Integrand size = 30, antiderivative size = 139 \[ \int (a+a \sec (e+f x))^{5/2} \sqrt {c-c \sec (e+f x)} \, dx=\frac {a^3 c \log (\cos (e+f x)) \tan (e+f x)}{f \sqrt {a+a \sec (e+f x)} \sqrt {c-c \sec (e+f x)}}-\frac {a^2 c \sqrt {a+a \sec (e+f x)} \tan (e+f x)}{f \sqrt {c-c \sec (e+f x)}}-\frac {a c (a+a \sec (e+f x))^{3/2} \tan (e+f x)}{2 f \sqrt {c-c \sec (e+f x)}} \]
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Time = 0.36 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {3991, 3990, 3556} \[ \int (a+a \sec (e+f x))^{5/2} \sqrt {c-c \sec (e+f x)} \, dx=\frac {a^3 c \tan (e+f x) \log (\cos (e+f x))}{f \sqrt {a \sec (e+f x)+a} \sqrt {c-c \sec (e+f x)}}-\frac {a^2 c \tan (e+f x) \sqrt {a \sec (e+f x)+a}}{f \sqrt {c-c \sec (e+f x)}}-\frac {a c \tan (e+f x) (a \sec (e+f x)+a)^{3/2}}{2 f \sqrt {c-c \sec (e+f x)}} \]
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Rule 3556
Rule 3990
Rule 3991
Rubi steps \begin{align*} \text {integral}& = -\frac {a c (a+a \sec (e+f x))^{3/2} \tan (e+f x)}{2 f \sqrt {c-c \sec (e+f x)}}+a \int (a+a \sec (e+f x))^{3/2} \sqrt {c-c \sec (e+f x)} \, dx \\ & = -\frac {a^2 c \sqrt {a+a \sec (e+f x)} \tan (e+f x)}{f \sqrt {c-c \sec (e+f x)}}-\frac {a c (a+a \sec (e+f x))^{3/2} \tan (e+f x)}{2 f \sqrt {c-c \sec (e+f x)}}+a^2 \int \sqrt {a+a \sec (e+f x)} \sqrt {c-c \sec (e+f x)} \, dx \\ & = -\frac {a^2 c \sqrt {a+a \sec (e+f x)} \tan (e+f x)}{f \sqrt {c-c \sec (e+f x)}}-\frac {a c (a+a \sec (e+f x))^{3/2} \tan (e+f x)}{2 f \sqrt {c-c \sec (e+f x)}}-\frac {\left (a^3 c \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^3 c \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^2 c \sqrt {a+a \sec (e+f x)} \tan (e+f x)}{f \sqrt {c-c \sec (e+f x)}}-\frac {a c (a+a \sec (e+f x))^{3/2} \tan (e+f x)}{2 f \sqrt {c-c \sec (e+f x)}} \\ \end{align*}
Time = 0.31 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.52 \[ \int (a+a \sec (e+f x))^{5/2} \sqrt {c-c \sec (e+f x)} \, dx=-\frac {a^3 c \left (-2 \log (\cos (e+f x))+4 \sec (e+f x)+\sec ^2(e+f x)\right ) \tan (e+f x)}{2 f \sqrt {a (1+\sec (e+f x))} \sqrt {c-c \sec (e+f x)}} \]
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Time = 2.24 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.98
method | result | size |
default | \(-\frac {a^{2} \sqrt {-c \left (\sec \left (f x +e \right )-1\right )}\, \sqrt {a \left (\sec \left (f x +e \right )+1\right )}\, \left (2 \ln \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )-1\right ) \cot \left (f x +e \right )+2 \ln \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1\right ) \cot \left (f x +e \right )-2 \cot \left (f x +e \right ) \ln \left (\frac {2}{\cos \left (f x +e \right )+1}\right )-3 \cot \left (f x +e \right )-4 \csc \left (f x +e \right )-\sec \left (f x +e \right ) \csc \left (f x +e \right )\right )}{2 f}\) | \(136\) |
