Optimal. Leaf size=145 \[ \frac {38733 \tan (c+d x)}{1024000 d (3 \sec (c+d x)+5)}+\frac {519 \tan (c+d x)}{12800 d (3 \sec (c+d x)+5)^2}+\frac {3 \tan (c+d x)}{80 d (3 \sec (c+d x)+5)^3}+\frac {278151 \log \left (2 \cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )}{20480000 d}-\frac {278151 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )+2 \cos \left (\frac {1}{2} (c+d x)\right )\right )}{20480000 d}+\frac {x}{625} \]
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Rubi [A] time = 0.18, antiderivative size = 145, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3785, 4060, 3919, 3831, 2659, 206} \[ \frac {38733 \tan (c+d x)}{1024000 d (3 \sec (c+d x)+5)}+\frac {519 \tan (c+d x)}{12800 d (3 \sec (c+d x)+5)^2}+\frac {3 \tan (c+d x)}{80 d (3 \sec (c+d x)+5)^3}+\frac {278151 \log \left (2 \cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )}{20480000 d}-\frac {278151 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )+2 \cos \left (\frac {1}{2} (c+d x)\right )\right )}{20480000 d}+\frac {x}{625} \]
Antiderivative was successfully verified.
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Rule 206
Rule 2659
Rule 3785
Rule 3831
Rule 3919
Rule 4060
Rubi steps
\begin {align*} \int \frac {1}{(5+3 \sec (c+d x))^4} \, dx &=\frac {3 \tan (c+d x)}{80 d (5+3 \sec (c+d x))^3}-\frac {1}{240} \int \frac {-48+45 \sec (c+d x)-18 \sec ^2(c+d x)}{(5+3 \sec (c+d x))^3} \, dx\\ &=\frac {3 \tan (c+d x)}{80 d (5+3 \sec (c+d x))^3}+\frac {519 \tan (c+d x)}{12800 d (5+3 \sec (c+d x))^2}+\frac {\int \frac {1536-4230 \sec (c+d x)+1557 \sec ^2(c+d x)}{(5+3 \sec (c+d x))^2} \, dx}{38400}\\ &=\frac {3 \tan (c+d x)}{80 d (5+3 \sec (c+d x))^3}+\frac {519 \tan (c+d x)}{12800 d (5+3 \sec (c+d x))^2}+\frac {38733 \tan (c+d x)}{1024000 d (5+3 \sec (c+d x))}-\frac {\int \frac {-24576+152145 \sec (c+d x)}{5+3 \sec (c+d x)} \, dx}{3072000}\\ &=\frac {x}{625}+\frac {3 \tan (c+d x)}{80 d (5+3 \sec (c+d x))^3}+\frac {519 \tan (c+d x)}{12800 d (5+3 \sec (c+d x))^2}+\frac {38733 \tan (c+d x)}{1024000 d (5+3 \sec (c+d x))}-\frac {278151 \int \frac {\sec (c+d x)}{5+3 \sec (c+d x)} \, dx}{5120000}\\ &=\frac {x}{625}+\frac {3 \tan (c+d x)}{80 d (5+3 \sec (c+d x))^3}+\frac {519 \tan (c+d x)}{12800 d (5+3 \sec (c+d x))^2}+\frac {38733 \tan (c+d x)}{1024000 d (5+3 \sec (c+d x))}-\frac {92717 \int \frac {1}{1+\frac {5}{3} \cos (c+d x)} \, dx}{5120000}\\ &=\frac {x}{625}+\frac {3 \tan (c+d x)}{80 d (5+3 \sec (c+d x))^3}+\frac {519 \tan (c+d x)}{12800 d (5+3 \sec (c+d x))^2}+\frac {38733 \tan (c+d x)}{1024000 d (5+3 \sec (c+d x))}-\frac {92717 \operatorname {Subst}\left (\int \frac {1}{\frac {8}{3}-\frac {2 x^2}{3}} \, dx,x,\tan \left (\frac {1}{2} (c+d x)\right )\right )}{2560000 d}\\ &=\frac {x}{625}+\frac {278151 \log \left (2 \cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )}{20480000 d}-\frac {278151 \log \left (2 \cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )}{20480000 d}+\frac {3 \tan (c+d x)}{80 d (5+3 \sec (c+d x))^3}+\frac {519 \tan (c+d x)}{12800 d (5+3 \sec (c+d x))^2}+\frac {38733 \tan (c+d x)}{1024000 d (5+3 \sec (c+d x))}\\ \end {align*}
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Mathematica [B] time = 0.52, size = 344, normalized size = 2.37 \[ \frac {52174260 \sin (c+d x)+51462000 \sin (2 (c+d x))+24286500 \sin (3 (c+d x))+4096000 c \cos (3 (c+d x))+4096000 d x \cos (3 (c+d x))+34768875 \cos (3 (c+d x)) \log \left (2 \cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+155208258 \log \left (2 \cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+915 \cos (c+d x) \left (32768 (c+d x)+278151 \log \left (2 \cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )-278151 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )+2 \cos \left (\frac {1}{2} (c+d x)\right )\right )\right )+450 \cos (2 (c+d x)) \left (32768 (c+d x)+278151 \log \left (2 \cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )-278151 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )+2 \cos \left (\frac {1}{2} (c+d x)\right )\right )\right )-34768875 \cos (3 (c+d x)) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )+2 \cos \left (\frac {1}{2} (c+d x)\right )\right )-155208258 