Optimal. Leaf size=115 \[ -\frac {\tan ^{-1}\left (\frac {\sqrt [3]{b}+2 \sqrt [3]{a} x}{\sqrt {3} \sqrt [3]{b}}\right )}{\sqrt {3} \sqrt [3]{a} b^{2/3}}+\frac {\log \left (\sqrt [3]{b}-\sqrt [3]{a} x\right )}{3 \sqrt [3]{a} b^{2/3}}-\frac {\log \left (b^{2/3}+\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2\right )}{6 \sqrt [3]{a} b^{2/3}} \]
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Rubi [A]
time = 0.05, antiderivative size = 115, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.546, Rules used = {206, 31, 648,
631, 210, 642} \begin {gather*} -\frac {\log \left (a^{2/3} x^2+\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}\right )}{6 \sqrt [3]{a} b^{2/3}}-\frac {\text {ArcTan}\left (\frac {2 \sqrt [3]{a} x+\sqrt [3]{b}}{\sqrt {3} \sqrt [3]{b}}\right )}{\sqrt {3} \sqrt [3]{a} b^{2/3}}+\frac {\log \left (\sqrt [3]{b}-\sqrt [3]{a} x\right )}{3 \sqrt [3]{a} b^{2/3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 206
Rule 210
Rule 631
Rule 642
Rule 648
Rubi steps
\begin {align*} \int \frac {1}{-b+a x^3} \, dx &=\frac {\int \frac {1}{-\sqrt [3]{b}+\sqrt [3]{a} x} \, dx}{3 b^{2/3}}+\frac {\int \frac {-2 \sqrt [3]{b}-\sqrt [3]{a} x}{b^{2/3}+\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2} \, dx}{3 b^{2/3}}\\ &=\frac {\log \left (\sqrt [3]{b}-\sqrt [3]{a} x\right )}{3 \sqrt [3]{a} b^{2/3}}-\frac {\int \frac {\sqrt [3]{a} \sqrt [3]{b}+2 a^{2/3} x}{b^{2/3}+\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2} \, dx}{6 \sqrt [3]{a} b^{2/3}}-\frac {\int \frac {1}{b^{2/3}+\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2} \, dx}{2 \sqrt [3]{b}}\\ &=\frac {\log \left (\sqrt [3]{b}-\sqrt [3]{a} x\right )}{3 \sqrt [3]{a} b^{2/3}}-\frac {\log \left (b^{2/3}+\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2\right )}{6 \sqrt [3]{a} b^{2/3}}+\frac {\text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2 \sqrt [3]{a} x}{\sqrt [3]{b}}\right )}{\sqrt [3]{a} b^{2/3}}\\ &=-\frac {\tan ^{-1}\left (\frac {\sqrt [3]{b}+2 \sqrt [3]{a} x}{\sqrt {3} \sqrt [3]{b}}\right )}{\sqrt {3} \sqrt [3]{a} b^{2/3}}+\frac {\log \left (\sqrt [3]{b}-\sqrt [3]{a} x\right )}{3 \sqrt [3]{a} b^{2/3}}-\frac {\log \left (b^{2/3}+\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2\right )}{6 \sqrt [3]{a} b^{2/3}}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 89, normalized size = 0.77 \begin {gather*} -\frac {2 \sqrt {3} \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{a} x}{\sqrt [3]{b}}}{\sqrt {3}}\right )-2 \log \left (\sqrt [3]{b}-\sqrt [3]{a} x\right )+\log \left (b^{2/3}+\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3} x^2\right )}{6 \sqrt [3]{a} b^{2/3}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.02, size = 92, normalized size = 0.80
method | result | size |
risch | \(\frac {\munderset {\textit {\_R} =\RootOf \left (a \,\textit {\_Z}^{3}-b \right )}{\sum }\frac {\ln \left (-\textit {\_R} +x \right )}{\textit {\_R}^{2}}}{3 a}\) | \(29\) |
default | \(\frac {\ln \left (x -\left (\frac {b}{a}\right )^{\frac {1}{3}}\right )}{3 a \left (\frac {b}{a}\right )^{\frac {2}{3}}}-\frac {\ln \left (x^{2}+\left (\frac {b}{a}\right )^{\frac {1}{3}} x +\left (\frac {b}{a}\right )^{\frac {2}{3}}\right )}{6 a \left (\frac {b}{a}\right )^{\frac {2}{3}}}-\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {b}{a}\right )^{\frac {1}{3}}}+1\right )}{3}\right )}{3 a \left (\frac {b}{a}\right )^{\frac {2}{3}}}\) | \(92\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 1.74, size = 97, normalized