Integrand size = 30, antiderivative size = 185 \[ \int \sqrt {a+a \sec (e+f x)} (c-c \sec (e+f x))^{7/2} \, dx=\frac {a c^4 \log (\cos (e+f x)) \tan (e+f x)}{f \sqrt {a+a \sec (e+f x)} \sqrt {c-c \sec (e+f x)}}-\frac {a c^3 \sqrt {c-c \sec (e+f x)} \tan (e+f x)}{f \sqrt {a+a \sec (e+f x)}}-\frac {a c^2 (c-c \sec (e+f x))^{3/2} \tan (e+f x)}{2 f \sqrt {a+a \sec (e+f x)}}-\frac {a c (c-c \sec (e+f x))^{5/2} \tan (e+f x)}{3 f \sqrt {a+a \sec (e+f x)}} \]
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Time = 0.48 (sec) , antiderivative size = 185, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {3991, 3990, 3556} \[ \int \sqrt {a+a \sec (e+f x)} (c-c \sec (e+f x))^{7/2} \, dx=\frac {a c^4 \tan (e+f x) \log (\cos (e+f x))}{f \sqrt {a \sec (e+f x)+a} \sqrt {c-c \sec (e+f x)}}-\frac {a c^3 \tan (e+f x) \sqrt {c-c \sec (e+f x)}}{f \sqrt {a \sec (e+f x)+a}}-\frac {a c^2 \tan (e+f x) (c-c \sec (e+f x))^{3/2}}{2 f \sqrt {a \sec (e+f x)+a}}-\frac {a c \tan (e+f x) (c-c \sec (e+f x))^{5/2}}{3 f \sqrt {a \sec (e+f x)+a}} \]
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Rule 3556
Rule 3990
Rule 3991
Rubi steps \begin{align*} \text {integral}& = -\frac {a c (c-c \sec (e+f x))^{5/2} \tan (e+f x)}{3 f \sqrt {a+a \sec (e+f x)}}+c \int \sqrt {a+a \sec (e+f x)} (c-c \sec (e+f x))^{5/2} \, dx \\ & = -\frac {a c^2 (c-c \sec (e+f x))^{3/2} \tan (e+f x)}{2 f \sqrt {a+a \sec (e+f x)}}-\frac {a c (c-c \sec (e+f x))^{5/2} \tan (e+f x)}{3 f \sqrt {a+a \sec (e+f x)}}+c^2 \int \sqrt {a+a \sec (e+f x)} (c-c \sec (e+f x))^{3/2} \, dx \\ & = -\frac {a c^3 \sqrt {c-c \sec (e+f x)} \tan (e+f x)}{f \sqrt {a+a \sec (e+f x)}}-\frac {a c^2 (c-c \sec (e+f x))^{3/2} \tan (e+f x)}{2 f \sqrt {a+a \sec (e+f x)}}-\frac {a c (c-c \sec (e+f x))^{5/2} \tan (e+f x)}{3 f \sqrt {a+a \sec (e+f x)}}+c^3 \int \sqrt {a+a \sec (e+f x)} \sqrt {c-c \sec (e+f x)} \, dx \\ & = -\frac {a c^3 \sqrt {c-c \sec (e+f x)} \tan (e+f x)}{f \sqrt {a+a \sec (e+f x)}}-\frac {a c^2 (c-c \sec (e+f x))^{3/2} \tan (e+f x)}{2 f \sqrt {a+a \sec (e+f x)}}-\frac {a c (c-c \sec (e+f x))^{5/2} \tan (e+f x)}{3 f \sqrt {a+a \sec (e+f x)}}-\frac {\left (a c^4 \tan (e+f x)\right ) \int \tan (e+f x) \, dx}{\sqrt {a+a \sec (e+f x)} \sqrt {c-c \sec (e+f x)}} \\ & = \frac {a c^4 \log (\cos (e+f x)) \tan (e+f x)}{f \sqrt {a+a \sec (e+f x)} \sqrt {c-c \sec (e+f x)}}-\frac {a c^3 \sqrt {c-c \sec (e+f x)} \tan (e+f x)}{f \sqrt {a+a \sec (e+f x)}}-\frac {a c^2 (c-c \sec (e+f x))^{3/2} \tan (e+f x)}{2 f \sqrt {a+a \sec (e+f x)}}-\frac {a c (c-c \sec (e+f x))^{5/2} \tan (e+f x)}{3 f \sqrt {a+a \sec (e+f x)}} \\ \end{align*}
Time = 1.82 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.45 \[ \int \sqrt {a+a \sec (e+f x)} (c-c \sec (e+f x))^{7/2} \, dx=\frac {a c^4 \left (6 \log (\cos (e+f x))+18 \sec (e+f x)-9 \sec ^2(e+f x)+2 \sec ^3(e+f x)\right ) \tan (e+f x)}{6 f \sqrt {a (1+\sec (e+f x))} \sqrt {c-c \sec (e+f x)}} \]
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Time = 2.49 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.90
method | result | size |
default | \(\frac {c^{3} \sqrt {a \left (\sec \left (f x +e \right )+1\right )}\, \sqrt {-c \left (\sec \left (f x +e \right )-1\right )}\, \left (\sec \left (f x +e \right )-1\right )^{3} \left (6 \cos \left (f x +e \right )^{3} \ln \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1\right )-6 \cos \left (f x +e \right )^{3} \ln \left (\frac {2}{\cos \left (f x +e \right )+1}\right )+6 \cos \left (f x +e \right )^{3} \ln \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )-1\right )+29 \cos \left (f x +e \right )^{3}+18 \cos \left (f x +e \right )^{2}-9 \cos \left (f x +e \right )+2\right ) \cot \left (f x +e \right )}{6 f \left (\cos \left (f x +e \right )-1\right )^{3}}\) | \(167\) |
