Integrand size = 12, antiderivative size = 50 \[ \int \frac {1}{(5+3 \tan (c+d x))^2} \, dx=\frac {4 x}{289}+\frac {15 \log (5 \cos (c+d x)+3 \sin (c+d x))}{578 d}-\frac {3}{34 d (5+3 \tan (c+d x))} \]
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Time = 0.09 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3564, 3612, 3611} \[ \int \frac {1}{(5+3 \tan (c+d x))^2} \, dx=-\frac {3}{34 d (3 \tan (c+d x)+5)}+\frac {15 \log (3 \sin (c+d x)+5 \cos (c+d x))}{578 d}+\frac {4 x}{289} \]
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Rule 3564
Rule 3611
Rule 3612
Rubi steps \begin{align*} \text {integral}& = -\frac {3}{34 d (5+3 \tan (c+d x))}+\frac {1}{34} \int \frac {5-3 \tan (c+d x)}{5+3 \tan (c+d x)} \, dx \\ & = \frac {4 x}{289}-\frac {3}{34 d (5+3 \tan (c+d x))}+\frac {15}{578} \int \frac {3-5 \tan (c+d x)}{5+3 \tan (c+d x)} \, dx \\ & = \frac {4 x}{289}+\frac {15 \log (5 \cos (c+d x)+3 \sin (c+d x))}{578 d}-\frac {3}{34 d (5+3 \tan (c+d x))} \\ \end{align*}
Result contains complex when optimal does not.
Time = 0.28 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.34 \[ \int \frac {1}{(5+3 \tan (c+d x))^2} \, dx=-\frac {(15+8 i) \log (i-\tan (c+d x))+(15-8 i) \log (i+\tan (c+d x))-30 \log (5+3 \tan (c+d x))+\frac {102}{5+3 \tan (c+d x)}}{1156 d} \]
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Time = 0.31 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.10
method | result | size |
derivativedivides | \(\frac {-\frac {3}{34 \left (5+3 \tan \left (d x +c \right )\right )}+\frac {15 \ln \left (5+3 \tan \left (d x +c \right )\right )}{578}-\frac {15 \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{1156}+\frac {4 \arctan \left (\tan \left (d x +c \right )\right )}{289}}{d}\) | \(55\) |
default | \(\frac {-\frac {3}{34 \left (5+3 \tan \left (d x +c \right )\right )}+\frac {15 \ln \left (5+3 \tan \left (d x +c \right )\right )}{578}-\frac {15 \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{1156}+\frac {4 \arctan \left (\tan \left (d x +c \right )\right )}{289}}{d}\) | \(55\) |
norman | \(\frac {\frac {20 x}{289}+\frac {12 x \tan \left (d x +c \right )}{289}-\frac {3}{34 d}}{5+3 \tan \left (d x +c \right )}+\frac {15 \ln \left (5+3 \tan \left (d x +c \right )\right )}{578 d}-\frac {15 \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{1156 d}\) | \(65\) |
risch | \(\frac {4 x}{289}-\frac {15 i x}{578}-\frac {15 i c}{289 d}-\frac {135}{578 d \left (17 \,{\mathrm e}^{2 i \left (d x +c \right )}+8+15 i\right )}+\frac {36 i}{289 d \left (17 \,{\mathrm e}^{2 i \left (d x +c \right )}+8+15 i\right )}+\frac {15 \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+\frac {8}{17}+\frac {15 i}{17}\right )}{578 d}\) | \(80\) |
parallelrisch | \(\frac {144 \tan \left (d x +c \right ) x d -306+270 \ln \left (\frac {5}{3}+\tan \left (d x +c \right )\right ) \tan \left (d x +c \right )-135 \ln \left (1+\tan ^{2}\left (d x +c \right )\right ) \tan \left (d x +c \right )+240 d x +450 \ln \left (\frac {5}{3}+\tan \left (d x +c \right )\right )-225 \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{3468 d \left (5+3 \tan \left (d x +c \right )\right )}\) | \(94\) |
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Time = 0.24 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.66 \[ \int \frac {1}{(5+3 \tan (c+d x))^2} \, dx=\frac {80 \, d x + 15 \, {\left (3 \, \tan \left (d x + c\right ) + 5\right )} \log \left (\frac {9 \, \tan \left (d x + c\right )^{2} + 30 \, \tan \left (d x + c\right ) + 25}{\tan \left (d x + c\right )^{2} + 1}\right ) + 3 \, {\left (16 \, d x + 15\right )} \tan \left (d x + c\right ) - 27}{1156 \, {\left (3 \, d \tan \left (d x + c\right ) + 5 \, d\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 190 vs. \(2 (42) = 84\).
Time = 0.27 (sec) , antiderivative size = 190, normalized size of antiderivative = 3.80 \[ \int \frac {1}{(5+3 \tan (c+d x))^2} \, dx=\begin {cases} \frac {48 d x \tan {\left (c + d x \right )}}{3468 d \tan {\left (c + d x \right )} + 5780 d} + \frac {80 d x}{3468 d \tan {\left (c + d x \right )} + 5780 d} + \frac {90 \log {\left (3 \tan {\left (c + d x \right )} + 5 \right )} \tan {\left (c + d x \right )}}{3468 d \tan {\left (c + d x \right )} + 5780 d} + \frac {150 \log {\left (3 \tan {\left (c + d x \right )} + 5 \right )}}{3468 d \tan {\left (c + d x \right )} + 5780 d} - \frac {45 \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )} \tan {\left (c + d x \right )}}{3468 d \tan {\left (c + d x \right )} + 5780 d} - \frac {75 \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{3468 d \tan {\left (c + d x \right )} + 5780 d} - \frac {102}{3468 d \tan {\left (c + d x \right )} + 5780 d} & \text {for}\: d \neq 0 \\\frac {x}{\left (3 \tan {\left (c \right )} + 5\right )^{2}} & \text {otherwise} \end {cases} \]
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Time = 0.40 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.06 \[ \int \frac {1}{(5+3 \tan (c+d x))^2} \, dx=\frac {16 \, d x + 16 \, c - \frac {102}{3 \, \tan \left (d x + c\right ) + 5} - 15 \, \log \left (\tan \left (d x + c\right )^{2} + 1\right ) + 30 \, \log \left (3 \, \tan \left (d x + c\right ) + 5\right )}{1156 \, d} \]
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Time = 0.31 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.28 \[ \int \frac {1}{(5+3 \tan (c+d x))^2} \, dx=\frac {16 \, d x + 16 \, c - \frac {18 \, {\left (5 \, \tan \left (d x + c\right ) + 14\right )}}{3 \, \tan \left (d x + c\right ) + 5} - 15 \, \log \left (\tan \left (d x + c\right )^{2} + 1\right ) + 30 \, \log \left ({\left | 3 \, \tan \left (d x + c\right ) + 5 \right |}\right )}{1156 \, d} \]
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Time = 5.15 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.28 \[ \int \frac {1}{(5+3 \tan (c+d x))^2} \, dx=\frac {15\,\ln \left (\mathrm {tan}\left (c+d\,x\right )+\frac {5}{3}\right )}{578\,d}-\frac {1}{34\,d\,\left (\mathrm {tan}\left (c+d\,x\right )+\frac {5}{3}\right )}+\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )\,\left (-\frac {15}{1156}-\frac {2}{289}{}\mathrm {i}\right )}{d}+\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,\left (-\frac {15}{1156}+\frac {2}{289}{}\mathrm {i}\right )}{d} \]
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