Optimal. Leaf size=186 \[ -\frac{\log \left (\sqrt [3]{2 c^3+d^3 x^3}-d x\right )}{4 c d}+\frac{3 \log \left (d (2 c+d x)-d \sqrt [3]{2 c^3+d^3 x^3}\right )}{4 c d}+\frac{\tan ^{-1}\left (\frac{\frac{2 d x}{\sqrt [3]{2 c^3+d^3 x^3}}+1}{\sqrt{3}}\right )}{2 \sqrt{3} c d}-\frac{\sqrt{3} \tan ^{-1}\left (\frac{\frac{2 (2 c+d x)}{\sqrt [3]{2 c^3+d^3 x^3}}+1}{\sqrt{3}}\right )}{2 c d}-\frac{\log (c+d x)}{2 c d} \]
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Rubi [A] time = 0.203521, antiderivative size = 186, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12, Rules used = {2149, 239, 2151} \[ -\frac{\log \left (\sqrt [3]{2 c^3+d^3 x^3}-d x\right )}{4 c d}+\frac{3 \log \left (d (2 c+d x)-d \sqrt [3]{2 c^3+d^3 x^3}\right )}{4 c d}+\frac{\tan ^{-1}\left (\frac{\frac{2 d x}{\sqrt [3]{2 c^3+d^3 x^3}}+1}{\sqrt{3}}\right )}{2 \sqrt{3} c d}-\frac{\sqrt{3} \tan ^{-1}\left (\frac{\frac{2 (2 c+d x)}{\sqrt [3]{2 c^3+d^3 x^3}}+1}{\sqrt{3}}\right )}{2 c d}-\frac{\log (c+d x)}{2 c d} \]
Antiderivative was successfully verified.
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Rule 2149
Rule 239
Rule 2151
Rubi steps
\begin{align*} \int \frac{1}{(c+d x) \sqrt [3]{2 c^3+d^3 x^3}} \, dx &=\frac{\int \frac{1}{\sqrt [3]{2 c^3+d^3 x^3}} \, dx}{2 c}+\frac{\int \frac{c-d x}{(c+d x) \sqrt [3]{2 c^3+d^3 x^3}} \, dx}{2 c}\\ &=\frac{\tan ^{-1}\left (\frac{1+\frac{2 d x}{\sqrt [3]{2 c^3+d^3 x^3}}}{\sqrt{3}}\right )}{2 \sqrt{3} c d}-\frac{\sqrt{3} \tan ^{-1}\left (\frac{1+\frac{2 (2 c+d x)}{\sqrt [3]{2 c^3+d^3 x^3}}}{\sqrt{3}}\right )}{2 c d}-\frac{\log (c+d x)}{2 c d}-\frac{\log \left (-d x+\sqrt [3]{2 c^3+d^3 x^3}\right )}{4 c d}+\frac{3 \log \left (d (2 c+d x)-d \sqrt [3]{2 c^3+d^3 x^3}\right )}{4 c d}\\ \end{align*}
Mathematica [F] time = 0.0633519, size = 0, normalized size = 0. \[ \int \frac{1}{(c+d x) \sqrt [3]{2 c^3+d^3 x^3}} \, dx \]
Verification is Not applicable to the result.
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Maple [F] time = 0.036, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{dx+c}{\frac{1}{\sqrt [3]{{d}^{3}{x}^{3}+2\,{c}^{3}}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (d^{3} x^{3} + 2 \, c^{3}\right )}^{\frac{1}{3}}{\left (d x + c\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (c + d x\right ) \sqrt [3]{2 c^{3} + d^{3} x^{3}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (d^{3} x^{3} + 2 \, c^{3}\right )}^{\frac{1}{3}}{\left (d x + c\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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