Integrand size = 21, antiderivative size = 225 \[ \int \cos ^2(e+f x) (d \tan (e+f x))^{5/2} \, dx=-\frac {3 d^{5/2} \arctan \left (1-\frac {\sqrt {2} \sqrt {d \tan (e+f x)}}{\sqrt {d}}\right )}{4 \sqrt {2} f}+\frac {3 d^{5/2} \arctan \left (1+\frac {\sqrt {2} \sqrt {d \tan (e+f x)}}{\sqrt {d}}\right )}{4 \sqrt {2} f}+\frac {3 d^{5/2} \log \left (\sqrt {d}+\sqrt {d} \tan (e+f x)-\sqrt {2} \sqrt {d \tan (e+f x)}\right )}{8 \sqrt {2} f}-\frac {3 d^{5/2} \log \left (\sqrt {d}+\sqrt {d} \tan (e+f x)+\sqrt {2} \sqrt {d \tan (e+f x)}\right )}{8 \sqrt {2} f}-\frac {d \cos ^2(e+f x) (d \tan (e+f x))^{3/2}}{2 f} \]
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Time = 0.20 (sec) , antiderivative size = 225, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {2687, 294, 335, 303, 1176, 631, 210, 1179, 642} \[ \int \cos ^2(e+f x) (d \tan (e+f x))^{5/2} \, dx=-\frac {3 d^{5/2} \arctan \left (1-\frac {\sqrt {2} \sqrt {d \tan (e+f x)}}{\sqrt {d}}\right )}{4 \sqrt {2} f}+\frac {3 d^{5/2} \arctan \left (\frac {\sqrt {2} \sqrt {d \tan (e+f x)}}{\sqrt {d}}+1\right )}{4 \sqrt {2} f}+\frac {3 d^{5/2} \log \left (\sqrt {d} \tan (e+f x)-\sqrt {2} \sqrt {d \tan (e+f x)}+\sqrt {d}\right )}{8 \sqrt {2} f}-\frac {3 d^{5/2} \log \left (\sqrt {d} \tan (e+f x)+\sqrt {2} \sqrt {d \tan (e+f x)}+\sqrt {d}\right )}{8 \sqrt {2} f}-\frac {d \cos ^2(e+f x) (d \tan (e+f x))^{3/2}}{2 f} \]
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Rule 210
Rule 294
Rule 303
Rule 335
Rule 631
Rule 642
Rule 1176
Rule 1179
Rule 2687
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {(d x)^{5/2}}{\left (1+x^2\right )^2} \, dx,x,\tan (e+f x)\right )}{f} \\ & = -\frac {d \cos ^2(e+f x) (d \tan (e+f x))^{3/2}}{2 f}+\frac {\left (3 d^2\right ) \text {Subst}\left (\int \frac {\sqrt {d x}}{1+x^2} \, dx,x,\tan (e+f x)\right )}{4 f} \\ & = -\frac {d \cos ^2(e+f x) (d \tan (e+f x))^{3/2}}{2 f}+\frac {(3 d) \text {Subst}\left (\int \frac {x^2}{1+\frac {x^4}{d^2}} \, dx,x,\sqrt {d \tan (e+f x)}\right )}{2 f} \\ & = -\frac {d \cos ^2(e+f x) (d \tan (e+f x))^{3/2}}{2 f}-\frac {(3 d) \text {Subst}\left (\int \frac {d-x^2}{1+\frac {x^4}{d^2}} \, dx,x,\sqrt {d \tan (e+f x)}\right )}{4 f}+\frac {(3 d) \text {Subst}\left (\int \frac {d+x^2}{1+\frac {x^4}{d^2}} \, dx,x,\sqrt {d \tan (e+f x)}\right )}{4 f} \\ & = -\frac {d \cos ^2(e+f x) (d \tan (e+f x))^{3/2}}{2 f}+\frac {\left (3 d^{5/2}\right ) \text {Subst}\left (\int \frac {\sqrt {2} \sqrt {d}+2 x}{-d-\sqrt {2} \sqrt {d} x-x^2} \, dx,x,\sqrt {d \tan (e+f x)}\right )}{8 \sqrt {2} f}+\frac {\left (3 d^{5/2}\right ) \text {Subst}\left (\int \frac {\sqrt {2} \sqrt {d}-2 x}{-d+\sqrt {2} \sqrt {d} x-x^2} \, dx,x,\sqrt {d \tan (e+f x)}\right )}{8 \sqrt {2} f}+\frac {\left (3 d^3\right ) \text {Subst}\left (\int \frac {1}{d-\sqrt {2} \sqrt {d} x+x^2} \, dx,x,\sqrt {d \tan (e+f x)}\right )}{8 f}+\frac {\left (3 d^3\right ) \text {Subst}\left (\int \frac {1}{d+\sqrt {2} \sqrt {d} x+x^2} \, dx,x,\sqrt {d \tan (e+f x)}\right )}{8 f} \\ & = \frac {3 d^{5/2} \log \left (\sqrt {d}+\sqrt {d} \tan (e+f x)-\sqrt {2} \sqrt {d \tan (e+f x)}\right )}{8 \sqrt {2} f}-\frac {3 d^{5/2} \log \left (\sqrt {d}+\sqrt {d} \tan (e+f x)+\sqrt {2} \sqrt {d \tan (e+f x)}\right )}{8 \sqrt {2} f}-\frac {d \cos ^2(e+f x) (d \tan (e+f x))^{3/2}}{2 f}+\frac {\left (3 d^{5/2}\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt {d \tan (e+f x)}}{\sqrt {d}}\right )}{4 \sqrt {2} f}-\frac {\left (3 d^{5/2}\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt {d \tan (e+f x)}}{\sqrt {d}}\right )}{4 \sqrt {2} f} \\ & = -\frac {3 d^{5/2} \arctan \left (1-\frac {\sqrt {2} \sqrt {d \tan (e+f x)}}{\sqrt {d}}\right )}{4 \sqrt {2} f}+\frac {3 d^{5/2} \arctan \left (1+\frac {\sqrt {2} \sqrt {d \tan (e+f x)}}{\sqrt {d}}\right )}{4 \sqrt {2} f}+\frac {3 d^{5/2} \log \left (\sqrt {d}+\sqrt {d} \tan (e+f x)-\sqrt {2} \sqrt {d \tan (e+f x)}\right )}{8 \sqrt {2} f}-\frac {3 d^{5/2} \log \left (\sqrt {d}+\sqrt {d} \tan (e+f x)+\sqrt {2} \sqrt {d \tan (e+f x)}\right )}{8 \sqrt {2} f}-\frac {d \cos ^2(e+f x) (d \tan (e+f x))^{3/2}}{2 f} \\ \end{align*}
Time = 0.40 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.48 \[ \int \cos ^2(e+f x) (d \tan (e+f x))^{5/2} \, dx=-\frac {d^2 \left (3 \arcsin (\cos (e+f x)-\sin (e+f x)) \csc (e+f x)+3 \csc (e+f x) \log \left (\cos (e+f x)+\sin (e+f x)+\sqrt {\sin (2 (e+f x))}\right )+2 \sqrt {\sin (2 (e+f x))}\right ) \sqrt {\sin (2 (e+f x))} \sqrt {d \tan (e+f x)}}{8 f} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(550\) vs. \(2(169)=338\).
