Optimal. Leaf size=115 \[ -\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{d} x+d^{2/3} x^2\right )}{6 a^{2/3} \sqrt [3]{d}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{d} x\right )}{3 a^{2/3} \sqrt [3]{d}}-\frac {\tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{d} x}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} a^{2/3} \sqrt [3]{d}} \]
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Rubi [A] time = 0.06, antiderivative size = 115, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {200, 31, 634, 617, 204, 628} \[ -\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{d} x+d^{2/3} x^2\right )}{6 a^{2/3} \sqrt [3]{d}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{d} x\right )}{3 a^{2/3} \sqrt [3]{d}}-\frac {\tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{d} x}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} a^{2/3} \sqrt [3]{d}} \]
Antiderivative was successfully verified.
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Rule 31
Rule 200
Rule 204
Rule 617
Rule 628
Rule 634
Rubi steps
\begin {align*} \int \frac {1}{a+d x^3} \, dx &=\frac {\int \frac {1}{\sqrt [3]{a}+\sqrt [3]{d} x} \, dx}{3 a^{2/3}}+\frac {\int \frac {2 \sqrt [3]{a}-\sqrt [3]{d} x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{d} x+d^{2/3} x^2} \, dx}{3 a^{2/3}}\\ &=\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{d} x\right )}{3 a^{2/3} \sqrt [3]{d}}+\frac {\int \frac {1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{d} x+d^{2/3} x^2} \, dx}{2 \sqrt [3]{a}}-\frac {\int \frac {-\sqrt [3]{a} \sqrt [3]{d}+2 d^{2/3} x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{d} x+d^{2/3} x^2} \, dx}{6 a^{2/3} \sqrt [3]{d}}\\ &=\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{d} x\right )}{3 a^{2/3} \sqrt [3]{d}}-\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{d} x+d^{2/3} x^2\right )}{6 a^{2/3} \sqrt [3]{d}}+\frac {\operatorname {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 \sqrt [3]{d} x}{\sqrt [3]{a}}\right )}{a^{2/3} \sqrt [3]{d}}\\ &=-\frac {\tan ^{-1}\left (\frac {\sqrt [3]{a}-2 \sqrt [3]{d} x}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} a^{2/3} \sqrt [3]{d}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{d} x\right )}{3 a^{2/3} \sqrt [3]{d}}-\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{d} x+d^{2/3} x^2\right )}{6 a^{2/3} \sqrt [3]{d}}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 89, normalized size = 0.77 \[ -\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{d} x+d^{2/3} x^2\right )-2 \log \left (\sqrt [3]{a}+\sqrt [3]{d} x\right )+2 \sqrt {3} \tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{d} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{6 a^{2/3} \sqrt [3]{d}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.83, size = 299, normalized size = 2.60 \[ \left [\frac {3 \, \sqrt {\frac {1}{3}} a d \sqrt {-\frac {\left (a^{2} d\right )^{\frac {1}{3}}}{d}} \log \left (\frac {2 \, a d x^{3} - 3 \, \left (a^{2} d\right )^{\frac {1}{3}} a x - a^{2} + 3 \, \sqrt {\frac {1}{3}} {\left (2 \, a d x^{2} + \left (a^{2} d\right )^{\frac {2}{3}} x - \left (a^{2} d\right )^{\frac {1}{3}} a\right )} \sqrt {-\frac {\left (a^{2} d\right )^{\frac {1}{3}}}{d}}}{d x^{3} + a}\right ) - \left (a^{2} d\right )^{\frac {2}{3}} \log \left (a d x^{2} - \left (a^{2} d\right )^{\frac {2}{3}} x + \left (a^{2} d\right )^{\frac {1}{3}} a\right ) + 2 \, \left (a^{2} d\right )^{\frac {2}{3}} \log \left (a d x + \left (a^{2} d\right )^{\frac {2}{3}}\right )}{6 \, a^{2} d}, \frac {6 \, \sqrt {\frac {1}{3}} a d \sqrt {\frac {\left (a^{2} d\right )^{\frac {1}{3}}}{d}} \arctan \left (\frac {\sqrt {\frac {1}{3}} {\left (2 \, \left (a^{2} d\right )^{\frac {2}{3}} x - \left (a^{2} d\right )^{\frac {1}{3}} a\right )} \sqrt {\frac {\left (a^{2} d\right )^{\frac {1}{3}}}{d}}}{a^{2}}\right ) - \left (a^{2} d\right )^{\frac {2}{3}} \log \left (a d x^{2} - \left (a^{2} d\right )^{\frac {2}{3}} x + \left (a^{2} d\right )^{\frac {1}{3}} a\right ) + 2 \, \left (a^{2} d\right )^{\frac {2}{3}} \log \left (a d x + \left (a^{2} d\right )^{\frac {2}{3}}\right )}{6 \, a^{2} d}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.26, size = 112, normalized size = 0.97 \[ -\frac {\left (-\frac {a}{d}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{d}\right )^{\frac {1}{3}} \right |}\right )}{3 \, a} + \frac {\sqrt {3} \left (-a d^{2}\right )^{\frac {1}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{d}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{d}\right )^{\frac {1}{3}}}\right )}{3 \, a d} + \frac {\left (-a d^{2}\right )^{\frac {1}{3}} \log \left (x^{2} + x \left (-\frac {a}{d}\right )^{\frac {1}{3}} + \left (-\frac {a}{d}\right )^{\frac {2}{3}}\right )}{6 \, a d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 91, normalized size = 0.79 \[ \frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{d}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 \left (\frac {a}{d}\right )^{\frac {2}{3}} d}+\frac {\ln \left (x +\left (\frac {a}{d}\right )^{\frac {1}{3}}\right )}{3 \left (\frac {a}{d}\right )^{\frac {2}{3}} d}-\frac {\ln \left (x^{2}-\left (\frac {a}{d}\right )^{\frac {1}{3}} x +\left (\frac {a}{d}\right )^{\frac {2}{3}}\right )}{6 \left (\frac {a}{d}\right )^{\frac {2}{3}} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.28, size = 98, normalized size = 0.85 \[ \frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{d}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{d}\right )^{\frac {1}{3}}}\right )}{3 \, d \left (\frac {a}{d}\right )^{\frac {2}{3}}} - \frac {\log \left (x^{2} - x \left (\frac {a}{d}\right )^{\frac {1}{3}} + \left (\frac {a}{d}\right )^{\frac {2}{3}}\right )}{6 \, d \left (\frac {a}{d}\right )^{\frac {2}{3}}} + \frac {\log \left (x + \left (\frac {a}{d}\right )^{\frac {1}{3}}\right )}{3 \, d \left (\frac {a}{d}\right )^{\frac {2}{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.23, size = 99, normalized size = 0.86 \[ \frac {\ln \left (d^{1/3}\,x+a^{1/3}\right )}{3\,a^{2/3}\,d^{1/3}}+\frac {\ln \left (3\,d^2\,x+\frac {3\,a^{1/3}\,d^{5/3}\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{2}\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{6\,a^{2/3}\,d^{1/3}}-\frac {\ln \left (3\,d^2\,x-\frac {3\,a^{1/3}\,d^{5/3}\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{2}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{6\,a^{2/3}\,d^{1/3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.16, size = 20, normalized size = 0.17 \[ \operatorname {RootSum} {\left (27 t^{3} a^{2} d - 1, \left (t \mapsto t \log {\left (3 t a + x \right )} \right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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