Integrand size = 9, antiderivative size = 17 \[ \int \sec ^3(x) \tan ^3(x) \, dx=-\frac {1}{3} \sec ^3(x)+\frac {\sec ^5(x)}{5} \]
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Time = 0.02 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2686, 14} \[ \int \sec ^3(x) \tan ^3(x) \, dx=\frac {\sec ^5(x)}{5}-\frac {\sec ^3(x)}{3} \]
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Rule 14
Rule 2686
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int x^2 \left (-1+x^2\right ) \, dx,x,\sec (x)\right ) \\ & = \text {Subst}\left (\int \left (-x^2+x^4\right ) \, dx,x,\sec (x)\right ) \\ & = -\frac {1}{3} \sec ^3(x)+\frac {\sec ^5(x)}{5} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00 \[ \int \sec ^3(x) \tan ^3(x) \, dx=-\frac {1}{3} \sec ^3(x)+\frac {\sec ^5(x)}{5} \]
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Time = 1.01 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.82
method | result | size |
derivativedivides | \(-\frac {\left (\sec ^{3}\left (x \right )\right )}{3}+\frac {\left (\sec ^{5}\left (x \right )\right )}{5}\) | \(14\) |
default | \(-\frac {\left (\sec ^{3}\left (x \right )\right )}{3}+\frac {\left (\sec ^{5}\left (x \right )\right )}{5}\) | \(14\) |
risch | \(-\frac {8 \left (5 \,{\mathrm e}^{7 i x}-2 \,{\mathrm e}^{5 i x}+5 \,{\mathrm e}^{3 i x}\right )}{15 \left ({\mathrm e}^{2 i x}+1\right )^{5}}\) | \(34\) |
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Time = 0.24 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.82 \[ \int \sec ^3(x) \tan ^3(x) \, dx=-\frac {5 \, \cos \left (x\right )^{2} - 3}{15 \, \cos \left (x\right )^{5}} \]
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Time = 0.05 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.82 \[ \int \sec ^3(x) \tan ^3(x) \, dx=\frac {3 - 5 \cos ^{2}{\left (x \right )}}{15 \cos ^{5}{\left (x \right )}} \]
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Time = 0.19 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.82 \[ \int \sec ^3(x) \tan ^3(x) \, dx=-\frac {5 \, \cos \left (x\right )^{2} - 3}{15 \, \cos \left (x\right )^{5}} \]
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Time = 0.26 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.82 \[ \int \sec ^3(x) \tan ^3(x) \, dx=-\frac {5 \, \cos \left (x\right )^{2} - 3}{15 \, \cos \left (x\right )^{5}} \]
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Time = 0.00 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.76 \[ \int \sec ^3(x) \tan ^3(x) \, dx=\frac {1}{5\,{\cos \left (x\right )}^5}-\frac {1}{3\,{\cos \left (x\right )}^3} \]
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