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

Optimal. Leaf size=181 \[ \frac{b \left (24 a^2-64 a b+35 b^2\right ) \sinh ^3(c+d x) \cosh (c+d x)}{192 d}+\frac{\left (-376 a^2 b+96 a^3+360 a b^2-105 b^3\right ) \sinh (c+d x) \cosh (c+d x)}{384 d}-\frac{1}{128} x \left (-144 a^2 b+64 a^3+120 a b^2-35 b^3\right )+\frac{\sinh (c+d x) \cosh (c+d x) \left (a+b \sinh ^2(c+d x)\right )^3}{8 d}+\frac{(6 a-7 b) \sinh (c+d x) \cosh (c+d x) \left (a+b \sinh ^2(c+d x)\right )^2}{48 d} \]

[Out]

-((64*a^3 - 144*a^2*b + 120*a*b^2 - 35*b^3)*x)/128 + ((96*a^3 - 376*a^2*b + 360*a*b^2 - 105*b^3)*Cosh[c + d*x]
*Sinh[c + d*x])/(384*d) + (b*(24*a^2 - 64*a*b + 35*b^2)*Cosh[c + d*x]*Sinh[c + d*x]^3)/(192*d) + ((6*a - 7*b)*
Cosh[c + d*x]*Sinh[c + d*x]*(a + b*Sinh[c + d*x]^2)^2)/(48*d) + (Cosh[c + d*x]*Sinh[c + d*x]*(a + b*Sinh[c + d
*x]^2)^3)/(8*d)

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Rubi [A]  time = 0.192777, antiderivative size = 181, 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 = {3170, 3169} \[ \frac{b \left (24 a^2-64 a b+35 b^2\right ) \sinh ^3(c+d x) \cosh (c+d x)}{192 d}+\frac{\left (-376 a^2 b+96 a^3+360 a b^2-105 b^3\right ) \sinh (c+d x) \cosh (c+d x)}{384 d}-\frac{1}{128} x \left (-144 a^2 b+64 a^3+120 a b^2-35 b^3\right )+\frac{\sinh (c+d x) \cosh (c+d x) \left (a+b \sinh ^2(c+d x)\right )^3}{8 d}+\frac{(6 a-7 b) \sinh (c+d x) \cosh (c+d x) \left (a+b \sinh ^2(c+d x)\right )^2}{48 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((64*a^3 - 144*a^2*b + 120*a*b^2 - 35*b^3)*x)/128 + ((96*a^3 - 376*a^2*b + 360*a*b^2 - 105*b^3)*Cosh[c + d*x]
*Sinh[c + d*x])/(384*d) + (b*(24*a^2 - 64*a*b + 35*b^2)*Cosh[c + d*x]*Sinh[c + d*x]^3)/(192*d) + ((6*a - 7*b)*
Cosh[c + d*x]*Sinh[c + d*x]*(a + b*Sinh[c + d*x]^2)^2)/(48*d) + (Cosh[c + d*x]*Sinh[c + d*x]*(a + b*Sinh[c + d
*x]^2)^3)/(8*d)

Rule 3170

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> -Sim
p[(B*Cos[e + f*x]*Sin[e + f*x]*(a + b*Sin[e + f*x]^2)^p)/(2*f*(p + 1)), x] + Dist[1/(2*(p + 1)), Int[(a + b*Si
n[e + f*x]^2)^(p - 1)*Simp[a*B + 2*a*A*(p + 1) + (2*A*b*(p + 1) + B*(b + 2*a*p + 2*b*p))*Sin[e + f*x]^2, x], x
], x] /; FreeQ[{a, b, e, f, A, B}, x] && GtQ[p, 0]

Rule 3169

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[((4*
A*(2*a + b) + B*(4*a + 3*b))*x)/8, x] + (-Simp[(b*B*Cos[e + f*x]*Sin[e + f*x]^3)/(4*f), x] - Simp[((4*A*b + B*
(4*a + 3*b))*Cos[e + f*x]*Sin[e + f*x])/(8*f), x]) /; FreeQ[{a, b, e, f, A, B}, x]

