Optimal. Leaf size=187 \[ -\frac{\log \left (d x-\sqrt [3]{2 c^3+d^3 x^3}\right )}{4 c^2 d}+\frac{3 \log \left (d (2 c+d x)-d \sqrt [3]{2 c^3+d^3 x^3}\right )}{4 c^2 d}-\frac{\tan ^{-1}\left (\frac{\frac{2 d x}{\sqrt [3]{2 c^3+d^3 x^3}}+1}{\sqrt{3}}\right )}{2 \sqrt{3} c^2 d}+\frac{\sqrt{3} \tan ^{-1}\left (\frac{\frac{2 (2 c+d x)}{\sqrt [3]{2 c^3+d^3 x^3}}+1}{\sqrt{3}}\right )}{2 c^2 d}-\frac{\log (c+d x)}{2 c^2 d} \]
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Rubi [F] time = 0.105546, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{1}{(c+d x) \left (2 c^3+d^3 x^3\right )^{2/3}} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin{align*} \int \frac{1}{(c+d x) \left (2 c^3+d^3 x^3\right )^{2/3}} \, dx &=\int \frac{1}{(c+d x) \left (2 c^3+d^3 x^3\right )^{2/3}} \, dx\\ \end{align*}
Mathematica [F] time = 0.0625508, size = 0, normalized size = 0. \[ \int \frac{1}{(c+d x) \left (2 c^3+d^3 x^3\right )^{2/3}} \, dx \]
Verification is Not applicable to the result.
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Maple [F] time = 0.044, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{dx+c} \left ({d}^{3}{x}^{3}+2\,{c}^{3} \right ) ^{-{\frac{2}{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{2}{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 ) \left (2 c^{3} + d^{3} x^{3}\right )^{\frac{2}{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{2}{3}}{\left (d x + c\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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