Integrand size = 36, antiderivative size = 145 \[ \int \frac {\sec (e+f x) (a+a \sec (e+f x))^{5/2}}{(c-c \sec (e+f x))^{5/2}} \, dx=-\frac {a (a+a \sec (e+f x))^{3/2} \tan (e+f x)}{2 f (c-c \sec (e+f x))^{5/2}}+\frac {a^2 \sqrt {a+a \sec (e+f x)} \tan (e+f x)}{c f (c-c \sec (e+f x))^{3/2}}+\frac {a^3 \log (1-\sec (e+f x)) \tan (e+f x)}{c^2 f \sqrt {a+a \sec (e+f x)} \sqrt {c-c \sec (e+f x)}} \]
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Time = 0.57 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {4039, 4037} \[ \int \frac {\sec (e+f x) (a+a \sec (e+f x))^{5/2}}{(c-c \sec (e+f x))^{5/2}} \, dx=\frac {a^3 \tan (e+f x) \log (1-\sec (e+f x))}{c^2 f \sqrt {a \sec (e+f x)+a} \sqrt {c-c \sec (e+f x)}}+\frac {a^2 \tan (e+f x) \sqrt {a \sec (e+f x)+a}}{c f (c-c \sec (e+f x))^{3/2}}-\frac {a \tan (e+f x) (a \sec (e+f x)+a)^{3/2}}{2 f (c-c \sec (e+f x))^{5/2}} \]
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Rule 4037
Rule 4039
Rubi steps \begin{align*} \text {integral}& = -\frac {a (a+a \sec (e+f x))^{3/2} \tan (e+f x)}{2 f (c-c \sec (e+f x))^{5/2}}-\frac {a \int \frac {\sec (e+f x) (a+a \sec (e+f x))^{3/2}}{(c-c \sec (e+f x))^{3/2}} \, dx}{c} \\ & = -\frac {a (a+a \sec (e+f x))^{3/2} \tan (e+f x)}{2 f (c-c \sec (e+f x))^{5/2}}+\frac {a^2 \sqrt {a+a \sec (e+f x)} \tan (e+f x)}{c f (c-c \sec (e+f x))^{3/2}}+\frac {a^2 \int \frac {\sec (e+f x) \sqrt {a+a \sec (e+f x)}}{\sqrt {c-c \sec (e+f x)}} \, dx}{c^2} \\ & = -\frac {a (a+a \sec (e+f x))^{3/2} \tan (e+f x)}{2 f (c-c \sec (e+f x))^{5/2}}+\frac {a^2 \sqrt {a+a \sec (e+f x)} \tan (e+f x)}{c f (c-c \sec (e+f x))^{3/2}}+\frac {a^3 \log (1-\sec (e+f x)) \tan (e+f x)}{c^2 f \sqrt {a+a \sec (e+f x)} \sqrt {c-c \sec (e+f x)}} \\ \end{align*}
Time = 1.30 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.56 \[ \int \frac {\sec (e+f x) (a+a \sec (e+f x))^{5/2}}{(c-c \sec (e+f x))^{5/2}} \, dx=-\frac {a^3 \left (-\log (1-\sec (e+f x))+\frac {-2+4 \sec (e+f x)}{(-1+\sec (e+f x))^2}\right ) \tan (e+f x)}{c^2 f \sqrt {a (1+\sec (e+f x))} \sqrt {c-c \sec (e+f x)}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(271\) vs. \(2(131)=262\).
