Integrand size = 9, antiderivative size = 8 \[ \int \sec ^2(x) \tan ^4(x) \, dx=\frac {\tan ^5(x)}{5} \]
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Time = 0.02 (sec) , antiderivative size = 8, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2687, 30} \[ \int \sec ^2(x) \tan ^4(x) \, dx=\frac {\tan ^5(x)}{5} \]
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Rule 30
Rule 2687
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int x^4 \, dx,x,\tan (x)\right ) \\ & = \frac {\tan ^5(x)}{5} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 8, normalized size of antiderivative = 1.00 \[ \int \sec ^2(x) \tan ^4(x) \, dx=\frac {\tan ^5(x)}{5} \]
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Time = 0.55 (sec) , antiderivative size = 7, normalized size of antiderivative = 0.88
method | result | size |
derivativedivides | \(\frac {\left (\tan ^{5}\left (x \right )\right )}{5}\) | \(7\) |
default | \(\frac {\left (\tan ^{5}\left (x \right )\right )}{5}\) | \(7\) |
risch | \(\frac {2 i \left (5 \,{\mathrm e}^{8 i x}+10 \,{\mathrm e}^{4 i x}+1\right )}{5 \left ({\mathrm e}^{2 i x}+1\right )^{5}}\) | \(29\) |
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Leaf count of result is larger than twice the leaf count of optimal. 20 vs. \(2 (6) = 12\).
Time = 0.25 (sec) , antiderivative size = 20, normalized size of antiderivative = 2.50 \[ \int \sec ^2(x) \tan ^4(x) \, dx=\frac {{\left (\cos \left (x\right )^{4} - 2 \, \cos \left (x\right )^{2} + 1\right )} \sin \left (x\right )}{5 \, \cos \left (x\right )^{5}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 29 vs. \(2 (5) = 10\).
Time = 0.02 (sec) , antiderivative size = 29, normalized size of antiderivative = 3.62 \[ \int \sec ^2(x) \tan ^4(x) \, dx=\frac {\sin {\left (x \right )}}{5 \cos {\left (x \right )}} - \frac {2 \sin {\left (x \right )}}{5 \cos ^{3}{\left (x \right )}} + \frac {\sin {\left (x \right )}}{5 \cos ^{5}{\left (x \right )}} \]
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none
Time = 0.19 (sec) , antiderivative size = 6, normalized size of antiderivative = 0.75 \[ \int \sec ^2(x) \tan ^4(x) \, dx=\frac {1}{5} \, \tan \left (x\right )^{5} \]
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none
Time = 0.31 (sec) , antiderivative size = 6, normalized size of antiderivative = 0.75 \[ \int \sec ^2(x) \tan ^4(x) \, dx=\frac {1}{5} \, \tan \left (x\right )^{5} \]
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Time = 0.02 (sec) , antiderivative size = 6, normalized size of antiderivative = 0.75 \[ \int \sec ^2(x) \tan ^4(x) \, dx=\frac {{\mathrm {tan}\left (x\right )}^5}{5} \]
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