Integrand size = 15, antiderivative size = 70 \[ \int \frac {\cot (x)}{\sqrt {a+b \tan ^4(x)}} \, dx=\frac {\text {arctanh}\left (\frac {a-b \tan ^2(x)}{\sqrt {a+b} \sqrt {a+b \tan ^4(x)}}\right )}{2 \sqrt {a+b}}-\frac {\text {arctanh}\left (\frac {\sqrt {a+b \tan ^4(x)}}{\sqrt {a}}\right )}{2 \sqrt {a}} \]
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Time = 0.16 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.533, Rules used = {3751, 1266, 975, 739, 212, 272, 65, 214} \[ \int \frac {\cot (x)}{\sqrt {a+b \tan ^4(x)}} \, dx=\frac {\text {arctanh}\left (\frac {a-b \tan ^2(x)}{\sqrt {a+b} \sqrt {a+b \tan ^4(x)}}\right )}{2 \sqrt {a+b}}-\frac {\text {arctanh}\left (\frac {\sqrt {a+b \tan ^4(x)}}{\sqrt {a}}\right )}{2 \sqrt {a}} \]
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Rule 65
Rule 212
Rule 214
Rule 272
Rule 739
Rule 975
Rule 1266
Rule 3751
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {1}{x \left (1+x^2\right ) \sqrt {a+b x^4}} \, dx,x,\tan (x)\right ) \\ & = \frac {1}{2} \text {Subst}\left (\int \frac {1}{x (1+x) \sqrt {a+b x^2}} \, dx,x,\tan ^2(x)\right ) \\ & = \frac {1}{2} \text {Subst}\left (\int \left (\frac {1}{(-1-x) \sqrt {a+b x^2}}+\frac {1}{x \sqrt {a+b x^2}}\right ) \, dx,x,\tan ^2(x)\right ) \\ & = \frac {1}{2} \text {Subst}\left (\int \frac {1}{(-1-x) \sqrt {a+b x^2}} \, dx,x,\tan ^2(x)\right )+\frac {1}{2} \text {Subst}\left (\int \frac {1}{x \sqrt {a+b x^2}} \, dx,x,\tan ^2(x)\right ) \\ & = \frac {1}{4} \text {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,\tan ^4(x)\right )-\frac {1}{2} \text {Subst}\left (\int \frac {1}{a+b-x^2} \, dx,x,\frac {-a+b \tan ^2(x)}{\sqrt {a+b \tan ^4(x)}}\right ) \\ & = \frac {\text {arctanh}\left (\frac {a-b \tan ^2(x)}{\sqrt {a+b} \sqrt {a+b \tan ^4(x)}}\right )}{2 \sqrt {a+b}}+\frac {\text {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \tan ^4(x)}\right )}{2 b} \\ & = \frac {\text {arctanh}\left (\frac {a-b \tan ^2(x)}{\sqrt {a+b} \sqrt {a+b \tan ^4(x)}}\right )}{2 \sqrt {a+b}}-\frac {\text {arctanh}\left (\frac {\sqrt {a+b \tan ^4(x)}}{\sqrt {a}}\right )}{2 \sqrt {a}} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.00 \[ \int \frac {\cot (x)}{\sqrt {a+b \tan ^4(x)}} \, dx=\frac {\text {arctanh}\left (\frac {a-b \tan ^2(x)}{\sqrt {a+b} \sqrt {a+b \tan ^4(x)}}\right )}{2 \sqrt {a+b}}-\frac {\text {arctanh}\left (\frac {\sqrt {a+b \tan ^4(x)}}{\sqrt {a}}\right )}{2 \sqrt {a}} \]
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\[\int \frac {\cot \left (x \right )}{\sqrt {a +b \tan \left (x \right )^{4}}}d x\]
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none
Time = 0.37 (sec) , antiderivative size = 475, normalized size of antiderivative = 6.79 \[ \int \frac {\cot (x)}{\sqrt {a+b \tan ^4(x)}} \, dx=\left [\frac {\sqrt {a + b} a \log \left (\frac {{\left (a b + 2 \, b^{2}\right )} \tan \left (x\right )^{4} - 2 \, a b \tan \left (x\right )^{2} - 2 \, \sqrt {b \tan \left (x\right )^{4} + a} {\left (b \tan \left (x\right )^{2} - a\right )} \sqrt {a + b} + 2 \, a^{2} + a b}{\tan \left (x\right )^{4} + 2 \, \tan \left (x\right )^{2} + 1}\right ) + {\left (a + b\right )} \sqrt {a} \log \left (-\frac {b \tan \left (x\right )^{4} - 2 \, \sqrt {b \tan \left (x\right )^{4} + a} \sqrt {a} + 2 \, a}{\tan \left (x\right )^{4}}\right )}{4 \, {\left (a^{2} + a b\right )}}, \frac {2 \, \sqrt {-a} {\left (a + b\right )} \arctan \left (\frac {\sqrt {b \tan \left (x\right )^{4} + a} \sqrt {-a}}{a}\right ) + \sqrt {a + b} a \log \left (\frac {{\left (a b + 2 \, b^{2}\right )} \tan \left (x\right )^{4} - 2 \, a b \tan \left (x\right )^{2} - 2 \, \sqrt {b \tan \left (x\right )^{4} + a} {\left (b \tan \left (x\right )^{2} - a\right )} \sqrt {a + b} + 2 \, a^{2} + a b}{\tan \left (x\right )^{4} + 2 \, \tan \left (x\right )^{2} + 1}\right )}{4 \, {\left (a^{2} + a b\right )}}, \frac {2 \, a \sqrt {-a - b} \arctan \left (\frac {\sqrt {b \tan \left (x\right )^{4} + a} {\left (b \tan \left (x\right )^{2} - a\right )} \sqrt {-a - b}}{{\left (a b + b^{2}\right )} \tan \left (x\right )^{4} + a^{2} + a b}\right ) + {\left (a + b\right )} \sqrt {a} \log \left (-\frac {b \tan \left (x\right )^{4} - 2 \, \sqrt {b \tan \left (x\right )^{4} + a} \sqrt {a} + 2 \, a}{\tan \left (x\right )^{4}}\right )}{4 \, {\left (a^{2} + a b\right )}}, \frac {a \sqrt {-a - b} \arctan \left (\frac {\sqrt {b \tan \left (x\right )^{4} + a} {\left (b \tan \left (x\right )^{2} - a\right )} \sqrt {-a - b}}{{\left (a b + b^{2}\right )} \tan \left (x\right )^{4} + a^{2} + a b}\right ) + \sqrt {-a} {\left (a + b\right )} \arctan \left (\frac {\sqrt {b \tan \left (x\right )^{4} + a} \sqrt {-a}}{a}\right )}{2 \, {\left (a^{2} + a b\right )}}\right ] \]
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\[ \int \frac {\cot (x)}{\sqrt {a+b \tan ^4(x)}} \, dx=\int \frac {\cot {\left (x \right )}}{\sqrt {a + b \tan ^{4}{\left (x \right )}}}\, dx \]
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\[ \int \frac {\cot (x)}{\sqrt {a+b \tan ^4(x)}} \, dx=\int { \frac {\cot \left (x\right )}{\sqrt {b \tan \left (x\right )^{4} + a}} \,d x } \]
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Exception generated. \[ \int \frac {\cot (x)}{\sqrt {a+b \tan ^4(x)}} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {\cot (x)}{\sqrt {a+b \tan ^4(x)}} \, dx=\int \frac {\mathrm {cot}\left (x\right )}{\sqrt {b\,{\mathrm {tan}\left (x\right )}^4+a}} \,d x \]
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