Optimal. Leaf size=187 \[ -\frac {\log (c+d x)}{2 c^2 d}-\frac {\log \left (d x-\sqrt [3]{2 c^3+d^3 x^3}\right )}{4 c^2 d}+\frac {3 \log \left (d (2 c+d x)-d \sqrt [3]{2 c^3+d^3 x^3}\right )}{4 c^2 d}-\frac {\tan ^{-1}\left (\frac {\frac {2 d x}{\sqrt [3]{2 c^3+d^3 x^3}}+1}{\sqrt {3}}\right )}{2 \sqrt {3} c^2 d}+\frac {\sqrt {3} \tan ^{-1}\left (\frac {\frac {2 (2 c+d x)}{\sqrt [3]{2 c^3+d^3 x^3}}+1}{\sqrt {3}}\right )}{2 c^2 d} \]
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Rubi [F] time = 0.11, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, 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*}
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Mathematica [F] time = 0.06, size = 0, normalized size = 0.00 \[ \int \frac {1}{(c+d x) \left (2 c^3+d^3 x^3\right )^{2/3}} \, dx \]
Verification is Not applicable to the result.
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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (d^{3} x^{3} + 2 \, c^{3}\right )}^{\frac {2}{3}} {\left (d x + c\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.06, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (d x +c \right ) \left (d^{3} x^{3}+2 c^{3}\right )^{\frac {2}{3}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (d^{3} x^{3} + 2 \, c^{3}\right )}^{\frac {2}{3}} {\left (d x + c\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{{\left (2\,c^3+d^3\,x^3\right )}^{2/3}\,\left (c+d\,x\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (c + d x\right ) \left (2 c^{3} + d^{3} x^{3}\right )^{\frac {2}{3}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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