Integrand size = 12, antiderivative size = 192 \[ \int \frac {1}{\sqrt {c \cot (a+b x)}} \, dx=\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {c \cot (a+b x)}}{\sqrt {c}}\right )}{\sqrt {2} b \sqrt {c}}-\frac {\arctan \left (1+\frac {\sqrt {2} \sqrt {c \cot (a+b x)}}{\sqrt {c}}\right )}{\sqrt {2} b \sqrt {c}}+\frac {\log \left (\sqrt {c}+\sqrt {c} \cot (a+b x)-\sqrt {2} \sqrt {c \cot (a+b x)}\right )}{2 \sqrt {2} b \sqrt {c}}-\frac {\log \left (\sqrt {c}+\sqrt {c} \cot (a+b x)+\sqrt {2} \sqrt {c \cot (a+b x)}\right )}{2 \sqrt {2} b \sqrt {c}} \]
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Time = 0.13 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {3557, 335, 217, 1179, 642, 1176, 631, 210} \[ \int \frac {1}{\sqrt {c \cot (a+b x)}} \, dx=\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {c \cot (a+b x)}}{\sqrt {c}}\right )}{\sqrt {2} b \sqrt {c}}-\frac {\arctan \left (\frac {\sqrt {2} \sqrt {c \cot (a+b x)}}{\sqrt {c}}+1\right )}{\sqrt {2} b \sqrt {c}}+\frac {\log \left (\sqrt {c} \cot (a+b x)-\sqrt {2} \sqrt {c \cot (a+b x)}+\sqrt {c}\right )}{2 \sqrt {2} b \sqrt {c}}-\frac {\log \left (\sqrt {c} \cot (a+b x)+\sqrt {2} \sqrt {c \cot (a+b x)}+\sqrt {c}\right )}{2 \sqrt {2} b \sqrt {c}} \]
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Rule 210
Rule 217
Rule 335
Rule 631
Rule 642
Rule 1176
Rule 1179
Rule 3557
Rubi steps \begin{align*} \text {integral}& = -\frac {c \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) \text {Subst}\left (\int \frac {1}{c^2+x^4} \, dx,x,\sqrt {c \cot (a+b x)}\right )}{b} \\ & = -\frac {\text {Subst}\left (\int \frac {c-x^2}{c^2+x^4} \, dx,x,\sqrt {c \cot (a+b x)}\right )}{b}-\frac {\text {Subst}\left (\int \frac {c+x^2}{c^2+x^4} \, dx,x,\sqrt {c \cot (a+b x)}\right )}{b} \\ & = -\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}-\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}+\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 \sqrt {c}}+\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 \sqrt {c}} \\ & = \frac {\log \left (\sqrt {c}+\sqrt {c} \cot (a+b x)-\sqrt {2} \sqrt {c \cot (a+b x)}\right )}{2 \sqrt {2} b \sqrt {c}}-\frac {\log \left (\sqrt {c}+\sqrt {c} \cot (a+b x)+\sqrt {2} \sqrt {c \cot (a+b x)}\right )}{2 \sqrt {2} b \sqrt {c}}-\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 \sqrt {c}}+\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 \sqrt {c}} \\ & = \frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {c \cot (a+b x)}}{\sqrt {c}}\right )}{\sqrt {2} b \sqrt {c}}-\frac {\arctan \left (1+\frac {\sqrt {2} \sqrt {c \cot (a+b x)}}{\sqrt {c}}\right )}{\sqrt {2} b \sqrt {c}}+\frac {\log \left (\sqrt {c}+\sqrt {c} \cot (a+b x)-\sqrt {2} \sqrt {c \cot (a+b x)}\right )}{2 \sqrt {2} b \sqrt {c}}-\frac {\log \left (\sqrt {c}+\sqrt {c} \cot (a+b x)+\sqrt {2} \sqrt {c \cot (a+b x)}\right )}{2 \sqrt {2} b \sqrt {c}} \\ \end{align*}
Time = 0.11 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.68 \[ \int \frac {1}{\sqrt {c \cot (a+b x)}} \, dx=\frac {\sqrt {\cot (a+b x)} \left (2 \arctan \left (1-\sqrt {2} \sqrt {\cot (a+b x)}\right )-2 \arctan \left (1+\sqrt {2} \sqrt {\cot (a+b x)}\right )+\log \left (1-\sqrt {2} \sqrt {\cot (a+b x)}+\cot (a+b x)\right )-\log \left (1+\sqrt {2} \sqrt {\cot (a+b x)}+\cot (a+b x)\right )\right )}{2 \sqrt {2} b \sqrt {c \cot (a+b x)}} \]
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Time = 0.06 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.72
method | result | size |
derivativedivides | \(-\frac {\left (c^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {c \cot \left (b x +a \right )+\left (c^{2}\right )^{\frac {1}{4}} \sqrt {c \cot \left (b x +a \right )}\, \sqrt {2}+\sqrt {c^{2}}}{c \cot \left (b x +a \right )-\left (c^{2}\right )^{\frac {1}{4}} \sqrt {c \cot \left (b x +a \right )}\, \sqrt {2}+\sqrt {c^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {c \cot \left (b x +a \right )}}{\left (c^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {c \cot \left (b x +a \right )}}{\left (c^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{4 b c}\) | \(138\) |
default | \(-\frac {\left (c^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {c \cot \left (b x +a \right )+\left (c^{2}\right )^{\frac {1}{4}} \sqrt {c \cot \left (b x +a \right )}\, \sqrt {2}+\sqrt {c^{2}}}{c \cot \left (b x +a \right )-\left (c^{2}\right )^{\frac {1}{4}} \sqrt {c \cot \left (b x +a \right )}\, \sqrt {2}+\sqrt {c^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {c \cot \left (b x +a \right )}}{\left (c^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {c \cot \left (b x +a \right )}}{\left (c^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{4 b c}\) | \(138\) |
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Result contains complex when optimal does not.
