Integrand size = 12, antiderivative size = 15 \[ \int \left (A+C \sec ^2(c+d x)\right ) \, dx=A x+\frac {C \tan (c+d x)}{d} \]
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Time = 0.02 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {3852, 8} \[ \int \left (A+C \sec ^2(c+d x)\right ) \, dx=A x+\frac {C \tan (c+d x)}{d} \]
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Rule 8
Rule 3852
Rubi steps \begin{align*} \text {integral}& = A x+C \int \sec ^2(c+d x) \, dx \\ & = A x-\frac {C \text {Subst}(\int 1 \, dx,x,-\tan (c+d x))}{d} \\ & = A x+\frac {C \tan (c+d x)}{d} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00 \[ \int \left (A+C \sec ^2(c+d x)\right ) \, dx=A x+\frac {C \tan (c+d x)}{d} \]
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Time = 0.04 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.07
method | result | size |
default | \(A x +\frac {C \tan \left (d x +c \right )}{d}\) | \(16\) |
parts | \(A x +\frac {C \tan \left (d x +c \right )}{d}\) | \(16\) |
derivativedivides | \(\frac {\left (d x +c \right ) A +C \tan \left (d x +c \right )}{d}\) | \(21\) |
risch | \(A x +\frac {2 i C}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}\) | \(25\) |
parallelrisch | \(-\frac {2 C \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right )}+A x\) | \(35\) |
norman | \(\frac {A x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-A x -\frac {2 C \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d}}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1}\) | \(51\) |
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Leaf count of result is larger than twice the leaf count of optimal. 31 vs. \(2 (15) = 30\).
Time = 0.25 (sec) , antiderivative size = 31, normalized size of antiderivative = 2.07 \[ \int \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {A d x \cos \left (d x + c\right ) + C \sin \left (d x + c\right )}{d \cos \left (d x + c\right )} \]
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\[ \int \left (A+C \sec ^2(c+d x)\right ) \, dx=\int \left (A + C \sec ^{2}{\left (c + d x \right )}\right )\, dx \]
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none
Time = 0.19 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00 \[ \int \left (A+C \sec ^2(c+d x)\right ) \, dx=A x + \frac {C \tan \left (d x + c\right )}{d} \]
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none
Time = 0.27 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00 \[ \int \left (A+C \sec ^2(c+d x)\right ) \, dx=A x + \frac {C \tan \left (d x + c\right )}{d} \]
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Time = 16.02 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.13 \[ \int \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {C\,\mathrm {tan}\left (c+d\,x\right )+A\,d\,x}{d} \]
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