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

Optimal. Leaf size=180 \[ \frac{a^2 \sinh (c+d x) \cosh (c+d x)}{2 d}-\frac{a^2 x}{2}+\frac{2 a b \cosh ^5(c+d x)}{5 d}-\frac{4 a b \cosh ^3(c+d x)}{3 d}+\frac{2 a b \cosh (c+d x)}{d}+\frac{b^2 \sinh ^7(c+d x) \cosh (c+d x)}{8 d}-\frac{7 b^2 \sinh ^5(c+d x) \cosh (c+d x)}{48 d}+\frac{35 b^2 \sinh ^3(c+d x) \cosh (c+d x)}{192 d}-\frac{35 b^2 \sinh (c+d x) \cosh (c+d x)}{128 d}+\frac{35 b^2 x}{128} \]

[Out]

-(a^2*x)/2 + (35*b^2*x)/128 + (2*a*b*Cosh[c + d*x])/d - (4*a*b*Cosh[c + d*x]^3)/(3*d) + (2*a*b*Cosh[c + d*x]^5
)/(5*d) + (a^2*Cosh[c + d*x]*Sinh[c + d*x])/(2*d) - (35*b^2*Cosh[c + d*x]*Sinh[c + d*x])/(128*d) + (35*b^2*Cos
h[c + d*x]*Sinh[c + d*x]^3)/(192*d) - (7*b^2*Cosh[c + d*x]*Sinh[c + d*x]^5)/(48*d) + (b^2*Cosh[c + d*x]*Sinh[c
+ d*x]^7)/(8*d)

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Rubi [A]  time = 0.154268, antiderivative size = 180, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174, Rules used = {3220, 2635, 8, 2633} \[ \frac{a^2 \sinh (c+d x) \cosh (c+d x)}{2 d}-\frac{a^2 x}{2}+\frac{2 a b \cosh ^5(c+d x)}{5 d}-\frac{4 a b \cosh ^3(c+d x)}{3 d}+\frac{2 a b \cosh (c+d x)}{d}+\frac{b^2 \sinh ^7(c+d x) \cosh (c+d x)}{8 d}-\frac{7 b^2 \sinh ^5(c+d x) \cosh (c+d x)}{48 d}+\frac{35 b^2 \sinh ^3(c+d x) \cosh (c+d x)}{192 d}-\frac{35 b^2 \sinh (c+d x) \cosh (c+d x)}{128 d}+\frac{35 b^2 x}{128} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(a^2*x)/2 + (35*b^2*x)/128 + (2*a*b*Cosh[c + d*x])/d - (4*a*b*Cosh[c + d*x]^3)/(3*d) + (2*a*b*Cosh[c + d*x]^5
)/(5*d) + (a^2*Cosh[c + d*x]*Sinh[c + d*x])/(2*d) - (35*b^2*Cosh[c + d*x]*Sinh[c + d*x])/(128*d) + (35*b^2*Cos
h[c + d*x]*Sinh[c + d*x]^3)/(192*d) - (7*b^2*Cosh[c + d*x]*Sinh[c + d*x]^5)/(48*d) + (b^2*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 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 ^2(c+d x) \left (a+b \sinh ^3(c+d x)\right )^2 \, dx &=-\int \left (-a^2 \sinh ^2(c+d x)-2 a b \sinh ^5(c+d x)-b^2 \sinh ^8(c+d x)\right ) \, dx\\ &=a^2 \int \sinh ^2(c+d x) \, dx+(2 a b) \int \sinh ^5(c+d x) \, dx+b^2 \int \sinh ^8(c+d x) \, dx\\ &=\frac{a^2 \cosh (c+d x) \sinh (c+d x)}{2 d}+\frac{b^2 \cosh (c+d x) \sinh ^7(c+d x)}{8 d}-\frac{1}{2} a^2 \int 1 \, dx-\frac{1}{8} \left (7 b^2\right ) \int \sinh ^6(c+d x) \, dx+\frac{(2 a b) \operatorname{Subst}\left (\int \left (1-2 x^2+x^4\right ) \, dx,x,\cosh (c+d x)\right )}{d}\\ &=-\frac{a^2 x}{2}+\frac{2 a b \cosh (c+d x)}{d}-\frac{4 a b \cosh ^3(c+d x)}{3 d}+\frac{2 a b \cosh ^5(c+d x)}{5 d}+\frac{a^2 \cosh (c+d x) \sinh (c+d x)}{2 d}-\frac{7 b^2 \cosh (c+d x) \sinh ^5(c+d x)}{48 d}+\frac{b^2 \cosh (c+d x) \sinh ^7(c+d x)}{8 d}+\frac{1}{48} \left (35 b^2\right ) \int \sinh ^4(c+d x) \, dx\\ &=-\frac{a^2 x}{2}+\frac{2 a b \cosh (c+d x)}{d}-\frac{4 a b \cosh ^3(c+d x)}{3 d}+\frac{2 a b \cosh ^5(c+d x)}{5 d}+\frac{a^2 \cosh (c+d x) \sinh (c+d x)}{2 d}+\frac{35 b^2 \cosh (c+d x) \sinh ^3(c+d x)}{192 d}-\frac{7 b^2 \cosh (c+d x) \sinh ^5(c+d x)}{48 d}+\frac{b^2 \cosh (c+d x) \sinh ^7(c+d x)}{8 d}-\frac{1}{64} \left (35 b^2\right ) \int \sinh ^2(c+d x) \, dx\\ &=-\frac{a^2 x}{2}+\frac{2 a b \cosh (c+d x)}{d}-\frac{4 a b \cosh ^3(c+d x)}{3 d}+\frac{2 a b \cosh ^5(c+d x)}{5 d}+\frac{a^2 \cosh (c+d x) \sinh (c+d x)}{2 d}-\frac{35 b^2 \cosh (c+d x) \sinh (c+d x)}{128 d}+\frac{35 b^2 \cosh (c+d x) \sinh ^3(c+d x)}{192 d}-\frac{7 b^2 \cosh (c+d x) \sinh ^5(c+d x)}{48 d}+\frac{b^2 \cosh (c+d x) \sinh ^7(c+d x)}{8 d}+\frac{1}{128} \left (35 b^2\right ) \int 1 \, dx\\ &=-\frac{a^2 x}{2}+\frac{35 b^2 x}{128}+\frac{2 a b \cosh (c+d x)}{d}-\frac{4 a b \cosh ^3(c+d x)}{3 d}+\frac{2 a b \cosh ^5(c+d x)}{5 d}+\frac{a^2 \cosh (c+d x) \sinh (c+d x)}{2 d}-\frac{35 b^2 \cosh (c+d x) \sinh (c+d x)}{128 d}+\frac{35 b^2 \cosh (c+d x) \sinh ^3(c+d x)}{192 d}-\frac{7 b^2 \cosh (c+d x) \sinh ^5(c+d x)}{48 d}+\frac{b^2 \cosh (c+d x) \sinh ^7(c+d x)}{8 d}\\ \end{align*}

