3.23.97 \(\int (4+8 x+(18+8 x) \log (2)+(8+2 x) \log ^2(2)+(-4-2 \log (2)) \log (3)+(-4-2 \log (2)) \log (\log (3))) \, dx\)

Optimal. Leaf size=22 \[ (1+2 x+(4+x) \log (2)-\log (3)-\log (\log (3)))^2 \]

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Rubi [B]  time = 0.03, antiderivative size = 50, normalized size of antiderivative = 2.27, number of steps used = 1, number of rules used = 0, integrand size = 42, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} 4 x^2+(x+4)^2 \log ^2(2)+2 x (2-\log (9)-2 \log (\log (3))-\log (2) \log (\log (27)))+\frac {1}{4} (4 x+9)^2 \log (2) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[4 + 8*x + (18 + 8*x)*Log[2] + (8 + 2*x)*Log[2]^2 + (-4 - 2*Log[2])*Log[3] + (-4 - 2*Log[2])*Log[Log[3]],x]

[Out]

4*x^2 + ((9 + 4*x)^2*Log[2])/4 + (4 + x)^2*Log[2]^2 + 2*x*(2 - Log[9] - 2*Log[Log[3]] - Log[2]*Log[Log[27]])

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=4 x^2+\frac {1}{4} (9+4 x)^2 \log (2)+(4+x)^2 \log ^2(2)+2 x (2-\log (9)-2 \log (\log (3))-\log (2) \log (\log (27)))\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 29, normalized size = 1.32 \begin {gather*} 2 (2+\log (2)) \left (x+x^2+\frac {1}{2} x^2 \log (2)+x \log \left (\frac {16}{\log (27)}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[4 + 8*x + (18 + 8*x)*Log[2] + (8 + 2*x)*Log[2]^2 + (-4 - 2*Log[2])*Log[3] + (-4 - 2*Log[2])*Log[Log[
3]],x]

[Out]

2*(2 + Log[2])*(x + x^2 + (x^2*Log[2])/2 + x*Log[16/Log[27]])

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fricas [B]  time = 1.02, size = 59, normalized size = 2.68 \begin {gather*} {\left (x^{2} + 8 \, x\right )} \log \relax (2)^{2} + 4 \, x^{2} - 2 \, {\left (x \log \relax (2) + 2 \, x\right )} \log \relax (3) + 2 \, {\left (2 \, x^{2} + 9 \, x\right )} \log \relax (2) - 2 \, {\left (x \log \relax (2) + 2 \, x\right )} \log \left (\log \relax (3)\right ) + 4 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*log(2)-4)*log(log(3))+(-2*log(2)-4)*log(3)+(2*x+8)*log(2)^2+(8*x+18)*log(2)+8*x+4,x, algorithm="
fricas")

[Out]

(x^2 + 8*x)*log(2)^2 + 4*x^2 - 2*(x*log(2) + 2*x)*log(3) + 2*(2*x^2 + 9*x)*log(2) - 2*(x*log(2) + 2*x)*log(log
(3)) + 4*x

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giac [B]  time = 0.27, size = 53, normalized size = 2.41 \begin {gather*} -2 \, x {\left (\log \relax (2) + 2\right )} \log \relax (3) + {\left (x^{2} + 8 \, x\right )} \log \relax (2)^{2} - 2 \, x {\left (\log \relax (2) + 2\right )} \log \left (\log \relax (3)\right ) + 4 \, x^{2} + 2 \, {\left (2 \, x^{2} + 9 \, x\right )} \log \relax (2) + 4 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*log(2)-4)*log(log(3))+(-2*log(2)-4)*log(3)+(2*x+8)*log(2)^2+(8*x+18)*log(2)+8*x+4,x, algorithm="
giac")

[Out]

-2*x*(log(2) + 2)*log(3) + (x^2 + 8*x)*log(2)^2 - 2*x*(log(2) + 2)*log(log(3)) + 4*x^2 + 2*(2*x^2 + 9*x)*log(2
) + 4*x

