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

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

[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 = {2633} \[ \frac {\sinh ^3(a+b x)}{3 b}+\frac {\sinh (a+b x)}{b} \]

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 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 \cosh ^3(a+b x) \, dx &=\frac {i \operatorname {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 \[ \frac {\sinh ^3(a+b x)}{3 b}+\frac {\sinh (a+b x)}{b} \]

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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fricas [A]  time = 0.73, size = 32, normalized size = 1.23 \[ \frac {\sinh \left (b x + a\right )^{3} + 3 \, {\left (\cosh \left (b x + a\right )^{2} + 3\right )} \sinh \left (b x + a\right )}{12 \, b} \]

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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giac [B]  time = 0.12, size = 54, normalized size = 2.08 \[ \frac {e^{\left (3 \, b x + 3 \, a\right )}}{24 \, b} + \frac {3 \, e^{\left (b x + a\right )}}{8 \, b} - \frac {3 \, e^{\left (-b x - a\right )}}{8 \, b} - \frac {e^{\left (-3 \, b x - 3 \, a\right )}}{24 \, b} \]

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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maple [A]  time = 0.20, size = 23, normalized size = 0.88 \[ \frac {\left (\frac {2}{3}+\frac {\left (\cosh ^{2}\left (b x +a \right )\right )}{3}\right ) \sinh \left (b x +a \right )}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [B]  time = 0.31, size = 54, normalized size = 2.08 \[ \frac {e^{\left (3 \, b x + 3 \, a\right )}}{24 \, b} + \frac {3 \, e^{\left (b x + a\right )}}{8 \, b} - \frac {3 \, e^{\left (-b x - a\right )}}{8 \, b} - \frac {e^{\left (-3 \, b x - 3 \, a\right )}}{24 \, b} \]

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

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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sympy [A]  time = 0.41, size = 36, normalized size = 1.38 \[ \begin {cases} - \frac {2 \sinh ^{3}{\left (a + b x \right )}}{3 b} + \frac {\sinh {\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{b} & \text {for}\: b \neq 0 \\x \cosh ^{3}{\relax (a )} & \text {otherwise} \end {cases} \]

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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