Integrand size = 19, antiderivative size = 209 \[ \int \frac {\tan (e+f x)}{\sqrt {d \cot (e+f x)}} \, dx=-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {d \cot (e+f x)}}{\sqrt {d}}\right )}{\sqrt {2} \sqrt {d} f}+\frac {\arctan \left (1+\frac {\sqrt {2} \sqrt {d \cot (e+f x)}}{\sqrt {d}}\right )}{\sqrt {2} \sqrt {d} f}+\frac {2}{f \sqrt {d \cot (e+f x)}}+\frac {\log \left (\sqrt {d}+\sqrt {d} \cot (e+f x)-\sqrt {2} \sqrt {d \cot (e+f x)}\right )}{2 \sqrt {2} \sqrt {d} f}-\frac {\log \left (\sqrt {d}+\sqrt {d} \cot (e+f x)+\sqrt {2} \sqrt {d \cot (e+f x)}\right )}{2 \sqrt {2} \sqrt {d} f} \]
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Time = 0.19 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.526, Rules used = {16, 3555, 3557, 335, 303, 1176, 631, 210, 1179, 642} \[ \int \frac {\tan (e+f x)}{\sqrt {d \cot (e+f x)}} \, dx=-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {d \cot (e+f x)}}{\sqrt {d}}\right )}{\sqrt {2} \sqrt {d} f}+\frac {\arctan \left (\frac {\sqrt {2} \sqrt {d \cot (e+f x)}}{\sqrt {d}}+1\right )}{\sqrt {2} \sqrt {d} f}+\frac {2}{f \sqrt {d \cot (e+f x)}}+\frac {\log \left (\sqrt {d} \cot (e+f x)-\sqrt {2} \sqrt {d \cot (e+f x)}+\sqrt {d}\right )}{2 \sqrt {2} \sqrt {d} f}-\frac {\log \left (\sqrt {d} \cot (e+f x)+\sqrt {2} \sqrt {d \cot (e+f x)}+\sqrt {d}\right )}{2 \sqrt {2} \sqrt {d} f} \]
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Rule 16
Rule 210
Rule 303
Rule 335
Rule 631
Rule 642
Rule 1176
Rule 1179
Rule 3555
Rule 3557
Rubi steps \begin{align*} \text {integral}& = d \int \frac {1}{(d \cot (e+f x))^{3/2}} \, dx \\ & = \frac {2}{f \sqrt {d \cot (e+f x)}}-\frac {\int \sqrt {d \cot (e+f x)} \, dx}{d} \\ & = \frac {2}{f \sqrt {d \cot (e+f x)}}+\frac {\text {Subst}\left (\int \frac {\sqrt {x}}{d^2+x^2} \, dx,x,d \cot (e+f x)\right )}{f} \\ & = \frac {2}{f \sqrt {d \cot (e+f x)}}+\frac {2 \text {Subst}\left (\int \frac {x^2}{d^2+x^4} \, dx,x,\sqrt {d \cot (e+f x)}\right )}{f} \\ & = \frac {2}{f \sqrt {d \cot (e+f x)}}-\frac {\text {Subst}\left (\int \frac {d-x^2}{d^2+x^4} \, dx,x,\sqrt {d \cot (e+f x)}\right )}{f}+\frac {\text {Subst}\left (\int \frac {d+x^2}{d^2+x^4} \, dx,x,\sqrt {d \cot (e+f x)}\right )}{f} \\ & = \frac {2}{f \sqrt {d \cot (e+f x)}}+\frac {\text {Subst}\left (\int \frac {1}{d-\sqrt {2} \sqrt {d} x+x^2} \, dx,x,\sqrt {d \cot (e+f x)}\right )}{2 f}+\frac {\text {Subst}\left (\int \frac {1}{d+\sqrt {2} \sqrt {d} x+x^2} \, dx,x,\sqrt {d \cot (e+f x)}\right )}{2 f}+\frac {\text {Subst}\left (\int \frac {\sqrt {2} \sqrt {d}+2 x}{-d-\sqrt {2} \sqrt {d} x-x^2} \, dx,x,\sqrt {d \cot (e+f x)}\right )}{2 \sqrt {2} \sqrt {d} f}+\frac {\text {Subst}\left (\int \frac {\sqrt {2} \sqrt {d}-2 x}{-d+\sqrt {2} \sqrt {d} x-x^2} \, dx,x,\sqrt {d \cot (e+f x)}\right )}{2 \sqrt {2} \sqrt {d} f} \\ & = \frac {2}{f \sqrt {d \cot (e+f x)}}+\frac {\log \left (\sqrt {d}+\sqrt {d} \cot (e+f x)-\sqrt {2} \sqrt {d \cot (e+f x)}\right )}{2 \sqrt {2} \sqrt {d} f}-\frac {\log \left (\sqrt {d}+\sqrt {d} \cot (e+f x)+\sqrt {2} \sqrt {d \cot (e+f x)}\right )}{2 \sqrt {2} \sqrt {d} f}+\frac {\text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt {d \cot (e+f x)}}{\sqrt {d}}\right )}{\sqrt {2} \sqrt {d} f}-\frac {\text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt {d \cot (e+f x)}}{\sqrt {d}}\right )}{\sqrt {2} \sqrt {d} f} \\ & = -\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {d \cot (e+f x)}}{\sqrt {d}}\right )}{\sqrt {2} \sqrt {d} f}+\frac {\arctan \left (1+\frac {\sqrt {2} \sqrt {d \cot (e+f x)}}{\sqrt {d}}\right )}{\sqrt {2} \sqrt {d} f}+\frac {2}{f \sqrt {d \cot (e+f