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

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

[Out]

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

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Rubi [A]  time = 0.01, antiderivative size = 17, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.00, size = 17, normalized size = 1.00 \[ \frac {1}{3} t^3 \log (t)-\frac {t^3}{9} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/9*t^3 + (t^3*Log[t])/3

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fricas [A]  time = 0.40, size = 13, normalized size = 0.76 \[ \frac {1}{3} \, t^{3} \log \relax (t) - \frac {1}{9} \, t^{3} \]

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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giac [A]  time = 0.82, size = 13, normalized size = 0.76 \[ \frac {1}{3} \, t^{3} \log \relax (t) - \frac {1}{9} \, t^{3} \]

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

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maple [A]  time = 0.00, size = 14, normalized size = 0.82 \[ \frac {t^{3} \ln \relax (t )}{3}-\frac {t^{3}}{9} \]

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 = 0.42, size = 13, normalized size = 0.76 \[ \frac {1}{3} \, t^{3} \log \relax (t) - \frac {1}{9} \, t^{3} \]

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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mupad [B]  time = 0.03, size = 9, normalized size = 0.53 \[ \frac {t^3\,\left (\ln \relax (t)-\frac {1}{3}\right )}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(t^3*(log(t) - 1/3))/3

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sympy [A]  time = 0.09, size = 12, normalized size = 0.71 \[ \frac {t^{3} \log {\relax (t )}}{3} - \frac {t^{3}}{9} \]

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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