Integrand size = 12, antiderivative size = 106 \[ \int \frac {1}{(3+5 \sec (c+d x))^4} \, dx=\frac {21553 x}{2654208}+\frac {11215 \arctan \left (\frac {\sin (c+d x)}{3+\cos (c+d x)}\right )}{1327104 d}-\frac {25 \tan (c+d x)}{144 d (3+5 \sec (c+d x))^3}-\frac {25 \tan (c+d x)}{4608 d (3+5 \sec (c+d x))^2}-\frac {16925 \tan (c+d x)}{221184 d (3+5 \sec (c+d x))} \]
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Time = 0.19 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {3870, 4145, 4004, 3916, 2736} \[ \int \frac {1}{(3+5 \sec (c+d x))^4} \, dx=\frac {11215 \arctan \left (\frac {\sin (c+d x)}{\cos (c+d x)+3}\right )}{1327104 d}-\frac {16925 \tan (c+d x)}{221184 d (5 \sec (c+d x)+3)}-\frac {25 \tan (c+d x)}{4608 d (5 \sec (c+d x)+3)^2}-\frac {25 \tan (c+d x)}{144 d (5 \sec (c+d x)+3)^3}+\frac {21553 x}{2654208} \]
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Rule 2736
Rule 3870
Rule 3916
Rule 4004
Rule 4145
Rubi steps \begin{align*} \text {integral}& = -\frac {25 \tan (c+d x)}{144 d (3+5 \sec (c+d x))^3}+\frac {1}{144} \int \frac {48+45 \sec (c+d x)-50 \sec ^2(c+d x)}{(3+5 \sec (c+d x))^3} \, dx \\ & = -\frac {25 \tan (c+d x)}{144 d (3+5 \sec (c+d x))^3}-\frac {25 \tan (c+d x)}{4608 d (3+5 \sec (c+d x))^2}+\frac {\int \frac {1536-870 \sec (c+d x)-75 \sec ^2(c+d x)}{(3+5 \sec (c+d x))^2} \, dx}{13824} \\ & = -\frac {25 \tan (c+d x)}{144 d (3+5 \sec (c+d x))^3}-\frac {25 \tan (c+d x)}{4608 d (3+5 \sec (c+d x))^2}-\frac {16925 \tan (c+d x)}{221184 d (3+5 \sec (c+d x))}+\frac {\int \frac {24576+29745 \sec (c+d x)}{3+5 \sec (c+d x)} \, dx}{663552} \\ & = \frac {x}{81}-\frac {25 \tan (c+d x)}{144 d (3+5 \sec (c+d x))^3}-\frac {25 \tan (c+d x)}{4608 d (3+5 \sec (c+d x))^2}-\frac {16925 \tan (c+d x)}{221184 d (3+5 \sec (c+d x))}-\frac {11215 \int \frac {\sec (c+d x)}{3+5 \sec (c+d x)} \, dx}{663552} \\ & = \frac {x}{81}-\frac {25 \tan (c+d x)}{144 d (3+5 \sec (c+d x))^3}-\frac {25 \tan (c+d x)}{4608 d (3+5 \sec (c+d x))^2}-\frac {16925 \tan (c+d x)}{221184 d (3+5 \sec (c+d x))}-\frac {2243 \int \frac {1}{1+\frac {3}{5} \cos (c+d x)} \, dx}{663552} \\ & = \frac {21553 x}{2654208}+\frac {11215 \arctan \left (\frac {\sin (c+d x)}{3+\cos (c+d x)}\right )}{1327104 d}-\frac {25 \tan (c+d x)}{144 d (3+5 \sec (c+d x))^3}-\frac {25 \tan (c+d x)}{4608 d (3+5 \sec (c+d x))^2}-\frac {16925 \tan (c+d x)}{221184 d (3+5 \sec (c+d x))} \\ \end{align*}
Time = 0.61 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.33 \[ \int \frac {1}{(3+5 \sec (c+d x))^4} \, dx=\frac {6307840 c+6307840 d x+8036352 (c+d x) \cos (c+d x)+22430 \arctan \left (2 \cot \left (\frac {1}{2} (c+d x)\right )\right ) (5+3 \cos (c+d x))^3+2211840 c \cos (2 (c+d x))+2211840 d x \cos (2 (c+d x))+221184 c \cos (3 (c+d x))+221184 d x \cos (3 (c+d x))-5660475 \sin (c+d x)-3082500 \sin (2 (c+d x))-582975 \sin (3 (c+d x))}{2654208 d (5+3 \cos (c+d x))^3} \]
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Time = 0.55 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.82
method | result | size |
derivativedivides | \(\frac {\frac {2 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{81}+\frac {-\frac {25925 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{221184}-\frac {3575 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{6912}-\frac {17675 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{13824}}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+4\right )^{3}}-\frac {11215 \arctan \left (\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2}\right )}{1327104}}{d}\) | \(87\) |
default | \(\frac {\frac {2 \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{81}+\frac {-\frac {25925 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{221184}-\frac {3575 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{6912}-\frac {17675 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{13824}}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+4\right )^{3}}-\frac {11215 \arctan \left (\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2}\right )}{1327104}}{d}\) | \(87\) |
