Integrand size = 17, antiderivative size = 100 \[ \int \frac {1}{\left (a-a \sec ^2(c+d x)\right )^{5/2}} \, dx=\frac {\cot (c+d x)}{2 a^2 d \sqrt {-a \tan ^2(c+d x)}}-\frac {\cot ^3(c+d x)}{4 a^2 d \sqrt {-a \tan ^2(c+d x)}}+\frac {\log (\sin (c+d x)) \tan (c+d x)}{a^2 d \sqrt {-a \tan ^2(c+d x)}} \]
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Time = 0.07 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {4206, 3739, 3554, 3556} \[ \int \frac {1}{\left (a-a \sec ^2(c+d x)\right )^{5/2}} \, dx=-\frac {\cot ^3(c+d x)}{4 a^2 d \sqrt {-a \tan ^2(c+d x)}}+\frac {\cot (c+d x)}{2 a^2 d \sqrt {-a \tan ^2(c+d x)}}+\frac {\tan (c+d x) \log (\sin (c+d x))}{a^2 d \sqrt {-a \tan ^2(c+d x)}} \]
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Rule 3554
Rule 3556
Rule 3739
Rule 4206
Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{\left (-a \tan ^2(c+d x)\right )^{5/2}} \, dx \\ & = \frac {\tan (c+d x) \int \cot ^5(c+d x) \, dx}{a^2 \sqrt {-a \tan ^2(c+d x)}} \\ & = -\frac {\cot ^3(c+d x)}{4 a^2 d \sqrt {-a \tan ^2(c+d x)}}-\frac {\tan (c+d x) \int \cot ^3(c+d x) \, dx}{a^2 \sqrt {-a \tan ^2(c+d x)}} \\ & = \frac {\cot (c+d x)}{2 a^2 d \sqrt {-a \tan ^2(c+d x)}}-\frac {\cot ^3(c+d x)}{4 a^2 d \sqrt {-a \tan ^2(c+d x)}}+\frac {\tan (c+d x) \int \cot (c+d x) \, dx}{a^2 \sqrt {-a \tan ^2(c+d x)}} \\ & = \frac {\cot (c+d x)}{2 a^2 d \sqrt {-a \tan ^2(c+d x)}}-\frac {\cot ^3(c+d x)}{4 a^2 d \sqrt {-a \tan ^2(c+d x)}}+\frac {\log (\sin (c+d x)) \tan (c+d x)}{a^2 d \sqrt {-a \tan ^2(c+d x)}} \\ \end{align*}
Time = 0.31 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.67 \[ \int \frac {1}{\left (a-a \sec ^2(c+d x)\right )^{5/2}} \, dx=\frac {2 \cot (c+d x)-\cot ^3(c+d x)+4 (\log (\cos (c+d x))+\log (\tan (c+d x))) \tan (c+d x)}{4 a^2 d \sqrt {-a \tan ^2(c+d x)}} \]
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Time = 0.87 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.07
method | result | size |
default | \(-\frac {\left (-32 \ln \left (-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right ) \sin \left (d x +c \right )^{4}+32 \ln \left (\frac {2}{\cos \left (d x +c \right )+1}\right ) \sin \left (d x +c \right )^{4}+13 \cos \left (d x +c \right )^{4}+6 \cos \left (d x +c \right )^{2}-11\right ) \sec \left (d x +c \right ) \csc \left (d x +c \right )^{3}}{32 d \sqrt {-a \tan \left (d x +c \right )^{2}}\, a^{2}}\) | \(107\) |
risch | \(\frac {\left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right ) x}{a^{2} \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right ) \sqrt {\frac {a \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{2}}{\left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{2}}}}-\frac {2 \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right ) \left (d x +c \right )}{a^{2} \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right ) \sqrt {\frac {a \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{2}}{\left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{2}}}\, d}+\frac {4 i \left ({\mathrm e}^{6 i \left (d x +c \right )}-{\mathrm e}^{4 i \left (d x +c \right )}+{\mathrm e}^{2 i \left (d x +c \right )}\right )}{a^{2} \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{3} \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right ) \sqrt {\frac {a \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{2}}{\left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{2}}}\, d}-\frac {i \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right ) \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{a^{2} \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right ) \sqrt {\frac {a \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{2}}{\left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{2}}}\, d}\) | \(298\) |
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Time = 0.28 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.25 \[ \int \frac {1}{\left (a-a \sec ^2(c+d x)\right )^{5/2}} \, dx=\frac {{\left (4 \, \cos \left (d x + c\right )^{3} - 4 \, {\left (\cos \left (d x + c\right )^{5} - 2 \, \cos \left (d x + c\right )^{3} + \cos \left (d x + c\right )\right )} \log \left (\frac {1}{2} \, \sin \left (d x + c\right )\right ) - 3 \, \cos \left (d x + c\right )\right )} \sqrt {\frac {a \cos \left (d x + c\right )^{2} - a}{\cos \left (d x + c\right )^{2}}}}{4 \, {\left (a^{3} d \cos \left (d x + c\right )^{4} - 2 \, a^{3} d \cos \left (d x + c\right )^{2} + a^{3} d\right )} \sin \left (d x + c\right )} \]
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\[ \int \frac {1}{\left (a-a \sec ^2(c+d x)\right )^{5/2}} \, dx=\int \frac {1}{\left (- a \sec ^{2}{\left (c + d x \right )} + a\right )^{\frac {5}{2}}}\, dx \]
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Time = 0.28 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.79 \[ \int \frac {1}{\left (a-a \sec ^2(c+d x)\right )^{5/2}} \, dx=-\frac {\frac {2 \, \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{\sqrt {-a} a^{2}} - \frac {4 \, \log \left (\tan \left (d x + c\right )\right )}{\sqrt {-a} a^{2}} + \frac {2 \, \sqrt {-a} \tan \left (d x + c\right )^{2} - \sqrt {-a}}{a^{3} \tan \left (d x + c\right )^{4}}}{4 \, d} \]
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Time = 0.61 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.47 \[ \int \frac {1}{\left (a-a \sec ^2(c+d x)\right )^{5/2}} \, dx=\frac {\frac {64 \, \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}{\sqrt {-a} a^{2}} - \frac {32 \, \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2}\right )}{\sqrt {-a} a^{2}} - \frac {\sqrt {-a} a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 12 \, \sqrt {-a} a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2}}{a^{5}} + \frac {48 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 12 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1}{\sqrt {-a} a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4}}}{64 \, d} \]
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Timed out. \[ \int \frac {1}{\left (a-a \sec ^2(c+d x)\right )^{5/2}} \, dx=\int \frac {1}{{\left (a-\frac {a}{{\cos \left (c+d\,x\right )}^2}\right )}^{5/2}} \,d x \]
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