Integrand size = 30, antiderivative size = 101 \[ \int \frac {1}{(a+a \sec (e+f x))^{3/2} (c-c \sec (e+f x))^{3/2}} \, dx=\frac {\cot (e+f x)}{2 a c f \sqrt {a+a \sec (e+f x)} \sqrt {c-c \sec (e+f x)}}+\frac {\log (\sin (e+f x)) \tan (e+f x)}{a c f \sqrt {a+a \sec (e+f x)} \sqrt {c-c \sec (e+f x)}} \]
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Time = 0.13 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {3990, 3554, 3556} \[ \int \frac {1}{(a+a \sec (e+f x))^{3/2} (c-c \sec (e+f x))^{3/2}} \, dx=\frac {\cot (e+f x)}{2 a c f \sqrt {a \sec (e+f x)+a} \sqrt {c-c \sec (e+f x)}}+\frac {\tan (e+f x) \log (\sin (e+f x))}{a c f \sqrt {a \sec (e+f x)+a} \sqrt {c-c \sec (e+f x)}} \]
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Rule 3554
Rule 3556
Rule 3990
Rubi steps \begin{align*} \text {integral}& = -\frac {\tan (e+f x) \int \cot ^3(e+f x) \, dx}{a c \sqrt {a+a \sec (e+f x)} \sqrt {c-c \sec (e+f x)}} \\ & = \frac {\cot (e+f x)}{2 a c f \sqrt {a+a \sec (e+f x)} \sqrt {c-c \sec (e+f x)}}+\frac {\tan (e+f x) \int \cot (e+f x) \, dx}{a c \sqrt {a+a \sec (e+f x)} \sqrt {c-c \sec (e+f x)}} \\ & = \frac {\cot (e+f x)}{2 a c f \sqrt {a+a \sec (e+f x)} \sqrt {c-c \sec (e+f x)}}+\frac {\log (\sin (e+f x)) \tan (e+f x)}{a c f \sqrt {a+a \sec (e+f x)} \sqrt {c-c \sec (e+f x)}} \\ \end{align*}
Time = 0.46 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.71 \[ \int \frac {1}{(a+a \sec (e+f x))^{3/2} (c-c \sec (e+f x))^{3/2}} \, dx=\frac {\cot (e+f x)+2 (\log (\cos (e+f x))+\log (\tan (e+f x))) \tan (e+f x)}{2 a c f \sqrt {a (1+\sec (e+f x))} \sqrt {c-c \sec (e+f x)}} \]
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Time = 2.26 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.55
method | result | size |
default | \(-\frac {\left (4 \cos \left (f x +e \right )^{2} \ln \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )-4 \cos \left (f x +e \right )^{2} \ln \left (\frac {2}{\cos \left (f x +e \right )+1}\right )-\cos \left (f x +e \right )^{2}-4 \ln \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )+4 \ln \left (\frac {2}{\cos \left (f x +e \right )+1}\right )-1\right ) \sqrt {a \left (\sec \left (f x +e \right )+1\right )}\, \tan \left (f x +e \right )}{4 f \,a^{2} \sqrt {-c \left (\sec \left (f x +e \right )-1\right )}\, c \left (\sec \left (f x +e \right )-1\right ) \left (\cos \left (f x +e \right )+1\right )^{2}}\) | \(157\) |
risch | \(-\frac {i {\mathrm e}^{4 i \left (f x +e \right )} \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}-1\right )+{\mathrm e}^{4 i \left (f x +e \right )} f x +2 \,{\mathrm e}^{4 i \left (f x +e \right )} e -2 i {\mathrm e}^{2 i \left (f x +e \right )} \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}-1\right )-2 \,{\mathrm e}^{2 i \left (f x +e \right )} f x -4 \,{\mathrm e}^{2 i \left (f x +e \right )} e -2 i {\mathrm e}^{2 i \left (f x +e \right )}+i \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}-1\right )+f x +2 e}{a c \left ({\mathrm e}^{i \left (f x +e \right )}+1\right ) \sqrt {\frac {a \left ({\mathrm e}^{i \left (f x +e \right )}+1\right )^{2}}{1+{\mathrm e}^{2 i \left (f x +e \right )}}}\, \left (1+{\mathrm e}^{2 i \left (f x +e \right )}\right ) \left ({\mathrm e}^{i \left (f x +e \right )}-1\right ) \sqrt {\frac {c \left ({\mathrm e}^{i \left (f x +e \right )}-1\right )^{2}}{1+{\mathrm e}^{2 i \left (f x +e \right )}}}\, f}\) | \(242\) |
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Leaf count of result is larger than twice the leaf count of optimal. 223 vs. \(2 (91) = 182\).
