Integrand size = 12, antiderivative size = 210 \[ \int (c \cot (a+b x))^{3/2} \, dx=-\frac {c^{3/2} \arctan \left (1-\frac {\sqrt {2} \sqrt {c \cot (a+b x)}}{\sqrt {c}}\right )}{\sqrt {2} b}+\frac {c^{3/2} \arctan \left (1+\frac {\sqrt {2} \sqrt {c \cot (a+b x)}}{\sqrt {c}}\right )}{\sqrt {2} b}-\frac {2 c \sqrt {c \cot (a+b x)}}{b}-\frac {c^{3/2} \log \left (\sqrt {c}+\sqrt {c} \cot (a+b x)-\sqrt {2} \sqrt {c \cot (a+b x)}\right )}{2 \sqrt {2} b}+\frac {c^{3/2} \log \left (\sqrt {c}+\sqrt {c} \cot (a+b x)+\sqrt {2} \sqrt {c \cot (a+b x)}\right )}{2 \sqrt {2} b} \]
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Time = 0.16 (sec) , antiderivative size = 210, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {3554, 3557, 335, 217, 1179, 642, 1176, 631, 210} \[ \int (c \cot (a+b x))^{3/2} \, dx=-\frac {c^{3/2} \arctan \left (1-\frac {\sqrt {2} \sqrt {c \cot (a+b x)}}{\sqrt {c}}\right )}{\sqrt {2} b}+\frac {c^{3/2} \arctan \left (\frac {\sqrt {2} \sqrt {c \cot (a+b x)}}{\sqrt {c}}+1\right )}{\sqrt {2} b}-\frac {c^{3/2} \log \left (\sqrt {c} \cot (a+b x)-\sqrt {2} \sqrt {c \cot (a+b x)}+\sqrt {c}\right )}{2 \sqrt {2} b}+\frac {c^{3/2} \log \left (\sqrt {c} \cot (a+b x)+\sqrt {2} \sqrt {c \cot (a+b x)}+\sqrt {c}\right )}{2 \sqrt {2} b}-\frac {2 c \sqrt {c \cot (a+b x)}}{b} \]
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Rule 210
Rule 217
Rule 335
Rule 631
Rule 642
Rule 1176
Rule 1179
Rule 3554
Rule 3557
Rubi steps \begin{align*} \text {integral}& = -\frac {2 c \sqrt {c \cot (a+b x)}}{b}-c^2 \int \frac {1}{\sqrt {c \cot (a+b x)}} \, dx \\ & = -\frac {2 c \sqrt {c \cot (a+b x)}}{b}+\frac {c^3 \text {Subst}\left (\int \frac {1}{\sqrt {x} \left (c^2+x^2\right )} \, dx,x,c \cot (a+b x)\right )}{b} \\ & = -\frac {2 c \sqrt {c \cot (a+b x)}}{b}+\frac {\left (2 c^3\right ) \text {Subst}\left (\int \frac {1}{c^2+x^4} \, dx,x,\sqrt {c \cot (a+b x)}\right )}{b} \\ & = -\frac {2 c \sqrt {c \cot (a+b x)}}{b}+\frac {c^2 \text {Subst}\left (\int \frac {c-x^2}{c^2+x^4} \, dx,x,\sqrt {c \cot (a+b x)}\right )}{b}+\frac {c^2 \text {Subst}\left (\int \frac {c+x^2}{c^2+x^4} \, dx,x,\sqrt {c \cot (a+b x)}\right )}{b} \\ & = -\frac {2 c \sqrt {c \cot (a+b x)}}{b}-\frac {c^{3/2} \text {Subst}\left (\int \frac {\sqrt {2} \sqrt {c}+2 x}{-c-\sqrt {2} \sqrt {c} x-x^2} \, dx,x,\sqrt {c \cot (a+b x)}\right )}{2 \sqrt {2} b}-\frac {c^{3/2} \text {Subst}\left (\int \frac {\sqrt {2} \sqrt {c}-2 x}{-c+\sqrt {2} \sqrt {c} x-x^2} \, dx,x,\sqrt {c \cot (a+b x)}\right )}{2 \sqrt {2} b}+\frac {c^2 \text {Subst}\left (\int \frac {1}{c-\sqrt {2} \sqrt {c} x+x^2} \, dx,x,\sqrt {c \cot (a+b x)}\right )}{2 b}+\frac {c^2 \text {Subst}\left (\int \frac {1}{c+\sqrt {2} \sqrt {c} x+x^2} \, dx,x,\sqrt {c \cot (a+b x)}\right )}{2 b} \\ & = -\frac {2 c \sqrt {c \cot (a+b x)}}{b}-\frac {c^{3/2} \log \left (\sqrt {c}+\sqrt {c} \cot (a+b x)-\sqrt {2} \sqrt {c \cot (a+b x)}\right )}{2 \sqrt {2} b}+\frac {c^{3/2} \log \left (\sqrt {c}+\sqrt {c} \cot (a+b x)+\sqrt {2} \sqrt {c \cot (a+b x)}\right )}{2 \sqrt {2} b}+\frac {c^{3/2} \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt {c \cot (a+b x)}}{\sqrt {c}}\right )}{\sqrt {2} b}-\frac {c^{3/2} \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt {c \cot (a+b x)}}{\sqrt {c}}\right )}{\sqrt {2} b} \\ & = -\frac {c^{3/2} \arctan \left (1-\frac {\sqrt {2} \sqrt {c \cot (a+b x)}}{\sqrt {c}}\right )}{\sqrt {2} b}+\frac {c^{3/2} \arctan \left (1+\frac {\sqrt {2} \sqrt {c \cot (a+b x)}}{\sqrt {c}}\right )}{\sqrt {2} b}-\frac {2 c \sqrt {c \cot (a+b x)}}{b}-\frac {c^{3/2} \log \left (\sqrt {c}+\sqrt {c} \cot (a+b x)-\sqrt {2} \sqrt {c \cot (a+b x)}\right )}{2 \sqrt {2} b}+\frac {c^{3/2} \log \left (\sqrt {c}+\sqrt {c} \cot (a+b x)+\sqrt {2} \sqrt {c \cot (a+b x)}\right )}{2 \sqrt {2} b} \\ \end{align*}
Time = 0.22 (sec) , antiderivative size = 161, normalized size of antiderivative = 0.77 \[ \int (c \cot (a+b x))^{3/2} \, dx=-\frac {(c \cot (a+b x))^{3/2} \left (\frac {\arctan \left (1-\sqrt {2} \sqrt {\cot (a+b x)}\right )}{\sqrt {2}}-\frac {\arctan \left (1+\sqrt {2} \sqrt {\cot (a+b x)}\right )}{\sqrt {2}}+2 \sqrt {\cot (a+b x)}+\frac {\log \left (1-\sqrt {2} \sqrt {\cot (a+b x)}+\cot (a+b x)\right )}{2 \sqrt {2}}-\frac {\log \left (1+\sqrt {2} \sqrt {\cot (a+b x)}+\cot (a+b x)\right )}{2 \sqrt {2}}\right )}{b \cot ^{\frac {3}{2}}(a+b x)} \]
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Time = 0.03 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.71
method | result | size |
derivativedivides | \(-\frac {2 c \left (\sqrt {c \cot \left (b x +a \right )}-\frac {\left (c^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {c \cot \left (b x +a \right )+\left (c^{2}\right )^{\frac {1}{4}} \sqrt {c \cot \left (b x +a \right )}\, \sqrt {2}+\sqrt {c^{2}}}{c \cot \left (b x +a \right )-\left (c^{2}\right )^{\frac {1}{4}} \sqrt {c \cot \left (b x +a \right )}\, \sqrt {2}+\sqrt {c^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {c \cot \left (b x +a \right )}}{\left (c^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {c \cot \left (b x +a \right )}}{\left (c^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8}\right )}{b}\) | \(149\) |
default | \(-\frac {2 c \left (\sqrt {c \cot \left (b x +a \right )}-\frac {\left (c^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {c \cot \left (b x +a \right )+\left (c^{2}\right )^{\frac {1}{4}} \sqrt {c \cot \left (b x +a \right )}\, \sqrt {2}+\sqrt {c^{2}}}{c \cot \left (b x +a \right )-\left (c^{2}\right )^{\frac {1}{4}} \sqrt {c \cot \left (b x +a \right )}\, \sqrt {2}+\sqrt {c^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {c \cot \left (b x +a \right )}}{\left (c^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {c \cot \left (b x +a \right )}}{\left (c^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8}\right )}{b}\) | \(149\) |
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Result contains complex when optimal does not.
