3.141 \(\int \sinh ^4(c+d x) (a+b \sinh ^3(c+d x)) \, dx\)

Optimal. Leaf size=106 \[ \frac{a \sinh ^3(c+d x) \cosh (c+d x)}{4 d}-\frac{3 a \sinh (c+d x) \cosh (c+d x)}{8 d}+\frac{3 a x}{8}+\frac{b \cosh ^7(c+d x)}{7 d}-\frac{3 b \cosh ^5(c+d x)}{5 d}+\frac{b \cosh ^3(c+d x)}{d}-\frac{b \cosh (c+d x)}{d} \]

[Out]

(3*a*x)/8 - (b*Cosh[c + d*x])/d + (b*Cosh[c + d*x]^3)/d - (3*b*Cosh[c + d*x]^5)/(5*d) + (b*Cosh[c + d*x]^7)/(7
*d) - (3*a*Cosh[c + d*x]*Sinh[c + d*x])/(8*d) + (a*Cosh[c + d*x]*Sinh[c + d*x]^3)/(4*d)

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Rubi [A]  time = 0.0943742, antiderivative size = 106, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {3220, 2635, 8, 2633} \[ \frac{a \sinh ^3(c+d x) \cosh (c+d x)}{4 d}-\frac{3 a \sinh (c+d x) \cosh (c+d x)}{8 d}+\frac{3 a x}{8}+\frac{b \cosh ^7(c+d x)}{7 d}-\frac{3 b \cosh ^5(c+d x)}{5 d}+\frac{b \cosh ^3(c+d x)}{d}-\frac{b \cosh (c+d x)}{d} \]

Antiderivative was successfully verified.

[In]

Int[Sinh[c + d*x]^4*(a + b*Sinh[c + d*x]^3),x]

[Out]

(3*a*x)/8 - (b*Cosh[c + d*x])/d + (b*Cosh[c + d*x]^3)/d - (3*b*Cosh[c + d*x]^5)/(5*d) + (b*Cosh[c + d*x]^7)/(7
*d) - (3*a*Cosh[c + d*x]*Sinh[c + d*x])/(8*d) + (a*Cosh[c + d*x]*Sinh[c + d*x]^3)/(4*d)

Rule 3220

Int[sin[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^(n_))^(p_.), x_Symbol] :> Int[ExpandTr
ig[sin[e + f*x]^m*(a + b*sin[e + f*x]^n)^p, x], x] /; FreeQ[{a, b, e, f}, x] && IntegersQ[m, p] && (EqQ[n, 4]
|| GtQ[p, 0] || (EqQ[p, -1] && IntegerQ[n]))

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]

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]

Rubi steps

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

Mathematica [A]  time = 0.102702, size = 81, normalized size = 0.76 \[ \frac{-560 a \sinh (2 (c+d x))+70 a \sinh (4 (c+d x))+840 a c+840 a d x-1225 b \cosh (c+d x)+245 b \cosh (3 (c+d x))-49 b \cosh (5 (c+d x))+5 b \cosh (7 (c+d x))}{2240 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Sinh[c + d*x]^4*(a + b*Sinh[c + d*x]^3),x]

[Out]

(840*a*c + 840*a*d*x - 1225*b*Cosh[c + d*x] + 245*b*Cosh[3*(c + d*x)] - 49*b*Cosh[5*(c + d*x)] + 5*b*Cosh[7*(c
+ d*x)] - 560*a*Sinh[2*(c + d*x)] + 70*a*Sinh[4*(c + d*x)])/(2240*d)

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Maple [A]  time = 0.016, size = 82, normalized size = 0.8 \begin{align*}{\frac{1}{d} \left ( b \left ( -{\frac{16}{35}}+{\frac{ \left ( \sinh \left ( dx+c \right ) \right ) ^{6}}{7}}-{\frac{6\, \left ( \sinh \left ( dx+c \right ) \right ) ^{4}}{35}}+{\frac{8\, \left ( \sinh \left ( dx+c \right ) \right ) ^{2}}{35}} \right ) \cosh \left ( dx+c \right ) +a \left ( \left ({\frac{ \left ( \sinh \left ( dx+c \right ) \right ) ^{3}}{4}}-{\frac{3\,\sinh \left ( dx+c \right ) }{8}} \right ) \cosh \left ( dx+c \right ) +{\frac{3\,dx}{8}}+{\frac{3\,c}{8}} \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sinh(d*x+c)^4*(a+b*sinh(d*x+c)^3),x)

[Out]

1/d*(b*(-16/35+1/7*sinh(d*x+c)^6-6/35*sinh(d*x+c)^4+8/35*sinh(d*x+c)^2)*cosh(d*x+c)+a*((1/4*sinh(d*x+c)^3-3/8*
sinh(d*x+c))*cosh(d*x+c)+3/8*d*x+3/8*c))

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Maxima [A]  time = 1.19747, size = 221, normalized size = 2.08 \begin{align*} \frac{1}{64} \, a{\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}{4480} \, b{\left (\frac{{\left (49 \, e^{\left (-2 \, d x - 2 \, c\right )} - 245 \, e^{\left (-4 \, d x - 4 \, c\right )} + 1225 \, e^{\left (-6 \, d x - 6 \, c\right )} - 5\right )} e^{\left (7 \, d x + 7 \, c\right )}}{d} + \frac{1225 \, e^{\left (-d x - c\right )} - 245 \, e^{\left (-3 \, d x - 3 \, c\right )} + 49 \, e^{\left (-5 \, d x - 5 \, c\right )} - 5 \, e^{\left (-7 \, d x - 7 \, c\right )}}{d}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(d*x+c)^4*(a+b*sinh(d*x+c)^3),x, algorithm="maxima")