Time = 0.49 (sec) , antiderivative size = 492, normalized size of antiderivative = 4.87 \[ \int \frac {1}{(a+a \sec (e+f x))^{3/2} (c-c \sec (e+f x))^{3/2}} \, dx=\left [-\frac {9 \, \sqrt {-a c} {\left (\cos \left (f x + e\right )^{2} - 1\right )} \log \left (-\frac {8 \, {\left ({\left (256 \, \cos \left (f x + e\right )^{5} - 512 \, \cos \left (f x + e\right )^{3} + 175 \, \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 )}} - {\left (256 \, a c \cos \left (f x + e\right )^{4} - 512 \, a c \cos \left (f x + e\right )^{2} + 337 \, a c\right )} \sin \left (f x + e\right )\right )}}{{\left (\cos \left (f x + e\right )^{2} - 1\right )} \sin \left (f x + e\right )}\right ) \sin \left (f x + e\right ) + {\left (16 \, \cos \left (f x + e\right )^{3} - 25 \, \cos \left (f x + e\right )\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 )}}}{18 \, {\left (a^{2} c^{2} f \cos \left (f x + e\right )^{2} - a^{2} c^{2} f\right )} \sin \left (f x + e\right )}, -\frac {18 \, \sqrt {a c} {\left (\cos \left (f x + e\right )^{2} - 1\right )} \arctan \left (\frac {{\left (16 \, \cos \left (f x + e\right )^{3} - 7 \, \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 )}}}{{\left (16 \, a c \cos \left (f x + e\right )^{2} - 25 \, a c\right )} \sin \left (f x + e\right )}\right ) \sin \left (f x + e\right ) + {\left (16 \, \cos \left (f x + e\right )^{3} - 25 \, \cos \left (f x + e\right )\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 )}}}{18 \, {\left (a^{2} c^{2} f \cos \left (f x + e\right )^{2} - a^{2} c^{2} f\right )} \sin \left (f x + e\right )}\right ] \]
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\[ \int \frac {1}{(a+a \sec (e+f x))^{3/2} (c-c \sec (e+f x))^{3/2}} \, dx=\int \frac {1}{\left (a \left (\sec {\left (e + f x \right )} + 1\right )\right )^{\frac {3}{2}} \left (- c \left (\sec {\left (e + f x \right )} - 1\right )\right )^{\frac {3}{2}}}\, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. 486 vs. \(2 (91) = 182\).
Time = 0.38 (sec) , antiderivative size = 486, normalized size of antiderivative = 4.81 \[ \int \frac {1}{(a+a \sec (e+f x))^{3/2} (c-c \sec (e+f x))^{3/2}} \, dx=-\frac {{\left ({\left (f x + e\right )} \cos \left (4 \, f x + 4 \, e\right )^{2} + 4 \, {\left (f x + e\right )} \cos \left (2 \, f x + 2 \, e\right )^{2} + {\left (f x + e\right )} \sin \left (4 \, f x + 4 \, e\right )^{2} + 4 \, {\left (f x + e\right )} \sin \left (2 \, f x + 2 \, e\right )^{2} + f x + {\left (2 \, {\left (2 \, \cos \left (2 \, f x + 2 \, e\right ) - 1\right )} \cos \left (4 \, f x + 4 \, e\right ) - \cos \left (4 \, f x + 4 \, e\right )^{2} - 4 \, \cos \left (2 \, f x + 2 \, e\right )^{2} - \sin \left (4 \, f x + 4 \, e\right )^{2} + 4 \, \sin \left (4 \, f x + 4 \, e\right ) \sin \left (2 \, f x + 2 \, e\right ) - 4 \, \sin \left (2 \, f x + 2 \, e\right )^{2} + 4 \, \cos \left (2 \, f x + 2 \, e\right ) - 1\right )} \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right ) - 1\right ) + 2 \, {\left (f x - 2 \, {\left (f x + e\right )} \cos \left (2 \, f x + 2 \, e\right ) + e + \sin \left (2 \, f x + 2 \, e\right )\right )} \cos \left (4 \, f x + 4 \, e\right ) - 4 \, {\left (f x + e\right )} \cos \left (2 \, f x + 2 \, e\right ) - 2 \, {\left (2 \, {\left (f x + e\right )} \sin \left (2 \, f x + 2 \, e\right ) + \cos \left (2 \, f x + 2 \, e\right )\right )} \sin \left (4 \, f x + 4 \, e\right ) + e + 2 \, \sin \left (2 \, f x + 2 \, e\right )\right )} \sqrt {a} \sqrt {c}}{{\left (a^{2} c^{2} \cos \left (4 \, f x + 4 \, e\right )^{2} + 4 \, a^{2} c^{2} \cos \left (2 \, f x + 2 \, e\right )^{2} + a^{2} c^{2} \sin \left (4 \, f x + 4 \, e\right )^{2} - 4 \, a^{2} c^{2} \sin \left (4 \, f x + 4 \, e\right ) \sin \left (2 \, f x + 2 \, e\right ) + 4 \, a^{2} c^{2} \sin \left (2 \, f x + 2 \, e\right )^{2} - 4 \, a^{2} c^{2} \cos \left (2 \, f x + 2 \, e\right ) + a^{2} c^{2} - 2 \, {\left (2 \, a^{2} c^{2} \cos \left (2 \, f x + 2 \, e\right ) - a^{2} c^{2}\right )} \cos \left (4 \, f x + 4 \, e\right )\right )} f} \]
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Time = 2.16 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.48 \[ \int \frac {1}{(a+a \sec (e+f x))^{3/2} (c-c \sec (e+f x))^{3/2}} \, dx=-\frac {\frac {4 \, \log \left ({\left | c \right |} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2}\right )}{\sqrt {-a c} a {\left | c \right |}} - \frac {8 \, \log \left ({\left | c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + c \right |}\right )}{\sqrt {-a c} a {\left | c \right |}} + \frac {c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - c}{\sqrt {-a c} a c {\left | c \right |}} - \frac {4 \, c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - c}{\sqrt {-a c} a c {\left | c \right |} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2}}}{8 \, f} \]
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Timed out. \[ \int \frac {1}{(a+a \sec (e+f x))^{3/2} (c-c \sec (e+f x))^{3/2}} \, dx=\int \frac {1}{{\left (a+\frac {a}{\cos \left (e+f\,x\right )}\right )}^{3/2}\,{\left (c-\frac {c}{\cos \left (e+f\,x\right )}\right )}^{3/2}} \,d x \]
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