x)}}+\frac {\log \left (\sqrt {d}+\sqrt {d} \cot (e+f x)-\sqrt {2} \sqrt {d \cot (e+f x)}\right )}{2 \sqrt {2} \sqrt {d} f}-\frac {\log \left (\sqrt {d}+\sqrt {d} \cot (e+f x)+\sqrt {2} \sqrt {d \cot (e+f x)}\right )}{2 \sqrt {2} \sqrt {d} f} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.38 \[ \int \frac {\tan (e+f x)}{\sqrt {d \cot (e+f x)}} \, dx=\frac {2+\arctan \left (\sqrt [4]{-\cot ^2(e+f x)}\right ) \sqrt [4]{-\cot ^2(e+f x)}-\text {arctanh}\left (\sqrt [4]{-\cot ^2(e+f x)}\right ) \sqrt [4]{-\cot ^2(e+f x)}}{f \sqrt {d \cot (e+f x)}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(662\) vs. \(2(160)=320\).
Time = 3.13 (sec) , antiderivative size = 663, normalized size of antiderivative = 3.17
method | result | size |
default | \(-\frac {\left (\left (\csc ^{2}\left (f x +e \right )\right ) \left (1-\cos \left (f x +e \right )\right )^{2}-1\right ) \left (\ln \left (\frac {\csc \left (f x +e \right ) \left (1-\cos \left (f x +e \right )\right )^{2}+2 \sin \left (f x +e \right ) \sqrt {\left (\csc ^{3}\left (f x +e \right )\right ) \left (1-\cos \left (f x +e \right )\right )^{3}-\csc \left (f x +e \right )+\cot \left (f x +e \right )}+2-2 \cos \left (f x +e \right )-\sin \left (f x +e \right )}{1-\cos \left (f x +e \right )}\right ) \sqrt {\left (\csc ^{3}\left (f x +e \right )\right ) \left (1-\cos \left (f x +e \right )\right )^{3}-\csc \left (f x +e \right )+\cot \left (f x +e \right )}-2 \arctan \left (\frac {\sin \left (f x +e \right ) \sqrt {\left (\csc ^{3}\left (f x +e \right )\right ) \left (1-\cos \left (f x +e \right )\right )^{3}-\csc \left (f x +e \right )+\cot \left (f x +e \right )}+1-\cos \left (f x +e \right )}{1-\cos \left (f x +e \right )}\right ) \sqrt {\left (\csc ^{3}\left (f x +e \right )\right ) \left (1-\cos \left (f x +e \right )\right )^{3}-\csc \left (f x +e \right )+\cot \left (f x +e \right )}-\ln \left (-\frac {-\csc \left (f x +e \right ) \left (1-\cos \left (f x +e \right )\right )^{2}+2 \sin \left (f x +e \right ) \sqrt {\left (\csc ^{3}\left (f x +e \right )\right ) \left (1-\cos \left (f x +e \right )\right )^{3}-\csc \left (f x +e \right )+\cot \left (f x +e \right )}-2+2 \cos \left (f x +e \right )+\sin \left (f x +e \right )}{1-\cos \left (f x +e \right )}\right ) \sqrt {\left (\csc ^{3}\left (f x +e \right )\right ) \left (1-\cos \left (f x +e \right )\right )^{3}-\csc \left (f x +e \right )+\cot \left (f x +e \right )}-2 \arctan \left (\frac {\sin \left (f x +e \right ) \sqrt {\left (\csc ^{3}\left (f x +e \right )\right ) \left (1-\cos \left (f x +e \right )\right )^{3}-\csc \left (f x +e \right )+\cot \left (f x +e \right )}-1+\cos \left (f x +e \right )}{1-\cos \left (f x +e \right )}\right ) \sqrt {\left (\csc ^{3}\left (f x +e \right )\right ) \left (1-\cos \left (f x +e \right )\right )^{3}-\csc \left (f x +e \right )+\cot \left (f x +e \right )}-8 \csc \left (f x +e \right )+8 \cot \left (f x +e \right )\right ) \sqrt {2}}{4 f \sqrt {-\frac {d \left (\csc \left (f x +e \right ) \left (1-\cos \left (f x +e \right )\right )^{2}-\sin \left (f x +e \right )\right )}{1-\cos \left (f x +e \right )}}\, \sqrt {\csc \left (f x +e \right ) \left (\left (\csc ^{2}\left (f x +e \right )\right ) \left (1-\cos \left (f x +e \right )\right )^{2}-1\right ) \left (1-\cos \left (f x +e \right )\right )}\, \sqrt {\left (\csc ^{3}\left (f x +e \right )\right ) \left (1-\cos \left (f x +e \right )\right )^{3}-\csc \left (f x +e \right )+\cot \left (f x +e \right )}}\) | \(663\) |
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Result contains complex when optimal does not.
