3.95.68 \(\int \frac {36 x^2+(12-36 x+12 x^2) \log (4)+(8-6 x+x^2) \log ^2(4)}{36 x^2+(-36 x+12 x^2) \log (4)+(9-6 x+x^2) \log ^2(4)} \, dx\)

Optimal. Leaf size=21 \[ x-\frac {2-x}{-3+x+\frac {6 x}{\log (4)}} \]

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Rubi [A]  time = 0.07, antiderivative size = 31, normalized size of antiderivative = 1.48, number of steps used = 5, number of rules used = 4, integrand size = 66, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.061, Rules used = {1984, 27, 6, 683} \begin {gather*} x-\frac {(12-\log (4)) \log (4)}{(6+\log (4)) (x (6+\log (4))-\log (64))} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(36*x^2 + (12 - 36*x + 12*x^2)*Log[4] + (8 - 6*x + x^2)*Log[4]^2)/(36*x^2 + (-36*x + 12*x^2)*Log[4] + (9 -
6*x + x^2)*Log[4]^2),x]

[Out]

x - ((12 - Log[4])*Log[4])/((6 + Log[4])*(x*(6 + Log[4]) - Log[64]))

Rule 6

Int[(u_.)*((w_.) + (a_.)*(v_) + (b_.)*(v_))^(p_.), x_Symbol] :> Int[u*((a + b)*v + w)^p, x] /; FreeQ[{a, b}, x
] &&  !FreeQ[v, x]

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 683

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d +
e*x)^m*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[2*c*d - b*e,
0] && IGtQ[p, 0] &&  !(EqQ[m, 3] && NeQ[p, 1])

Rule 1984

Int[(u_)^(p_.)*(v_)^(q_.), x_Symbol] :> Int[ExpandToSum[u, x]^p*ExpandToSum[v, x]^q, x] /; FreeQ[{p, q}, x] &&
QuadraticQ[{u, v}, x] &&  !QuadraticMatchQ[{u, v}, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-6 x \log (4) (6+\log (4))+x^2 (6+\log (4))^2+4 \log (4) (3+\log (16))}{9 \log ^2(4)-6 x \log (4) (6+\log (4))+x^2 (6+\log (4))^2} \, dx\\ &=\int \frac {-6 x \log (4) (6+\log (4))+x^2 (6+\log (4))^2+4 \log (4) (3+\log (16))}{(6 x-3 \log (4)+x \log (4))^2} \, dx\\ &=\int \frac {-6 x \log (4) (6+\log (4))+x^2 (6+\log (4))^2+4 \log (4) (3+\log (16))}{(-3 \log (4)+x (6+\log (4)))^2} \, dx\\ &=\int \left (1+\frac {(12-\log (4)) \log (4)}{(3 \log (4)-x (6+\log (4)))^2}\right ) \, dx\\ &=x-\frac {(12-\log (4)) \log (4)}{(6+\log (4)) (x (6+\log (4))-\log (64))}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.04, size = 28, normalized size = 1.33 \begin {gather*} x+\frac {(-12+\log (4)) \log (4)}{(6+\log (4)) (-3 \log (4)+x (6+\log (4)))} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(36*x^2 + (12 - 36*x + 12*x^2)*Log[4] + (8 - 6*x + x^2)*Log[4]^2)/(36*x^2 + (-36*x + 12*x^2)*Log[4]
+ (9 - 6*x + x^2)*Log[4]^2),x]

[Out]

x + ((-12 + Log[4])*Log[4])/((6 + Log[4])*(-3*Log[4] + x*(6 + Log[4])))

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fricas [B]  time = 0.53, size = 57, normalized size = 2.71 \begin {gather*} \frac {{\left (x^{2} - 3 \, x + 1\right )} \log \relax (2)^{2} + 9 \, x^{2} + 3 \, {\left (2 \, x^{2} - 3 \, x - 2\right )} \log \relax (2)}{{\left (x - 3\right )} \log \relax (2)^{2} + 3 \, {\left (2 \, x - 3\right )} \log \relax (2) + 9 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*(x^2-6*x+8)*log(2)^2+2*(12*x^2-36*x+12)*log(2)+36*x^2)/(4*(x^2-6*x+9)*log(2)^2+2*(12*x^2-36*x)*lo
g(2)+36*x^2),x, algorithm="fricas")

