Integrand size = 34, antiderivative size = 11 \[ \int \frac {(a B+b B \cos (c+d x)) \sec ^2(c+d x)}{a+b \cos (c+d x)} \, dx=\frac {B \tan (c+d x)}{d} \]
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Time = 0.01 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.088, Rules used = {21, 3852, 8} \[ \int \frac {(a B+b B \cos (c+d x)) \sec ^2(c+d x)}{a+b \cos (c+d x)} \, dx=\frac {B \tan (c+d x)}{d} \]
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Rule 8
Rule 21
Rule 3852
Rubi steps \begin{align*} \text {integral}& = B \int \sec ^2(c+d x) \, dx \\ & = -\frac {B \text {Subst}(\int 1 \, dx,x,-\tan (c+d x))}{d} \\ & = \frac {B \tan (c+d x)}{d} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.00 \[ \int \frac {(a B+b B \cos (c+d x)) \sec ^2(c+d x)}{a+b \cos (c+d x)} \, dx=\frac {B \tan (c+d x)}{d} \]
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Time = 1.83 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.09
method | result | size |
derivativedivides | \(\frac {B \tan \left (d x +c \right )}{d}\) | \(12\) |
default | \(\frac {B \tan \left (d x +c \right )}{d}\) | \(12\) |
risch | \(\frac {2 i B}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}\) | \(21\) |
parallelrisch | \(-\frac {2 B \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}\) | \(31\) |
norman | \(\frac {-\frac {2 B \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d}-\frac {2 B \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}\) | \(65\) |
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none
Time = 0.28 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.73 \[ \int \frac {(a B+b B \cos (c+d x)) \sec ^2(c+d x)}{a+b \cos (c+d x)} \, dx=\frac {B \sin \left (d x + c\right )}{d \cos \left (d x + c\right )} \]
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Leaf count of result is larger than twice the leaf count of optimal. 32 vs. \(2 (8) = 16\).
Time = 1.44 (sec) , antiderivative size = 32, normalized size of antiderivative = 2.91 \[ \int \frac {(a B+b B \cos (c+d x)) \sec ^2(c+d x)}{a+b \cos (c+d x)} \, dx=\begin {cases} \frac {B \tan {\left (c + d x \right )}}{d} & \text {for}\: d \neq 0 \\\frac {x \left (B a + B b \cos {\left (c \right )}\right ) \sec ^{2}{\left (c \right )}}{a + b \cos {\left (c \right )}} & \text {otherwise} \end {cases} \]
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Exception generated. \[ \int \frac {(a B+b B \cos (c+d x)) \sec ^2(c+d x)}{a+b \cos (c+d x)} \, dx=\text {Exception raised: ValueError} \]
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none
Time = 0.29 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.00 \[ \int \frac {(a B+b B \cos (c+d x)) \sec ^2(c+d x)}{a+b \cos (c+d x)} \, dx=\frac {B \tan \left (d x + c\right )}{d} \]
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Time = 0.51 (sec) , antiderivative size = 30, normalized size of antiderivative = 2.73 \[ \int \frac {(a B+b B \cos (c+d x)) \sec ^2(c+d x)}{a+b \cos (c+d x)} \, dx=-\frac {2\,B\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-1\right )} \]
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