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

Optimal. Leaf size=79 \[ \frac{3 b^2 (a-b) \cosh ^5(c+d x)}{5 d}+\frac{b (a-b)^2 \cosh ^3(c+d x)}{d}+\frac{(a-b)^3 \cosh (c+d x)}{d}+\frac{b^3 \cosh ^7(c+d x)}{7 d} \]

[Out]

((a - b)^3*Cosh[c + d*x])/d + ((a - b)^2*b*Cosh[c + d*x]^3)/d + (3*(a - b)*b^2*Cosh[c + d*x]^5)/(5*d) + (b^3*C
osh[c + d*x]^7)/(7*d)

________________________________________________________________________________________

Rubi [A]  time = 0.0873647, antiderivative size = 79, 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 = {3186, 194} \[ \frac{3 b^2 (a-b) \cosh ^5(c+d x)}{5 d}+\frac{b (a-b)^2 \cosh ^3(c+d x)}{d}+\frac{(a-b)^3 \cosh (c+d x)}{d}+\frac{b^3 \cosh ^7(c+d x)}{7 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((a - b)^3*Cosh[c + d*x])/d + ((a - b)^2*b*Cosh[c + d*x]^3)/d + (3*(a - b)*b^2*Cosh[c + d*x]^5)/(5*d) + (b^3*C
osh[c + d*x]^7)/(7*d)

Rule 3186

Int[sin[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(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 - b*ff^2*x^2)^p, x], x, Cos
[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[(m - 1)/2]

Rule 194

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

Rubi steps

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

Mathematica [A]  time = 0.303804, size = 94, normalized size = 1.19 \[ \frac{\cosh (c+d x) \left (b \left (560 a^2-784 a b+299 b^2\right ) \cosh (2 (c+d x))-2800 a^2 b+1120 a^3+6 b^2 (14 a-9 b) \cosh (4 (c+d x))+2492 a b^2+5 b^3 \cosh (6 (c+d x))-762 b^3\right )}{1120 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Cosh[c + d*x]*(1120*a^3 - 2800*a^2*b + 2492*a*b^2 - 762*b^3 + b*(560*a^2 - 784*a*b + 299*b^2)*Cosh[2*(c + d*x
)] + 6*(14*a - 9*b)*b^2*Cosh[4*(c + d*x)] + 5*b^3*Cosh[6*(c + d*x)]))/(1120*d)

________________________________________________________________________________________

Maple [A]  time = 0.016, size = 116, normalized size = 1.5 \begin{align*}{\frac{1}{d} \left ({b}^{3} \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 ) +3\,a{b}^{2} \left ({\frac{8}{15}}+1/5\, \left ( \sinh \left ( dx+c \right ) \right ) ^{4}-{\frac{4\, \left ( \sinh \left ( dx+c \right ) \right ) ^{2}}{15}} \right ) \cosh \left ( dx+c \right ) +3\,{a}^{2}b \left ( -2/3+1/3\, \left ( \sinh \left ( dx+c \right ) \right ) ^{2} \right ) \cosh \left ( dx+c \right ) +{a}^{3}\cosh \left ( dx+c \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [B]  time = 1.05319, size = 355, normalized size = 4.49 \begin{align*} -\frac{1}{4480} \, b^{3}{\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}{160} \, a b^{2}{\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}{8} \, a^{2} 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 )} + \frac{a^{3} \cosh \left (d x + c\right )}{d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4480*b^3*((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 + (122
5*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) + 1/160*a*b^2*(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) + 1/8*a^2*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) + a^3*co
sh(d*x + c)/d

________________________________________________________________________________________

Fricas [B]  time = 1.77414, size = 589, normalized size = 7.46 \begin{align*} \frac{5 \, b^{3} \cosh \left (d x + c\right )^{7} + 35 \, b^{3} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{6} + 7 \,{\left (12 \, a b^{2} - 7 \, b^{3}\right )} \cosh \left (d x + c\right )^{5} + 35 \,{\left (5 \, b^{3} \cosh \left (d x + c\right )^{3} +{\left (12 \, a b^{2} - 7 \, b^{3}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{4} + 35 \,{\left (16 \, a^{2} b - 20 \, a b^{2} + 7 \, b^{3}\right )} \cosh \left (d x + c\right )^{3} + 35 \,{\left (3 \, b^{3} \cosh \left (d x + c\right )^{5} + 2 \,{\left (12 \, a b^{2} - 7 \, b^{3}\right )} \cosh \left (d x + c\right )^{3} + 3 \,{\left (16 \, a^{2} b - 20 \, a b^{2} + 7 \, b^{3}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{2} + 35 \,{\left (64 \, a^{3} - 144 \, a^{2} b + 120 \, a b^{2} - 35 \, b^{3}\right )} \cosh \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)*(a+b*sinh(d*x+c)^2)^3,x, algorithm="fricas")

[Out]

