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

Optimal. Leaf size=26 \[ \frac {\sinh (a+b x)}{b}+\frac {\sinh ^3(a+b x)}{3 b} \]

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 26, normalized size of antiderivative = 1.00, 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 = {2713} \begin {gather*} \frac {\sinh ^3(a+b x)}{3 b}+\frac {\sinh (a+b x)}{b} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Cosh[a + b*x]^3,x]

[Out]

Sinh[a + b*x]/b + Sinh[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*} \int \cosh ^3(a+b x) \, dx &=\frac {i \text {Subst}\left (\int \left (1-x^2\right ) \, dx,x,-i \sinh (a+b x)\right )}{b}\\ &=\frac {\sinh (a+b x)}{b}+\frac {\sinh ^3(a+b x)}{3 b}\\ \end {align*}

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Mathematica [A]
time = 0.01, size = 26, normalized size = 1.00 \begin {gather*} \frac {\sinh (a+b x)}{b}+\frac {\sinh ^3(a+b x)}{3 b} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Cosh[a + b*x]^3,x]

[Out]

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

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Maple [A]
time = 1.18, size = 27, normalized size = 1.04

method	result	size
default	\(\frac {3 \sinh \left (b x +a \right )}{4 b}+\frac {\sinh \left (3 b x +3 a \right )}{12 b}\)	\(27\)
risch	\(\frac {{\mathrm e}^{3 b x +3 a}}{24 b}+\frac {3 \,{\mathrm e}^{b x +a}}{8 b}-\frac {3 \,{\mathrm e}^{-b x -a}}{8 b}-\frac {{\mathrm e}^{-3 b x -3 a}}{24 b}\)	\(55\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/4*sinh(b*x+a)/b+1/12/b*sinh(3*b*x+3*a)

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 54 vs. \(2 (24) = 48\).
time = 0.25, size = 54, normalized size = 2.08 \begin {gather*} \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 {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(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 = 0.39, size = 32, normalized size = 1.23 \begin {gather*} \frac {\sinh \left (b x + a\right )^{3} + 3 \, {\left (\cosh \left (b x + a\right )^{2} + 3\right )} \sinh \left (b x + a\right )}{12 \, b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]
time = 0.12, size = 36, normalized size = 1.38 \begin {gather*} \begin {cases} - \frac {2 \sinh ^{3}{\left (a + b x \right )}}{3 b} + \frac {\sinh {\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{b} & \text {for}\: b \neq 0 \\x \cosh ^{3}{\left (a \right )} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 54 vs. \(2 (24) = 48\).
time = 0.42, size = 54, normalized size = 2.08 \begin {gather*} \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 {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(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

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Mupad [B]
time = 0.89, size = 22, normalized size = 0.85 \begin {gather*} \frac {{\mathrm {sinh}\left (a+b\,x\right )}^3+3\,\mathrm {sinh}\left (a+b\,x\right )}{3\,b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(a + b*x)^3,x)

[Out]

(3*sinh(a + b*x) + sinh(a + b*x)^3)/(3*b)

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