Optimal. Leaf size=17 \[ 5 \left (x-\frac {x^2}{\log (1-x)}\right ) \]
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Rubi [B] time = 0.37, antiderivative size = 46, normalized size of antiderivative = 2.71, number of steps used = 19, number of rules used = 13, integrand size = 50, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.260, Rules used = {6688, 2411, 2353, 2297, 2298, 2302, 30, 2306, 2309, 2178, 2399, 2389, 2390} \begin {gather*} 5 x-\frac {5 (1-x)^2}{\log (1-x)}+\frac {10 (1-x)}{\log (1-x)}-\frac {5}{\log (1-x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 30
Rule 2178
Rule 2297
Rule 2298
Rule 2302
Rule 2306
Rule 2309
Rule 2353
Rule 2389
Rule 2390
Rule 2399
Rule 2411
Rule 6688
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \left (5+\frac {5 x^2}{(-1+x) \log ^2(1-x)}-\frac {10 x}{\log (1-x)}\right ) \, dx\\ &=5 x+5 \int \frac {x^2}{(-1+x) \log ^2(1-x)} \, dx-10 \int \frac {x}{\log (1-x)} \, dx\\ &=5 x+5 \operatorname {Subst}\left (\int \frac {(1-x)^2}{x \log ^2(x)} \, dx,x,1-x\right )-10 \int \left (\frac {1}{\log (1-x)}-\frac {1-x}{\log (1-x)}\right ) \, dx\\ &=5 x+5 \operatorname {Subst}\left (\int \left (-\frac {2}{\log ^2(x)}+\frac {1}{x \log ^2(x)}+\frac {x}{\log ^2(x)}\right ) \, dx,x,1-x\right )-10 \int \frac {1}{\log (1-x)} \, dx+10 \int \frac {1-x}{\log (1-x)} \, dx\\ &=5 x+5 \operatorname {Subst}\left (\int \frac {1}{x \log ^2(x)} \, dx,x,1-x\right )+5 \operatorname {Subst}\left (\int \frac {x}{\log ^2(x)} \, dx,x,1-x\right )-10 \operatorname {Subst}\left (\int \frac {1}{\log ^2(x)} \, dx,x,1-x\right )+10 \operatorname {Subst}\left (\int \frac {1}{\log (x)} \, dx,x,1-x\right )-10 \operatorname {Subst}\left (\int \frac {x}{\log (x)} \, dx,x,1-x\right )\\ &=5 x+\frac {10 (1-x)}{\log (1-x)}-\frac {5 (1-x)^2}{\log (1-x)}+10 \text {li}(1-x)+5 \operatorname {Subst}\left (\int \frac {1}{x^2} \, dx,x,\log (1-x)\right )-10 \operatorname {Subst}\left (\int \frac {e^{2 x}}{x} \, dx,x,\log (1-x)\right )-10 \operatorname {Subst}\left (\int \frac {1}{\log (x)} \, dx,x,1-x\right )+10 \operatorname {Subst}\left (\int \frac {x}{\log (x)} \, dx,x,1-x\right )\\ &=5 x-10 \text {Ei}(2 \log (1-x))-\frac {5}{\log (1-x)}+\frac {10 (1-x)}{\log (1-x)}-\frac {5 (1-x)^2}{\log (1-x)}+10 \operatorname {Subst}\left (\int \frac {e^{2 x}}{x} \, dx,x,\log (1-x)\right )\\ &=5 x-\frac {5}{\log (1-x)}+\frac {10 (1-x)}{\log (1-x)}-\frac {5 (1-x)^2}{\log (1-x)}\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.09, size = 16, normalized size = 0.94 \begin {gather*} 5 x \left (1-\frac {x}{\log (1-x)}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.53, size = 23, normalized size = 1.35 \begin {gather*} -\frac {5 \, {\left (x^{2} - x \log \left (-x + 1\right )\right )}}{\log \left (-x + 1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.30, size = 17, normalized size = 1.00 \begin {gather*} 5 \, x - \frac {5 \, x^{2}}{\log \left (-x + 1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.25, size = 18, normalized size = 1.06
method | result | size |
risch | \(5 x -\frac {5 x^{2}}{\ln \left (1-x \right )}\) | \(18\) |
norman | \(\frac {-5 x^{2}+5 x \ln \left (1-x \right )}{\ln \left (1-x \right )}\) | \(25\) |
derivativedivides | \(-5+5 x -\frac {5 \left (1-x \right )^{2}}{\ln \left (1-x \right )}+\frac {-10 x +10}{\ln \left (1-x \right )}-\frac {5}{\ln \left (1-x \right )}\) | \(48\) |
default | \(-5+5 x -\frac {5 \left (1-x \right )^{2}}{\ln \left (1-x \right )}+\frac {-10 x +10}{\ln \left (1-x \right )}-\frac {5}{\ln \left (1-x \right )}\) | \(48\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.68, size = 38, normalized size = 2.24 \begin {gather*} -\frac {5 \, {\left (x^{2} - x \log \left (-x + 1\right )\right )}}{\log \left (-x + 1\right )} + 5 \, \log \left (x - 1\right ) - 5 \, \log \left (-x + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.13, size = 17, normalized size = 1.00 \begin {gather*} 5\,x-\frac {5\,x^2}{\ln \left (1-x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.10, size = 12, normalized size = 0.71 \begin {gather*} - \frac {5 x^{2}}{\log {\left (1 - x \right )}} + 5 x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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