Time = 2.48 (sec) , antiderivative size = 551, normalized size of antiderivative = 2.45
method | result | size |
default | \(\frac {\left (\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right ) \left (4 \sqrt {2}\, \cos \left (f x +e \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}}}\, \sin \left (f x +e \right )+4 \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 )-3 \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 )+3 \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 )-6 \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 \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 )\right ) \sqrt {d \tan \left (f x +e \right )}\, d^{2} \sqrt {2}}{16 f \left (\cos \left (f x +e \right )-1\right ) \left (\cos \left (f x +e \right )+1\right )^{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}}}}\) | \(551\) |
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Result contains complex when optimal does not.
Time = 0.42 (sec) , antiderivative size = 969, normalized size of antiderivative = 4.31 \[ \int \cos ^2(e+f x) (d \tan (e+f x))^{5/2} \, dx=\text {Too large to display} \]
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Timed out. \[ \int \cos ^2(e+f x) (d \tan (e+f x))^{5/2} \, dx=\text {Timed out} \]
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Time = 0.30 (sec) , antiderivative size = 194, normalized size of antiderivative = 0.86 \[ \int \cos ^2(e+f x) (d \tan (e+f x))^{5/2} \, dx=\frac {3 \, d^{4} {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {d} + 2 \, \sqrt {d \tan \left (f x + e\right )}\right )}}{2 \, \sqrt {d}}\right )}{\sqrt {d}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {d} - 2 \, \sqrt {d \tan \left (f x + e\right )}\right )}}{2 \, \sqrt {d}}\right )}{\sqrt {d}} - \frac {\sqrt {2} \log \left (d \tan \left (f x + e\right ) + \sqrt {2} \sqrt {d \tan \left (f x + e\right )} \sqrt {d} + d\right )}{\sqrt {d}} + \frac {\sqrt {2} \log \left (d \tan \left (f x + e\right ) - \sqrt {2} \sqrt {d \tan \left (f x + e\right )} \sqrt {d} + d\right )}{\sqrt {d}}\right )} - \frac {8 \, \left (d \tan \left (f x + e\right )\right )^{\frac {3}{2}} d^{4}}{d^{2} \tan \left (f x + e\right )^{2} + d^{2}}}{16 \, d f} \]
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Time = 0.42 (sec) , antiderivative size = 231, normalized size of antiderivative = 1.03 \[ \int \cos ^2(e+f x) (d \tan (e+f x))^{5/2} \, dx=-\frac {1}{16} \, {\left (\frac {8 \, \sqrt {d \tan \left (f x + e\right )} d^{2} \tan \left (f x + e\right )}{{\left (d^{2} \tan \left (f x + e\right )^{2} + d^{2}\right )} f} - \frac {6 \, \sqrt {2} {\left | d \right |}^{\frac {3}{2}} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {{\left | d \right |}} + 2 \, \sqrt {d \tan \left (f x + e\right )}\right )}}{2 \, \sqrt {{\left | d \right |}}}\right )}{d f} - \frac {6 \, \sqrt {2} {\left | d \right |}^{\frac {3}{2}} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {{\left | d \right |}} - 2 \, \sqrt {d \tan \left (f x + e\right )}\right )}}{2 \, \sqrt {{\left | d \right |}}}\right )}{d f} + \frac {3 \, \sqrt {2} {\left | d \right |}^{\frac {3}{2}} \log \left (d \tan \left (f x + e\right ) + \sqrt {2} \sqrt {d \tan \left (f x + e\right )} \sqrt {{\left | d \right |}} + {\left | d \right |}\right )}{d f} - \frac {3 \, \sqrt {2} {\left | d \right |}^{\frac {3}{2}} \log \left (d \tan \left (f x + e\right ) - \sqrt {2} \sqrt {d \tan \left (f x + e\right )} \sqrt {{\left | d \right |}} + {\left | d \right |}\right )}{d f}\right )} d^{2} \]
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Timed out. \[ \int \cos ^2(e+f x) (d \tan (e+f x))^{5/2} \, dx=\int {\cos \left (e+f\,x\right )}^2\,{\left (d\,\mathrm {tan}\left (e+f\,x\right )\right )}^{5/2} \,d x \]
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