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

Optimal. Leaf size=46 \[ \frac{(a+b) \cosh (c+d x)}{d}+\frac{b \cosh ^5(c+d x)}{5 d}-\frac{2 b \cosh ^3(c+d x)}{3 d} \]

[Out]

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

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Rubi [A]  time = 0.0342356, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.053, Rules used = {3215} \[ \frac{(a+b) \cosh (c+d x)}{d}+\frac{b \cosh ^5(c+d x)}{5 d}-\frac{2 b \cosh ^3(c+d x)}{3 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 0.0223954, size = 69, normalized size = 1.5 \[ \frac{a \sinh (c) \sinh (d x)}{d}+\frac{a \cosh (c) \cosh (d x)}{d}+\frac{5 b \cosh (c+d x)}{8 d}-\frac{5 b \cosh (3 (c+d x))}{48 d}+\frac{b \cosh (5 (c+d x))}{80 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.013, size = 44, normalized size = 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\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)^4),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*cosh(d*x+c))

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Maxima [B]  time = 1.05633, size = 131, normalized size = 2.85 \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{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)^4),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) + a*cosh(d*x + c)/d

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Fricas [B]  time = 1.66974, size = 250, normalized size = 5.43 \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} - 25 \, b \cosh \left (d x + c\right )^{3} + 15 \,{\left (2 \, b \cosh \left (d x + c\right )^{3} - 5 \, b \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{2} + 30 \,{\left (8 \, 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)*(a+b*sinh(d*x+c)^4),x, algorithm="fricas")

[Out]

1/240*(3*b*cosh(d*x + c)^5 + 15*b*cosh(d*x + c)*sinh(d*x + c)^4 - 25*b*cosh(d*x + c)^3 + 15*(2*b*cosh(d*x + c)
^3 - 5*b*cosh(d*x + c))*sinh(d*x + c)^2 + 30*(8*a + 5*b)*cosh(d*x + c))/d

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Sympy [A]  time = 2.53825, size = 80, normalized size = 1.74 \begin{align*} \begin{cases} \frac{a \cosh{\left (c + d x \right )}}{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 ^{4}{\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)**4),x)

[Out]

Piecewise((a*cosh(c + d*x)/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)**4)*sinh(c), True))

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Giac [B]  time = 1.18581, size = 132, normalized size = 2.87 \begin{align*} \frac{3 \, b e^{\left (5 \, d x + 5 \, c\right )} - 25 \, b e^{\left (3 \, d x + 3 \, c\right )} + 240 \, a e^{\left (d x + c\right )} + 150 \, b e^{\left (d x + c\right )} +{\left (240 \, a e^{\left (4 \, d x + 4 \, c\right )} + 150 \, b e^{\left (4 \, d x + 4 \, 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)*(a+b*sinh(d*x+c)^4),x, algorithm="giac")

[Out]

1/480*(3*b*e^(5*d*x + 5*c) - 25*b*e^(3*d*x + 3*c) + 240*a*e^(d*x + c) + 150*b*e^(d*x + c) + (240*a*e^(4*d*x +
4*c) + 150*b*e^(4*d*x + 4*c) - 25*b*e^(2*d*x + 2*c) + 3*b)*e^(-5*d*x - 5*c))/d