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

Optimal. Leaf size=114 \[ a^2 x+\frac{2 a b \cosh ^3(c+d x)}{3 d}-\frac{2 a b \cosh (c+d x)}{d}+\frac{b^2 \sinh ^5(c+d x) \cosh (c+d x)}{6 d}-\frac{5 b^2 \sinh ^3(c+d x) \cosh (c+d x)}{24 d}+\frac{5 b^2 \sinh (c+d x) \cosh (c+d x)}{16 d}-\frac{5 b^2 x}{16} \]

[Out]

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

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Rubi [A]  time = 0.0832559, antiderivative size = 114, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 4, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {3213, 2633, 2635, 8} \[ a^2 x+\frac{2 a b \cosh ^3(c+d x)}{3 d}-\frac{2 a b \cosh (c+d x)}{d}+\frac{b^2 \sinh ^5(c+d x) \cosh (c+d x)}{6 d}-\frac{5 b^2 \sinh ^3(c+d x) \cosh (c+d x)}{24 d}+\frac{5 b^2 \sinh (c+d x) \cosh (c+d x)}{16 d}-\frac{5 b^2 x}{16} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Sinh[c + d*x]^3)^2,x]

[Out]

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

Rule 3213

Int[((a_) + (b_.)*((c_.)*sin[(e_.) + (f_.)*(x_)])^(n_))^(p_), x_Symbol] :> Int[ExpandTrig[(a + b*(c*sin[e + f*
x])^n)^p, x], x] /; FreeQ[{a, b, c, e, f, n}, x] && (IGtQ[p, 0] || (EqQ[p, -1] && IntegerQ[n]))

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]

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]

Rubi steps

\begin{align*} \int \left (a+b \sinh ^3(c+d x)\right )^2 \, dx &=\int \left (a^2+2 a b \sinh ^3(c+d x)+b^2 \sinh ^6(c+d x)\right ) \, dx\\ &=a^2 x+(2 a b) \int \sinh ^3(c+d x) \, dx+b^2 \int \sinh ^6(c+d x) \, dx\\ &=a^2 x+\frac{b^2 \cosh (c+d x) \sinh ^5(c+d x)}{6 d}-\frac{1}{6} \left (5 b^2\right ) \int \sinh ^4(c+d x) \, dx-\frac{(2 a b) \operatorname{Subst}\left (\int \left (1-x^2\right ) \, dx,x,\cosh (c+d x)\right )}{d}\\ &=a^2 x-\frac{2 a b \cosh (c+d x)}{d}+\frac{2 a b \cosh ^3(c+d x)}{3 d}-\frac{5 b^2 \cosh (c+d x) \sinh ^3(c+d x)}{24 d}+\frac{b^2 \cosh (c+d x) \sinh ^5(c+d x)}{6 d}+\frac{1}{8} \left (5 b^2\right ) \int \sinh ^2(c+d x) \, dx\\ &=a^2 x-\frac{2 a b \cosh (c+d x)}{d}+\frac{2 a b \cosh ^3(c+d x)}{3 d}+\frac{5 b^2 \cosh (c+d x) \sinh (c+d x)}{16 d}-\frac{5 b^2 \cosh (c+d x) \sinh ^3(c+d x)}{24 d}+\frac{b^2 \cosh (c+d x) \sinh ^5(c+d x)}{6 d}-\frac{1}{16} \left (5 b^2\right ) \int 1 \, dx\\ &=a^2 x-\frac{5 b^2 x}{16}-\frac{2 a b \cosh (c+d x)}{d}+\frac{2 a b \cosh ^3(c+d x)}{3 d}+\frac{5 b^2 \cosh (c+d x) \sinh (c+d x)}{16 d}-\frac{5 b^2 \cosh (c+d x) \sinh ^3(c+d x)}{24 d}+\frac{b^2 \cosh (c+d x) \sinh ^5(c+d x)}{6 d}\\ \end{align*}

Mathematica [A]  time = 0.146479, size = 94, normalized size = 0.82 \[ \frac{192 a^2 c+192 a^2 d x-288 a b \cosh (c+d x)+32 a b \cosh (3 (c+d x))+45 b^2 \sinh (2 (c+d x))-9 b^2 \sinh (4 (c+d x))+b^2 \sinh (6 (c+d x))-60 b^2 c-60 b^2 d x}{192 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(192*a^2*c - 60*b^2*c + 192*a^2*d*x - 60*b^2*d*x - 288*a*b*Cosh[c + d*x] + 32*a*b*Cosh[3*(c + d*x)] + 45*b^2*S
inh[2*(c + d*x)] - 9*b^2*Sinh[4*(c + d*x)] + b^2*Sinh[6*(c + d*x)])/(192*d)

