Integrand size = 30, antiderivative size = 152 \[ \int \frac {(a+a \sec (e+f x))^{5/2}}{\sqrt {c-c \sec (e+f x)}} \, dx=\frac {a^3 \log (\cos (e+f x)) \tan (e+f x)}{f \sqrt {a+a \sec (e+f x)} \sqrt {c-c \sec (e+f x)}}+\frac {4 a^3 \log (1-\sec (e+f x)) \tan (e+f x)}{f \sqrt {a+a \sec (e+f x)} \sqrt {c-c \sec (e+f x)}}+\frac {a^3 \sec (e+f x) \tan (e+f x)}{f \sqrt {a+a \sec (e+f x)} \sqrt {c-c \sec (e+f x)}} \]
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Time = 0.14 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {3997, 84} \[ \int \frac {(a+a \sec (e+f x))^{5/2}}{\sqrt {c-c \sec (e+f x)}} \, dx=\frac {a^3 \tan (e+f x) \sec (e+f x)}{f \sqrt {a \sec (e+f x)+a} \sqrt {c-c \sec (e+f x)}}+\frac {4 a^3 \tan (e+f x) \log (1-\sec (e+f x))}{f \sqrt {a \sec (e+f x)+a} \sqrt {c-c \sec (e+f x)}}+\frac {a^3 \tan (e+f x) \log (\cos (e+f x))}{f \sqrt {a \sec (e+f x)+a} \sqrt {c-c \sec (e+f x)}} \]
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Rule 84
Rule 3997
Rubi steps \begin{align*} \text {integral}& = -\frac {(a c \tan (e+f x)) \text {Subst}\left (\int \frac {(a+a x)^2}{x (c-c x)} \, dx,x,\sec (e+f x)\right )}{f \sqrt {a+a \sec (e+f x)} \sqrt {c-c \sec (e+f x)}} \\ & = -\frac {(a c \tan (e+f x)) \text {Subst}\left (\int \left (-\frac {a^2}{c}-\frac {4 a^2}{c (-1+x)}+\frac {a^2}{c x}\right ) \, dx,x,\sec (e+f x)\right )}{f \sqrt {a+a \sec (e+f x)} \sqrt {c-c \sec (e+f x)}} \\ & = \frac {a^3 \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 {4 a^3 \log (1-\sec (e+f x)) \tan (e+f x)}{f \sqrt {a+a \sec (e+f x)} \sqrt {c-c \sec (e+f x)}}+\frac {a^3 \sec (e+f x) \tan (e+f x)}{f \sqrt {a+a \sec (e+f x)} \sqrt {c-c \sec (e+f x)}} \\ \end{align*}
Time = 0.37 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.45 \[ \int \frac {(a+a \sec (e+f x))^{5/2}}{\sqrt {c-c \sec (e+f x)}} \, dx=\frac {a^3 (\log (\cos (e+f x))+4 \log (1-\sec (e+f x))+\sec (e+f x)) \tan (e+f x)}{f \sqrt {a (1+\sec (e+f x))} \sqrt {c-c \sec (e+f x)}} \]
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Time = 2.29 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.02
method | result | size |
default | \(-\frac {a^{2} \sqrt {a \left (\sec \left (f x +e \right )+1\right )}\, \left (\ln \left (\frac {2}{\cos \left (f x +e \right )+1}\right ) \sin \left (f x +e \right )-8 \ln \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right ) \sin \left (f x +e \right )+3 \ln \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )-1\right ) \sin \left (f x +e \right )+3 \ln \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1\right ) \sin \left (f x +e \right )-\sin \left (f x +e \right )-\tan \left (f x +e \right )\right )}{f \left (\cos \left (f x +e \right )+1\right ) \sqrt {-c \left (\sec \left (f x +e \right )-1\right )}}\) | \(155\) |
risch | \(-\frac {a^{2} \sqrt {\frac {a \left ({\mathrm e}^{i \left (f x +e \right )}+1\right )^{2}}{1+{\mathrm e}^{2 i \left (f x +e \right )}}}\, \left (-2 i {\mathrm e}^{i \left (f x +e \right )}+3 i {\mathrm e}^{2 i \left (f x +e \right )} \ln \left (1+{\mathrm e}^{2 i \left (f x +e \right )}\right )+{\mathrm e}^{3 i \left (f x +e \right )} f x +8 i {\mathrm e}^{3 i \left (f x +e \right )} \ln \left ({\mathrm e}^{i \left (f x +e \right )}-1\right )+2 i {\mathrm e}^{2 i \left (f x +e \right )}-3 i {\mathrm e}^{i \left (f x +e \right )} \ln \left (1+{\mathrm e}^{2 i \left (f x +e \right )}\right )+8 i {\mathrm e}^{i \left (f x +e \right )} \ln \left ({\mathrm e}^{i \left (f x +e \right )}-1\right )+2 \,{\mathrm e}^{3 i \left (f x +e \right )} e +{\mathrm e}^{i \left (f x +e \right )} f x -{\mathrm e}^{2 i \left (f x +e \right )} f x -3 i \ln \left (1+{\mathrm e}^{2 i \left (f x +e \right )}\right ) {\mathrm e}^{3 i \left (f x +e \right )}-8 i {\mathrm e}^{2 i \left (f x +e \right )} \ln \left ({\mathrm e}^{i \left (f x +e \right )}-1\right )+3 i \ln \left (1+{\mathrm e}^{2 i \left (f x +e \right )}\right )-8 i \ln \left ({\mathrm e}^{i \left (f x +e \right )}-1\right )+2 \,{\mathrm e}^{i \left (f x +e \right )} e -2 \,{\mathrm e}^{2 i \left (f x +e \right )} e -f x -2 e \right )}{\left ({\mathrm e}^{i \left (f x +e \right )}+1\right ) \sqrt {\frac {c \left ({\mathrm e}^{i \left (f x +e \right )}-1\right )^{2}}{1+{\mathrm e}^{2 i \left (f x +e \right )}}}\, f \left (1+{\mathrm e}^{2 i \left (f x +e \right )}\right )}\) | \(374\) |
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\[ \int \frac {(a+a \sec (e+f x))^{5/2}}{\sqrt {c-c \sec (e+f x)}} \, dx=\int { \frac {{\left (a \sec \left (f x + e\right ) + a\right )}^{\frac {5}{2}}}{\sqrt {-c \sec \left (f x + e\right ) + c}} \,d x } \]
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Timed out. \[ \int \frac {(a+a \sec (e+f x))^{5/2}}{\sqrt {c-c \sec (e+f x)}} \, dx=\text {Timed out} \]
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Exception generated. \[ \int \frac {(a+a \sec (e+f x))^{5/2}}{\sqrt {c-c \sec (e+f x)}} \, dx=\text {Exception raised: RuntimeError} \]
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\[ \int \frac {(a+a \sec (e+f x))^{5/2}}{\sqrt {c-c \sec (e+f x)}} \, dx=\int { \frac {{\left (a \sec \left (f x + e\right ) + a\right )}^{\frac {5}{2}}}{\sqrt {-c \sec \left (f x + e\right ) + c}} \,d x } \]
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Timed out. \[ \int \frac {(a+a \sec (e+f x))^{5/2}}{\sqrt {c-c \sec (e+f x)}} \, dx=\int \frac {{\left (a+\frac {a}{\cos \left (e+f\,x\right )}\right )}^{5/2}}{\sqrt {c-\frac {c}{\cos \left (e+f\,x\right )}}} \,d x \]
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