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

Optimal. Leaf size=128 \[ \frac{b \left (64 a^2-54 a b+15 b^2\right ) \sinh (c+d x) \cosh (c+d x)}{48 d}+\frac{1}{16} x (2 a-b) \left (8 a^2-8 a b+5 b^2\right )+\frac{5 b^2 (2 a-b) \sinh ^3(c+d x) \cosh (c+d x)}{24 d}+\frac{b \sinh (c+d x) \cosh (c+d x) \left (a+b \sinh ^2(c+d x)\right )^2}{6 d} \]

[Out]

((2*a - b)*(8*a^2 - 8*a*b + 5*b^2)*x)/16 + (b*(64*a^2 - 54*a*b + 15*b^2)*Cosh[c + d*x]*Sinh[c + d*x])/(48*d) +
(5*(2*a - b)*b^2*Cosh[c + d*x]*Sinh[c + d*x]^3)/(24*d) + (b*Cosh[c + d*x]*Sinh[c + d*x]*(a + b*Sinh[c + d*x]^
2)^2)/(6*d)

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Rubi [A]  time = 0.100085, antiderivative size = 128, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {3180, 3169} \[ \frac{b \left (64 a^2-54 a b+15 b^2\right ) \sinh (c+d x) \cosh (c+d x)}{48 d}+\frac{1}{16} x (2 a-b) \left (8 a^2-8 a b+5 b^2\right )+\frac{5 b^2 (2 a-b) \sinh ^3(c+d x) \cosh (c+d x)}{24 d}+\frac{b \sinh (c+d x) \cosh (c+d x) \left (a+b \sinh ^2(c+d x)\right )^2}{6 d} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Sinh[c + d*x]^2)^3,x]

[Out]

((2*a - b)*(8*a^2 - 8*a*b + 5*b^2)*x)/16 + (b*(64*a^2 - 54*a*b + 15*b^2)*Cosh[c + d*x]*Sinh[c + d*x])/(48*d) +
(5*(2*a - b)*b^2*Cosh[c + d*x]*Sinh[c + d*x]^3)/(24*d) + (b*Cosh[c + d*x]*Sinh[c + d*x]*(a + b*Sinh[c + d*x]^
2)^2)/(6*d)

Rule 3180

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

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 \left (a+b \sinh ^2(c+d x)\right )^3 \, dx &=\frac{b \cosh (c+d x) \sinh (c+d x) \left (a+b \sinh ^2(c+d x)\right )^2}{6 d}+\frac{1}{6} \int \left (a+b \sinh ^2(c+d x)\right ) \left (a (6 a-b)+5 (2 a-b) b \sinh ^2(c+d x)\right ) \, dx\\ &=\frac{1}{16} (2 a-b) \left (8 a^2-8 a b+5 b^2\right ) x+\frac{b \left (64 a^2-54 a b+15 b^2\right ) \cosh (c+d x) \sinh (c+d x)}{48 d}+\frac{5 (2 a-b) b^2 \cosh (c+d x) \sinh ^3(c+d x)}{24 d}+\frac{b \cosh (c+d x) \sinh (c+d x) \left (a+b \sinh ^2(c+d x)\right )^2}{6 d}\\ \end{align*}

Mathematica [A]  time = 0.259497, size = 95, normalized size = 0.74 \[ \frac{12 (2 a-b) \left (8 a^2-8 a b+5 b^2\right ) (c+d x)+9 b \left (16 a^2-16 a b+5 b^2\right ) \sinh (2 (c+d x))+9 b^2 (2 a-b) \sinh (4 (c+d x))+b^3 \sinh (6 (c+d x))}{192 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(12*(2*a - b)*(8*a^2 - 8*a*b + 5*b^2)*(c + d*x) + 9*b*(16*a^2 - 16*a*b + 5*b^2)*Sinh[2*(c + d*x)] + 9*(2*a - b
)*b^2*Sinh[4*(c + d*x)] + b^3*Sinh[6*(c + d*x)])/(192*d)

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Maple [A]  time = 0.013, size = 131, normalized size = 1. \begin{align*}{\frac{1}{d} \left ({b}^{3} \left ( \left ({\frac{ \left ( \sinh \left ( dx+c \right ) \right ) ^{5}}{6}}-{\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{b}^{2} \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 ) +3\,{a}^{2}b \left ( 1/2\,\cosh \left ( dx+c \right ) \sinh \left ( dx+c \right ) -1/2\,dx-c/2 \right ) +{a}^{3} \left ( dx+c \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/d*(b^3*((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*b^2*((1/4*s
inh(d*x+c)^3-3/8*sinh(d*x+c))*cosh(d*x+c)+3/8*d*x+3/8*c)+3*a^2*b*(1/2*cosh(d*x+c)*sinh(d*x+c)-1/2*d*x-1/2*c)+a
^3*(d*x+c))

