Integrand size = 21, antiderivative size = 212 \[ \int \frac {\tan ^2(e+f x)}{\sqrt {d \cot (e+f x)}} \, dx=-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {d \cot (e+f x)}}{\sqrt {d}}\right )}{\sqrt {2} \sqrt {d} f}+\frac {\arctan \left (1+\frac {\sqrt {2} \sqrt {d \cot (e+f x)}}{\sqrt {d}}\right )}{\sqrt {2} \sqrt {d} f}+\frac {2 d}{3 f (d \cot (e+f x))^{3/2}}-\frac {\log \left (\sqrt {d}+\sqrt {d} \cot (e+f x)-\sqrt {2} \sqrt {d \cot (e+f x)}\right )}{2 \sqrt {2} \sqrt {d} f}+\frac {\log \left (\sqrt {d}+\sqrt {d} \cot (e+f x)+\sqrt {2} \sqrt {d \cot (e+f x)}\right )}{2 \sqrt {2} \sqrt {d} f} \]
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Time = 0.21 (sec) , antiderivative size = 212, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.476, Rules used = {16, 3555, 3557, 335, 217, 1179, 642, 1176, 631, 210} \[ \int \frac {\tan ^2(e+f x)}{\sqrt {d \cot (e+f x)}} \, dx=-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {d \cot (e+f x)}}{\sqrt {d}}\right )}{\sqrt {2} \sqrt {d} f}+\frac {\arctan \left (\frac {\sqrt {2} \sqrt {d \cot (e+f x)}}{\sqrt {d}}+1\right )}{\sqrt {2} \sqrt {d} f}+\frac {2 d}{3 f (d \cot (e+f x))^{3/2}}-\frac {\log \left (\sqrt {d} \cot (e+f x)-\sqrt {2} \sqrt {d \cot (e+f x)}+\sqrt {d}\right )}{2 \sqrt {2} \sqrt {d} f}+\frac {\log \left (\sqrt {d} \cot (e+f x)+\sqrt {2} \sqrt {d \cot (e+f x)}+\sqrt {d}\right )}{2 \sqrt {2} \sqrt {d} f} \]
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Rule 16
Rule 210
Rule 217
Rule 335
Rule 631
Rule 642
Rule 1176
Rule 1179
Rule 3555
Rule 3557
Rubi steps \begin{align*} \text {integral}& = d^2 \int \frac {1}{(d \cot (e+f x))^{5/2}} \, dx \\ & = \frac {2 d}{3 f (d \cot (e+f x))^{3/2}}-\int \frac {1}{\sqrt {d \cot (e+f x)}} \, dx \\ & = \frac {2 d}{3 f (d \cot (e+f x))^{3/2}}+\frac {d \text {Subst}\left (\int \frac {1}{\sqrt {x} \left (d^2+x^2\right )} \, dx,x,d \cot (e+f x)\right )}{f} \\ & = \frac {2 d}{3 f (d \cot (e+f x))^{3/2}}+\frac {(2 d) \text {Subst}\left (\int \frac {1}{d^2+x^4} \, dx,x,\sqrt {d \cot (e+f x)}\right )}{f} \\ & = \frac {2 d}{3 f (d \cot (e+f x))^{3/2}}+\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 d}{3 f (d \cot (e+f x))^{3/2}}+\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 d}{3 f (d \cot (e+f x))^{3/2}}-\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 d}{3 f (d \cot (e+f x))^{3/2}}-\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 = 0.19 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.40 \[ \int \frac {\tan ^2(e+f x)}{\sqrt {d \cot (e+f x)}} \, dx=-\frac {d \left (-2+3 \arctan \left (\sqrt [4]{-\cot ^2(e+f x)}\right ) \left (-\cot ^2(e+f x)\right )^{3/4}+3 \text {arctanh}\left (\sqrt [4]{-\cot ^2(e+f x)}\right ) \left (-\cot ^2(e+f x)\right )^{3/4}\right )}{3 f (d \cot (e+f x))^{3/2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(483\) vs. \(2(161)=322\).
Time = 2.27 (sec) , antiderivative size = 484, normalized size of antiderivative = 2.28
method | result | size |
default | \(-\frac {\sec \left (f x +e \right ) \csc \left (f x +e \right ) \left (\cos \left (f x +e \right )+1\right ) \left (-6 \sqrt {-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}\, \cos \left (f x +e \right ) \arctan \left (\frac {\sqrt {2}\, \sqrt {-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}\, \sin \left (f x +e \right )+\cos \left (f x +e \right )-1}{\cos \left (f x +e \right )-1}\right )-6 \sqrt {-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}\, \cos \left (f x +e \right ) \arctan \left (\frac {\sqrt {2}\, \sqrt {-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}\, \sin \left (f x +e \right )-\cos \left (f x +e \right )+1}{\cos \left (f x +e \right )-1}\right )+3 \cos \left (f x +e \right ) \ln \left (2 \cot \left (f x +e \right ) \sqrt {2}\, \sqrt {-\left (\csc ^{2}\left (f x +e \right )\right ) \cot \left (f x +e \right ) \left (\cos \left (f x +e \right )-1\right )^{2}}+2 \csc \left (f x +e \right ) \sqrt {2}\, \sqrt {-\left (\csc ^{2}\left (f x +e \right )\right ) \cot \left (f x +e \right ) \left (\cos \left (f x +e \right )-1\right )^{2}}-2 \cot \left (f x +e \right )+2\right ) \sqrt {-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}-3 \cos \left (f x +e \right ) \ln \left (-2 \cot \left (f x +e \right ) \sqrt {2}\, \sqrt {-\left (\csc ^{2}\left (f x +e \right )\right ) \cot \left (f x +e \right ) \left (\cos \left (f x +e \right )-1\right )^{2}}-2 \csc \left (f x +e \right ) \sqrt {2}\, \sqrt {-\left (\csc ^{2}\left (f x +e \right )\right ) \cot \left (f x +e \right ) \left (\cos \left (f x +e \right )-1\right )^{2}}-2 \cot \left (f x +e \right )+2\right ) \sqrt {-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}+4 \sqrt {2}\, \cos \left (f x +e \right )-4 \sqrt {2}\right ) \sqrt {2}}{12 f \sqrt {\cot \left (f x +e \right ) d}}\) | \(484\) |
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Result contains complex when optimal does not.