Mathematica [A]  time = 0.129859, size = 133, normalized size = 0.74 \[ \frac{3840 a^2 \sinh (2 (c+d x))-7680 a^2 c-7680 a^2 d x+19200 a b \cosh (c+d x)-3200 a b \cosh (3 (c+d x))+384 a b \cosh (5 (c+d x))-3360 b^2 \sinh (2 (c+d x))+840 b^2 \sinh (4 (c+d x))-160 b^2 \sinh (6 (c+d x))+15 b^2 \sinh (8 (c+d x))+4200 b^2 c+4200 b^2 d x}{15360 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-7680*a^2*c + 4200*b^2*c - 7680*a^2*d*x + 4200*b^2*d*x + 19200*a*b*Cosh[c + d*x] - 3200*a*b*Cosh[3*(c + d*x)]
+ 384*a*b*Cosh[5*(c + d*x)] + 3840*a^2*Sinh[2*(c + d*x)] - 3360*b^2*Sinh[2*(c + d*x)] + 840*b^2*Sinh[4*(c + d
*x)] - 160*b^2*Sinh[6*(c + d*x)] + 15*b^2*Sinh[8*(c + d*x)])/(15360*d)

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Maple [A]  time = 0.021, size = 122, normalized size = 0.7 \begin{align*}{\frac{1}{d} \left ({b}^{2} \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 ) +2\,ab \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}^{2} \left ({\frac{\cosh \left ( dx+c \right ) \sinh \left ( dx+c \right ) }{2}}-{\frac{dx}{2}}-{\frac{c}{2}} \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.15235, size = 320, normalized size = 1.78 \begin{align*} -\frac{1}{8} \, a^{2}{\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^{2}{\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}{240} \, a 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 )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8*a^2*(4*x - e^(2*d*x + 2*c)/d + e^(-2*d*x - 2*c)/d) - 1/6144*b^2*((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) + 1/240*a*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)

