Optimal. Leaf size=88 \[ -\frac {161 x}{334084}-\frac {60 \log (3 \cos (c+d x)+5 \sin (c+d x))}{83521 d}-\frac {5}{102 d (3+5 \tan (c+d x))^3}-\frac {15}{1156 d (3+5 \tan (c+d x))^2}-\frac {5}{19652 d (3+5 \tan (c+d x))} \]
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Rubi [A]
time = 0.09, antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps
used = 5, 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 {5}{19652 d (5 \tan (c+d x)+3)}-\frac {15}{1156 d (5 \tan (c+d x)+3)^2}-\frac {5}{102 d (5 \tan (c+d x)+3)^3}-\frac {60 \log (5 \sin (c+d x)+3 \cos (c+d x))}{83521 d}-\frac {161 x}{334084} \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))^4} \, dx &=-\frac {5}{102 d (3+5 \tan (c+d x))^3}+\frac {1}{34} \int \frac {3-5 \tan (c+d x)}{(3+5 \tan (c+d x))^3} \, dx\\ &=-\frac {5}{102 d (3+5 \tan (c+d x))^3}-\frac {15}{1156 d (3+5 \tan (c+d x))^2}+\frac {\int \frac {-16-30 \tan (c+d x)}{(3+5 \tan (c+d x))^2} \, dx}{1156}\\ &=-\frac {5}{102 d (3+5 \tan (c+d x))^3}-\frac {15}{1156 d (3+5 \tan (c+d x))^2}-\frac {5}{19652 d (3+5 \tan (c+d x))}+\frac {\int \frac {-198-10 \tan (c+d x)}{3+5 \tan (c+d x)} \, dx}{39304}\\ &=-\frac {161 x}{334084}-\frac {5}{102 d (3+5 \tan (c+d x))^3}-\frac {15}{1156 d (3+5 \tan (c+d x))^2}-\frac {5}{19652 d (3+5 \tan (c+d x))}-\frac {60 \int \frac {5-3 \tan (c+d x)}{3+5 \tan (c+d x)} \, dx}{83521}\\ &=-\frac {161 x}{334084}-\frac {60 \log (3 \cos (c+d x)+5 \sin (c+d x))}{83521 d}-\frac {5}{102 d (3+5 \tan (c+d x))^3}-\frac {15}{1156 d (3+5 \tan (c+d x))^2}-\frac {5}{19652 d (3+5 \tan (c+d x))}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 1.18, size = 87, normalized size = 0.99 \begin {gather*} \frac {(720+483 i) \log (i-\tan (c+d x))+(720-483 i) \log (i+\tan (c+d x))-1440 \log (3+5 \tan (c+d x))-\frac {170 \left (1064+855 \tan (c+d x)+75 \tan ^2(c+d x)\right )}{(3+5 \tan (c+d x))^3}}{2004504 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.10, size = 83, normalized size = 0.94
method | result | size |
derivativedivides | \(\frac {\frac {30 \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{83521}-\frac {161 \arctan \left (\tan \left (d x +c \right )\right )}{334084}-\frac {5}{102 \left (3+5 \tan \left (d x +c \right )\right )^{3}}-\frac {15}{1156 \left (3+5 \tan \left (d x +c \right )\right )^{2}}-\frac {5}{19652 \left (3+5 \tan \left (d x +c \right )\right )}-\frac {60 \ln \left (3+5 \tan \left (d x +c \right )\right )}{83521}}{d}\) | \(83\) |
default | \(\frac {\frac {30 \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{83521}-\frac {161 \arctan \left (\tan \left (d x +c \right )\right )}{334084}-\frac {5}{102 \left (3+5 \tan \left (d x +c \right )\right )^{3}}-\frac {15}{1156 \left (3+5 \tan \left (d x +c \right )\right )^{2}}-\frac {5}{19652 \left (3+5 \tan \left (d x +c \right )\right )}-\frac {60 \ln \left (3+5 \tan \left (d x +c \right )\right )}{83521}}{d}\) | \(83\) |
risch | \(-\frac {161 x}{334084}+\frac {60 i x}{83521}+\frac {120 i c}{83521 d}+\frac {\left (\frac {875}{75502984}-\frac {5825 i}{226508952}\right ) \left (391884 \,{\mathrm e}^{4 i \left (d x +c \right )}+531675 i {\mathrm e}^{2 i \left (d x +c \right )}+114393 \,{\mathrm e}^{2 i \left (d x +c \right )}-116591+67425 i\right )}{d \left (17 \,{\mathrm e}^{2 i \left (d x +c \right )}-8+15 i\right )^{3}}-\frac {60 \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {8}{17}+\frac {15 i}{17}\right )}{83521 d}\) | \(97\) |
