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

Optimal. Leaf size=67 \[ \frac{(a+3 b) \cosh ^3(c+d x)}{3 d}-\frac{(a+b) \cosh (c+d x)}{d}+\frac{b \cosh ^7(c+d x)}{7 d}-\frac{3 b \cosh ^5(c+d x)}{5 d} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0661279, antiderivative size = 67, 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 = {3215, 1153} \[ \frac{(a+3 b) \cosh ^3(c+d x)}{3 d}-\frac{(a+b) \cosh (c+d x)}{d}+\frac{b \cosh ^7(c+d x)}{7 d}-\frac{3 b \cosh ^5(c+d x)}{5 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 3215

Int[sin[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^4)^(p_.), x_Symbol] :> With[{ff = Free
Factors[Cos[e + f*x], x]}, -Dist[ff/f, Subst[Int[(1 - ff^2*x^2)^((m - 1)/2)*(a + b - 2*b*ff^2*x^2 + b*ff^4*x^4
)^p, x], x, Cos[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[(m - 1)/2]

Rule 1153

Int[((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(
d + e*x^2)^q*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 -
b*d*e + a*e^2, 0] && IGtQ[p, 0] && IGtQ[q, -2]

Rubi steps

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

Mathematica [A]  time = 0.0288673, size = 93, normalized size = 1.39 \[ -\frac{3 a \cosh (c+d x)}{4 d}+\frac{a \cosh (3 (c+d x))}{12 d}-\frac{35 b \cosh (c+d x)}{64 d}+\frac{7 b \cosh (3 (c+d x))}{64 d}-\frac{7 b \cosh (5 (c+d x))}{320 d}+\frac{b \cosh (7 (c+d x))}{448 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*a*Cosh[c + d*x])/(4*d) - (35*b*Cosh[c + d*x])/(64*d) + (a*Cosh[3*(c + d*x)])/(12*d) + (7*b*Cosh[3*(c + d*x
)])/(64*d) - (7*b*Cosh[5*(c + d*x)])/(320*d) + (b*Cosh[7*(c + d*x)])/(448*d)

________________________________________________________________________________________

Maple [A]  time = 0.014, size = 66, normalized size = 1. \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 ( -{\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)^4),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*(-2/3+1/3*sinh(d*x+c)^2)
*cosh(d*x+c))

________________________________________________________________________________________

Maxima [B]  time = 1.12945, size = 212, normalized size = 3.16 \begin{align*} -\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 )} + \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)^4),x, algorithm="maxima")

[Out]

-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) + 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)

________________________________________________________________________________________

Fricas [B]  time = 1.6346, size = 439, normalized size = 6.55 \begin{align*} \frac{15 \, b \cosh \left (d x + c\right )^{7} + 105 \, b \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{6} - 147 \, b \cosh \left (d x + c\right )^{5} + 105 \,{\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} + 35 \,{\left (16 \, a + 21 \, b\right )} \cosh \left (d x + c\right )^{3} + 105 \,{\left (3 \, b \cosh \left (d x + c\right )^{5} - 14 \, b \cosh \left (d x + c\right )^{3} +{\left (16 \, a + 21 \, b\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{2} - 105 \,{\left (48 \, a + 35 \, b\right )} \cosh \left (d x + c\right )}{6720 \, 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)^4),x, algorithm="fricas")

[Out]

1/6720*(15*b*cosh(d*x + c)^7 + 105*b*cosh(d*x + c)*sinh(d*x + c)^6 - 147*b*cosh(d*x + c)^5 + 105*(5*b*cosh(d*x
+ c)^3 - 7*b*cosh(d*x + c))*sinh(d*x + c)^4 + 35*(16*a + 21*b)*cosh(d*x + c)^3 + 105*(3*b*cosh(d*x + c)^5 - 1
4*b*cosh(d*x + c)^3 + (16*a + 21*b)*cosh(d*x + c))*sinh(d*x + c)^2 - 105*(48*a + 35*b)*cosh(d*x + c))/d

________________________________________________________________________________________

Sympy [A]  time = 8.48735, size = 128, normalized size = 1.91 \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 ^{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 ^{4}{\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)**4),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)**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)**4)*sinh(c)**3, True))

________________________________________________________________________________________

Giac [B]  time = 1.16218, size = 198, normalized size = 2.96 \begin{align*} \frac{15 \, b e^{\left (7 \, d x + 7 \, c\right )} - 147 \, b e^{\left (5 \, d x + 5 \, c\right )} + 560 \, a e^{\left (3 \, d x + 3 \, c\right )} + 735 \, b e^{\left (3 \, d x + 3 \, c\right )} - 5040 \, a e^{\left (d x + c\right )} - 3675 \, b e^{\left (d x + c\right )} -{\left (5040 \, a e^{\left (6 \, d x + 6 \, c\right )} + 3675 \, b e^{\left (6 \, d x + 6 \, c\right )} - 560 \, a e^{\left (4 \, d x + 4 \, c\right )} - 735 \, b e^{\left (4 \, d x + 4 \, c\right )} + 147 \, b e^{\left (2 \, d x + 2 \, c\right )} - 15 \, b\right )} e^{\left (-7 \, d x - 7 \, c\right )}}{13440 \, 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)^4),x, algorithm="giac")

[Out]

1/13440*(15*b*e^(7*d*x + 7*c) - 147*b*e^(5*d*x + 5*c) + 560*a*e^(3*d*x + 3*c) + 735*b*e^(3*d*x + 3*c) - 5040*a
*e^(d*x + c) - 3675*b*e^(d*x + c) - (5040*a*e^(6*d*x + 6*c) + 3675*b*e^(6*d*x + 6*c) - 560*a*e^(4*d*x + 4*c) -
735*b*e^(4*d*x + 4*c) + 147*b*e^(2*d*x + 2*c) - 15*b)*e^(-7*d*x - 7*c))/d