Integrand size = 12, antiderivative size = 244 \[ \int \frac {1}{(c \cot (a+b x))^{4/3}} \, dx=\frac {\arctan \left (\frac {\sqrt [3]{c \cot (a+b x)}}{\sqrt [3]{c}}\right )}{b c^{4/3}}-\frac {\arctan \left (\sqrt {3}-\frac {2 \sqrt [3]{c \cot (a+b x)}}{\sqrt [3]{c}}\right )}{2 b c^{4/3}}+\frac {\arctan \left (\sqrt {3}+\frac {2 \sqrt [3]{c \cot (a+b x)}}{\sqrt [3]{c}}\right )}{2 b c^{4/3}}+\frac {3}{b c \sqrt [3]{c \cot (a+b x)}}+\frac {\sqrt {3} \log \left (c^{2/3}-\sqrt {3} \sqrt [3]{c} \sqrt [3]{c \cot (a+b x)}+(c \cot (a+b x))^{2/3}\right )}{4 b c^{4/3}}-\frac {\sqrt {3} \log \left (c^{2/3}+\sqrt {3} \sqrt [3]{c} \sqrt [3]{c \cot (a+b x)}+(c \cot (a+b x))^{2/3}\right )}{4 b c^{4/3}} \]
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Time = 0.52 (sec) , antiderivative size = 244, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {3555, 3557, 335, 301, 648, 632, 210, 642, 209} \[ \int \frac {1}{(c \cot (a+b x))^{4/3}} \, dx=\frac {\arctan \left (\frac {\sqrt [3]{c \cot (a+b x)}}{\sqrt [3]{c}}\right )}{b c^{4/3}}-\frac {\arctan \left (\sqrt {3}-\frac {2 \sqrt [3]{c \cot (a+b x)}}{\sqrt [3]{c}}\right )}{2 b c^{4/3}}+\frac {\arctan \left (\frac {2 \sqrt [3]{c \cot (a+b x)}}{\sqrt [3]{c}}+\sqrt {3}\right )}{2 b c^{4/3}}+\frac {\sqrt {3} \log \left (-\sqrt {3} \sqrt [3]{c} \sqrt [3]{c \cot (a+b x)}+(c \cot (a+b x))^{2/3}+c^{2/3}\right )}{4 b c^{4/3}}-\frac {\sqrt {3} \log \left (\sqrt {3} \sqrt [3]{c} \sqrt [3]{c \cot (a+b x)}+(c \cot (a+b x))^{2/3}+c^{2/3}\right )}{4 b c^{4/3}}+\frac {3}{b c \sqrt [3]{c \cot (a+b x)}} \]
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Rule 209
Rule 210
Rule 301
Rule 335
Rule 632
Rule 642
Rule 648
Rule 3555
Rule 3557
Rubi steps \begin{align*} \text {integral}& = \frac {3}{b c \sqrt [3]{c \cot (a+b x)}}-\frac {\int (c \cot (a+b x))^{2/3} \, dx}{c^2} \\ & = \frac {3}{b c \sqrt [3]{c \cot (a+b x)}}+\frac {\text {Subst}\left (\int \frac {x^{2/3}}{c^2+x^2} \, dx,x,c \cot (a+b x)\right )}{b c} \\ & = \frac {3}{b c \sqrt [3]{c \cot (a+b x)}}+\frac {3 \text {Subst}\left (\int \frac {x^4}{c^2+x^6} \, dx,x,\sqrt [3]{c \cot (a+b x)}\right )}{b c} \\ & = \frac {3}{b c \sqrt [3]{c \cot (a+b x)}}+\frac {\text {Subst}\left (\int \frac {-\frac {\sqrt [3]{c}}{2}+\frac {\sqrt {3} x}{2}}{c^{2/3}-\sqrt {3} \sqrt [3]{c} x+x^2} \, dx,x,\sqrt [3]{c \cot (a+b x)}\right )}{b c^{4/3}}+\frac {\text {Subst}\left (\int \frac {-\frac {\sqrt [3]{c}}{2}-\frac {\sqrt {3} x}{2}}{c^{2/3}+\sqrt {3} \sqrt [3]{c} x+x^2} \, dx,x,\sqrt [3]{c \cot (a+b x)}\right )}{b c^{4/3}}+\frac {\text {Subst}\left (\int \frac {1}{c^{2/3}+x^2} \, dx,x,\sqrt [3]{c \cot (a+b x)}\right )}{b c} \\ & = \frac {\arctan \left (\frac {\sqrt [3]{c \cot (a+b x)}}{\sqrt [3]{c}}\right )}{b c^{4/3}}+\frac {3}{b c \sqrt [3]{c \cot (a+b x)}}+\frac {\sqrt {3} \text {Subst}\left (\int \frac {-\sqrt {3} \sqrt [3]{c}+2 x}{c^{2/3}-\sqrt {3} \sqrt [3]{c} x+x^2} \, dx,x,\sqrt [3]{c \cot (a+b x)}\right )}{4 b c^{4/3}}-\frac {\sqrt {3} \text {Subst}\left (\int \frac {\sqrt {3} \sqrt [3]{c}+2 x}{c^{2/3}+\sqrt {3} \sqrt [3]{c} x+x^2} \, dx,x,\sqrt [3]{c \cot (a+b x)}\right )}{4 b c^{4/3}}+\frac {\text {Subst}\left (\int \frac {1}{c^{2/3}-\sqrt {3} \sqrt [3]{c} x+x^2} \, dx,x,\sqrt [3]{c \cot (a+b x)}\right )}{4 b c}+\frac {\text {Subst}\left (\int \frac {1}{c^{2/3}+\sqrt {3} \sqrt [3]{c} x+x^2} \, dx,x,\sqrt [3]{c \cot (a+b x)}\right )}{4 b c} \\ & = \frac {\arctan \left (\frac {\sqrt [3]{c \cot (a+b x)}}{\sqrt [3]{c}}\right )}{b c^{4/3}}+\frac {3}{b c \sqrt [3]{c \cot (a+b x)}}+\frac {\sqrt {3} \log \left (c^{2/3}-\sqrt {3} \sqrt [3]{c} \sqrt [3]{c \cot (a+b x)}+(c \cot (a+b x))^{2/3}\right )}{4 b c^{4/3}}-\frac {\sqrt {3} \log \left (c^{2/3}+\sqrt {3} \sqrt [3]{c} \sqrt [3]{c \cot (a+b x)}+(c \cot (a+b x))^{2/3}\right )}{4 b c^{4/3}}+\frac {\text {Subst}\left (\int \frac {1}{-\frac {1}{3}-x^2} \, dx,x,1-\frac {2 \sqrt [3]{c \cot (a+b x)}}{\sqrt {3} \sqrt [3]{c}}\right )}{2 \sqrt {3} b c^{4/3}}-\frac {\text {Subst}\left (\int \frac {1}{-\frac {1}{3}-x^2} \, dx,x,1+\frac {2 \sqrt [3]{c \cot (a+b x)}}{\sqrt {3} \sqrt [3]{c}}\right )}{2 \sqrt {3} b c^{4/3}} \\ & = \frac {\arctan \left (\frac {\sqrt [3]{c \cot (a+b x)}}{\sqrt [3]{c}}\right )}{b c^{4/3}}-\frac {\arctan \left (\frac {1}{3} \left (3 \sqrt {3}-\frac {6 \sqrt [3]{c \cot (a+b x)}}{\sqrt [3]{c}}\right )\right )}{2 b c^{4/3}}+\frac {\arctan \left (\frac {1}{3} \left (3 \sqrt {3}+\frac {6 \sqrt [3]{c \cot (a+b x)}}{\sqrt [3]{c}}\right )\right )}{2 b c^{4/3}}+\frac {3}{b c \sqrt [3]{c \cot (a+b x)}}+\frac {\sqrt {3} \log \left (c^{2/3}-\sqrt {3} \sqrt [3]{c} \sqrt [3]{c \cot (a+b x)}+(c \cot (a+b x))^{2/3}\right )}{4 b c^{4/3}}-\frac {\sqrt {3} \log \left (c^{2/3}+\sqrt {3} \sqrt [3]{c} \sqrt [3]{c \cot (a+b x)}+(c \cot (a+b x))^{2/3}\right )}{4 b c^{4/3}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 0.26 (sec) , antiderivative size = 254, normalized size of antiderivative = 1.04 \[ \int \frac {1}{(c \cot (a+b x))^{4/3}} \, dx=\frac {6+i \sqrt [6]{\cot ^2(a+b x)} \log \left (1-i \sqrt [6]{\cot ^2(a+b x)}\right )-i \sqrt [6]{\cot ^2(a+b x)} \log \left (1+i \sqrt [6]{\cot ^2(a+b x)}\right )+\sqrt [6]{-1} \sqrt [6]{\cot ^2(a+b x)} \log \left (1-\sqrt [6]{-1} \sqrt [6]{\cot ^2(a+b x)}\right )-\sqrt [6]{-1} \sqrt [6]{\cot ^2(a+b x)} \log \left (1+\sqrt [6]{-1} \sqrt [6]{\cot ^2(a+b x)}\right )+(-1)^{5/6} \sqrt [6]{\cot ^2(a+b x)} \log \left (1-(-1)^{5/6} \sqrt [6]{\cot ^2(a+b x)}\right )-(-1)^{5/6} \sqrt [6]{\cot ^2(a+b x)} \log \left (1+(-1)^{5/6} \sqrt [6]{\cot ^2(a+b x)}\right )}{2 b c \sqrt [3]{c \cot (a+b x)}} \]
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Time = 0.07 (sec) , antiderivative size = 215, normalized size of antiderivative = 0.88
method | result | size |
derivativedivides | \(-\frac {3 c \left (-\frac {\frac {\sqrt {3}\, \left (c^{2}\right )^{\frac {5}{6}} \ln \left (-\left (c \cot \left (b x +a \right )\right )^{\frac {2}{3}}+\sqrt {3}\, \left (c^{2}\right )^{\frac {1}{6}} \left (c \cot \left (b x +a \right )\right )^{\frac {1}{3}}-\left (c^{2}\right )^{\frac {1}{3}}\right )}{12 c^{2}}+\frac {\arctan \left (\frac {2 \left (c \cot \left (b x +a \right )\right )^{\frac {1}{3}}}{\left (c^{2}\right )^{\frac {1}{6}}}-\sqrt {3}\right )}{6 \left (c^{2}\right )^{\frac {1}{6}}}+\frac {\arctan \left (\frac {\left (c \cot \left (b x +a \right )\right )^{\frac {1}{3}}}{\left (c^{2}\right )^{\frac {1}{6}}}\right )}{3 \left (c^{2}\right )^{\frac {1}{6}}}-\frac {\sqrt {3}\, \left (c^{2}\right )^{\frac {5}{6}} \ln \left (\left (c \cot \left (b x +a \right )\right )^{\frac {2}{3}}+\sqrt {3}\, \left (c^{2}\right )^{\frac {1}{6}} \left (c \cot \left (b x +a \right )\right )^{\frac {1}{3}}+\left (c^{2}\right )^{\frac {1}{3}}\right )}{12 c^{2}}+\frac {\arctan \left (\frac {2 \left (c \cot \left (b x +a \right )\right )^{\frac {1}{3}}}{\left (c^{2}\right )^{\frac {1}{6}}}+\sqrt {3}\right )}{6 \left (c^{2}\right )^{\frac {1}{6}}}}{c^{2}}-\frac {1}{c^{2} \left (c \cot \left (b x +a \right )\right )^{\frac {1}{3}}}\right )}{b}\) | \(215\) |
