Integrand size = 12, antiderivative size = 131 \[ \int \frac {1}{\sqrt [3]{c \cot (a+b x)}} \, dx=\frac {\sqrt {3} \arctan \left (\frac {c^{2/3}-2 (c \cot (a+b x))^{2/3}}{\sqrt {3} c^{2/3}}\right )}{2 b \sqrt [3]{c}}-\frac {\log \left (c^{2/3}+(c \cot (a+b x))^{2/3}\right )}{2 b \sqrt [3]{c}}+\frac {\log \left (c^{4/3}-c^{2/3} (c \cot (a+b x))^{2/3}+(c \cot (a+b x))^{4/3}\right )}{4 b \sqrt [3]{c}} \]
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Time = 0.12 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {3557, 335, 281, 206, 31, 648, 631, 210, 642} \[ \int \frac {1}{\sqrt [3]{c \cot (a+b x)}} \, dx=\frac {\sqrt {3} \arctan \left (\frac {c^{2/3}-2 (c \cot (a+b x))^{2/3}}{\sqrt {3} c^{2/3}}\right )}{2 b \sqrt [3]{c}}-\frac {\log \left ((c \cot (a+b x))^{2/3}+c^{2/3}\right )}{2 b \sqrt [3]{c}}+\frac {\log \left (-c^{2/3} (c \cot (a+b x))^{2/3}+(c \cot (a+b x))^{4/3}+c^{4/3}\right )}{4 b \sqrt [3]{c}} \]
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Rule 31
Rule 206
Rule 210
Rule 281
Rule 335
Rule 631
Rule 642
Rule 648
Rule 3557
Rubi steps \begin{align*} \text {integral}& = -\frac {c \text {Subst}\left (\int \frac {1}{\sqrt [3]{x} \left (c^2+x^2\right )} \, dx,x,c \cot (a+b x)\right )}{b} \\ & = -\frac {(3 c) \text {Subst}\left (\int \frac {x}{c^2+x^6} \, dx,x,\sqrt [3]{c \cot (a+b x)}\right )}{b} \\ & = -\frac {(3 c) \text {Subst}\left (\int \frac {1}{c^2+x^3} \, dx,x,(c \cot (a+b x))^{2/3}\right )}{2 b} \\ & = -\frac {\text {Subst}\left (\int \frac {1}{c^{2/3}+x} \, dx,x,(c \cot (a+b x))^{2/3}\right )}{2 b \sqrt [3]{c}}-\frac {\text {Subst}\left (\int \frac {2 c^{2/3}-x}{c^{4/3}-c^{2/3} x+x^2} \, dx,x,(c \cot (a+b x))^{2/3}\right )}{2 b \sqrt [3]{c}} \\ & = -\frac {\log \left (c^{2/3}+(c \cot (a+b x))^{2/3}\right )}{2 b \sqrt [3]{c}}+\frac {\text {Subst}\left (\int \frac {-c^{2/3}+2 x}{c^{4/3}-c^{2/3} x+x^2} \, dx,x,(c \cot (a+b x))^{2/3}\right )}{4 b \sqrt [3]{c}}-\frac {\left (3 \sqrt [3]{c}\right ) \text {Subst}\left (\int \frac {1}{c^{4/3}-c^{2/3} x+x^2} \, dx,x,(c \cot (a+b x))^{2/3}\right )}{4 b} \\ & = -\frac {\log \left (c^{2/3}+(c \cot (a+b x))^{2/3}\right )}{2 b \sqrt [3]{c}}+\frac {\log \left (c^{4/3}-c^{2/3} (c \cot (a+b x))^{2/3}+(c \cot (a+b x))^{4/3}\right )}{4 b \sqrt [3]{c}}-\frac {3 \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1-\frac {2 (c \cot (a+b x))^{2/3}}{c^{2/3}}\right )}{2 b \sqrt [3]{c}} \\ & = \frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 (c \cot (a+b x))^{2/3}}{c^{2/3}}}{\sqrt {3}}\right )}{2 b \sqrt [3]{c}}-\frac {\log \left (c^{2/3}+(c \cot (a+b x))^{2/3}\right )}{2 b \sqrt [3]{c}}+\frac {\log \left (c^{4/3}-c^{2/3} (c \cot (a+b x))^{2/3}+(c \cot (a+b x))^{4/3}\right )}{4 b \sqrt [3]{c}} \\ \end{align*}
Time = 0.15 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.75 \[ \int \frac {1}{\sqrt [3]{c \cot (a+b x)}} \, dx=\frac {\sqrt [3]{\cot (a+b x)} \left (-2 \sqrt {3} \arctan \left (\frac {-1+2 \cot ^{\frac {2}{3}}(a+b x)}{\sqrt {3}}\right )-2 \log \left (1+\cot ^{\frac {2}{3}}(a+b x)\right )+\log \left (1-\cot ^{\frac {2}{3}}(a+b x)+\cot ^{\frac {4}{3}}(a+b x)\right )\right )}{4 b \sqrt [3]{c \cot (a+b x)}} \]
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Time = 0.04 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.82
method | result | size |
