Integrand size = 21, antiderivative size = 92 \[ \int \frac {\cot (c+d x)}{\sqrt {a+a \sec (c+d x)}} \, dx=\frac {2 \text {arctanh}\left (\frac {\sqrt {a+a \sec (c+d x)}}{\sqrt {a}}\right )}{\sqrt {a} d}-\frac {\text {arctanh}\left (\frac {\sqrt {a+a \sec (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{\sqrt {2} \sqrt {a} d}-\frac {1}{d \sqrt {a+a \sec (c+d x)}} \]
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Time = 0.12 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {3965, 87, 162, 65, 213} \[ \int \frac {\cot (c+d x)}{\sqrt {a+a \sec (c+d x)}} \, dx=\frac {2 \text {arctanh}\left (\frac {\sqrt {a \sec (c+d x)+a}}{\sqrt {a}}\right )}{\sqrt {a} d}-\frac {\text {arctanh}\left (\frac {\sqrt {a \sec (c+d x)+a}}{\sqrt {2} \sqrt {a}}\right )}{\sqrt {2} \sqrt {a} d}-\frac {1}{d \sqrt {a \sec (c+d x)+a}} \]
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Rule 65
Rule 87
Rule 162
Rule 213
Rule 3965
Rubi steps \begin{align*} \text {integral}& = \frac {a^2 \text {Subst}\left (\int \frac {1}{x (-a+a x) (a+a x)^{3/2}} \, dx,x,\sec (c+d x)\right )}{d} \\ & = -\frac {1}{d \sqrt {a+a \sec (c+d x)}}+\frac {\text {Subst}\left (\int \frac {2 a^2-a^2 x}{x (-a+a x) \sqrt {a+a x}} \, dx,x,\sec (c+d x)\right )}{2 a d} \\ & = -\frac {1}{d \sqrt {a+a \sec (c+d x)}}-\frac {\text {Subst}\left (\int \frac {1}{x \sqrt {a+a x}} \, dx,x,\sec (c+d x)\right )}{d}+\frac {a \text {Subst}\left (\int \frac {1}{(-a+a x) \sqrt {a+a x}} \, dx,x,\sec (c+d x)\right )}{2 d} \\ & = -\frac {1}{d \sqrt {a+a \sec (c+d x)}}+\frac {\text {Subst}\left (\int \frac {1}{-2 a+x^2} \, dx,x,\sqrt {a+a \sec (c+d x)}\right )}{d}-\frac {2 \text {Subst}\left (\int \frac {1}{-1+\frac {x^2}{a}} \, dx,x,\sqrt {a+a \sec (c+d x)}\right )}{a d} \\ & = \frac {2 \text {arctanh}\left (\frac {\sqrt {a+a \sec (c+d x)}}{\sqrt {a}}\right )}{\sqrt {a} d}-\frac {\text {arctanh}\left (\frac {\sqrt {a+a \sec (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{\sqrt {2} \sqrt {a} d}-\frac {1}{d \sqrt {a+a \sec (c+d x)}} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.06 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.62 \[ \int \frac {\cot (c+d x)}{\sqrt {a+a \sec (c+d x)}} \, dx=\frac {\operatorname {Hypergeometric2F1}\left (-\frac {1}{2},1,\frac {1}{2},\frac {1}{2} (1+\sec (c+d x))\right )-2 \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},1,\frac {1}{2},1+\sec (c+d x)\right )}{d \sqrt {a (1+\sec (c+d x))}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(228\) vs. \(2(75)=150\).
Time = 1.80 (sec) , antiderivative size = 229, normalized size of antiderivative = 2.49
method | result | size |
default | \(-\frac {\sqrt {-\frac {2 a}{\left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )^{2}-1}}\, \sqrt {\left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )^{2}-1}\, \left (-\left (\left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )^{2}-1\right )^{\frac {3}{2}}+\left (1-\cos \left (d x +c \right )\right )^{2} \sqrt {\left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )^{2}-1}\, \csc \left (d x +c \right )^{2}+6 \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {\left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )^{2}-1}}{2}\right )+3 \arctan \left (\frac {1}{\sqrt {\left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )^{2}-1}}\right )-4 \sqrt {\left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )^{2}-1}\right )}{6 d a}\) | \(229\) |
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Leaf count of result is larger than twice the leaf count of optimal. 180 vs. \(2 (75) = 150\).
Time = 0.35 (sec) , antiderivative size = 384, normalized size of antiderivative = 4.17 \[ \int \frac {\cot (c+d x)}{\sqrt {a+a \sec (c+d x)}} \, dx=\left [\frac {2 \, \sqrt {a} {\left (\cos \left (d x + c\right ) + 1\right )} \log \left (-8 \, a \cos \left (d x + c\right )^{2} - 4 \, {\left (2 \, \cos \left (d x + c\right )^{2} + \cos \left (d x + c\right )\right )} \sqrt {a} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} - 8 \, a \cos \left (d x + c\right ) - a\right ) + \frac {\sqrt {2} {\left (a \cos \left (d x + c\right ) + a\right )} \log \left (-\frac {\frac {2 \, \sqrt {2} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \cos \left (d x + c\right )}{\sqrt {a}} - 3 \, \cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) - 1}\right )}{\sqrt {a}} - 4 \, \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \cos \left (d x + c\right )}{4 \, {\left (a d \cos \left (d x + c\right ) + a d\right )}}, \frac {\sqrt {2} {\left (a \cos \left (d x + c\right ) + a\right )} \sqrt {-\frac {1}{a}} \arctan \left (\frac {\sqrt {2} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \sqrt {-\frac {1}{a}} \cos \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right ) - 2 \, \sqrt {-a} {\left (\cos \left (d x + c\right ) + 1\right )} \arctan \left (\frac {2 \, \sqrt {-a} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \cos \left (d x + c\right )}{2 \, a \cos \left (d x + c\right ) + a}\right ) - 2 \, \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \cos \left (d x + c\right )}{2 \, {\left (a d \cos \left (d x + c\right ) + a d\right )}}\right ] \]
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\[ \int \frac {\cot (c+d x)}{\sqrt {a+a \sec (c+d x)}} \, dx=\int \frac {\cot {\left (c + d x \right )}}{\sqrt {a \left (\sec {\left (c + d x \right )} + 1\right )}}\, dx \]
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\[ \int \frac {\cot (c+d x)}{\sqrt {a+a \sec (c+d x)}} \, dx=\int { \frac {\cot \left (d x + c\right )}{\sqrt {a \sec \left (d x + c\right ) + a}} \,d x } \]
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none
Time = 0.75 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.21 \[ \int \frac {\cot (c+d x)}{\sqrt {a+a \sec (c+d x)}} \, dx=-\frac {\sqrt {2} {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a}}{2 \, \sqrt {-a}}\right )}{\sqrt {-a}} - \frac {\arctan \left (\frac {\sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a}}{\sqrt {-a}}\right )}{\sqrt {-a}} + \frac {\sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a}}{a}\right )}}{2 \, d \mathrm {sgn}\left (\cos \left (d x + c\right )\right )} \]
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Timed out. \[ \int \frac {\cot (c+d x)}{\sqrt {a+a \sec (c+d x)}} \, dx=\int \frac {\mathrm {cot}\left (c+d\,x\right )}{\sqrt {a+\frac {a}{\cos \left (c+d\,x\right )}}} \,d x \]
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