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Fricas [A]  time = 1.90907, size = 734, normalized size = 4.08 \begin{align*} \frac{15 \, b^{2} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{7} + 48 \, a b \cosh \left (d x + c\right )^{5} + 240 \, a b \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{4} + 15 \,{\left (7 \, b^{2} \cosh \left (d x + c\right )^{3} - 8 \, b^{2} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{5} - 400 \, a b \cosh \left (d x + c\right )^{3} + 5 \,{\left (21 \, b^{2} \cosh \left (d x + c\right )^{5} - 80 \, b^{2} \cosh \left (d x + c\right )^{3} + 84 \, b^{2} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{3} - 15 \,{\left (64 \, a^{2} - 35 \, b^{2}\right )} d x + 2400 \, a b \cosh \left (d x + c\right ) + 240 \,{\left (2 \, a b \cosh \left (d x + c\right )^{3} - 5 \, a b \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{2} + 15 \,{\left (b^{2} \cosh \left (d x + c\right )^{7} - 8 \, b^{2} \cosh \left (d x + c\right )^{5} + 28 \, b^{2} \cosh \left (d x + c\right )^{3} + 8 \,{\left (8 \, a^{2} - 7 \, b^{2}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )}{1920 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/1920*(15*b^2*cosh(d*x + c)*sinh(d*x + c)^7 + 48*a*b*cosh(d*x + c)^5 + 240*a*b*cosh(d*x + c)*sinh(d*x + c)^4
+ 15*(7*b^2*cosh(d*x + c)^3 - 8*b^2*cosh(d*x + c))*sinh(d*x + c)^5 - 400*a*b*cosh(d*x + c)^3 + 5*(21*b^2*cosh(
d*x + c)^5 - 80*b^2*cosh(d*x + c)^3 + 84*b^2*cosh(d*x + c))*sinh(d*x + c)^3 - 15*(64*a^2 - 35*b^2)*d*x + 2400*
a*b*cosh(d*x + c) + 240*(2*a*b*cosh(d*x + c)^3 - 5*a*b*cosh(d*x + c))*sinh(d*x + c)^2 + 15*(b^2*cosh(d*x + c)^
7 - 8*b^2*cosh(d*x + c)^5 + 28*b^2*cosh(d*x + c)^3 + 8*(8*a^2 - 7*b^2)*cosh(d*x + c))*sinh(d*x + c))/d

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Sympy [A]  time = 15.3381, size = 340, normalized size = 1.89 \begin{align*} \begin{cases} \frac{a^{2} x \sinh ^{2}{\left (c + d x \right )}}{2} - \frac{a^{2} x \cosh ^{2}{\left (c + d x \right )}}{2} + \frac{a^{2} \sinh{\left (c + d x \right )} \cosh{\left (c + d x \right )}}{2 d} + \frac{2 a b \sinh ^{4}{\left (c + d x \right )} \cosh{\left (c + d x \right )}}{d} - \frac{8 a b \sinh ^{2}{\left (c + d x \right )} \cosh ^{3}{\left (c + d x \right )}}{3 d} + \frac{16 a b \cosh ^{5}{\left (c + d x \right )}}{15 d} + \frac{35 b^{2} x \sinh ^{8}{\left (c + d x \right )}}{128} - \frac{35 b^{2} x \sinh ^{6}{\left (c + d x \right )} \cosh ^{2}{\left (c + d x \right )}}{32} + \frac{105 b^{2} x \sinh ^{4}{\left (c + d x \right )} \cosh ^{4}{\left (c + d x \right )}}{64} - \frac{35 b^{2} x \sinh ^{2}{\left (c + d x \right )} \cosh ^{6}{\left (c + d x \right )}}{32} + \frac{35 b^{2} x \cosh ^{8}{\left (c + d x \right )}}{128} + \frac{93 b^{2} \sinh ^{7}{\left (c + d x \right )} \cosh{\left (c + d x \right )}}{128 d} - \frac{511 b^{2} \sinh ^{5}{\left (c + d x \right )} \cosh ^{3}{\left (c + d x \right )}}{384 d} + \frac{385 b^{2} \sinh ^{3}{\left (c + d x \right )} \cosh ^{5}{\left (c + d x \right )}}{384 d} - \frac{35 b^{2} \sinh{\left (c + d x \right )} \cosh ^{7}{\left (c + d x \right )}}{128 d} & \text{for}\: d \neq 0 \\x \left (a + b \sinh ^{3}{\left (c \right )}\right )^{2} \sinh ^{2}{\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)**2*(a+b*sinh(d*x+c)**3)**2,x)

