Integrand size = 21, antiderivative size = 31 \[ \int \frac {\cot (c+d x)}{\sqrt {a \cos ^2(c+d x)}} \, dx=-\frac {\text {arctanh}\left (\frac {\sqrt {a \cos ^2(c+d x)}}{\sqrt {a}}\right )}{\sqrt {a} d} \]
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Time = 0.03 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3284, 65, 212} \[ \int \frac {\cot (c+d x)}{\sqrt {a \cos ^2(c+d x)}} \, dx=-\frac {\text {arctanh}\left (\frac {\sqrt {a \cos ^2(c+d x)}}{\sqrt {a}}\right )}{\sqrt {a} d} \]
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Rule 65
Rule 212
Rule 3284
Rubi steps \begin{align*} \text {integral}& = -\frac {\text {Subst}\left (\int \frac {1}{(1-x) \sqrt {a x}} \, dx,x,\cos ^2(c+d x)\right )}{2 d} \\ & = -\frac {\text {Subst}\left (\int \frac {1}{1-\frac {x^2}{a}} \, dx,x,\sqrt {a \cos ^2(c+d x)}\right )}{a d} \\ & = -\frac {\text {arctanh}\left (\frac {\sqrt {a \cos ^2(c+d x)}}{\sqrt {a}}\right )}{\sqrt {a} d} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00 \[ \int \frac {\cot (c+d x)}{\sqrt {a \cos ^2(c+d x)}} \, dx=-\frac {\text {arctanh}\left (\frac {\sqrt {a \cos ^2(c+d x)}}{\sqrt {a}}\right )}{\sqrt {a} d} \]
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Time = 0.72 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00
method | result | size |
default | \(-\frac {\cos \left (d x +c \right ) \operatorname {arctanh}\left (\cos \left (d x +c \right )\right )}{\sqrt {a \cos \left (d x +c \right )^{2}}\, d}\) | \(31\) |
risch | \(\frac {2 \ln \left ({\mathrm e}^{i d x}-{\mathrm e}^{-i c}\right ) \cos \left (d x +c \right )}{d \sqrt {a \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{2} {\mathrm e}^{-2 i \left (d x +c \right )}}}-\frac {2 \ln \left ({\mathrm e}^{i d x}+{\mathrm e}^{-i c}\right ) \cos \left (d x +c \right )}{d \sqrt {a \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{2} {\mathrm e}^{-2 i \left (d x +c \right )}}}\) | \(104\) |
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Time = 0.27 (sec) , antiderivative size = 84, normalized size of antiderivative = 2.71 \[ \int \frac {\cot (c+d x)}{\sqrt {a \cos ^2(c+d x)}} \, dx=\left [-\frac {\sqrt {a \cos \left (d x + c\right )^{2}} \log \left (-\frac {\cos \left (d x + c\right ) + 1}{\cos \left (d x + c\right ) - 1}\right )}{2 \, a d \cos \left (d x + c\right )}, \frac {\sqrt {-a} \arctan \left (\frac {\sqrt {a \cos \left (d x + c\right )^{2}} \sqrt {-a}}{a}\right )}{a d}\right ] \]
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\[ \int \frac {\cot (c+d x)}{\sqrt {a \cos ^2(c+d x)}} \, dx=\int \frac {\cot {\left (c + d x \right )}}{\sqrt {a \cos ^{2}{\left (c + d x \right )}}}\, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. 51 vs. \(2 (25) = 50\).
Time = 0.24 (sec) , antiderivative size = 51, normalized size of antiderivative = 1.65 \[ \int \frac {\cot (c+d x)}{\sqrt {a \cos ^2(c+d x)}} \, dx=-\frac {\log \left (\frac {2 \, \sqrt {-a \sin \left (d x + c\right )^{2} + a} \sqrt {a}}{{\left | \sin \left (d x + c\right ) \right |}} + \frac {2 \, a}{{\left | \sin \left (d x + c\right ) \right |}}\right )}{\sqrt {a} d} \]
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Time = 0.28 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.87 \[ \int \frac {\cot (c+d x)}{\sqrt {a \cos ^2(c+d x)}} \, dx=\frac {\log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right )}{\sqrt {a} d \mathrm {sgn}\left (\cos \left (d x + c\right )\right )} \]
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Timed out. \[ \int \frac {\cot (c+d x)}{\sqrt {a \cos ^2(c+d x)}} \, dx=\int \frac {\mathrm {cot}\left (c+d\,x\right )}{\sqrt {a\,{\cos \left (c+d\,x\right )}^2}} \,d x \]
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