - 1001*b^3*cosh(d*x + c)^2 + 624*a*b^2*cosh(d*x + c) + 64*b^3)*sinh(d*x + c)^
12 + 4*(1092*b^3*cosh(d*x + c)^5 - 1001*b^3*cosh(d*x + c)^3 + 936*a*b^2*cosh(d*x + c)^2 + 192*b^3*cosh(d*x + c
) - 132*a*b^2)*sinh(d*x + c)^11 - 48*(48*a^2*b - 5*b^3)*d*x*cosh(d*x + c)^8 - 3*(33*b^3 - 8*(48*a^2*b - 5*b^3)
*d*x)*cosh(d*x + c)^10 + (8008*b^3*cosh(d*x + c)^6 - 11011*b^3*cosh(d*x + c)^4 + 13728*a*b^2*cosh(d*x + c)^3 +
4224*b^3*cosh(d*x + c)^2 - 5808*a*b^2*cosh(d*x + c) - 99*b^3 + 24*(48*a^2*b - 5*b^3)*d*x)*sinh(d*x + c)^10 -
96*(4*a^3 - 5*a*b^2)*cosh(d*x + c)^9 + 2*(5720*b^3*cosh(d*x + c)^7 - 11011*b^3*cosh(d*x + c)^5 + 17160*a*b^2*c
osh(d*x + c)^4 + 7040*b^3*cosh(d*x + c)^3 - 14520*a*b^2*cosh(d*x + c)^2 - 192*a^3 + 240*a*b^2 - 15*(33*b^3 - 8
*(48*a^2*b - 5*b^3)*d*x)*cosh(d*x + c))*sinh(d*x + c)^9 + 3*(4290*b^3*cosh(d*x + c)^8 - 11011*b^3*cosh(d*x + c
)^6 + 20592*a*b^2*cosh(d*x + c)^5 + 10560*b^3*cosh(d*x + c)^4 - 29040*a*b^2*cosh(d*x + c)^3 - 16*(48*a^2*b - 5
*b^3)*d*x - 45*(33*b^3 - 8*(48*a^2*b - 5*b^3)*d*x)*cosh(d*x + c)^2 - 288*(4*a^3 - 5*a*b^2)*cosh(d*x + c))*sinh
(d*x + c)^8 - 528*a*b^2*cosh(d*x + c)^5 - 96*(4*a^3 - 5*a*b^2)*cosh(d*x + c)^7 + 8*(1430*b^3*cosh(d*x + c)^9 -
4719*b^3*cosh(d*x + c)^7 + 10296*a*b^2*cosh(d*x + c)^6 + 6336*b^3*cosh(d*x + c)^5 - 21780*a*b^2*cosh(d*x + c)
^4 - 48*(48*a^2*b - 5*b^3)*d*x*cosh(d*x + c) - 45*(33*b^3 - 8*(48*a^2*b - 5*b^3)*d*x)*cosh(d*x + c)^3 - 48*a^3
+ 60*a*b^2 - 432*(4*a^3 - 5*a*b^2)*cosh(d*x + c)^2)*sinh(d*x + c)^7 - 64*b^3*cosh(d*x + c)^4 + 3*(33*b^3 + 8*
(48*a^2*b - 5*b^3)*d*x)*cosh(d*x + c)^6 + (8008*b^3*cosh(d*x + c)^10 - 33033*b^3*cosh(d*x + c)^8 + 82368*a*b^2
*cosh(d*x + c)^7 + 59136*b^3*cosh(d*x + c)^6 - 243936*a*b^2*cosh(d*x + c)^5 - 1344*(48*a^2*b - 5*b^3)*d*x*cosh
(d*x + c)^2 - 630*(33*b^3 - 8*(48*a^2*b - 5*b^3)*d*x)*cosh(d*x + c)^4 - 8064*(4*a^3 - 5*a*b^2)*cosh(d*x + c)^3
+ 99*b^3 + 24*(48*a^2*b - 5*b^3)*d*x - 672*(4*a^3 - 5*a*b^2)*cosh(d*x + c))*sinh(d*x + c)^6 + 48*a*b^2*cosh(d
*x + c)^3 + 2*(2184*b^3*cosh(d*x + c)^11 - 11011*b^3*cosh(d*x + c)^9 + 30888*a*b^2*cosh(d*x + c)^8 + 25344*b^3
*cosh(d*x + c)^7 - 121968*a*b^2*cosh(d*x + c)^6 - 1344*(48*a^2*b - 5*b^3)*d*x*cosh(d*x + c)^3 - 378*(33*b^3 -