Rubi steps

\begin{align*} \int \sinh ^2(c+d x) \left (a+b \sinh ^2(c+d x)\right )^3 \, dx &=\frac{\cosh (c+d x) \sinh (c+d x) \left (a+b \sinh ^2(c+d x)\right )^3}{8 d}-\frac{1}{8} \int \left (a-(6 a-7 b) \sinh ^2(c+d x)\right ) \left (a+b \sinh ^2(c+d x)\right )^2 \, dx\\ &=\frac{(6 a-7 b) \cosh (c+d x) \sinh (c+d x) \left (a+b \sinh ^2(c+d x)\right )^2}{48 d}+\frac{\cosh (c+d x) \sinh (c+d x) \left (a+b \sinh ^2(c+d x)\right )^3}{8 d}-\frac{1}{48} \int \left (a+b \sinh ^2(c+d x)\right ) \left (a (12 a-7 b)-\left (24 a^2-64 a b+35 b^2\right ) \sinh ^2(c+d x)\right ) \, dx\\ &=-\frac{1}{128} \left (64 a^3-144 a^2 b+120 a b^2-35 b^3\right ) x+\frac{\left (96 a^3-376 a^2 b+360 a b^2-105 b^3\right ) \cosh (c+d x) \sinh (c+d x)}{384 d}+\frac{b \left (24 a^2-64 a b+35 b^2\right ) \cosh (c+d x) \sinh ^3(c+d x)}{192 d}+\frac{(6 a-7 b) \cosh (c+d x) \sinh (c+d x) \left (a+b \sinh ^2(c+d x)\right )^2}{48 d}+\frac{\cosh (c+d x) \sinh (c+d x) \left (a+b \sinh ^2(c+d x)\right )^3}{8 d}\\ \end{align*}

Mathematica [A]  time = 0.29103, size = 130, normalized size = 0.72 \[ \frac{-24 \left (-144 a^2 b+64 a^3+120 a b^2-35 b^3\right ) (c+d x)+24 b \left (12 a^2-18 a b+7 b^2\right ) \sinh (4 (c+d x))+48 \left (-48 a^2 b+16 a^3+45 a b^2-14 b^3\right ) \sinh (2 (c+d x))+16 b^2 (3 a-2 b) \sinh (6 (c+d x))+3 b^3 \sinh (8 (c+d x))}{3072 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-24*(64*a^3 - 144*a^2*b + 120*a*b^2 - 35*b^3)*(c + d*x) + 48*(16*a^3 - 48*a^2*b + 45*a*b^2 - 14*b^3)*Sinh[2*(
c + d*x)] + 24*b*(12*a^2 - 18*a*b + 7*b^2)*Sinh[4*(c + d*x)] + 16*(3*a - 2*b)*b^2*Sinh[6*(c + d*x)] + 3*b^3*Si
nh[8*(c + d*x)])/(3072*d)

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Maple [A]  time = 0.014, size = 180, normalized size = 1. \begin{align*}{\frac{1}{d} \left ({b}^{3} \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 ) +3\,a{b}^{2} \left ( \left ( 1/6\, \left ( \sinh \left ( dx+c \right ) \right ) ^{5}-{\frac{5\, \left ( \sinh \left ( dx+c \right ) \right ) ^{3}}{24}}+{\frac{5\,\sinh \left ( dx+c \right ) }{16}} \right ) \cosh \left ( dx+c \right ) -{\frac{5\,dx}{16}}-{\frac{5\,c}{16}} \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} \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)^2)^3,x)

[Out]

1/d*(b^3*((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)+3*a*b^2*((1/6*sinh(d*x+c)^5-5/24*sinh(d*x+c)^3+5/16*sinh(d*x+c))*cosh(d*x+c)-5/16*d*x-5/16*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*(1/2*cosh(d*x+c)*sinh(d*x+c)-1/2*d*x-1
/2*c))

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Maxima [A]  time = 1.10491, size = 413, normalized size = 2.28 \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}{8} \, a^{3}{\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}{128} \, a b^{2}{\left (\frac{{\left (9 \, e^{\left (-2 \, d x - 2 \, c\right )} - 45 \, e^{\left (-4 \, d x - 4 \, c\right )} - 1\right )} e^{\left (6 \, d x + 6 \, c\right )}}{d} + \frac{120 \,{\left (d x + c\right )}}{d} + \frac{45 \, e^{\left (-2 \, d x - 2 \, c\right )} - 9 \, e^{\left (-4 \, d x - 4 \, c\right )} + e^{\left (-6 \, d x - 6 \, 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)^2)^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) - 1/8*
a^3*(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) - 1/128*a*b^2*((9*e^(-2*d*x - 2*c) - 45*e^(-4*d*x - 4*c) - 1)*
e^(6*d*x + 6*c)/d + 120*(d*x + c)/d + (45*e^(-2*d*x - 2*c) - 9*e^(-4*d*x - 4*c) + e^(-6*d*x - 6*c))/d)