risch | \(\frac {c^{3} \sqrt {\frac {a \left ({\mathrm e}^{i \left (f x +e \right )}+1\right )^{2}}{1+{\mathrm e}^{2 i \left (f x +e \right )}}}\, \sqrt {\frac {c \left ({\mathrm e}^{i \left (f x +e \right )}-1\right )^{2}}{1+{\mathrm e}^{2 i \left (f x +e \right )}}}\, \left (-18 i {\mathrm e}^{5 i \left (f x +e \right )}-3 \,{\mathrm e}^{6 i \left (f x +e \right )} f x -18 i {\mathrm e}^{i \left (f x +e \right )}-6 \,{\mathrm e}^{6 i \left (f x +e \right )} e -9 \,{\mathrm e}^{4 i \left (f x +e \right )} f x -9 i {\mathrm e}^{4 i \left (f x +e \right )} \ln \left (1+{\mathrm e}^{2 i \left (f x +e \right )}\right )-18 \,{\mathrm e}^{4 i \left (f x +e \right )} e -9 \,{\mathrm e}^{2 i \left (f x +e \right )} f x +18 i {\mathrm e}^{4 i \left (f x +e \right )}-3 i {\mathrm e}^{6 i \left (f x +e \right )} \ln \left (1+{\mathrm e}^{2 i \left (f x +e \right )}\right )+18 i {\mathrm e}^{2 i \left (f x +e \right )}-9 i {\mathrm e}^{2 i \left (f x +e \right )} \ln \left (1+{\mathrm e}^{2 i \left (f x +e \right )}\right )-44 i {\mathrm e}^{3 i \left (f x +e \right )}-3 i \ln \left (1+{\mathrm e}^{2 i \left (f x +e \right )}\right )-18 \,{\mathrm e}^{2 i \left (f x +e \right )} e -3 f x -6 e \right )}{3 \left ({\mathrm e}^{i \left (f x +e \right )}+1\right ) \left ({\mathrm e}^{i \left (f x +e \right )}-1\right ) \left (1+{\mathrm e}^{2 i \left (f x +e \right )}\right )^{2} f}\) | \(338\) |
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Time = 0.33 (sec) , antiderivative size = 459, normalized size of antiderivative = 2.48 \[ \int \sqrt {a+a \sec (e+f x)} (c-c \sec (e+f x))^{7/2} \, dx=\left [-\frac {{\left (11 \, c^{3} \cos \left (f x + e\right )^{2} - 7 \, c^{3} \cos \left (f x + e\right ) + 2 \, c^{3}\right )} \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \sqrt {\frac {c \cos \left (f x + e\right ) - c}{\cos \left (f x + e\right )}} \sin \left (f x + e\right ) - 3 \, {\left (c^{3} \cos \left (f x + e\right )^{3} + c^{3} \cos \left (f x + e\right )^{2}\right )} \sqrt {-a c} \log \left (\frac {a c \cos \left (f x + e\right )^{4} - {\left (\cos \left (f x + e\right )^{3} + \cos \left (f x + e\right )\right )} \sqrt {-a c} \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \sqrt {\frac {c \cos \left (f x + e\right ) - c}{\cos \left (f x + e\right )}} \sin \left (f x + e\right ) + a c}{2 \, \cos \left (f x + e\right )^{2}}\right )}{6 \, {\left (f \cos \left (f x + e\right )^{3} + f \cos \left (f x + e\right )^{2}\right )}}, -\frac {{\left (11 \, c^{3} \cos \left (f x + e\right )^{2} - 7 \, c^{3} \cos \left (f x + e\right ) + 2 \, c^{3}\right )} \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \sqrt {\frac {c \cos \left (f x + e\right ) - c}{\cos \left (f x + e\right )}} \sin \left (f x + e\right ) - 6 \, {\left (c^{3} \cos \left (f x + e\right )^{3} + c^{3} \cos \left (f x + e\right )^{2}\right )} \sqrt {a c} \arctan \left (\frac {\sqrt {a c} \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \sqrt {\frac {c \cos \left (f x + e\right ) - c}{\cos \left (f x + e\right )}} \cos \left (f x + e\right ) \sin \left (f x + e\right )}{a c \cos \left (f x + e\right )^{2} + a c}\right )}{6 \, {\left (f \cos \left (f x + e\right )^{3} + f \cos \left (f x + e\right )^{2}\right )}}\right ] \]
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Timed out. \[ \int \sqrt {a+a \sec (e+f x)} (c-c \sec (e+f x))^{7/2} \, dx=\text {Timed out} \]
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Leaf count of result is larger than twice the leaf count of optimal. 1289 vs. \(2 (165) = 330\).
Time = 0.46 (sec) , antiderivative size = 1289, normalized size of antiderivative = 6.97 \[ \int \sqrt {a+a \sec (e+f x)} (c-c \sec (e+f x))^{7/2} \, dx=\text {Too large to display} \]
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\[ \int \sqrt {a+a \sec (e+f x)} (c-c \sec (e+f x))^{7/2} \, dx=\int { \sqrt {a \sec \left (f x + e\right ) + a} {\left (-c \sec \left (f x + e\right ) + c\right )}^{\frac {7}{2}} \,d x } \]
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Timed out. \[ \int \sqrt {a+a \sec (e+f x)} (c-c \sec (e+f x))^{7/2} \, dx=\int \sqrt {a+\frac {a}{\cos \left (e+f\,x\right )}}\,{\left (c-\frac {c}{\cos \left (e+f\,x\right )}\right )}^{7/2} \,d x \]
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