3.2.64 \(\int \text {csch}(c+d x) (a+b \sinh ^3(c+d x))^3 \, dx\) [164]

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

[Out]

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

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Rubi [A]
time = 0.14, antiderivative size = 201, normalized size of antiderivative = 1.00, 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 = {3299, 3855, 2715, 8, 2713} \begin {gather*} -\frac {a^3 \tanh ^{-1}(\cosh (c+d x))}{d}+\frac {3 a^2 b \sinh (c+d x) \cosh (c+d x)}{2 d}-\frac {3}{2} a^2 b x+\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} \end {gather*}

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 8

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

Rule 2713

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 2715

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] && Integ
erQ[2*n]

Rule 3299

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 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}(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 ) \text {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*}

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Mathematica [A]
time = 0.19, size = 158, normalized size = 0.79 \begin {gather*} \frac {-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 \left (\tanh \left (\frac {1}{2} (c+d x)\right )\right )+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} \end {gather*}

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 = 2.08, size = 325, normalized size = 1.62

method	result	size
risch	\(-\frac {3 a^{2} b x}{2}+\frac {35 b^{3} x}{128}+\frac {b^{3} {\mathrm e}^{8 d x +8 c}}{2048 d}-\frac {b^{3} {\mathrm e}^{6 d x +6 c}}{192 d}+\frac {3 a \,b^{2} {\mathrm e}^{5 d x +5 c}}{160 d}+\frac {7 b^{3} {\mathrm e}^{4 d x +4 c}}{256 d}-\frac {5 a \,b^{2} {\mathrm e}^{3 d x +3 c}}{32 d}+\frac {3 b \,{\mathrm e}^{2 d x +2 c} a^{2}}{8 d}-\frac {7 b^{3} {\mathrm e}^{2 d x +2 c}}{64 d}+\frac {15 a \,{\mathrm e}^{d x +c} b^{2}}{16 d}+\frac {15 a \,{\mathrm e}^{-d x -c} b^{2}}{16 d}-\frac {3 b \,{\mathrm e}^{-2 d x -2 c} a^{2}}{8 d}+\frac {7 b^{3} {\mathrm e}^{-2 d x -2 c}}{64 d}-\frac {5 a \,b^{2} {\mathrm e}^{-3 d x -3 c}}{32 d}-\frac {7 b^{3} {\mathrm e}^{-4 d x -4 c}}{256 d}+\frac {3 a \,b^{2} {\mathrm e}^{-5 d x -5 c}}{160 d}+\frac {b^{3} {\mathrm e}^{-6 d x -6 c}}{192 d}-\frac {b^{3} {\mathrm e}^{-8 d x -8 c}}{2048 d}+\frac {a^{3} \ln \left ({\mathrm e}^{d x +c}-1\right )}{d}-\frac {a^{3} \ln \left ({\mathrm e}^{d x +c}+1\right )}{d}\)	\(325\)

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,method=_RETURNVERBOSE)

[Out]

-3/2*a^2*b*x+35/128*b^3*x+1/2048*b^3/d*exp(8*d*x+8*c)-1/192*b^3/d*exp(6*d*x+6*c)+3/160*a*b^2/d*exp(5*d*x+5*c)+
7/256*b^3/d*exp(4*d*x+4*c)-5/32*a*b^2/d*exp(3*d*x+3*c)+3/8*b/d*exp(2*d*x+2*c)*a^2-7/64*b^3/d*exp(2*d*x+2*c)+15
/16*a/d*exp(d*x+c)*b^2+15/16*a/d*exp(-d*x-c)*b^2-3/8*b/d*exp(-2*d*x-2*c)*a^2+7/64*b^3/d*exp(-2*d*x-2*c)-5/32*a
*b^2/d*exp(-3*d*x-3*c)-7/256*b^3/d*exp(-4*d*x-4*c)+3/160*a*b^2/d*exp(-5*d*x-5*c)+1/192*b^3/d*exp(-6*d*x-6*c)-1
/2048*b^3/d*exp(-8*d*x-8*c)+a^3/d*ln(exp(d*x+c)-1)-a^3/d*ln(exp(d*x+c)+1)

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Maxima [A]
time = 0.29, size = 257, normalized size = 1.28 \begin {gather*} -\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 {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 2609 vs. \(2 (185) = 370\).
time = 0.45, size = 2609, normalized size = 12.98 \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)*(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...

