Integrand size = 12, antiderivative size = 108 \[ \int \frac {1}{(-5+3 \sin (c+d x))^4} \, dx=\frac {385 x}{32768}-\frac {385 \arctan \left (\frac {\cos (c+d x)}{3-\sin (c+d x)}\right )}{16384 d}-\frac {\cos (c+d x)}{16 d (5-3 \sin (c+d x))^3}-\frac {25 \cos (c+d x)}{512 d (5-3 \sin (c+d x))^2}-\frac {311 \cos (c+d x)}{8192 d (5-3 \sin (c+d x))} \]
[Out]
Time = 0.07 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2743, 2833, 12, 2737} \[ \int \frac {1}{(-5+3 \sin (c+d x))^4} \, dx=-\frac {385 \arctan \left (\frac {\cos (c+d x)}{3-\sin (c+d x)}\right )}{16384 d}-\frac {311 \cos (c+d x)}{8192 d (5-3 \sin (c+d x))}-\frac {25 \cos (c+d x)}{512 d (5-3 \sin (c+d x))^2}-\frac {\cos (c+d x)}{16 d (5-3 \sin (c+d x))^3}+\frac {385 x}{32768} \]
[In]
[Out]
Rule 12
Rule 2737
Rule 2743
Rule 2833
Rubi steps \begin{align*} \text {integral}& = -\frac {\cos (c+d x)}{16 d (5-3 \sin (c+d x))^3}-\frac {1}{48} \int \frac {15+6 \sin (c+d x)}{(-5+3 \sin (c+d x))^3} \, dx \\ & = -\frac {\cos (c+d x)}{16 d (5-3 \sin (c+d x))^3}-\frac {25 \cos (c+d x)}{512 d (5-3 \sin (c+d x))^2}+\frac {\int \frac {186+75 \sin (c+d x)}{(-5+3 \sin (c+d x))^2} \, dx}{1536} \\ & = -\frac {\cos (c+d x)}{16 d (5-3 \sin (c+d x))^3}-\frac {25 \cos (c+d x)}{512 d (5-3 \sin (c+d x))^2}-\frac {311 \cos (c+d x)}{8192 d (5-3 \sin (c+d x))}-\frac {\int \frac {1155}{-5+3 \sin (c+d x)} \, dx}{24576} \\ & = -\frac {\cos (c+d x)}{16 d (5-3 \sin (c+d x))^3}-\frac {25 \cos (c+d x)}{512 d (5-3 \sin (c+d x))^2}-\frac {311 \cos (c+d x)}{8192 d (5-3 \sin (c+d x))}-\frac {385 \int \frac {1}{-5+3 \sin (c+d x)} \, dx}{8192} \\ & = \frac {385 x}{32768}-\frac {385 \arctan \left (\frac {\cos (c+d x)}{3-\sin (c+d x)}\right )}{16384 d}-\frac {\cos (c+d x)}{16 d (5-3 \sin (c+d x))^3}-\frac {25 \cos (c+d x)}{512 d (5-3 \sin (c+d x))^2}-\frac {311 \cos (c+d x)}{8192 d (5-3 \sin (c+d x))} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.23 \[ \int \frac {1}{(-5+3 \sin (c+d x))^4} \, dx=\frac {-1925 \arctan \left (\frac {2 \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )}{\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )}\right )+\frac {-239470+219735 \cos (c+d x)+83970 \cos (2 (c+d x))-13995 \cos (3 (c+d x))+305091 \sin (c+d x)-105300 \sin (2 (c+d x))-8397 \sin (3 (c+d x))}{2 (-5+3 \sin (c+d x))^3}}{81920 d} \]
[In]
[Out]
Time = 0.36 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.08
method | result | size |
derivativedivides | \(\frac {\frac {\frac {39933 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{20480}-\frac {672723 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{102400}+\frac {2870073 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{256000}-\frac {604899 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{51200}+\frac {145233 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{20480}-\frac {10287}{4096}}{{\left (5 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-6 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+5\right )}^{3}}+\frac {385 \arctan \left (\frac {5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4}-\frac {3}{4}\right )}{16384}}{d}\) | \(117\) |
default | \(\frac {\frac {\frac {39933 \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{20480}-\frac {672723 \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{102400}+\frac {2870073 \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{256000}-\frac {604899 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{51200}+\frac {145233 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{20480}-\frac {10287}{4096}}{{\left (5 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-6 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+5\right )}^{3}}+\frac {385 \arctan \left (\frac {5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{4}-\frac {3}{4}\right )}{16384}}{d}\) | \(117\) |
risch | \(-\frac {-239470 \,{\mathrm e}^{3 i \left (d x +c \right )}-86625 i {\mathrm e}^{4 i \left (d x +c \right )}+218466 i {\mathrm e}^{2 i \left (d x +c \right )}+10395 \,{\mathrm e}^{5 i \left (d x +c \right )}+73575 \,{\mathrm e}^{i \left (d x +c \right )}-8397 i}{12288 \left (3 \,{\mathrm e}^{2 i \left (d x +c \right )}-3-10 i {\mathrm e}^{i \left (d x +c \right )}\right )^{3} d}-\frac {385 i \ln \left ({\mathrm e}^{i \left (d x +c \right )}-\frac {i}{3}\right )}{32768 d}+\frac {385 i \ln \left ({\mathrm e}^{i \left (d x +c \right )}-3 i\right )}{32768 d}\) | \(132\) |
