Integrand size = 48, antiderivative size = 57 \[ \int \frac {\csc ^2(x) \left (\sec ^2(x)-3 \tan (x) \sqrt {4 \sec ^2(x)+5 \tan ^2(x)}\right )}{\left (4 \sec ^2(x)+5 \tan ^2(x)\right )^{3/2}} \, dx=-\frac {3}{4} \log (\tan (x))+\frac {3}{8} \log \left (4+9 \tan ^2(x)\right )-\frac {\cot (x)}{4 \sqrt {4+9 \tan ^2(x)}}-\frac {7 \tan (x)}{8 \sqrt {4+9 \tan ^2(x)}} \]
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Time = 0.62 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.146, Rules used = {6874, 197, 277, 272, 36, 29, 31} \[ \int \frac {\csc ^2(x) \left (\sec ^2(x)-3 \tan (x) \sqrt {4 \sec ^2(x)+5 \tan ^2(x)}\right )}{\left (4 \sec ^2(x)+5 \tan ^2(x)\right )^{3/2}} \, dx=-\frac {7 \tan (x)}{8 \sqrt {9 \tan ^2(x)+4}}+\frac {3}{8} \log \left (9 \tan ^2(x)+4\right )-\frac {3}{4} \log (\tan (x))-\frac {\cot (x)}{4 \sqrt {9 \tan ^2(x)+4}} \]
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Rule 29
Rule 31
Rule 36
Rule 197
Rule 272
Rule 277
Rule 6874
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {1+x^2-3 x \sqrt {4+9 x^2}}{x^2 \left (4+9 x^2\right )^{3/2}} \, dx,x,\tan (x)\right ) \\ & = \text {Subst}\left (\int \left (\frac {1}{\left (4+9 x^2\right )^{3/2}}+\frac {1}{x^2 \left (4+9 x^2\right )^{3/2}}-\frac {3}{x \left (4+9 x^2\right )}\right ) \, dx,x,\tan (x)\right ) \\ & = -\left (3 \text {Subst}\left (\int \frac {1}{x \left (4+9 x^2\right )} \, dx,x,\tan (x)\right )\right )+\text {Subst}\left (\int \frac {1}{\left (4+9 x^2\right )^{3/2}} \, dx,x,\tan (x)\right )+\text {Subst}\left (\int \frac {1}{x^2 \left (4+9 x^2\right )^{3/2}} \, dx,x,\tan (x)\right ) \\ & = -\frac {\cot (x)}{4 \sqrt {4+9 \tan ^2(x)}}+\frac {\tan (x)}{4 \sqrt {4+9 \tan ^2(x)}}-\frac {3}{2} \text {Subst}\left (\int \frac {1}{x (4+9 x)} \, dx,x,\tan ^2(x)\right )-\frac {9}{2} \text {Subst}\left (\int \frac {1}{\left (4+9 x^2\right )^{3/2}} \, dx,x,\tan (x)\right ) \\ & = -\frac {\cot (x)}{4 \sqrt {4+9 \tan ^2(x)}}-\frac {7 \tan (x)}{8 \sqrt {4+9 \tan ^2(x)}}-\frac {3}{8} \text {Subst}\left (\int \frac {1}{x} \, dx,x,\tan ^2(x)\right )+\frac {27}{8} \text {Subst}\left (\int \frac {1}{4+9 x} \, dx,x,\tan ^2(x)\right ) \\ & = -\frac {3}{4} \log (\tan (x))+\frac {3}{8} \log \left (4+9 \tan ^2(x)\right )-\frac {\cot (x)}{4 \sqrt {4+9 \tan ^2(x)}}-\frac {7 \tan (x)}{8 \sqrt {4+9 \tan ^2(x)}} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(116\) vs. \(2(57)=114\).
Time = 5.35 (sec) , antiderivative size = 116, normalized size of antiderivative = 2.04 \[ \int \frac {\csc ^2(x) \left (\sec ^2(x)-3 \tan (x) \sqrt {4 \sec ^2(x)+5 \tan ^2(x)}\right )}{\left (4 \sec ^2(x)+5 \tan ^2(x)\right )^{3/2}} \, dx=\frac {5 \cot (x)+6 \sqrt {\frac {13-5 \cos (2 x)}{1+\cos (2 x)}} \log \left (1+7 \tan ^2\left (\frac {x}{2}\right )+\tan ^4\left (\frac {x}{2}\right )\right )-9 \csc (x) \sec (x)-5 \tan (x)-6 \sqrt {2} \log \left (\tan \left (\frac {x}{2}\right )\right ) \sqrt {-5+13 \sec ^2(x)+5 \tan ^2(x)}}{16 \sqrt {\frac {13-5 \cos (2 x)}{1+\cos (2 x)}}} \]
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Time = 3.32 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.09
method | result | size |
parts | \(-\frac {\left (\sec ^{3}\left (x \right )\right ) \csc \left (x \right ) \left (25 \left (\cos ^{4}\left (x \right )\right )-80 \left (\cos ^{2}\left (x \right )\right )+63\right ) \sqrt {4}}{16 {\left (-5+9 \left (\sec ^{2}\left (x \right )\right )\right )}^{\frac {3}{2}}}-\frac {3 \ln \left (\cos \left (x \right )+1\right )}{8}+\frac {3 \ln \left (5 \left (\cos ^{2}\left (x \right )\right )-9\right )}{8}-\frac {3 \ln \left (-1+\cos \left (x \right )\right )}{8}\) | \(62\) |
