Integrand size = 25, antiderivative size = 325 \[ \int \frac {1}{(e \cot (c+d x))^{5/2} (a+a \sec (c+d x))} \, dx=\frac {2 \cos (c+d x) \cot (c+d x)}{a d (e \cot (c+d x))^{5/2}}-\frac {2 \cos (c+d x) \cot ^2(c+d x) E\left (\left .c-\frac {\pi }{4}+d x\right |2\right )}{a d (e \cot (c+d x))^{5/2} \sqrt {\sin (2 c+2 d x)}}+\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} a d (e \cot (c+d x))^{5/2} \tan ^{\frac {5}{2}}(c+d x)}-\frac {\arctan \left (1+\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} a d (e \cot (c+d x))^{5/2} \tan ^{\frac {5}{2}}(c+d x)}-\frac {\log \left (1-\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} a d (e \cot (c+d x))^{5/2} \tan ^{\frac {5}{2}}(c+d x)}+\frac {\log \left (1+\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} a d (e \cot (c+d x))^{5/2} \tan ^{\frac {5}{2}}(c+d x)} \]
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Time = 0.41 (sec) , antiderivative size = 325, normalized size of antiderivative = 1.00, number of steps used = 18, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {3985, 3973, 3969, 3557, 335, 303, 1176, 631, 210, 1179, 642, 2693, 2695, 2652, 2719} \[ \int \frac {1}{(e \cot (c+d x))^{5/2} (a+a \sec (c+d x))} \, dx=\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} a d \tan ^{\frac {5}{2}}(c+d x) (e \cot (c+d x))^{5/2}}-\frac {\arctan \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2} a d \tan ^{\frac {5}{2}}(c+d x) (e \cot (c+d x))^{5/2}}+\frac {2 \cos (c+d x) \cot (c+d x)}{a d (e \cot (c+d x))^{5/2}}-\frac {\log \left (\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2} a d \tan ^{\frac {5}{2}}(c+d x) (e \cot (c+d x))^{5/2}}+\frac {\log \left (\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2} a d \tan ^{\frac {5}{2}}(c+d x) (e \cot (c+d x))^{5/2}}-\frac {2 \cos (c+d x) \cot ^2(c+d x) E\left (\left .c+d x-\frac {\pi }{4}\right |2\right )}{a d \sqrt {\sin (2 c+2 d x)} (e \cot (c+d x))^{5/2}} \]
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Rule 210
Rule 303
Rule 335
Rule 631
Rule 642
Rule 1176
Rule 1179
Rule 2652
Rule 2693
Rule 2695
Rule 2719
Rule 3557
Rule 3969
Rule 3973
Rule 3985
Rubi steps \begin{align*} \text {integral}& = \frac {\int \frac {\tan ^{\frac {5}{2}}(c+d x)}{a+a \sec (c+d x)} \, dx}{(e \cot (c+d x))^{5/2} \tan ^{\frac {5}{2}}(c+d x)} \\ & = \frac {\int (-a+a \sec (c+d x)) \sqrt {\tan (c+d x)} \, dx}{a^2 (e \cot (c+d x))^{5/2} \tan ^{\frac {5}{2}}(c+d x)} \\ & = -\frac {\int \sqrt {\tan (c+d x)} \, dx}{a (e \cot (c+d x))^{5/2} \tan ^{\frac {5}{2}}(c+d x)}+\frac {\int \sec (c+d x) \sqrt {\tan (c+d x)} \, dx}{a (e \cot (c+d x))^{5/2} \tan ^{\frac {5}{2}}(c+d x)} \\ & = \frac {2 \cos (c+d x) \cot (c+d x)}{a d (e \cot (c+d x))^{5/2}}-\frac {2 \int \cos (c+d x) \sqrt {\tan (c+d x)} \, dx}{a (e \cot (c+d x))^{5/2} \tan ^{\frac {5}{2}}(c+d x)}-\frac {\text {Subst}\left (\int \frac {\sqrt {x}}{1+x^2} \, dx,x,\tan (c+d x)\right )}{a d (e \cot (c+d x))^{5/2} \tan ^{\frac {5}{2}}(c+d x)} \\ & = \frac {2 \cos (c+d x) \cot (c+d x)}{a d (e \cot (c+d x))^{5/2}}-\frac {\left (2 \cos ^{\frac {5}{2}}(c+d x)\right ) \int \sqrt {\cos (c+d x)} \sqrt {\sin (c+d x)} \, dx}{a (e \cot (c+d x))^{5/2} \sin ^{\frac {5}{2}}(c+d x)}-\frac {2 \text {Subst}\left (\int \frac {x^2}{1+x^4} \, dx,x,\sqrt {\tan (c+d x)}\right )}{a d (e \cot (c+d x))^{5/2} \tan ^{\frac {5}{2}}(c+d x)} \\ & = \frac {2 \cos (c+d x) \cot (c+d x)}{a d (e \cot (c+d x))^{5/2}}-\frac {\left (2 \cos (c+d x) \cot ^2(c+d x)\right ) \int \sqrt {\sin (2 c+2 d x)} \, dx}{a (e \cot (c+d x))^{5/2} \sqrt {\sin (2 c+2 d x)}}+\frac {\text {Subst}\left (\int \frac {1-x^2}{1+x^4} \, dx,x,\sqrt {\tan (c+d x)}\right )}{a d (e \cot (c+d x))^{5/2} \tan ^{\frac {5}{2}}(c+d x)}-\frac {\text {Subst}\left (\int \frac {1+x^2}{1+x^4} \, dx,x,\sqrt {\tan (c+d x)}\right )}{a d (e \cot (c+d x))^{5/2} \tan ^{\frac {5}{2}}(c+d x)} \\ & = \frac {2 \cos (c+d x) \cot (c+d x)}{a d (e \cot (c+d x))^{5/2}}-\frac {2 \cos (c+d x) \cot ^2(c+d x) E\left (\left .c-\frac {\pi }{4}+d x\right |2\right )}{a d (e \cot (c+d x))^{5/2} \sqrt {\sin (2 c+2 d x)}}-\frac {\text {Subst}\left (\int \frac {1}{1-\sqrt {2} x+x^2} \, dx,x,\sqrt {\tan (c+d x)}\right )}{2 a d (e \cot (c+d x))^{5/2} \tan ^{\frac {5}{2}}(c+d x)}-\frac {\text {Subst}\left (\int \frac {1}{1+\sqrt {2} x+x^2} \, dx,x,\sqrt {\tan (c+d x)}\right )}{2 a d (e \cot (c+d x))^{5/2} \tan ^{\frac {5}{2}}(c+d x)}-\frac {\text {Subst}\left (\int \frac {\sqrt {2}+2 x}{-1-\sqrt {2} x-x^2} \, dx,x,\sqrt {\tan (c+d x)}\right )}{2 \sqrt {2} a d (e \cot (c+d x))^{5/2} \tan ^{\frac {5}{2}}(c+d x)}-\frac {\text {Subst}\left (\int \frac {\sqrt {2}-2 x}{-1+\sqrt {2} x-x^2} \, dx,x,\sqrt {\tan (c+d x)}\right )}{2 \sqrt {2} a d (e \cot (c+d x))^{5/2} \tan ^{\frac {5}{2}}(c+d x)} \\ & = \frac {2 \cos (c+d x) \cot (c+d x)}{a d (e \cot (c+d x))^{5/2}}-\frac {2 \cos (c+d x) \cot ^2(c+d x) E\left (\left .c-\frac {\pi }{4}+d x\right |2\right )}{a d (e \cot (c+d x))^{5/2} \sqrt {\sin (2 c+2 d x)}}-\frac {\log \left (1-\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} a d (e \cot (c+d x))^{5/2} \tan ^{\frac {5}{2}}(c+d x)}+\frac {\log \left (1+\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} a d (e \cot (c+d x))^{5/2} \tan ^{\frac {5}{2}}(c+d x)}-\frac {\text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} a d (e \cot (c+d x))^{5/2} \tan ^{\frac {5}{2}}(c+d x)}+\frac {\text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} a d (e \cot (c+d x))^{5/2} \tan ^{\frac {5}{2}}(c+d x)} \\ & = \frac {2 \cos (c+d x) \cot (c+d x)}{a d (e \cot (c+d x))^{5/2}}-\frac {2 \cos (c+d x) \cot ^2(c+d x) E\left (\left .c-\frac {\pi }{4}+d x\right |2\right )}{a d (e \cot (c+d