Optimal. Leaf size=71 \[ \frac {\sqrt {3} \tan ^{-1}\left (\frac {a+2 \sqrt [3]{a^3+b^3 x}}{\sqrt {3} a}\right )}{a}-\frac {\log (x)}{2 a}+\frac {3 \log \left (a-\sqrt [3]{a^3+b^3 x}\right )}{2 a} \]
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Rubi [A]
time = 0.02, antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {57, 631, 210,
31} \begin {gather*} \frac {\sqrt {3} \text {ArcTan}\left (\frac {2 \sqrt [3]{a^3+b^3 x}+a}{\sqrt {3} a}\right )}{a}+\frac {3 \log \left (a-\sqrt [3]{a^3+b^3 x}\right )}{2 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 = 0.06, size = 97, normalized size = 1.37 \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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Maple [A]
time = 0.12, size = 86, normalized size = 1.21
method | result | size |
derivativedivides | \(\frac {\ln \left (a -\left (b^{3} x +a^{3}\right )^{\frac {1}{3}}\right )}{a}+\frac {-\frac {\ln \left (a^{2}+a \left (b^{3} x +a^{3}\right )^{\frac {1}{3}}+\left (b^{3} x +a^{3}\right )^{\frac {2}{3}}\right )}{2}+\sqrt {3}\, \arctan \left (\frac {\left (a +2 \left (b^{3} x +a^{3}\right )^{\frac {1}{3}}\right ) \sqrt {3}}{3 a}\right )}{a}\) | \(86\) |
default | \(\frac {\ln \left (a -\left (b^{3} x +a^{3}\right )^{\frac {1}{3}}\right )}{a}+\frac {-\frac {\ln \left (a^{2}+a \left (b^{3} x +a^{3}\right )^{\frac {1}{3}}+\left (b^{3} x +a^{3}\right )^{\frac {2}{3}}\right )}{2}+\sqrt {3}\, \arctan \left (\frac {\left (a +2 \left (b^{3} x +a^{3}\right )^{\frac {1}{3}}\right ) \sqrt {3}}{3 a}\right )}{a}\) | \(86\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.48, size = 86, normalized size = 1.21 \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 = 1.02, size = 88, normalized size = 1.24 \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 = 0.88, size = 138, normalized size = 1.94 \begin {gather*} \frac {e^{\frac {i \pi }{3}} \log {\left (- \frac {a e^{\frac {2 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^{\frac {4 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 {\log {\left (- \frac {a e^{2 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 )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.56, size = 87, 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 ({\left | -a + {\left (b^{3} x + a^{3}\right )}^{\frac {1}{3}} \right |}\right )}{a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.10, size = 105, normalized size = 1.48 \begin {gather*} \frac {\ln \left (9\,{\left (a^3+x\,b^3\right )}^{1/3}-9\,a\right )}{a}+\frac {\ln \left (9\,{\left (a^3+x\,b^3\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+x\,b^3\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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