Integrand size = 34, antiderivative size = 16 \[ \int \frac {\tan ^2(c+d x) (a B+b B \tan (c+d x))}{a+b \tan (c+d x)} \, dx=-B x+\frac {B \tan (c+d x)}{d} \]
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Time = 0.01 (sec) , antiderivative size = 16, 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, 3554, 8} \[ \int \frac {\tan ^2(c+d x) (a B+b B \tan (c+d x))}{a+b \tan (c+d x)} \, dx=\frac {B \tan (c+d x)}{d}-B x \]
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Rule 8
Rule 21
Rule 3554
Rubi steps \begin{align*} \text {integral}& = B \int \tan ^2(c+d x) \, dx \\ & = \frac {B \tan (c+d x)}{d}-B \int 1 \, dx \\ & = -B x+\frac {B \tan (c+d x)}{d} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.56 \[ \int \frac {\tan ^2(c+d x) (a B+b B \tan (c+d x))}{a+b \tan (c+d x)} \, dx=B \left (-\frac {\arctan (\tan (c+d x))}{d}+\frac {\tan (c+d x)}{d}\right ) \]
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Time = 0.03 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.06
method | result | size |
norman | \(-B x +\frac {B \tan \left (d x +c \right )}{d}\) | \(17\) |
parallelrisch | \(-\frac {B d x -B \tan \left (d x +c \right )}{d}\) | \(20\) |
derivativedivides | \(\frac {B \left (\tan \left (d x +c \right )-\arctan \left (\tan \left (d x +c \right )\right )\right )}{d}\) | \(22\) |
default | \(\frac {B \left (\tan \left (d x +c \right )-\arctan \left (\tan \left (d x +c \right )\right )\right )}{d}\) | \(22\) |
risch | \(-B x +\frac {2 i B}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}\) | \(26\) |
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Time = 0.25 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.19 \[ \int \frac {\tan ^2(c+d x) (a B+b B \tan (c+d x))}{a+b \tan (c+d x)} \, dx=-\frac {B d x - B \tan \left (d x + c\right )}{d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 36 vs. \(2 (12) = 24\).
Time = 0.40 (sec) , antiderivative size = 36, normalized size of antiderivative = 2.25 \[ \int \frac {\tan ^2(c+d x) (a B+b B \tan (c+d x))}{a+b \tan (c+d x)} \, dx=\begin {cases} - B x + \frac {B \tan {\left (c + d x \right )}}{d} & \text {for}\: d \neq 0 \\\frac {x \left (B a + B b \tan {\left (c \right )}\right ) \tan ^{2}{\left (c \right )}}{a + b \tan {\left (c \right )}} & \text {otherwise} \end {cases} \]
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Time = 0.38 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.38 \[ \int \frac {\tan ^2(c+d x) (a B+b B \tan (c+d x))}{a+b \tan (c+d x)} \, dx=-\frac {{\left (d x + c\right )} B - B \tan \left (d x + c\right )}{d} \]
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Time = 0.44 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.38 \[ \int \frac {\tan ^2(c+d x) (a B+b B \tan (c+d x))}{a+b \tan (c+d x)} \, dx=-\frac {{\left (d x + c\right )} B - B \tan \left (d x + c\right )}{d} \]
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Time = 7.98 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int \frac {\tan ^2(c+d x) (a B+b B \tan (c+d x))}{a+b \tan (c+d x)} \, dx=\frac {B\,\mathrm {tan}\left (c+d\,x\right )}{d}-B\,x \]
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