Integrand size = 21, antiderivative size = 232 \[ \int \frac {\tan ^2(e+f x)}{(d \cot (e+f x))^{3/2}} \, dx=\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {d \cot (e+f x)}}{\sqrt {d}}\right )}{\sqrt {2} d^{3/2} f}-\frac {\arctan \left (1+\frac {\sqrt {2} \sqrt {d \cot (e+f x)}}{\sqrt {d}}\right )}{\sqrt {2} d^{3/2} f}+\frac {2 d}{5 f (d \cot (e+f x))^{5/2}}-\frac {2}{d f \sqrt {d \cot (e+f x)}}-\frac {\log \left (\sqrt {d}+\sqrt {d} \cot (e+f x)-\sqrt {2} \sqrt {d \cot (e+f x)}\right )}{2 \sqrt {2} d^{3/2} f}+\frac {\log \left (\sqrt {d}+\sqrt {d} \cot (e+f x)+\sqrt {2} \sqrt {d \cot (e+f x)}\right )}{2 \sqrt {2} d^{3/2} f} \]
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Time = 0.24 (sec) , antiderivative size = 232, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.476, Rules used = {16, 3555, 3557, 335, 303, 1176, 631, 210, 1179, 642} \[ \int \frac {\tan ^2(e+f x)}{(d \cot (e+f x))^{3/2}} \, dx=\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {d \cot (e+f x)}}{\sqrt {d}}\right )}{\sqrt {2} d^{3/2} f}-\frac {\arctan \left (\frac {\sqrt {2} \sqrt {d \cot (e+f x)}}{\sqrt {d}}+1\right )}{\sqrt {2} d^{3/2} f}-\frac {\log \left (\sqrt {d} \cot (e+f x)-\sqrt {2} \sqrt {d \cot (e+f x)}+\sqrt {d}\right )}{2 \sqrt {2} d^{3/2} f}+\frac {\log \left (\sqrt {d} \cot (e+f x)+\sqrt {2} \sqrt {d \cot (e+f x)}+\sqrt {d}\right )}{2 \sqrt {2} d^{3/2} f}+\frac {2 d}{5 f (d \cot (e+f x))^{5/2}}-\frac {2}{d f \sqrt {d \cot (e+f x)}} \]
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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^2 \int \frac {1}{(d \cot (e+f x))^{7/2}} \, dx \\ & = \frac {2 d}{5 f (d \cot (e+f x))^{5/2}}-\int \frac {1}{(d \cot (e+f x))^{3/2}} \, dx \\ & = \frac {2 d}{5 f (d \cot (e+f x))^{5/2}}-\frac {2}{d f \sqrt {d \cot (e+f x)}}+\frac {\int \sqrt {d \cot (e+f x)} \, dx}{d^2} \\ & = \frac {2 d}{5 f (d \cot (e+f x))^{5/2}}-\frac {2}{d 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 )}{d f} \\ & = \frac {2 d}{5 f (d \cot (e+f x))^{5/2}}-\frac {2}{d 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 )}{d f} \\ & = \frac {2 d}{5 f (d \cot (e+f x))^{5/2}}-\frac {2}{d 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 )}{d f}-\frac {\text {Subst}\left (\int \frac {d+x^2}{d^2+x^4} \, dx,x,\sqrt {d \cot (e+f x)}\right )}{d f} \\ & = \frac {2 d}{5 f (d \cot (e+f x))^{5/2}}-\frac {2}{d f \sqrt {d \cot (e+f x)}}-\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} d^{3/2} 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} d^{3/2} 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 d 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 d f} \\ & = \frac {2 d}{5 f (d \cot (e+f x))^{5/2}}-\frac {2}{d 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} d^{3/2} f}+\frac {\log \left (\sqrt {d}+\sqrt {d} \cot (e+f x)+\sqrt {2} \sqrt {d \cot (e+f x)}\right )}{2 \sqrt {2} d^{3/2} 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} d^{3/2} 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} d^{3/2} f} \\ & = \frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {d \cot (e+f x)}}{\sqrt {d}}\right )}{\sqrt {2} d^{3/2} f}-\frac {\arctan \left (1+\frac {\sqrt {2} \sqrt {d \cot (e+f x)}}{\sqrt {d}}\right )}{\sqrt {2} d^{3/2} f}+\frac {2 d}{5 f (d \cot (e+f x))^{5/2}}-\frac {2}{d 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} d^{3/2} f}+\frac {\log \left (\sqrt {d}+\sqrt {d} \cot (e+f x)+\sqrt {2} \sqrt {d \cot (e+f x)}\right )}{2 \sqrt {2} d^{3/2} f} \\ \end{align*}
Time = 0.22 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.42 \[ \int \frac {\tan ^2(e+f x)}{(d \cot (e+f x))^{3/2}} \, dx=\frac {-5 \arctan \left (\sqrt [4]{-\cot ^2(e+f x)}\right ) \sqrt [4]{-\cot ^2(e+f x)}+5 \text {arctanh}\left (\sqrt [4]{-\cot ^2(e+f x)}\right ) \sqrt [4]{-\cot ^2(e+f x)}+2 \left (-5+\tan ^2(e+f x)\right )}{5 d f \sqrt {d \cot (e+f x)}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(733\) vs. \(2(179)=358\).