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Fricas [A]  time = 1.81714, size = 667, normalized size = 3.69 \begin{align*} \frac{3 \, b^{3} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{7} + 3 \,{\left (7 \, b^{3} \cosh \left (d x + c\right )^{3} + 4 \,{\left (3 \, a b^{2} - 2 \, b^{3}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{5} +{\left (21 \, b^{3} \cosh \left (d x + c\right )^{5} + 40 \,{\left (3 \, a b^{2} - 2 \, b^{3}\right )} \cosh \left (d x + c\right )^{3} + 12 \,{\left (12 \, a^{2} b - 18 \, a b^{2} + 7 \, b^{3}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{3} - 3 \,{\left (64 \, a^{3} - 144 \, a^{2} b + 120 \, a b^{2} - 35 \, b^{3}\right )} d x + 3 \,{\left (b^{3} \cosh \left (d x + c\right )^{7} + 4 \,{\left (3 \, a b^{2} - 2 \, b^{3}\right )} \cosh \left (d x + c\right )^{5} + 4 \,{\left (12 \, a^{2} b - 18 \, a b^{2} + 7 \, b^{3}\right )} \cosh \left (d x + c\right )^{3} + 4 \,{\left (16 \, a^{3} - 48 \, a^{2} b + 45 \, a b^{2} - 14 \, b^{3}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )}{384 \, 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)^2)^3,x, algorithm="fricas")

[Out]

1/384*(3*b^3*cosh(d*x + c)*sinh(d*x + c)^7 + 3*(7*b^3*cosh(d*x + c)^3 + 4*(3*a*b^2 - 2*b^3)*cosh(d*x + c))*sin
h(d*x + c)^5 + (21*b^3*cosh(d*x + c)^5 + 40*(3*a*b^2 - 2*b^3)*cosh(d*x + c)^3 + 12*(12*a^2*b - 18*a*b^2 + 7*b^
3)*cosh(d*x + c))*sinh(d*x + c)^3 - 3*(64*a^3 - 144*a^2*b + 120*a*b^2 - 35*b^3)*d*x + 3*(b^3*cosh(d*x + c)^7 +
4*(3*a*b^2 - 2*b^3)*cosh(d*x + c)^5 + 4*(12*a^2*b - 18*a*b^2 + 7*b^3)*cosh(d*x + c)^3 + 4*(16*a^3 - 48*a^2*b
+ 45*a*b^2 - 14*b^3)*cosh(d*x + c))*sinh(d*x + c))/d

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Sympy [A]  time = 14.3627, size = 561, normalized size = 3.1 \begin{align*} \text{result too large to display} \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)**2)**3,x)

[Out]

Piecewise((a**3*x*sinh(c + d*x)**2/2 - a**3*x*cosh(c + d*x)**2/2 + a**3*sinh(c + d*x)*cosh(c + d*x)/(2*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)*cosh(c + d*x)**3/(8*d) + 15*a*b**2*x*
sinh(c + d*x)**6/16 - 45*a*b**2*x*sinh(c + d*x)**4*cosh(c + d*x)**2/16 + 45*a*b**2*x*sinh(c + d*x)**2*cosh(c +
d*x)**4/16 - 15*a*b**2*x*cosh(c + d*x)**6/16 + 33*a*b**2*sinh(c + d*x)**5*cosh(c + d*x)/(16*d) - 5*a*b**2*sin
h(c + d*x)**3*cosh(c + d*x)**3/(2*d) + 15*a*b**2*sinh(c + d*x)*cosh(c + d*x)**5/(16*d) + 35*b**3*x*sinh(c + d*
x)**8/128 - 35*b**3*x*sinh(c + d*x)**6*cosh(c + d*x)**2/32 + 105*b**3*x*sinh(c + d*x)**4*cosh(c + d*x)**4/64 -
35*b**3*x*sinh(c + d*x)**2*cosh(c + d*x)**6/32 + 35*b**3*x*cosh(c + d*x)**8/128 + 93*b**3*sinh(c + d*x)**7*co
sh(c + d*x)/(128*d) - 511*b**3*sinh(c + d*x)**5*cosh(c + d*x)**3/(384*d) + 385*b**3*sinh(c + d*x)**3*cosh(c +
d*x)**5/(384*d) - 35*b**3*sinh(c + d*x)*cosh(c + d*x)**7/(128*d), Ne(d, 0)), (x*(a + b*sinh(c)**2)**3*sinh(c)*
*2, True))

