3.1.30 \(\int t^2 \log (t) \, dt\) [30]

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

[Out]

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

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Rubi [A]
time = 0.00, 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 = {2341} \begin {gather*} \frac {1}{3} t^3 \log (t)-\frac {t^3}{9} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/9*t^3 + (t^3*Log[t])/3

Rule 2341

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 \begin {gather*} -\frac {t^3}{9}+\frac {1}{3} t^3 \log (t) \end {gather*}

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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Mathics [A]
time = 1.74, size = 11, normalized size = 0.65 \begin {gather*} \frac {t^3 \left (-1+3 \text {Log}\left [t\right ]\right )}{9} \end {gather*}

Antiderivative was successfully verified.

[In]

mathics('Integrate[t^2*Log[t],t]')

[Out]

t ^ 3 (-1 + 3 Log[t]) / 9

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Maple [A]
time = 0.01, size = 14, normalized size = 0.82

method	result	size
default	\(-\frac {t^{3}}{9}+\frac {t^{3} \ln \left (t \right )}{3}\)	\(14\)
norman	\(-\frac {t^{3}}{9}+\frac {t^{3} \ln \left (t \right )}{3}\)	\(14\)
risch	\(-\frac {t^{3}}{9}+\frac {t^{3} \ln \left (t \right )}{3}\)	\(14\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(t^2*ln(t),t,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.26, size = 13, normalized size = 0.76 \begin {gather*} \frac {1}{3} \, t^{3} \log \left (t\right ) - \frac {1}{9} \, t^{3} \end {gather*}

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 = 0.33, size = 13, normalized size = 0.76 \begin {gather*} \frac {1}{3} \, t^{3} \log \left (t\right ) - \frac {1}{9} \, t^{3} \end {gather*}

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 = 0.04, size = 12, normalized size = 0.71 \begin {gather*} \frac {t^{3} \log {\left (t \right )}}{3} - \frac {t^{3}}{9} \end {gather*}

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 = 0.00, size = 16, normalized size = 0.94 \begin {gather*} -\frac {t^{3}}{9}+\frac {1}{3} t^{3} \ln t \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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 \begin {gather*} \frac {t^3\,\left (\ln \left (t\right )-\frac {1}{3}\right )}{3} \end {gather*}

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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