Time = 0.27 (sec) , antiderivative size = 220, normalized size of antiderivative = 1.15 \[ \int \frac {1}{\sqrt {c \cot (a+b x)}} \, dx=-\frac {1}{2} \, \left (-\frac {1}{b^{4} c^{2}}\right )^{\frac {1}{4}} \log \left (b c \left (-\frac {1}{b^{4} c^{2}}\right )^{\frac {1}{4}} + \sqrt {\frac {c \cos \left (2 \, b x + 2 \, a\right ) + c}{\sin \left (2 \, b x + 2 \, a\right )}}\right ) - \frac {1}{2} i \, \left (-\frac {1}{b^{4} c^{2}}\right )^{\frac {1}{4}} \log \left (i \, b c \left (-\frac {1}{b^{4} c^{2}}\right )^{\frac {1}{4}} + \sqrt {\frac {c \cos \left (2 \, b x + 2 \, a\right ) + c}{\sin \left (2 \, b x + 2 \, a\right )}}\right ) + \frac {1}{2} i \, \left (-\frac {1}{b^{4} c^{2}}\right )^{\frac {1}{4}} \log \left (-i \, b c \left (-\frac {1}{b^{4} c^{2}}\right )^{\frac {1}{4}} + \sqrt {\frac {c \cos \left (2 \, b x + 2 \, a\right ) + c}{\sin \left (2 \, b x + 2 \, a\right )}}\right ) + \frac {1}{2} \, \left (-\frac {1}{b^{4} c^{2}}\right )^{\frac {1}{4}} \log \left (-b c \left (-\frac {1}{b^{4} c^{2}}\right )^{\frac {1}{4}} + \sqrt {\frac {c \cos \left (2 \, b x + 2 \, a\right ) + c}{\sin \left (2 \, b x + 2 \, a\right )}}\right ) \]
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\[ \int \frac {1}{\sqrt {c \cot (a+b x)}} \, dx=\int \frac {1}{\sqrt {c \cot {\left (a + b x \right )}}}\, dx \]
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none
Time = 0.31 (sec) , antiderivative size = 165, normalized size of antiderivative = 0.86 \[ \int \frac {1}{\sqrt {c \cot (a+b x)}} \, dx=-\frac {c {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {c} + 2 \, \sqrt {\frac {c}{\tan \left (b x + a\right )}}\right )}}{2 \, \sqrt {c}}\right )}{c^{\frac {3}{2}}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {c} - 2 \, \sqrt {\frac {c}{\tan \left (b x + a\right )}}\right )}}{2 \, \sqrt {c}}\right )}{c^{\frac {3}{2}}} + \frac {\sqrt {2} \log \left (\sqrt {2} \sqrt {c} \sqrt {\frac {c}{\tan \left (b x + a\right )}} + c + \frac {c}{\tan \left (b x + a\right )}\right )}{c^{\frac {3}{2}}} - \frac {\sqrt {2} \log \left (-\sqrt {2} \sqrt {c} \sqrt {\frac {c}{\tan \left (b x + a\right )}} + c + \frac {c}{\tan \left (b x + a\right )}\right )}{c^{\frac {3}{2}}}\right )}}{4 \, b} \]
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\[ \int \frac {1}{\sqrt {c \cot (a+b x)}} \, dx=\int { \frac {1}{\sqrt {c \cot \left (b x + a\right )}} \,d x } \]
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Time = 12.21 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.30 \[ \int \frac {1}{\sqrt {c \cot (a+b x)}} \, dx=\frac {{\left (-1\right )}^{1/4}\,\mathrm {atan}\left (\frac {{\left (-1\right )}^{1/4}\,\sqrt {c\,\mathrm {cot}\left (a+b\,x\right )}}{\sqrt {c}}\right )\,1{}\mathrm {i}}{b\,\sqrt {c}}+\frac {{\left (-1\right )}^{1/4}\,\mathrm {atanh}\left (\frac {{\left (-1\right )}^{1/4}\,\sqrt {c\,\mathrm {cot}\left (a+b\,x\right )}}{\sqrt {c}}\right )\,1{}\mathrm {i}}{b\,\sqrt {c}} \]
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