3.14.52 \(\int ((-2-x) \log (4)-\log (4) \log (e^x x^2)) \, dx\)

Optimal. Leaf size=16 \[ 5+e-x \log (4) \log \left (e^x x^2\right ) \]

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Rubi [B]  time = 0.02, antiderivative size = 39, normalized size of antiderivative = 2.44, number of steps used = 3, number of rules used = 1, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {2548} \begin {gather*} \frac {1}{2} x^2 \log (4)-x \log (4) \log \left (e^x x^2\right )+2 x \log (4)-\frac {1}{2} (x+2)^2 \log (4) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-2 - x)*Log[4] - Log[4]*Log[E^x*x^2],x]

[Out]

2*x*Log[4] + (x^2*Log[4])/2 - ((2 + x)^2*Log[4])/2 - x*Log[4]*Log[E^x*x^2]

Rule 2548

Int[Log[u_], x_Symbol] :> Simp[x*Log[u], x] - Int[SimplifyIntegrand[(x*D[u, x])/u, x], x] /; InverseFunctionFr
eeQ[u, x]

Rubi steps

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

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Mathematica [A]  time = 0.01, size = 13, normalized size = 0.81 \begin {gather*} -x \log (4) \log \left (e^x x^2\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-2 - x)*Log[4] - Log[4]*Log[E^x*x^2],x]

[Out]

-(x*Log[4]*Log[E^x*x^2])

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fricas [A]  time = 0.91, size = 12, normalized size = 0.75 \begin {gather*} -2 \, x \log \relax (2) \log \left (x^{2} e^{x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*x*log(2)*log(x^2*e^x)

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giac [A]  time = 0.41, size = 32, normalized size = 2.00 \begin {gather*} {\left (x^{2} - 2 \, x \log \left (x^{2} e^{x}\right ) + 4 \, x\right )} \log \relax (2) - {\left (x^{2} + 4 \, x\right )} \log \relax (2) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(x^2 - 2*x*log(x^2*e^x) + 4*x)*log(2) - (x^2 + 4*x)*log(2)

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maple [A]  time = 0.06, size = 13, normalized size = 0.81

method	result	size
default	\(-2 \ln \left ({\mathrm e}^{x} x^{2}\right ) \ln \relax (2) x\)	\(13\)
norman	\(-2 \ln \left ({\mathrm e}^{x} x^{2}\right ) \ln \relax (2) x\)	\(13\)
risch	\(-2 \ln \relax (2) x \ln \left ({\mathrm e}^{x}\right )-4 x \ln \relax (2) \ln \relax (x )+i \ln \relax (2) \pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right ) x -2 i \ln \relax (2) \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2} x +i \ln \relax (2) \pi \mathrm {csgn}\left (i x^{2}\right )^{3} x +i \ln \relax (2) \pi \,\mathrm {csgn}\left (i x^{2}\right ) \mathrm {csgn}\left (i {\mathrm e}^{x}\right ) \mathrm {csgn}\left (i x^{2} {\mathrm e}^{x}\right ) x -i \ln \relax (2) \pi \,\mathrm {csgn}\left (i x^{2}\right ) \mathrm {csgn}\left (i x^{2} {\mathrm e}^{x}\right )^{2} x -i \ln \relax (2) \pi \,\mathrm {csgn}\left (i {\mathrm e}^{x}\right ) \mathrm {csgn}\left (i x^{2} {\mathrm e}^{x}\right )^{2} x +i \ln \relax (2) \pi \mathrm {csgn}\left (i x^{2} {\mathrm e}^{x}\right )^{3} x\)	\(171\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*ln(exp(x)*x^2)*ln(2)*x

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maxima [A]  time = 0.34, size = 12, normalized size = 0.75 \begin {gather*} -2 \, x \log \relax (2) \log \left (x^{2} e^{x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*x*log(2)*log(x^2*e^x)

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mupad [B]  time = 0.95, size = 11, normalized size = 0.69 \begin {gather*} -2\,x\,\ln \relax (2)\,\left (x+\ln \left (x^2\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*x*log(2)*(x + log(x^2))

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sympy [A]  time = 0.17, size = 15, normalized size = 0.94 \begin {gather*} - 2 x \log {\relax (2 )} \log {\left (x^{2} e^{x} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*x*log(2)*log(x**2*exp(x))

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