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

Optimal. Leaf size=53 \[ \frac{(a-2 b) \cosh ^3(c+d x)}{3 d}-\frac{(a-b) \cosh (c+d x)}{d}+\frac{b \cosh ^5(c+d x)}{5 d} \]

[Out]

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

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Rubi [A]  time = 0.0557225, antiderivative size = 53, 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 = {3013, 373} \[ \frac{(a-2 b) \cosh ^3(c+d x)}{3 d}-\frac{(a-b) \cosh (c+d x)}{d}+\frac{b \cosh ^5(c+d x)}{5 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 3013

Int[sin[(e_.) + (f_.)*(x_)]^(m_.)*((A_) + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> -Dist[f^(-1), Subst[I
nt[(1 - x^2)^((m - 1)/2)*(A + C - C*x^2), x], x, Cos[e + f*x]], x] /; FreeQ[{e, f, A, C}, x] && IGtQ[(m + 1)/2
, 0]

Rule 373

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

Rubi steps

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

Mathematica [A]  time = 0.0255729, size = 77, normalized size = 1.45 \[ -\frac{3 a \cosh (c+d x)}{4 d}+\frac{a \cosh (3 (c+d x))}{12 d}+\frac{5 b \cosh (c+d x)}{8 d}-\frac{5 b \cosh (3 (c+d x))}{48 d}+\frac{b \cosh (5 (c+d x))}{80 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*a*Cosh[c + d*x])/(4*d) + (5*b*Cosh[c + d*x])/(8*d) + (a*Cosh[3*(c + d*x)])/(12*d) - (5*b*Cosh[3*(c + d*x)]
)/(48*d) + (b*Cosh[5*(c + d*x)])/(80*d)

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Maple [A]  time = 0.013, size = 56, normalized size = 1.1 \begin{align*}{\frac{1}{d} \left ( b \left ({\frac{8}{15}}+{\frac{ \left ( \sinh \left ( dx+c \right ) \right ) ^{4}}{5}}-{\frac{4\, \left ( \sinh \left ( dx+c \right ) \right ) ^{2}}{15}} \right ) \cosh \left ( dx+c \right ) +a \left ( -{\frac{2}{3}}+{\frac{ \left ( \sinh \left ( dx+c \right ) \right ) ^{2}}{3}} \right ) \cosh \left ( dx+c \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 1.04109, size = 190, normalized size = 3.58 \begin{align*} \frac{1}{480} \, 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{1}{24} \, a{\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(sinh(d*x+c)^3*(a+b*sinh(d*x+c)^2),x, algorithm="maxima")

[Out]

1/480*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) + 1/24*a*(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 [B]  time = 1.82341, size = 273, normalized size = 5.15 \begin{align*} \frac{3 \, b \cosh \left (d x + c\right )^{5} + 15 \, b \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{4} + 5 \,{\left (4 \, a - 5 \, b\right )} \cosh \left (d x + c\right )^{3} + 15 \,{\left (2 \, b \cosh \left (d x + c\right )^{3} +{\left (4 \, a - 5 \, b\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{2} - 30 \,{\left (6 \, a - 5 \, b\right )} \cosh \left (d x + c\right )}{240 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/240*(3*b*cosh(d*x + c)^5 + 15*b*cosh(d*x + c)*sinh(d*x + c)^4 + 5*(4*a - 5*b)*cosh(d*x + c)^3 + 15*(2*b*cosh
(d*x + c)^3 + (4*a - 5*b)*cosh(d*x + c))*sinh(d*x + c)^2 - 30*(6*a - 5*b)*cosh(d*x + c))/d

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Sympy [A]  time = 2.47998, size = 105, normalized size = 1.98 \begin{align*} \begin{cases} \frac{a \sinh ^{2}{\left (c + d x \right )} \cosh{\left (c + d x \right )}}{d} - \frac{2 a \cosh ^{3}{\left (c + d x \right )}}{3 d} + \frac{b \sinh ^{4}{\left (c + d x \right )} \cosh{\left (c + d x \right )}}{d} - \frac{4 b \sinh ^{2}{\left (c + d x \right )} \cosh ^{3}{\left (c + d x \right )}}{3 d} + \frac{8 b \cosh ^{5}{\left (c + d x \right )}}{15 d} & \text{for}\: d \neq 0 \\x \left (a + b \sinh ^{2}{\left (c \right )}\right ) \sinh ^{3}{\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)**3*(a+b*sinh(d*x+c)**2),x)

[Out]

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

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Giac [B]  time = 1.25488, size = 166, normalized size = 3.13 \begin{align*} \frac{3 \, b e^{\left (5 \, d x + 5 \, c\right )} + 20 \, a e^{\left (3 \, d x + 3 \, c\right )} - 25 \, b e^{\left (3 \, d x + 3 \, c\right )} - 180 \, a e^{\left (d x + c\right )} + 150 \, b e^{\left (d x + c\right )} -{\left (180 \, a e^{\left (4 \, d x + 4 \, c\right )} - 150 \, b e^{\left (4 \, d x + 4 \, c\right )} - 20 \, a e^{\left (2 \, d x + 2 \, c\right )} + 25 \, b e^{\left (2 \, d x + 2 \, c\right )} - 3 \, b\right )} e^{\left (-5 \, d x - 5 \, c\right )}}{480 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/480*(3*b*e^(5*d*x + 5*c) + 20*a*e^(3*d*x + 3*c) - 25*b*e^(3*d*x + 3*c) - 180*a*e^(d*x + c) + 150*b*e^(d*x +
c) - (180*a*e^(4*d*x + 4*c) - 150*b*e^(4*d*x + 4*c) - 20*a*e^(2*d*x + 2*c) + 25*b*e^(2*d*x + 2*c) - 3*b)*e^(-5
*d*x - 5*c))/d