parallelrisch | \(\frac {31683960+48125 i \left (770+27 \sin \left (3 d x +3 c \right )-981 \sin \left (d x +c \right )-270 \cos \left (2 d x +2 c \right )\right ) \ln \left (5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-3-4 i\right )+48125 i \left (-27 \sin \left (3 d x +3 c \right )+981 \sin \left (d x +c \right )-770+270 \cos \left (2 d x +2 c \right )\right ) \ln \left (5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-3+4 i\right )+21973500 \cos \left (d x +c \right )-11109960 \cos \left (2 d x +2 c \right )-1399500 \cos \left (3 d x +3 c \right )-40366188 \sin \left (d x +c \right )-10530000 \sin \left (2 d x +2 c \right )+1110996 \sin \left (3 d x +3 c \right )}{4096000 d \left (-27 \sin \left (3 d x +3 c \right )+981 \sin \left (d x +c \right )-770+270 \cos \left (2 d x +2 c \right )\right )}\) | \(204\) |
[In]
[Out]
none
Time = 0.29 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.20 \[ \int \frac {1}{(-5+3 \sin (c+d x))^4} \, dx=-\frac {11196 \, \cos \left (d x + c\right )^{3} - 385 \, {\left (135 \, \cos \left (d x + c\right )^{2} - 9 \, {\left (3 \, \cos \left (d x + c\right )^{2} - 28\right )} \sin \left (d x + c\right ) - 260\right )} \arctan \left (\frac {5 \, \sin \left (d x + c\right ) - 3}{4 \, \cos \left (d x + c\right )}\right ) + 42120 \, \cos \left (d x + c\right ) \sin \left (d x + c\right ) - 52344 \, \cos \left (d x + c\right )}{32768 \, {\left (135 \, d \cos \left (d x + c\right )^{2} - 9 \, {\left (3 \, d \cos \left (d x + c\right )^{2} - 28 \, d\right )} \sin \left (d x + c\right ) - 260 \, d\right )}} \]
[In]
[Out]
Result contains complex when optimal does not.
Time = 4.32 (sec) , antiderivative size = 1690, normalized size of antiderivative = 15.65 \[ \int \frac {1}{(-5+3 \sin (c+d x))^4} \, dx=\text {Too large to display} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 253 vs. \(2 (97) = 194\).
Time = 0.27 (sec) , antiderivative size = 253, normalized size of antiderivative = 2.34 \[ \int \frac {1}{(-5+3 \sin (c+d x))^4} \, dx=-\frac {\frac {36 \, {\left (\frac {403425 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {672110 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {637794 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {373735 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {110925 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - 142875\right )}}{\frac {450 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {915 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {1116 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {915 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {450 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {125 \, \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - 125} - 48125 \, \arctan \left (\frac {5 \, \sin \left (d x + c\right )}{4 \, {\left (\cos \left (d x + c\right ) + 1\right )}} - \frac {3}{4}\right )}{2048000 \, d} \]
[In]
[Out]
none
Time = 0.32 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.37 \[ \int \frac {1}{(-5+3 \sin (c+d x))^4} \, dx=\frac {48125 \, d x + 48125 \, c + \frac {72 \, {\left (110925 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 373735 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 637794 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 672110 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 403425 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 142875\right )}}{{\left (5 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 6 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 5\right )}^{3}} + 96250 \, \arctan \left (\frac {3 \, \cos \left (d x + c\right ) - \sin \left (d x + c\right ) + 3}{\cos \left (d x + c\right ) + 3 \, \sin \left (d x + c\right ) - 9}\right )}{4096000 \, d} \]
[In]
[Out]
Time = 0.00 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.73 \[ \int \frac {1}{(-5+3 \sin (c+d x))^4} \, dx=\frac {385\,\mathrm {atan}\left (\frac {5\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{4}-\frac {3}{4}\right )}{16384\,d}-\frac {385\,\left (\mathrm {atan}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )-\frac {d\,x}{2}\right )}{16384\,d}+\frac {\frac {39933\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{2560000}-\frac {672723\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{12800000}+\frac {2870073\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{32000000}-\frac {604899\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{6400000}+\frac {145233\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2560000}-\frac {10287}{512000}}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-\frac {18\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{5}+\frac {183\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{25}-\frac {1116\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{125}+\frac {183\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{25}-\frac {18\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{5}+1\right )} \]
[In]
[Out]