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maple [A]  time = 0.06, size = 29, normalized size = 1.32

method	result	size
gosper	\(\left (\ln \relax (2)+2\right ) x \left (x \ln \relax (2)+8 \ln \relax (2)-2 \ln \left (\ln \relax (3)\right )-2 \ln \relax (3)+2 x +2\right )\)	\(29\)
norman	\(\left (\ln \relax (2)^{2}+4 \ln \relax (2)+4\right ) x^{2}+\left (8 \ln \relax (2)^{2}-2 \ln \left (\ln \relax (3)\right ) \ln \relax (2)-2 \ln \relax (2) \ln \relax (3)+18 \ln \relax (2)-4 \ln \left (\ln \relax (3)\right )-4 \ln \relax (3)+4\right ) x\)	\(52\)
default	\(\left (-2 \ln \relax (2)-4\right ) \ln \left (\ln \relax (3)\right ) x +\left (-2 \ln \relax (2)-4\right ) \ln \relax (3) x +\left (x^{2}+8 x \right ) \ln \relax (2)^{2}+\ln \relax (2) \left (4 x^{2}+18 x \right )+4 x^{2}+4 x\)	\(55\)
risch	\(x^{2} \ln \relax (2)^{2}+8 x \ln \relax (2)^{2}-2 x \ln \left (\ln \relax (3)\right ) \ln \relax (2)-2 x \ln \relax (2) \ln \relax (3)+4 x^{2} \ln \relax (2)+18 x \ln \relax (2)-4 \ln \left (\ln \relax (3)\right ) x -4 x \ln \relax (3)+4 x^{2}+4 x\)	\(63\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-2*ln(2)-4)*ln(ln(3))+(-2*ln(2)-4)*ln(3)+(2*x+8)*ln(2)^2+(8*x+18)*ln(2)+8*x+4,x,method=_RETURNVERBOSE)

[Out]

(ln(2)+2)*x*(x*ln(2)+8*ln(2)-2*ln(ln(3))-2*ln(3)+2*x+2)

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maxima [B]  time = 0.35, size = 53, normalized size = 2.41 \begin {gather*} -2 \, x {\left (\log \relax (2) + 2\right )} \log \relax (3) + {\left (x^{2} + 8 \, x\right )} \log \relax (2)^{2} - 2 \, x {\left (\log \relax (2) + 2\right )} \log \left (\log \relax (3)\right ) + 4 \, x^{2} + 2 \, {\left (2 \, x^{2} + 9 \, x\right )} \log \relax (2) + 4 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*log(2)-4)*log(log(3))+(-2*log(2)-4)*log(3)+(2*x+8)*log(2)^2+(8*x+18)*log(2)+8*x+4,x, algorithm="
maxima")

[Out]

-2*x*(log(2) + 2)*log(3) + (x^2 + 8*x)*log(2)^2 - 2*x*(log(2) + 2)*log(log(3)) + 4*x^2 + 2*(2*x^2 + 9*x)*log(2
) + 4*x

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mupad [B]  time = 0.08, size = 42, normalized size = 1.91 \begin {gather*} x^2\,\left (\ln \left (16\right )+{\ln \relax (2)}^2+4\right )-x\,\left (\ln \left (\frac {81\,{\ln \relax (3)}^4}{262144}\right )+\ln \relax (3)\,\ln \relax (4)+\ln \relax (4)\,\ln \left (\ln \relax (3)\right )-8\,{\ln \relax (2)}^2-4\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(8*x + log(2)*(8*x + 18) + log(2)^2*(2*x + 8) - log(log(3))*(2*log(2) + 4) - log(3)*(2*log(2) + 4) + 4,x)

[Out]

x^2*(log(16) + log(2)^2 + 4) - x*(log((81*log(3)^4)/262144) + log(3)*log(4) + log(4)*log(log(3)) - 8*log(2)^2
- 4)

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sympy [B]  time = 0.07, size = 60, normalized size = 2.73 \begin {gather*} x^{2} \left (\log {\relax (2 )}^{2} + 4 \log {\relax (2 )} + 4\right ) + x \left (- 4 \log {\relax (3 )} - 2 \log {\relax (2 )} \log {\relax (3 )} - 4 \log {\left (\log {\relax (3 )} \right )} - 2 \log {\relax (2 )} \log {\left (\log {\relax (3 )} \right )} + 8 \log {\relax (2 )}^{2} + 4 + 18 \log {\relax (2 )}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*ln(2)-4)*ln(ln(3))+(-2*ln(2)-4)*ln(3)+(2*x+8)*ln(2)**2+(8*x+18)*ln(2)+8*x+4,x)

[Out]

x**2*(log(2)**2 + 4*log(2) + 4) + x*(-4*log(3) - 2*log(2)*log(3) - 4*log(log(3)) - 2*log(2)*log(log(3)) + 8*lo
g(2)**2 + 4 + 18*log(2))

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