3.33.70 \(\int \frac {(-12+2 x) \log (3)+\log (4)}{\log (4)} \, dx\)

Optimal. Leaf size=19 \[ 2+x+\frac {\left (-1+(6-x)^2\right ) \log (3)}{\log (4)} \]

________________________________________________________________________________________

Rubi [A]  time = 0.01, antiderivative size = 16, normalized size of antiderivative = 0.84, number of steps used = 2, number of rules used = 1, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {12} \begin {gather*} x+\frac {(6-x)^2 \log (3)}{\log (4)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((-12 + 2*x)*Log[3] + Log[4])/Log[4],x]

[Out]

x + ((6 - x)^2*Log[3])/Log[4]

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} &=\frac {\int ((-12+2 x) \log (3)+\log (4)) \, dx}{\log (4)}\\ &=x+\frac {(6-x)^2 \log (3)}{\log (4)}\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.00, size = 21, normalized size = 1.11 \begin {gather*} x-\frac {12 x \log (3)}{\log (4)}+\frac {x^2 \log (3)}{\log (4)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((-12 + 2*x)*Log[3] + Log[4])/Log[4],x]

[Out]

x - (12*x*Log[3])/Log[4] + (x^2*Log[3])/Log[4]

________________________________________________________________________________________

fricas [A]  time = 0.67, size = 22, normalized size = 1.16 \begin {gather*} \frac {{\left (x^{2} - 12 \, x\right )} \log \relax (3) + 2 \, x \log \relax (2)}{2 \, \log \relax (2)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*(2*log(2)+(2*x-12)*log(3))/log(2),x, algorithm="fricas")

[Out]

1/2*((x^2 - 12*x)*log(3) + 2*x*log(2))/log(2)

________________________________________________________________________________________

giac [A]  time = 0.29, size = 22, normalized size = 1.16 \begin {gather*} \frac {{\left (x^{2} - 12 \, x\right )} \log \relax (3) + 2 \, x \log \relax (2)}{2 \, \log \relax (2)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*(2*log(2)+(2*x-12)*log(3))/log(2),x, algorithm="giac")

[Out]

1/2*((x^2 - 12*x)*log(3) + 2*x*log(2))/log(2)

________________________________________________________________________________________

maple [A]  time = 0.02, size = 21, normalized size = 1.11

method	result	size
gosper	\(\frac {x \left (x \ln \relax (3)+2 \ln \relax (2)-12 \ln \relax (3)\right )}{2 \ln \relax (2)}\)	\(21\)
default	\(\frac {2 x \ln \relax (2)+\ln \relax (3) \left (x^{2}-12 x \right )}{2 \ln \relax (2)}\)	\(23\)
risch	\(\frac {\ln \relax (3) x^{2}}{2 \ln \relax (2)}+x -\frac {6 x \ln \relax (3)}{\ln \relax (2)}\)	\(23\)
norman	\(\frac {\left (\ln \relax (2)-6 \ln \relax (3)\right ) x}{\ln \relax (2)}+\frac {\ln \relax (3) x^{2}}{2 \ln \relax (2)}\)	\(26\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/2*(2*ln(2)+(2*x-12)*ln(3))/ln(2),x,method=_RETURNVERBOSE)

[Out]

1/2*x*(x*ln(3)+2*ln(2)-12*ln(3))/ln(2)

________________________________________________________________________________________

maxima [A]  time = 0.36, size = 22, normalized size = 1.16 \begin {gather*} \frac {{\left (x^{2} - 12 \, x\right )} \log \relax (3) + 2 \, x \log \relax (2)}{2 \, \log \relax (2)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*(2*log(2)+(2*x-12)*log(3))/log(2),x, algorithm="maxima")

[Out]

1/2*((x^2 - 12*x)*log(3) + 2*x*log(2))/log(2)

________________________________________________________________________________________

mupad [B]  time = 0.56, size = 24, normalized size = 1.26 \begin {gather*} \frac {{\left (\ln \relax (2)+\frac {\ln \relax (3)\,\left (2\,x-12\right )}{2}\right )}^2}{2\,\ln \relax (2)\,\ln \relax (3)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(2) + (log(3)*(2*x - 12))/2)/log(2),x)

[Out]

(log(2) + (log(3)*(2*x - 12))/2)^2/(2*log(2)*log(3))

________________________________________________________________________________________

sympy [A]  time = 0.06, size = 24, normalized size = 1.26 \begin {gather*} \frac {x^{2} \log {\relax (3 )}}{2 \log {\relax (2 )}} + \frac {x \left (- 6 \log {\relax (3 )} + \log {\relax (2 )}\right )}{\log {\relax (2 )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*(2*ln(2)+(2*x-12)*ln(3))/ln(2),x)

[Out]

x**2*log(3)/(2*log(2)) + x*(-6*log(3) + log(2))/log(2)

________________________________________________________________________________________