default | \(-\frac {3 c \left (-\frac {\frac {\sqrt {3}\, \left (c^{2}\right )^{\frac {5}{6}} \ln \left (-\left (c \cot \left (b x +a \right )\right )^{\frac {2}{3}}+\sqrt {3}\, \left (c^{2}\right )^{\frac {1}{6}} \left (c \cot \left (b x +a \right )\right )^{\frac {1}{3}}-\left (c^{2}\right )^{\frac {1}{3}}\right )}{12 c^{2}}+\frac {\arctan \left (\frac {2 \left (c \cot \left (b x +a \right )\right )^{\frac {1}{3}}}{\left (c^{2}\right )^{\frac {1}{6}}}-\sqrt {3}\right )}{6 \left (c^{2}\right )^{\frac {1}{6}}}+\frac {\arctan \left (\frac {\left (c \cot \left (b x +a \right )\right )^{\frac {1}{3}}}{\left (c^{2}\right )^{\frac {1}{6}}}\right )}{3 \left (c^{2}\right )^{\frac {1}{6}}}-\frac {\sqrt {3}\, \left (c^{2}\right )^{\frac {5}{6}} \ln \left (\left (c \cot \left (b x +a \right )\right )^{\frac {2}{3}}+\sqrt {3}\, \left (c^{2}\right )^{\frac {1}{6}} \left (c \cot \left (b x +a \right )\right )^{\frac {1}{3}}+\left (c^{2}\right )^{\frac {1}{3}}\right )}{12 c^{2}}+\frac {\arctan \left (\frac {2 \left (c \cot \left (b x +a \right )\right )^{\frac {1}{3}}}{\left (c^{2}\right )^{\frac {1}{6}}}+\sqrt {3}\right )}{6 \left (c^{2}\right )^{\frac {1}{6}}}}{c^{2}}-\frac {1}{c^{2} \left (c \cot \left (b x +a \right )\right )^{\frac {1}{3}}}\right )}{b}\) | \(215\) |
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Leaf count of result is larger than twice the leaf count of optimal. 660 vs. \(2 (186) = 372\).
Time = 0.27 (sec) , antiderivative size = 660, normalized size of antiderivative = 2.70 \[ \int \frac {1}{(c \cot (a+b x))^{4/3}} \, dx=\frac {2 \, {\left (b c^{2} \cos \left (2 \, b x + 2 \, a\right ) + b c^{2}\right )} \left (-\frac {1}{b^{6} c^{8}}\right )^{\frac {1}{6}} \log \left (b^{5} c^{7} \left (-\frac {1}{b^{6} c^{8}}\right )^{\frac {5}{6}} + \left (\frac {c \cos \left (2 \, b x + 2 \, a\right ) + c}{\sin \left (2 \, b x + 2 \, a\right )}\right )^{\frac {1}{3}}\right ) - 2 \, {\left (b c^{2} \cos \left (2 \, b x + 2 \, a\right ) + b c^{2}\right )} \left (-\frac {1}{b^{6} c^{8}}\right )^{\frac {1}{6}} \log \left (-b^{5} c^{7} \left (-\frac {1}{b^{6} c^{8}}\right )^{\frac {5}{6}} + \left (\frac {c \cos \left (2 \, b x + 2 \, a\right ) + c}{\sin \left (2 \, b x + 2 \, a\right )}\right )^{\frac {1}{3}}\right ) - {\left (\sqrt {-3} b c^{2} - b c^{2} + {\left (\sqrt {-3} b c^{2} - b c^{2}\right )} \cos \left (2 \, b x + 2 \, a\right )\right )} \left (-\frac {1}{b^{6} c^{8}}\right )^{\frac {1}{6}} \log \left (\frac {1}{2} \, {\left (\sqrt {-3} b^{5} c^{7} + b^{5} c^{7}\right )} \left (-\frac {1}{b^{6} c^{8}}\right )^{\frac {5}{6}} + \left (\frac {c \cos \left (2 \, b x + 2 \, a\right ) + c}{\sin \left (2 \, b x + 2 \, a\right )}\right )^{\frac {1}{3}}\right ) + {\left (\sqrt {-3} b c^{2} - b c^{2} + {\left (\sqrt {-3} b c^{2} - b c^{2}\right )} \cos \left (2 \, b