size = 0.84 \begin {gather*} -\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (\frac {b}{a}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {b}{a}\right )^{\frac {1}{3}}}\right )}{3 \, a \left (\frac {b}{a}\right )^{\frac {2}{3}}} - \frac {\log \left (x^{2} + x \left (\frac {b}{a}\right )^{\frac {1}{3}} + \left (\frac {b}{a}\right )^{\frac {2}{3}}\right )}{6 \, a \left (\frac {b}{a}\right )^{\frac {2}{3}}} + \frac {\log \left (x - \left (\frac {b}{a}\right )^{\frac {1}{3}}\right )}{3 \, a \left (\frac {b}{a}\right )^{\frac {2}{3}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.87, size = 300, normalized size = 2.61 \begin {gather*} \left [\frac {3 \, \sqrt {\frac {1}{3}} a b \sqrt {-\frac {\left (a b^{2}\right )^{\frac {1}{3}}}{a}} \log \left (\frac {2 \, a b x^{3} - 3 \, \left (a b^{2}\right )^{\frac {1}{3}} b x + b^{2} - 3 \, \sqrt {\frac {1}{3}} {\left (2 \, a b x^{2} - \left (a b^{2}\right )^{\frac {2}{3}} x - \left (a b^{2}\right )^{\frac {1}{3}} b\right )} \sqrt {-\frac {\left (a b^{2}\right )^{\frac {1}{3}}}{a}}}{a x^{3} - b}\right ) - \left (a b^{2}\right )^{\frac {2}{3}} \log \left (a b x^{2} + \left (a b^{2}\right )^{\frac {2}{3}} x + \left (a b^{2}\right )^{\frac {1}{3}} b\right ) + 2 \, \left (a b^{2}\right )^{\frac {2}{3}} \log \left (a b x - \left (a b^{2}\right )^{\frac {2}{3}}\right )}{6 \, a b^{2}}, -\frac {6 \, \sqrt {\frac {1}{3}} a b \sqrt {\frac {\left (a b^{2}\right )^{\frac {1}{3}}}{a}} \arctan \left (\frac {\sqrt {\frac {1}{3}} {\left (2 \, \left (a b^{2}\right )^{\frac {2}{3}} x + \left (a b^{2}\right )^{\frac {1}{3}} b\right )} \sqrt {\frac {\left (a b^{2}\right )^{\frac {1}{3}}}{a}}}{b^{2}}\right ) + \left (a b^{2}\right )^{\frac {2}{3}} \log \left (a b x^{2} + \left (a b^{2}\right )^{\frac {2}{3}} x + \left (a b^{2}\right )^{\frac {1}{3}} b\right ) - 2 \, \left (a b^{2}\right )^{\frac {2}{3}} \log \left (a b x - \left (a b^{2}\right )^{\frac {2}{3}}\right )}{6 \, a b^{2}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.06, size = 20, normalized size = 0.17 \begin {gather*} \operatorname {RootSum} {\left (27 t^{3} a b^{2} - 1, \left ( t \mapsto t \log {\left (- 3 t b + x \right )} \right )\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.46, size = 104, normalized size = 0.90 \begin {gather*} \frac {\left (\frac {b}{a}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (\frac {b}{a}\right )^{\frac {1}{3}} \right |}\right )}{3 \, b} - \frac {\sqrt {3} \left (a^{2} b\right )^{\frac {1}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (\frac {b}{a}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {b}{a}\right )^{\frac {1}{3}}}\right )}{3 \, a b} - \frac {\left (a^{2} b\right )^{\frac {1}{3}} \log \left (x^{2} + x \left (\frac {b}{a}\right )^{\frac {1}{3}} + \left (\frac {b}{a}\right )^{\frac {2}{3}}\right )}{6 \, a b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.31, size = 101, normalized size = 0.88 \begin {gather*} \frac {\ln \left (a^{1/3}\,x-b^{1/3}\right )}{3\,a^{1/3}\,b^{2/3}}+\frac {\ln \left (3\,a^2\,x-\frac {3\,a^{5/3}\,b^{1/3}\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{2}\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{6\,a^{1/3}\,b^{2/3}}-\frac {\ln \left (3\,a^2\,x+\frac {3\,a^{5/3}\,b^{1/3}\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{2}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{6\,a^{1/3}\,b^{2/3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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