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

Optimal. Leaf size=60 \[ \frac{a \cosh (c+d x)}{d}+\frac{b \sinh ^3(c+d x) \cosh (c+d x)}{4 d}-\frac{3 b \sinh (c+d x) \cosh (c+d x)}{8 d}+\frac{3 b x}{8} \]

[Out]

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

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Rubi [A]  time = 0.0684057, antiderivative size = 60, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21, Rules used = {3220, 2638, 2635, 8} \[ \frac{a \cosh (c+d x)}{d}+\frac{b \sinh ^3(c+d x) \cosh (c+d x)}{4 d}-\frac{3 b \sinh (c+d x) \cosh (c+d x)}{8 d}+\frac{3 b x}{8} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(3*b*x)/8 + (a*Cosh[c + d*x])/d - (3*b*Cosh[c + d*x]*Sinh[c + d*x])/(8*d) + (b*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 2638

Int[sin[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[Cos[c + d*x]/d, x] /; FreeQ[{c, d}, x]

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

Mathematica [A]  time = 0.125911, size = 45, normalized size = 0.75 \[ \frac{32 a \cosh (c+d x)+b (12 (c+d x)-8 \sinh (2 (c+d x))+\sinh (4 (c+d x)))}{32 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(32*a*Cosh[c + d*x] + b*(12*(c + d*x) - 8*Sinh[2*(c + d*x)] + Sinh[4*(c + d*x)]))/(32*d)

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Maple [A]  time = 0.014, size = 50, normalized size = 0.8 \begin{align*}{\frac{1}{d} \left ( b \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 ) +a\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)^3),x)

[Out]

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

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Maxima [A]  time = 1.15149, size = 100, normalized size = 1.67 \begin{align*} \frac{1}{64} \, b{\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{a \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)^3),x, algorithm="maxima")

[Out]

1/64*b*(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) + a*cosh(d
*x + c)/d

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Fricas [A]  time = 1.80175, size = 171, normalized size = 2.85 \begin{align*} \frac{b \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + 3 \, b d x + 8 \, a \cosh \left (d x + c\right ) +{\left (b \cosh \left (d x + c\right )^{3} - 4 \, b \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )}{8 \, 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)^3),x, algorithm="fricas")

[Out]

1/8*(b*cosh(d*x + c)*sinh(d*x + c)^3 + 3*b*d*x + 8*a*cosh(d*x + c) + (b*cosh(d*x + c)^3 - 4*b*cosh(d*x + c))*s
inh(d*x + c))/d

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Sympy [A]  time = 1.24827, size = 121, normalized size = 2.02 \begin{align*} \begin{cases} \frac{a \cosh{\left (c + d x \right )}}{d} + \frac{3 b x \sinh ^{4}{\left (c + d x \right )}}{8} - \frac{3 b x \sinh ^{2}{\left (c + d x \right )} \cosh ^{2}{\left (c + d x \right )}}{4} + \frac{3 b x \cosh ^{4}{\left (c + d x \right )}}{8} + \frac{5 b \sinh ^{3}{\left (c + d x \right )} \cosh{\left (c + d x \right )}}{8 d} - \frac{3 b \sinh{\left (c + d x \right )} \cosh ^{3}{\left (c + d x \right )}}{8 d} & \text{for}\: d \neq 0 \\x \left (a + b \sinh ^{3}{\left (c \right )}\right ) \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)**3),x)

[Out]

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

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Giac [A]  time = 1.17231, size = 113, normalized size = 1.88 \begin{align*} \frac{24 \,{\left (d x + c\right )} b + b e^{\left (4 \, d x + 4 \, c\right )} - 8 \, b e^{\left (2 \, d x + 2 \, c\right )} + 32 \, a e^{\left (d x + c\right )} +{\left (32 \, a e^{\left (3 \, d x + 3 \, c\right )} + 8 \, b e^{\left (2 \, d x + 2 \, c\right )} - b\right )} e^{\left (-4 \, d x - 4 \, c\right )}}{64 \, 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)^3),x, algorithm="giac")

[Out]

1/64*(24*(d*x + c)*b + b*e^(4*d*x + 4*c) - 8*b*e^(2*d*x + 2*c) + 32*a*e^(d*x + c) + (32*a*e^(3*d*x + 3*c) + 8*
b*e^(2*d*x + 2*c) - b)*e^(-4*d*x - 4*c))/d