3.1.23 \(\int x^2 \cosh (x^3) \, dx\) [23]

Optimal. Leaf size=8 \[ \frac {\sinh \left (x^3\right )}{3} \]

[Out]

1/3*sinh(x^3)

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Rubi [A]
time = 0.01, antiderivative size = 8, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {5429, 2717} \begin {gather*} \frac {\sinh \left (x^3\right )}{3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^2*Cosh[x^3],x]

[Out]

Sinh[x^3]/3

Rule 2717

Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rule 5429

Int[((a_.) + Cosh[(c_.) + (d_.)*(x_)^(n_)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simpli
fy[(m + 1)/n] - 1)*(a + b*Cosh[c + d*x])^p, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p}, x] && IntegerQ[Sim
plify[(m + 1)/n]] && (EqQ[p, 1] || EqQ[m, n - 1] || (IntegerQ[p] && GtQ[Simplify[(m + 1)/n], 0]))

Rubi steps

\begin {align*} \int x^2 \cosh \left (x^3\right ) \, dx &=\frac {1}{3} \text {Subst}\left (\int \cosh (x) \, dx,x,x^3\right )\\ &=\frac {\sinh \left (x^3\right )}{3}\\ \end {align*}

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Mathematica [A]
time = 0.01, size = 8, normalized size = 1.00 \begin {gather*} \frac {\sinh \left (x^3\right )}{3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^2*Cosh[x^3],x]

[Out]

Sinh[x^3]/3

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Maple [A]
time = 0.46, size = 7, normalized size = 0.88

method	result	size
derivativedivides	\(\frac {\sinh \left (x^{3}\right )}{3}\)	\(7\)
default	\(\frac {\sinh \left (x^{3}\right )}{3}\)	\(7\)
meijerg	\(\frac {\sinh \left (x^{3}\right )}{3}\)	\(7\)
risch	\(\frac {{\mathrm e}^{x^{3}}}{6}-\frac {{\mathrm e}^{-x^{3}}}{6}\)	\(16\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*cosh(x^3),x,method=_RETURNVERBOSE)

[Out]

1/3*sinh(x^3)

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Maxima [A]
time = 0.27, size = 6, normalized size = 0.75 \begin {gather*} \frac {1}{3} \, \sinh \left (x^{3}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*cosh(x^3),x, algorithm="maxima")

[Out]

1/3*sinh(x^3)

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Fricas [A]
time = 0.38, size = 6, normalized size = 0.75 \begin {gather*} \frac {1}{3} \, \sinh \left (x^{3}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*cosh(x^3),x, algorithm="fricas")

[Out]

1/3*sinh(x^3)

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Sympy [A]
time = 0.12, size = 5, normalized size = 0.62 \begin {gather*} \frac {\sinh {\left (x^{3} \right )}}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2*cosh(x**3),x)

[Out]

sinh(x**3)/3

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 15 vs. \(2 (6) = 12\).
time = 0.42, size = 15, normalized size = 1.88 \begin {gather*} -\frac {1}{6} \, e^{\left (-x^{3}\right )} + \frac {1}{6} \, e^{\left (x^{3}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*cosh(x^3),x, algorithm="giac")

[Out]

-1/6*e^(-x^3) + 1/6*e^(x^3)

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Mupad [B]
time = 0.88, size = 6, normalized size = 0.75 \begin {gather*} \frac {\mathrm {sinh}\left (x^3\right )}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*cosh(x^3),x)

[Out]

sinh(x^3)/3

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