Time = 0.27 (sec) , antiderivative size = 198, normalized size of antiderivative = 0.93 \[ \int \frac {\tan ^2(e+f x)}{\sqrt {d \cot (e+f x)}} \, dx=\frac {3 \, d f \left (-\frac {1}{d^{2} f^{4}}\right )^{\frac {1}{4}} \log \left (d f \left (-\frac {1}{d^{2} f^{4}}\right )^{\frac {1}{4}} + \sqrt {\frac {d}{\tan \left (f x + e\right )}}\right ) + 3 i \, d f \left (-\frac {1}{d^{2} f^{4}}\right )^{\frac {1}{4}} \log \left (i \, d f \left (-\frac {1}{d^{2} f^{4}}\right )^{\frac {1}{4}} + \sqrt {\frac {d}{\tan \left (f x + e\right )}}\right ) - 3 i \, d f \left (-\frac {1}{d^{2} f^{4}}\right )^{\frac {1}{4}} \log \left (-i \, d f \left (-\frac {1}{d^{2} f^{4}}\right )^{\frac {1}{4}} + \sqrt {\frac {d}{\tan \left (f x + e\right )}}\right ) - 3 \, d f \left (-\frac {1}{d^{2} f^{4}}\right )^{\frac {1}{4}} \log \left (-d f \left (-\frac {1}{d^{2} f^{4}}\right )^{\frac {1}{4}} + \sqrt {\frac {d}{\tan \left (f x + e\right )}}\right ) + 4 \, \sqrt {\frac {d}{\tan \left (f x + e\right )}} \tan \left (f x + e\right )^{2}}{6 \, d f} \]
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\[ \int \frac {\tan ^2(e+f x)}{\sqrt {d \cot (e+f x)}} \, dx=\int \frac {\tan ^{2}{\left (e + f x \right )}}{\sqrt {d \cot {\left (e + f x \right )}}}\, dx \]
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none
Time = 0.35 (sec) , antiderivative size = 190, normalized size of antiderivative = 0.90 \[ \int \frac {\tan ^2(e+f x)}{\sqrt {d \cot (e+f x)}} \, dx=\frac {d^{3} {\left (\frac {3 \, {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {d} + 2 \, \sqrt {\frac {d}{\tan \left (f x + e\right )}}\right )}}{2 \, \sqrt {d}}\right )}{d^{\frac {3}{2}}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {d} - 2 \, \sqrt {\frac {d}{\tan \left (f x + e\right )}}\right )}}{2 \, \sqrt {d}}\right )}{d^{\frac {3}{2}}} + \frac {\sqrt {2} \log \left (\sqrt {2} \sqrt {d} \sqrt {\frac {d}{\tan \left (f x + e\right )}} + d + \frac {d}{\tan \left (f x + e\right )}\right )}{d^{\frac {3}{2}}} - \frac {\sqrt {2} \log \left (-\sqrt {2} \sqrt {d} \sqrt {\frac {d}{\tan \left (f x + e\right )}} + d + \frac {d}{\tan \left (f x + e\right )}\right )}{d^{\frac {3}{2}}}\right )}}{d^{2}} + \frac {8}{d^{2} \left (\frac {d}{\tan \left (f x + e\right )}\right )^{\frac {3}{2}}}\right )}}{12 \, f} \]
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\[ \int \frac {\tan ^2(e+f x)}{\sqrt {d \cot (e+f x)}} \, dx=\int { \frac {\tan \left (f x + e\right )^{2}}{\sqrt {d \cot \left (f x + e\right )}} \,d x } \]
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Time = 2.99 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.38 \[ \int \frac {\tan ^2(e+f x)}{\sqrt {d \cot (e+f x)}} \, dx=\frac {2\,d}{3\,f\,{\left (\frac {d}{\mathrm {tan}\left (e+f\,x\right )}\right )}^{3/2}}-\frac {{\left (-1\right )}^{1/4}\,\mathrm {atan}\left (\frac {{\left (-1\right )}^{1/4}\,\sqrt {\frac {d}{\mathrm {tan}\left (e+f\,x\right )}}}{\sqrt {d}}\right )\,1{}\mathrm {i}}{\sqrt {d}\,f}-\frac {{\left (-1\right )}^{1/4}\,\mathrm {atanh}\left (\frac {{\left (-1\right )}^{1/4}\,\sqrt {\frac {d}{\mathrm {tan}\left (e+f\,x\right )}}}{\sqrt {d}}\right )\,1{}\mathrm {i}}{\sqrt {d}\,f} \]
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