[Out]

((x^2 - 3*x + 1)*log(2)^2 + 9*x^2 + 3*(2*x^2 - 3*x - 2)*log(2))/((x - 3)*log(2)^2 + 3*(2*x - 3)*log(2) + 9*x)

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giac [B]  time = 0.22, size = 59, normalized size = 2.81 \begin {gather*} \frac {x \log \relax (2)^{2} + 6 \, x \log \relax (2) + 9 \, x}{\log \relax (2)^{2} + 6 \, \log \relax (2) + 9} + \frac {\log \relax (2)^{2} - 6 \, \log \relax (2)}{{\left (x \log \relax (2) + 3 \, x - 3 \, \log \relax (2)\right )} {\left (\log \relax (2) + 3\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*(x^2-6*x+8)*log(2)^2+2*(12*x^2-36*x+12)*log(2)+36*x^2)/(4*(x^2-6*x+9)*log(2)^2+2*(12*x^2-36*x)*lo
g(2)+36*x^2),x, algorithm="giac")

[Out]

(x*log(2)^2 + 6*x*log(2) + 9*x)/(log(2)^2 + 6*log(2) + 9) + (log(2)^2 - 6*log(2))/((x*log(2) + 3*x - 3*log(2))
*(log(2) + 3))

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maple [A]  time = 0.70, size = 30, normalized size = 1.43

method	result	size
default	\(x +\frac {\ln \relax (2) \left (\ln \relax (2)-6\right )}{\left (3+\ln \relax (2)\right ) \left (x \ln \relax (2)-3 \ln \relax (2)+3 x \right )}\)	\(30\)
gosper	\(\frac {x \left (3 x \ln \relax (2)-8 \ln \relax (2)+9 x -6\right )}{3 x \ln \relax (2)-9 \ln \relax (2)+9 x}\)	\(32\)
norman	\(\frac {\left (3+\ln \relax (2)\right ) x^{2}+\left (-\frac {8 \ln \relax (2)}{3}-2\right ) x}{x \ln \relax (2)-3 \ln \relax (2)+3 x}\)	\(33\)
risch	\(x +\frac {\ln \relax (2)^{2}}{\left (3+\ln \relax (2)\right ) \left (x \ln \relax (2)-3 \ln \relax (2)+3 x \right )}-\frac {6 \ln \relax (2)}{\left (3+\ln \relax (2)\right ) \left (x \ln \relax (2)-3 \ln \relax (2)+3 x \right )}\)	\(52\)
meijerg	\(-\frac {3 \ln \relax (2) \left (-\frac {x \left (3+\ln \relax (2)\right ) \left (-\frac {x \left (3+\ln \relax (2)\right )}{\ln \relax (2)}+6\right )}{9 \ln \relax (2) \left (1-\frac {x \left (3+\ln \relax (2)\right )}{3 \ln \relax (2)}\right )}-2 \ln \left (1-\frac {x \left (3+\ln \relax (2)\right )}{3 \ln \relax (2)}\right )\right )}{3+\ln \relax (2)}+\frac {\left (-24 \ln \relax (2)^{2}-72 \ln \relax (2)\right ) \left (\frac {x \left (3+\ln \relax (2)\right )}{3 \ln \relax (2) \left (1-\frac {x \left (3+\ln \relax (2)\right )}{3 \ln \relax (2)}\right )}+\ln \left (1-\frac {x \left (3+\ln \relax (2)\right )}{3 \ln \relax (2)}\right )\right )}{4 \ln \relax (2)^{2}+24 \ln \relax (2)+36}+\frac {32 \left (3+\ln \relax (2)\right )^{2} x}{9 \left (4 \ln \relax (2)^{2}+24 \ln \relax (2)+36\right ) \left (1-\frac {x \left (3+\ln \relax (2)\right )}{3 \ln \relax (2)}\right )}+\frac {8 \left (3+\ln \relax (2)\right )^{2} x}{3 \ln \relax (2) \left (4 \ln \relax (2)^{2}+24 \ln \relax (2)+36\right ) \left (1-\frac {x \left (3+\ln \relax (2)\right )}{3 \ln \relax (2)}\right )}\)	\(215\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((4*(x^2-6*x+8)*ln(2)^2+2*(12*x^2-36*x+12)*ln(2)+36*x^2)/(4*(x^2-6*x+9)*ln(2)^2+2*(12*x^2-36*x)*ln(2)+36*x^
2),x,method=_RETURNVERBOSE)