Time = 0.23 (sec) , antiderivative size = 211, normalized size of antiderivative = 1.01 \[ \int \frac {\tan (e+f x)}{\sqrt {d \cot (e+f x)}} \, dx=\frac {d f \left (-\frac {1}{d^{2} f^{4}}\right )^{\frac {1}{4}} \log \left (d^{2} f^{3} \left (-\frac {1}{d^{2} f^{4}}\right )^{\frac {3}{4}} + \sqrt {\frac {d}{\tan \left (f x + e\right )}}\right ) - i \, d f \left (-\frac {1}{d^{2} f^{4}}\right )^{\frac {1}{4}} \log \left (i \, d^{2} f^{3} \left (-\frac {1}{d^{2} f^{4}}\right )^{\frac {3}{4}} + \sqrt {\frac {d}{\tan \left (f x + e\right )}}\right ) + i \, d f \left (-\frac {1}{d^{2} f^{4}}\right )^{\frac {1}{4}} \log \left (-i \, d^{2} f^{3} \left (-\frac {1}{d^{2} f^{4}}\right )^{\frac {3}{4}} + \sqrt {\frac {d}{\tan \left (f x + e\right )}}\right ) - d f \left (-\frac {1}{d^{2} f^{4}}\right )^{\frac {1}{4}} \log \left (-d^{2} f^{3} \left (-\frac {1}{d^{2} f^{4}}\right )^{\frac {3}{4}} + \sqrt {\frac {d}{\tan \left (f x + e\right )}}\right ) + 4 \, \sqrt {\frac {d}{\tan \left (f x + e\right )}} \tan \left (f x + e\right )}{2 \, d f} \]
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\[ \int \frac {\tan (e+f x)}{\sqrt {d \cot (e+f x)}} \, dx=\int \frac {\tan {\left (e + f x \right )}}{\sqrt {d \cot {\left (e + f x \right )}}}\, dx \]
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none
Time = 0.33 (sec) , antiderivative size = 189, normalized size of antiderivative = 0.90 \[ \int \frac {\tan (e+f x)}{\sqrt {d \cot (e+f x)}} \, dx=\frac {d^{2} {\left (\frac {\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {d} + 2 \, \sqrt {\frac {d}{\tan \left (f x + e\right )}}\right )}}{2 \, \sqrt {d}}\right )}{\sqrt {d}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {d} - 2 \, \sqrt {\frac {d}{\tan \left (f x + e\right )}}\right )}}{2 \, \sqrt {d}}\right )}{\sqrt {d}} - \frac {\sqrt {2} \log \left (\sqrt {2} \sqrt {d} \sqrt {\frac {d}{\tan \left (f x + e\right )}} + d + \frac {d}{\tan \left (f x + e\right )}\right )}{\sqrt {d}} + \frac {\sqrt {2} \log \left (-\sqrt {2} \sqrt {d} \sqrt {\frac {d}{\tan \left (f x + e\right )}} + d + \frac {d}{\tan \left (f x + e\right )}\right )}{\sqrt {d}}}{d^{2}} + \frac {8}{d^{2} \sqrt {\frac {d}{\tan \left (f x + e\right )}}}\right )}}{4 \, f} \]
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\[ \int \frac {\tan (e+f x)}{\sqrt {d \cot (e+f x)}} \, dx=\int { \frac {\tan \left (f x + e\right )}{\sqrt {d \cot \left (f x + e\right )}} \,d x } \]
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Time = 0.22 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.38 \[ \int \frac {\tan (e+f x)}{\sqrt {d \cot (e+f x)}} \, dx=\frac {2}{f\,\sqrt {\frac {d}{\mathrm {tan}\left (e+f\,x\right )}}}+\frac {{\left (-1\right )}^{1/4}\,\mathrm {atan}\left (\frac {{\left (-1\right )}^{1/4}\,\sqrt {\frac {d}{\mathrm {tan}\left (e+f\,x\right )}}}{\sqrt {d}}\right )}{\sqrt {d}\,f}-\frac {{\left (-1\right )}^{1/4}\,\mathrm {atanh}\left (\frac {{\left (-1\right )}^{1/4}\,\sqrt {\frac {d}{\mathrm {tan}\left (e+f\,x\right )}}}{\sqrt {d}}\right )}{\sqrt {d}\,f} \]
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