Integrand size = 30, antiderivative size = 103 \[ \int (a+a \sec (e+f x))^{3/2} (c-c \sec (e+f x))^{3/2} \, dx=\frac {a^2 c^2 \log (\cos (e+f x)) \tan (e+f x)}{f \sqrt {a+a \sec (e+f x)} \sqrt {c-c \sec (e+f x)}}+\frac {a^2 c^2 \tan ^3(e+f x)}{2 f \sqrt {a+a \sec (e+f x)} \sqrt {c-c \sec (e+f x)}} \]
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Time = 0.15 (sec) , antiderivative size = 103, 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} \[ \int (a+a \sec (e+f x))^{3/2} (c-c \sec (e+f x))^{3/2} \, dx=\frac {a^2 c^2 \tan ^3(e+f x)}{2 f \sqrt {a \sec (e+f x)+a} \sqrt {c-c \sec (e+f x)}}+\frac {a^2 c^2 \tan (e+f x) \log (\cos (e+f x))}{f \sqrt {a \sec (e+f x)+a} \sqrt {c-c \sec (e+f x)}} \]
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Rule 3554
Rule 3556
Rule 3990
Rubi steps \begin{align*} \text {integral}& = \frac {\left (a^2 c^2 \tan (e+f x)\right ) \int \tan ^3(e+f x) \, dx}{\sqrt {a+a \sec (e+f x)} \sqrt {c-c \sec (e+f x)}} \\ & = \frac {a^2 c^2 \tan ^3(e+f x)}{2 f \sqrt {a+a \sec (e+f x)} \sqrt {c-c \sec (e+f x)}}-\frac {\left (a^2 c^2 \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^2 c^2 \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^2 \tan ^3(e+f x)}{2 f \sqrt {a+a \sec (e+f x)} \sqrt {c-c \sec (e+f x)}} \\ \end{align*}
Time = 0.58 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.64 \[ \int (a+a \sec (e+f x))^{3/2} (c-c \sec (e+f x))^{3/2} \, dx=\frac {a^2 c^2 \left (2 \log (\cos (e+f x))+\sec ^2(e+f x)\right ) \tan (e+f x)}{2 f \sqrt {a (1+\sec (e+f x))} \sqrt {c-c \sec (e+f x)}} \]
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Time = 2.30 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.39
method | result | size |
default | \(\frac {a \left (2 \cos \left (f x +e \right )^{2} \ln \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )-1\right )+2 \cos \left (f x +e \right )^{2} \ln \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1\right )-2 \cos \left (f x +e \right )^{2} \ln \left (\frac {2}{\cos \left (f x +e \right )+1}\right )+\sin \left (f x +e \right )^{2}\right ) \sqrt {-c \left (\sec \left (f x +e \right )-1\right )}\, \left (\sec \left (f x +e \right )-1\right ) c \sqrt {a \left (\sec \left (f x +e \right )+1\right )}\, \csc \left (f x +e \right )}{2 f \left (\cos \left (f x +e \right )-1\right )}\) | \(143\) |
risch | \(-\frac {a c \sqrt {\frac {a \left ({\mathrm e}^{i \left (f x +e \right )}+1\right )^{2}}{1+{\mathrm e}^{2 i \left (f x +e \right )}}}\, \sqrt {\frac {c \left ({\mathrm e}^{i \left (f x +e \right )}-1\right )^{2}}{1+{\mathrm e}^{2 i \left (f x +e \right )}}}\, \left (i {\mathrm e}^{4 i \left (f x +e \right )} \ln \left (1+{\mathrm e}^{2 i \left (f x +e \right )}\right )+{\mathrm e}^{4 i \left (f x +e \right )} f x +2 \,{\mathrm e}^{4 i \left (f x +e \right )} e +2 i {\mathrm e}^{2 i \left (f x +e \right )} \ln \left (1+{\mathrm e}^{2 i \left (f x +e \right )}\right )+2 \,{\mathrm e}^{2 i \left (f x +e \right )} f x +2 i {\mathrm e}^{2 i \left (f x +e \right )}+4 \,{\mathrm e}^{2 i \left (f x +e \right )} e +i \ln \left (1+{\mathrm e}^{2 i \left (f x +e \right )}\right )+f x +2 e \right )}{\left (1+{\mathrm e}^{2 i \left (f x +e \right )}\right ) \left ({\mathrm e}^{i \left (f x +e \right )}+1\right ) \left ({\mathrm e}^{i \left (f x +e \right )}-1\right ) f}\) | \(238\) |
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Time = 0.32 (sec) , antiderivative size = 346, normalized size of antiderivative = 3.36 \[ \int (a+a \sec (e+f x))^{3/2} (c-c \sec (e+f x))^{3/2} \, dx=\left [\frac {\sqrt {-a c} a c \cos \left (f x + e\right ) \log \left (\frac {a c \cos \left (f x + e\right )^{4} - {\left (\cos \left (f x + e\right )^{3} + \cos \left (f x + e\right )\right )} \sqrt {-a c} \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \sqrt {\frac {c \cos \left (f x + e\right ) - c}{\cos \left (f x + e\right )}} \sin \left (f x + e\right ) + a c}{2 \, \cos \left (f x + e\right )^{2}}\right ) - a c \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \sqrt {\frac {c \cos \left (f x + e\right ) - c}{\cos \left (f x + e\right )}} \sin \left (f x + e\right )}{2 \, f \cos \left (f x + e\right )}, \frac {2 \, \sqrt {a c} a c \arctan \left (\frac {\sqrt {a c} \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \sqrt {\frac {c \cos \left (f x + e\right ) - c}{\cos \left (f x + e\right )}} \cos \left (f x + e\right ) \sin \left (f x + e\right )}{a c \cos \left (f x + e\right )^{2} + a c}\right ) \cos \left (f x + e\right ) - a c \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \sqrt {\frac {c \cos \left (f x + e\right ) - c}{\cos \left (f x + e\right )}} \sin \left (f x + e\right )}{2 \, f \cos \left (f x + e\right )}\right ] \]
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\[ \int (a+a \sec (e+f x))^{3/2} (c-c \sec (e+f x))^{3/2} \, dx=\int \left (a \left (\sec {\left (e + f x \right )} + 1\right )\right )^{\frac {3}{2}} \left (- c \left (\sec {\left (e + f x \right )} - 1\right )\right )^{\frac {3}{2}}\, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. 477 vs. \(2 (93) = 186\).
