3.73.28 \(\int (-9 x^2+2 x \log (x)+2 x \log ^2(x)-e^x \log (1+\log (4))) \, dx\)

Optimal. Leaf size=24 \[ -3 x^3+x^2 \log ^2(x)-e^x \log (1+\log (4)) \]

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Rubi [A]  time = 0.02, antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.107, Rules used = {2304, 2305, 2194} \begin {gather*} -3 x^3+x^2 \log ^2(x)-e^x \log (1+\log (4)) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[-9*x^2 + 2*x*Log[x] + 2*x*Log[x]^2 - E^x*Log[1 + Log[4]],x]

[Out]

-3*x^3 + x^2*Log[x]^2 - E^x*Log[1 + Log[4]]

Rule 2194

Int[((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.), x_Symbol] :> Simp[(F^(c*(a + b*x)))^n/(b*c*n*Log[F]), x] /; Fre
eQ[{F, a, b, c, n}, x]

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
]

Rule 2305

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*Lo
g[c*x^n])^p)/(d*(m + 1)), x] - Dist[(b*n*p)/(m + 1), Int[(d*x)^m*(a + b*Log[c*x^n])^(p - 1), x], x] /; FreeQ[{
a, b, c, d, m, n}, x] && NeQ[m, -1] && GtQ[p, 0]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=-3 x^3+2 \int x \log (x) \, dx+2 \int x \log ^2(x) \, dx-\log (1+\log (4)) \int e^x \, dx\\ &=-\frac {x^2}{2}-3 x^3+x^2 \log (x)+x^2 \log ^2(x)-e^x \log (1+\log (4))-2 \int x \log (x) \, dx\\ &=-3 x^3+x^2 \log ^2(x)-e^x \log (1+\log (4))\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 24, normalized size = 1.00 \begin {gather*} -3 x^3+x^2 \log ^2(x)-e^x \log (1+\log (4)) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[-9*x^2 + 2*x*Log[x] + 2*x*Log[x]^2 - E^x*Log[1 + Log[4]],x]

[Out]

-3*x^3 + x^2*Log[x]^2 - E^x*Log[1 + Log[4]]

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fricas [A]  time = 1.14, size = 25, normalized size = 1.04 \begin {gather*} x^{2} \log \relax (x)^{2} - 3 \, x^{3} - e^{x} \log \left (2 \, \log \relax (2) + 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-exp(x)*log(1+2*log(2))+2*x*log(x)^2+2*x*log(x)-9*x^2,x, algorithm="fricas")

[Out]

x^2*log(x)^2 - 3*x^3 - e^x*log(2*log(2) + 1)

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giac [A]  time = 0.19, size = 25, normalized size = 1.04 \begin {gather*} x^{2} \log \relax (x)^{2} - 3 \, x^{3} - e^{x} \log \left (2 \, \log \relax (2) + 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-exp(x)*log(1+2*log(2))+2*x*log(x)^2+2*x*log(x)-9*x^2,x, algorithm="giac")

[Out]

x^2*log(x)^2 - 3*x^3 - e^x*log(2*log(2) + 1)

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maple [A]  time = 0.04, size = 26, normalized size = 1.08

method	result	size
default	\(x^{2} \ln \relax (x )^{2}-{\mathrm e}^{x} \ln \left (1+2 \ln \relax (2)\right )-3 x^{3}\)	\(26\)
norman	\(x^{2} \ln \relax (x )^{2}-{\mathrm e}^{x} \ln \left (1+2 \ln \relax (2)\right )-3 x^{3}\)	\(26\)
risch	\(x^{2} \ln \relax (x )^{2}-{\mathrm e}^{x} \ln \left (1+2 \ln \relax (2)\right )-3 x^{3}\)	\(26\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-exp(x)*ln(1+2*ln(2))+2*x*ln(x)^2+2*x*ln(x)-9*x^2,x,method=_RETURNVERBOSE)

[Out]

x^2*ln(x)^2-exp(x)*ln(1+2*ln(2))-3*x^3

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maxima [A]  time = 0.36, size = 45, normalized size = 1.88 \begin {gather*} \frac {1}{2} \, {\left (2 \, \log \relax (x)^{2} - 2 \, \log \relax (x) + 1\right )} x^{2} - 3 \, x^{3} + x^{2} \log \relax (x) - \frac {1}{2} \, x^{2} - e^{x} \log \left (2 \, \log \relax (2) + 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-exp(x)*log(1+2*log(2))+2*x*log(x)^2+2*x*log(x)-9*x^2,x, algorithm="maxima")

[Out]

1/2*(2*log(x)^2 - 2*log(x) + 1)*x^2 - 3*x^3 + x^2*log(x) - 1/2*x^2 - e^x*log(2*log(2) + 1)

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mupad [B]  time = 4.32, size = 25, normalized size = 1.04 \begin {gather*} x^2\,{\ln \relax (x)}^2-{\mathrm {e}}^x\,\ln \left (2\,\ln \relax (2)+1\right )-3\,x^3 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(2*x*log(x)^2 + 2*x*log(x) - exp(x)*log(2*log(2) + 1) - 9*x^2,x)

[Out]

x^2*log(x)^2 - exp(x)*log(2*log(2) + 1) - 3*x^3

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sympy [A]  time = 0.30, size = 24, normalized size = 1.00 \begin {gather*} - 3 x^{3} + x^{2} \log {\relax (x )}^{2} - e^{x} \log {\left (1 + 2 \log {\relax (2 )} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-exp(x)*ln(1+2*ln(2))+2*x*ln(x)**2+2*x*ln(x)-9*x**2,x)

[Out]

-3*x**3 + x**2*log(x)**2 - exp(x)*log(1 + 2*log(2))

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