Integrand size = 25, antiderivative size = 139 \[ \int \frac {1}{(c+d x) \sqrt [3]{-c^3+d^3 x^3}} \, dx=\frac {\sqrt {3} \arctan \left (\frac {1-\frac {\sqrt [3]{2} (c-d x)}{\sqrt [3]{-c^3+d^3 x^3}}}{\sqrt {3}}\right )}{2 \sqrt [3]{2} c d}+\frac {\log \left ((c-d x) (c+d x)^2\right )}{4 \sqrt [3]{2} c d}-\frac {3 \log \left (d (c-d x)+2^{2/3} d \sqrt [3]{-c^3+d^3 x^3}\right )}{4 \sqrt [3]{2} c d} \]
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Time = 0.05 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.040, Rules used = {2174} \[ \int \frac {1}{(c+d x) \sqrt [3]{-c^3+d^3 x^3}} \, dx=\frac {\sqrt {3} \arctan \left (\frac {1-\frac {\sqrt [3]{2} (c-d x)}{\sqrt [3]{d^3 x^3-c^3}}}{\sqrt {3}}\right )}{2 \sqrt [3]{2} c d}-\frac {3 \log \left (2^{2/3} d \sqrt [3]{d^3 x^3-c^3}+d (c-d x)\right )}{4 \sqrt [3]{2} c d}+\frac {\log \left ((c-d x) (c+d x)^2\right )}{4 \sqrt [3]{2} c d} \]
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Rule 2174
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {3} \tan ^{-1}\left (\frac {1-\frac {\sqrt [3]{2} (c-d x)}{\sqrt [3]{-c^3+d^3 x^3}}}{\sqrt {3}}\right )}{2 \sqrt [3]{2} c d}+\frac {\log \left ((c-d x) (c+d x)^2\right )}{4 \sqrt [3]{2} c d}-\frac {3 \log \left (d (c-d x)+2^{2/3} d \sqrt [3]{-c^3+d^3 x^3}\right )}{4 \sqrt [3]{2} c d} \\ \end{align*}
Result contains complex when optimal does not.
Time = 2.07 (sec) , antiderivative size = 311, normalized size of antiderivative = 2.24 \[ \int \frac {1}{(c+d x) \sqrt [3]{-c^3+d^3 x^3}} \, dx=\frac {\sqrt [3]{-\frac {1}{2}} \left (2 i \sqrt {3} \text {arctanh}\left (\frac {\sqrt [3]{2} \left (3+i \sqrt {3}\right ) c+\sqrt [3]{2} \left (-3-i \sqrt {3}\right ) d x+2 i \sqrt {3} \sqrt [3]{-c^3+d^3 x^3}}{6 \sqrt [3]{-c^3+d^3 x^3}}\right )+2 \log \left (\sqrt {c} \sqrt {d} \left (-c+i \sqrt {3} c+d x-i \sqrt {3} d x+2\ 2^{2/3} \sqrt [3]{-c^3+d^3 x^3}\right )\right )-\log \left (-c d \left (\left (1+i \sqrt {3}\right ) c^2+\left (1+i \sqrt {3}\right ) d^2 x^2-2 (-2)^{2/3} d x \sqrt [3]{-c^3+d^3 x^3}-4 \sqrt [3]{2} \left (-c^3+d^3 x^3\right )^{2/3}+2 c \left (\left (-1-i \sqrt {3}\right ) d x+(-2)^{2/3} \sqrt [3]{-c^3+d^3 x^3}\right )\right )\right )\right )}{4 c d} \]
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\[\int \frac {1}{\left (d x +c \right ) \left (d^{3} x^{3}-c^{3}\right )^{\frac {1}{3}}}d x\]
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Timed out. \[ \int \frac {1}{(c+d x) \sqrt [3]{-c^3+d^3 x^3}} \, dx=\text {Timed out} \]
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\[ \int \frac {1}{(c+d x) \sqrt [3]{-c^3+d^3 x^3}} \, dx=\int \frac {1}{\sqrt [3]{\left (- c + d x\right ) \left (c^{2} + c d x + d^{2} x^{2}\right )} \left (c + d x\right )}\, dx \]
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\[ \int \frac {1}{(c+d x) \sqrt [3]{-c^3+d^3 x^3}} \, dx=\int { \frac {1}{{\left (d^{3} x^{3} - c^{3}\right )}^{\frac {1}{3}} {\left (d x + c\right )}} \,d x } \]
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\[ \int \frac {1}{(c+d x) \sqrt [3]{-c^3+d^3 x^3}} \, dx=\int { \frac {1}{{\left (d^{3} x^{3} - c^{3}\right )}^{\frac {1}{3}} {\left (d x + c\right )}} \,d x } \]
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Timed out. \[ \int \frac {1}{(c+d x) \sqrt [3]{-c^3+d^3 x^3}} \, dx=\int \frac {1}{{\left (d^3\,x^3-c^3\right )}^{1/3}\,\left (c+d\,x\right )} \,d x \]
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