Integrand size = 20, antiderivative size = 54 \[ \int \frac {\cot (x)}{\sqrt [4]{a^4-b^4 \csc ^2(x)}} \, dx=-\frac {\arctan \left (\frac {\sqrt [4]{a^4-b^4 \csc ^2(x)}}{a}\right )}{a}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{a^4-b^4 \csc ^2(x)}}{a}\right )}{a} \]
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Time = 0.07 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {4224, 272, 65, 304, 209, 212} \[ \int \frac {\cot (x)}{\sqrt [4]{a^4-b^4 \csc ^2(x)}} \, dx=\frac {\text {arctanh}\left (\frac {\sqrt [4]{a^4-b^4 \csc ^2(x)}}{a}\right )}{a}-\frac {\arctan \left (\frac {\sqrt [4]{a^4-b^4 \csc ^2(x)}}{a}\right )}{a} \]
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Rule 65
Rule 209
Rule 212
Rule 272
Rule 304
Rule 4224
Rubi steps \begin{align*} \text {integral}& = -\text {Subst}\left (\int \frac {1}{x \sqrt [4]{a^4-b^4 x^2}} \, dx,x,\csc (x)\right ) \\ & = -\left (\frac {1}{2} \text {Subst}\left (\int \frac {1}{x \sqrt [4]{a^4-b^4 x}} \, dx,x,\csc ^2(x)\right )\right ) \\ & = \frac {2 \text {Subst}\left (\int \frac {x^2}{\frac {a^4}{b^4}-\frac {x^4}{b^4}} \, dx,x,\sqrt [4]{a^4-b^4 \csc ^2(x)}\right )}{b^4} \\ & = \text {Subst}\left (\int \frac {1}{a^2-x^2} \, dx,x,\sqrt [4]{a^4-b^4 \csc ^2(x)}\right )-\text {Subst}\left (\int \frac {1}{a^2+x^2} \, dx,x,\sqrt [4]{a^4-b^4 \csc ^2(x)}\right ) \\ & = -\frac {\arctan \left (\frac {\sqrt [4]{a^4-b^4 \csc ^2(x)}}{a}\right )}{a}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{a^4-b^4 \csc ^2(x)}}{a}\right )}{a} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(245\) vs. \(2(54)=108\).
Time = 0.30 (sec) , antiderivative size = 245, normalized size of antiderivative = 4.54 \[ \int \frac {\cot (x)}{\sqrt [4]{a^4-b^4 \csc ^2(x)}} \, dx=\frac {\sqrt [4]{-a^4+2 b^4+a^4 \cos (2 x)} \left (-2 \arctan \left (1-\frac {\sqrt {2} a \sqrt {\sin (x)}}{\sqrt [4]{b^4-a^4 \sin ^2(x)}}\right )+2 \arctan \left (1+\frac {\sqrt {2} a \sqrt {\sin (x)}}{\sqrt [4]{b^4-a^4 \sin ^2(x)}}\right )-\log \left (1+\frac {a^2 \sin (x)}{\sqrt {b^4-a^4 \sin ^2(x)}}-\frac {\sqrt {2} a \sqrt {\sin (x)}}{\sqrt [4]{b^4-a^4 \sin ^2(x)}}\right )+\log \left (1+\frac {a^2 \sin (x)}{\sqrt {b^4-a^4 \sin ^2(x)}}+\frac {\sqrt {2} a \sqrt {\sin (x)}}{\sqrt [4]{b^4-a^4 \sin ^2(x)}}\right )\right )}{2\ 2^{3/4} a \sqrt [4]{a^4-b^4 \csc ^2(x)} \sqrt {\sin (x)}} \]
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\[\int \frac {\cot \left (x \right )}{\left (a^{4}-b^{4} \left (\csc ^{2}\left (x \right )\right )\right )^{\frac {1}{4}}}d x\]
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Timed out. \[ \int \frac {\cot (x)}{\sqrt [4]{a^4-b^4 \csc ^2(x)}} \, dx=\text {Timed out} \]
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\[ \int \frac {\cot (x)}{\sqrt [4]{a^4-b^4 \csc ^2(x)}} \, dx=\int \frac {\cot {\left (x \right )}}{\sqrt [4]{\left (a^{2} - b^{2} \csc {\left (x \right )}\right ) \left (a^{2} + b^{2} \csc {\left (x \right )}\right )}}\, dx \]
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Time = 0.29 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.37 \[ \int \frac {\cot (x)}{\sqrt [4]{a^4-b^4 \csc ^2(x)}} \, dx=-\frac {\arctan \left (\frac {{\left (a^{4} - \frac {b^{4}}{\sin \left (x\right )^{2}}\right )}^{\frac {1}{4}}}{a}\right )}{a} + \frac {\log \left (a + {\left (a^{4} - \frac {b^{4}}{\sin \left (x\right )^{2}}\right )}^{\frac {1}{4}}\right )}{2 \, a} - \frac {\log \left (-a + {\left (a^{4} - \frac {b^{4}}{\sin \left (x\right )^{2}}\right )}^{\frac {1}{4}}\right )}{2 \, a} \]
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Time = 0.41 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.41 \[ \int \frac {\cot (x)}{\sqrt [4]{a^4-b^4 \csc ^2(x)}} \, dx=-\frac {\arctan \left (\frac {{\left (a^{4} - \frac {b^{4}}{\sin \left (x\right )^{2}}\right )}^{\frac {1}{4}}}{a}\right )}{a} + \frac {\log \left ({\left | a + {\left (a^{4} - \frac {b^{4}}{\sin \left (x\right )^{2}}\right )}^{\frac {1}{4}} \right |}\right )}{2 \, a} - \frac {\log \left ({\left | -a + {\left (a^{4} - \frac {b^{4}}{\sin \left (x\right )^{2}}\right )}^{\frac {1}{4}} \right |}\right )}{2 \, a} \]
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Timed out. \[ \int \frac {\cot (x)}{\sqrt [4]{a^4-b^4 \csc ^2(x)}} \, dx=\int \frac {\mathrm {cot}\left (x\right )}{{\left (a^4-\frac {b^4}{{\sin \left (x\right )}^2}\right )}^{1/4}} \,d x \]
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