Integrand size = 12, antiderivative size = 212 \[ \int \frac {1}{(c \cot (a+b x))^{3/2}} \, dx=-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {c \cot (a+b x)}}{\sqrt {c}}\right )}{\sqrt {2} b c^{3/2}}+\frac {\arctan \left (1+\frac {\sqrt {2} \sqrt {c \cot (a+b x)}}{\sqrt {c}}\right )}{\sqrt {2} b c^{3/2}}+\frac {2}{b c \sqrt {c \cot (a+b x)}}+\frac {\log \left (\sqrt {c}+\sqrt {c} \cot (a+b x)-\sqrt {2} \sqrt {c \cot (a+b x)}\right )}{2 \sqrt {2} b c^{3/2}}-\frac {\log \left (\sqrt {c}+\sqrt {c} \cot (a+b x)+\sqrt {2} \sqrt {c \cot (a+b x)}\right )}{2 \sqrt {2} b c^{3/2}} \]
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Time = 0.17 (sec) , antiderivative size = 212, 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 = {3555, 3557, 335, 303, 1176, 631, 210, 1179, 642} \[ \int \frac {1}{(c \cot (a+b x))^{3/2}} \, dx=-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {c \cot (a+b x)}}{\sqrt {c}}\right )}{\sqrt {2} b c^{3/2}}+\frac {\arctan \left (\frac {\sqrt {2} \sqrt {c \cot (a+b x)}}{\sqrt {c}}+1\right )}{\sqrt {2} b c^{3/2}}+\frac {\log \left (\sqrt {c} \cot (a+b x)-\sqrt {2} \sqrt {c \cot (a+b x)}+\sqrt {c}\right )}{2 \sqrt {2} b c^{3/2}}-\frac {\log \left (\sqrt {c} \cot (a+b x)+\sqrt {2} \sqrt {c \cot (a+b x)}+\sqrt {c}\right )}{2 \sqrt {2} b c^{3/2}}+\frac {2}{b c \sqrt {c \cot (a+b x)}} \]
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Rule 210
Rule 303
Rule 335
Rule 631
Rule 642
Rule 1176
Rule 1179
Rule 3555
Rule 3557
Rubi steps \begin{align*} \text {integral}& = \frac {2}{b c \sqrt {c \cot (a+b x)}}-\frac {\int \sqrt {c \cot (a+b x)} \, dx}{c^2} \\ & = \frac {2}{b c \sqrt {c \cot (a+b x)}}+\frac {\text {Subst}\left (\int \frac {\sqrt {x}}{c^2+x^2} \, dx,x,c \cot (a+b x)\right )}{b c} \\ & = \frac {2}{b c \sqrt {c \cot (a+b x)}}+\frac {2 \text {Subst}\left (\int \frac {x^2}{c^2+x^4} \, dx,x,\sqrt {c \cot (a+b x)}\right )}{b c} \\ & = \frac {2}{b c \sqrt {c \cot (a+b x)}}-\frac {\text {Subst}\left (\int \frac {c-x^2}{c^2+x^4} \, dx,x,\sqrt {c \cot (a+b x)}\right )}{b c}+\frac {\text {Subst}\left (\int \frac {c+x^2}{c^2+x^4} \, dx,x,\sqrt {c \cot (a+b x)}\right )}{b c} \\ & = \frac {2}{b c \sqrt {c \cot (a+b x)}}+\frac {\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 c^{3/2}}+\frac {\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 c^{3/2}}+\frac {\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 c}+\frac {\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 c} \\ & = \frac {2}{b c \sqrt {c \cot (a+b x)}}+\frac {\log \left (\sqrt {c}+\sqrt {c} \cot (a+b x)-\sqrt {2} \sqrt {c \cot (a+b x)}\right )}{2 \sqrt {2} b c^{3/2}}-\frac {\log \left (\sqrt {c}+\sqrt {c} \cot (a+b x)+\sqrt {2} \sqrt {c \cot (a+b x)}\right )}{2 \sqrt {2} b c^{3/2}}+\frac {\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 c^{3/2}}-\frac {\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 c^{3/2}} \\ & = -\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {c \cot (a+b x)}}{\sqrt {c}}\right )}{\sqrt {2} b c^{3/2}}+\frac {\arctan \left (1+\frac {\sqrt {2} \sqrt {c \cot (a+b x)}}{\sqrt {c}}\right )}{\sqrt {2} b c^{3/2}}+\frac {2}{b c \sqrt {c \cot (a+b x)}}+\frac {\log \left (\sqrt {c}+\sqrt {c} \cot (a+b x)-\sqrt {2} \sqrt {c \cot (a+b x)}\right )}{2 \sqrt {2} b c^{3/2}}-\frac {\log \left (\sqrt {c}+\sqrt {c} \cot (a+b x)+\sqrt {2} \sqrt {c \cot (a+b x)}\right )}{2 \sqrt {2} b c^{3/2}} \\ \end{align*}
Time = 0.14 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.39 \[ \int \frac {1}{(c \cot (a+b x))^{3/2}} \, dx=\frac {2+\arctan \left (\sqrt [4]{-\cot ^2(a+b x)}\right ) \sqrt [4]{-\cot ^2(a+b x)}-\text {arctanh}\left (\sqrt [4]{-\cot ^2(a+b x)}\right ) \sqrt [4]{-\cot ^2(a+b x)}}{b c \sqrt {c \cot (a+b x)}} \]
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Time = 0.03 (sec) , antiderivative size = 157, normalized size of antiderivative = 0.74
method | result | size |
derivativedivides | \(-\frac {2 c \left (-\frac {\sqrt {2}\, \left (\ln \left (\frac {c \cot \left (b x +a \right )-\left (c^{2}\right )^{\frac {1}{4}} \sqrt {c \cot \left (b x +a \right )}\, \sqrt {2}+\sqrt {c^{2}}}{c \cot \left (b x +a \right )+\left (c^{2}\right )^{\frac {1}{4}} \sqrt {c \cot \left (b x +a \right )}\, \sqrt {2}+\sqrt {c^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {c \cot \left (b x +a \right )}}{\left (c^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {c \cot \left (b x +a \right )}}{\left (c^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 c^{2} \left (c^{2}\right )^{\frac {1}{4}}}-\frac {1}{c^{2} \sqrt {c \cot \left (b x +a \right )}}\right )}{b}\) | \(157\) |
default | \(-\frac {2 c \left (-\frac {\sqrt {2}\, \left (\ln \left (\frac {c \cot \left (b x +a \right )-\left (c^{2}\right )^{\frac {1}{4}} \sqrt {c \cot \left (b x +a \right )}\, \sqrt {2}+\sqrt {c^{2}}}{c \cot \left (b x +a \right )+\left (c^{2}\right )^{\frac {1}{4}} \sqrt {c \cot \left (b x +a \right )}\, \sqrt {2}+\sqrt {c^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {c \cot \left (b x +a \right )}}{\left (c^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {c \cot \left (b x +a \right )}}{\left (c^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 c^{2} \left (c^{2}\right )^{\frac {1}{4}}}-\frac {1}{c^{2} \sqrt {c \cot \left (b x +a \right )}}\right )}{b}\) | \(157\) |
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Result contains complex when optimal does not.
