3.14.12 \(\int (8 e^4 x-12 x^2-\log (2 e^3)) \log (\log (5)) \, dx\)

Optimal. Leaf size=23 \[ \left (e^4-x\right ) \left (4 x^2+\log \left (2 e^3\right )\right ) \log (\log (5)) \]

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Rubi [A]  time = 0.01, antiderivative size = 30, normalized size of antiderivative = 1.30, number of steps used = 2, number of rules used = 1, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {12} \begin {gather*} -4 x^3 \log (\log (5))+4 e^4 x^2 \log (\log (5))-x (3+\log (2)) \log (\log (5)) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(8*E^4*x - 12*x^2 - Log[2*E^3])*Log[Log[5]],x]

[Out]

4*E^4*x^2*Log[Log[5]] - 4*x^3*Log[Log[5]] - x*(3 + Log[2])*Log[Log[5]]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\log (\log (5)) \int \left (8 e^4 x-12 x^2-\log \left (2 e^3\right )\right ) \, dx\\ &=4 e^4 x^2 \log (\log (5))-4 x^3 \log (\log (5))-x (3+\log (2)) \log (\log (5))\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 26, normalized size = 1.13 \begin {gather*} \left (-3 x+4 e^4 x^2-4 x^3-x \log (2)\right ) \log (\log (5)) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(8*E^4*x - 12*x^2 - Log[2*E^3])*Log[Log[5]],x]

[Out]

(-3*x + 4*E^4*x^2 - 4*x^3 - x*Log[2])*Log[Log[5]]

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fricas [A]  time = 0.81, size = 25, normalized size = 1.09 \begin {gather*} -{\left (4 \, x^{3} - 4 \, x^{2} e^{4} + x \log \relax (2) + 3 \, x\right )} \log \left (\log \relax (5)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-log(2*exp(3))+8*x*exp(4)-12*x^2)*log(log(5)),x, algorithm="fricas")

[Out]

-(4*x^3 - 4*x^2*e^4 + x*log(2) + 3*x)*log(log(5))

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giac [A]  time = 0.34, size = 25, normalized size = 1.09 \begin {gather*} -{\left (4 \, x^{3} - 4 \, x^{2} e^{4} + x \log \left (2 \, e^{3}\right )\right )} \log \left (\log \relax (5)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-log(2*exp(3))+8*x*exp(4)-12*x^2)*log(log(5)),x, algorithm="giac")

[Out]

-(4*x^3 - 4*x^2*e^4 + x*log(2*e^3))*log(log(5))

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maple [A]  time = 0.02, size = 23, normalized size = 1.00

method	result	size
gosper	\(-\ln \left (\ln \relax (5)\right ) x \left (-4 x \,{\mathrm e}^{4}+4 x^{2}+\ln \left (2 \,{\mathrm e}^{3}\right )\right )\)	\(23\)
default	\(\ln \left (\ln \relax (5)\right ) \left (-\ln \left (2 \,{\mathrm e}^{3}\right ) x +4 x^{2} {\mathrm e}^{4}-4 x^{3}\right )\)	\(26\)
risch	\(4 \ln \left (\ln \relax (5)\right ) {\mathrm e}^{4} x^{2}-4 \ln \left (\ln \relax (5)\right ) x^{3}-\ln \left (\ln \relax (5)\right ) x \ln \relax (2)-3 x \ln \left (\ln \relax (5)\right )\)	\(34\)
norman	\(\left (-3 \ln \left (\ln \relax (5)\right )-\ln \left (\ln \relax (5)\right ) \ln \relax (2)\right ) x -4 \ln \left (\ln \relax (5)\right ) x^{3}+4 \ln \left (\ln \relax (5)\right ) {\mathrm e}^{4} x^{2}\)	\(35\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-ln(2*exp(3))+8*x*exp(4)-12*x^2)*ln(ln(5)),x,method=_RETURNVERBOSE)

[Out]

-ln(ln(5))*x*(-4*x*exp(4)+4*x^2+ln(2*exp(3)))

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maxima [A]  time = 0.66, size = 25, normalized size = 1.09 \begin {gather*} -{\left (4 \, x^{3} - 4 \, x^{2} e^{4} + x \log \left (2 \, e^{3}\right )\right )} \log \left (\log \relax (5)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-log(2*exp(3))+8*x*exp(4)-12*x^2)*log(log(5)),x, algorithm="maxima")

[Out]

-(4*x^3 - 4*x^2*e^4 + x*log(2*e^3))*log(log(5))

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mupad [B]  time = 0.91, size = 20, normalized size = 0.87 \begin {gather*} -x\,\ln \left (\ln \relax (5)\right )\,\left (4\,x^2-4\,{\mathrm {e}}^4\,x+\ln \relax (2)+3\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-log(log(5))*(log(2*exp(3)) - 8*x*exp(4) + 12*x^2),x)

[Out]

-x*log(log(5))*(log(2) - 4*x*exp(4) + 4*x^2 + 3)

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sympy [B]  time = 0.06, size = 41, normalized size = 1.78 \begin {gather*} - 4 x^{3} \log {\left (\log {\relax (5 )} \right )} + 4 x^{2} e^{4} \log {\left (\log {\relax (5 )} \right )} + x \left (- 3 \log {\left (\log {\relax (5 )} \right )} - \log {\relax (2 )} \log {\left (\log {\relax (5 )} \right )}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-ln(2*exp(3))+8*x*exp(4)-12*x**2)*ln(ln(5)),x)

[Out]

-4*x**3*log(log(5)) + 4*x**2*exp(4)*log(log(5)) + x*(-3*log(log(5)) - log(2)*log(log(5)))

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