3.162 \(\int \sinh (c+d x) (a+b \sinh ^3(c+d x))^3 \, dx\)

Optimal. Leaf size=267 \[ \frac{3 a^2 b \sinh ^3(c+d x) \cosh (c+d x)}{4 d}-\frac{9 a^2 b \sinh (c+d x) \cosh (c+d x)}{8 d}+\frac{9}{8} a^2 b x+\frac{a^3 \cosh (c+d x)}{d}+\frac{3 a b^2 \cosh ^7(c+d x)}{7 d}-\frac{9 a b^2 \cosh ^5(c+d x)}{5 d}+\frac{3 a b^2 \cosh ^3(c+d x)}{d}-\frac{3 a b^2 \cosh (c+d x)}{d}+\frac{b^3 \sinh ^9(c+d x) \cosh (c+d x)}{10 d}-\frac{9 b^3 \sinh ^7(c+d x) \cosh (c+d x)}{80 d}+\frac{21 b^3 \sinh ^5(c+d x) \cosh (c+d x)}{160 d}-\frac{21 b^3 \sinh ^3(c+d x) \cosh (c+d x)}{128 d}+\frac{63 b^3 \sinh (c+d x) \cosh (c+d x)}{256 d}-\frac{63 b^3 x}{256} \]

[Out]

(9*a^2*b*x)/8 - (63*b^3*x)/256 + (a^3*Cosh[c + d*x])/d - (3*a*b^2*Cosh[c + d*x])/d + (3*a*b^2*Cosh[c + d*x]^3)
/d - (9*a*b^2*Cosh[c + d*x]^5)/(5*d) + (3*a*b^2*Cosh[c + d*x]^7)/(7*d) - (9*a^2*b*Cosh[c + d*x]*Sinh[c + d*x])
/(8*d) + (63*b^3*Cosh[c + d*x]*Sinh[c + d*x])/(256*d) + (3*a^2*b*Cosh[c + d*x]*Sinh[c + d*x]^3)/(4*d) - (21*b^
3*Cosh[c + d*x]*Sinh[c + d*x]^3)/(128*d) + (21*b^3*Cosh[c + d*x]*Sinh[c + d*x]^5)/(160*d) - (9*b^3*Cosh[c + d*
x]*Sinh[c + d*x]^7)/(80*d) + (b^3*Cosh[c + d*x]*Sinh[c + d*x]^9)/(10*d)

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Rubi [A]  time = 0.222053, antiderivative size = 267, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238, Rules used = {3220, 2638, 2635, 8, 2633} \[ \frac{3 a^2 b \sinh ^3(c+d x) \cosh (c+d x)}{4 d}-\frac{9 a^2 b \sinh (c+d x) \cosh (c+d x)}{8 d}+\frac{9}{8} a^2 b x+\frac{a^3 \cosh (c+d x)}{d}+\frac{3 a b^2 \cosh ^7(c+d x)}{7 d}-\frac{9 a b^2 \cosh ^5(c+d x)}{5 d}+\frac{3 a b^2 \cosh ^3(c+d x)}{d}-\frac{3 a b^2 \cosh (c+d x)}{d}+\frac{b^3 \sinh ^9(c+d x) \cosh (c+d x)}{10 d}-\frac{9 b^3 \sinh ^7(c+d x) \cosh (c+d x)}{80 d}+\frac{21 b^3 \sinh ^5(c+d x) \cosh (c+d x)}{160 d}-\frac{21 b^3 \sinh ^3(c+d x) \cosh (c+d x)}{128 d}+\frac{63 b^3 \sinh (c+d x) \cosh (c+d x)}{256 d}-\frac{63 b^3 x}{256} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(9*a^2*b*x)/8 - (63*b^3*x)/256 + (a^3*Cosh[c + d*x])/d - (3*a*b^2*Cosh[c + d*x])/d + (3*a*b^2*Cosh[c + d*x]^3)
/d - (9*a*b^2*Cosh[c + d*x]^5)/(5*d) + (3*a*b^2*Cosh[c + d*x]^7)/(7*d) - (9*a^2*b*Cosh[c + d*x]*Sinh[c + d*x])
/(8*d) + (63*b^3*Cosh[c + d*x]*Sinh[c + d*x])/(256*d) + (3*a^2*b*Cosh[c + d*x]*Sinh[c + d*x]^3)/(4*d) - (21*b^