8*(48*a^2*b - 5*b^3)*d*x)*cosh(d*x + c)^5 - 6048*(4*a^3 - 5*a*b^2)*cosh(d*x + c)^4 - 264*a*b^2 - 1008*(4*a^3 -
5*a*b^2)*cosh(d*x + c)^2 + 9*(33*b^3 + 8*(48*a^2*b - 5*b^3)*d*x)*cosh(d*x + c))*sinh(d*x + c)^5 + 11*b^3*cosh
(d*x + c)^2 + (1820*b^3*cosh(d*x + c)^12 - 11011*b^3*cosh(d*x + c)^10 + 34320*a*b^2*cosh(d*x + c)^9 + 31680*b^
3*cosh(d*x + c)^8 - 174240*a*b^2*cosh(d*x + c)^7 - 3360*(48*a^2*b - 5*b^3)*d*x*cosh(d*x + c)^4 - 630*(33*b^3 -
8*(48*a^2*b - 5*b^3)*d*x)*cosh(d*x + c)^6 - 12096*(4*a^3 - 5*a*b^2)*cosh(d*x + c)^5 - 2640*a*b^2*cosh(d*x + c
) - 3360*(4*a^3 - 5*a*b^2)*cosh(d*x + c)^3 - 64*b^3 + 45*(33*b^3 + 8*(48*a^2*b - 5*b^3)*d*x)*cosh(d*x + c)^2)*
sinh(d*x + c)^4 + 4*(140*b^3*cosh(d*x + c)^13 - 1001*b^3*cosh(d*x + c)^11 + 3432*a*b^2*cosh(d*x + c)^10 + 3520
*b^3*cosh(d*x + c)^9 - 21780*a*b^2*cosh(d*x + c)^8 - 672*(48*a^2*b - 5*b^3)*d*x*cosh(d*x + c)^5 - 90*(33*b^3 -
8*(48*a^2*b - 5*b^3)*d*x)*cosh(d*x + c)^7 - 2016*(4*a^3 - 5*a*b^2)*cosh(d*x + c)^6 - 1320*a*b^2*cosh(d*x + c)
^2 - 840*(4*a^3 - 5*a*b^2)*cosh(d*x + c)^4 - 64*b^3*cosh(d*x + c) + 15*(33*b^3 + 8*(48*a^2*b - 5*b^3)*d*x)*cos
h(d*x + c)^3 + 12*a*b^2)*sinh(d*x + c)^3 - b^3 + (120*b^3*cosh(d*x + c)^14 - 1001*b^3*cosh(d*x + c)^12 + 3744*
a*b^2*cosh(d*x + c)^11 + 4224*b^3*cosh(d*x + c)^10 - 29040*a*b^2*cosh(d*x + c)^9 - 1344*(48*a^2*b - 5*b^3)*d*x
*cosh(d*x + c)^6 - 135*(33*b^3 - 8*(48*a^2*b - 5*b^3)*d*x)*cosh(d*x + c)^8 - 3456*(4*a^3 - 5*a*b^2)*cosh(d*x +
c)^7 - 5280*a*b^2*cosh(d*x + c)^3 - 2016*(4*a^3 - 5*a*b^2)*cosh(d*x + c)^5 - 384*b^3*cosh(d*x + c)^2 + 45*(33
*b^3 + 8*(48*a^2*b - 5*b^3)*d*x)*cosh(d*x + c)^4 + 144*a*b^2*cosh(d*x + c) + 11*b^3)*sinh(d*x + c)^2 + 192*(a^
3*cosh(d*x + c)^10 + 10*a^3*cosh(d*x + c)*sinh(d*x + c)^9 + a^3*sinh(d*x + c)^10 - 2*a^3*cosh(d*x + c)^8 + a^3
*cosh(d*x + c)^6 + (45*a^3*cosh(d*x + c)^2 - 2*a^3)*sinh(d*x + c)^8 + 8*(15*a^3*cosh(d*x + c)^3 - 2*a^3*cosh(d
*x + c))*sinh(d*x + c)^7 + (210*a^3*cosh(d*x + c)^4 - 56*a^3*cosh(d*x + c)^2 + a^3)*sinh(d*x + c)^6 + 2*(126*a
^3*cosh(d*x + c)^5 - 56*a^3*cosh(d*x + c)^3 + 3*a^3*cosh(d*x + c))*sinh(d*x + c)^5 + 5*(42*a^3*cosh(d*x + c)^6
