Integrand size = 10, antiderivative size = 61 \[ \int \frac {1}{\left (a \cos ^2(x)\right )^{5/2}} \, dx=\frac {3 \text {arctanh}(\sin (x)) \cos (x)}{8 a^2 \sqrt {a \cos ^2(x)}}+\frac {\tan (x)}{4 a \left (a \cos ^2(x)\right )^{3/2}}+\frac {3 \tan (x)}{8 a^2 \sqrt {a \cos ^2(x)}} \]
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Time = 0.06 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {3283, 3286, 3855} \[ \int \frac {1}{\left (a \cos ^2(x)\right )^{5/2}} \, dx=\frac {3 \cos (x) \text {arctanh}(\sin (x))}{8 a^2 \sqrt {a \cos ^2(x)}}+\frac {3 \tan (x)}{8 a^2 \sqrt {a \cos ^2(x)}}+\frac {\tan (x)}{4 a \left (a \cos ^2(x)\right )^{3/2}} \]
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Rule 3283
Rule 3286
Rule 3855
Rubi steps \begin{align*} \text {integral}& = \frac {\tan (x)}{4 a \left (a \cos ^2(x)\right )^{3/2}}+\frac {3 \int \frac {1}{\left (a \cos ^2(x)\right )^{3/2}} \, dx}{4 a} \\ & = \frac {\tan (x)}{4 a \left (a \cos ^2(x)\right )^{3/2}}+\frac {3 \tan (x)}{8 a^2 \sqrt {a \cos ^2(x)}}+\frac {3 \int \frac {1}{\sqrt {a \cos ^2(x)}} \, dx}{8 a^2} \\ & = \frac {\tan (x)}{4 a \left (a \cos ^2(x)\right )^{3/2}}+\frac {3 \tan (x)}{8 a^2 \sqrt {a \cos ^2(x)}}+\frac {(3 \cos (x)) \int \sec (x) \, dx}{8 a^2 \sqrt {a \cos ^2(x)}} \\ & = \frac {3 \text {arctanh}(\sin (x)) \cos (x)}{8 a^2 \sqrt {a \cos ^2(x)}}+\frac {\tan (x)}{4 a \left (a \cos ^2(x)\right )^{3/2}}+\frac {3 \tan (x)}{8 a^2 \sqrt {a \cos ^2(x)}} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.59 \[ \int \frac {1}{\left (a \cos ^2(x)\right )^{5/2}} \, dx=\frac {3 \text {arctanh}(\sin (x)) \cos (x)+\left (3+2 \sec ^2(x)\right ) \tan (x)}{8 a^2 \sqrt {a \cos ^2(x)}} \]
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Time = 1.08 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.46
method | result | size |
default | \(\frac {\sqrt {a \left (\sin ^{2}\left (x \right )\right )}\, \left (3 \ln \left (\frac {2 \sqrt {a}\, \sqrt {a \left (\sin ^{2}\left (x \right )\right )}+2 a}{\cos \left (x \right )}\right ) a \left (\cos ^{4}\left (x \right )\right )+3 \sqrt {a \left (\sin ^{2}\left (x \right )\right )}\, \left (\cos ^{2}\left (x \right )\right ) \sqrt {a}+2 \sqrt {a}\, \sqrt {a \left (\sin ^{2}\left (x \right )\right )}\right )}{8 a^{\frac {7}{2}} \cos \left (x \right )^{3} \sin \left (x \right ) \sqrt {a \left (\cos ^{2}\left (x \right )\right )}}\) | \(89\) |
risch | \(-\frac {i \left (3 \,{\mathrm e}^{6 i x}+11 \,{\mathrm e}^{4 i x}-11 \,{\mathrm e}^{2 i x}-3\right )}{4 a^{2} \left ({\mathrm e}^{2 i x}+1\right )^{3} \sqrt {a \left ({\mathrm e}^{2 i x}+1\right )^{2} {\mathrm e}^{-2 i x}}}+\frac {3 \ln \left ({\mathrm e}^{i x}+i\right ) \cos \left (x \right )}{4 a^{2} \sqrt {a \left ({\mathrm e}^{2 i x}+1\right )^{2} {\mathrm e}^{-2 i x}}}-\frac {3 \ln \left ({\mathrm e}^{i x}-i\right ) \cos \left (x \right )}{4 a^{2} \sqrt {a \left ({\mathrm e}^{2 i x}+1\right )^{2} {\mathrm e}^{-2 i x}}}\) | \(126\) |
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Time = 0.30 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.80 \[ \int \frac {1}{\left (a \cos ^2(x)\right )^{5/2}} \, dx=-\frac {{\left (3 \, \cos \left (x\right )^{4} \log \left (-\frac {\sin \left (x\right ) - 1}{\sin \left (x\right ) + 1}\right ) - 2 \, {\left (3 \, \cos \left (x\right )^{2} + 2\right )} \sin \left (x\right )\right )} \sqrt {a \cos \left (x\right )^{2}}}{16 \, a^{3} \cos \left (x\right )^{5}} \]
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Timed out. \[ \int \frac {1}{\left (a \cos ^2(x)\right )^{5/2}} \, dx=\text {Timed out} \]
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Leaf count of result is larger than twice the leaf count of optimal. 933 vs. \(2 (49) = 98\).
Time = 0.62 (sec) , antiderivative size = 933, normalized size of antiderivative = 15.30 \[ \int \frac {1}{\left (a \cos ^2(x)\right )^{5/2}} \, dx=\text {Too large to display} \]
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Time = 0.31 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.48 \[ \int \frac {1}{\left (a \cos ^2(x)\right )^{5/2}} \, dx=-\frac {3 \, \sin \left (x\right )^{3} - 5 \, \sin \left (x\right )}{8 \, {\left (\sin \left (x\right )^{2} - 1\right )}^{2} a^{\frac {5}{2}} \mathrm {sgn}\left (\cos \left (x\right )\right )} \]
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Timed out. \[ \int \frac {1}{\left (a \cos ^2(x)\right )^{5/2}} \, dx=\int \frac {1}{{\left (a\,{\cos \left (x\right )}^2\right )}^{5/2}} \,d x \]
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