Time = 3.28 (sec) , antiderivative size = 734, normalized size of antiderivative = 3.16
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 )}^{2} \left (-40 \left (\csc ^{7}\left (f x +e \right )\right ) \left (1-\cos \left (f x +e \right )\right )^{7}+5 \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 ) {\left (\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 )\right )}^{\frac {5}{2}}-10 \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 ) {\left (\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 )\right )}^{\frac {5}{2}}-5 \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 ) {\left (\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 )\right )}^{\frac {5}{2}}-10 \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 ) {\left (\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 )\right )}^{\frac {5}{2}}+112 \left (\csc ^{5}\left (f x +e \right )\right ) \left (1-\cos \left (f x +e \right )\right )^{5}-40 \left (\csc ^{3}\left (f x +e \right )\right ) \left (1-\cos \left (f x +e \right )\right )^{3}\right ) \sin \left (f x +e \right ) \sqrt {2}}{20 f {\left (-\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 )}\right )}^{\frac {3}{2}} \left (1-\cos \left (f x +e \right )\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 )}\, {\left (\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 )\right )}^{\frac {5}{2}}}\) | \(734\) |
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Result contains complex when optimal does not.
Time = 0.25 (sec) , antiderivative size = 231, normalized size of antiderivative = 1.00 \[ \int \frac {\tan ^2(e+f x)}{(d \cot (e+f x))^{3/2}} \, dx=-\frac {5 \, d^{2} f \left (-\frac {1}{d^{6} f^{4}}\right )^{\frac {1}{4}} \log \left (d^{5} f^{3} \left (-\frac {1}{d^{6} f^{4}}\right )^{\frac {3}{4}} + \sqrt {\frac {d}{\tan \left (f x + e\right )}}\right ) - 5 i \, d^{2} f \left (-\frac {1}{d^{6} f^{4}}\right )^{\frac {1}{4}} \log \left (i \, d^{5} f^{3} \left (-\frac {1}{d^{6} f^{4}}\right )^{\frac {3}{4}} + \sqrt {\frac {d}{\tan \left (f x + e\right )}}\right ) + 5 i \, d^{2} f \left (-\frac {1}{d^{6} f^{4}}\right )^{\frac {1}{4}} \log \left (-i \, d^{5} f^{3} \left (-\frac {1}{d^{6} f^{4}}\right )^{\frac {3}{4}} + \sqrt {\frac {d}{\tan \left (f x + e\right )}}\right ) - 5 \, d^{2} f \left (-\frac {1}{d^{6} f^{4}}\right )^{\frac {1}{4}} \log \left (-d^{5} f^{3} \left (-\frac {1}{d^{6} f^{4}}\right )^{\frac {3}{4}} + \sqrt {\frac {d}{\tan \left (f x + e\right )}}\right ) - 4 \, {\left (\tan \left (f x + e\right )^{3} - 5 \, \tan \left (f x + e\right )\right )} \sqrt {\frac {d}{\tan \left (f x + e\right )}}}{10 \, d^{2} f} \]
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\[ \int \frac {\tan ^2(e+f x)}{(d \cot (e+f x))^{3/2}} \, dx=\int \frac {\tan ^{2}{\left (e + f x \right )}}{\left (d \cot {\left (e + f x \right )}\right )^{\frac {3}{2}}}\, dx \]
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Time = 0.34 (sec) , antiderivative size = 207, normalized size of antiderivative = 0.89 \[ \int \frac {\tan ^2(e+f x)}{(d \cot (e+f x))^{3/2}} \, dx=-\frac {d^{3} {\left (\frac {5 \, {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {d} + 2 \, \sqrt {\frac {d}{\tan \left (f x + e\right )}}\right )}}{2 \, \sqrt {d}}\right )}{\sqrt {d}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {d} - 2 \, \sqrt {\frac {d}{\tan \left (f x + e\right )}}\right )}}{2 \, \sqrt {d}}\right )}{\sqrt {d}} - \frac {\sqrt {2} \log \left (\sqrt {2} \sqrt {d} \sqrt {\frac {d}{\tan \left (f x + e\right )}} + d + \frac {d}{\tan \left (f x + e\right )}\right )}{\sqrt {d}} + \frac {\sqrt {2} \log \left (-\sqrt {2} \sqrt {d} \sqrt {\frac {d}{\tan \left (f x + e\right )}} + d + \frac {d}{\tan \left (f x + e\right )}\right )}{\sqrt {d}}\right )}}{d^{4}} - \frac {8 \, {\left (d^{2} - \frac {5 \, d^{2}}{\tan \left (f x + e\right )^{2}}\right )}}{d^{4} \left (\frac {d}{\tan \left (f x + e\right )}\right )^{\frac {5}{2}}}\right )}}{20 \, f} \]
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\[ \int \frac {\tan ^2(e+f x)}{(d \cot (e+f x))^{3/2}} \, dx=\int { \frac {\tan \left (f x + e\right )^{2}}{\left (d \cot \left (f x + e\right )\right )^{\frac {3}{2}}} \,d x } \]
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Time = 3.08 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.40 \[ \int \frac {\tan ^2(e+f x)}{(d \cot (e+f x))^{3/2}} \, dx=\frac {\frac {2\,d}{5}-\frac {2\,d}{{\mathrm {tan}\left (e+f\,x\right )}^2}}{f\,{\left (\frac {d}{\mathrm {tan}\left (e+f\,x\right )}\right )}^{5/2}}-\frac {{\left (-1\right )}^{1/4}\,\mathrm {atan}\left (\frac {{\left (-1\right )}^{1/4}\,\sqrt {\frac {d}{\mathrm {tan}\left (e+f\,x\right )}}}{\sqrt {d}}\right )}{d^{3/2}\,f}+\frac {{\left (-1\right )}^{1/4}\,\mathrm {atanh}\left (\frac {{\left (-1\right )}^{1/4}\,\sqrt {\frac {d}{\mathrm {tan}\left (e+f\,x\right )}}}{\sqrt {d}}\right )}{d^{3/2}\,f} \]
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