- 28*a^3*cosh(d*x + c)^4 + 3*a^3*cosh(d*x + c)^2)*sinh(d*x + c)^4 + 4*(30*a^3*cosh(d*x + c)^7 - 28*a^3*cosh(d
*x + c)^5 + 5*a^3*cosh(d*x + c)^3)*sinh(d*x + c)^3 + (45*a^3*cosh(d*x + c)^8 - 56*a^3*cosh(d*x + c)^6 + 15*a^3
*cosh(d*x + c)^4)*sinh(d*x + c)^2 + 2*(5*a^3*cosh(d*x + c)^9 - 8*a^3*cosh(d*x + c)^7 + 3*a^3*cosh(d*x + c)^5)*
sinh(d*x + c))*log(cosh(d*x + c) + sinh(d*x + c) + 1) - 192*(a^3*cosh(d*x + c)^10 + 10*a^3*cosh(d*x + c)*sinh(
d*x + c)^9 + a^3*sinh(d*x + c)^10 - 2*a^3*cosh(d*x + c)^8 + a^3*cosh(d*x + c)^6 + (45*a^3*cosh(d*x + c)^2 - 2*
a^3)*sinh(d*x + c)^8 + 8*(15*a^3*cosh(d*x + c)^3 - 2*a^3*cosh(d*x + c))*sinh(d*x + c)^7 + (210*a^3*cosh(d*x +
c)^4 - 56*a^3*cosh(d*x + c)^2 + a^3)*sinh(d*x + c)^6 + 2*(126*a^3*cosh(d*x + c)^5 - 56*a^3*cosh(d*x + c)^3 + 3
*a^3*cosh(d*x + c))*sinh(d*x + c)^5 + 5*(42*a^3*cosh(d*x + c)^6 - 28*a^3*cosh(d*x + c)^4 + 3*a^3*cosh(d*x + c)
^2)*sinh(d*x + c)^4 + 4*(30*a^3*cosh(d*x + c)^7 - 28*a^3*cosh(d*x + c)^5 + 5*a^3*cosh(d*x + c)^3)*sinh(d*x + c
)^3 + (45*a^3*cosh(d*x + c)^8 - 56*a^3*cosh(d*x + c)^6 + 15*a^3*cosh(d*x + c)^4)*sinh(d*x + c)^2 + 2*(5*a^3*co
sh(d*x + c)^9 - 8*a^3*cosh(d*x + c)^7 + 3*a^3*cosh(d*x + c)^5)*sinh(d*x + c))*log(cosh(d*x + c) + sinh(d*x + c
) - 1) + 2*(8*b^3*cosh(d*x + c)^15 - 77*b^3*cosh(d*x + c)^13 + 312*a*b^2*cosh(d*x + c)^12 + 384*b^3*cosh(d*x +
c)^11 - 2904*a*b^2*cosh(d*x + c)^10 - 192*(48*a^2*b - 5*b^3)*d*x*cosh(d*x + c)^7 - 15*(33*b^3 - 8*(48*a^2*b -
5*b^3)*d*x)*cosh(d*x + c)^9 - 432*(4*a^3 - 5*a*b^2)*cosh(d*x + c)^8 - 1320*a*b^2*cosh(d*x + c)^4 - 336*(4*a^3
- 5*a*b^2)*cosh(d*x + c)^6 - 128*b^3*cosh(d*x + c)^3 + 9*(33*b^3 + 8*(48*a^2*b - 5*b^3)*d*x)*cosh(d*x + c)^5
+ 72*a*b^2*cosh(d*x + c)^2 + 11*b^3*cosh(d*x + c))*sinh(d*x + c))/(d*cosh(d*x + c)^10 + 10*d*cosh(d*x + c)*sin
h(d*x + c)^9 + d*sinh(d*x + c)^10 - 2*d*cosh(d*x + c)^8 + (45*d*cosh(d*x + c)^2 - 2*d)*sinh(d*x + c)^8 + 8*(15
*d*cosh(d*x + c)^3 - 2*d*cosh(d*x + c))*sinh(d*x + c)^7 + d*cosh(d*x + c)^6 + (210*d*cosh(d*x + c)^4 - 56*d*co
sh(d*x + c)^2 + d)*sinh(d*x + c)^6 + 2*(126*d*cosh(d*x + c)^5 - 56*d*cosh(d*x + c)^3 + 3*d*cosh(d*x + c))*sinh
