Integrand size = 28, antiderivative size = 249 \[ \int \frac {1}{(a+a \sec (e+f x))^{3/2} (c-c \sec (e+f x))^3} \, dx=\frac {2 \arctan \left (\frac {\sqrt {a} \tan (e+f x)}{\sqrt {a+a \sec (e+f x)}}\right )}{a^{3/2} c^3 f}-\frac {11 \arctan \left (\frac {\sqrt {a} \tan (e+f x)}{\sqrt {2} \sqrt {a+a \sec (e+f x)}}\right )}{16 \sqrt {2} a^{3/2} c^3 f}+\frac {21 \cot (e+f x) \sqrt {a+a \sec (e+f x)}}{16 a^2 c^3 f}-\frac {5 \cot ^3(e+f x) (a+a \sec (e+f x))^{3/2}}{24 a^3 c^3 f}-\frac {3 \cot ^5(e+f x) (a+a \sec (e+f x))^{5/2}}{20 a^4 c^3 f}+\frac {\cos (e+f x) \cot ^5(e+f x) \sec ^2\left (\frac {1}{2} (e+f x)\right ) (a+a \sec (e+f x))^{5/2}}{4 a^4 c^3 f} \]
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Time = 0.44 (sec) , antiderivative size = 249, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {3989, 3972, 483, 597, 536, 209} \[ \int \frac {1}{(a+a \sec (e+f x))^{3/2} (c-c \sec (e+f x))^3} \, dx=\frac {2 \arctan \left (\frac {\sqrt {a} \tan (e+f x)}{\sqrt {a \sec (e+f x)+a}}\right )}{a^{3/2} c^3 f}-\frac {11 \arctan \left (\frac {\sqrt {a} \tan (e+f x)}{\sqrt {2} \sqrt {a \sec (e+f x)+a}}\right )}{16 \sqrt {2} a^{3/2} c^3 f}-\frac {3 \cot ^5(e+f x) (a \sec (e+f x)+a)^{5/2}}{20 a^4 c^3 f}+\frac {\cos (e+f x) \cot ^5(e+f x) \sec ^2\left (\frac {1}{2} (e+f x)\right ) (a \sec (e+f x)+a)^{5/2}}{4 a^4 c^3 f}-\frac {5 \cot ^3(e+f x) (a \sec (e+f x)+a)^{3/2}}{24 a^3 c^3 f}+\frac {21 \cot (e+f x) \sqrt {a \sec (e+f x)+a}}{16 a^2 c^3 f} \]
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Rule 209
Rule 483
Rule 536
Rule 597
Rule 3972
Rule 3989
Rubi steps \begin{align*} \text {integral}& = -\frac {\int \cot ^6(e+f x) (a+a \sec (e+f x))^{3/2} \, dx}{a^3 c^3} \\ & = \frac {2 \text {Subst}\left (\int \frac {1}{x^6 \left (1+a x^2\right ) \left (2+a x^2\right )^2} \, dx,x,-\frac {\tan (e+f x)}{\sqrt {a+a \sec (e+f x)}}\right )}{a^4 c^3 f} \\ & = \frac {\cos (e+f x) \cot ^5(e+f x) \sec ^2\left (\frac {1}{2} (e+f x)\right ) (a+a \sec (e+f x))^{5/2}}{4 a^4 c^3 f}+\frac {\text {Subst}\left (\int \frac {-3 a-7 a^2 x^2}{x^6 \left (1+a x^2\right ) \left (2+a x^2\right )} \, dx,x,-\frac {\tan (e+f x)}{\sqrt {a+a \sec (e+f x)}}\right )}{2 a^5 c^3 f} \\ & = -\frac {3 \cot ^5(e+f x) (a+a \sec (e+f x))^{5/2}}{20 a^4 c^3 f}+\frac {\cos (e+f x) \cot ^5(e+f x) \sec ^2\left (\frac {1}{2} (e+f x)\right ) (a+a \sec (e+f x))^{5/2}}{4 a^4 c^3 f}-\frac {\text {Subst}\left (\int \frac {25 a^2-15 a^3 x^2}{x^4 \left (1+a x^2\right ) \left (2+a x^2\right )} \, dx,x,-\frac {\tan (e+f x)}{\sqrt {a+a \sec (e+f x)}}\right )}{20 a^5 c^3 f} \\ & = -\frac {5 \cot ^3(e+f x) (a+a \sec (e+f x))^{3/2}}{24 a^3 c^3 f}-\frac {3 \cot ^5(e+f x) (a+a \sec (e+f x))^{5/2}}{20 a^4 c^3 f}+\frac {\cos (e+f x) \cot ^5(e+f x) \sec ^2\left (\frac {1}{2} (e+f x)\right ) (a+a \sec (e+f x))^{5/2}}{4 a^4 c^3 f}+\frac {\text {Subst}\left (\int \frac {315 a^3+75 a^4 x^2}{x^2 \left (1+a x^2\right ) \left (2+a x^2\right )} \, dx,x,-\frac {\tan (e+f x)}{\sqrt {a+a \sec (e+f x)}}\right )}{120 a^5 c^3 f} \\ & = \frac {21 \cot (e+f x) \sqrt {a+a \sec (e+f x)}}{16 a^2 c^3 f}-\frac {5 \cot ^3(e+f x) (a+a \sec (e+f x))^{3/2}}{24 a^3 c^3 f}-\frac {3 \cot ^5(e+f x) (a+a \sec (e+f x))^{5/2}}{20 a^4 c^3 f}+\frac {\cos (e+f x) \cot ^5(e+f x) \sec ^2\left (\frac {1}{2} (e+f x)\right ) (a+a \sec (e+f x))^{5/2}}{4 a^4 c^3 f}-\frac {\text {Subst}\left (\int \frac {795 a^4+315 a^5 x^2}{\left (1+a x^2\right ) \left (2+a x^2\right )} \, dx,x,-\frac {\tan (e+f x)}{\sqrt {a+a \sec (e+f x)}}\right )}{240 a^5 c^3 f} \\ & = \frac {21 \cot (e+f x) \sqrt {a+a \sec (e+f x)}}{16 a^2 c^3 f}-\frac {5 \cot ^3(e+f x) (a+a \sec (e+f x))^{3/2}}{24 a^3 c^3 f}-\frac {3 \cot ^5(e+f x) (a+a \sec (e+f x))^{5/2}}{20 a^4 c^3 f}+\frac {\cos (e+f x) \cot ^5(e+f x) \sec ^2\left (\frac {1}{2} (e+f x)\right ) (a+a \sec (e+f x))^{5/2}}{4 a^4 c^3 f}+\frac {11 \text {Subst}\left (\int \frac {1}{2+a x^2} \, dx,x,-\frac {\tan (e+f x)}{\sqrt {a+a \sec (e+f x)}}\right )}{16 a c^3 f}-\frac {2 \text {Subst}\left (\int \frac {1}{1+a x^2} \, dx,x,-\frac {\tan (e+f x)}{\sqrt {a+a \sec (e+f x)}}\right )}{a c^3 f} \\ & = \frac {2 \arctan \left (\frac {\sqrt {a} \tan (e+f x)}{\sqrt {a+a \sec (e+f x)}}\right )}{a^{3/2} c^3 f}-\frac {11 \arctan \left (\frac {\sqrt {a} \tan (e+f x)}{\sqrt {2} \sqrt {a+a \sec (e+f x)}}\right )}{16 \sqrt {2} a^{3/2} c^3 f}+\frac {21 \cot (e+f x) \sqrt {a+a \sec (e+f x)}}{16 a^2 c^3 f}-\frac {5 \cot ^3(e+f x) (a+a \sec (e+f x))^{3/2}}{24 a^3 c^3 f}-\frac {3 \cot ^5(e+f x) (a+a \sec (e+f x))^{5/2}}{20 a^4 c^3 f}+\frac {\cos (e+f x) \cot ^5(e+f x) \sec ^2\left (\frac {1}{2} (e+f x)\right ) (a+a \sec (e+f x))^{5/2}}{4 a^4 c^3 f} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.60 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.41 \[ \int \frac {1}{(a+a \sec (e+f x))^{3/2} (c-c \sec (e+f x))^3} \, dx=-\frac {\left (-10+11 \operatorname {Hypergeometric2F1}\left (-\frac {5}{2},1,-\frac {3}{2},\frac {1}{2} (1-\sec (e+f x))\right ) (1+\sec (e+f x))-8 \operatorname {Hypergeometric2F1}\left (-\frac {5}{2},1,-\frac {3}{2},1-\sec (e+f x)\right ) (1+\sec (e+f x))\right ) \tan (e+f x)}{20 c^3 f (-1+\sec (e+f x))^3 (a (1+\sec (e+f x)))^{3/2}} \]
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Time = 2.75 (sec) , antiderivative size = 232, normalized size of antiderivative = 0.93
method | result | size |
default | \(-\frac {\left (165 \sin \left (f x +e \right )^{5} \sqrt {2}\, \sqrt {-\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \ln \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )+\sqrt {\cot \left (f x +e \right )^{2}-2 \csc \left (f x +e \right ) \cot \left (f x +e \right )+\csc \left (f x +e \right )^{2}-1}\right )-960 \sin \left (f x +e \right )^{5} \sqrt {-\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \operatorname {arctanh}\left (\frac {\sin \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right ) \sqrt {-\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}}\right )-898 \cos \left (f x +e \right )^{5}-196 \cos \left (f x +e \right )^{4}+1432 \cos \left (f x +e \right )^{3}+100 \cos \left (f x +e \right )^{2}-630 \cos \left (f x +e \right )\right ) \sqrt {a \left (\sec \left (f x +e \right )+1\right )}\, \csc \left (f x +e \right )^{5}}{480 c^{3} f \,a^{2}}\) | \(232\) |
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Time = 0.40 (sec) , antiderivative size = 714, normalized size of antiderivative = 2.87 \[ \int \frac {1}{(a+a \sec (e+f x))^{3/2} (c-c \sec (e+f x))^3} \, dx=\left [-\frac {165 \, \sqrt {2} {\left (\cos \left (f x + e\right )^{3} - \cos \left (f x + e\right )^{2} - \cos \left (f x + e\right ) + 1\right )} \sqrt {-a} \log \left (-\frac {2 \, \sqrt {2} \sqrt {-a} \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \cos \left (f x + e\right ) \sin \left (f x + e\right ) - 3 \, a \cos \left (f x + e\right )^{2} - 2 \, a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )^{2} + 