Integrand size = 12, antiderivative size = 140 \[ \int \frac {1}{(3-5 \sin (c+d x))^4} \, dx=\frac {279 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-3 \sin \left (\frac {1}{2} (c+d x)\right )\right )}{32768 d}-\frac {279 \log \left (3 \cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )}{32768 d}+\frac {5 \cos (c+d x)}{48 d (3-5 \sin (c+d x))^3}-\frac {25 \cos (c+d x)}{512 d (3-5 \sin (c+d x))^2}+\frac {995 \cos (c+d x)}{24576 d (3-5 \sin (c+d x))} \]
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Time = 0.08 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {2743, 2833, 12, 2739, 630, 31} \[ \int \frac {1}{(3-5 \sin (c+d x))^4} \, dx=\frac {995 \cos (c+d x)}{24576 d (3-5 \sin (c+d x))}-\frac {25 \cos (c+d x)}{512 d (3-5 \sin (c+d x))^2}+\frac {5 \cos (c+d x)}{48 d (3-5 \sin (c+d x))^3}+\frac {279 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-3 \sin \left (\frac {1}{2} (c+d x)\right )\right )}{32768 d}-\frac {279 \log \left (3 \cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )}{32768 d} \]
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Rule 12
Rule 31
Rule 630
Rule 2739
Rule 2743
Rule 2833
Rubi steps \begin{align*} \text {integral}& = \frac {5 \cos (c+d x)}{48 d (3-5 \sin (c+d x))^3}+\frac {1}{48} \int \frac {-9-10 \sin (c+d x)}{(3-5 \sin (c+d x))^3} \, dx \\ & = \frac {5 \cos (c+d x)}{48 d (3-5 \sin (c+d x))^3}-\frac {25 \cos (c+d x)}{512 d (3-5 \sin (c+d x))^2}+\frac {\int \frac {154+75 \sin (c+d x)}{(3-5 \sin (c+d x))^2} \, dx}{1536} \\ & = \frac {5 \cos (c+d x)}{48 d (3-5 \sin (c+d x))^3}-\frac {25 \cos (c+d x)}{512 d (3-5 \sin (c+d x))^2}+\frac {995 \cos (c+d x)}{24576 d (3-5 \sin (c+d x))}+\frac {\int -\frac {837}{3-5 \sin (c+d x)} \, dx}{24576} \\ & = \frac {5 \cos (c+d x)}{48 d (3-5 \sin (c+d x))^3}-\frac {25 \cos (c+d x)}{512 d (3-5 \sin (c+d x))^2}+\frac {995 \cos (c+d x)}{24576 d (3-5 \sin (c+d x))}-\frac {279 \int \frac {1}{3-5 \sin (c+d x)} \, dx}{8192} \\ & = \frac {5 \cos (c+d x)}{48 d (3-5 \sin (c+d x))^3}-\frac {25 \cos (c+d x)}{512 d (3-5 \sin (c+d x))^2}+\frac {995 \cos (c+d x)}{24576 d (3-5 \sin (c+d x))}-\frac {279 \text {Subst}\left (\int \frac {1}{3-10 x+3 x^2} \, dx,x,\tan \left (\frac {1}{2} (c+d x)\right )\right )}{4096 d} \\ & = \frac {5 \cos (c+d x)}{48 d (3-5 \sin (c+d x))^3}-\frac {25 \cos (c+d x)}{512 d (3-5 \sin (c+d x))^2}+\frac {995 \cos (c+d x)}{24576 d (3-5 \sin (c+d x))}-\frac {837 \text {Subst}\left (\int \frac {1}{-9+3 x} \, dx,x,\tan \left (\frac {1}{2} (c+d x)\right )\right )}{32768 d}+\frac {837 \text {Subst}\left (\int \frac {1}{-1+3 x} \, dx,x,\tan \left (\frac {1}{2} (c+d x)\right )\right )}{32768 d} \\ & = \frac {279 \log \left (1-3 \tan \left (\frac {1}{2} (c+d x)\right )\right )}{32768 d}-\frac {279 \log \left (3-\tan \left (\frac {1}{2} (c+d x)\right )\right )}{32768 d}+\frac {5 \cos (c+d x)}{48 d (3-5 \sin (c+d x))^3}-\frac {25 \cos (c+d x)}{512 d (3-5 \sin (c+d x))^2}+\frac {995 \cos (c+d x)}{24576 d (3-5 \sin (c+d x))} \\ \end{align*}
Time = 0.71 (sec) , antiderivative size = 241, normalized size of antiderivative = 1.72 \[ \int \frac {1}{(3-5 \sin (c+d x))^4} \, dx=\frac {2511 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-3 \sin \left (\frac {1}{2} (c+d x)\right )\right )-2511 \log \left (3 \cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )-\frac {720}{\left (\cos \left (\frac {1}{2} (c+d x)\right )-3 \sin \left (\frac {1}{2} (c+d x)\right )\right )^2}+20 \left (\frac {240}{\left (\cos \left (\frac {1}{2} (c+d x)\right )-3 \sin \left (\frac {1}{2} (c+d x)\right )\right )^3}+\frac {597}{\cos \left (\frac {1}{2} (c+d x)\right )-3 \sin \left (\frac {1}{2} (c+d x)\right )}+\frac {80}{\left (3 \cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^3}+\frac {199}{3 \cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )}\right ) \sin \left (\frac {1}{2} (c+d x)\right )+\frac {2320}{\left (-3 \cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^2}}{294912 d} \]
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Time = 0.34 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.94