[Out]

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

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maxima [B]  time = 0.36, size = 37, normalized size = 1.76 \begin {gather*} x + \frac {\log \relax (2)^{2} - 6 \, \log \relax (2)}{{\left (\log \relax (2)^{2} + 6 \, \log \relax (2) + 9\right )} x - 3 \, \log \relax (2)^{2} - 9 \, \log \relax (2)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*(x^2-6*x+8)*log(2)^2+2*(12*x^2-36*x+12)*log(2)+36*x^2)/(4*(x^2-6*x+9)*log(2)^2+2*(12*x^2-36*x)*lo
g(2)+36*x^2),x, algorithm="maxima")

[Out]

x + (log(2)^2 - 6*log(2))/((log(2)^2 + 6*log(2) + 9)*x - 3*log(2)^2 - 9*log(2))

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mupad [B]  time = 7.64, size = 112, normalized size = 5.33 \begin {gather*} x-\frac {\mathrm {atan}\left (\frac {\frac {\left (18\,\ln \relax (2)+6\,{\ln \relax (2)}^2\right )\,\left (\ln \left (64\right )-{\ln \relax (2)}^2\right )}{6\,\ln \relax (2)\,\sqrt {\ln \left (64\right )-6\,\ln \relax (2)}}-\frac {x\,\left (\ln \left (64\right )-{\ln \relax (2)}^2\right )\,\left (2\,\ln \left (64\right )+2\,{\ln \relax (2)}^2+18\right )}{6\,\ln \relax (2)\,\sqrt {\ln \left (64\right )-6\,\ln \relax (2)}}}{\ln \left (64\right )-{\ln \relax (2)}^2}\right )\,\left (\ln \left (64\right )-{\ln \relax (2)}^2\right )}{3\,\ln \relax (2)\,\sqrt {\ln \left (64\right )-6\,\ln \relax (2)}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*log(2)*(12*x^2 - 36*x + 12) + 4*log(2)^2*(x^2 - 6*x + 8) + 36*x^2)/(4*log(2)^2*(x^2 - 6*x + 9) - 2*log(
2)*(36*x - 12*x^2) + 36*x^2),x)

[Out]

x - (atan((((18*log(2) + 6*log(2)^2)*(log(64) - log(2)^2))/(6*log(2)*(log(64) - 6*log(2))^(1/2)) - (x*(log(64)
- log(2)^2)*(2*log(64) + 2*log(2)^2 + 18))/(6*log(2)*(log(64) - 6*log(2))^(1/2)))/(log(64) - log(2)^2))*(log(
64) - log(2)^2))/(3*log(2)*(log(64) - 6*log(2))^(1/2))

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sympy [B]  time = 0.31, size = 36, normalized size = 1.71 \begin {gather*} x + \frac {- 6 \log {\relax (2 )} + \log {\relax (2 )}^{2}}{x \left (\log {\relax (2 )}^{2} + 6 \log {\relax (2 )} + 9\right ) - 9 \log {\relax (2 )} - 3 \log {\relax (2 )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*(x**2-6*x+8)*ln(2)**2+2*(12*x**2-36*x+12)*ln(2)+36*x**2)/(4*(x**2-6*x+9)*ln(2)**2+2*(12*x**2-36*x
)*ln(2)+36*x**2),x)

[Out]

x + (-6*log(2) + log(2)**2)/(x*(log(2)**2 + 6*log(2) + 9) - 9*log(2) - 3*log(2)**2)

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