Integrand size = 30, antiderivative size = 142 \[ \int \frac {\sqrt {a+a \sec (e+f x)}}{(c-c \sec (e+f x))^{5/2}} \, dx=-\frac {a \tan (e+f x)}{2 f \sqrt {a+a \sec (e+f x)} (c-c \sec (e+f x))^{5/2}}-\frac {a \tan (e+f x)}{c f \sqrt {a+a \sec (e+f x)} (c-c \sec (e+f x))^{3/2}}+\frac {a \log (1-\cos (e+f x)) \tan (e+f x)}{c^2 f \sqrt {a+a \sec (e+f x)} \sqrt {c-c \sec (e+f x)}} \]
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Time = 0.37 (sec) , antiderivative size = 142, 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 = {3992, 3996, 31} \[ \int \frac {\sqrt {a+a \sec (e+f x)}}{(c-c \sec (e+f x))^{5/2}} \, dx=\frac {a \tan (e+f x) \log (1-\cos (e+f x))}{c^2 f \sqrt {a \sec (e+f x)+a} \sqrt {c-c \sec (e+f x)}}-\frac {a \tan (e+f x)}{c f \sqrt {a \sec (e+f x)+a} (c-c \sec (e+f x))^{3/2}}-\frac {a \tan (e+f x)}{2 f \sqrt {a \sec (e+f x)+a} (c-c \sec (e+f x))^{5/2}} \]
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Rule 31
Rule 3992
Rule 3996
Rubi steps \begin{align*} \text {integral}& = -\frac {a \tan (e+f x)}{2 f \sqrt {a+a \sec (e+f x)} (c-c \sec (e+f x))^{5/2}}+\frac {\int \frac {\sqrt {a+a \sec (e+f x)}}{(c-c \sec (e+f x))^{3/2}} \, dx}{c} \\ & = -\frac {a \tan (e+f x)}{2 f \sqrt {a+a \sec (e+f x)} (c-c \sec (e+f x))^{5/2}}-\frac {a \tan (e+f x)}{c f \sqrt {a+a \sec (e+f x)} (c-c \sec (e+f x))^{3/2}}+\frac {\int \frac {\sqrt {a+a \sec (e+f x)}}{\sqrt {c-c \sec (e+f x)}} \, dx}{c^2} \\ & = -\frac {a \tan (e+f x)}{2 f \sqrt {a+a \sec (e+f x)} (c-c \sec (e+f x))^{5/2}}-\frac {a \tan (e+f x)}{c f \sqrt {a+a \sec (e+f x)} (c-c \sec (e+f x))^{3/2}}+\frac {(a \tan (e+f x)) \text {Subst}\left (\int \frac {1}{-c+c x} \, dx,x,\cos (e+f x)\right )}{c f \sqrt {a+a \sec (e+f x)} \sqrt {c-c \sec (e+f x)}} \\ & = -\frac {a \tan (e+f x)}{2 f \sqrt {a+a \sec (e+f x)} (c-c \sec (e+f x))^{5/2}}-\frac {a \tan (e+f x)}{c f \sqrt {a+a \sec (e+f x)} (c-c \sec (e+f x))^{3/2}}+\frac {a \log (1-\cos (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 = 0.82 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.63 \[ \int \frac {\sqrt {a+a \sec (e+f x)}}{(c-c \sec (e+f x))^{5/2}} \, dx=-\frac {a \left (-2 \log (\cos (e+f x))-2 \log (1-\sec (e+f x))+\frac {3-2 \sec (e+f x)}{(-1+\sec (e+f x))^2}\right ) \tan (e+f x)}{2 c^2 f \sqrt {a (1+\sec (e+f x))} \sqrt {c-c \sec (e+f x)}} \]
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Time = 2.38 (sec) , antiderivative size = 237, normalized size of antiderivative = 1.67
method | result | size |
default | \(-\frac {\sqrt {2}\, \sqrt {-\frac {2 a}{\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}-1}}\, \left (1-\cos \left (f x +e \right )\right ) \left (8 \ln \left (\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}+1\right ) \left (1-\cos \left (f x +e \right )\right )^{4} \csc \left (f x +e \right )^{4}-16 \ln \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right ) \left (1-\cos \left (f x +e \right )\right )^{4} \csc \left (f x +e \right )^{4}-6 \left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}+1\right ) \csc \left (f x +e \right )}{16 f \left (\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}-1\right )^{2} \left (\frac {c \left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}}{\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}-1}\right )^{\frac {5}{2}}}\) | \(237\) |
