Optimal. Leaf size=73 \[ \frac {\sqrt {3} \tan ^{-1}\left (\frac {a+2 \sqrt [3]{a^3-b^3 x}}{\sqrt {3} a}\right )}{a}-\frac {\log (x)}{2 a}+\frac {3 \log \left (a-\sqrt [3]{a^3-b^3 x}\right )}{2 a} \]
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Rubi [A]
time = 0.02, antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {57, 631, 210,
31} \begin {gather*} \frac {3 \log \left (a-\sqrt [3]{a^3-b^3 x}\right )}{2 a}+\frac {\sqrt {3} \tan ^{-1}\left (\frac {2 \sqrt [3]{a^3-b^3 x}+a}{\sqrt {3} a}\right )}{a}-\frac {\log (x)}{2 a} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 57
Rule 210
Rule 631
Rubi steps
\begin {align*} \int \frac {1}{x \sqrt [3]{a^3-b^3 x}} \, dx &=-\frac {\log (x)}{2 a}+\frac {3}{2} \text {Subst}\left (\int \frac {1}{a^2+a x+x^2} \, dx,x,\sqrt [3]{a^3-b^3 x}\right )-\frac {3 \text {Subst}\left (\int \frac {1}{a-x} \, dx,x,\sqrt [3]{a^3-b^3 x}\right )}{2 a}\\ &=-\frac {\log (x)}{2 a}+\frac {3 \log \left (a-\sqrt [3]{a^3-b^3 x}\right )}{2 a}-\frac {3 \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2 \sqrt [3]{a^3-b^3 x}}{a}\right )}{a}\\ &=\frac {\sqrt {3} \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{a^3-b^3 x}}{a}}{\sqrt {3}}\right )}{a}-\frac {\log (x)}{2 a}+\frac {3 \log \left (a-\sqrt [3]{a^3-b^3 x}\right )}{2 a}\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 101, normalized size = 1.38 \begin {gather*} \frac {2 \sqrt {3} \tan ^{-1}\left (\frac {a+2 \sqrt [3]{a^3-b^3 x}}{\sqrt {3} a}\right )+2 \log \left (a-\sqrt [3]{a^3-b^3 x}\right )-\log \left (a^2+a \sqrt [3]{a^3-b^3 x}+\left (a^3-b^3 x\right )^{2/3}\right )}{2 a} \end {gather*}
Antiderivative was successfully verified.
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Mathics [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in
optimal.
time = 3.75, size = 98, normalized size = 1.34 \begin {gather*} \frac {\text {Gamma}\left [-\frac {1}{3}\right ] \left (-\text {Log}\left [1-\frac {a \text {exp\_polar}\left [\frac {I}{3} \text {Pi}\right ]}{b {\left (-\frac {a^3}{b^3}+x\right )}^{\frac {1}{3}}}\right ]-\text {Log}\left [1-\frac {a \text {exp\_polar}\left [\frac {5 I}{3} \text {Pi}\right ]}{b {\left (-\frac {a^3}{b^3}+x\right )}^{\frac {1}{3}}}\right ]\right )}{3 a \text {Gamma}\left [\frac {2}{3}\right ]} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 0.12, size = 90, normalized size = 1.23
method | result | size |
derivativedivides | \(\frac {\ln \left (a -\left (-b^{3} x +a^{3}\right )^{\frac {1}{3}}\right )}{a}+\frac {-\frac {\ln \left (a^{2}+a \left (-b^{3} x +a^{3}\right )^{\frac {1}{3}}+\left (-b^{3} x +a^{3}\right )^{\frac {2}{3}}\right )}{2}+\sqrt {3}\, \arctan \left (\frac {\left (a +2 \left (-b^{3} x +a^{3}\right )^{\frac {1}{3}}\right ) \sqrt {3}}{3 a}\right )}{a}\) | \(90\) |