2 \, \cos \left (f x + e\right ) + 1}\right ) \sin \left (f x + e\right ) + 480 \, {\left (\cos \left (f x + e\right )^{3} - \cos \left (f x + e\right )^{2} - \cos \left (f x + e\right ) + 1\right )} \sqrt {-a} \log \left (-\frac {8 \, a \cos \left (f x + e\right )^{3} + 4 \, {\left (2 \, \cos \left (f x + e\right )^{2} - \cos \left (f x + e\right )\right )} \sqrt {-a} \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \sin \left (f x + e\right ) - 7 \, a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right ) + 1}\right ) \sin \left (f x + e\right ) - 4 \, {\left (449 \, \cos \left (f x + e\right )^{4} - 351 \, \cos \left (f x + e\right )^{3} - 365 \, \cos \left (f x + e\right )^{2} + 315 \, \cos \left (f x + e\right )\right )} \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}}}{960 \, {\left (a^{2} c^{3} f \cos \left (f x + e\right )^{3} - a^{2} c^{3} f \cos \left (f x + e\right )^{2} - a^{2} c^{3} f \cos \left (f x + e\right ) + a^{2} c^{3} f\right )} \sin \left (f x + e\right )}, \frac {165 \, \sqrt {2} {\left (\cos \left (f x + e\right )^{3} - \cos \left (f x + e\right )^{2} - \cos \left (f x + e\right ) + 1\right )} \sqrt {a} \arctan \left (\frac {\sqrt {2} \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \cos \left (f x + e\right )}{\sqrt {a} \sin \left (f x + e\right )}\right ) \sin \left (f x + e\right ) + 480 \, {\left (\cos \left (f x + e\right )^{3} - \cos \left (f x + e\right )^{2} - \cos \left (f x + e\right ) + 1\right )} \sqrt {a} \arctan \left (\frac {2 \, \sqrt {a} \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \cos \left (f x + e\right ) \sin \left (f x + e\right )}{2 \, a \cos \left (f x + e\right )^{2} + a \cos \left (f x + e\right ) - a}\right ) \sin \left (f x + e\right ) + 2 \, {\left (449 \, \cos \left (f x + e\right )^{4} - 351 \, \cos \left (f x + e\right )^{3} - 365 \, \cos \left (f x + e\right )^{2} + 315 \, \cos \left (f x + e\right )\right )} \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}}}{480 \, {\left (a^{2} c^{3} f \cos \left (f x + e\right )^{3} - a^{2} c^{3} f \cos \left (f x + e\right )^{2} - a^{2} c^{3} f \cos \left (f x + e\right ) + a^{2} c^{3} f\right )} \sin \left (f x + e\right )}\right ] \]
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\[ \int \frac {1}{(a+a \sec (e+f x))^{3/2} (c-c \sec (e+f x))^3} \, dx=- \frac {\int \frac {1}{a \sqrt {a \sec {\left (e + f x \right )} + a} \sec ^{4}{\left (e + f x \right )} - 2 a \sqrt {a \sec {\left (e + f x \right )} + a} \sec ^{3}{\left (e + f x \right )} + 2 a \sqrt {a \sec {\left (e + f x \right )} + a} \sec {\left (e + f x \right )} - a \sqrt {a \sec {\left (e + f x \right )} + a}}\, dx}{c^{3}} \]
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\[ \int \frac {1}{(a+a \sec (e+f x))^{3/2} (c-c \sec (e+f x))^3} \, dx=\int { -\frac {1}{{\left (a \sec \left (f x + e\right ) + a\right )}^{\frac {3}{2}} {\left (c \sec \left (f x + e\right ) - c\right )}^{3}} \,d x } \]
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Exception generated. \[ \int \frac {1}{(a+a \sec (e+f x))^{3/2} (c-c \sec (e+f x))^3} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {1}{(a+a \sec (e+f x))^{3/2} (c-c \sec (e+f x))^3} \, dx=\int \frac {1}{{\left (a+\frac {a}{\cos \left (e+f\,x\right )}\right )}^{3/2}\,{\left (c-\frac {c}{\cos \left (e+f\,x\right )}\right )}^3} \,d x \]
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