3.198 \(\int \sinh (c+d x) (a+b \sinh ^4(c+d x))^2 \, dx\)

Optimal. Leaf size=92 \[ \frac{2 b (a+3 b) \cosh ^5(c+d x)}{5 d}-\frac{4 b (a+b) \cosh ^3(c+d x)}{3 d}+\frac{(a+b)^2 \cosh (c+d x)}{d}+\frac{b^2 \cosh ^9(c+d x)}{9 d}-\frac{4 b^2 \cosh ^7(c+d x)}{7 d} \]

[Out]

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

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Rubi [A]  time = 0.0860677, antiderivative size = 92, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {3215, 1090} \[ \frac{2 b (a+3 b) \cosh ^5(c+d x)}{5 d}-\frac{4 b (a+b) \cosh ^3(c+d x)}{3 d}+\frac{(a+b)^2 \cosh (c+d x)}{d}+\frac{b^2 \cosh ^9(c+d x)}{9 d}-\frac{4 b^2 \cosh ^7(c+d x)}{7 d} \]

Antiderivative was successfully verified.

[In]

Int[Sinh[c + d*x]*(a + b*Sinh[c + d*x]^4)^2,x]

[Out]

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

Rule 3215

Int[sin[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^4)^(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 - 2*b*ff^2*x^2 + b*ff^4*x^4
)^p, x], x, Cos[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[(m - 1)/2]

Rule 1090

Int[((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Int[ExpandIntegrand[(a + b*x^2 + c*x^4)^p, x], x]
/; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && IGtQ[p, 0]

Rubi steps

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

Mathematica [A]  time = 0.0440235, size = 164, normalized size = 1.78 \[ \frac{a^2 \sinh (c) \sinh (d x)}{d}+\frac{a^2 \cosh (c) \cosh (d x)}{d}+\frac{5 a b \cosh (c+d x)}{4 d}-\frac{5 a b \cosh (3 (c+d x))}{24 d}+\frac{a b \cosh (5 (c+d x))}{40 d}+\frac{63 b^2 \cosh (c+d x)}{128 d}-\frac{7 b^2 \cosh (3 (c+d x))}{64 d}+\frac{9 b^2 \cosh (5 (c+d x))}{320 d}-\frac{9 b^2 \cosh (7 (c+d x))}{1792 d}+\frac{b^2 \cosh (9 (c+d x))}{2304 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Sinh[c + d*x]*(a + b*Sinh[c + d*x]^4)^2,x]

[Out]

(a^2*Cosh[c]*Cosh[d*x])/d + (5*a*b*Cosh[c + d*x])/(4*d) + (63*b^2*Cosh[c + d*x])/(128*d) - (5*a*b*Cosh[3*(c +
d*x)])/(24*d) - (7*b^2*Cosh[3*(c + d*x)])/(64*d) + (a*b*Cosh[5*(c + d*x)])/(40*d) + (9*b^2*Cosh[5*(c + d*x)])/
(320*d) - (9*b^2*Cosh[7*(c + d*x)])/(1792*d) + (b^2*Cosh[9*(c + d*x)])/(2304*d) + (a^2*Sinh[c]*Sinh[d*x])/d

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Maple [A]  time = 0.021, size = 100, normalized size = 1.1 \begin{align*}{\frac{1}{d} \left ({b}^{2} \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 ) +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}\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)^4)^2,x)

[Out]

1/d*(b^2*(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)+
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*cosh(d*x+c))

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Maxima [B]  time = 1.04689, size = 305, normalized size = 3.32 \begin{align*} -\frac{1}{161280} \, b^{2}{\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{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 )} + \frac{a^{2} \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)^4)^2,x, algorithm="maxima")

[Out]

-1/161280*b^2*((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) + 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) + a^2*cosh(d*x + c)/d

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Fricas [B]  time = 1.63248, size = 740, normalized size = 8.04 \begin{align*} \frac{35 \, b^{2} \cosh \left (d x + c\right )^{9} + 315 \, b^{2} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{8} - 405 \, b^{2} \cosh \left (d x + c\right )^{7} + 105 \,{\left (28 \, b^{2} \cosh \left (d x + c\right )^{3} - 27 \, b^{2} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{6} + 252 \,{\left (8 \, a b + 9 \, b^{2}\right )} \cosh \left (d x + c\right )^{5} + 315 \,{\left (14 \, b^{2} \cosh \left (d x + c\right )^{5} - 45 \, b^{2} \cosh \left (d x + c\right )^{3} + 4 \,{\left (8 \, a b + 9 \, b^{2}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{4} - 420 \,{\left (40 \, a b + 21 \, b^{2}\right )} \cosh \left (d x + c\right )^{3} + 315 \,{\left (4 \, b^{2} \cosh \left (d x + c\right )^{7} - 27 \, b^{2} \cosh \left (d x + c\right )^{5} + 8 \,{\left (8 \, a b + 9 \, b^{2}\right )} \cosh \left (d x + c\right )^{3} - 4 \,{\left (40 \, a b + 21 \, b^{2}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{2} + 630 \,{\left (128 \, a^{2} + 160 \, a b + 63 \, b^{2}\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)*(a+b*sinh(d*x+c)^4)^2,x, algorithm="fricas")

