Optimal. Leaf size=42 \[ -\frac {67 x}{250}-\frac {28}{125} \log (\cos (x)+3 \sin (x))-\frac {7}{10 (1+3 \tan (x))^2}-\frac {29}{50 (1+3 \tan (x))} \]
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Rubi [A]
time = 0.07, antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {3709, 3610,
3612, 3611} \begin {gather*} -\frac {67 x}{250}-\frac {29}{50 (3 \tan (x)+1)}-\frac {7}{10 (3 \tan (x)+1)^2}-\frac {28}{125} \log (3 \sin (x)+\cos (x)) \end {gather*}
Antiderivative was successfully verified.
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Rule 3610
Rule 3611
Rule 3612
Rule 3709
Rubi steps
\begin {align*} \int \frac {5-\tan (x)-6 \tan ^2(x)}{(1+3 \tan (x))^3} \, dx &=-\frac {7}{10 (1+3 \tan (x))^2}+\frac {1}{10} \int \frac {8-34 \tan (x)}{(1+3 \tan (x))^2} \, dx\\ &=-\frac {7}{10 (1+3 \tan (x))^2}-\frac {29}{50 (1+3 \tan (x))}+\frac {1}{100} \int \frac {-94-58 \tan (x)}{1+3 \tan (x)} \, dx\\ &=-\frac {67 x}{250}-\frac {7}{10 (1+3 \tan (x))^2}-\frac {29}{50 (1+3 \tan (x))}-\frac {28}{125} \int \frac {3-\tan (x)}{1+3 \tan (x)} \, dx\\ &=-\frac {67 x}{250}-\frac {28}{125} \log (\cos (x)+3 \sin (x))-\frac {7}{10 (1+3 \tan (x))^2}-\frac {29}{50 (1+3 \tan (x))}\\ \end {align*}
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Mathematica [A]
time = 0.15, size = 70, normalized size = 1.67 \begin {gather*} -\frac {-1305+670 x+560 \log (\cos (x)+3 \sin (x))-4 \cos (2 x) (-405+134 x+112 \log (\cos (x)+3 \sin (x)))+6 (-90+67 x+56 \log (\cos (x)+3 \sin (x))) \sin (2 x)}{500 (\cos (x)+3 \sin (x))^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.05, size = 45, normalized size = 1.07
method | result | size |
derivativedivides | \(\frac {14 \ln \left (1+\tan ^{2}\left (x \right )\right )}{125}-\frac {67 \arctan \left (\tan \left (x \right )\right )}{250}-\frac {7}{10 \left (1+3 \tan \left (x \right )\right )^{2}}-\frac {29}{50 \left (1+3 \tan \left (x \right )\right )}-\frac {28 \ln \left (1+3 \tan \left (x \right )\right )}{125}\) | \(45\) |
default | \(\frac {14 \ln \left (1+\tan ^{2}\left (x \right )\right )}{125}-\frac {67 \arctan \left (\tan \left (x \right )\right )}{250}-\frac {7}{10 \left (1+3 \tan \left (x \right )\right )^{2}}-\frac {29}{50 \left (1+3 \tan \left (x \right )\right )}-\frac {28 \ln \left (1+3 \tan \left (x \right )\right )}{125}\) | \(45\) |
risch | \(-\frac {67 x}{250}+\frac {28 i x}{125}+\frac {\left (-\frac {36}{24125}-\frac {621 i}{48250}\right ) \left (965 \,{\mathrm e}^{2 i x}-324+768 i\right )}{\left (5 \,{\mathrm e}^{2 i x}-4+3 i\right )^{2}}-\frac {28 \ln \left ({\mathrm e}^{2 i x}-\frac {4}{5}+\frac {3 i}{5}\right )}{125}\) | \(49\) |
norman | \(\frac {\frac {297 \tan \left (x \right )}{50}+\frac {288 \left (\tan ^{2}\left (x \right )\right )}{25}-\frac {67 x}{250}-\frac {201 x \tan \left (x \right )}{125}-\frac {603 x \left (\tan ^{2}\left (x \right )\right )}{250}}{\left (1+3 \tan \left (x \right )\right )^{2}}-\frac {28 \ln \left (1+3 \tan \left (x \right )\right )}{125}+\frac {14 \ln \left (1+\tan ^{2}\left (x \right )\right )}{125}\) | \(55\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 1.58, size = 44, normalized size = 1.05 \begin {gather*} -\frac {67}{250} \, x - \frac {87 \, \tan \left (x\right ) + 64}{50 \, {\left (9 \, \tan \left (x\right )^{2} + 6 \, \tan \left (x\right ) + 1\right )}} + \frac {14}{125} \, \log \left (\tan \left (x\right )^{2} + 1\right ) - \frac {28}{125} \, \log \left (3 \, \tan \left (x\right ) + 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 77 vs.