Time = 0.26 (sec) , antiderivative size = 262, normalized size of antiderivative = 1.25 \[ \int (c \cot (a+b x))^{3/2} \, dx=\frac {\left (-\frac {c^{6}}{b^{4}}\right )^{\frac {1}{4}} b \log \left (c \sqrt {\frac {c \cos \left (2 \, b x + 2 \, a\right ) + c}{\sin \left (2 \, b x + 2 \, a\right )}} + \left (-\frac {c^{6}}{b^{4}}\right )^{\frac {1}{4}} b\right ) + i \, \left (-\frac {c^{6}}{b^{4}}\right )^{\frac {1}{4}} b \log \left (c \sqrt {\frac {c \cos \left (2 \, b x + 2 \, a\right ) + c}{\sin \left (2 \, b x + 2 \, a\right )}} + i \, \left (-\frac {c^{6}}{b^{4}}\right )^{\frac {1}{4}} b\right ) - i \, \left (-\frac {c^{6}}{b^{4}}\right )^{\frac {1}{4}} b \log \left (c \sqrt {\frac {c \cos \left (2 \, b x + 2 \, a\right ) + c}{\sin \left (2 \, b x + 2 \, a\right )}} - i \, \left (-\frac {c^{6}}{b^{4}}\right )^{\frac {1}{4}} b\right ) - \left (-\frac {c^{6}}{b^{4}}\right )^{\frac {1}{4}} b \log \left (c \sqrt {\frac {c \cos \left (2 \, b x + 2 \, a\right ) + c}{\sin \left (2 \, b x + 2 \, a\right )}} - \left (-\frac {c^{6}}{b^{4}}\right )^{\frac {1}{4}} b\right ) - 4 \, c \sqrt {\frac {c \cos \left (2 \, b x + 2 \, a\right ) + c}{\sin \left (2 \, b x + 2 \, a\right )}}}{2 \, b} \]
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\[ \int (c \cot (a+b x))^{3/2} \, dx=\int \left (c \cot {\left (a + b x \right )}\right )^{\frac {3}{2}}\, dx \]
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none
Time = 0.34 (sec) , antiderivative size = 179, normalized size of antiderivative = 0.85 \[ \int (c \cot (a+b x))^{3/2} \, dx=\frac {{\left (2 \, \sqrt {2} \sqrt {c} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {c} + 2 \, \sqrt {\frac {c}{\tan \left (b x + a\right )}}\right )}}{2 \, \sqrt {c}}\right ) + 2 \, \sqrt {2} \sqrt {c} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {c} - 2 \, \sqrt {\frac {c}{\tan \left (b x + a\right )}}\right )}}{2 \, \sqrt {c}}\right ) + \sqrt {2} \sqrt {c} \log \left (\sqrt {2} \sqrt {c} \sqrt {\frac {c}{\tan \left (b x + a\right )}} + c + \frac {c}{\tan \left (b x + a\right )}\right ) - \sqrt {2} \sqrt {c} \log \left (-\sqrt {2} \sqrt {c} \sqrt {\frac {c}{\tan \left (b x + a\right )}} + c + \frac {c}{\tan \left (b x + a\right )}\right ) - 8 \, \sqrt {\frac {c}{\tan \left (b x + a\right )}}\right )} c}{4 \, b} \]
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\[ \int (c \cot (a+b x))^{3/2} \, dx=\int { \left (c \cot \left (b x + a\right )\right )^{\frac {3}{2}} \,d x } \]
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Time = 12.37 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.36 \[ \int (c \cot (a+b x))^{3/2} \, dx=-\frac {2\,c\,\sqrt {c\,\mathrm {cot}\left (a+b\,x\right )}}{b}-\frac {{\left (-1\right )}^{1/4}\,c^{3/2}\,\mathrm {atan}\left (\frac {{\left (-1\right )}^{1/4}\,\sqrt {c\,\mathrm {cot}\left (a+b\,x\right )}}{\sqrt {c}}\right )\,1{}\mathrm {i}}{b}-\frac {{\left (-1\right )}^{1/4}\,c^{3/2}\,\mathrm {atanh}\left (\frac {{\left (-1\right )}^{1/4}\,\sqrt {c\,\mathrm {cot}\left (a+b\,x\right )}}{\sqrt {c}}\right )\,1{}\mathrm {i}}{b} \]
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