Time = 0.28 (sec) , antiderivative size = 379, normalized size of antiderivative = 1.79 \[ \int \frac {1}{(c \cot (a+b x))^{3/2}} \, dx=\frac {{\left (b c^{2} \cos \left (2 \, b x + 2 \, a\right ) + b c^{2}\right )} \left (-\frac {1}{b^{4} c^{6}}\right )^{\frac {1}{4}} \log \left (b^{3} c^{5} \left (-\frac {1}{b^{4} c^{6}}\right )^{\frac {3}{4}} + \sqrt {\frac {c \cos \left (2 \, b x + 2 \, a\right ) + c}{\sin \left (2 \, b x + 2 \, a\right )}}\right ) + {\left (-i \, b c^{2} \cos \left (2 \, b x + 2 \, a\right ) - i \, b c^{2}\right )} \left (-\frac {1}{b^{4} c^{6}}\right )^{\frac {1}{4}} \log \left (i \, b^{3} c^{5} \left (-\frac {1}{b^{4} c^{6}}\right )^{\frac {3}{4}} + \sqrt {\frac {c \cos \left (2 \, b x + 2 \, a\right ) + c}{\sin \left (2 \, b x + 2 \, a\right )}}\right ) + {\left (i \, b c^{2} \cos \left (2 \, b x + 2 \, a\right ) + i \, b c^{2}\right )} \left (-\frac {1}{b^{4} c^{6}}\right )^{\frac {1}{4}} \log \left (-i \, b^{3} c^{5} \left (-\frac {1}{b^{4} c^{6}}\right )^{\frac {3}{4}} + \sqrt {\frac {c \cos \left (2 \, b x + 2 \, a\right ) + c}{\sin \left (2 \, b x + 2 \, a\right )}}\right ) - {\left (b c^{2} \cos \left (2 \, b x + 2 \, a\right ) + b c^{2}\right )} \left (-\frac {1}{b^{4} c^{6}}\right )^{\frac {1}{4}} \log \left (-b^{3} c^{5} \left (-\frac {1}{b^{4} c^{6}}\right )^{\frac {3}{4}} + \sqrt {\frac {c \cos \left (2 \, b x + 2 \, a\right ) + c}{\sin \left (2 \, b x + 2 \, a\right )}}\right ) + 4 \, \sqrt {\frac {c \cos \left (2 \, b x + 2 \, a\right ) + c}{\sin \left (2 \, b x + 2 \, a\right )}} \sin \left (2 \, b x + 2 \, a\right )}{2 \, {\left (b c^{2} \cos \left (2 \, b x + 2 \, a\right ) + b c^{2}\right )}} \]
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\[ \int \frac {1}{(c \cot (a+b x))^{3/2}} \, dx=\int \frac {1}{\left (c \cot {\left (a + b x \right )}\right )^{\frac {3}{2}}}\, dx \]
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none
Time = 0.31 (sec) , antiderivative size = 187, normalized size of antiderivative = 0.88 \[ \int \frac {1}{(c \cot (a+b x))^{3/2}} \, dx=\frac {c {\left (\frac {\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {c} + 2 \, \sqrt {\frac {c}{\tan \left (b x + a\right )}}\right )}}{2 \, \sqrt {c}}\right )}{\sqrt {c}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {c} - 2 \, \sqrt {\frac {c}{\tan \left (b x + a\right )}}\right )}}{2 \, \sqrt {c}}\right )}{\sqrt {c}} - \frac {\sqrt {2} \log \left (\sqrt {2} \sqrt {c} \sqrt {\frac {c}{\tan \left (b x + a\right )}} + c + \frac {c}{\tan \left (b x + a\right )}\right )}{\sqrt {c}} + \frac {\sqrt {2} \log \left (-\sqrt {2} \sqrt {c} \sqrt {\frac {c}{\tan \left (b x + a\right )}} + c + \frac {c}{\tan \left (b x + a\right )}\right )}{\sqrt {c}}}{c^{2}} + \frac {8}{c^{2} \sqrt {\frac {c}{\tan \left (b x + a\right )}}}\right )}}{4 \, b} \]
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\[ \int \frac {1}{(c \cot (a+b x))^{3/2}} \, dx=\int { \frac {1}{\left (c \cot \left (b x + a\right )\right )^{\frac {3}{2}}} \,d x } \]
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Time = 12.02 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.36 \[ \int \frac {1}{(c \cot (a+b x))^{3/2}} \, dx=\frac {2}{b\,c\,\sqrt {c\,\mathrm {cot}\left (a+b\,x\right )}}+\frac {{\left (-1\right )}^{1/4}\,\mathrm {atan}\left (\frac {{\left (-1\right )}^{1/4}\,\sqrt {c\,\mathrm {cot}\left (a+b\,x\right )}}{\sqrt {c}}\right )}{b\,c^{3/2}}-\frac {{\left (-1\right )}^{1/4}\,\mathrm {atanh}\left (\frac {{\left (-1\right )}^{1/4}\,\sqrt {c\,\mathrm {cot}\left (a+b\,x\right )}}{\sqrt {c}}\right )}{b\,c^{3/2}} \]
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