default | \(\frac {\ln \left (a -\left (-b^{3} x +a^{3}\right )^{\frac {1}{3}}\right )}{a}+\frac {-\frac {\ln \left (a^{2}+a \left (-b^{3} x +a^{3}\right )^{\frac {1}{3}}+\left (-b^{3} x +a^{3}\right )^{\frac {2}{3}}\right )}{2}+\sqrt {3}\, \arctan \left (\frac {\left (a +2 \left (-b^{3} x +a^{3}\right )^{\frac {1}{3}}\right ) \sqrt {3}}{3 a}\right )}{a}\) | \(90\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.36, size = 90, normalized size = 1.23 \begin {gather*} \frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (a + 2 \, {\left (-b^{3} x + a^{3}\right )}^{\frac {1}{3}}\right )}}{3 \, a}\right )}{a} - \frac {\log \left (a^{2} + {\left (-b^{3} x + a^{3}\right )}^{\frac {1}{3}} a + {\left (-b^{3} x + a^{3}\right )}^{\frac {2}{3}}\right )}{2 \, a} + \frac {\log \left (-a + {\left (-b^{3} x + a^{3}\right )}^{\frac {1}{3}}\right )}{a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 92, normalized size = 1.26 \begin {gather*} \frac {2 \, \sqrt {3} \arctan \left (\frac {\sqrt {3} a + 2 \, \sqrt {3} {\left (-b^{3} x + a^{3}\right )}^{\frac {1}{3}}}{3 \, a}\right ) - \log \left (a^{2} + {\left (-b^{3} x + a^{3}\right )}^{\frac {1}{3}} a + {\left (-b^{3} x + a^{3}\right )}^{\frac {2}{3}}\right ) + 2 \, \log \left (-a + {\left (-b^{3} x + a^{3}\right )}^{\frac {1}{3}}\right )}{2 \, a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 1.04, size = 136, normalized size = 1.86 \begin {gather*} - \frac {e^{- \frac {2 i \pi }{3}} \log {\left (- \frac {a e^{\frac {i \pi }{3}}}{b \sqrt [3]{- \frac {a^{3}}{b^{3}} + x}} + 1 \right )} \Gamma \left (- \frac {1}{3}\right )}{3 a \Gamma \left (\frac {2}{3}\right )} + \frac {e^{- \frac {i \pi }{3}} \log {\left (- \frac {a e^{i \pi }}{b \sqrt [3]{- \frac {a^{3}}{b^{3}} + x}} + 1 \right )} \Gamma \left (- \frac {1}{3}\right )}{3 a \Gamma \left (\frac {2}{3}\right )} - \frac {\log {\left (- \frac {a e^{\frac {5 i \pi }{3}}}{b \sqrt [3]{- \frac {a^{3}}{b^{3}} + x}} + 1 \right )} \Gamma \left (- \frac {1}{3}\right )}{3 a \Gamma \left (\frac {2}{3}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.00, size = 107, normalized size = 1.47 \begin {gather*} 3 \left (\frac {\ln \left |\left (a^{3}-b^{3} x\right )^{\frac {1}{3}}-a\right |}{3 a}-\frac {\ln \left (\left (\left (a^{3}-b^{3} x\right )^{\frac {1}{3}}\right )^{2}+\left (a^{3}-b^{3} x\right )^{\frac {1}{3}} a+a^{2}\right )}{6 a}+\frac {\arctan \left (\frac {a+2 \left (a^{3}-b^{3} x\right )^{\frac {1}{3}}}{a \sqrt {3}}\right )}{\sqrt {3} a}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.13, size = 108, normalized size = 1.48 \begin {gather*} \frac {\ln \left (9\,{\left (a^3-b^3\,x\right )}^{1/3}-9\,a\right )}{a}+\frac {\ln \left (9\,{\left (a^3-b^3\,x\right )}^{1/3}-\frac {9\,a\,{\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}^2}{4}\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{2\,a}-\frac {\ln \left (9\,{\left (a^3-b^3\,x\right )}^{1/3}-\frac {9\,a\,{\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}^2}{4}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{2\,a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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