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Maple [A]  time = 0.017, size = 85, normalized size = 0.8 \begin{align*}{\frac{1}{d} \left ({b}^{2} \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 ) +2\,ab \left ( -2/3+1/3\, \left ( \sinh \left ( dx+c \right ) \right ) ^{2} \right ) \cosh \left ( dx+c \right ) +{a}^{2} \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)^3)^2,x)

[Out]

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

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Maxima [A]  time = 1.14157, size = 204, normalized size = 1.79 \begin{align*} a^{2} x - \frac{1}{384} \, 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 )} + \frac{1}{12} \, a b{\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((a+b*sinh(d*x+c)^3)^2,x, algorithm="maxima")

[Out]

a^2*x - 1/384*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) + 1/12*a*b*(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 [A]  time = 1.98662, size = 421, normalized size = 3.69 \begin{align*} \frac{3 \, b^{2} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{5} + 16 \, a b \cosh \left (d x + c\right )^{3} + 48 \, a b \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{2} + 2 \,{\left (5 \, b^{2} \cosh \left (d x + c\right )^{3} - 9 \, b^{2} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{3} + 6 \,{\left (16 \, a^{2} - 5 \, b^{2}\right )} d x - 144 \, a b \cosh \left (d x + c\right ) + 3 \,{\left (b^{2} \cosh \left (d x + c\right )^{5} - 6 \, b^{2} \cosh \left (d x + c\right )^{3} + 15 \, b^{2} \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)^3)^2,x, algorithm="fricas")

[Out]

1/96*(3*b^2*cosh(d*x + c)*sinh(d*x + c)^5 + 16*a*b*cosh(d*x + c)^3 + 48*a*b*cosh(d*x + c)*sinh(d*x + c)^2 + 2*
(5*b^2*cosh(d*x + c)^3 - 9*b^2*cosh(d*x + c))*sinh(d*x + c)^3 + 6*(16*a^2 - 5*b^2)*d*x - 144*a*b*cosh(d*x + c)
+ 3*(b^2*cosh(d*x + c)^5 - 6*b^2*cosh(d*x + c)^3 + 15*b^2*cosh(d*x + c))*sinh(d*x + c))/d

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Sympy [A]  time = 5.48219, size = 212, normalized size = 1.86 \begin{align*} \begin{cases} a^{2} x + \frac{2 a b \sinh ^{2}{\left (c + d x \right )} \cosh{\left (c + d x \right )}}{d} - \frac{4 a b \cosh ^{3}{\left (c + d x \right )}}{3 d} + \frac{5 b^{2} x \sinh ^{6}{\left (c + d x \right )}}{16} - \frac{15 b^{2} x \sinh ^{4}{\left (c + d x \right )} \cosh ^{2}{\left (c + d x \right )}}{16} + \frac{15 b^{2} x \sinh ^{2}{\left (c + d x \right )} \cosh ^{4}{\left (c + d x \right )}}{16} - \frac{5 b^{2} x \cosh ^{6}{\left (c + d x \right )}}{16} + \frac{11 b^{2} \sinh ^{5}{\left (c + d x \right )} \cosh{\left (c + d x \right )}}{16 d} - \frac{5 b^{2} \sinh ^{3}{\left (c + d x \right )} \cosh ^{3}{\left (c + d x \right )}}{6 d} + \frac{5 b^{2} \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 ^{3}{\left (c \right )}\right )^{2} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.12821, size = 212, normalized size = 1.86 \begin{align*} \frac{b^{2} e^{\left (6 \, d x + 6 \, c\right )} - 9 \, b^{2} e^{\left (4 \, d x + 4 \, c\right )} + 32 \, a b e^{\left (3 \, d x + 3 \, c\right )} + 45 \, b^{2} e^{\left (2 \, d x + 2 \, c\right )} - 288 \, a b e^{\left (d x + c\right )} + 24 \,{\left (16 \, a^{2} - 5 \, b^{2}\right )}{\left (d x + c\right )} -{\left (288 \, a b e^{\left (5 \, d x + 5 \, c\right )} + 45 \, b^{2} e^{\left (4 \, d x + 4 \, c\right )} - 32 \, a b e^{\left (3 \, d x + 3 \, c\right )} - 9 \, b^{2} e^{\left (2 \, d x + 2 \, c\right )} + b^{2}\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)^3)^2,x, algorithm="giac")

[Out]

1/384*(b^2*e^(6*d*x + 6*c) - 9*b^2*e^(4*d*x + 4*c) + 32*a*b*e^(3*d*x + 3*c) + 45*b^2*e^(2*d*x + 2*c) - 288*a*b
*e^(d*x + c) + 24*(16*a^2 - 5*b^2)*(d*x + c) - (288*a*b*e^(5*d*x + 5*c) + 45*b^2*e^(4*d*x + 4*c) - 32*a*b*e^(3
*d*x + 3*c) - 9*b^2*e^(2*d*x + 2*c) + b^2)*e^(-6*d*x - 6*c))/d