risch | \(\frac {a^{2} \sqrt {\frac {a \left ({\mathrm e}^{i \left (f x +e \right )}+1\right )^{2}}{1+{\mathrm e}^{2 i \left (f x +e \right )}}}\, \sqrt {\frac {c \left ({\mathrm e}^{i \left (f x +e \right )}-1\right )^{2}}{1+{\mathrm e}^{2 i \left (f x +e \right )}}}\, \left (-i {\mathrm e}^{4 i \left (f x +e \right )} \ln \left (1+{\mathrm e}^{2 i \left (f x +e \right )}\right )-{\mathrm e}^{4 i \left (f x +e \right )} f x -2 i {\mathrm e}^{2 i \left (f x +e \right )} \ln \left (1+{\mathrm e}^{2 i \left (f x +e \right )}\right )-2 \,{\mathrm e}^{4 i \left (f x +e \right )} e -2 \,{\mathrm e}^{2 i \left (f x +e \right )} f x +4 i {\mathrm e}^{3 i \left (f x +e \right )}-i \ln \left (1+{\mathrm e}^{2 i \left (f x +e \right )}\right )+2 i {\mathrm e}^{2 i \left (f x +e \right )}+4 i {\mathrm e}^{i \left (f x +e \right )}-4 \,{\mathrm e}^{2 i \left (f x +e \right )} e -f x -2 e \right )}{\left (1+{\mathrm e}^{2 i \left (f x +e \right )}\right ) \left ({\mathrm e}^{i \left (f x +e \right )}+1\right ) \left ({\mathrm e}^{i \left (f x +e \right )}-1\right ) f}\) | \(264\) |
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Time = 0.34 (sec) , antiderivative size = 420, normalized size of antiderivative = 3.02 \[ \int (a+a \sec (e+f x))^{5/2} \sqrt {c-c \sec (e+f x)} \, dx=\left [\frac {{\left (5 \, a^{2} \cos \left (f x + e\right ) + a^{2}\right )} \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \sqrt {\frac {c \cos \left (f x + e\right ) - c}{\cos \left (f x + e\right )}} \sin \left (f x + e\right ) + {\left (a^{2} \cos \left (f x + e\right )^{2} + a^{2} \cos \left (f x + e\right )\right )} \sqrt {-a c} \log \left (\frac {a c \cos \left (f x + e\right )^{4} - {\left (\cos \left (f x + e\right )^{3} + \cos \left (f x + e\right )\right )} \sqrt {-a c} \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \sqrt {\frac {c \cos \left (f x + e\right ) - c}{\cos \left (f x + e\right )}} \sin \left (f x + e\right ) + a c}{2 \, \cos \left (f x + e\right )^{2}}\right )}{2 \, {\left (f \cos \left (f x + e\right )^{2} + f \cos \left (f x + e\right )\right )}}, \frac {{\left (5 \, a^{2} \cos \left (f x + e\right ) + a^{2}\right )} \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \sqrt {\frac {c \cos \left (f x + e\right ) - c}{\cos \left (f x + e\right )}} \sin \left (f x + e\right ) + 2 \, {\left (a^{2} \cos \left (f x + e\right )^{2} + a^{2} \cos \left (f x + e\right )\right )} \sqrt {a c} \arctan \left (\frac {\sqrt {a c} \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \sqrt {\frac {c \cos \left (f x + e\right ) - c}{\cos \left (f x + e\right )}} \cos \left (f x + e\right ) \sin \left (f x + e\right )}{a c \cos \left (f x + e\right )^{2} + a c}\right )}{2 \, {\left (f \cos \left (f x + e\right )^{2} + f \cos \left (f x + e\right )\right )}}\right ] \]
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Timed out. \[ \int (a+a \sec (e+f x))^{5/2} \sqrt {c-c \sec (e+f x)} \, dx=\text {Timed out} \]
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Leaf count of result is larger than twice the leaf count of optimal. 710 vs. \(2 (125) = 250\).
