Integrand size = 17, antiderivative size = 45 \[ \int \cot ^3(x) \sqrt {a+a \tan ^2(x)} \, dx=\frac {1}{2} \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a \sec ^2(x)}}{\sqrt {a}}\right )-\frac {1}{2} \cot ^2(x) \sqrt {a \sec ^2(x)} \]
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Time = 0.11 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.294, Rules used = {3738, 4209, 44, 65, 213} \[ \int \cot ^3(x) \sqrt {a+a \tan ^2(x)} \, dx=\frac {1}{2} \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a \sec ^2(x)}}{\sqrt {a}}\right )-\frac {1}{2} \cot ^2(x) \sqrt {a \sec ^2(x)} \]
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Rule 44
Rule 65
Rule 213
Rule 3738
Rule 4209
Rubi steps \begin{align*} \text {integral}& = \int \cot ^3(x) \sqrt {a \sec ^2(x)} \, dx \\ & = \frac {1}{2} a \text {Subst}\left (\int \frac {1}{(-1+x)^2 \sqrt {a x}} \, dx,x,\sec ^2(x)\right ) \\ & = -\frac {1}{2} \cot ^2(x) \sqrt {a \sec ^2(x)}-\frac {1}{4} a \text {Subst}\left (\int \frac {1}{(-1+x) \sqrt {a x}} \, dx,x,\sec ^2(x)\right ) \\ & = -\frac {1}{2} \cot ^2(x) \sqrt {a \sec ^2(x)}-\frac {1}{2} \text {Subst}\left (\int \frac {1}{-1+\frac {x^2}{a}} \, dx,x,\sqrt {a \sec ^2(x)}\right ) \\ & = \frac {1}{2} \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a \sec ^2(x)}}{\sqrt {a}}\right )-\frac {1}{2} \cot ^2(x) \sqrt {a \sec ^2(x)} \\ \end{align*}
Time = 0.18 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.91 \[ \int \cot ^3(x) \sqrt {a+a \tan ^2(x)} \, dx=\frac {\frac {a \text {arctanh}\left (\sqrt {\cos ^2(x)}\right )}{\sqrt {\cos ^2(x)}}-a \csc ^2(x)}{2 \sqrt {a \sec ^2(x)}} \]
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Time = 0.33 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.60
method | result | size |
default | \(-\frac {\sqrt {a \sec \left (x \right )^{2}}\, \left (\ln \left (-\cot \left (x \right )+\csc \left (x \right )\right ) \cos \left (x \right )+\cot \left (x \right )^{2}\right )}{2}\) | \(27\) |
risch | \(\frac {\sqrt {\frac {a \,{\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}+1\right )^{2}}}\, \left ({\mathrm e}^{2 i x}+1\right )^{2}}{\left ({\mathrm e}^{2 i x}-1\right )^{2}}+\sqrt {\frac {a \,{\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}+1\right )^{2}}}\, \ln \left ({\mathrm e}^{i x}+1\right ) \cos \left (x \right )-\sqrt {\frac {a \,{\mathrm e}^{2 i x}}{\left ({\mathrm e}^{2 i x}+1\right )^{2}}}\, \ln \left ({\mathrm e}^{i x}-1\right ) \cos \left (x \right )\) | \(98\) |
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Time = 0.28 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.29 \[ \int \cot ^3(x) \sqrt {a+a \tan ^2(x)} \, dx=\frac {\sqrt {a} \log \left (\frac {a \tan \left (x\right )^{2} + 2 \, \sqrt {a \tan \left (x\right )^{2} + a} \sqrt {a} + 2 \, a}{\tan \left (x\right )^{2}}\right ) \tan \left (x\right )^{2} - 2 \, \sqrt {a \tan \left (x\right )^{2} + a}}{4 \, \tan \left (x\right )^{2}} \]
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\[ \int \cot ^3(x) \sqrt {a+a \tan ^2(x)} \, dx=\int \sqrt {a \left (\tan ^{2}{\left (x \right )} + 1\right )} \cot ^{3}{\left (x \right )}\, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. 303 vs. \(2 (33) = 66\).
