Integrand size = 21, antiderivative size = 234 \[ \int (d \cot (e+f x))^{3/2} \tan ^5(e+f x) \, dx=\frac {d^{3/2} \arctan \left (1-\frac {\sqrt {2} \sqrt {d \cot (e+f x)}}{\sqrt {d}}\right )}{\sqrt {2} f}-\frac {d^{3/2} \arctan \left (1+\frac {\sqrt {2} \sqrt {d \cot (e+f x)}}{\sqrt {d}}\right )}{\sqrt {2} f}+\frac {2 d^4}{5 f (d \cot (e+f x))^{5/2}}-\frac {2 d^2}{f \sqrt {d \cot (e+f x)}}-\frac {d^{3/2} \log \left (\sqrt {d}+\sqrt {d} \cot (e+f x)-\sqrt {2} \sqrt {d \cot (e+f x)}\right )}{2 \sqrt {2} f}+\frac {d^{3/2} \log \left (\sqrt {d}+\sqrt {d} \cot (e+f x)+\sqrt {2} \sqrt {d \cot (e+f x)}\right )}{2 \sqrt {2} f} \]
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Time = 0.27 (sec) , antiderivative size = 234, 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} \[ \int (d \cot (e+f x))^{3/2} \tan ^5(e+f x) \, dx=\frac {d^{3/2} \arctan \left (1-\frac {\sqrt {2} \sqrt {d \cot (e+f x)}}{\sqrt {d}}\right )}{\sqrt {2} f}-\frac {d^{3/2} \arctan \left (\frac {\sqrt {2} \sqrt {d \cot (e+f x)}}{\sqrt {d}}+1\right )}{\sqrt {2} f}-\frac {d^{3/2} \log \left (\sqrt {d} \cot (e+f x)-\sqrt {2} \sqrt {d \cot (e+f x)}+\sqrt {d}\right )}{2 \sqrt {2} f}+\frac {d^{3/2} \log \left (\sqrt {d} \cot (e+f x)+\sqrt {2} \sqrt {d \cot (e+f x)}+\sqrt {d}\right )}{2 \sqrt {2} f}+\frac {2 d^4}{5 f (d \cot (e+f x))^{5/2}}-\frac {2 d^2}{f \sqrt {d \cot (e+f x)}} \]
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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^5 \int \frac {1}{(d \cot (e+f x))^{7/2}} \, dx \\ & = \frac {2 d^4}{5 f (d \cot (e+f x))^{5/2}}-d^3 \int \frac {1}{(d \cot (e+f x))^{3/2}} \, dx \\ & = \frac {2 d^4}{5 f (d \cot (e+f x))^{5/2}}-\frac {2 d^2}{f \sqrt {d \cot (e+f x)}}+d \int \sqrt {d \cot (e+f x)} \, dx \\ & = \frac {2 d^4}{5 f (d \cot (e+f x))^{5/2}}-\frac {2 d^2}{f \sqrt {d \cot (e+f x)}}-\frac {d^2 \text {Subst}\left (\int \frac {\sqrt {x}}{d^2+x^2} \, dx,x,d \cot (e+f x)\right )}{f} \\ & = \frac {2 d^4}{5 f (d \cot (e+f x))^{5/2}}-\frac {2 d^2}{f \sqrt {d \cot (e+f x)}}-\frac {\left (2 d^2\right ) \text {Subst}\left (\int \frac {x^2}{d^2+x^4} \, dx,x,\sqrt {d \cot (e+f x)}\right )}{f} \\ & = \frac {2 d^4}{5 f (d \cot (e+f x))^{5/2}}-\frac {2 d^2}{f \sqrt {d \cot (e+f x)}}+\frac {d^2 \text {Subst}\left (\int \frac {d-x^2}{d^2+x^4} \, dx,x,\sqrt {d \cot (e+f x)}\right )}{f}-\frac {d^2 \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^4}{5 f (d \cot (e+f x))^{5/2}}-\frac {2 d^2}{f \sqrt {d \cot (e+f x)}}-\frac {d^{3/2} \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} f}-\frac {d^{3/2} \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} f}-\frac {d^2 \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 {d^2 \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 {2 d^4}{5 f (d \cot (e+f x))^{5/2}}-\frac {2 d^2}{f \sqrt {d \cot (e+f x)}}-\frac {d^{3/2} \log \left (\sqrt {d}+\sqrt {d} \cot (e+f x)-\sqrt {2} \sqrt {d \cot (e+f x)}\right )}{2 \sqrt {2} f}+\frac {d^{3/2} \log \left (\sqrt {d}+\sqrt {d} \cot (e+f x)+\sqrt {2} \sqrt {d \cot (e+f x)}\right )}{2 \sqrt {2} f}-\frac {d^{3/2} \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} f}+\frac {d^{3/2} \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} f} \\ & = \frac {d^{3/2} \arctan \left (1-\frac {\sqrt {2} \sqrt {d \cot (e+f x)}}{\sqrt {d}}\right )}{\sqrt {2} f}-\frac {d^{3/2} \arctan \left (1+\frac {\sqrt {2} \sqrt {d \cot (e+f x)}}{\sqrt {d}}\right )}{\sqrt {2} f}+\frac {2 d^4}{5 f (d \cot (e+f x))^{5/2}}-\frac {2 d^2}{f \sqrt {d \cot (e+f x)}}-\frac {d^{3/2} \log \left (\sqrt {d}+\sqrt {d} \cot (e+f x)-\sqrt {2} \sqrt {d \cot (e+f x)}\right )}{2 \sqrt {2} f}+\frac {d^{3/2} \log \left (\sqrt {d}+\sqrt {d} \cot (e+f x)+\sqrt {2} \sqrt {d \cot (e+f x)}\right )}{2 \sqrt {2} f} \\ \end{align*}
