Optimal. Leaf size=69 \[ -\frac {99 x}{19652}+\frac {5 \log (3 \cos (c+d x)+5 \sin (c+d x))}{19652 d}-\frac {5}{68 d (3+5 \tan (c+d x))^2}-\frac {15}{578 d (3+5 \tan (c+d x))} \]
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Rubi [A]
time = 0.07, antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3564, 3610,
3612, 3611} \begin {gather*} -\frac {15}{578 d (5 \tan (c+d x)+3)}-\frac {5}{68 d (5 \tan (c+d x)+3)^2}+\frac {5 \log (5 \sin (c+d x)+3 \cos (c+d x))}{19652 d}-\frac {99 x}{19652} \end {gather*}
Antiderivative was successfully verified.
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Rule 3564
Rule 3610
Rule 3611
Rule 3612
Rubi steps
\begin {align*} \int \frac {1}{(3+5 \tan (c+d x))^3} \, dx &=-\frac {5}{68 d (3+5 \tan (c+d x))^2}+\frac {1}{34} \int \frac {3-5 \tan (c+d x)}{(3+5 \tan (c+d x))^2} \, dx\\ &=-\frac {5}{68 d (3+5 \tan (c+d x))^2}-\frac {15}{578 d (3+5 \tan (c+d x))}+\frac {\int \frac {-16-30 \tan (c+d x)}{3+5 \tan (c+d x)} \, dx}{1156}\\ &=-\frac {99 x}{19652}-\frac {5}{68 d (3+5 \tan (c+d x))^2}-\frac {15}{578 d (3+5 \tan (c+d x))}+\frac {5 \int \frac {5-3 \tan (c+d x)}{3+5 \tan (c+d x)} \, dx}{19652}\\ &=-\frac {99 x}{19652}+\frac {5 \log (3 \cos (c+d x)+5 \sin (c+d x))}{19652 d}-\frac {5}{68 d (3+5 \tan (c+d x))^2}-\frac {15}{578 d (3+5 \tan (c+d x))}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 0.61, size = 84, normalized size = 1.22 \begin {gather*} \frac {\left (\frac {1}{39304}+\frac {i}{39304}\right ) \left ((47+52 i) \log (i-\tan (c+d x))-(52+47 i) \log (i+\tan (c+d x))+(5-5 i) \left (\log (3+5 \tan (c+d x))-\frac {85 (7+6 \tan (c+d x))}{(3+5 \tan (c+d x))^2}\right )\right )}{d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.11, size = 69, normalized size = 1.00
method | result | size |
derivativedivides | \(\frac {-\frac {5 \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{39304}-\frac {99 \arctan \left (\tan \left (d x +c \right )\right )}{19652}-\frac {5}{68 \left (3+5 \tan \left (d x +c \right )\right )^{2}}-\frac {15}{578 \left (3+5 \tan \left (d x +c \right )\right )}+\frac {5 \ln \left (3+5 \tan \left (d x +c \right )\right )}{19652}}{d}\) | \(69\) |
default | \(\frac {-\frac {5 \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{39304}-\frac {99 \arctan \left (\tan \left (d x +c \right )\right )}{19652}-\frac {5}{68 \left (3+5 \tan \left (d x +c \right )\right )^{2}}-\frac {15}{578 \left (3+5 \tan \left (d x +c \right )\right )}+\frac {5 \ln \left (3+5 \tan \left (d x +c \right )\right )}{19652}}{d}\) | \(69\) |
risch | \(-\frac {99 x}{19652}-\frac {5 i x}{19652}-\frac {5 i c}{9826 d}+\frac {\left (-\frac {2375}{2083112}-\frac {3675 i}{2083112}\right ) \left (1802 \,{\mathrm e}^{2 i \left (d x +c \right )}+27+1575 i\right )}{d \left (17 \,{\mathrm e}^{2 i \left (d x +c \right )}-8+15 i\right )^{2}}+\frac {5 \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {8}{17}+\frac {15 i}{17}\right )}{19652 d}\) | \(74\) |
norman | \(\frac {-\frac {891 x}{19652}-\frac {1485 x \tan \left (d x +c \right )}{9826}-\frac {2475 x \left (\tan ^{2}\left (d x +c \right )\right )}{19652}-\frac {175}{1156 d}-\frac {75 \tan \left (d x +c \right )}{578 d}}{\left (3+5 \tan \left (d x +c \right )\right )^{2}}+\frac {5 \ln \left (3+5 \tan \left (d x +c \right )\right )}{19652 d}-\frac {5 \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{39304 d}\) | \(87\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.49, size = 73, normalized size = 1.06 \begin {gather*} -\frac {198 \, d x + 198 \, c + \frac {850 \, {\left (6 \, \tan \left (d x + c\right ) + 7\right )}}{25 \, \tan \left (d x + c\right )^{2} + 30 \, \tan \left (d x + c\right ) + 9} + 5 \, \log \left (\tan \left (d x + c\right )^{2} + 1\right ) - 10 \, \log \left (5 \, \tan \left (d x + c\right ) + 3\right )}{39304 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.00, size = 120, normalized size = 1.74 \begin {gather*} -\frac {50 \, {\left (99 \, d x - 25\right )} \tan \left (d x + c\right )^{2} + 1782 \, d x - 5 \, {\left (25 \, \tan \left (d x + c\right )^{2} + 30 \, \tan \left (d x + c\right ) + 9\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 ) + 180 \, {\left (33 \, d x + 20\right )} \tan \left (d x + c\right ) + 5500}{39304 \, {\left (25 \, d \tan \left (d x + c\right )^{2} + 30 \, d \tan \left (d x + c\right ) + 9 \, 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. 442 vs.