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Maxima [A]  time = 1.04539, size = 266, normalized size = 2.08 \begin{align*} \frac{3}{64} \, a b^{2}{\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{3}{8} \, a^{2} b{\left (4 \, x - \frac{e^{\left (2 \, d x + 2 \, c\right )}}{d} + \frac{e^{\left (-2 \, d x - 2 \, c\right )}}{d}\right )} + a^{3} x - \frac{1}{384} \, b^{3}{\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((a+b*sinh(d*x+c)^2)^3,x, algorithm="maxima")

[Out]

3/64*a*b^2*(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) - 3/8*
a^2*b*(4*x - e^(2*d*x + 2*c)/d + e^(-2*d*x - 2*c)/d) + a^3*x - 1/384*b^3*((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 = 1.82293, size = 398, normalized size = 3.11 \begin{align*} \frac{3 \, b^{3} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{5} + 2 \,{\left (5 \, b^{3} \cosh \left (d x + c\right )^{3} + 9 \,{\left (2 \, a b^{2} - b^{3}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{3} + 6 \,{\left (16 \, a^{3} - 24 \, a^{2} b + 18 \, a b^{2} - 5 \, b^{3}\right )} d x + 3 \,{\left (b^{3} \cosh \left (d x + c\right )^{5} + 6 \,{\left (2 \, a b^{2} - b^{3}\right )} \cosh \left (d x + c\right )^{3} + 3 \,{\left (16 \, a^{2} b - 16 \, a b^{2} + 5 \, b^{3}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )}{96 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/96*(3*b^3*cosh(d*x + c)*sinh(d*x + c)^5 + 2*(5*b^3*cosh(d*x + c)^3 + 9*(2*a*b^2 - b^3)*cosh(d*x + c))*sinh(d
*x + c)^3 + 6*(16*a^3 - 24*a^2*b + 18*a*b^2 - 5*b^3)*d*x + 3*(b^3*cosh(d*x + c)^5 + 6*(2*a*b^2 - b^3)*cosh(d*x
+ c)^3 + 3*(16*a^2*b - 16*a*b^2 + 5*b^3)*cosh(d*x + c))*sinh(d*x + c))/d

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Sympy [A]  time = 5.04416, size = 350, normalized size = 2.73 \begin{align*} \begin{cases} a^{3} x + \frac{3 a^{2} b x \sinh ^{2}{\left (c + d x \right )}}{2} - \frac{3 a^{2} b x \cosh ^{2}{\left (c + d x \right )}}{2} + \frac{3 a^{2} b \sinh{\left (c + d x \right )} \cosh{\left (c + d x \right )}}{2 d} + \frac{9 a b^{2} x \sinh ^{4}{\left (c + d x \right )}}{8} - \frac{9 a b^{2} x \sinh ^{2}{\left (c + d x \right )} \cosh ^{2}{\left (c + d x \right )}}{4} + \frac{9 a b^{2} x \cosh ^{4}{\left (c + d x \right )}}{8} + \frac{15 a b^{2} \sinh ^{3}{\left (c + d x \right )} \cosh{\left (c + d x \right )}}{8 d} - \frac{9 a b^{2} \sinh{\left (c + d x \right )} \cosh ^{3}{\left (c + d x \right )}}{8 d} + \frac{5 b^{3} x \sinh ^{6}{\left (c + d x \right )}}{16} - \frac{15 b^{3} x \sinh ^{4}{\left (c + d x \right )} \cosh ^{2}{\left (c + d x \right )}}{16} + \frac{15 b^{3} x \sinh ^{2}{\left (c + d x \right )} \cosh ^{4}{\left (c + d x \right )}}{16} - \frac{5 b^{3} x \cosh ^{6}{\left (c + d x \right )}}{16} + \frac{11 b^{3} \sinh ^{5}{\left (c + d x \right )} \cosh{\left (c + d x \right )}}{16 d} - \frac{5 b^{3} \sinh ^{3}{\left (c + d x \right )} \cosh ^{3}{\left (c + d x \right )}}{6 d} + \frac{5 b^{3} \sinh{\left (c + d x \right )} \cosh ^{5}{\left (c + d x \right )}}{16 d} & \text{for}\: d \neq 0 \\x \left (a + b \sinh ^{2}{\left (c \right )}\right )^{3} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((a**3*x + 3*a**2*b*x*sinh(c + d*x)**2/2 - 3*a**2*b*x*cosh(c + d*x)**2/2 + 3*a**2*b*sinh(c + d*x)*cos
h(c + d*x)/(2*d) + 9*a*b**2*x*sinh(c + d*x)**4/8 - 9*a*b**2*x*sinh(c + d*x)**2*cosh(c + d*x)**2/4 + 9*a*b**2*x
*cosh(c + d*x)**4/8 + 15*a*b**2*sinh(c + d*x)**3*cosh(c + d*x)/(8*d) - 9*a*b**2*sinh(c + d*x)*cosh(c + d*x)**3
/(8*d) + 5*b**3*x*sinh(c + d*x)**6/16 - 15*b**3*x*sinh(c + d*x)**4*cosh(c + d*x)**2/16 + 15*b**3*x*sinh(c + d*
x)**2*cosh(c + d*x)**4/16 - 5*b**3*x*cosh(c + d*x)**6/16 + 11*b**3*sinh(c + d*x)**5*cosh(c + d*x)/(16*d) - 5*b
**3*sinh(c + d*x)**3*cosh(c + d*x)**3/(6*d) + 5*b**3*sinh(c + d*x)*cosh(c + d*x)**5/(16*d), Ne(d, 0)), (x*(a +
b*sinh(c)**2)**3, True))