[Out]

Piecewise((a**2*x*sinh(c + d*x)**2/2 - a**2*x*cosh(c + d*x)**2/2 + a**2*sinh(c + d*x)*cosh(c + d*x)/(2*d) + 2*
a*b*sinh(c + d*x)**4*cosh(c + d*x)/d - 8*a*b*sinh(c + d*x)**2*cosh(c + d*x)**3/(3*d) + 16*a*b*cosh(c + d*x)**5
/(15*d) + 35*b**2*x*sinh(c + d*x)**8/128 - 35*b**2*x*sinh(c + d*x)**6*cosh(c + d*x)**2/32 + 105*b**2*x*sinh(c
+ d*x)**4*cosh(c + d*x)**4/64 - 35*b**2*x*sinh(c + d*x)**2*cosh(c + d*x)**6/32 + 35*b**2*x*cosh(c + d*x)**8/12
8 + 93*b**2*sinh(c + d*x)**7*cosh(c + d*x)/(128*d) - 511*b**2*sinh(c + d*x)**5*cosh(c + d*x)**3/(384*d) + 385*
b**2*sinh(c + d*x)**3*cosh(c + d*x)**5/(384*d) - 35*b**2*sinh(c + d*x)*cosh(c + d*x)**7/(128*d), Ne(d, 0)), (x
*(a + b*sinh(c)**3)**2*sinh(c)**2, True))

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Giac [A]  time = 1.25071, size = 317, normalized size = 1.76 \begin{align*} \frac{15 \, b^{2} e^{\left (8 \, d x + 8 \, c\right )} - 160 \, b^{2} e^{\left (6 \, d x + 6 \, c\right )} + 384 \, a b e^{\left (5 \, d x + 5 \, c\right )} + 840 \, b^{2} e^{\left (4 \, d x + 4 \, c\right )} - 3200 \, a b e^{\left (3 \, d x + 3 \, c\right )} + 3840 \, a^{2} e^{\left (2 \, d x + 2 \, c\right )} - 3360 \, b^{2} e^{\left (2 \, d x + 2 \, c\right )} + 19200 \, a b e^{\left (d x + c\right )} - 240 \,{\left (64 \, a^{2} - 35 \, b^{2}\right )}{\left (d x + c\right )} +{\left (19200 \, a b e^{\left (7 \, d x + 7 \, c\right )} - 3200 \, a b e^{\left (5 \, d x + 5 \, c\right )} - 840 \, b^{2} e^{\left (4 \, d x + 4 \, c\right )} + 384 \, a b e^{\left (3 \, d x + 3 \, c\right )} + 160 \, b^{2} e^{\left (2 \, d x + 2 \, c\right )} - 15 \, b^{2} - 480 \,{\left (8 \, a^{2} - 7 \, b^{2}\right )} e^{\left (6 \, d x + 6 \, c\right )}\right )} e^{\left (-8 \, d x - 8 \, c\right )}}{30720 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/30720*(15*b^2*e^(8*d*x + 8*c) - 160*b^2*e^(6*d*x + 6*c) + 384*a*b*e^(5*d*x + 5*c) + 840*b^2*e^(4*d*x + 4*c)
- 3200*a*b*e^(3*d*x + 3*c) + 3840*a^2*e^(2*d*x + 2*c) - 3360*b^2*e^(2*d*x + 2*c) + 19200*a*b*e^(d*x + c) - 240
*(64*a^2 - 35*b^2)*(d*x + c) + (19200*a*b*e^(7*d*x + 7*c) - 3200*a*b*e^(5*d*x + 5*c) - 840*b^2*e^(4*d*x + 4*c)
+ 384*a*b*e^(3*d*x + 3*c) + 160*b^2*e^(2*d*x + 2*c) - 15*b^2 - 480*(8*a^2 - 7*b^2)*e^(6*d*x + 6*c))*e^(-8*d*x
- 8*c))/d