3.3 \(\int \sinh ^3(a+b x) \, dx\)

Optimal. Leaf size=27 \[ \frac{\cosh ^3(a+b x)}{3 b}-\frac{\cosh (a+b x)}{b} \]

[Out]

-(Cosh[a + b*x]/b) + Cosh[a + b*x]^3/(3*b)

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Rubi [A]  time = 0.0139676, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {2633} \[ \frac{\cosh ^3(a+b x)}{3 b}-\frac{\cosh (a+b x)}{b} \]

Antiderivative was successfully verified.

[In]

Int[Sinh[a + b*x]^3,x]

[Out]

-(Cosh[a + b*x]/b) + Cosh[a + b*x]^3/(3*b)

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 ^3(a+b x) \, dx &=-\frac{\operatorname{Subst}\left (\int \left (1-x^2\right ) \, dx,x,\cosh (a+b x)\right )}{b}\\ &=-\frac{\cosh (a+b x)}{b}+\frac{\cosh ^3(a+b x)}{3 b}\\ \end{align*}

Mathematica [A]  time = 0.0117888, size = 29, normalized size = 1.07 \[ \frac{\cosh (3 (a+b x))}{12 b}-\frac{3 \cosh (a+b x)}{4 b} \]

Antiderivative was successfully verified.

[In]

Integrate[Sinh[a + b*x]^3,x]

[Out]

(-3*Cosh[a + b*x])/(4*b) + Cosh[3*(a + b*x)]/(12*b)

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Maple [A]  time = 0.009, size = 23, normalized size = 0.9 \begin{align*}{\frac{\cosh \left ( bx+a \right ) }{b} \left ( -{\frac{2}{3}}+{\frac{ \left ( \sinh \left ( bx+a \right ) \right ) ^{2}}{3}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/b*(-2/3+1/3*sinh(b*x+a)^2)*cosh(b*x+a)

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Maxima [B]  time = 1.09844, size = 73, normalized size = 2.7 \begin{align*} \frac{e^{\left (3 \, b x + 3 \, a\right )}}{24 \, b} - \frac{3 \, e^{\left (b x + a\right )}}{8 \, b} - \frac{3 \, e^{\left (-b x - a\right )}}{8 \, b} + \frac{e^{\left (-3 \, b x - 3 \, a\right )}}{24 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/24*e^(3*b*x + 3*a)/b - 3/8*e^(b*x + a)/b - 3/8*e^(-b*x - a)/b + 1/24*e^(-3*b*x - 3*a)/b

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Fricas [A]  time = 1.95594, size = 105, normalized size = 3.89 \begin{align*} \frac{\cosh \left (b x + a\right )^{3} + 3 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{2} - 9 \, \cosh \left (b x + a\right )}{12 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*(cosh(b*x + a)^3 + 3*cosh(b*x + a)*sinh(b*x + a)^2 - 9*cosh(b*x + a))/b

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Sympy [A]  time = 0.56425, size = 36, normalized size = 1.33 \begin{align*} \begin{cases} \frac{\sinh ^{2}{\left (a + b x \right )} \cosh{\left (a + b x \right )}}{b} - \frac{2 \cosh ^{3}{\left (a + b x \right )}}{3 b} & \text{for}\: b \neq 0 \\x \sinh ^{3}{\left (a \right )} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sinh(b*x+a)**3,x)

[Out]

Piecewise((sinh(a + b*x)**2*cosh(a + b*x)/b - 2*cosh(a + b*x)**3/(3*b), Ne(b, 0)), (x*sinh(a)**3, True))

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Giac [A]  time = 1.46189, size = 65, normalized size = 2.41 \begin{align*} -\frac{{\left (9 \, e^{\left (2 \, b x + 2 \, a\right )} - 1\right )} e^{\left (-3 \, b x - 3 \, a\right )} - e^{\left (3 \, b x + 3 \, a\right )} + 9 \, e^{\left (b x + a\right )}}{24 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/24*((9*e^(2*b*x + 2*a) - 1)*e^(-3*b*x - 3*a) - e^(3*b*x + 3*a) + 9*e^(b*x + a))/b