Time = 0.56 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.47 \[ \int (d \cot (e+f x))^{3/2} \tan ^5(e+f x) \, dx=-\frac {(d \cot (e+f x))^{3/2} \left (-2+10 \cot ^2(e+f x)+5 \arctan \left (\sqrt [4]{-\cot ^2(e+f x)}\right ) \sqrt [4]{-\cot (e+f x)} \cot ^{\frac {9}{4}}(e+f x)+5 \text {arctanh}\left (\sqrt [4]{-\cot ^2(e+f x)}\right ) \left (-\cot ^2(e+f x)\right )^{5/4}\right ) \tan ^4(e+f x)}{5 f} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(631\) vs. \(2(181)=362\).
Time = 3.41 (sec) , antiderivative size = 632, normalized size of antiderivative = 2.70
method | result | size |
default | \(\frac {\left (\sec ^{3}\left (f x +e \right )\right ) \csc \left (f x +e \right ) \left (5 \left (\cos ^{2}\left (f x +e \right )\right ) \sin \left (f x +e \right ) \ln \left (-\frac {\cot \left (f x +e \right ) \cos \left (f x +e \right )-2 \cot \left (f x +e \right )+2 \sin \left (f x +e \right ) \sqrt {-\left (\cot ^{3}\left (f x +e \right )\right )+3 \left (\cot ^{2}\left (f x +e \right )\right ) \csc \left (f x +e \right )-3 \left (\csc ^{2}\left (f x +e \right )\right ) \cot \left (f x +e \right )+\csc ^{3}\left (f x +e \right )+\cot \left (f x +e \right )-\csc \left (f x +e \right )}-2 \cos \left (f x +e \right )-\sin \left (f x +e \right )+\csc \left (f x +e \right )+2}{\cos \left (f x +e \right )-1}\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}}}-5 \left (\cos ^{2}\left (f x +e \right )\right ) \sin \left (f x +e \right ) \ln \left (-\frac {\cot \left (f x +e \right ) \cos \left (f x +e \right )-2 \cot \left (f x +e \right )-2 \sin \left (f x +e \right ) \sqrt {-\left (\cot ^{3}\left (f x +e \right )\right )+3 \left (\cot ^{2}\left (f x +e \right )\right ) \csc \left (f x +e \right )-3 \left (\csc ^{2}\left (f x +e \right )\right ) \cot \left (f x +e \right )+\csc ^{3}\left (f x +e \right )+\cot \left (f x +e \right )-\csc \left (f x +e \right )}-2 \cos \left (f x +e \right )-\sin \left (f x +e \right )+\csc \left (f x +e \right )+2}{\cos \left (f x +e \right )-1}\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}}}+10 \left (\cos ^{2}\left (f x +e \right )\right ) \sin \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 ) \sqrt {-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}-10 \left (\cos ^{2}\left (f x +e \right )\right ) \sin \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 ) \sqrt {-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}+24 \sqrt {2}\, \left (\cos ^{3}\left (f x +e \right )\right )-24 \sqrt {2}\, \left (\cos ^{2}\left (f x +e \right )\right )-4 \sqrt {2}\, \cos \left (f x +e \right )+4 \sqrt {2}\right ) \left (\cos \left (f x +e \right )+1\right ) d \sqrt {\cot \left (f x +e \right ) d}\, \sqrt {2}}{20 f}\) | \(632\) |
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Result contains complex when optimal does not.
