Integrand size = 17, antiderivative size = 15 \[ \int \sec ^2(a+b x) \tan ^3(a+b x) \, dx=\frac {\tan ^4(a+b x)}{4 b} \]
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Time = 0.02 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {2687, 30} \[ \int \sec ^2(a+b x) \tan ^3(a+b x) \, dx=\frac {\tan ^4(a+b x)}{4 b} \]
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Rule 30
Rule 2687
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int x^3 \, dx,x,\tan (a+b x)\right )}{b} \\ & = \frac {\tan ^4(a+b x)}{4 b} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00 \[ \int \sec ^2(a+b x) \tan ^3(a+b x) \, dx=\frac {\tan ^4(a+b x)}{4 b} \]
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Time = 0.09 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.73
method | result | size |
derivativedivides | \(\frac {\frac {\left (\sec ^{4}\left (b x +a \right )\right )}{4}-\frac {\left (\sec ^{2}\left (b x +a \right )\right )}{2}}{b}\) | \(26\) |
default | \(\frac {\frac {\left (\sec ^{4}\left (b x +a \right )\right )}{4}-\frac {\left (\sec ^{2}\left (b x +a \right )\right )}{2}}{b}\) | \(26\) |
norman | \(\frac {4 \left (\tan ^{4}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{b \left (\tan ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )-1\right )^{4}}\) | \(32\) |
risch | \(-\frac {2 \left ({\mathrm e}^{6 i \left (b x +a \right )}+{\mathrm e}^{2 i \left (b x +a \right )}\right )}{b \left ({\mathrm e}^{2 i \left (b x +a \right )}+1\right )^{4}}\) | \(38\) |
parallelrisch | \(\frac {4 \left (\tan ^{4}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{b \left (\tan \left (\frac {b x}{2}+\frac {a}{2}\right )-1\right )^{4} \left (\tan \left (\frac {b x}{2}+\frac {a}{2}\right )+1\right )^{4}}\) | \(43\) |
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Time = 0.31 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.67 \[ \int \sec ^2(a+b x) \tan ^3(a+b x) \, dx=-\frac {2 \, \cos \left (b x + a\right )^{2} - 1}{4 \, b \cos \left (b x + a\right )^{4}} \]
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Timed out. \[ \int \sec ^2(a+b x) \tan ^3(a+b x) \, dx=\text {Timed out} \]
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Leaf count of result is larger than twice the leaf count of optimal. 39 vs. \(2 (13) = 26\).
Time = 0.18 (sec) , antiderivative size = 39, normalized size of antiderivative = 2.60 \[ \int \sec ^2(a+b x) \tan ^3(a+b x) \, dx=\frac {2 \, \sin \left (b x + a\right )^{2} - 1}{4 \, {\left (\sin \left (b x + a\right )^{4} - 2 \, \sin \left (b x + a\right )^{2} + 1\right )} b} \]
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Time = 0.33 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.67 \[ \int \sec ^2(a+b x) \tan ^3(a+b x) \, dx=-\frac {2 \, \cos \left (b x + a\right )^{2} - 1}{4 \, b \cos \left (b x + a\right )^{4}} \]
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Time = 0.11 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.87 \[ \int \sec ^2(a+b x) \tan ^3(a+b x) \, dx=\frac {{\mathrm {tan}\left (a+b\,x\right )}^4}{4\,b} \]
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