(d*x + c)^5 + 5*(42*d*cosh(d*x + c)^6 - 28*d*cosh(d*x + c)^4 + 3*d*cosh(d*x + c)^2)*sinh(d*x + c)^4 + 4*(30*d*
cosh(d*x + c)^7 - 28*d*cosh(d*x + c)^5 + 5*d*cosh(d*x + c)^3)*sinh(d*x + c)^3 + (45*d*cosh(d*x + c)^8 - 56*d*c
osh(d*x + c)^6 + 15*d*cosh(d*x + c)^4)*sinh(d*x + c)^2 + 2*(5*d*cosh(d*x + c)^9 - 8*d*cosh(d*x + c)^7 + 3*d*co
sh(d*x + c)^5)*sinh(d*x + c))

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giac [B]  time = 0.33, size = 289, normalized size = 1.85 \[ \frac {b^{3} e^{\left (6 \, d x + 6 \, c\right )} - 9 \, b^{3} e^{\left (4 \, d x + 4 \, c\right )} + 48 \, a b^{2} e^{\left (3 \, d x + 3 \, c\right )} + 45 \, b^{3} e^{\left (2 \, d x + 2 \, c\right )} - 432 \, a b^{2} e^{\left (d x + c\right )} + 192 \, a^{3} \log \left (e^{\left (d x + c\right )} + 1\right ) - 192 \, a^{3} \log \left ({\left | e^{\left (d x + c\right )} - 1 \right |}\right ) + 24 \, {\left (48 \, a^{2} b - 5 \, b^{3}\right )} {\left (d x + c\right )} - \frac {{\left (45 \, b^{3} e^{\left (8 \, d x + 8 \, c\right )} - 99 \, b^{3} e^{\left (6 \, d x + 6 \, c\right )} + 528 \, a b^{2} e^{\left (5 \, d x + 5 \, c\right )} + 64 \, b^{3} e^{\left (4 \, d x + 4 \, c\right )} - 48 \, a b^{2} e^{\left (3 \, d x + 3 \, c\right )} - 11 \, b^{3} e^{\left (2 \, d x + 2 \, c\right )} + b^{3} + 48 \, {\left (8 \, a^{3} + 9 \, a b^{2}\right )} e^{\left (9 \, d x + 9 \, c\right )} + 48 \, {\left (8 \, a^{3} - 19 \, a b^{2}\right )} e^{\left (7 \, d x + 7 \, c\right )}\right )} e^{\left (-6 \, d x - 6 \, c\right )}}{{\left (e^{\left (d x + c\right )} + 1\right )}^{2} {\left (e^{\left (d x + c\right )} - 1\right )}^{2}}}{384 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/384*(b^3*e^(6*d*x + 6*c) - 9*b^3*e^(4*d*x + 4*c) + 48*a*b^2*e^(3*d*x + 3*c) + 45*b^3*e^(2*d*x + 2*c) - 432*a
*b^2*e^(d*x + c) + 192*a^3*log(e^(d*x + c) + 1) - 192*a^3*log(abs(e^(d*x + c) - 1)) + 24*(48*a^2*b - 5*b^3)*(d
*x + c) - (45*b^3*e^(8*d*x + 8*c) - 99*b^3*e^(6*d*x + 6*c) + 528*a*b^2*e^(5*d*x + 5*c) + 64*b^3*e^(4*d*x + 4*c
) - 48*a*b^2*e^(3*d*x + 3*c) - 11*b^3*e^(2*d*x + 2*c) + b^3 + 48*(8*a^3 + 9*a*b^2)*e^(9*d*x + 9*c) + 48*(8*a^3