[Out]

1/80640*(35*b^2*cosh(d*x + c)^9 + 315*b^2*cosh(d*x + c)*sinh(d*x + c)^8 - 405*b^2*cosh(d*x + c)^7 + 105*(28*b^
2*cosh(d*x + c)^3 - 27*b^2*cosh(d*x + c))*sinh(d*x + c)^6 + 252*(8*a*b + 9*b^2)*cosh(d*x + c)^5 + 315*(14*b^2*
cosh(d*x + c)^5 - 45*b^2*cosh(d*x + c)^3 + 4*(8*a*b + 9*b^2)*cosh(d*x + c))*sinh(d*x + c)^4 - 420*(40*a*b + 21
*b^2)*cosh(d*x + c)^3 + 315*(4*b^2*cosh(d*x + c)^7 - 27*b^2*cosh(d*x + c)^5 + 8*(8*a*b + 9*b^2)*cosh(d*x + c)^
3 - 4*(40*a*b + 21*b^2)*cosh(d*x + c))*sinh(d*x + c)^2 + 630*(128*a^2 + 160*a*b + 63*b^2)*cosh(d*x + c))/d

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Sympy [A]  time = 43.7265, size = 204, normalized size = 2.22 \begin{align*} \begin{cases} \frac{a^{2} \cosh{\left (c + d x \right )}}{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{b^{2} \sinh ^{8}{\left (c + d x \right )} \cosh{\left (c + d x \right )}}{d} - \frac{8 b^{2} \sinh ^{6}{\left (c + d x \right )} \cosh ^{3}{\left (c + d x \right )}}{3 d} + \frac{16 b^{2} \sinh ^{4}{\left (c + d x \right )} \cosh ^{5}{\left (c + d x \right )}}{5 d} - \frac{64 b^{2} \sinh ^{2}{\left (c + d x \right )} \cosh ^{7}{\left (c + d x \right )}}{35 d} + \frac{128 b^{2} \cosh ^{9}{\left (c + d x \right )}}{315 d} & \text{for}\: d \neq 0 \\x \left (a + b \sinh ^{4}{\left (c \right )}\right )^{2} \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)**4)**2,x)

[Out]

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

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Giac [B]  time = 1.27739, size = 331, normalized size = 3.6 \begin{align*} \frac{35 \, b^{2} e^{\left (9 \, d x + 9 \, c\right )} - 405 \, b^{2} e^{\left (7 \, d x + 7 \, c\right )} + 2016 \, a b e^{\left (5 \, d x + 5 \, c\right )} + 2268 \, b^{2} e^{\left (5 \, d x + 5 \, c\right )} - 16800 \, a b e^{\left (3 \, d x + 3 \, c\right )} - 8820 \, b^{2} e^{\left (3 \, d x + 3 \, c\right )} + 80640 \, a^{2} e^{\left (d x + c\right )} + 100800 \, a b e^{\left (d x + c\right )} + 39690 \, b^{2} e^{\left (d x + c\right )} +{\left (80640 \, a^{2} e^{\left (8 \, d x + 8 \, c\right )} + 100800 \, a b e^{\left (8 \, d x + 8 \, c\right )} + 39690 \, b^{2} e^{\left (8 \, d x + 8 \, c\right )} - 16800 \, a b e^{\left (6 \, d x + 6 \, c\right )} - 8820 \, b^{2} e^{\left (6 \, d x + 6 \, c\right )} + 2016 \, a b e^{\left (4 \, d x + 4 \, c\right )} + 2268 \, b^{2} e^{\left (4 \, d x + 4 \, c\right )} - 405 \, b^{2} e^{\left (2 \, d x + 2 \, c\right )} + 35 \, b^{2}\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)*(a+b*sinh(d*x+c)^4)^2,x, algorithm="giac")

[Out]

1/161280*(35*b^2*e^(9*d*x + 9*c) - 405*b^2*e^(7*d*x + 7*c) + 2016*a*b*e^(5*d*x + 5*c) + 2268*b^2*e^(5*d*x + 5*
c) - 16800*a*b*e^(3*d*x + 3*c) - 8820*b^2*e^(3*d*x + 3*c) + 80640*a^2*e^(d*x + c) + 100800*a*b*e^(d*x + c) + 3
9690*b^2*e^(d*x + c) + (80640*a^2*e^(8*d*x + 8*c) + 100800*a*b*e^(8*d*x + 8*c) + 39690*b^2*e^(8*d*x + 8*c) - 1
6800*a*b*e^(6*d*x + 6*c) - 8820*b^2*e^(6*d*x + 6*c) + 2016*a*b*e^(4*d*x + 4*c) + 2268*b^2*e^(4*d*x + 4*c) - 40
5*b^2*e^(2*d*x + 2*c) + 35*b^2)*e^(-9*d*x - 9*c))/d