Integrand size = 33, antiderivative size = 33 \[ \int \frac {\cot ^{-1}(a+b x)}{\sqrt [3]{\left (1+a^2\right ) c+2 a b c x+b^2 c x^2}} \, dx=\text {Int}\left (\frac {\cot ^{-1}(a+b x)}{\sqrt [3]{c+c (a+b x)^2}},x\right ) \]
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Not integrable
Time = 0.04 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 0, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {\cot ^{-1}(a+b x)}{\sqrt [3]{\left (1+a^2\right ) c+2 a b c x+b^2 c x^2}} \, dx=\int \frac {\cot ^{-1}(a+b x)}{\sqrt [3]{\left (1+a^2\right ) c+2 a b c x+b^2 c x^2}} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {\cot ^{-1}(x)}{\sqrt [3]{c+c x^2}} \, dx,x,a+b x\right )}{b} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(180\) vs. \(2(25)=50\).
Time = 0.14 (sec) , antiderivative size = 180, normalized size of antiderivative = 5.45 \[ \int \frac {\cot ^{-1}(a+b x)}{\sqrt [3]{\left (1+a^2\right ) c+2 a b c x+b^2 c x^2}} \, dx=\frac {c \left (6 \operatorname {Gamma}\left (\frac {11}{6}\right ) \operatorname {Gamma}\left (\frac {7}{3}\right ) \left (5 \left (1+a^2+2 a b x+b^2 x^2\right ) \left (-3+2 (a+b x) \cot ^{-1}(a+b x)\right )+4 (a+b x) \cot ^{-1}(a+b x) \operatorname {Hypergeometric2F1}\left (1,\frac {4}{3},\frac {11}{6},\frac {1}{1+a^2+2 a b x+b^2 x^2}\right )\right )-5 \sqrt [3]{2} \sqrt {\pi } \operatorname {Gamma}\left (\frac {5}{3}\right ) \, _3F_2\left (1,\frac {4}{3},\frac {4}{3};\frac {11}{6},\frac {7}{3};\frac {1}{1+a^2+2 a b x+b^2 x^2}\right )\right )}{20 b \left (c \left (1+a^2+2 a b x+b^2 x^2\right )\right )^{4/3} \operatorname {Gamma}\left (\frac {11}{6}\right ) \operatorname {Gamma}\left (\frac {7}{3}\right )} \]
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Not integrable
Time = 0.78 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.94
\[\int \frac {\operatorname {arccot}\left (b x +a \right )}{{\left (\left (a^{2}+1\right ) c +2 a b c x +b^{2} c \,x^{2}\right )}^{\frac {1}{3}}}d x\]
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Not integrable
Time = 0.28 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.00 \[ \int \frac {\cot ^{-1}(a+b x)}{\sqrt [3]{\left (1+a^2\right ) c+2 a b c x+b^2 c x^2}} \, dx=\int { \frac {\operatorname {arccot}\left (b x + a\right )}{{\left (b^{2} c x^{2} + 2 \, a b c x + {\left (a^{2} + 1\right )} c\right )}^{\frac {1}{3}}} \,d x } \]
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Not integrable
Time = 5.94 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.94 \[ \int \frac {\cot ^{-1}(a+b x)}{\sqrt [3]{\left (1+a^2\right ) c+2 a b c x+b^2 c x^2}} \, dx=\int \frac {\operatorname {acot}{\left (a + b x \right )}}{\sqrt [3]{c \left (a^{2} + 2 a b x + b^{2} x^{2} + 1\right )}}\, dx \]
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Not integrable
Time = 0.30 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.00 \[ \int \frac {\cot ^{-1}(a+b x)}{\sqrt [3]{\left (1+a^2\right ) c+2 a b c x+b^2 c x^2}} \, dx=\int { \frac {\operatorname {arccot}\left (b x + a\right )}{{\left (b^{2} c x^{2} + 2 \, a b c x + {\left (a^{2} + 1\right )} c\right )}^{\frac {1}{3}}} \,d x } \]
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Not integrable
Time = 0.45 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.00 \[ \int \frac {\cot ^{-1}(a+b x)}{\sqrt [3]{\left (1+a^2\right ) c+2 a b c x+b^2 c x^2}} \, dx=\int { \frac {\operatorname {arccot}\left (b x + a\right )}{{\left (b^{2} c x^{2} + 2 \, a b c x + {\left (a^{2} + 1\right )} c\right )}^{\frac {1}{3}}} \,d x } \]
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Not integrable
Time = 0.84 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.00 \[ \int \frac {\cot ^{-1}(a+b x)}{\sqrt [3]{\left (1+a^2\right ) c+2 a b c x+b^2 c x^2}} \, dx=\int \frac {\mathrm {acot}\left (a+b\,x\right )}{{\left (c\,b^2\,x^2+2\,a\,c\,b\,x+c\,\left (a^2+1\right )\right )}^{1/3}} \,d x \]
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