x))^{5/2} \sqrt {\sin (2 c+2 d x)}}+\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} a d (e \cot (c+d x))^{5/2} \tan ^{\frac {5}{2}}(c+d x)}-\frac {\arctan \left (1+\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} a d (e \cot (c+d x))^{5/2} \tan ^{\frac {5}{2}}(c+d x)}-\frac {\log \left (1-\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} a d (e \cot (c+d x))^{5/2} \tan ^{\frac {5}{2}}(c+d x)}+\frac {\log \left (1+\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} a d (e \cot (c+d x))^{5/2} \tan ^{\frac {5}{2}}(c+d x)} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 23.23 (sec) , antiderivative size = 194, normalized size of antiderivative = 0.60 \[ \int \frac {1}{(e \cot (c+d x))^{5/2} (a+a \sec (c+d x))} \, dx=-\frac {\sqrt {e \cot (c+d x)} \left (-8 \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},-\tan ^2(c+d x)\right )+3 \sqrt {2} \cot ^{\frac {3}{2}}(c+d x) \left (2 \arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )-2 \arctan \left (1+\sqrt {2} \sqrt {\cot (c+d x)}\right )+\log \left (1-\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )-\log \left (1+\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )\right )\right ) \sec (c+d x) \left (1+\sqrt {\sec ^2(c+d x)}\right ) \sin ^2\left (\frac {1}{2} (c+d x)\right )}{6 a d e^3} \]
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Result contains complex when optimal does not.
Time = 9.77 (sec) , antiderivative size = 1100, normalized size of antiderivative = 3.38
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Timed out. \[ \int \frac {1}{(e \cot (c+d x))^{5/2} (a+a \sec (c+d x))} \, dx=\text {Timed out} \]
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\[ \int \frac {1}{(e \cot (c+d x))^{5/2} (a+a \sec (c+d x))} \, dx=\frac {\int \frac {1}{\left (e \cot {\left (c + d x \right )}\right )^{\frac {5}{2}} \sec {\left (c + d x \right )} + \left (e \cot {\left (c + d x \right )}\right )^{\frac {5}{2}}}\, dx}{a} \]
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\[ \int \frac {1}{(e \cot (c+d x))^{5/2} (a+a \sec (c+d x))} \, dx=\int { \frac {1}{\left (e \cot \left (d x + c\right )\right )^{\frac {5}{2}} {\left (a \sec \left (d x + c\right ) + a\right )}} \,d x } \]
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\[ \int \frac {1}{(e \cot (c+d x))^{5/2} (a+a \sec (c+d x))} \, dx=\int { \frac {1}{\left (e \cot \left (d x + c\right )\right )^{\frac {5}{2}} {\left (a \sec \left (d x + c\right ) + a\right )}} \,d x } \]
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Timed out. \[ \int \frac {1}{(e \cot (c+d x))^{5/2} (a+a \sec (c+d x))} \, dx=\int \frac {\cos \left (c+d\,x\right )}{a\,{\left (e\,\mathrm {cot}\left (c+d\,x\right )\right )}^{5/2}\,\left (\cos \left (c+d\,x\right )+1\right )} \,d x \]
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