method | result | size |
derivativedivides | \(\frac {-\frac {125}{768 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-3\right )^{3}}-\frac {75}{1024 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-3\right )^{2}}-\frac {345}{8192 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-3\right )}-\frac {279 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-3\right )}{32768}-\frac {125}{20736 \left (3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {275}{27648 \left (3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {3505}{221184 \left (3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {279 \ln \left (3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{32768}}{d}\) | \(132\) |
default | \(\frac {-\frac {125}{768 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-3\right )^{3}}-\frac {75}{1024 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-3\right )^{2}}-\frac {345}{8192 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-3\right )}-\frac {279 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-3\right )}{32768}-\frac {125}{20736 \left (3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {275}{27648 \left (3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {3505}{221184 \left (3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {279 \ln \left (3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{32768}}{d}\) | \(132\) |
risch | \(\frac {-111042 \,{\mathrm e}^{3 i \left (d x +c \right )}-62775 i {\mathrm e}^{4 i \left (d x +c \right )}+119310 i {\mathrm e}^{2 i \left (d x +c \right )}+20925 \,{\mathrm e}^{5 i \left (d x +c \right )}+68625 \,{\mathrm e}^{i \left (d x +c \right )}-24875 i}{12288 \left (5 \,{\mathrm e}^{2 i \left (d x +c \right )}-5-6 i {\mathrm e}^{i \left (d x +c \right )}\right )^{3} d}+\frac {279 \ln \left (-\frac {4}{5}-\frac {3 i}{5}+{\mathrm e}^{i \left (d x +c \right )}\right )}{32768 d}-\frac {279 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {4}{5}-\frac {3 i}{5}\right )}{32768 d}\) | \(132\) |
norman | \(\frac {\frac {7915}{12288 d}-\frac {15725 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{12288 d}-\frac {3047275 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{165888 d}-\frac {63425 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{12288 d}+\frac {296245 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{18432 d}+\frac {270245 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{36864 d}}{{\left (3 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-10 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+3\right )}^{3}}-\frac {279 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-3\right )}{32768 d}+\frac {279 \ln \left (3 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{32768 d}\) | \(151\) |
parallelrisch | \(\frac {\left (10169550 \cos \left (2 d x +2 c \right )+20678085 \sin \left (d x +c \right )-2824875 \sin \left (3 d x +3 c \right )-12610242\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {1}{3}\right )+\left (-10169550 \cos \left (2 d x +2 c \right )-20678085 \sin \left (d x +c \right )+2824875 \sin \left (3 d x +3 c \right )+12610242\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-3\right )-6105780 \cos \left (d x +c \right )+14247000 \cos \left (2 d x +2 c \right )+2686500 \cos \left (3 d x +3 c \right )+28968900 \sin \left (d x +c \right )+5151600 \sin \left (2 d x +2 c \right )-3957500 \sin \left (3 d x +3 c \right )-17666280}{2654208 d \left (-125 \sin \left (3 d x +3 c \right )+915 \sin \left (d x +c \right )-558+450 \cos \left (2 d x +2 c \right )\right )}\) | \(192\) |
