Integrand size = 14, antiderivative size = 40 \[ \int (c+d x) \tan ^2(a+b x) \, dx=-c x-\frac {d x^2}{2}+\frac {d \log (\cos (a+b x))}{b^2}+\frac {(c+d x) \tan (a+b x)}{b} \]
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Time = 0.04 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3801, 3556} \[ \int (c+d x) \tan ^2(a+b x) \, dx=\frac {d \log (\cos (a+b x))}{b^2}+\frac {(c+d x) \tan (a+b x)}{b}-c x-\frac {d x^2}{2} \]
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Rule 3556
Rule 3801
Rubi steps \begin{align*} \text {integral}& = \frac {(c+d x) \tan (a+b x)}{b}-\frac {d \int \tan (a+b x) \, dx}{b}-\int (c+d x) \, dx \\ & = -c x-\frac {d x^2}{2}+\frac {d \log (\cos (a+b x))}{b^2}+\frac {(c+d x) \tan (a+b x)}{b} \\ \end{align*}
Time = 0.42 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.90 \[ \int (c+d x) \tan ^2(a+b x) \, dx=-\frac {c \arctan (\tan (a+b x))}{b}+\frac {d \log (\cos (a+b x))}{b^2}-\frac {d x \sec (a) (b x \cos (a)-2 \sin (a))}{2 b}+\frac {d x \sec (a) \sec (a+b x) \sin (b x)}{b}+\frac {c \tan (a+b x)}{b} \]
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Time = 1.18 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.30
method | result | size |
norman | \(\frac {c \tan \left (x b +a \right )}{b}+\frac {d x \tan \left (x b +a \right )}{b}-c x -\frac {d \,x^{2}}{2}-\frac {d \ln \left (1+\tan \left (x b +a \right )^{2}\right )}{2 b^{2}}\) | \(52\) |
parallelrisch | \(-\frac {d \,x^{2} b^{2}-2 d \tan \left (x b +a \right ) x b +2 c x \,b^{2}-2 c \tan \left (x b +a \right ) b +d \ln \left (1+\tan \left (x b +a \right )^{2}\right )}{2 b^{2}}\) | \(56\) |
default | \(-\frac {d \,x^{2}}{2}-c x +\frac {-\frac {d a \tan \left (x b +a \right )}{b}+c \tan \left (x b +a \right )+\frac {d \left (\left (x b +a \right ) \tan \left (x b +a \right )+\ln \left (\cos \left (x b +a \right )\right )\right )}{b}}{b}\) | \(63\) |
risch | \(-\frac {d \,x^{2}}{2}-c x -\frac {2 i d x}{b}-\frac {2 i d a}{b^{2}}+\frac {2 i \left (d x +c \right )}{b \left ({\mathrm e}^{2 i \left (x b +a \right )}+1\right )}+\frac {d \ln \left ({\mathrm e}^{2 i \left (x b +a \right )}+1\right )}{b^{2}}\) | \(69\) |
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none
Time = 0.24 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.32 \[ \int (c+d x) \tan ^2(a+b x) \, dx=-\frac {b^{2} d x^{2} + 2 \, b^{2} c x - d \log \left (\frac {1}{\tan \left (b x + a\right )^{2} + 1}\right ) - 2 \, {\left (b d x + b c\right )} \tan \left (b x + a\right )}{2 \, b^{2}} \]
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Time = 0.16 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.62 \[ \int (c+d x) \tan ^2(a+b x) \, dx=\begin {cases} - c x - \frac {d x^{2}}{2} + \frac {c \tan {\left (a + b x \right )}}{b} + \frac {d x \tan {\left (a + b x \right )}}{b} - \frac {d \log {\left (\tan ^{2}{\left (a + b x \right )} + 1 \right )}}{2 b^{2}} & \text {for}\: b \neq 0 \\\left (c x + \frac {d x^{2}}{2}\right ) \tan ^{2}{\left (a \right )} & \text {otherwise} \end {cases} \]
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Leaf count of result is larger than twice the leaf count of optimal. 237 vs. \(2 (38) = 76\).
