Optimal. Leaf size=22 \[ 3-\frac {2+x}{x-x \log ^2\left (-\frac {3}{4}+x\right )} \]
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Rubi [F]
time = 1.01, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps
used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {}
\begin {gather*} \int \frac {-6+8 x+\left (-16 x-8 x^2\right ) \log \left (\frac {1}{4} (-3+4 x)\right )+(6-8 x) \log ^2\left (\frac {1}{4} (-3+4 x)\right )}{-3 x^2+4 x^3+\left (6 x^2-8 x^3\right ) \log ^2\left (\frac {1}{4} (-3+4 x)\right )+\left (-3 x^2+4 x^3\right ) \log ^4\left (\frac {1}{4} (-3+4 x)\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {2 \left (3-4 x+4 x (2+x) \log \left (-\frac {3}{4}+x\right )+(-3+4 x) \log ^2\left (-\frac {3}{4}+x\right )\right )}{(3-4 x) x^2 \left (1-\log ^2\left (-\frac {3}{4}+x\right )\right )^2} \, dx\\ &=2 \int \frac {3-4 x+4 x (2+x) \log \left (-\frac {3}{4}+x\right )+(-3+4 x) \log ^2\left (-\frac {3}{4}+x\right )}{(3-4 x) x^2 \left (1-\log ^2\left (-\frac {3}{4}+x\right )\right )^2} \, dx\\ &=2 \int \left (\frac {-2-x}{x (-3+4 x) \left (-1+\log \left (-\frac {3}{4}+x\right )\right )^2}-\frac {1}{2 x^2 \left (-1+\log \left (-\frac {3}{4}+x\right )\right )}+\frac {2+x}{x (-3+4 x) \left (1+\log \left (-\frac {3}{4}+x\right )\right )^2}+\frac {1}{2 x^2 \left (1+\log \left (-\frac {3}{4}+x\right )\right )}\right ) \, dx\\ &=2 \int \frac {-2-x}{x (-3+4 x) \left (-1+\log \left (-\frac {3}{4}+x\right )\right )^2} \, dx+2 \int \frac {2+x}{x (-3+4 x) \left (1+\log \left (-\frac {3}{4}+x\right )\right )^2} \, dx-\int \frac {1}{x^2 \left (-1+\log \left (-\frac {3}{4}+x\right )\right )} \, dx+\int \frac {1}{x^2 \left (1+\log \left (-\frac {3}{4}+x\right )\right )} \, dx\\ &=2 \int \left (\frac {2}{3 x \left (-1+\log \left (-\frac {3}{4}+x\right )\right )^2}-\frac {11}{3 (-3+4 x) \left (-1+\log \left (-\frac {3}{4}+x\right )\right )^2}\right ) \, dx+2 \int \left (-\frac {2}{3 x \left (1+\log \left (-\frac {3}{4}+x\right )\right )^2}+\frac {11}{3 (-3+4 x) \left (1+\log \left (-\frac {3}{4}+x\right )\right )^2}\right ) \, dx-\int \frac {1}{x^2 \left (-1+\log \left (-\frac {3}{4}+x\right )\right )} \, dx+\int \frac {1}{x^2 \left (1+\log \left (-\frac {3}{4}+x\right )\right )} \, dx\\ &=\frac {4}{3} \int \frac {1}{x \left (-1+\log \left (-\frac {3}{4}+x\right )\right )^2} \, dx-\frac {4}{3} \int \frac {1}{x \left (1+\log \left (-\frac {3}{4}+x\right )\right )^2} \, dx-\frac {22}{3} \int \frac {1}{(-3+4 x) \left (-1+\log \left (-\frac {3}{4}+x\right )\right )^2} \, dx+\frac {22}{3} \int \frac {1}{(-3+4 x) \left (1+\log \left (-\frac {3}{4}+x\right )\right )^2} \, dx-\int \frac {1}{x^2 \left (-1+\log \left (-\frac {3}{4}+x\right )\right )} \, dx+\int \frac {1}{x^2 \left (1+\log \left (-\frac {3}{4}+x\right )\right )} \, dx\\ &=\frac {4}{3} \int \frac {1}{x \left (-1+\log \left (-\frac {3}{4}+x\right )\right )^2} \, dx-\frac {4}{3} \int \frac {1}{x \left (1+\log \left (-\frac {3}{4}+x\right )\right )^2} \, dx-\frac {22}{3} \text {Subst}\left (\int \frac {1}{4 x (-1+\log (x))^2} \, dx,x,-\frac {3}{4}+x\right )+\frac {22}{3} \text {Subst}\left (\int \frac {1}{4 x (1+\log (x))^2} \, dx,x,-\frac {3}{4}+x\right )-\int \frac {1}{x^2 \left (-1+\log \left (-\frac {3}{4}+x\right )\right )} \, dx+\int \frac {1}{x^2 \left (1+\log \left (-\frac {3}{4}+x\right )\right )} \, dx\\ &=\frac {4}{3} \int \frac {1}{x \left (-1+\log \left (-\frac {3}{4}+x\right )\right )^2} \, dx-\frac {4}{3} \int \frac {1}{x \left (1+\log \left (-\frac {3}{4}+x\right )\right )^2} \, dx-\frac {11}{6} \text {Subst}\left (\int \frac {1}{x (-1+\log (x))^2} \, dx,x,-\frac {3}{4}+x\right )+\frac {11}{6} \text {Subst}\left (\int \frac {1}{x (1+\log (x))^2} \, dx,x,-\frac {3}{4}+x\right )-\int \frac {1}{x^2 \left (-1+\log \left (-\frac {3}{4}+x\right )\right )} \, dx+\int \frac {1}{x^2 \left (1+\log \left (-\frac {3}{4}+x\right )\right )} \, dx\\ &=\frac {4}{3} \int \frac {1}{x \left (-1+\log \left (-\frac {3}{4}+x\right )\right )^2} \, dx-\frac {4}{3} \int \frac {1}{x \left (1+\log \left (-\frac {3}{4}+x\right )\right )^2} \, dx-\frac {11}{6} \text {Subst}\left (\int \frac {1}{x^2} \, dx,x,-1+\log \left (-\frac {3}{4}+x\right )\right )+\frac {11}{6} \text {Subst}\left (\int \frac {1}{x^2} \, dx,x,1+\log \left (-\frac {3}{4}+x\right )\right )-\int \frac {1}{x^2 \left (-1+\log \left (-\frac {3}{4}+x\right )\right )} \, dx+\int \frac {1}{x^2 \left (1+\log \left (-\frac {3}{4}+x\right )\right )} \, dx\\ &=-\frac {11}{6 \left (1-\log \left (-\frac {3}{4}+x\right )\right )}-\frac {11}{6 \left (1+\log \left (-\frac {3}{4}+x\right )\right )}+\frac {4}{3} \int \frac {1}{x \left (-1+\log \left (-\frac {3}{4}+x\right )\right )^2} \, dx-\frac {4}{3} \int \frac {1}{x \left (1+\log \left (-\frac {3}{4}+x\right )\right )^2} \, dx-\int \frac {1}{x^2 \left (-1+\log \left (-\frac {3}{4}+x\right )\right )} \, dx+\int \frac {1}{x^2 \left (1+\log \left (-\frac {3}{4}+x\right )\right )} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.08, size = 22, normalized size = 1.00 \begin {gather*} -\frac {-2-x}{x \left (-1+\log ^2\left (-\frac {3}{4}+x\right )\right )} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.60, size = 37, normalized size = 1.68
method | result | size |
norman | \(\frac {2+x}{x \left (\ln \left (x -\frac {3}{4}\right )^{2}-1\right )}\) | \(18\) |
risch | \(\frac {2+x}{x \left (\ln \left (x -\frac {3}{4}\right )^{2}-1\right )}\) | \(18\) |
derivativedivides | \(-\frac {-\frac {3 \ln \left (x -\frac {3}{4}\right )^{2}}{2}-2 \ln \left (x -\frac {3}{4}\right )^{2} \left (x -\frac {3}{4}\right )-4}{2 x \left (\ln \left (x -\frac {3}{4}\right )^{2}-1\right )}\) | \(37\) |
default | \(-\frac {-\frac {3 \ln \left (x -\frac {3}{4}\right )^{2}}{2}-2 \ln \left (x -\frac {3}{4}\right )^{2} \left (x -\frac {3}{4}\right )-4}{2 x \left (\ln \left (x -\frac {3}{4}\right )^{2}-1\right )}\) | \(37\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 41 vs.
\(2 (20) = 40\).
time = 0.52, size = 41, normalized size = 1.86 \begin {gather*} -\frac {x + 2}{4 \, x \log \left (2\right ) \log \left (4 \, x - 3\right ) - x \log \left (4 \, x - 3\right )^{2} - {\left (4 \, \log \left (2\right )^{2} - 1\right )} x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 18, normalized size = 0.82 \begin {gather*} \frac {x + 2}{x \log \left (x - \frac {3}{4}\right )^{2} - x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.04, size = 14, normalized size = 0.64 \begin {gather*} \frac {x + 2}{x \log {\left (x - \frac {3}{4} \right )}^{2} - x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.41, size = 18, normalized size = 0.82 \begin {gather*} \frac {x + 2}{x \log \left (x - \frac {3}{4}\right )^{2} - x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 2.09, size = 17, normalized size = 0.77 \begin {gather*} \frac {x+2}{x\,\left ({\ln \left (x-\frac {3}{4}\right )}^2-1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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