3*Cosh[c + d*x]*Sinh[c + d*x]^3)/(128*d) + (21*b^3*Cosh[c + d*x]*Sinh[c + d*x]^5)/(160*d) - (9*b^3*Cosh[c + d*
x]*Sinh[c + d*x]^7)/(80*d) + (b^3*Cosh[c + d*x]*Sinh[c + d*x]^9)/(10*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 2638

Int[sin[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[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 \sinh (c+d x) \left (a+b \sinh ^3(c+d x)\right )^3 \, dx &=-\left (i \int \left (i a^3 \sinh (c+d x)+3 i a^2 b \sinh ^4(c+d x)+3 i a b^2 \sinh ^7(c+d x)+i b^3 \sinh ^{10}(c+d x)\right ) \, dx\right )\\ &=a^3 \int \sinh (c+d x) \, dx+\left (3 a^2 b\right ) \int \sinh ^4(c+d x) \, dx+\left (3 a b^2\right ) \int \sinh ^7(c+d x) \, dx+b^3 \int \sinh ^{10}(c+d x) \, dx\\ &=\frac{a^3 \cosh (c+d x)}{d}+\frac{3 a^2 b \cosh (c+d x) \sinh ^3(c+d x)}{4 d}+\frac{b^3 \cosh (c+d x) \sinh ^9(c+d x)}{10 d}-\frac{1}{4} \left (9 a^2 b\right ) \int \sinh ^2(c+d x) \, dx-\frac{1}{10} \left (9 b^3\right ) \int \sinh ^8(c+d x) \, dx-\frac{\left (3 a b^2\right ) \operatorname{Subst}\left (\int \left (1-3 x^2+3 x^4-x^6\right ) \, dx,x,\cosh (c+d x)\right )}{d}\\ &=\frac{a^3 \cosh (c+d x)}{d}-\frac{3 a b^2 \cosh (c+d x)}{d}+\frac{3 a b^2 \cosh ^3(c+d x)}{d}-\frac{9 a b^2 \cosh ^5(c+d x)}{5 d}+\frac{3 a b^2 \cosh ^7(c+d x)}{7 d}-\frac{9 a^2 b \cosh (c+d x) \sinh (c+d x)}{8 d}+\frac{3 a^2 b \cosh (c+d x) \sinh ^3(c+d x)}{4 d}-\frac{9 b^3 \cosh (c+d x) \sinh ^7(c+d x)}{80 d}+\frac{b^3 \cosh (c+d x) \sinh ^9(c+d x)}{10 d}+\frac{1}{8} \left (9 a^2 b\right ) \int 1 \, dx+\frac{1}{80} \left (63 b^3\right ) \int \sinh ^6(c+d x) \, dx\\ &=\frac{9}{8} a^2 b x+\frac{a^3 \cosh (c+d x)}{d}-\frac{3 a b^2 \cosh (c+d x)}{d}+\frac{3 a b^2 \cosh ^3(c+d x)}{d}-\frac{9 a b^2 \cosh ^5(c+d x)}{5 d}+\frac{3 a b^2 \cosh ^7(c+d x)}{7 d}-\frac{9 a^2 b \cosh (c+d x) \sinh (c+d x)}{8 d}+\frac{3 a^2 b \cosh (c+d x) \sinh ^3(c+d x)}{4 d}+\frac{21 b^3 \cosh (c+d x) \sinh ^5(c+d x)}{160 d}-\frac{9 b^3 \cosh (c+d x) \sinh ^7(c+d x)}{80 d}+\frac{b^3 \cosh (c+d x) \sinh ^9(c+d x)}{10 d}-\frac{1}{32} \left (21 b^3\right ) \int \sinh ^4(c+d x) \, dx\\ &=\frac{9}{8} a^2 b x+\frac{a^3 \cosh (c+d x)}{d}-\frac{3 a b^2 \cosh (c+d x)}{d}+\frac{3 a b^2 \cosh ^3(c+d x)}{d}-\frac{9 a b^2 \cosh ^5(c+d x)}{5 d}+\frac{3 a b^2 \cosh ^7(c+d x)}{7 d}-\frac{9 a^2 b \cosh (c+d x) \sinh (c+d x)}{8 d}+\frac{3 a^2 b \cosh (c+d x) \sinh ^3(c+d x)}{4 d}-\frac{21 b^3 \cosh (c+d x) \sinh ^3(c+d x)}{128 d}+\frac{21 b^3 \cosh (c+d x) \sinh ^5(c+d x)}{160 d}-\frac{9 b^3 \cosh (c+d x) \sinh ^7(c+d x)}{80 d}+\frac{b^3 \cosh (c+d x) \sinh ^9(c+d x)}{10 d}+\frac{1}{128} \left (63 b^3\right ) \int \sinh ^2(c+d x) \, dx\\ &=\frac{9}{8} a^2 b x+\frac{a^3 \cosh (c+d x)}{d}-\frac{3 a b^2 \cosh (c+d x)}{d}+\frac{3 a b^2 \cosh ^3(c+d x)}{d}-\frac{9 a b^2 \cosh ^5(c+d x)}{5 d}+\frac{3 a b^2 \cosh ^7(c+d x)}{7 d}-\frac{9 a^2 b \cosh (c+d x) \sinh (c+d x)}{8 d}+\frac{63 b^3 \cosh (c+d x) \sinh (c+d x)}{256 d}+\frac{3 a^2 b \cosh (c+d x) \sinh ^3(c+d x)}{4 d}-\frac{21 b^3 \cosh (c+d x) \sinh ^3(c+d x)}{128 d}+\frac{21 b^3 \cosh (c+d x) \sinh ^5(c+d x)}{160 d}-\frac{9 b^3 \cosh (c+d x) \sinh ^7(c+d x)}{80 d}+\frac{b^3 \cosh (c+d x) \sinh ^9(c+d x)}{10 d}-\frac{1}{256} \left (63 b^3\right ) \int 1 \, dx\\ &=\frac{9}{8} a^2 b x-\frac{63 b^3 x}{256}+\frac{a^3 \cosh (c+d x)}{d}-\frac{3 a b^2 \cosh (c+d x)}{d}+\frac{3 a b^2 \cosh ^3(c+d x)}{d}-\frac{9 a b^2 \cosh ^5(c+d x)}{5 d}+\frac{3 a b^2 \cosh ^7(c+d x)}{7 d}-\frac{9 a^2 b \cosh (c+d x) \sinh (c+d x)}{8 d}+\frac{63 b^3 \cosh (c+d x) \sinh (c+d x)}{256 d}+\frac{3 a^2 b \cosh (c+d x) \sinh ^3(c+d x)}{4 d}-\frac{21 b^3 \cosh (c+d x) \sinh ^3(c+d x)}{128 d}+\frac{21 b^3 \cosh (c+d x) \sinh ^5(c+d x)}{160 d}-\frac{9 b^3 \cosh (c+d x) \sinh ^7(c+d x)}{80 d}+\frac{b^3 \cosh (c+d x) \sinh ^9(c+d x)}{10 d}\\ \end{align*}

Mathematica [A]  time = 0.459003, size = 184, normalized size = 0.69 \[ \frac{1120 a \left (64 a^2-105 b^2\right ) \cosh (c+d x)+b \left (-53760 a^2 \sinh (2 (c+d x))+6720 a^2 \sinh (4 (c+d x))+80640 a^2 c+80640 a^2 d x+23520 a b \cosh (3 (c+d x))-4704 a b \cosh (5 (c+d x))+480 a b \cosh (7 (c+d x))+14700 b^2 \sinh (2 (c+d x))-4200 b^2 \sinh (4 (c+d x))+1050 b^2 \sinh (6 (c+d x))-175 b^2 \sinh (8 (c+d x))+14 b^2 \sinh (10 (c+d x))-17640 b^2 c-17640 b^2 d x\right )}{71680 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(1120*a*(64*a^2 - 105*b^2)*Cosh[c + d*x] + b*(80640*a^2*c - 17640*b^2*c + 80640*a^2*d*x - 17640*b^2*d*x + 2352
0*a*b*Cosh[3*(c + d*x)] - 4704*a*b*Cosh[5*(c + d*x)] + 480*a*b*Cosh[7*(c + d*x)] - 53760*a^2*Sinh[2*(c + d*x)]
+ 14700*b^2*Sinh[2*(c + d*x)] + 6720*a^2*Sinh[4*(c + d*x)] - 4200*b^2*Sinh[4*(c + d*x)] + 1050*b^2*Sinh[6*(c