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Time = 0.30 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.29 \[ \int \frac {1}{(3-5 \sin (c+d x))^4} \, dx=\frac {199000 \, \cos \left (d x + c\right )^{3} - 837 \, {\left (225 \, \cos \left (d x + c\right )^{2} - 5 \, {\left (25 \, \cos \left (d x + c\right )^{2} - 52\right )} \sin \left (d x + c\right ) - 252\right )} \log \left (4 \, \cos \left (d x + c\right ) - 3 \, \sin \left (d x + c\right ) + 5\right ) + 837 \, {\left (225 \, \cos \left (d x + c\right )^{2} - 5 \, {\left (25 \, \cos \left (d x + c\right )^{2} - 52\right )} \sin \left (d x + c\right ) - 252\right )} \log \left (-4 \, \cos \left (d x + c\right ) - 3 \, \sin \left (d x + c\right ) + 5\right ) + 190800 \, \cos \left (d x + c\right ) \sin \left (d x + c\right ) - 262320 \, \cos \left (d x + c\right )}{196608 \, {\left (225 \, d \cos \left (d x + c\right )^{2} - 5 \, {\left (25 \, d \cos \left (d x + c\right )^{2} - 52 \, d\right )} \sin \left (d x + c\right ) - 252 \, d\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 2353 vs. \(2 (126) = 252\).
Time = 3.15 (sec) , antiderivative size = 2353, normalized size of antiderivative = 16.81 \[ \int \frac {1}{(3-5 \sin (c+d x))^4} \, dx=\text {Too large to display} \]
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Leaf count of result is larger than twice the leaf count of optimal. 275 vs. \(2 (126) = 252\).
Time = 0.19 (sec) , antiderivative size = 275, normalized size of antiderivative = 1.96 \[ \int \frac {1}{(3-5 \sin (c+d x))^4} \, dx=\frac {\frac {40 \, {\left (\frac {342495 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {1066482 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {1218910 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {486441 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {84915 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - 42741\right )}}{\frac {270 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {981 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {1540 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {981 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {270 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {27 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - 27} + 22599 \, \log \left (\frac {3 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - 1\right ) - 22599 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - 3\right )}{2654208 \, d} \]
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Time = 0.32 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.95 \[ \int \frac {1}{(3-5 \sin (c+d x))^4} \, dx=-\frac {\frac {40 \, {\left (84915 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 486441 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 1218910 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 1066482 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 342495 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 42741\right )}}{{\left (3 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 10 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 3\right )}^{3}} - 22599 \, \log \left ({\left | 3 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) + 22599 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 3 \right |}\right )}{2654208 \, d} \]
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Time = 9.10 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.20 \[ \int \frac {1}{(3-5 \sin (c+d x))^4} \, dx=\frac {279\,\mathrm {atanh}\left (\frac {3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{4}-\frac {5}{4}\right )}{16384\,d}-\frac {\frac {15725\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{331776}-\frac {270245\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{995328}+\frac {3047275\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{4478976}-\frac {296245\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{497664}+\frac {63425\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{331776}-\frac {7915}{331776}}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-10\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\frac {109\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{3}-\frac {1540\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{27}+\frac {109\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{3}-10\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+1\right )} \]
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