Integrand size = 28, antiderivative size = 6 \[ \int \frac {\frac {a B}{b}+B \sec (c+d x)}{a+b \sec (c+d x)} \, dx=\frac {B x}{b} \]
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Time = 0.00 (sec) , antiderivative size = 6, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {21, 8} \[ \int \frac {\frac {a B}{b}+B \sec (c+d x)}{a+b \sec (c+d x)} \, dx=\frac {B x}{b} \]
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Rule 8
Rule 21
Rubi steps \begin{align*} \text {integral}& = \frac {B \int 1 \, dx}{b} \\ & = \frac {B x}{b} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 6, normalized size of antiderivative = 1.00 \[ \int \frac {\frac {a B}{b}+B \sec (c+d x)}{a+b \sec (c+d x)} \, dx=\frac {B x}{b} \]
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Time = 0.36 (sec) , antiderivative size = 7, normalized size of antiderivative = 1.17
method | result | size |
default | \(\frac {B x}{b}\) | \(7\) |
norman | \(\frac {B x}{b}\) | \(7\) |
risch | \(\frac {B x}{b}\) | \(7\) |
derivativedivides | \(\frac {B \left (d x +c \right )}{d b}\) | \(14\) |
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none
Time = 0.25 (sec) , antiderivative size = 6, normalized size of antiderivative = 1.00 \[ \int \frac {\frac {a B}{b}+B \sec (c+d x)}{a+b \sec (c+d x)} \, dx=\frac {B x}{b} \]
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Time = 2.83 (sec) , antiderivative size = 3, normalized size of antiderivative = 0.50 \[ \int \frac {\frac {a B}{b}+B \sec (c+d x)}{a+b \sec (c+d x)} \, dx=\frac {B x}{b} \]
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Exception generated. \[ \int \frac {\frac {a B}{b}+B \sec (c+d x)}{a+b \sec (c+d x)} \, dx=\text {Exception raised: ValueError} \]
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Leaf count of result is larger than twice the leaf count of optimal. 13 vs. \(2 (6) = 12\).
Time = 0.30 (sec) , antiderivative size = 13, normalized size of antiderivative = 2.17 \[ \int \frac {\frac {a B}{b}+B \sec (c+d x)}{a+b \sec (c+d x)} \, dx=\frac {{\left (d x + c\right )} B}{b d} \]
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Time = 14.48 (sec) , antiderivative size = 6, normalized size of antiderivative = 1.00 \[ \int \frac {\frac {a B}{b}+B \sec (c+d x)}{a+b \sec (c+d x)} \, dx=\frac {B\,x}{b} \]
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