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Giac [B]  time = 1.40784, size = 521, normalized size = 2.88 \begin{align*} \frac{3 \, b^{3} e^{\left (8 \, d x + 8 \, c\right )} + 48 \, a b^{2} e^{\left (6 \, d x + 6 \, c\right )} - 32 \, b^{3} e^{\left (6 \, d x + 6 \, c\right )} + 288 \, a^{2} b e^{\left (4 \, d x + 4 \, c\right )} - 432 \, a b^{2} e^{\left (4 \, d x + 4 \, c\right )} + 168 \, b^{3} e^{\left (4 \, d x + 4 \, c\right )} + 768 \, a^{3} e^{\left (2 \, d x + 2 \, c\right )} - 2304 \, a^{2} b e^{\left (2 \, d x + 2 \, c\right )} + 2160 \, a b^{2} e^{\left (2 \, d x + 2 \, c\right )} - 672 \, b^{3} e^{\left (2 \, d x + 2 \, c\right )} - 48 \,{\left (64 \, a^{3} - 144 \, a^{2} b + 120 \, a b^{2} - 35 \, b^{3}\right )}{\left (d x + c\right )} +{\left (3200 \, a^{3} e^{\left (8 \, d x + 8 \, c\right )} - 7200 \, a^{2} b e^{\left (8 \, d x + 8 \, c\right )} + 6000 \, a b^{2} e^{\left (8 \, d x + 8 \, c\right )} - 1750 \, b^{3} e^{\left (8 \, d x + 8 \, c\right )} - 768 \, a^{3} e^{\left (6 \, d x + 6 \, c\right )} + 2304 \, a^{2} b e^{\left (6 \, d x + 6 \, c\right )} - 2160 \, a b^{2} e^{\left (6 \, d x + 6 \, c\right )} + 672 \, b^{3} e^{\left (6 \, d x + 6 \, c\right )} - 288 \, a^{2} b e^{\left (4 \, d x + 4 \, c\right )} + 432 \, a b^{2} e^{\left (4 \, d x + 4 \, c\right )} - 168 \, b^{3} e^{\left (4 \, d x + 4 \, c\right )} - 48 \, a b^{2} e^{\left (2 \, d x + 2 \, c\right )} + 32 \, b^{3} e^{\left (2 \, d x + 2 \, c\right )} - 3 \, b^{3}\right )} e^{\left (-8 \, d x - 8 \, c\right )}}{6144 \, 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)^2)^3,x, algorithm="giac")

[Out]

1/6144*(3*b^3*e^(8*d*x + 8*c) + 48*a*b^2*e^(6*d*x + 6*c) - 32*b^3*e^(6*d*x + 6*c) + 288*a^2*b*e^(4*d*x + 4*c)
- 432*a*b^2*e^(4*d*x + 4*c) + 168*b^3*e^(4*d*x + 4*c) + 768*a^3*e^(2*d*x + 2*c) - 2304*a^2*b*e^(2*d*x + 2*c) +
2160*a*b^2*e^(2*d*x + 2*c) - 672*b^3*e^(2*d*x + 2*c) - 48*(64*a^3 - 144*a^2*b + 120*a*b^2 - 35*b^3)*(d*x + c)
+ (3200*a^3*e^(8*d*x + 8*c) - 7200*a^2*b*e^(8*d*x + 8*c) + 6000*a*b^2*e^(8*d*x + 8*c) - 1750*b^3*e^(8*d*x + 8
*c) - 768*a^3*e^(6*d*x + 6*c) + 2304*a^2*b*e^(6*d*x + 6*c) - 2160*a*b^2*e^(6*d*x + 6*c) + 672*b^3*e^(6*d*x + 6
*c) - 288*a^2*b*e^(4*d*x + 4*c) + 432*a*b^2*e^(4*d*x + 4*c) - 168*b^3*e^(4*d*x + 4*c) - 48*a*b^2*e^(2*d*x + 2*
c) + 32*b^3*e^(2*d*x + 2*c) - 3*b^3)*e^(-8*d*x - 8*c))/d