norman | \(\frac {-\frac {4347 x}{334084}-\frac {21735 x \tan \left (d x +c \right )}{334084}-\frac {36225 x \left (\tan ^{2}\left (d x +c \right )\right )}{334084}-\frac {20125 x \left (\tan ^{3}\left (d x +c \right )\right )}{334084}+\frac {166250 \left (\tan ^{3}\left (d x +c \right )\right )}{397953 d}+\frac {22325 \tan \left (d x +c \right )}{58956 d}+\frac {131875 \left (\tan ^{2}\left (d x +c \right )\right )}{176868 d}}{\left (3+5 \tan \left (d x +c \right )\right )^{3}}-\frac {60 \ln \left (3+5 \tan \left (d x +c \right )\right )}{83521 d}+\frac {30 \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{83521 d}\) | \(119\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.49, size = 93, normalized size = 1.06 \begin {gather*} -\frac {483 \, d x + 483 \, c + \frac {85 \, {\left (75 \, \tan \left (d x + c\right )^{2} + 855 \, \tan \left (d x + c\right ) + 1064\right )}}{125 \, \tan \left (d x + c\right )^{3} + 225 \, \tan \left (d x + c\right )^{2} + 135 \, \tan \left (d x + c\right ) + 27} - 360 \, \log \left (\tan \left (d x + c\right )^{2} + 1\right ) + 720 \, \log \left (5 \, \tan \left (d x + c\right ) + 3\right )}{1002252 \, d} \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. 157 vs.
\(2 (78) = 156\).
time = 1.34, size = 157, normalized size = 1.78 \begin {gather*} -\frac {375 \, {\left (161 \, d x + 135\right )} \tan \left (d x + c\right )^{3} + 75 \, {\left (1449 \, d x + 1300\right )} \tan \left (d x + c\right )^{2} + 13041 \, d x + 360 \, {\left (125 \, \tan \left (d x + c\right )^{3} + 225 \, \tan \left (d x + c\right )^{2} + 135 \, \tan \left (d x + c\right ) + 27\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 ) + 45 \, {\left (1449 \, d x + 2830\right )} \tan \left (d x + c\right ) + 101375}{1002252 \, {\left (125 \, d \tan \left (d x + c\right )^{3} + 225 \, d \tan \left (d x + c\right )^{2} + 135 \, d \tan \left (d x + c\right ) + 27 \, 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. 790 vs.
\(2 (78) = 156\).
time = 0.48, size = 790, normalized size = 8.98 \begin {gather*} \begin {cases} - \frac {60375 d x \tan ^{3}{\left (c + d x \right )}}{125281500 d \tan ^{3}{\left (c + d x \right )} + 225506700 d \tan ^{2}{\left (c + d x \right )} + 135304020 d \tan {\left (c + d x \right )} + 27060804 d} - \frac {108675 d x \tan ^{2}{\left (c + d x \right )}}{125281500 d \tan ^{3}{\left (c + d x \right )} + 225506700 d \tan ^{2}{\left (c + d x \right )} + 135304020 d \tan {\left (c + d x \right )} + 27060804 d} - \frac {65205 d x \tan {\left (c + d x \right )}}{125281500 d \tan ^{3}{\left (c + d x \right )} + 225506700 d \tan ^{2}{\left (c + d x \right )} + 135304020 d \tan {\left (c + d x \right )} + 27060804 d} - \frac {13041 d x}{125281500 d \tan ^{3}{\left (c + d x \right )} + 225506700 d \tan ^{2}{\left (c + d x \right )} + 135304020 d \tan {\left (c + d x \right )} + 27060804 d} - \frac {90000 \log {\left (5 \tan {\left (c + d x \right )} + 3 \right )} \tan ^{3}{\left (c + d x \right )}}{125281500 d \tan ^{3}{\left (c + d x \right )} + 225506700 d \tan ^{2}{\left (c + d x \right )} + 135304020 d \tan {\left (c + d x \right )} + 27060804 d} - \frac {162000 \log {\left (5 \tan {\left (c + d x \right )} + 3 \right )} \tan ^{2}{\left (c + d x \right )}}{125281500 d \tan ^{3}{\left (c + d x \right )} + 225506700 d \tan ^{2}{\left (c + d x \right )} + 135304020 d \tan {\left (c + d x \right )} + 27060804 d} - \frac {97200 \log {\left (5 \tan {\left (c + d x \right )} + 3 \right )} \tan {\left (c + d x \right )}}{125281500 d \tan ^{3}{\left (c + d x \right )} + 225506700 d \tan ^{2}{\left (c + d x \right )} + 135304020 d \tan {\left (c + d x \right )} + 27060804 d} - \frac {19440 \log {\left (5 \tan {\left (c + d x \right )} + 3 \right )}}{125281500 d \tan ^{3}{\left (c + d x \right )} + 225506700 d \tan ^{2}{\left (c + d x \right )} + 135304020 d \tan {\left (c + d x \right )} + 27060804 d} + \frac {45000 \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )} \tan ^{3}{\left (c + d x \right )}}{125281500 d \tan ^{3}{\left (c + d x \right )} + 225506700 d \tan ^{2}{\left (c + d x \right )} + 135304020 d \tan {\left (c + d x \right )} + 27060804 d} + \frac {81000 \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )} \tan ^{2}{\left (c + d x \right )}}{125281500 d \tan ^{3}{\left (c + d x \right )} + 225506700 d \tan ^{2}{\left (c + d x \right )} + 135304020 d \tan {\left (c + d x \right )} + 27060804 d} + \frac {48600 \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )} \tan {\left (c + d x \right )}}{125281500 d \tan ^{3}{\left (c + d x \right )} + 225506700 d \tan ^{2}{\left (c + d x \right )} + 135304020 d \tan {\left (c + d x \right )} + 27060804 d} + \frac {9720 \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{125281500 d \tan ^{3}{\left (c + d x \right )} + 225506700 d \tan ^{2}{\left (c + d x \right )} + 135304020 d \tan {\left (c + d x \right )} + 27060804 d} - \frac {6375 \tan ^{2}{\left (c + d x \right )}}{125281500 d \tan ^{3}{\left (c + d x \right )} + 225506700 d \tan ^{2}{\left (c + d x \right )} + 135304020 d \tan {\left (c + d x \right )} + 27060804 d} - \frac {72675 \tan {\left (c + d x \right )}}{125281500 d \tan ^{3}{\left (c + d x \right )} + 225506700 d \tan ^{2}{\left (c + d x \right )} + 135304020 d \tan {\left (c + d x \right )} + 27060804 d} - \frac {90440}{125281500 d \tan ^{3}{\left (c + d x \right )} + 225506700 d \tan ^{2}{\left (c + d x \right )} + 135304020 d \tan {\left (c + d x \right )} + 27060804 d} & \text {for}\: d \neq 0 \\\frac {x}{\left (5 \tan {\left (c \right )} + 3\right )^{4}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.59, size = 84, normalized size = 0.95 \begin {gather*} -\frac {483 \, d x + 483 \, c - \frac {25 \, {\left (6600 \, \tan \left (d x + c\right )^{3} + 11625 \, \tan \left (d x + c\right )^{2} + 4221 \, \tan \left (d x + c\right ) - 2192\right )}}{{\left (5 \, \tan \left (d x + c\right ) + 3\right )}^{3}} - 360 \, \log \left (\tan \left (d x + c\right )^{2} + 1\right ) + 720 \, \log \left ({\left | 5 \, \tan \left (d x + c\right ) + 3 \right |}\right )}{1002252 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.12, size = 104, normalized size = 1.18 \begin {gather*} -\frac {60\,\ln \left (\mathrm {tan}\left (c+d\,x\right )+\frac {3}{5}\right )}{83521\,d}-\frac {\frac {{\mathrm {tan}\left (c+d\,x\right )}^2}{19652}+\frac {57\,\mathrm {tan}\left (c+d\,x\right )}{98260}+\frac {266}{368475}}{d\,\left ({\mathrm {tan}\left (c+d\,x\right )}^3+\frac {9\,{\mathrm {tan}\left (c+d\,x\right )}^2}{5}+\frac {27\,\mathrm {tan}\left (c+d\,x\right )}{25}+\frac {27}{125}\right )}+\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )\,\left (\frac {30}{83521}+\frac {161}{668168}{}\mathrm {i}\right )}{d}+\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,\left (\frac {30}{83521}-\frac {161}{668168}{}\mathrm {i}\right )}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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