Time = 0.40 (sec) , antiderivative size = 477, normalized size of antiderivative = 4.63 \[ \int (a+a \sec (e+f x))^{3/2} (c-c \sec (e+f x))^{3/2} \, dx=-\frac {{\left ({\left (f x + e\right )} a c \cos \left (4 \, f x + 4 \, e\right )^{2} + 4 \, {\left (f x + e\right )} a c \cos \left (2 \, f x + 2 \, e\right )^{2} + {\left (f x + e\right )} a c \sin \left (4 \, f x + 4 \, e\right )^{2} + 4 \, {\left (f x + e\right )} a c \sin \left (2 \, f x + 2 \, e\right )^{2} + 4 \, {\left (f x + e\right )} a c \cos \left (2 \, f x + 2 \, e\right ) + {\left (f x + e\right )} a c - 2 \, a c \sin \left (2 \, f x + 2 \, e\right ) - {\left (a c \cos \left (4 \, f x + 4 \, e\right )^{2} + 4 \, a c \cos \left (2 \, f x + 2 \, e\right )^{2} + a c \sin \left (4 \, f x + 4 \, e\right )^{2} + 4 \, a c \sin \left (4 \, f x + 4 \, e\right ) \sin \left (2 \, f x + 2 \, e\right ) + 4 \, a c \sin \left (2 \, f x + 2 \, e\right )^{2} + 4 \, a c \cos \left (2 \, f x + 2 \, e\right ) + a c + 2 \, {\left (2 \, a c \cos \left (2 \, f x + 2 \, e\right ) + a c\right )} \cos \left (4 \, f x + 4 \, e\right )\right )} \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right ) + 1\right ) + 2 \, {\left (2 \, {\left (f x + e\right )} a c \cos \left (2 \, f x + 2 \, e\right ) + {\left (f x + e\right )} a c - a c \sin \left (2 \, f x + 2 \, e\right )\right )} \cos \left (4 \, f x + 4 \, e\right ) + 2 \, {\left (2 \, {\left (f x + e\right )} a c \sin \left (2 \, f x + 2 \, e\right ) + a c \cos \left (2 \, f x + 2 \, e\right )\right )} \sin \left (4 \, f x + 4 \, e\right )\right )} \sqrt {a} \sqrt {c}}{{\left (2 \, {\left (2 \, \cos \left (2 \, f x + 2 \, e\right ) + 1\right )} \cos \left (4 \, f x + 4 \, e\right ) + \cos \left (4 \, f x + 4 \, e\right )^{2} + 4 \, \cos \left (2 \, f x + 2 \, e\right )^{2} + \sin \left (4 \, f x + 4 \, e\right )^{2} + 4 \, \sin \left (4 \, f x + 4 \, e\right ) \sin \left (2 \, f x + 2 \, e\right ) + 4 \, \sin \left (2 \, f x + 2 \, e\right )^{2} + 4 \, \cos \left (2 \, f x + 2 \, e\right ) + 1\right )} f} \]
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\[ \int (a+a \sec (e+f x))^{3/2} (c-c \sec (e+f x))^{3/2} \, dx=\int { {\left (a \sec \left (f x + e\right ) + a\right )}^{\frac {3}{2}} {\left (-c \sec \left (f x + e\right ) + c\right )}^{\frac {3}{2}} \,d x } \]
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Timed out. \[ \int (a+a \sec (e+f x))^{3/2} (c-c \sec (e+f x))^{3/2} \, dx=\int {\left (a+\frac {a}{\cos \left (e+f\,x\right )}\right )}^{3/2}\,{\left (c-\frac {c}{\cos \left (e+f\,x\right )}\right )}^{3/2} \,d x \]
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