1/2240*(5*b^3*cosh(d*x + c)^7 + 35*b^3*cosh(d*x + c)*sinh(d*x + c)^6 + 7*(12*a*b^2 - 7*b^3)*cosh(d*x + c)^5 +
35*(5*b^3*cosh(d*x + c)^3 + (12*a*b^2 - 7*b^3)*cosh(d*x + c))*sinh(d*x + c)^4 + 35*(16*a^2*b - 20*a*b^2 + 7*b^
3)*cosh(d*x + c)^3 + 35*(3*b^3*cosh(d*x + c)^5 + 2*(12*a*b^2 - 7*b^3)*cosh(d*x + c)^3 + 3*(16*a^2*b - 20*a*b^2
+ 7*b^3)*cosh(d*x + c))*sinh(d*x + c)^2 + 35*(64*a^3 - 144*a^2*b + 120*a*b^2 - 35*b^3)*cosh(d*x + c))/d

________________________________________________________________________________________

Sympy [A]  time = 7.78788, size = 221, normalized size = 2.8 \begin{align*} \begin{cases} \frac{a^{3} \cosh{\left (c + d x \right )}}{d} + \frac{3 a^{2} b \sinh ^{2}{\left (c + d x \right )} \cosh{\left (c + d x \right )}}{d} - \frac{2 a^{2} b \cosh ^{3}{\left (c + d x \right )}}{d} + \frac{3 a b^{2} \sinh ^{4}{\left (c + d x \right )} \cosh{\left (c + d x \right )}}{d} - \frac{4 a b^{2} \sinh ^{2}{\left (c + d x \right )} \cosh ^{3}{\left (c + d x \right )}}{d} + \frac{8 a b^{2} \cosh ^{5}{\left (c + d x \right )}}{5 d} + \frac{b^{3} \sinh ^{6}{\left (c + d x \right )} \cosh{\left (c + d x \right )}}{d} - \frac{2 b^{3} \sinh ^{4}{\left (c + d x \right )} \cosh ^{3}{\left (c + d x \right )}}{d} + \frac{8 b^{3} \sinh ^{2}{\left (c + d x \right )} \cosh ^{5}{\left (c + d x \right )}}{5 d} - \frac{16 b^{3} \cosh ^{7}{\left (c + d x \right )}}{35 d} & \text{for}\: d \neq 0 \\x \left (a + b \sinh ^{2}{\left (c \right )}\right )^{3} \sinh{\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)*(a+b*sinh(d*x+c)**2)**3,x)

[Out]

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

________________________________________________________________________________________

Giac [B]  time = 1.31953, size = 386, normalized size = 4.89 \begin{align*} \frac{5 \, b^{3} e^{\left (7 \, d x + 7 \, c\right )} + 84 \, a b^{2} e^{\left (5 \, d x + 5 \, c\right )} - 49 \, b^{3} e^{\left (5 \, d x + 5 \, c\right )} + 560 \, a^{2} b e^{\left (3 \, d x + 3 \, c\right )} - 700 \, a b^{2} e^{\left (3 \, d x + 3 \, c\right )} + 245 \, b^{3} e^{\left (3 \, d x + 3 \, c\right )} + 2240 \, a^{3} e^{\left (d x + c\right )} - 5040 \, a^{2} b e^{\left (d x + c\right )} + 4200 \, a b^{2} e^{\left (d x + c\right )} - 1225 \, b^{3} e^{\left (d x + c\right )} +{\left (2240 \, a^{3} e^{\left (6 \, d x + 6 \, c\right )} - 5040 \, a^{2} b e^{\left (6 \, d x + 6 \, c\right )} + 4200 \, a b^{2} e^{\left (6 \, d x + 6 \, c\right )} - 1225 \, b^{3} e^{\left (6 \, d x + 6 \, c\right )} + 560 \, a^{2} b e^{\left (4 \, d x + 4 \, c\right )} - 700 \, a b^{2} e^{\left (4 \, d x + 4 \, c\right )} + 245 \, b^{3} e^{\left (4 \, d x + 4 \, c\right )} + 84 \, a b^{2} e^{\left (2 \, d x + 2 \, c\right )} - 49 \, b^{3} e^{\left (2 \, d x + 2 \, c\right )} + 5 \, b^{3}\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)*(a+b*sinh(d*x+c)^2)^3,x, algorithm="giac")

[Out]

1/4480*(5*b^3*e^(7*d*x + 7*c) + 84*a*b^2*e^(5*d*x + 5*c) - 49*b^3*e^(5*d*x + 5*c) + 560*a^2*b*e^(3*d*x + 3*c)
- 700*a*b^2*e^(3*d*x + 3*c) + 245*b^3*e^(3*d*x + 3*c) + 2240*a^3*e^(d*x + c) - 5040*a^2*b*e^(d*x + c) + 4200*a
*b^2*e^(d*x + c) - 1225*b^3*e^(d*x + c) + (2240*a^3*e^(6*d*x + 6*c) - 5040*a^2*b*e^(6*d*x + 6*c) + 4200*a*b^2*
e^(6*d*x + 6*c) - 1225*b^3*e^(6*d*x + 6*c) + 560*a^2*b*e^(4*d*x + 4*c) - 700*a*b^2*e^(4*d*x + 4*c) + 245*b^3*e
^(4*d*x + 4*c) + 84*a*b^2*e^(2*d*x + 2*c) - 49*b^3*e^(2*d*x + 2*c) + 5*b^3)*e^(-7*d*x - 7*c))/d