\(2 (60) = 120\).
time = 0.38, size = 442, normalized size = 6.41 \begin {gather*} \begin {cases} - \frac {4950 d x \tan ^{2}{\left (c + d x \right )}}{982600 d \tan ^{2}{\left (c + d x \right )} + 1179120 d \tan {\left (c + d x \right )} + 353736 d} - \frac {5940 d x \tan {\left (c + d x \right )}}{982600 d \tan ^{2}{\left (c + d x \right )} + 1179120 d \tan {\left (c + d x \right )} + 353736 d} - \frac {1782 d x}{982600 d \tan ^{2}{\left (c + d x \right )} + 1179120 d \tan {\left (c + d x \right )} + 353736 d} + \frac {250 \log {\left (5 \tan {\left (c + d x \right )} + 3 \right )} \tan ^{2}{\left (c + d x \right )}}{982600 d \tan ^{2}{\left (c + d x \right )} + 1179120 d \tan {\left (c + d x \right )} + 353736 d} + \frac {300 \log {\left (5 \tan {\left (c + d x \right )} + 3 \right )} \tan {\left (c + d x \right )}}{982600 d \tan ^{2}{\left (c + d x \right )} + 1179120 d \tan {\left (c + d x \right )} + 353736 d} + \frac {90 \log {\left (5 \tan {\left (c + d x \right )} + 3 \right )}}{982600 d \tan ^{2}{\left (c + d x \right )} + 1179120 d \tan {\left (c + d x \right )} + 353736 d} - \frac {125 \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )} \tan ^{2}{\left (c + d x \right )}}{982600 d \tan ^{2}{\left (c + d x \right )} + 1179120 d \tan {\left (c + d x \right )} + 353736 d} - \frac {150 \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )} \tan {\left (c + d x \right )}}{982600 d \tan ^{2}{\left (c + d x \right )} + 1179120 d \tan {\left (c + d x \right )} + 353736 d} - \frac {45 \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{982600 d \tan ^{2}{\left (c + d x \right )} + 1179120 d \tan {\left (c + d x \right )} + 353736 d} - \frac {5100 \tan {\left (c + d x \right )}}{982600 d \tan ^{2}{\left (c + d x \right )} + 1179120 d \tan {\left (c + d x \right )} + 353736 d} - \frac {5950}{982600 d \tan ^{2}{\left (c + d x \right )} + 1179120 d \tan {\left (c + d x \right )} + 353736 d} & \text {for}\: d \neq 0 \\\frac {x}{\left (5 \tan {\left (c \right )} + 3\right )^{3}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.67, size = 74, normalized size = 1.07 \begin {gather*} -\frac {198 \, d x + 198 \, c + \frac {5 \, {\left (75 \, \tan \left (d x + c\right )^{2} + 1110 \, \tan \left (d x + c\right ) + 1217\right )}}{{\left (5 \, \tan \left (d x + c\right ) + 3\right )}^{2}} + 5 \, \log \left (\tan \left (d x + c\right )^{2} + 1\right ) - 10 \, \log \left ({\left | 5 \, \tan \left (d x + c\right ) + 3 \right |}\right )}{39304 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.02, size = 84, normalized size = 1.22 \begin {gather*} \frac {5\,\ln \left (\mathrm {tan}\left (c+d\,x\right )+\frac {3}{5}\right )}{19652\,d}-\frac {\frac {3\,\mathrm {tan}\left (c+d\,x\right )}{578}+\frac {7}{1156}}{d\,\left ({\mathrm {tan}\left (c+d\,x\right )}^2+\frac {6\,\mathrm {tan}\left (c+d\,x\right )}{5}+\frac {9}{25}\right )}+\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )\,\left (-\frac {5}{39304}+\frac {99}{39304}{}\mathrm {i}\right )}{d}+\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,\left (-\frac {5}{39304}-\frac {99}{39304}{}\mathrm {i}\right )}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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