derivativedivides | \(-\frac {3 c \left (\frac {\ln \left (\left (c \cot \left (b x +a \right )\right )^{\frac {2}{3}}+\left (c^{2}\right )^{\frac {1}{3}}\right )}{6 \left (c^{2}\right )^{\frac {2}{3}}}-\frac {\ln \left (\left (c \cot \left (b x +a \right )\right )^{\frac {4}{3}}-\left (c \cot \left (b x +a \right )\right )^{\frac {2}{3}} \left (c^{2}\right )^{\frac {1}{3}}+\left (c^{2}\right )^{\frac {2}{3}}\right )}{12 \left (c^{2}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \left (c \cot \left (b x +a \right )\right )^{\frac {2}{3}}}{\left (c^{2}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{6 \left (c^{2}\right )^{\frac {2}{3}}}\right )}{b}\) | \(108\) |
default | \(-\frac {3 c \left (\frac {\ln \left (\left (c \cot \left (b x +a \right )\right )^{\frac {2}{3}}+\left (c^{2}\right )^{\frac {1}{3}}\right )}{6 \left (c^{2}\right )^{\frac {2}{3}}}-\frac {\ln \left (\left (c \cot \left (b x +a \right )\right )^{\frac {4}{3}}-\left (c \cot \left (b x +a \right )\right )^{\frac {2}{3}} \left (c^{2}\right )^{\frac {1}{3}}+\left (c^{2}\right )^{\frac {2}{3}}\right )}{12 \left (c^{2}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \left (c \cot \left (b x +a \right )\right )^{\frac {2}{3}}}{\left (c^{2}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{6 \left (c^{2}\right )^{\frac {2}{3}}}\right )}{b}\) | \(108\) |
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Leaf count of result is larger than twice the leaf count of optimal. 268 vs. \(2 (100) = 200\).
Time = 0.27 (sec) , antiderivative size = 639, normalized size of antiderivative = 4.88 \[ \int \frac {1}{\sqrt [3]{c \cot (a+b x)}} \, dx=\left [\frac {\sqrt {3} c \sqrt {\frac {\left (-c\right )^{\frac {1}{3}}}{c}} \log \left (\frac {1}{2} \, \sqrt {3} {\left (\left (-c\right )^{\frac {2}{3}} \left (\frac {c \cos \left (2 \, b x + 2 \, a\right ) + c}{\sin \left (2 \, b x + 2 \, a\right )}\right )^{\frac {2}{3}} {\left (\cos \left (2 \, b x + 2 \, a\right ) - 1\right )} - 2 \, c \left (\frac {c \cos \left (2 \, b x + 2 \, a\right ) + c}{\sin \left (2 \, b x + 2 \, a\right )}\right )^{\frac {1}{3}} \sin \left (2 \, b x + 2 \, a\right ) + {\left (c \cos \left (2 \, b x + 2 \, a\right ) - c\right )} \left (-c\right )^{\frac {1}{3}}\right )} \sqrt {\frac {\left (-c\right )^{\frac {1}{3}}}{c}} - \frac {3}{2} \, \left (-c\right )^{\frac {1}{3}} \left (\frac {c \cos \left (2 \, b x + 2 \, a\right ) + c}{\sin \left (2 \, b x + 2 \, a\right )}\right )^{\frac {2}{3}} {\left (\cos \left (2 \, b x + 2 \, a\right ) - 1\right )} + \frac {3}{2} \, c \cos \left (2 \, b x + 2 \, a\right ) + \frac {1}{2} \, c\right ) - 2 \, \left (-c\right )^{\frac {2}{3}} \log \left (\left (-c\right )^{\frac {2}{3}} + \left (\frac {c \cos \left (2 \, b x + 2 \, a\right ) + c}{\sin \left (2 \, b x + 2 \, a\right )}\right )^{\frac {2}{3}}\right ) + \left (-c\right )^{\frac {2}{3}} \log \left (-\frac {\left (-c\right )^{\frac {1}{3}} c \sin \left (2 \, b x + 2 \, a\right ) + \left (-c\right )^{\frac {2}{3}} \left (\frac {c \cos \left (2 \, b x + 2 \, a\right ) + c}{\sin \left (2 \, b x + 2 \, a\right )}\right )^{\frac {2}{3}} \sin \left (2 \, b x + 2 \, a\right ) - {\left (c \cos \left (2 \, b x + 2 \, a\right ) + c\right )} \left (\frac {c \cos \left (2 \, b x + 2 \, a\right ) + c}{\sin \left (2 \, b x + 2 \, a\right )}\right )^{\frac {1}{3}}}{\sin \left (2 \, b x + 2 \, a\right )}\right )}{4 \, b c}, -\frac {2 \, \sqrt {3} c \sqrt {-\frac {\left (-c\right )^{\frac {1}{3}}}{c}} \arctan \left (\frac {\sqrt {3} \left (-c\right )^{\frac {1}{3}} c \sqrt {-\frac {\left (-c\right )^{\frac {1}{3}}}{c}} + 2 \, \sqrt {3} \left (-c\right )^{\frac {2}{3}} \left (\frac {c \cos \left (2 \, b x + 2 \, a\right ) + c}{\sin \left (2 \, b x + 2 \, a\right )}\right )^{\frac {2}{3}} \sqrt {-\frac {\left (-c\right )^{\frac {1}{3}}}{c}}}{3 \, c}\right ) + 2 \, \left (-c\right )^{\frac {2}{3}} \log \left (\left (-c\right )^{\frac {2}{3}} + \left (\frac {c \cos \left (2 \, b x + 2 \, a\right ) + c}{\sin \left (2 \, b x + 2 \, a\right )}\right )^{\frac {2}{3}}\right ) - \left (-c\right )^{\frac {2}{3}} \log \left (-\frac {\left (-c\right )^{\frac {1}{3}} c \sin \left (2 \, b x + 2 \, a\right ) + \left (-c\right )^{\frac {2}{3}} \left (\frac {c \cos \left (2 \, b x + 2 \, a\right ) + c}{\sin \left (2 \, b x + 2 \, a\right )}\right )^{\frac {2}{3}} \sin \left (2 \, b x + 2 \, a\right ) - {\left (c \cos \left (2 \, b x + 2 \, a\right ) + c\right )} \left (\frac {c \cos \left (2 \, b x + 2 \, a\right ) + c}{\sin \left (2 \, b x + 2 \, a\right )}\right )^{\frac {1}{3}}}{\sin \left (2 \, b x + 2 \, a\right )}\right )}{4 \, b c}\right ] \]
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\[ \int \frac {1}{\sqrt [3]{c \cot (a+b x)}} \, dx=\int \frac {1}{\sqrt [3]{c \cot {\left (a + b x \right )}}}\, dx \]
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Time = 0.34 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.79 \[ \int \frac {1}{\sqrt [3]{c \cot (a+b x)}} \, dx=-\frac {c {\left (\frac {2 \, \sqrt {3} \arctan \left (-\frac {\sqrt {3} {\left (c^{\frac {2}{3}} - 2 \, \left (\frac {c}{\tan \left (b x + a\right )}\right )^{\frac {2}{3}}\right )}}{3 \, c^{\frac {2}{3}}}\right )}{c^{\frac {4}{3}}} - \frac {\log \left (c^{\frac {4}{3}} - c^{\frac {2}{3}} \left (\frac {c}{\tan \left (b x + a\right )}\right )^{\frac {2}{3}} + \left (\frac {c}{\tan \left (b x + a\right )}\right )^{\frac {4}{3}}\right )}{c^{\frac {4}{3}}} + \frac {2 \, \log \left (c^{\frac {2}{3}} + \left (\frac {c}{\tan \left (b x + a\right )}\right )^{\frac {2}{3}}\right )}{c^{\frac {4}{3}}}\right )}}{4 \, b} \]
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\[ \int \frac {1}{\sqrt [3]{c \cot (a+b x)}} \, dx=\int { \frac {1}{\left (c \cot \left (b x + a\right )\right )^{\frac {1}{3}}} \,d x } \]
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Time = 12.16 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.98 \[ \int \frac {1}{\sqrt [3]{c \cot (a+b x)}} \, dx=-\frac {\ln \left ({\left (c\,\mathrm {cot}\left (a+b\,x\right )\right )}^{2/3}+c^{2/3}\right )}{2\,b\,c^{1/3}}-\frac {\ln \left (\frac {81\,c^{11/3}\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{b^3}+\frac {162\,c^3\,{\left (c\,\mathrm {cot}\left (a+b\,x\right )\right )}^{2/3}}{b^3}\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{4\,b\,c^{1/3}}+\frac {\ln \left (\frac {81\,c^{11/3}\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{b^3}-\frac {162\,c^3\,{\left (c\,\mathrm {cot}\left (a+b\,x\right )\right )}^{2/3}}{b^3}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{4\,b\,c^{1/3}} \]
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