3.20 \(\int \sinh ^3(c+d x) (a+b \sinh ^2(c+d x))^3 \, dx\)

Optimal. Leaf size=115 \[ \frac{b^2 (3 a-4 b) \cosh ^7(c+d x)}{7 d}+\frac{3 b (a-2 b) (a-b) \cosh ^5(c+d x)}{5 d}+\frac{(a-4 b) (a-b)^2 \cosh ^3(c+d x)}{3 d}-\frac{(a-b)^3 \cosh (c+d x)}{d}+\frac{b^3 \cosh ^9(c+d x)}{9 d} \]

[Out]

-(((a - b)^3*Cosh[c + d*x])/d) + ((a - 4*b)*(a - b)^2*Cosh[c + d*x]^3)/(3*d) + (3*(a - 2*b)*(a - b)*b*Cosh[c +
d*x]^5)/(5*d) + ((3*a - 4*b)*b^2*Cosh[c + d*x]^7)/(7*d) + (b^3*Cosh[c + d*x]^9)/(9*d)

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Rubi [A]  time = 0.129173, antiderivative size = 115, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087, Rules used = {3186, 373} \[ \frac{b^2 (3 a-4 b) \cosh ^7(c+d x)}{7 d}+\frac{3 b (a-2 b) (a-b) \cosh ^5(c+d x)}{5 d}+\frac{(a-4 b) (a-b)^2 \cosh ^3(c+d x)}{3 d}-\frac{(a-b)^3 \cosh (c+d x)}{d}+\frac{b^3 \cosh ^9(c+d x)}{9 d} \]

Antiderivative was successfully verified.

[In]

Int[Sinh[c + d*x]^3*(a + b*Sinh[c + d*x]^2)^3,x]

[Out]

-(((a - b)^3*Cosh[c + d*x])/d) + ((a - 4*b)*(a - b)^2*Cosh[c + d*x]^3)/(3*d) + (3*(a - 2*b)*(a - b)*b*Cosh[c +
d*x]^5)/(5*d) + ((3*a - 4*b)*b^2*Cosh[c + d*x]^7)/(7*d) + (b^3*Cosh[c + d*x]^9)/(9*d)

Rule 3186

Int[sin[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_.), x_Symbol] :> With[{ff = Free
Factors[Cos[e + f*x], x]}, -Dist[ff/f, Subst[Int[(1 - ff^2*x^2)^((m - 1)/2)*(a + b - b*ff^2*x^2)^p, x], x, Cos
[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[(m - 1)/2]

Rule 373

Int[((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x^n
)^p*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[p, 0] && IGtQ[q, 0]

Rubi steps

\begin{align*} \int \sinh ^3(c+d x) \left (a+b \sinh ^2(c+d x)\right )^3 \, dx &=-\frac{\operatorname{Subst}\left (\int \left (1-x^2\right ) \left (a-b+b x^2\right )^3 \, dx,x,\cosh (c+d x)\right )}{d}\\ &=-\frac{\operatorname{Subst}\left (\int \left ((a-b)^3-(a-4 b) (a-b)^2 x^2+3 (a-2 b) b (-a+b) x^4-(3 a-4 b) b^2 x^6-b^3 x^8\right ) \, dx,x,\cosh (c+d x)\right )}{d}\\ &=-\frac{(a-b)^3 \cosh (c+d x)}{d}+\frac{(a-4 b) (a-b)^2 \cosh ^3(c+d x)}{3 d}+\frac{3 (a-2 b) (a-b) b \cosh ^5(c+d x)}{5 d}+\frac{(3 a-4 b) b^2 \cosh ^7(c+d x)}{7 d}+\frac{b^3 \cosh ^9(c+d x)}{9 d}\\ \end{align*}

Mathematica [A]  time = 0.796978, size = 127, normalized size = 1.1 \[ \frac{-1890 (4 a-3 b) \left (8 a^2-14 a b+7 b^2\right ) \cosh (c+d x)+420 \left (-60 a^2 b+16 a^3+63 a b^2-21 b^3\right ) \cosh (3 (c+d x))+135 b^2 (4 a-3 b) \cosh (7 (c+d x))+756 b (4 a-3 b) (a-b) \cosh (5 (c+d x))+35 b^3 \cosh (9 (c+d x))}{80640 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Sinh[c + d*x]^3*(a + b*Sinh[c + d*x]^2)^3,x]

[Out]

(-1890*(4*a - 3*b)*(8*a^2 - 14*a*b + 7*b^2)*Cosh[c + d*x] + 420*(16*a^3 - 60*a^2*b + 63*a*b^2 - 21*b^3)*Cosh[3
*(c + d*x)] + 756*(4*a - 3*b)*(a - b)*b*Cosh[5*(c + d*x)] + 135*(4*a - 3*b)*b^2*Cosh[7*(c + d*x)] + 35*b^3*Cos
h[9*(c + d*x)])/(80640*d)