Time = 0.40 (sec) , antiderivative size = 303, normalized size of antiderivative = 6.73 \[ \int \cot ^3(x) \sqrt {a+a \tan ^2(x)} \, dx=-\frac {{\left (4 \, {\left (\cos \left (3 \, x\right ) + \cos \left (x\right )\right )} \cos \left (4 \, x\right ) - 4 \, {\left (2 \, \cos \left (2 \, x\right ) - 1\right )} \cos \left (3 \, x\right ) - 8 \, \cos \left (2 \, x\right ) \cos \left (x\right ) - {\left (2 \, {\left (2 \, \cos \left (2 \, x\right ) - 1\right )} \cos \left (4 \, x\right ) - \cos \left (4 \, x\right )^{2} - 4 \, \cos \left (2 \, x\right )^{2} - \sin \left (4 \, x\right )^{2} + 4 \, \sin \left (4 \, x\right ) \sin \left (2 \, x\right ) - 4 \, \sin \left (2 \, x\right )^{2} + 4 \, \cos \left (2 \, x\right ) - 1\right )} \log \left (\cos \left (x\right )^{2} + \sin \left (x\right )^{2} + 2 \, \cos \left (x\right ) + 1\right ) + {\left (2 \, {\left (2 \, \cos \left (2 \, x\right ) - 1\right )} \cos \left (4 \, x\right ) - \cos \left (4 \, x\right )^{2} - 4 \, \cos \left (2 \, x\right )^{2} - \sin \left (4 \, x\right )^{2} + 4 \, \sin \left (4 \, x\right ) \sin \left (2 \, x\right ) - 4 \, \sin \left (2 \, x\right )^{2} + 4 \, \cos \left (2 \, x\right ) - 1\right )} \log \left (\cos \left (x\right )^{2} + \sin \left (x\right )^{2} - 2 \, \cos \left (x\right ) + 1\right ) + 4 \, {\left (\sin \left (3 \, x\right ) + \sin \left (x\right )\right )} \sin \left (4 \, x\right ) - 8 \, \sin \left (3 \, x\right ) \sin \left (2 \, x\right ) - 8 \, \sin \left (2 \, x\right ) \sin \left (x\right ) + 4 \, \cos \left (x\right )\right )} \sqrt {a}}{4 \, {\left (2 \, {\left (2 \, \cos \left (2 \, x\right ) - 1\right )} \cos \left (4 \, x\right ) - \cos \left (4 \, x\right )^{2} - 4 \, \cos \left (2 \, x\right )^{2} - \sin \left (4 \, x\right )^{2} + 4 \, \sin \left (4 \, x\right ) \sin \left (2 \, x\right ) - 4 \, \sin \left (2 \, x\right )^{2} + 4 \, \cos \left (2 \, x\right ) - 1\right )}} \]
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Time = 0.27 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.93 \[ \int \cot ^3(x) \sqrt {a+a \tan ^2(x)} \, dx=-\frac {a \arctan \left (\frac {\sqrt {a \tan \left (x\right )^{2} + a}}{\sqrt {-a}}\right )}{2 \, \sqrt {-a}} - \frac {\sqrt {a \tan \left (x\right )^{2} + a}}{2 \, \tan \left (x\right )^{2}} \]
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Time = 11.15 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.82 \[ \int \cot ^3(x) \sqrt {a+a \tan ^2(x)} \, dx=\frac {\sqrt {a}\,\mathrm {atanh}\left (\frac {\sqrt {a\,{\mathrm {tan}\left (x\right )}^2+a}}{\sqrt {a}}\right )}{2}-\frac {\sqrt {a\,{\mathrm {tan}\left (x\right )}^2+a}}{2\,{\mathrm {tan}\left (x\right )}^2} \]
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