default | \(-\frac {6 {\left (-5+9 \left (\sec ^{2}\left (x \right )\right )\right )}^{\frac {3}{2}} \ln \left (\csc \left (x \right )-\cot \left (x \right )\right )-3 {\left (-5+9 \left (\sec ^{2}\left (x \right )\right )\right )}^{\frac {3}{2}} \ln \left (-\frac {5 \left (\cos ^{2}\left (x \right )\right )-9}{\left (\cos \left (x \right )+1\right )^{2}}\right )+25 \cot \left (x \right )-80 \sec \left (x \right ) \csc \left (x \right )+63 \left (\sec ^{3}\left (x \right )\right ) \csc \left (x \right )}{8 {\left (-5+9 \left (\sec ^{2}\left (x \right )\right )\right )}^{\frac {3}{2}}}\) | \(81\) |
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Time = 0.28 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.47 \[ \int \frac {\csc ^2(x) \left (\sec ^2(x)-3 \tan (x) \sqrt {4 \sec ^2(x)+5 \tan ^2(x)}\right )}{\left (4 \sec ^2(x)+5 \tan ^2(x)\right )^{3/2}} \, dx=\frac {3 \, {\left (5 \, \cos \left (x\right )^{2} - 9\right )} \log \left (-\frac {5}{4} \, \cos \left (x\right )^{2} + \frac {9}{4}\right ) \sin \left (x\right ) - 6 \, {\left (5 \, \cos \left (x\right )^{2} - 9\right )} \log \left (\frac {1}{2} \, \sin \left (x\right )\right ) \sin \left (x\right ) - {\left (5 \, \cos \left (x\right )^{3} - 7 \, \cos \left (x\right )\right )} \sqrt {-\frac {5 \, \cos \left (x\right )^{2} - 9}{\cos \left (x\right )^{2}}}}{8 \, {\left (5 \, \cos \left (x\right )^{2} - 9\right )} \sin \left (x\right )} \]
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\[ \int \frac {\csc ^2(x) \left (\sec ^2(x)-3 \tan (x) \sqrt {4 \sec ^2(x)+5 \tan ^2(x)}\right )}{\left (4 \sec ^2(x)+5 \tan ^2(x)\right )^{3/2}} \, dx=\int \frac {- 3 \sqrt {5 \tan ^{2}{\left (x \right )} + 4 \sec ^{2}{\left (x \right )}} \tan {\left (x \right )} + \sec ^{2}{\left (x \right )}}{\left (5 \tan ^{2}{\left (x \right )} + 4 \sec ^{2}{\left (x \right )}\right )^{\frac {3}{2}} \sin ^{2}{\left (x \right )}}\, dx \]
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Time = 0.28 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.82 \[ \int \frac {\csc ^2(x) \left (\sec ^2(x)-3 \tan (x) \sqrt {4 \sec ^2(x)+5 \tan ^2(x)}\right )}{\left (4 \sec ^2(x)+5 \tan ^2(x)\right )^{3/2}} \, dx=-\frac {7 \, \tan \left (x\right )}{8 \, \sqrt {9 \, \tan \left (x\right )^{2} + 4}} - \frac {1}{4 \, \sqrt {9 \, \tan \left (x\right )^{2} + 4} \tan \left (x\right )} + \frac {3}{8} \, \log \left (9 \, \tan \left (x\right )^{2} + 4\right ) - \frac {3}{4} \, \log \left (\tan \left (x\right )\right ) \]
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\[ \int \frac {\csc ^2(x) \left (\sec ^2(x)-3 \tan (x) \sqrt {4 \sec ^2(x)+5 \tan ^2(x)}\right )}{\left (4 \sec ^2(x)+5 \tan ^2(x)\right )^{3/2}} \, dx=\int { \frac {\sec \left (x\right )^{2} - 3 \, \sqrt {4 \, \sec \left (x\right )^{2} + 5 \, \tan \left (x\right )^{2}} \tan \left (x\right )}{{\left (4 \, \sec \left (x\right )^{2} + 5 \, \tan \left (x\right )^{2}\right )}^{\frac {3}{2}} \sin \left (x\right )^{2}} \,d x } \]
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Time = 1.61 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.98 \[ \int \frac {\csc ^2(x) \left (\sec ^2(x)-3 \tan (x) \sqrt {4 \sec ^2(x)+5 \tan ^2(x)}\right )}{\left (4 \sec ^2(x)+5 \tan ^2(x)\right )^{3/2}} \, dx=\frac {3\,\ln \left (\left (\cos \left (2\,x\right )+\sin \left (2\,x\right )\,1{}\mathrm {i}\right )\,\left (5\,\cos \left (2\,x\right )-13\right )\right )}{8}-\frac {3\,\ln \left (\cos \left (2\,x\right )\,852930{}\mathrm {i}-852930\,\sin \left (2\,x\right )-852930{}\mathrm {i}\right )}{4}-\frac {\frac {18\,\sin \left (2\,x\right )\,\sqrt {13-5\,\cos \left (2\,x\right )}}{\sqrt {\cos \left (2\,x\right )+1}}-\frac {5\,\sin \left (4\,x\right )\,\sqrt {13-5\,\cos \left (2\,x\right )}}{\sqrt {\cos \left (2\,x\right )+1}}}{80\,{\cos \left (2\,x\right )}^2-288\,\cos \left (2\,x\right )+208} \]
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