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

Optimal result

Integrand size = 8, antiderivative size = 27 \[ \int \sinh ^3(a+b x) \, dx=-\frac {\cosh (a+b x)}{b}+\frac {\cosh ^3(a+b x)}{3 b} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2713} \[ \int \sinh ^3(a+b x) \, dx=\frac {\cosh ^3(a+b x)}{3 b}-\frac {\cosh (a+b x)}{b} \]

[In]

Int[Sinh[a + b*x]^3,x]

[Out]

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

Rule 2713

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*} \text {integral}& = -\frac {\text {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] (verified)

Time = 0.00 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.07 \[ \int \sinh ^3(a+b x) \, dx=-\frac {3 \cosh (a+b x)}{4 b}+\frac {\cosh (3 (a+b x))}{12 b} \]

[In]

Integrate[Sinh[a + b*x]^3,x]

[Out]

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

Maple [A] (verified)

Time = 1.49 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.85

method	result	size
derivativedivides	\(\frac {\left (-\frac {2}{3}+\frac {\sinh \left (b x +a \right )^{2}}{3}\right ) \cosh \left (b x +a \right )}{b}\)	\(23\)
default	\(\frac {\left (-\frac {2}{3}+\frac {\sinh \left (b x +a \right )^{2}}{3}\right ) \cosh \left (b x +a \right )}{b}\)	\(23\)
parallelrisch	\(\frac {\cosh \left (3 b x +3 a \right )-9 \cosh \left (b x +a \right )-8}{12 b}\)	\(25\)
risch	\(\frac {{\mathrm e}^{3 b x +3 a}}{24 b}-\frac {3 \,{\mathrm e}^{b x +a}}{8 b}-\frac {3 \,{\mathrm e}^{-b x -a}}{8 b}+\frac {{\mathrm e}^{-3 b x -3 a}}{24 b}\)	\(55\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.41 \[ \int \sinh ^3(a+b x) \, dx=\frac {\cosh \left (b x + a\right )^{3} + 3 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{2} - 9 \, \cosh \left (b x + a\right )}{12 \, b} \]

[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

Sympy [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.33 \[ \int \sinh ^3(a+b x) \, dx=\begin {cases} \frac {\sinh ^{2}{\left (a + b x \right )} \cosh {\left (a + b x \right )}}{b} - \frac {2 \cosh ^{3}{\left (a + b x \right )}}{3 b} & \text {for}\: b \neq 0 \\x \sinh ^{3}{\left (a \right )} & \text {otherwise} \end {cases} \]

[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))

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 54 vs. \(2 (25) = 50\).

Time = 0.19 (sec) , antiderivative size = 54, normalized size of antiderivative = 2.00 \[ \int \sinh ^3(a+b x) \, dx=\frac {e^{\left (3 \, b x + 3 \, a\right )}}{24 \, b} - \frac {3 \, e^{\left (b x + a\right )}}{8 \, b} - \frac {3 \, e^{\left (-b x - a\right )}}{8 \, b} + \frac {e^{\left (-3 \, b x - 3 \, a\right )}}{24 \, b} \]

[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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 54 vs. \(2 (25) = 50\).

Time = 0.27 (sec) , antiderivative size = 54, normalized size of antiderivative = 2.00 \[ \int \sinh ^3(a+b x) \, dx=\frac {e^{\left (3 \, b x + 3 \, a\right )}}{24 \, b} - \frac {3 \, e^{\left (b x + a\right )}}{8 \, b} - \frac {3 \, e^{\left (-b x - a\right )}}{8 \, b} + \frac {e^{\left (-3 \, b x - 3 \, a\right )}}{24 \, b} \]

[In]

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

[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

Mupad [B] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.89 \[ \int \sinh ^3(a+b x) \, dx=-\frac {3\,\mathrm {cosh}\left (a+b\,x\right )-{\mathrm {cosh}\left (a+b\,x\right )}^3}{3\,b} \]

[In]

int(sinh(a + b*x)^3,x)

[Out]

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