\(2 (34) = 68\).
time = 0.96, size = 77, normalized size = 1.83 \begin {gather*} -\frac {9 \, {\left (134 \, x - 1\right )} \tan \left (x\right )^{2} + 56 \, {\left (9 \, \tan \left (x\right )^{2} + 6 \, \tan \left (x\right ) + 1\right )} \log \left (\frac {9 \, \tan \left (x\right )^{2} + 6 \, \tan \left (x\right ) + 1}{\tan \left (x\right )^{2} + 1}\right ) + 12 \, {\left (67 \, x + 72\right )} \tan \left (x\right ) + 134 \, x + 639}{500 \, {\left (9 \, \tan \left (x\right )^{2} + 6 \, \tan \left (x\right ) + 1\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. 252 vs.
\(2 (39) = 78\).
time = 0.24, size = 252, normalized size = 6.00 \begin {gather*} - \frac {603 x \tan ^{2}{\left (x \right )}}{2250 \tan ^{2}{\left (x \right )} + 1500 \tan {\left (x \right )} + 250} - \frac {402 x \tan {\left (x \right )}}{2250 \tan ^{2}{\left (x \right )} + 1500 \tan {\left (x \right )} + 250} - \frac {67 x}{2250 \tan ^{2}{\left (x \right )} + 1500 \tan {\left (x \right )} + 250} - \frac {504 \log {\left (3 \tan {\left (x \right )} + 1 \right )} \tan ^{2}{\left (x \right )}}{2250 \tan ^{2}{\left (x \right )} + 1500 \tan {\left (x \right )} + 250} - \frac {336 \log {\left (3 \tan {\left (x \right )} + 1 \right )} \tan {\left (x \right )}}{2250 \tan ^{2}{\left (x \right )} + 1500 \tan {\left (x \right )} + 250} - \frac {56 \log {\left (3 \tan {\left (x \right )} + 1 \right )}}{2250 \tan ^{2}{\left (x \right )} + 1500 \tan {\left (x \right )} + 250} + \frac {252 \log {\left (\tan ^{2}{\left (x \right )} + 1 \right )} \tan ^{2}{\left (x \right )}}{2250 \tan ^{2}{\left (x \right )} + 1500 \tan {\left (x \right )} + 250} + \frac {168 \log {\left (\tan ^{2}{\left (x \right )} + 1 \right )} \tan {\left (x \right )}}{2250 \tan ^{2}{\left (x \right )} + 1500 \tan {\left (x \right )} + 250} + \frac {28 \log {\left (\tan ^{2}{\left (x \right )} + 1 \right )}}{2250 \tan ^{2}{\left (x \right )} + 1500 \tan {\left (x \right )} + 250} - \frac {435 \tan {\left (x \right )}}{2250 \tan ^{2}{\left (x \right )} + 1500 \tan {\left (x \right )} + 250} - \frac {320}{2250 \tan ^{2}{\left (x \right )} + 1500 \tan {\left (x \right )} + 250} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.67, size = 39, normalized size = 0.93 \begin {gather*} -\frac {67}{250} \, x - \frac {87 \, \tan \left (x\right ) + 64}{50 \, {\left (3 \, \tan \left (x\right ) + 1\right )}^{2}} + \frac {14}{125} \, \log \left (\tan \left (x\right )^{2} + 1\right ) - \frac {28}{125} \, \log \left ({\left | 3 \, \tan \left (x\right ) + 1 \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.28, size = 48, normalized size = 1.14 \begin {gather*} -\frac {28\,\ln \left (\mathrm {tan}\left (x\right )+\frac {1}{3}\right )}{125}-\frac {\frac {29\,\mathrm {tan}\left (x\right )}{150}+\frac {32}{225}}{{\mathrm {tan}\left (x\right )}^2+\frac {2\,\mathrm {tan}\left (x\right )}{3}+\frac {1}{9}}+\ln \left (\mathrm {tan}\left (x\right )-\mathrm {i}\right )\,\left (\frac {14}{125}+\frac {67}{500}{}\mathrm {i}\right )+\ln \left (\mathrm {tan}\left (x\right )+1{}\mathrm {i}\right )\,\left (\frac {14}{125}-\frac {67}{500}{}\mathrm {i}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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