Integrand size = 8, antiderivative size = 10 \[ \int \sec ^2(a+b x) \, dx=\frac {\tan (a+b x)}{b} \]
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Time = 0.01 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3852, 8} \[ \int \sec ^2(a+b x) \, dx=\frac {\tan (a+b x)}{b} \]
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Rule 8
Rule 3852
Rubi steps \begin{align*} \text {integral}& = -\frac {\text {Subst}(\int 1 \, dx,x,-\tan (a+b x))}{b} \\ & = \frac {\tan (a+b x)}{b} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00 \[ \int \sec ^2(a+b x) \, dx=\frac {\tan (a+b x)}{b} \]
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Time = 0.18 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.10
method | result | size |
derivativedivides | \(\frac {\tan \left (b x +a \right )}{b}\) | \(11\) |
default | \(\frac {\tan \left (b x +a \right )}{b}\) | \(11\) |
risch | \(\frac {2 i}{b \left ({\mathrm e}^{2 i \left (b x +a \right )}+1\right )}\) | \(20\) |
norman | \(-\frac {2 \tan \left (\frac {b x}{2}+\frac {a}{2}\right )}{b \left (\tan ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )-1\right )}\) | \(30\) |
parallelrisch | \(-\frac {2 \tan \left (\frac {b x}{2}+\frac {a}{2}\right )}{b \left (\tan ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )-1\right )}\) | \(30\) |
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none
Time = 0.25 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.80 \[ \int \sec ^2(a+b x) \, dx=\frac {\sin \left (b x + a\right )}{b \cos \left (b x + a\right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 58 vs. \(2 (7) = 14\).
Time = 0.62 (sec) , antiderivative size = 58, normalized size of antiderivative = 5.80 \[ \int \sec ^2(a+b x) \, dx=\begin {cases} \tilde {\infty } x & \text {for}\: \left (a = - \frac {\pi }{2} \vee a = - b x - \frac {\pi }{2}\right ) \wedge \left (a = - b x - \frac {\pi }{2} \vee b = 0\right ) \\\frac {x}{\cos ^{2}{\left (a \right )}} & \text {for}\: b = 0 \\- \frac {2 \tan {\left (\frac {a}{2} + \frac {b x}{2} \right )}}{b \tan ^{2}{\left (\frac {a}{2} + \frac {b x}{2} \right )} - b} & \text {otherwise} \end {cases} \]
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none
Time = 0.21 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00 \[ \int \sec ^2(a+b x) \, dx=\frac {\tan \left (b x + a\right )}{b} \]
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none
Time = 0.29 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00 \[ \int \sec ^2(a+b x) \, dx=\frac {\tan \left (b x + a\right )}{b} \]
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Time = 0.13 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00 \[ \int \sec ^2(a+b x) \, dx=\frac {\mathrm {tan}\left (a+b\,x\right )}{b} \]
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