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Maple [A]  time = 0.049, size = 158, normalized size = 1.4 \begin{align*}{\frac{1}{d} \left ({b}^{3} \left ({\frac{128}{315}}+{\frac{ \left ( \sinh \left ( dx+c \right ) \right ) ^{8}}{9}}-{\frac{8\, \left ( \sinh \left ( dx+c \right ) \right ) ^{6}}{63}}+{\frac{16\, \left ( \sinh \left ( dx+c \right ) \right ) ^{4}}{105}}-{\frac{64\, \left ( \sinh \left ( dx+c \right ) \right ) ^{2}}{315}} \right ) \cosh \left ( dx+c \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 ({\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 ) +{a}^{3} \left ( -{\frac{2}{3}}+{\frac{ \left ( \sinh \left ( dx+c \right ) \right ) ^{2}}{3}} \right ) \cosh \left ( dx+c \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sinh(d*x+c)^3*(a+b*sinh(d*x+c)^2)^3,x)

[Out]

1/d*(b^3*(128/315+1/9*sinh(d*x+c)^8-8/63*sinh(d*x+c)^6+16/105*sinh(d*x+c)^4-64/315*sinh(d*x+c)^2)*cosh(d*x+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)*cosh(d*x+c)+3*a^2*b*(8/15+1/5*sinh(d*
x+c)^4-4/15*sinh(d*x+c)^2)*cosh(d*x+c)+a^3*(-2/3+1/3*sinh(d*x+c)^2)*cosh(d*x+c))

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Maxima [B]  time = 1.08331, size = 508, normalized size = 4.42 \begin{align*} -\frac{1}{161280} \, b^{3}{\left (\frac{{\left (405 \, e^{\left (-2 \, d x - 2 \, c\right )} - 2268 \, e^{\left (-4 \, d x - 4 \, c\right )} + 8820 \, e^{\left (-6 \, d x - 6 \, c\right )} - 39690 \, e^{\left (-8 \, d x - 8 \, c\right )} - 35\right )} e^{\left (9 \, d x + 9 \, c\right )}}{d} - \frac{39690 \, e^{\left (-d x - c\right )} - 8820 \, e^{\left (-3 \, d x - 3 \, c\right )} + 2268 \, e^{\left (-5 \, d x - 5 \, c\right )} - 405 \, e^{\left (-7 \, d x - 7 \, c\right )} + 35 \, e^{\left (-9 \, d x - 9 \, 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{1}{160} \, a^{2} b{\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{1}{24} \, a^{3}{\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 )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(d*x+c)^3*(a+b*sinh(d*x+c)^2)^3,x, algorithm="maxima")

[Out]

-1/161280*b^3*((405*e^(-2*d*x - 2*c) - 2268*e^(-4*d*x - 4*c) + 8820*e^(-6*d*x - 6*c) - 39690*e^(-8*d*x - 8*c)
- 35)*e^(9*d*x + 9*c)/d - (39690*e^(-d*x - c) - 8820*e^(-3*d*x - 3*c) + 2268*e^(-5*d*x - 5*c) - 405*e^(-7*d*x
- 7*c) + 35*e^(-9*d*x - 9*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) + 1/160*a^2*b*(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) + 1/24*a^3*(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)

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Fricas [B]  time = 1.82041, size = 940, normalized size = 8.17 \begin{align*} \frac{35 \, b^{3} \cosh \left (d x + c\right )^{9} + 315 \, b^{3} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{8} + 135 \,{\left (4 \, a b^{2} - 3 \, b^{3}\right )} \cosh \left (d x + c\right )^{7} + 105 \,{\left (28 \, b^{3} \cosh \left (d x + c\right )^{3} + 9 \,{\left (4 \, a b^{2} - 3 \, b^{3}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{6} + 756 \,{\left (4 \, a^{2} b - 7 \, a b^{2} + 3 \, b^{3}\right )} \cosh \left (d x + c\right )^{5} + 315 \,{\left (14 \, b^{3} \cosh \left (d x + c\right )^{5} + 15 \,{\left (4 \, a b^{2} - 3 \, b^{3}\right )} \cosh \left (d x + c\right )^{3} + 12 \,{\left (4 \, a^{2} b - 7 \, a b^{2} + 3 \, b^{3}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{4} + 420 \,{\left (16 \, a^{3} - 60 \, a^{2} b + 63 \, a b^{2} - 21 \, b^{3}\right )} \cosh \left (d x + c\right )^{3} + 315 \,{\left (4 \, b^{3} \cosh \left (d x + c\right )^{7} + 9 \,{\left (4 \, a b^{2} - 3 \, b^{3}\right )} \cosh \left (d x + c\right )^{5} + 24 \,{\left (4 \, a^{2} b - 7 \, a b^{2} + 3 \, b^{3}\right )} \cosh \left (d x + c\right )^{3} + 4 \,{\left (16 \, a^{3} - 60 \, a^{2} b + 63 \, a b^{2} - 21 \, b^{3}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{2} - 1890 \,{\left (32 \, a^{3} - 80 \, a^{2} b + 70 \, a b^{2} - 21 \, b^{3}\right )} \cosh \left (d x + c\right )}{80640 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(d*x+c)^3*(a+b*sinh(d*x+c)^2)^3,x, algorithm="fricas")