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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*}

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 = 0.48, size = 279, normalized size = 1.39 \begin {gather*} \frac {15 \, b^{3} e^{\left (8 \, d x + 8 \, c\right )} - 160 \, b^{3} e^{\left (6 \, d x + 6 \, c\right )} + 576 \, a b^{2} e^{\left (5 \, d x + 5 \, c\right )} + 840 \, b^{3} e^{\left (4 \, d x + 4 \, c\right )} - 4800 \, a b^{2} e^{\left (3 \, d x + 3 \, c\right )} + 11520 \, a^{2} b e^{\left (2 \, d x + 2 \, c\right )} - 3360 \, b^{3} e^{\left (2 \, d x + 2 \, c\right )} + 28800 \, a b^{2} e^{\left (d x + c\right )} - 30720 \, a^{3} \log \left (e^{\left (d x + c\right )} + 1\right ) + 30720 \, a^{3} \log \left ({\left | e^{\left (d x + c\right )} - 1 \right |}\right ) - 240 \, {\left (192 \, a^{2} b - 35 \, b^{3}\right )} {\left (d x + c\right )} + {\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} \end {gather*}

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]

1/30720*(15*b^3*e^(8*d*x + 8*c) - 160*b^3*e^(6*d*x + 6*c) + 576*a*b^2*e^(5*d*x + 5*c) + 840*b^3*e^(4*d*x + 4*c
) - 4800*a*b^2*e^(3*d*x + 3*c) + 11520*a^2*b*e^(2*d*x + 2*c) - 3360*b^3*e^(2*d*x + 2*c) + 28800*a*b^2*e^(d*x +
c) - 30720*a^3*log(e^(d*x + c) + 1) + 30720*a^3*log(abs(e^(d*x + c) - 1)) - 240*(192*a^2*b - 35*b^3)*(d*x + c
) + (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

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Mupad [B]
time = 0.50, size = 315, normalized size = 1.57 \begin {gather*} \frac {7\,b^3\,{\mathrm {e}}^{4\,c+4\,d\,x}}{256\,d}-\frac {2\,\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 {7\,b^3\,{\mathrm {e}}^{-4\,c-4\,d\,x}}{256\,d}-x\,\left (\frac {3\,a^2\,b}{2}-\frac {35\,b^3}{128}\right )+\frac {b^3\,{\mathrm {e}}^{-6\,c-6\,d\,x}}{192\,d}-\frac {b^3\,{\mathrm {e}}^{6\,c+6\,d\,x}}{192\,d}-\frac {b^3\,{\mathrm {e}}^{-8\,c-8\,d\,x}}{2048\,d}+\frac {b^3\,{\mathrm {e}}^{8\,c+8\,d\,x}}{2048\,d}-\frac {{\mathrm {e}}^{-2\,c-2\,d\,x}\,\left (24\,a^2\,b-7\,b^3\right )}{64\,d}+\frac {{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (24\,a^2\,b-7\,b^3\right )}{64\,d}+\frac {15\,a\,b^2\,{\mathrm {e}}^{-c-d\,x}}{16\,d}-\frac {5\,a\,b^2\,{\mathrm {e}}^{-3\,c-3\,d\,x}}{32\,d}-\frac {5\,a\,b^2\,{\mathrm {e}}^{3\,c+3\,d\,x}}{32\,d}+\frac {3\,a\,b^2\,{\mathrm {e}}^{-5\,c-5\,d\,x}}{160\,d}+\frac {3\,a\,b^2\,{\mathrm {e}}^{5\,c+5\,d\,x}}{160\,d}+\frac {15\,a\,b^2\,{\mathrm {e}}^{c+d\,x}}{16\,d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(7*b^3*exp(4*c + 4*d*x))/(256*d) - (2*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) - (7*b^3*exp(- 4*c - 4*d*x))/(256*d) - x*((3*a^2*b)/2 - (35*b^3)/128) + (b^3*exp(- 6*c - 6*d*x))/(1
92*d) - (b^3*exp(6*c + 6*d*x))/(192*d) - (b^3*exp(- 8*c - 8*d*x))/(2048*d) + (b^3*exp(8*c + 8*d*x))/(2048*d) -
(exp(- 2*c - 2*d*x)*(24*a^2*b - 7*b^3))/(64*d) + (exp(2*c + 2*d*x)*(24*a^2*b - 7*b^3))/(64*d) + (15*a*b^2*exp
(- c - d*x))/(16*d) - (5*a*b^2*exp(- 3*c - 3*d*x))/(32*d) - (5*a*b^2*exp(3*c + 3*d*x))/(32*d) + (3*a*b^2*exp(-
5*c - 5*d*x))/(160*d) + (3*a*b^2*exp(5*c + 5*d*x))/(160*d) + (15*a*b^2*exp(c + d*x))/(16*d)

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