+ d*x)] - 175*b^2*Sinh[8*(c + d*x)] + 14*b^2*Sinh[10*(c + d*x)]))/(71680*d)

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Maple [A]  time = 0.023, size = 168, normalized size = 0.6 \begin{align*}{\frac{1}{d} \left ({b}^{3} \left ( \left ({\frac{ \left ( \sinh \left ( dx+c \right ) \right ) ^{9}}{10}}-{\frac{9\, \left ( \sinh \left ( dx+c \right ) \right ) ^{7}}{80}}+{\frac{21\, \left ( \sinh \left ( dx+c \right ) \right ) ^{5}}{160}}-{\frac{21\, \left ( \sinh \left ( dx+c \right ) \right ) ^{3}}{128}}+{\frac{63\,\sinh \left ( dx+c \right ) }{256}} \right ) \cosh \left ( dx+c \right ) -{\frac{63\,dx}{256}}-{\frac{63\,c}{256}} \right ) +3\,a{b}^{2} \left ( -{\frac{16}{35}}+1/7\, \left ( \sinh \left ( dx+c \right ) \right ) ^{6}-{\frac{6\, \left ( \sinh \left ( dx+c \right ) \right ) ^{4}}{35}}+{\frac{8\, \left ( \sinh \left ( dx+c \right ) \right ) ^{2}}{35}} \right ) \cosh \left ( dx+c \right ) +3\,{a}^{2}b \left ( \left ( 1/4\, \left ( \sinh \left ( dx+c \right ) \right ) ^{3}-3/8\,\sinh \left ( dx+c \right ) \right ) \cosh \left ( dx+c \right ) +3/8\,dx+3/8\,c \right ) +{a}^{3}\cosh \left ( dx+c \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/d*(b^3*((1/10*sinh(d*x+c)^9-9/80*sinh(d*x+c)^7+21/160*sinh(d*x+c)^5-21/128*sinh(d*x+c)^3+63/256*sinh(d*x+c))
*cosh(d*x+c)-63/256*d*x-63/256*c)+3*a*b^2*(-16/35+1/7*sinh(d*x+c)^6-6/35*sinh(d*x+c)^4+8/35*sinh(d*x+c)^2)*cos
h(d*x+c)+3*a^2*b*((1/4*sinh(d*x+c)^3-3/8*sinh(d*x+c))*cosh(d*x+c)+3/8*d*x+3/8*c)+a^3*cosh(d*x+c))

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Maxima [A]  time = 1.14679, size = 429, normalized size = 1.61 \begin{align*} \frac{3}{64} \, a^{2} b{\left (24 \, x + \frac{e^{\left (4 \, d x + 4 \, c\right )}}{d} - \frac{8 \, e^{\left (2 \, d x + 2 \, c\right )}}{d} + \frac{8 \, e^{\left (-2 \, d x - 2 \, c\right )}}{d} - \frac{e^{\left (-4 \, d x - 4 \, c\right )}}{d}\right )} - \frac{1}{20480} \, b^{3}{\left (\frac{{\left (25 \, e^{\left (-2 \, d x - 2 \, c\right )} - 150 \, e^{\left (-4 \, d x - 4 \, c\right )} + 600 \, e^{\left (-6 \, d x - 6 \, c\right )} - 2100 \, e^{\left (-8 \, d x - 8 \, c\right )} - 2\right )} e^{\left (10 \, d x + 10 \, c\right )}}{d} + \frac{5040 \,{\left (d x + c\right )}}{d} + \frac{2100 \, e^{\left (-2 \, d x - 2 \, c\right )} - 600 \, e^{\left (-4 \, d x - 4 \, c\right )} + 150 \, e^{\left (-6 \, d x - 6 \, c\right )} - 25 \, e^{\left (-8 \, d x - 8 \, c\right )} + 2 \, e^{\left (-10 \, d x - 10 \, c\right )}}{d}\right )} - \frac{3}{4480} \, a b^{2}{\left (\frac{{\left (49 \, e^{\left (-2 \, d x - 2 \, c\right )} - 245 \, e^{\left (-4 \, d x - 4 \, c\right )} + 1225 \, e^{\left (-6 \, d x - 6 \, c\right )} - 5\right )} e^{\left (7 \, d x + 7 \, c\right )}}{d} + \frac{1225 \, e^{\left (-d x - c\right )} - 245 \, e^{\left (-3 \, d x - 3 \, c\right )} + 49 \, e^{\left (-5 \, d x - 5 \, c\right )} - 5 \, e^{\left (-7 \, d x - 7 \, c\right )}}{d}\right )} + \frac{a^{3} \cosh \left (d x + c\right )}{d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/64*a^2*b*(24*x + e^(4*d*x + 4*c)/d - 8*e^(2*d*x + 2*c)/d + 8*e^(-2*d*x - 2*c)/d - e^(-4*d*x - 4*c)/d) - 1/20
480*b^3*((25*e^(-2*d*x - 2*c) - 150*e^(-4*d*x - 4*c) + 600*e^(-6*d*x - 6*c) - 2100*e^(-8*d*x - 8*c) - 2)*e^(10
*d*x + 10*c)/d + 5040*(d*x + c)/d + (2100*e^(-2*d*x - 2*c) - 600*e^(-4*d*x - 4*c) + 150*e^(-6*d*x - 6*c) - 25*
e^(-8*d*x - 8*c) + 2*e^(-10*d*x - 10*c))/d) - 3/4480*a*b^2*((49*e^(-2*d*x - 2*c) - 245*e^(-4*d*x - 4*c) + 1225
*e^(-6*d*x - 6*c) - 5)*e^(7*d*x + 7*c)/d + (1225*e^(-d*x - c) - 245*e^(-3*d*x - 3*c) + 49*e^(-5*d*x - 5*c) - 5
*e^(-7*d*x - 7*c))/d) + a^3*cosh(d*x + c)/d

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Fricas [A]  time = 2.00611, size = 1193, normalized size = 4.47 \begin{align*} \frac{35 \, b^{3} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{9} + 120 \, a b^{2} \cosh \left (d x + c\right )^{7} + 840 \, a b^{2} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{6} - 1176 \, a b^{2} \cosh \left (d x + c\right )^{5} + 70 \,{\left (6 \, b^{3} \cosh \left (d x + c\right )^{3} - 5 \, b^{3} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{7} + 5880 \, a b^{2} \cosh \left (d x + c\right )^{3} + 7 \,{\left (126 \, b^{3} \cosh \left (d x + c\right )^{5} - 350 \, b^{3} \cosh \left (d x + c\right )^{3} + 225 \, b^{3} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{5} + 840 \,{\left (5 \, a b^{2} \cosh \left (d x + c\right )^{3} - 7 \, a b^{2} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{4} + 70 \,{\left (6 \, b^{3} \cosh \left (d x + c\right )^{7} - 35 \, b^{3} \cosh \left (d x + c\right )^{5} + 75 \, b^{3} \cosh \left (d x + c\right )^{3} + 12 \,{\left (8 \, a^{2} b - 5 \, b^{3}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{3} + 630 \,{\left (32 \, a^{2} b - 7 \, b^{3}\right )} d x + 840 \,{\left (3 \, a b^{2} \cosh \left (d x + c\right )^{5} - 14 \, a b^{2} \cosh \left (d x + c\right )^{3} + 21 \, a b^{2} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{2} + 280 \,{\left (64 \, a^{3} - 105 \, a b^{2}\right )} \cosh \left (d x + c\right ) + 35 \,{\left (b^{3} \cosh \left (d x + c\right )^{9} - 10 \, b^{3} \cosh \left (d x + c\right )^{7} + 45 \, b^{3} \cosh \left (d x + c\right )^{5} + 24 \,{\left (8 \, a^{2} b - 5 \, b^{3}\right )} \cosh \left (d x + c\right )^{3} - 6 \,{\left (128 \, a^{2} b - 35 \, b^{3}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )}{17920 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/17920*(35*b^3*cosh(d*x + c)*sinh(d*x + c)^9 + 120*a*b^2*cosh(d*x + c)^7 + 840*a*b^2*cosh(d*x + c)*sinh(d*x +
c)^6 - 1176*a*b^2*cosh(d*x + c)^5 + 70*(6*b^3*cosh(d*x + c)^3 - 5*b^3*cosh(d*x + c))*sinh(d*x + c)^7 + 5880*a
*b^2*cosh(d*x + c)^3 + 7*(126*b^3*cosh(d*x + c)^5 - 350*b^3*cosh(d*x + c)^3 + 225*b^3*cosh(d*x + c))*sinh(d*x
+ c)^5 + 840*(5*a*b^2*cosh(d*x + c)^3 - 7*a*b^2*cosh(d*x + c))*sinh(d*x + c)^4 + 70*(6*b^3*cosh(d*x + c)^7 - 3
5*b^3*cosh(d*x + c)^5 + 75*b^3*cosh(d*x + c)^3 + 12*(8*a^2*b - 5*b^3)*cosh(d*x + c))*sinh(d*x + c)^3 + 630*(32
*a^2*b - 7*b^3)*d*x + 840*(3*a*b^2*cosh(d*x + c)^5 - 14*a*b^2*cosh(d*x + c)^3 + 21*a*b^2*cosh(d*x + c))*sinh(d
*x + c)^2 + 280*(64*a^3 - 105*a*b^2)*cosh(d*x + c) + 35*(b^3*cosh(d*x + c)^9 - 10*b^3*cosh(d*x + c)^7 + 45*b^3
*cosh(d*x + c)^5 + 24*(8*a^2*b - 5*b^3)*cosh(d*x + c)^3 - 6*(128*a^2*b - 35*b^3)*cosh(d*x + c))*sinh(d*x + c))
/d

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Sympy [A]  time = 97.2029, size = 496, normalized size = 1.86 \begin{align*} \begin{cases} \frac{a^{3} \cosh{\left (c + d x \right )}}{d} + \frac{9 a^{2} b x \sinh ^{4}{\left (c + d x \right )}}{8} - \frac{9 a^{2} b x \sinh ^{2}{\left (c + d x \right )} \cosh ^{2}{\left (c + d x \right )}}{4} + \frac{9 a^{2} b x \cosh ^{4}{\left (c + d x \right )}}{8} + \frac{15 a^{2} b \sinh ^{3}{\left (c + d x \right )} \cosh{\left (c + d x \right )}}{8 d} - \frac{9 a^{2} b \sinh{\left (c + d x \right )} \cosh ^{3}{\left (c + d x \right )}}{8 d} + \frac{3 a b^{2} \sinh ^{6}{\left (c + d x \right )} \cosh{\left (c + d x \right )}}{d} - \frac{6 a b^{2} \sinh ^{4}{\left (c + d x \right )} \cosh ^{3}{\left (c + d x \right )}}{d} + \frac{24 a b^{2} \sinh ^{2}{\left (c + d x \right )} \cosh ^{5}{\left (c + d x \right )}}{5 d} - \frac{48 a b^{2} \cosh ^{7}{\left (c + d x \right )}}{35 d} + \frac{63 b^{3} x \sinh ^{10}{\left (c + d x \right )}}{256} - \frac{315 b^{3} x \sinh ^{8}{\left (c + d x \right )} \cosh ^{2}{\left (c + d x \right )}}{256} + \frac{315 b^{3} x \sinh ^{6}{\left (c + d x \right )} \cosh ^{4}{\left (c + d x \right )}}{128} - \frac{315 b^{3} x \sinh ^{4}{\left (c + d x \right )} \cosh ^{6}{\left (c + d x \right )}}{128} + \frac{315 b^{3} x \sinh ^{2}{\left (c + d x \right )} \cosh ^{8}{\left (c + d x \right )}}{256} - \frac{63 b^{3} x \cosh ^{10}{\left (c + d x \right )}}{256} + \frac{193 b^{3} \sinh ^{9}{\left (c + d x \right )} \cosh{\left (c + d x \right )}}{256 d} - \frac{237 b^{3} \sinh ^{7}{\left (c + d x \right )} \cosh ^{3}{\left (c + d x \right )}}{128 d} + \frac{21 b^{3} \sinh ^{5}{\left (c + d x \right )} \cosh ^{5}{\left (c + d x \right )}}{10 d} - \frac{147 b^{3} \sinh ^{3}{\left (c + d x \right )} \cosh ^{7}{\left (c + d x \right )}}{128 d} + \frac{63 b^{3} \sinh{\left (c + d x \right )} \cosh ^{9}{\left (c + d x \right )}}{256 d} & \text{for}\: d \neq 0 \\x \left (a + b \sinh ^{3}{\left (c \right )}\right )^{3} \sinh{\left (c \right )} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((a**3*cosh(c + d*x)/d + 9*a**2*b*x*sinh(c + d*x)**4/8 - 9*a**2*b*x*sinh(c + d*x)**2*cosh(c + d*x)**2
/4 + 9*a**2*b*x*cosh(c + d*x)**4/8 + 15*a**2*b*sinh(c + d*x)**3*cosh(c + d*x)/(8*d) - 9*a**2*b*sinh(c + d*x)*c
osh(c + d*x)**3/(8*d) + 3*a*b**2*sinh(c + d*x)**6*cosh(c + d*x)/d - 6*a*b**2*sinh(c + d*x)**4*cosh(c + d*x)**3
/d + 24*a*b**2*sinh(c + d*x)**2*cosh(c + d*x)**5/(5*d) - 48*a*b**2*cosh(c + d*x)**7/(35*d) + 63*b**3*x*sinh(c
+ d*x)**10/256 - 315*b**3*x*sinh(c + d*x)**8*cosh(c + d*x)**2/256 + 315*b**3*x*sinh(c + d*x)**6*cosh(c + d*x)*
*4/128 - 315*b**3*x*sinh(c + d*x)**4*cosh(c + d*x)**6/128 + 315*b**3*x*sinh(c + d*x)**2*cosh(c + d*x)**8/256 -
63*b**3*x*cosh(c + d*x)**10/256 + 193*b**3*sinh(c + d*x)**9*cosh(c + d*x)/(256*d) - 237*b**3*sinh(c + d*x)**7
*cosh(c + d*x)**3/(128*d) + 21*b**3*sinh(c + d*x)**5*cosh(c + d*x)**5/(10*d) - 147*b**3*sinh(c + d*x)**3*cosh(