[Out]

1/80640*(35*b^3*cosh(d*x + c)^9 + 315*b^3*cosh(d*x + c)*sinh(d*x + c)^8 + 135*(4*a*b^2 - 3*b^3)*cosh(d*x + c)^
7 + 105*(28*b^3*cosh(d*x + c)^3 + 9*(4*a*b^2 - 3*b^3)*cosh(d*x + c))*sinh(d*x + c)^6 + 756*(4*a^2*b - 7*a*b^2
+ 3*b^3)*cosh(d*x + c)^5 + 315*(14*b^3*cosh(d*x + c)^5 + 15*(4*a*b^2 - 3*b^3)*cosh(d*x + c)^3 + 12*(4*a^2*b -
7*a*b^2 + 3*b^3)*cosh(d*x + c))*sinh(d*x + c)^4 + 420*(16*a^3 - 60*a^2*b + 63*a*b^2 - 21*b^3)*cosh(d*x + c)^3
+ 315*(4*b^3*cosh(d*x + c)^7 + 9*(4*a*b^2 - 3*b^3)*cosh(d*x + c)^5 + 24*(4*a^2*b - 7*a*b^2 + 3*b^3)*cosh(d*x +
c)^3 + 4*(16*a^3 - 60*a^2*b + 63*a*b^2 - 21*b^3)*cosh(d*x + c))*sinh(d*x + c)^2 - 1890*(32*a^3 - 80*a^2*b + 7
0*a*b^2 - 21*b^3)*cosh(d*x + c))/d

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Sympy [A]  time = 21.724, size = 330, normalized size = 2.87 \begin{align*} \begin{cases} \frac{a^{3} \sinh ^{2}{\left (c + d x \right )} \cosh{\left (c + d x \right )}}{d} - \frac{2 a^{3} \cosh ^{3}{\left (c + d x \right )}}{3 d} + \frac{3 a^{2} b \sinh ^{4}{\left (c + d x \right )} \cosh{\left (c + d x \right )}}{d} - \frac{4 a^{2} b \sinh ^{2}{\left (c + d x \right )} \cosh ^{3}{\left (c + d x \right )}}{d} + \frac{8 a^{2} b \cosh ^{5}{\left (c + d x \right )}}{5 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{b^{3} \sinh ^{8}{\left (c + d x \right )} \cosh{\left (c + d x \right )}}{d} - \frac{8 b^{3} \sinh ^{6}{\left (c + d x \right )} \cosh ^{3}{\left (c + d x \right )}}{3 d} + \frac{16 b^{3} \sinh ^{4}{\left (c + d x \right )} \cosh ^{5}{\left (c + d x \right )}}{5 d} - \frac{64 b^{3} \sinh ^{2}{\left (c + d x \right )} \cosh ^{7}{\left (c + d x \right )}}{35 d} + \frac{128 b^{3} \cosh ^{9}{\left (c + d x \right )}}{315 d} & \text{for}\: d \neq 0 \\x \left (a + b \sinh ^{2}{\left (c \right )}\right )^{3} \sinh ^{3}{\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)**3*(a+b*sinh(d*x+c)**2)**3,x)

[Out]

Piecewise((a**3*sinh(c + d*x)**2*cosh(c + d*x)/d - 2*a**3*cosh(c + d*x)**3/(3*d) + 3*a**2*b*sinh(c + d*x)**4*c
osh(c + d*x)/d - 4*a**2*b*sinh(c + d*x)**2*cosh(c + d*x)**3/d + 8*a**2*b*cosh(c + d*x)**5/(5*d) + 3*a*b**2*sin
h(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) + b**3*sinh(c + d*x)**8*cosh(c + d*x)/d - 8*b**3*sinh(c
+ d*x)**6*cosh(c + d*x)**3/(3*d) + 16*b**3*sinh(c + d*x)**4*cosh(c + d*x)**5/(5*d) - 64*b**3*sinh(c + d*x)**2
*cosh(c + d*x)**7/(35*d) + 128*b**3*cosh(c + d*x)**9/(315*d), Ne(d, 0)), (x*(a + b*sinh(c)**2)**3*sinh(c)**3,
True))