- 19*a*b^2)*e^(7*d*x + 7*c))*e^(-6*d*x - 6*c)/((e^(d*x + c) + 1)^2*(e^(d*x + c) - 1)^2))/d

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maple [A]  time = 0.14, size = 115, normalized size = 0.74 \[ \frac {a^{3} \left (-\frac {\mathrm {csch}\left (d x +c \right ) \coth \left (d x +c \right )}{2}+\arctanh \left ({\mathrm e}^{d x +c}\right )\right )+3 a^{2} b \left (d x +c \right )+3 a \,b^{2} \left (-\frac {2}{3}+\frac {\left (\sinh ^{2}\left (d x +c \right )\right )}{3}\right ) \cosh \left (d x +c \right )+b^{3} \left (\left (\frac {\left (\sinh ^{5}\left (d x +c \right )\right )}{6}-\frac {5 \left (\sinh ^{3}\left (d x +c \right )\right )}{24}+\frac {5 \sinh \left (d x +c \right )}{16}\right ) \cosh \left (d x +c \right )-\frac {5 d x}{16}-\frac {5 c}{16}\right )}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/d*(a^3*(-1/2*csch(d*x+c)*coth(d*x+c)+arctanh(exp(d*x+c)))+3*a^2*b*(d*x+c)+3*a*b^2*(-2/3+1/3*sinh(d*x+c)^2)*c
osh(d*x+c)+b^3*((1/6*sinh(d*x+c)^5-5/24*sinh(d*x+c)^3+5/16*sinh(d*x+c))*cosh(d*x+c)-5/16*d*x-5/16*c))

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maxima [A]  time = 0.33, size = 244, normalized size = 1.56 \[ 3 \, a^{2} b x - \frac {1}{384} \, b^{3} {\left (\frac {{\left (9 \, e^{\left (-2 \, d x - 2 \, c\right )} - 45 \, e^{\left (-4 \, d x - 4 \, c\right )} - 1\right )} e^{\left (6 \, d x + 6 \, c\right )}}{d} + \frac {120 \, {\left (d x + c\right )}}{d} + \frac {45 \, e^{\left (-2 \, d x - 2 \, c\right )} - 9 \, e^{\left (-4 \, d x - 4 \, c\right )} + e^{\left (-6 \, d x - 6 \, c\right )}}{d}\right )} + \frac {1}{8} \, a b^{2} {\left (\frac {e^{\left (3 \, d x + 3 \, c\right )}}{d} - \frac {9 \, e^{\left (d x + c\right )}}{d} - \frac {9 \, e^{\left (-d x - c\right )}}{d} + \frac {e^{\left (-3 \, d x - 3 \, c\right )}}{d}\right )} + \frac {1}{2} \, a^{3} {\left (\frac {\log \left (e^{\left (-d x - c\right )} + 1\right )}{d} - \frac {\log \left (e^{\left (-d x - c\right )} - 1\right )}{d} + \frac {2 \, {\left (e^{\left (-d x - c\right )} + e^{\left (-3 \, d x - 3 \, c\right )}\right )}}{d {\left (2 \, e^{\left (-2 \, d x - 2 \, c\right )} - e^{\left (-4 \, d x - 4 \, c\right )} - 1\right )}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3*a^2*b*x - 1/384*b^3*((9*e^(-2*d*x - 2*c) - 45*e^(-4*d*x - 4*c) - 1)*e^(6*d*x + 6*c)/d + 120*(d*x + c)/d + (4