c + d*x)**7/(128*d) + 63*b**3*sinh(c + d*x)*cosh(c + d*x)**9/(256*d), Ne(d, 0)), (x*(a + b*sinh(c)**3)**3*sinh
(c), True))

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Giac [A]  time = 1.47622, size = 474, normalized size = 1.78 \begin{align*} \frac{14 \, b^{3} e^{\left (10 \, d x + 10 \, c\right )} - 175 \, b^{3} e^{\left (8 \, d x + 8 \, c\right )} + 480 \, a b^{2} e^{\left (7 \, d x + 7 \, c\right )} + 1050 \, b^{3} e^{\left (6 \, d x + 6 \, c\right )} - 4704 \, a b^{2} e^{\left (5 \, d x + 5 \, c\right )} + 6720 \, a^{2} b e^{\left (4 \, d x + 4 \, c\right )} - 4200 \, b^{3} e^{\left (4 \, d x + 4 \, c\right )} + 23520 \, a b^{2} e^{\left (3 \, d x + 3 \, c\right )} - 53760 \, a^{2} b e^{\left (2 \, d x + 2 \, c\right )} + 14700 \, b^{3} e^{\left (2 \, d x + 2 \, c\right )} + 71680 \, a^{3} e^{\left (d x + c\right )} - 117600 \, a b^{2} e^{\left (d x + c\right )} + 5040 \,{\left (32 \, a^{2} b - 7 \, b^{3}\right )}{\left (d x + c\right )} +{\left (23520 \, a b^{2} e^{\left (7 \, d x + 7 \, c\right )} - 4704 \, a b^{2} e^{\left (5 \, d x + 5 \, c\right )} - 1050 \, b^{3} e^{\left (4 \, d x + 4 \, c\right )} + 480 \, a b^{2} e^{\left (3 \, d x + 3 \, c\right )} + 175 \, b^{3} e^{\left (2 \, d x + 2 \, c\right )} - 14 \, b^{3} + 1120 \,{\left (64 \, a^{3} - 105 \, a b^{2}\right )} e^{\left (9 \, d x + 9 \, c\right )} + 420 \,{\left (128 \, a^{2} b - 35 \, b^{3}\right )} e^{\left (8 \, d x + 8 \, c\right )} - 840 \,{\left (8 \, a^{2} b - 5 \, b^{3}\right )} e^{\left (6 \, d x + 6 \, c\right )}\right )} e^{\left (-10 \, d x - 10 \, c\right )}}{143360 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/143360*(14*b^3*e^(10*d*x + 10*c) - 175*b^3*e^(8*d*x + 8*c) + 480*a*b^2*e^(7*d*x + 7*c) + 1050*b^3*e^(6*d*x +
6*c) - 4704*a*b^2*e^(5*d*x + 5*c) + 6720*a^2*b*e^(4*d*x + 4*c) - 4200*b^3*e^(4*d*x + 4*c) + 23520*a*b^2*e^(3*
d*x + 3*c) - 53760*a^2*b*e^(2*d*x + 2*c) + 14700*b^3*e^(2*d*x + 2*c) + 71680*a^3*e^(d*x + c) - 117600*a*b^2*e^
(d*x + c) + 5040*(32*a^2*b - 7*b^3)*(d*x + c) + (23520*a*b^2*e^(7*d*x + 7*c) - 4704*a*b^2*e^(5*d*x + 5*c) - 10
50*b^3*e^(4*d*x + 4*c) + 480*a*b^2*e^(3*d*x + 3*c) + 175*b^3*e^(2*d*x + 2*c) - 14*b^3 + 1120*(64*a^3 - 105*a*b
^2)*e^(9*d*x + 9*c) + 420*(128*a^2*b - 35*b^3)*e^(8*d*x + 8*c) - 840*(8*a^2*b - 5*b^3)*e^(6*d*x + 6*c))*e^(-10
*d*x - 10*c))/d