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Giac [B]  time = 1.3218, size = 362, normalized size = 2.83 \begin{align*} \frac{b^{3} e^{\left (6 \, d x + 6 \, c\right )} + 18 \, a b^{2} e^{\left (4 \, d x + 4 \, c\right )} - 9 \, b^{3} e^{\left (4 \, d x + 4 \, c\right )} + 144 \, a^{2} b e^{\left (2 \, d x + 2 \, c\right )} - 144 \, a b^{2} e^{\left (2 \, d x + 2 \, c\right )} + 45 \, b^{3} e^{\left (2 \, d x + 2 \, c\right )} + 24 \,{\left (16 \, a^{3} - 24 \, a^{2} b + 18 \, a b^{2} - 5 \, b^{3}\right )}{\left (d x + c\right )} -{\left (352 \, a^{3} e^{\left (6 \, d x + 6 \, c\right )} - 528 \, a^{2} b e^{\left (6 \, d x + 6 \, c\right )} + 396 \, a b^{2} e^{\left (6 \, d x + 6 \, c\right )} - 110 \, b^{3} e^{\left (6 \, d x + 6 \, c\right )} + 144 \, a^{2} b e^{\left (4 \, d x + 4 \, c\right )} - 144 \, a b^{2} e^{\left (4 \, d x + 4 \, c\right )} + 45 \, b^{3} e^{\left (4 \, d x + 4 \, c\right )} + 18 \, a b^{2} e^{\left (2 \, d x + 2 \, c\right )} - 9 \, b^{3} e^{\left (2 \, d x + 2 \, c\right )} + b^{3}\right )} e^{\left (-6 \, d x - 6 \, c\right )}}{384 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/384*(b^3*e^(6*d*x + 6*c) + 18*a*b^2*e^(4*d*x + 4*c) - 9*b^3*e^(4*d*x + 4*c) + 144*a^2*b*e^(2*d*x + 2*c) - 14
4*a*b^2*e^(2*d*x + 2*c) + 45*b^3*e^(2*d*x + 2*c) + 24*(16*a^3 - 24*a^2*b + 18*a*b^2 - 5*b^3)*(d*x + c) - (352*
a^3*e^(6*d*x + 6*c) - 528*a^2*b*e^(6*d*x + 6*c) + 396*a*b^2*e^(6*d*x + 6*c) - 110*b^3*e^(6*d*x + 6*c) + 144*a^
2*b*e^(4*d*x + 4*c) - 144*a*b^2*e^(4*d*x + 4*c) + 45*b^3*e^(4*d*x + 4*c) + 18*a*b^2*e^(2*d*x + 2*c) - 9*b^3*e^
(2*d*x + 2*c) + b^3)*e^(-6*d*x - 6*c))/d