Time = 3.60 (sec) , antiderivative size = 272, normalized size of antiderivative = 1.88
method | result | size |
default | \(-\frac {\sqrt {2}\, a^{2} \sqrt {-\frac {2 a}{\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}-1}}\, \left (1-\cos \left (f x +e \right )\right ) \left (2 \ln \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1\right ) \left (1-\cos \left (f x +e \right )\right )^{4} \csc \left (f x +e \right )^{4}+2 \ln \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )-1\right ) \left (1-\cos \left (f x +e \right )\right )^{4} \csc \left (f x +e \right )^{4}-4 \ln \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right ) \left (1-\cos \left (f x +e \right )\right )^{4} \csc \left (f x +e \right )^{4}+2 \left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}+1\right ) \csc \left (f x +e \right )}{4 f \left (\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}-1\right )^{2} \left (\frac {c \left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}}{\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}-1}\right )^{\frac {5}{2}}}\) | \(272\) |
risch | \(\frac {8 i a^{2} \sqrt {\frac {a \left ({\mathrm e}^{i \left (f x +e \right )}+1\right )^{2}}{1+{\mathrm e}^{2 i \left (f x +e \right )}}}\, {\mathrm e}^{2 i \left (f x +e \right )}}{c^{2} \left ({\mathrm e}^{i \left (f x +e \right )}+1\right ) \left ({\mathrm e}^{i \left (f x +e \right )}-1\right )^{3} \sqrt {\frac {c \left ({\mathrm e}^{i \left (f x +e \right )}-1\right )^{2}}{1+{\mathrm e}^{2 i \left (f x +e \right )}}}\, f}-\frac {2 i a^{2} \sqrt {\frac {a \left ({\mathrm e}^{i \left (f x +e \right )}+1\right )^{2}}{1+{\mathrm e}^{2 i \left (f x +e \right )}}}\, \left ({\mathrm e}^{i \left (f x +e \right )}-1\right ) \ln \left ({\mathrm e}^{i \left (f x +e \right )}-1\right )}{c^{2} \left ({\mathrm e}^{i \left (f x +e \right )}+1\right ) \sqrt {\frac {c \left ({\mathrm e}^{i \left (f x +e \right )}-1\right )^{2}}{1+{\mathrm e}^{2 i \left (f x +e \right )}}}\, f}+\frac {i a^{2} \sqrt {\frac {a \left ({\mathrm e}^{i \left (f x +e \right )}+1\right )^{2}}{1+{\mathrm e}^{2 i \left (f x +e \right )}}}\, \left ({\mathrm e}^{i \left (f x +e \right )}-1\right ) \ln \left (1+{\mathrm e}^{2 i \left (f x +e \right )}\right )}{c^{2} \left ({\mathrm e}^{i \left (f x +e \right )}+1\right ) \sqrt {\frac {c \left ({\mathrm e}^{i \left (f x +e \right )}-1\right )^{2}}{1+{\mathrm e}^{2 i \left (f x +e \right )}}}\, f}\) | \(325\) |
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\[ \int \frac {\sec (e+f x) (a+a \sec (e+f x))^{5/2}}{(c-c \sec (e+f x))^{5/2}} \, dx=\int { \frac {{\left (a \sec \left (f x + e\right ) + a\right )}^{\frac {5}{2}} \sec \left (f x + e\right )}{{\left (-c \sec \left (f x + e\right ) + c\right )}^{\frac {5}{2}}} \,d x } \]
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Timed out. \[ \int \frac {\sec (e+f x) (a+a \sec (e+f x))^{5/2}}{(c-c \sec (e+f x))^{5/2}} \, dx=\text {Timed out} \]
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none
Time = 0.31 (sec) , antiderivative size = 169, normalized size of antiderivative = 1.17 \[ \int \frac {\sec (e+f x) (a+a \sec (e+f x))^{5/2}}{(c-c \sec (e+f x))^{5/2}} \, dx=-\frac {\frac {2 \, \sqrt {-a} a^{2} \log \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + 1\right )}{c^{\frac {5}{2}}} + \frac {2 \, \sqrt {-a} a^{2} \log \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - 1\right )}{c^{\frac {5}{2}}} - \frac {4 \, \sqrt {-a} a^{2} \log \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1}\right )}{c^{\frac {5}{2}}} + \frac {{\left (\sqrt {-a} a^{2} \sqrt {c} + \frac {2 \, \sqrt {-a} a^{2} \sqrt {c} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}}\right )} {\left (\cos \left (f x + e\right ) + 1\right )}^{4}}{c^{3} \sin \left (f x + e\right )^{4}}}{2 \, f} \]
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\[ \int \frac {\sec (e+f x) (a+a \sec (e+f x))^{5/2}}{(c-c \sec (e+f x))^{5/2}} \, dx=\int { \frac {{\left (a \sec \left (f x + e\right ) + a\right )}^{\frac {5}{2}} \sec \left (f x + e\right )}{{\left (-c \sec \left (f x + e\right ) + c\right )}^{\frac {5}{2}}} \,d x } \]
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Timed out. \[ \int \frac {\sec (e+f x) (a+a \sec (e+f x))^{5/2}}{(c-c \sec (e+f x))^{5/2}} \, dx=\int \frac {{\left (a+\frac {a}{\cos \left (e+f\,x\right )}\right )}^{5/2}}{\cos \left (e+f\,x\right )\,{\left (c-\frac {c}{\cos \left (e+f\,x\right )}\right )}^{5/2}} \,d x \]
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