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Giac [B]  time = 1.41424, size = 544, normalized size = 4.73 \begin{align*} \frac{35 \, b^{3} e^{\left (9 \, d x + 9 \, c\right )} + 540 \, a b^{2} e^{\left (7 \, d x + 7 \, c\right )} - 405 \, b^{3} e^{\left (7 \, d x + 7 \, c\right )} + 3024 \, a^{2} b e^{\left (5 \, d x + 5 \, c\right )} - 5292 \, a b^{2} e^{\left (5 \, d x + 5 \, c\right )} + 2268 \, b^{3} e^{\left (5 \, d x + 5 \, c\right )} + 6720 \, a^{3} e^{\left (3 \, d x + 3 \, c\right )} - 25200 \, a^{2} b e^{\left (3 \, d x + 3 \, c\right )} + 26460 \, a b^{2} e^{\left (3 \, d x + 3 \, c\right )} - 8820 \, b^{3} e^{\left (3 \, d x + 3 \, c\right )} - 60480 \, a^{3} e^{\left (d x + c\right )} + 151200 \, a^{2} b e^{\left (d x + c\right )} - 132300 \, a b^{2} e^{\left (d x + c\right )} + 39690 \, b^{3} e^{\left (d x + c\right )} -{\left (60480 \, a^{3} e^{\left (8 \, d x + 8 \, c\right )} - 151200 \, a^{2} b e^{\left (8 \, d x + 8 \, c\right )} + 132300 \, a b^{2} e^{\left (8 \, d x + 8 \, c\right )} - 39690 \, b^{3} e^{\left (8 \, d x + 8 \, c\right )} - 6720 \, a^{3} e^{\left (6 \, d x + 6 \, c\right )} + 25200 \, a^{2} b e^{\left (6 \, d x + 6 \, c\right )} - 26460 \, a b^{2} e^{\left (6 \, d x + 6 \, c\right )} + 8820 \, b^{3} e^{\left (6 \, d x + 6 \, c\right )} - 3024 \, a^{2} b e^{\left (4 \, d x + 4 \, c\right )} + 5292 \, a b^{2} e^{\left (4 \, d x + 4 \, c\right )} - 2268 \, b^{3} e^{\left (4 \, d x + 4 \, c\right )} - 540 \, a b^{2} e^{\left (2 \, d x + 2 \, c\right )} + 405 \, b^{3} e^{\left (2 \, d x + 2 \, c\right )} - 35 \, b^{3}\right )} e^{\left (-9 \, d x - 9 \, c\right )}}{161280 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(d*x+c)^3*(a+b*sinh(d*x+c)^2)^3,x, algorithm="giac")

[Out]

1/161280*(35*b^3*e^(9*d*x + 9*c) + 540*a*b^2*e^(7*d*x + 7*c) - 405*b^3*e^(7*d*x + 7*c) + 3024*a^2*b*e^(5*d*x +
5*c) - 5292*a*b^2*e^(5*d*x + 5*c) + 2268*b^3*e^(5*d*x + 5*c) + 6720*a^3*e^(3*d*x + 3*c) - 25200*a^2*b*e^(3*d*
x + 3*c) + 26460*a*b^2*e^(3*d*x + 3*c) - 8820*b^3*e^(3*d*x + 3*c) - 60480*a^3*e^(d*x + c) + 151200*a^2*b*e^(d*
x + c) - 132300*a*b^2*e^(d*x + c) + 39690*b^3*e^(d*x + c) - (60480*a^3*e^(8*d*x + 8*c) - 151200*a^2*b*e^(8*d*x
+ 8*c) + 132300*a*b^2*e^(8*d*x + 8*c) - 39690*b^3*e^(8*d*x + 8*c) - 6720*a^3*e^(6*d*x + 6*c) + 25200*a^2*b*e^
(6*d*x + 6*c) - 26460*a*b^2*e^(6*d*x + 6*c) + 8820*b^3*e^(6*d*x + 6*c) - 3024*a^2*b*e^(4*d*x + 4*c) + 5292*a*b
^2*e^(4*d*x + 4*c) - 2268*b^3*e^(4*d*x + 4*c) - 540*a*b^2*e^(2*d*x + 2*c) + 405*b^3*e^(2*d*x + 2*c) - 35*b^3)*
e^(-9*d*x - 9*c))/d