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )+2 \cos \left (\frac {1}{2} (c+d x)\right )\right )+18284544 c+18284544 d x}{81920000 d (5 \cos (c+d x)+3)^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.53, size = 208, normalized size = 1.43 \[ \frac {8192000 \, d x \cos \left (d x + c\right )^{3} + 14745600 \, d x \cos \left (d x + c\right )^{2} + 8847360 \, d x \cos \left (d x + c\right ) + 1769472 \, d x - 278151 \, {\left (125 \, \cos \left (d x + c\right )^{3} + 225 \, \cos \left (d x + c\right )^{2} + 135 \, \cos \left (d x + c\right ) + 27\right )} \log \left (\frac {3}{2} \, \cos \left (d x + c\right ) + 2 \, \sin \left (d x + c\right ) + \frac {5}{2}\right ) + 278151 \, {\left (125 \, \cos \left (d x + c\right )^{3} + 225 \, \cos \left (d x + c\right )^{2} + 135 \, \cos \left (d x + c\right ) + 27\right )} \log \left (\frac {3}{2} \, \cos \left (d x + c\right ) - 2 \, \sin \left (d x + c\right ) + \frac {5}{2}\right ) + 1080 \, {\left (44975 \, \cos \left (d x + c\right )^{2} + 47650 \, \cos \left (d x + c\right ) + 12911\right )} \sin \left (d x + c\right )}{40960000 \, {\left (125 \, d \cos \left (d x + c\right )^{3} + 225 \, d \cos \left (d x + c\right )^{2} + 135 \, d \cos \left (d x + c\right ) + 27 \, d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 98, normalized size = 0.68 \[ \frac {32768 \, d x + 32768 \, c - \frac {540 \, {\left (2559 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 16032 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 26384 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 4\right )}^{3}} - 278151 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 2 \right |}\right ) + 278151 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 2 \right |}\right )}{20480000 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.55, size = 159, normalized size = 1.10 \[ -\frac {27}{10240 d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-2\right )^{3}}-\frac {1431}{102400 d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-2\right )^{2}}-\frac {69093}{2048000 d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-2\right )}+\frac {278151 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-2\right )}{20480000 d}-\frac {27}{10240 d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2\right )^{3}}+\frac {1431}{102400 d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2\right )^{2}}-\frac {69093}{2048000 d \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2\right )}-\frac {278151 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2\right )}{20480000 d}+\frac {2 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{625 d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.52, size = 194, normalized size = 1.34 \[ -\frac {\frac {540 \, {\left (\frac {26384 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {16032 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {2559 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}}\right )}}{\frac {48 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {12 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {\sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - 64} - 65536 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right ) + 278151 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 2\right ) - 278151 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - 2\right )}{20480000 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.10, size = 105, normalized size = 0.72 \[ \frac {x}{625}-\frac {278151\,\mathrm {atanh}\left (\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2}\right )}{10240000\,d}-\frac {\frac {69093\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{1024000}-\frac {13527\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{32000}+\frac {44523\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{64000}}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-12\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+48\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-64\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (3 \sec {\left (c + d x \right )} + 5\right )^{4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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