x + 2 \, a\right )\right )} \left (-\frac {1}{b^{6} c^{8}}\right )^{\frac {1}{6}} \log \left (-\frac {1}{2} \, {\left (\sqrt {-3} b^{5} c^{7} + b^{5} c^{7}\right )} \left (-\frac {1}{b^{6} c^{8}}\right )^{\frac {5}{6}} + \left (\frac {c \cos \left (2 \, b x + 2 \, a\right ) + c}{\sin \left (2 \, b x + 2 \, a\right )}\right )^{\frac {1}{3}}\right ) - {\left (\sqrt {-3} b c^{2} + b c^{2} + {\left (\sqrt {-3} b c^{2} + b c^{2}\right )} \cos \left (2 \, b x + 2 \, a\right )\right )} \left (-\frac {1}{b^{6} c^{8}}\right )^{\frac {1}{6}} \log \left (\frac {1}{2} \, {\left (\sqrt {-3} b^{5} c^{7} - b^{5} c^{7}\right )} \left (-\frac {1}{b^{6} c^{8}}\right )^{\frac {5}{6}} + \left (\frac {c \cos \left (2 \, b x + 2 \, a\right ) + c}{\sin \left (2 \, b x + 2 \, a\right )}\right )^{\frac {1}{3}}\right ) + {\left (\sqrt {-3} b c^{2} + b c^{2} + {\left (\sqrt {-3} b c^{2} + b c^{2}\right )} \cos \left (2 \, b x + 2 \, a\right )\right )} \left (-\frac {1}{b^{6} c^{8}}\right )^{\frac {1}{6}} \log \left (-\frac {1}{2} \, {\left (\sqrt {-3} b^{5} c^{7} - b^{5} c^{7}\right )} \left (-\frac {1}{b^{6} c^{8}}\right )^{\frac {5}{6}} + \left (\frac {c \cos \left (2 \, b x + 2 \, a\right ) + c}{\sin \left (2 \, b x + 2 \, a\right )}\right )^{\frac {1}{3}}\right ) + 12 \, \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 )}{4 \, {\left (b c^{2} \cos \left (2 \, b x + 2 \, a\right ) + b c^{2}\right )}} \]
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\[ \int \frac {1}{(c \cot (a+b x))^{4/3}} \, dx=\int \frac {1}{\left (c \cot {\left (a + b x \right )}\right )^{\frac {4}{3}}}\, dx \]
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Time = 0.32 (sec) , antiderivative size = 204, normalized size of antiderivative = 0.84 \[ \int \frac {1}{(c \cot (a+b x))^{4/3}} \, dx=-\frac {c {\left (\frac {\frac {\sqrt {3} \log \left (\sqrt {3} c^{\frac {1}{3}} \left (\frac {c}{\tan \left (b x + a\right )}\right )^{\frac {1}{3}} + c^{\frac {2}{3}} + \left (\frac {c}{\tan \left (b x + a\right )}\right )^{\frac {2}{3}}\right )}{c^{\frac {1}{3}}} - \frac {\sqrt {3} \log \left (-\sqrt {3} c^{\frac {1}{3}} \left (\frac {c}{\tan \left (b x + a\right )}\right )^{\frac {1}{3}} + c^{\frac {2}{3}} + \left (\frac {c}{\tan \left (b x + a\right )}\right )^{\frac {2}{3}}\right )}{c^{\frac {1}{3}}} - \frac {2 \, \arctan \left (\frac {\sqrt {3} c^{\frac {1}{3}} + 2 \, \left (\frac {c}{\tan \left (b x + a\right )}\right )^{\frac {1}{3}}}{c^{\frac {1}{3}}}\right )}{c^{\frac {1}{3}}} - \frac {2 \, \arctan \left (-\frac {\sqrt {3} c^{\frac {1}{3}} - 2 \, \left (\frac {c}{\tan \left (b x + a\right )}\right )^{\frac {1}{3}}}{c^{\frac {1}{3}}}\right )}{c^{\frac {1}{3}}} - \frac {4 \, \arctan \left (\frac {\left (\frac {c}{\tan \left (b x + a\right )}\right )^{\frac {1}{3}}}{c^{\frac {1}{3}}}\right )}{c^{\frac {1}{3}}}}{c^{2}} - \frac {12}{c^{2} \left (\frac {c}{\tan \left (b x + a\right )}\right )^{\frac {1}{3}}}\right )}}{4 \, b} \]
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\[ \int \frac {1}{(c \cot (a+b x))^{4/3}} \, dx=\int { \frac {1}{\left (c \cot \left (b x + a\right )\right )^{\frac {4}{3}}} \,d x } \]
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Time = 12.35 (sec) , antiderivative size = 277, normalized size of antiderivative = 1.14 \[ \int \frac {1}{(c \cot (a+b x))^{4/3}} \, dx=\frac {3}{b\,c\,{\left (c\,\mathrm {cot}\left (a+b\,x\right )\right )}^{1/3}}+\frac {{\left (-1\right )}^{1/6}\,\mathrm {atan}\left (\frac {{\left (-1\right )}^{2/3}\,{\left (c\,\mathrm {cot}\left (a+b\,x\right )\right )}^{1/3}}{c^{1/3}}\right )\,1{}\mathrm {i}}{b\,c^{4/3}}-\frac {{\left (-1\right )}^{1/6}\,\ln \left (972\,b^6\,c^{12}+972\,{\left (-1\right )}^{1/6}\,b^6\,c^{35/3}\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,{\left (c\,\mathrm {cot}\left (a+b\,x\right )\right )}^{1/3}\right )\,\left (-\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{2\,b\,c^{4/3}}-\frac {{\left (-1\right )}^{1/6}\,\ln \left (972\,b^6\,c^{12}+972\,{\left (-1\right )}^{1/6}\,b^6\,c^{35/3}\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,{\left (c\,\mathrm {cot}\left (a+b\,x\right )\right )}^{1/3}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )}{2\,b\,c^{4/3}}+\frac {{\left (-1\right )}^{1/6}\,\ln \left (972\,b^6\,c^{12}-1944\,{\left (-1\right )}^{1/6}\,b^6\,c^{35/3}\,\left (-\frac {1}{4}+\frac {\sqrt {3}\,1{}\mathrm {i}}{4}\right )\,{\left (c\,\mathrm {cot}\left (a+b\,x\right )\right )}^{1/3}\right )\,\left (-\frac {1}{4}+\frac {\sqrt {3}\,1{}\mathrm {i}}{4}\right )}{b\,c^{4/3}}+\frac {{\left (-1\right )}^{1/6}\,\ln \left (972\,b^6\,c^{12}-1944\,{\left (-1\right )}^{1/6}\,b^6\,c^{35/3}\,\left (\frac {1}{4}+\frac {\sqrt {3}\,1{}\mathrm {i}}{4}\right )\,{\left (c\,\mathrm {cot}\left (a+b\,x\right )\right )}^{1/3}\right )\,\left (\frac {1}{4}+\frac {\sqrt {3}\,1{}\mathrm {i}}{4}\right )}{b\,c^{4/3}} \]
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