[Out]

1/64*a*(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) - 1/4480*b
*((49*e^(-2*d*x - 2*c) - 245*e^(-4*d*x - 4*c) + 1225*e^(-6*d*x - 6*c) - 5)*e^(7*d*x + 7*c)/d + (1225*e^(-d*x -
c) - 245*e^(-3*d*x - 3*c) + 49*e^(-5*d*x - 5*c) - 5*e^(-7*d*x - 7*c))/d)

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Fricas [A]  time = 1.93103, size = 539, normalized size = 5.08 \begin{align*} \frac{5 \, b \cosh \left (d x + c\right )^{7} + 35 \, b \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{6} - 49 \, b \cosh \left (d x + c\right )^{5} + 280 \, a \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + 35 \,{\left (5 \, b \cosh \left (d x + c\right )^{3} - 7 \, b \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{4} + 245 \, b \cosh \left (d x + c\right )^{3} + 840 \, a d x + 35 \,{\left (3 \, b \cosh \left (d x + c\right )^{5} - 14 \, b \cosh \left (d x + c\right )^{3} + 21 \, b \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{2} - 1225 \, b \cosh \left (d x + c\right ) + 280 \,{\left (a \cosh \left (d x + c\right )^{3} - 4 \, a \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )}{2240 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(d*x+c)^4*(a+b*sinh(d*x+c)^3),x, algorithm="fricas")

[Out]

1/2240*(5*b*cosh(d*x + c)^7 + 35*b*cosh(d*x + c)*sinh(d*x + c)^6 - 49*b*cosh(d*x + c)^5 + 280*a*cosh(d*x + c)*
sinh(d*x + c)^3 + 35*(5*b*cosh(d*x + c)^3 - 7*b*cosh(d*x + c))*sinh(d*x + c)^4 + 245*b*cosh(d*x + c)^3 + 840*a
*d*x + 35*(3*b*cosh(d*x + c)^5 - 14*b*cosh(d*x + c)^3 + 21*b*cosh(d*x + c))*sinh(d*x + c)^2 - 1225*b*cosh(d*x
+ c) + 280*(a*cosh(d*x + c)^3 - 4*a*cosh(d*x + c))*sinh(d*x + c))/d

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Sympy [A]  time = 8.10677, size = 192, normalized size = 1.81 \begin{align*} \begin{cases} \frac{3 a x \sinh ^{4}{\left (c + d x \right )}}{8} - \frac{3 a x \sinh ^{2}{\left (c + d x \right )} \cosh ^{2}{\left (c + d x \right )}}{4} + \frac{3 a x \cosh ^{4}{\left (c + d x \right )}}{8} + \frac{5 a \sinh ^{3}{\left (c + d x \right )} \cosh{\left (c + d x \right )}}{8 d} - \frac{3 a \sinh{\left (c + d x \right )} \cosh ^{3}{\left (c + d x \right )}}{8 d} + \frac{b \sinh ^{6}{\left (c + d x \right )} \cosh{\left (c + d x \right )}}{d} - \frac{2 b \sinh ^{4}{\left (c + d x \right )} \cosh ^{3}{\left (c + d x \right )}}{d} + \frac{8 b \sinh ^{2}{\left (c + d x \right )} \cosh ^{5}{\left (c + d x \right )}}{5 d} - \frac{16 b \cosh ^{7}{\left (c + d x \right )}}{35 d} & \text{for}\: d \neq 0 \\x \left (a + b \sinh ^{3}{\left (c \right )}\right ) \sinh ^{4}{\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)**4*(a+b*sinh(d*x+c)**3),x)

[Out]

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

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Giac [A]  time = 1.21197, size = 213, normalized size = 2.01 \begin{align*} \frac{1680 \,{\left (d x + c\right )} a + 5 \, b e^{\left (7 \, d x + 7 \, c\right )} - 49 \, b e^{\left (5 \, d x + 5 \, c\right )} + 70 \, a e^{\left (4 \, d x + 4 \, c\right )} + 245 \, b e^{\left (3 \, d x + 3 \, c\right )} - 560 \, a e^{\left (2 \, d x + 2 \, c\right )} - 1225 \, b e^{\left (d x + c\right )} -{\left (1225 \, b e^{\left (6 \, d x + 6 \, c\right )} - 560 \, a e^{\left (5 \, d x + 5 \, c\right )} - 245 \, b e^{\left (4 \, d x + 4 \, c\right )} + 70 \, a e^{\left (3 \, d x + 3 \, c\right )} + 49 \, b e^{\left (2 \, d x + 2 \, c\right )} - 5 \, b\right )} e^{\left (-7 \, d x - 7 \, c\right )}}{4480 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(d*x+c)^4*(a+b*sinh(d*x+c)^3),x, algorithm="giac")

[Out]

1/4480*(1680*(d*x + c)*a + 5*b*e^(7*d*x + 7*c) - 49*b*e^(5*d*x + 5*c) + 70*a*e^(4*d*x + 4*c) + 245*b*e^(3*d*x
+ 3*c) - 560*a*e^(2*d*x + 2*c) - 1225*b*e^(d*x + c) - (1225*b*e^(6*d*x + 6*c) - 560*a*e^(5*d*x + 5*c) - 245*b*
e^(4*d*x + 4*c) + 70*a*e^(3*d*x + 3*c) + 49*b*e^(2*d*x + 2*c) - 5*b)*e^(-7*d*x - 7*c))/d