risch | \(\frac {x}{81}-\frac {25 i \left (164835 \,{\mathrm e}^{5 i \left (d x +c \right )}+931257 \,{\mathrm e}^{4 i \left (d x +c \right )}+1995070 \,{\mathrm e}^{3 i \left (d x +c \right )}+1610514 \,{\mathrm e}^{2 i \left (d x +c \right )}+534735 \,{\mathrm e}^{i \left (d x +c \right )}+69957\right )}{995328 d \left (3 \,{\mathrm e}^{2 i \left (d x +c \right )}+10 \,{\mathrm e}^{i \left (d x +c \right )}+3\right )^{3}}-\frac {11215 i \ln \left ({\mathrm e}^{i \left (d x +c \right )}+3\right )}{2654208 d}+\frac {11215 i \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {1}{3}\right )}{2654208 d}\) | \(130\) |
parallelrisch | \(\frac {11215 i \left (770+27 \cos \left (3 d x +3 c \right )+981 \cos \left (d x +c \right )+270 \cos \left (2 d x +2 c \right )\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-2 i\right )+11215 i \left (-770-27 \cos \left (3 d x +3 c \right )-981 \cos \left (d x +c \right )-270 \cos \left (2 d x +2 c \right )\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 i\right )+32145408 d x \cos \left (d x +c \right )+8847360 d x \cos \left (2 d x +2 c \right )+884736 d x \cos \left (3 d x +3 c \right )+25231360 d x -22641900 \sin \left (d x +c \right )-12330000 \sin \left (2 d x +2 c \right )-2331900 \sin \left (3 d x +3 c \right )}{2654208 d \left (770+27 \cos \left (3 d x +3 c \right )+981 \cos \left (d x +c \right )+270 \cos \left (2 d x +2 c \right )\right )}\) | \(207\) |
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Time = 0.29 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.50 \[ \int \frac {1}{(3+5 \sec (c+d x))^4} \, dx=\frac {884736 \, d x \cos \left (d x + c\right )^{3} + 4423680 \, d x \cos \left (d x + c\right )^{2} + 7372800 \, d x \cos \left (d x + c\right ) + 4096000 \, d x + 11215 \, {\left (27 \, \cos \left (d x + c\right )^{3} + 135 \, \cos \left (d x + c\right )^{2} + 225 \, \cos \left (d x + c\right ) + 125\right )} \arctan \left (\frac {5 \, \cos \left (d x + c\right ) + 3}{4 \, \sin \left (d x + c\right )}\right ) - 300 \, {\left (7773 \, \cos \left (d x + c\right )^{2} + 20550 \, \cos \left (d x + c\right ) + 16925\right )} \sin \left (d x + c\right )}{2654208 \, {\left (27 \, d \cos \left (d x + c\right )^{3} + 135 \, d \cos \left (d x + c\right )^{2} + 225 \, d \cos \left (d x + c\right ) + 125 \, d\right )}} \]
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\[ \int \frac {1}{(3+5 \sec (c+d x))^4} \, dx=\int \frac {1}{\left (5 \sec {\left (c + d x \right )} + 3\right )^{4}}\, dx \]
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Time = 0.29 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.61 \[ \int \frac {1}{(3+5 \sec (c+d x))^4} \, dx=-\frac {\frac {150 \, {\left (\frac {11312 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {4576 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {1037 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}}\right )}}{\frac {48 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {12 \, \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {\sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + 64} - 32768 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right ) + 11215 \, \arctan \left (\frac {\sin \left (d x + c\right )}{2 \, {\left (\cos \left (d x + c\right ) + 1\right )}}\right )}{1327104 \, d} \]
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Time = 0.31 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.83 \[ \int \frac {1}{(3+5 \sec (c+d x))^4} \, dx=\frac {21553 \, d x + 21553 \, c - \frac {300 \, {\left (1037 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 4576 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 11312 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 4\right )}^{3}} + 22430 \, \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 3}\right )}{2654208 \, d} \]
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Time = 14.43 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.99 \[ \int \frac {1}{(3+5 \sec (c+d x))^4} \, dx=\frac {x}{81}-\frac {11215\,\mathrm {atan}\left (\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2}\right )}{1327104\,d}-\frac {\frac {25925\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{221184}+\frac {3575\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{6912}+\frac {17675\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{13824}}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+12\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+48\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+64\right )} \]
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