Time = 0.26 (sec) , antiderivative size = 223, normalized size of antiderivative = 0.95 \[ \int (d \cot (e+f x))^{3/2} \tan ^5(e+f x) \, dx=-\frac {5 \, \left (-\frac {d^{6}}{f^{4}}\right )^{\frac {1}{4}} f \log \left (d^{4} \sqrt {\frac {d}{\tan \left (f x + e\right )}} + \left (-\frac {d^{6}}{f^{4}}\right )^{\frac {3}{4}} f^{3}\right ) - 5 i \, \left (-\frac {d^{6}}{f^{4}}\right )^{\frac {1}{4}} f \log \left (d^{4} \sqrt {\frac {d}{\tan \left (f x + e\right )}} + i \, \left (-\frac {d^{6}}{f^{4}}\right )^{\frac {3}{4}} f^{3}\right ) + 5 i \, \left (-\frac {d^{6}}{f^{4}}\right )^{\frac {1}{4}} f \log \left (d^{4} \sqrt {\frac {d}{\tan \left (f x + e\right )}} - i \, \left (-\frac {d^{6}}{f^{4}}\right )^{\frac {3}{4}} f^{3}\right ) - 5 \, \left (-\frac {d^{6}}{f^{4}}\right )^{\frac {1}{4}} f \log \left (d^{4} \sqrt {\frac {d}{\tan \left (f x + e\right )}} - \left (-\frac {d^{6}}{f^{4}}\right )^{\frac {3}{4}} f^{3}\right ) - 4 \, {\left (d \tan \left (f x + e\right )^{3} - 5 \, d \tan \left (f x + e\right )\right )} \sqrt {\frac {d}{\tan \left (f x + e\right )}}}{10 \, f} \]
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\[ \int (d \cot (e+f x))^{3/2} \tan ^5(e+f x) \, dx=\int \left (d \cot {\left (e + f x \right )}\right )^{\frac {3}{2}} \tan ^{5}{\left (e + f x \right )}\, dx \]
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none
Time = 0.47 (sec) , antiderivative size = 207, normalized size of antiderivative = 0.88 \[ \int (d \cot (e+f x))^{3/2} \tan ^5(e+f x) \, dx=-\frac {d^{6} {\left (\frac {5 \, {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {d} + 2 \, \sqrt {\frac {d}{\tan \left (f x + e\right )}}\right )}}{2 \, \sqrt {d}}\right )}{\sqrt {d}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {d} - 2 \, \sqrt {\frac {d}{\tan \left (f x + e\right )}}\right )}}{2 \, \sqrt {d}}\right )}{\sqrt {d}} - \frac {\sqrt {2} \log \left (\sqrt {2} \sqrt {d} \sqrt {\frac {d}{\tan \left (f x + e\right )}} + d + \frac {d}{\tan \left (f x + e\right )}\right )}{\sqrt {d}} + \frac {\sqrt {2} \log \left (-\sqrt {2} \sqrt {d} \sqrt {\frac {d}{\tan \left (f x + e\right )}} + d + \frac {d}{\tan \left (f x + e\right )}\right )}{\sqrt {d}}\right )}}{d^{4}} - \frac {8 \, {\left (d^{2} - \frac {5 \, d^{2}}{\tan \left (f x + e\right )^{2}}\right )}}{d^{4} \left (\frac {d}{\tan \left (f x + e\right )}\right )^{\frac {5}{2}}}\right )}}{20 \, f} \]
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\[ \int (d \cot (e+f x))^{3/2} \tan ^5(e+f x) \, dx=\int { \left (d \cot \left (f x + e\right )\right )^{\frac {3}{2}} \tan \left (f x + e\right )^{5} \,d x } \]
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Time = 2.78 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.41 \[ \int (d \cot (e+f x))^{3/2} \tan ^5(e+f x) \, dx=\frac {\frac {2\,d^4}{5}-\frac {2\,d^4}{{\mathrm {tan}\left (e+f\,x\right )}^2}}{f\,{\left (\frac {d}{\mathrm {tan}\left (e+f\,x\right )}\right )}^{5/2}}-\frac {{\left (-1\right )}^{1/4}\,d^{3/2}\,\mathrm {atan}\left (\frac {{\left (-1\right )}^{1/4}\,\sqrt {\frac {d}{\mathrm {tan}\left (e+f\,x\right )}}}{\sqrt {d}}\right )}{f}+\frac {{\left (-1\right )}^{1/4}\,d^{3/2}\,\mathrm {atanh}\left (\frac {{\left (-1\right )}^{1/4}\,\sqrt {\frac {d}{\mathrm {tan}\left (e+f\,x\right )}}}{\sqrt {d}}\right )}{f} \]
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