Time = 0.40 (sec) , antiderivative size = 710, normalized size of antiderivative = 5.11 \[ \int (a+a \sec (e+f x))^{5/2} \sqrt {c-c \sec (e+f x)} \, dx=-\frac {{\left ({\left (f x + e\right )} a^{2} \cos \left (4 \, f x + 4 \, e\right )^{2} + 4 \, {\left (f x + e\right )} a^{2} \cos \left (2 \, f x + 2 \, e\right )^{2} + {\left (f x + e\right )} a^{2} \sin \left (4 \, f x + 4 \, e\right )^{2} + 4 \, {\left (f x + e\right )} a^{2} \sin \left (2 \, f x + 2 \, e\right )^{2} + 4 \, {\left (f x + e\right )} a^{2} \cos \left (2 \, f x + 2 \, e\right ) + {\left (f x + e\right )} a^{2} + 2 \, a^{2} \sin \left (2 \, f x + 2 \, e\right ) - {\left (a^{2} \cos \left (4 \, f x + 4 \, e\right )^{2} + 4 \, a^{2} \cos \left (2 \, f x + 2 \, e\right )^{2} + a^{2} \sin \left (4 \, f x + 4 \, e\right )^{2} + 4 \, a^{2} \sin \left (4 \, f x + 4 \, e\right ) \sin \left (2 \, f x + 2 \, e\right ) + 4 \, a^{2} \sin \left (2 \, f x + 2 \, e\right )^{2} + 4 \, a^{2} \cos \left (2 \, f x + 2 \, e\right ) + a^{2} + 2 \, {\left (2 \, a^{2} \cos \left (2 \, f x + 2 \, e\right ) + a^{2}\right )} \cos \left (4 \, f x + 4 \, e\right )\right )} \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right ) + 1\right ) + 2 \, {\left (2 \, {\left (f x + e\right )} a^{2} \cos \left (2 \, f x + 2 \, e\right ) + {\left (f x + e\right )} a^{2} + a^{2} \sin \left (2 \, f x + 2 \, e\right )\right )} \cos \left (4 \, f x + 4 \, e\right ) - 4 \, {\left (a^{2} \sin \left (4 \, f x + 4 \, e\right ) + 2 \, a^{2} \sin \left (2 \, f x + 2 \, e\right )\right )} \cos \left (\frac {3}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right ) - 4 \, {\left (a^{2} \sin \left (4 \, f x + 4 \, e\right ) + 2 \, a^{2} \sin \left (2 \, f x + 2 \, e\right )\right )} \cos \left (\frac {1}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right ) + 2 \, {\left (2 \, {\left (f x + e\right )} a^{2} \sin \left (2 \, f x + 2 \, e\right ) - a^{2} \cos \left (2 \, f x + 2 \, e\right )\right )} \sin \left (4 \, f x + 4 \, e\right ) + 4 \, {\left (a^{2} \cos \left (4 \, f x + 4 \, e\right ) + 2 \, a^{2} \cos \left (2 \, f x + 2 \, e\right ) + a^{2}\right )} \sin \left (\frac {3}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right ) + 4 \, {\left (a^{2} \cos \left (4 \, f x + 4 \, e\right ) + 2 \, a^{2} \cos \left (2 \, f x + 2 \, e\right ) + a^{2}\right )} \sin \left (\frac {1}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right )\right )} \sqrt {a} \sqrt {c}}{{\left (2 \, {\left (2 \, \cos \left (2 \, f x + 2 \, e\right ) + 1\right )} \cos \left (4 \, f x + 4 \, e\right ) + \cos \left (4 \, f x + 4 \, e\right )^{2} + 4 \, \cos \left (2 \, f x + 2 \, e\right )^{2} + \sin \left (4 \, f x + 4 \, e\right )^{2} + 4 \, \sin \left (4 \, f x + 4 \, e\right ) \sin \left (2 \, f x + 2 \, e\right ) + 4 \, \sin \left (2 \, f x + 2 \, e\right )^{2} + 4 \, \cos \left (2 \, f x + 2 \, e\right ) + 1\right )} f} \]
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\[ \int (a+a \sec (e+f x))^{5/2} \sqrt {c-c \sec (e+f x)} \, dx=\int { {\left (a \sec \left (f x + e\right ) + a\right )}^{\frac {5}{2}} \sqrt {-c \sec \left (f x + e\right ) + c} \,d x } \]
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Timed out. \[ \int (a+a \sec (e+f x))^{5/2} \sqrt {c-c \sec (e+f x)} \, dx=\int {\left (a+\frac {a}{\cos \left (e+f\,x\right )}\right )}^{5/2}\,\sqrt {c-\frac {c}{\cos \left (e+f\,x\right )}} \,d x \]
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