risch | \(-\frac {\sqrt {\frac {a \left ({\mathrm e}^{i \left (f x +e \right )}+1\right )^{2}}{1+{\mathrm e}^{2 i \left (f x +e \right )}}}\, \left (-4 i {\mathrm e}^{i \left (f x +e \right )}+{\mathrm e}^{4 i \left (f x +e \right )} f x -8 i {\mathrm e}^{3 i \left (f x +e \right )} \ln \left ({\mathrm e}^{i \left (f x +e \right )}-1\right )+2 \,{\mathrm e}^{4 i \left (f x +e \right )} e -4 \,{\mathrm e}^{3 i \left (f x +e \right )} f x +12 i {\mathrm e}^{2 i \left (f x +e \right )} \ln \left ({\mathrm e}^{i \left (f x +e \right )}-1\right )-8 \,{\mathrm e}^{3 i \left (f x +e \right )} e +6 \,{\mathrm e}^{2 i \left (f x +e \right )} f x +2 i {\mathrm e}^{4 i \left (f x +e \right )} \ln \left ({\mathrm e}^{i \left (f x +e \right )}-1\right )+12 \,{\mathrm e}^{2 i \left (f x +e \right )} e -4 \,{\mathrm e}^{i \left (f x +e \right )} f x +6 i {\mathrm e}^{2 i \left (f x +e \right )}-8 i {\mathrm e}^{i \left (f x +e \right )} \ln \left ({\mathrm e}^{i \left (f x +e \right )}-1\right )-4 i {\mathrm e}^{3 i \left (f x +e \right )}+2 i \ln \left ({\mathrm e}^{i \left (f x +e \right )}-1\right )-8 \,{\mathrm e}^{i \left (f x +e \right )} e +f x +2 e \right )}{c^{2} \left ({\mathrm e}^{i \left (f x +e \right )}+1\right ) \left ({\mathrm e}^{i \left (f x +e \right )}-1\right )^{3} \sqrt {\frac {c \left ({\mathrm e}^{i \left (f x +e \right )}-1\right )^{2}}{1+{\mathrm e}^{2 i \left (f x +e \right )}}}\, f}\) | \(348\) |
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\[ \int \frac {\sqrt {a+a \sec (e+f x)}}{(c-c \sec (e+f x))^{5/2}} \, dx=\int { \frac {\sqrt {a \sec \left (f x + e\right ) + a}}{{\left (-c \sec \left (f x + e\right ) + c\right )}^{\frac {5}{2}}} \,d x } \]
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\[ \int \frac {\sqrt {a+a \sec (e+f x)}}{(c-c \sec (e+f x))^{5/2}} \, dx=\int \frac {\sqrt {a \left (\sec {\left (e + f x \right )} + 1\right )}}{\left (- c \left (\sec {\left (e + f x \right )} - 1\right )\right )^{\frac {5}{2}}}\, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. 1173 vs. \(2 (128) = 256\).
Time = 0.48 (sec) , antiderivative size = 1173, normalized size of antiderivative = 8.26 \[ \int \frac {\sqrt {a+a \sec (e+f x)}}{(c-c \sec (e+f x))^{5/2}} \, dx=\text {Too large to display} \]
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\[ \int \frac {\sqrt {a+a \sec (e+f x)}}{(c-c \sec (e+f x))^{5/2}} \, dx=\int { \frac {\sqrt {a \sec \left (f x + e\right ) + a}}{{\left (-c \sec \left (f x + e\right ) + c\right )}^{\frac {5}{2}}} \,d x } \]
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Timed out. \[ \int \frac {\sqrt {a+a \sec (e+f x)}}{(c-c \sec (e+f x))^{5/2}} \, dx=\int \frac {\sqrt {a+\frac {a}{\cos \left (e+f\,x\right )}}}{{\left (c-\frac {c}{\cos \left (e+f\,x\right )}\right )}^{5/2}} \,d x \]
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