5*e^(-2*d*x - 2*c) - 9*e^(-4*d*x - 4*c) + e^(-6*d*x - 6*c))/d) + 1/8*a*b^2*(e^(3*d*x + 3*c)/d - 9*e^(d*x + c)/
d - 9*e^(-d*x - c)/d + e^(-3*d*x - 3*c)/d) + 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)))

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mupad [B]  time = 0.93, size = 290, normalized size = 1.86 \[ x\,\left (3\,a^2\,b-\frac {5\,b^3}{16}\right )+\frac {\mathrm {atan}\left (\frac {a^3\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\sqrt {-d^2}}{d\,\sqrt {a^6}}\right )\,\sqrt {a^6}}{\sqrt {-d^2}}-\frac {15\,b^3\,{\mathrm {e}}^{-2\,c-2\,d\,x}}{128\,d}+\frac {15\,b^3\,{\mathrm {e}}^{2\,c+2\,d\,x}}{128\,d}+\frac {3\,b^3\,{\mathrm {e}}^{-4\,c-4\,d\,x}}{128\,d}-\frac {3\,b^3\,{\mathrm {e}}^{4\,c+4\,d\,x}}{128\,d}-\frac {b^3\,{\mathrm {e}}^{-6\,c-6\,d\,x}}{384\,d}+\frac {b^3\,{\mathrm {e}}^{6\,c+6\,d\,x}}{384\,d}-\frac {9\,a\,b^2\,{\mathrm {e}}^{-c-d\,x}}{8\,d}+\frac {a\,b^2\,{\mathrm {e}}^{-3\,c-3\,d\,x}}{8\,d}+\frac {a\,b^2\,{\mathrm {e}}^{3\,c+3\,d\,x}}{8\,d}-\frac {9\,a\,b^2\,{\mathrm {e}}^{c+d\,x}}{8\,d}-\frac {a^3\,{\mathrm {e}}^{c+d\,x}}{d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}-1\right )}-\frac {2\,a^3\,{\mathrm {e}}^{c+d\,x}}{d\,\left ({\mathrm {e}}^{4\,c+4\,d\,x}-2\,{\mathrm {e}}^{2\,c+2\,d\,x}+1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*(3*a^2*b - (5*b^3)/16) + (atan((a^3*exp(d*x)*exp(c)*(-d^2)^(1/2))/(d*(a^6)^(1/2)))*(a^6)^(1/2))/(-d^2)^(1/2)
- (15*b^3*exp(- 2*c - 2*d*x))/(128*d) + (15*b^3*exp(2*c + 2*d*x))/(128*d) + (3*b^3*exp(- 4*c - 4*d*x))/(128*d
) - (3*b^3*exp(4*c + 4*d*x))/(128*d) - (b^3*exp(- 6*c - 6*d*x))/(384*d) + (b^3*exp(6*c + 6*d*x))/(384*d) - (9*
a*b^2*exp(- c - d*x))/(8*d) + (a*b^2*exp(- 3*c - 3*d*x))/(8*d) + (a*b^2*exp(3*c + 3*d*x))/(8*d) - (9*a*b^2*exp
(c + d*x))/(8*d) - (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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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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