3.30 \(\int t^2 \log (t) \, dt\)

Optimal. Leaf size=17 \[ \frac{1}{3} t^3 \log (t)-\frac{t^3}{9} \]

[Out]

-t^3/9 + (t^3*Log[t])/3

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Rubi [A]  time = 0.0073478, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 6, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {2304} \[ \frac{1}{3} t^3 \log (t)-\frac{t^3}{9} \]

Antiderivative was successfully verified.

[In]

Int[t^2*Log[t],t]

[Out]

-t^3/9 + (t^3*Log[t])/3

Rule 2304

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*Log[c*x^
n]))/(d*(m + 1)), x] - Simp[(b*n*(d*x)^(m + 1))/(d*(m + 1)^2), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1
]

Rubi steps

\begin{align*} \int t^2 \log (t) \, dt &=-\frac{t^3}{9}+\frac{1}{3} t^3 \log (t)\\ \end{align*}

Mathematica [A]  time = 0.0009988, size = 17, normalized size = 1. \[ \frac{1}{3} t^3 \log (t)-\frac{t^3}{9} \]

Antiderivative was successfully verified.

[In]

Integrate[t^2*Log[t],t]

[Out]

-t^3/9 + (t^3*Log[t])/3

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Maple [A]  time = 0.001, size = 14, normalized size = 0.8 \begin{align*} -{\frac{{t}^{3}}{9}}+{\frac{{t}^{3}\ln \left ( t \right ) }{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(t^2*ln(t),t)

[Out]

-1/9*t^3+1/3*t^3*ln(t)

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Maxima [A]  time = 0.935756, size = 18, normalized size = 1.06 \begin{align*} \frac{1}{3} \, t^{3} \log \left (t\right ) - \frac{1}{9} \, t^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(t^2*log(t),t, algorithm="maxima")

[Out]

1/3*t^3*log(t) - 1/9*t^3

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Fricas [A]  time = 1.83429, size = 35, normalized size = 2.06 \begin{align*} \frac{1}{3} \, t^{3} \log \left (t\right ) - \frac{1}{9} \, t^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(t^2*log(t),t, algorithm="fricas")

[Out]

1/3*t^3*log(t) - 1/9*t^3

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Sympy [A]  time = 0.082899, size = 12, normalized size = 0.71 \begin{align*} \frac{t^{3} \log{\left (t \right )}}{3} - \frac{t^{3}}{9} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(t**2*ln(t),t)

[Out]

t**3*log(t)/3 - t**3/9

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Giac [A]  time = 1.05941, size = 18, normalized size = 1.06 \begin{align*} \frac{1}{3} \, t^{3} \log \left (t\right ) - \frac{1}{9} \, t^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(t^2*log(t),t, algorithm="giac")

[Out]

1/3*t^3*log(t) - 1/9*t^3