Optimal. Leaf size=50 \[ -\frac {4 x}{289}+\frac {15 \log (3 \cos (c+d x)+5 \sin (c+d x))}{578 d}-\frac {5}{34 d (3+5 \tan (c+d x))} \]
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Rubi [A]
time = 0.04, antiderivative size = 50, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3564, 3612,
3611} \begin {gather*} -\frac {5}{34 d (5 \tan (c+d x)+3)}+\frac {15 \log (5 \sin (c+d x)+3 \cos (c+d x))}{578 d}-\frac {4 x}{289} \end {gather*}
Antiderivative was successfully verified.
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Rule 3564
Rule 3611
Rule 3612
Rubi steps
\begin {align*} \int \frac {1}{(3+5 \tan (c+d x))^2} \, dx &=-\frac {5}{34 d (3+5 \tan (c+d x))}+\frac {1}{34} \int \frac {3-5 \tan (c+d x)}{3+5 \tan (c+d x)} \, dx\\ &=-\frac {4 x}{289}-\frac {5}{34 d (3+5 \tan (c+d x))}+\frac {15}{578} \int \frac {5-3 \tan (c+d x)}{3+5 \tan (c+d x)} \, dx\\ &=-\frac {4 x}{289}+\frac {15 \log (3 \cos (c+d x)+5 \sin (c+d x))}{578 d}-\frac {5}{34 d (3+5 \tan (c+d x))}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 0.28, size = 67, normalized size = 1.34 \begin {gather*} -\frac {(15-8 i) \log (i-\tan (c+d x))+(15+8 i) \log (i+\tan (c+d x))-30 \log (3+5 \tan (c+d x))+\frac {170}{3+5 \tan (c+d x)}}{1156 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.08, size = 55, normalized size = 1.10
method | result | size |
derivativedivides | \(\frac {-\frac {15 \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{1156}-\frac {4 \arctan \left (\tan \left (d x +c \right )\right )}{289}-\frac {5}{34 \left (3+5 \tan \left (d x +c \right )\right )}+\frac {15 \ln \left (3+5 \tan \left (d x +c \right )\right )}{578}}{d}\) | \(55\) |
default | \(\frac {-\frac {15 \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{1156}-\frac {4 \arctan \left (\tan \left (d x +c \right )\right )}{289}-\frac {5}{34 \left (3+5 \tan \left (d x +c \right )\right )}+\frac {15 \ln \left (3+5 \tan \left (d x +c \right )\right )}{578}}{d}\) | \(55\) |
norman | \(\frac {-\frac {12 x}{289}-\frac {20 x \tan \left (d x +c \right )}{289}+\frac {25 \tan \left (d x +c \right )}{102 d}}{3+5 \tan \left (d x +c \right )}+\frac {15 \ln \left (3+5 \tan \left (d x +c \right )\right )}{578 d}-\frac {15 \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{1156 d}\) | \(71\) |
risch | \(-\frac {4 x}{289}-\frac {15 i x}{578}-\frac {15 i c}{289 d}-\frac {375}{578 d \left (17 \,{\mathrm e}^{2 i \left (d x +c \right )}-8+15 i\right )}-\frac {100 i}{289 d \left (17 \,{\mathrm e}^{2 i \left (d x +c \right )}-8+15 i\right )}+\frac {15 \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {8}{17}+\frac {15 i}{17}\right )}{578 d}\) | \(80\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 53, normalized size = 1.06 \begin {gather*} -\frac {16 \, d x + 16 \, c + \frac {170}{5 \, \tan \left (d x + c\right ) + 3} + 15 \, \log \left (\tan \left (d x + c\right )^{2} + 1\right ) - 30 \, \log \left (5 \, \tan \left (d x + c\right ) + 3\right )}{1156 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.02, size = 83, normalized size = 1.66 \begin {gather*} -\frac {48 \, d x - 15 \, {\left (5 \, \tan \left (d x + c\right ) + 3\right )} \log \left (\frac {25 \, \tan \left (d x + c\right )^{2} + 30 \, \tan \left (d x + c\right ) + 9}{\tan \left (d x + c\right )^{2} + 1}\right ) + 5 \, {\left (16 \, d x - 15\right )} \tan \left (d x + c\right ) + 125}{1156 \, {\left (5 \, d \tan \left (d x + c\right ) + 3 \, d\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 190 vs.
\(2 (42) = 84\).
time = 0.29, size = 190, normalized size = 3.80 \begin {gather*} \begin {cases} - \frac {80 d x \tan {\left (c + d x \right )}}{5780 d \tan {\left (c + d x \right )} + 3468 d} - \frac {48 d x}{5780 d \tan {\left (c + d x \right )} + 3468 d} + \frac {150 \log {\left (5 \tan {\left (c + d x \right )} + 3 \right )} \tan {\left (c + d x \right )}}{5780 d \tan {\left (c + d x \right )} + 3468 d} + \frac {90 \log {\left (5 \tan {\left (c + d x \right )} + 3 \right )}}{5780 d \tan {\left (c + d x \right )} + 3468 d} - \frac {75 \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )} \tan {\left (c + d x \right )}}{5780 d \tan {\left (c + d x \right )} + 3468 d} - \frac {45 \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{5780 d \tan {\left (c + d x \right )} + 3468 d} - \frac {170}{5780 d \tan {\left (c + d x \right )} + 3468 d} & \text {for}\: d \neq 0 \\\frac {x}{\left (5 \tan {\left (c \right )} + 3\right )^{2}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.46, size = 64, normalized size = 1.28 \begin {gather*} -\frac {16 \, d x + 16 \, c + \frac {10 \, {\left (15 \, \tan \left (d x + c\right ) + 26\right )}}{5 \, \tan \left (d x + c\right ) + 3} + 15 \, \log \left (\tan \left (d x + c\right )^{2} + 1\right ) - 30 \, \log \left ({\left | 5 \, \tan \left (d x + c\right ) + 3 \right |}\right )}{1156 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.95, size = 64, normalized size = 1.28 \begin {gather*} \frac {15\,\ln \left (\mathrm {tan}\left (c+d\,x\right )+\frac {3}{5}\right )}{578\,d}-\frac {1}{34\,d\,\left (\mathrm {tan}\left (c+d\,x\right )+\frac {3}{5}\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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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