Time = 0.31 (sec) , antiderivative size = 237, normalized size of antiderivative = 5.92 \[ \int (c+d x) \tan ^2(a+b x) \, dx=-\frac {2 \, {\left (b x + a - \tan \left (b x + a\right )\right )} c - \frac {2 \, {\left (b x + a - \tan \left (b x + a\right )\right )} a d}{b} + \frac {{\left ({\left (b x + a\right )}^{2} \cos \left (2 \, b x + 2 \, a\right )^{2} + {\left (b x + a\right )}^{2} \sin \left (2 \, b x + 2 \, a\right )^{2} + 2 \, {\left (b x + a\right )}^{2} \cos \left (2 \, b x + 2 \, a\right ) + {\left (b x + a\right )}^{2} - {\left (\cos \left (2 \, b x + 2 \, a\right )^{2} + \sin \left (2 \, b x + 2 \, a\right )^{2} + 2 \, \cos \left (2 \, b x + 2 \, a\right ) + 1\right )} \log \left (\cos \left (2 \, b x + 2 \, a\right )^{2} + \sin \left (2 \, b x + 2 \, a\right )^{2} + 2 \, \cos \left (2 \, b x + 2 \, a\right ) + 1\right ) - 4 \, {\left (b x + a\right )} \sin \left (2 \, b x + 2 \, a\right )\right )} d}{{\left (\cos \left (2 \, b x + 2 \, a\right )^{2} + \sin \left (2 \, b x + 2 \, a\right )^{2} + 2 \, \cos \left (2 \, b x + 2 \, a\right ) + 1\right )} b}}{2 \, b} \]
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Leaf count of result is larger than twice the leaf count of optimal. 203 vs. \(2 (38) = 76\).
Time = 0.45 (sec) , antiderivative size = 203, normalized size of antiderivative = 5.08 \[ \int (c+d x) \tan ^2(a+b x) \, dx=-\frac {b^{2} d x^{2} \tan \left (b x\right ) \tan \left (a\right ) + 2 \, b^{2} c x \tan \left (b x\right ) \tan \left (a\right ) - b^{2} d x^{2} - 2 \, b^{2} c x + 2 \, b d x \tan \left (b x\right ) + 2 \, b d x \tan \left (a\right ) - d \log \left (\frac {4 \, {\left (\tan \left (b x\right )^{2} \tan \left (a\right )^{2} - 2 \, \tan \left (b x\right ) \tan \left (a\right ) + 1\right )}}{\tan \left (b x\right )^{2} \tan \left (a\right )^{2} + \tan \left (b x\right )^{2} + \tan \left (a\right )^{2} + 1}\right ) \tan \left (b x\right ) \tan \left (a\right ) + 2 \, b c \tan \left (b x\right ) + 2 \, b c \tan \left (a\right ) + d \log \left (\frac {4 \, {\left (\tan \left (b x\right )^{2} \tan \left (a\right )^{2} - 2 \, \tan \left (b x\right ) \tan \left (a\right ) + 1\right )}}{\tan \left (b x\right )^{2} \tan \left (a\right )^{2} + \tan \left (b x\right )^{2} + \tan \left (a\right )^{2} + 1}\right )}{2 \, {\left (b^{2} \tan \left (b x\right ) \tan \left (a\right ) - b^{2}\right )}} \]
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Time = 25.92 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.30 \[ \int (c+d x) \tan ^2(a+b x) \, dx=-c\,x-\frac {d\,x^2}{2}-\frac {\frac {d\,\ln \left ({\mathrm {tan}\left (a+b\,x\right )}^2+1\right )}{2}-b\,\left (c\,\mathrm {tan}\left (a+b\,x\right )+d\,x\,\mathrm {tan}\left (a+b\,x\right )\right )}{b^2} \]
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