Integrand size = 24, antiderivative size = 174 \[ \int \frac {\cos ^5(c+d x)}{(a+i a \tan (c+d x))^4} \, dx=\frac {56 \sin (c+d x)}{143 a^4 d}-\frac {56 \sin ^3(c+d x)}{143 a^4 d}+\frac {168 \sin ^5(c+d x)}{715 a^4 d}-\frac {8 \sin ^7(c+d x)}{143 a^4 d}+\frac {i \cos ^5(c+d x)}{13 d (a+i a \tan (c+d x))^4}+\frac {9 i \cos ^5(c+d x)}{143 a d (a+i a \tan (c+d x))^3}+\frac {16 i \cos ^7(c+d x)}{143 d \left (a^4+i a^4 \tan (c+d x)\right )} \]
[Out]
Time = 0.20 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {3583, 3581, 2713} \[ \int \frac {\cos ^5(c+d x)}{(a+i a \tan (c+d x))^4} \, dx=-\frac {8 \sin ^7(c+d x)}{143 a^4 d}+\frac {168 \sin ^5(c+d x)}{715 a^4 d}-\frac {56 \sin ^3(c+d x)}{143 a^4 d}+\frac {56 \sin (c+d x)}{143 a^4 d}+\frac {16 i \cos ^7(c+d x)}{143 d \left (a^4+i a^4 \tan (c+d x)\right )}+\frac {9 i \cos ^5(c+d x)}{143 a d (a+i a \tan (c+d x))^3}+\frac {i \cos ^5(c+d x)}{13 d (a+i a \tan (c+d x))^4} \]
[In]
[Out]
Rule 2713
Rule 3581
Rule 3583
Rubi steps \begin{align*} \text {integral}& = \frac {i \cos ^5(c+d x)}{13 d (a+i a \tan (c+d x))^4}+\frac {9 \int \frac {\cos ^5(c+d x)}{(a+i a \tan (c+d x))^3} \, dx}{13 a} \\ & = \frac {i \cos ^5(c+d x)}{13 d (a+i a \tan (c+d x))^4}+\frac {9 i \cos ^5(c+d x)}{143 a d (a+i a \tan (c+d x))^3}+\frac {72 \int \frac {\cos ^5(c+d x)}{(a+i a \tan (c+d x))^2} \, dx}{143 a^2} \\ & = \frac {i \cos ^5(c+d x)}{13 d (a+i a \tan (c+d x))^4}+\frac {9 i \cos ^5(c+d x)}{143 a d (a+i a \tan (c+d x))^3}+\frac {16 i \cos ^7(c+d x)}{143 d \left (a^4+i a^4 \tan (c+d x)\right )}+\frac {56 \int \cos ^7(c+d x) \, dx}{143 a^4} \\ & = \frac {i \cos ^5(c+d x)}{13 d (a+i a \tan (c+d x))^4}+\frac {9 i \cos ^5(c+d x)}{143 a d (a+i a \tan (c+d x))^3}+\frac {16 i \cos ^7(c+d x)}{143 d \left (a^4+i a^4 \tan (c+d x)\right )}-\frac {56 \text {Subst}\left (\int \left (1-3 x^2+3 x^4-x^6\right ) \, dx,x,-\sin (c+d x)\right )}{143 a^4 d} \\ & = \frac {56 \sin (c+d x)}{143 a^4 d}-\frac {56 \sin ^3(c+d x)}{143 a^4 d}+\frac {168 \sin ^5(c+d x)}{715 a^4 d}-\frac {8 \sin ^7(c+d x)}{143 a^4 d}+\frac {i \cos ^5(c+d x)}{13 d (a+i a \tan (c+d x))^4}+\frac {9 i \cos ^5(c+d x)}{143 a d (a+i a \tan (c+d x))^3}+\frac {16 i \cos ^7(c+d x)}{143 d \left (a^4+i a^4 \tan (c+d x)\right )} \\ \end{align*}
Time = 1.24 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.80 \[ \int \frac {\cos ^5(c+d x)}{(a+i a \tan (c+d x))^4} \, dx=-\frac {i \sec ^4(c+d x) (-24024 \cos (c+d x)-34320 \cos (3 (c+d x))+11440 \cos (5 (c+d x))+780 \cos (7 (c+d x))+44 \cos (9 (c+d x))-6006 i \sin (c+d x)-25740 i \sin (3 (c+d x))+14300 i \sin (5 (c+d x))+1365 i \sin (7 (c+d x))+99 i \sin (9 (c+d x)))}{183040 a^4 d (-i+\tan (c+d x))^4} \]
[In]
[Out]
Time = 0.75 (sec) , antiderivative size = 173, normalized size of antiderivative = 0.99
method | result | size |
risch | \(\frac {3 i {\mathrm e}^{-7 i \left (d x +c \right )}}{128 a^{4} d}+\frac {i {\mathrm e}^{-9 i \left (d x +c \right )}}{128 a^{4} d}+\frac {9 i {\mathrm e}^{-11 i \left (d x +c \right )}}{5632 a^{4} d}+\frac {i {\mathrm e}^{-13 i \left (d x +c \right )}}{6656 a^{4} d}+\frac {3 i \cos \left (d x +c \right )}{32 a^{4} d}+\frac {15 \sin \left (d x +c \right )}{64 a^{4} d}+\frac {25 i \cos \left (5 d x +5 c \right )}{512 a^{4} d}+\frac {127 \sin \left (5 d x +5 c \right )}{2560 a^{4} d}+\frac {39 i \cos \left (3 d x +3 c \right )}{512 a^{4} d}+\frac {45 \sin \left (3 d x +3 c \right )}{512 a^{4} d}\) | \(173\) |
derivativedivides | \(\frac {-\frac {135 i}{\left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{8}}+\frac {i}{32 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+i\right )^{4}}+\frac {1}{80 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+i\right )^{5}}-\frac {5}{64 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+i\right )^{3}}+\frac {23}{128 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+i\right )}-\frac {1375 i}{32 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}+\frac {465 i}{4 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{6}}-\frac {8 i}{\left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{12}}+\frac {825 i}{128 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}+\frac {62 i}{\left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{10}}-\frac {11 i}{128 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+i\right )^{2}}+\frac {16}{13 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{13}}-\frac {300}{11 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{11}}+\frac {104}{\left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{9}}-\frac {279}{2 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{7}}+\frac {6291}{80 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{5}}-\frac {1207}{64 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}+\frac {233}{128 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}}{a^{4} d}\) | \(306\) |
default | \(\frac {-\frac {135 i}{\left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{8}}+\frac {i}{32 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+i\right )^{4}}+\frac {1}{80 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+i\right )^{5}}-\frac {5}{64 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+i\right )^{3}}+\frac {23}{128 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+i\right )}-\frac {1375 i}{32 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}+\frac {465 i}{4 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{6}}-\frac {8 i}{\left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{12}}+\frac {825 i}{128 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}+\frac {62 i}{\left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{10}}-\frac {11 i}{128 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+i\right )^{2}}+\frac {16}{13 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{13}}-\frac {300}{11 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{11}}+\frac {104}{\left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{9}}-\frac {279}{2 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{7}}+\frac {6291}{80 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{5}}-\frac {1207}{64 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}+\frac {233}{128 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}}{a^{4} d}\) | \(306\) |
[In]
[Out]
none
Time = 0.24 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.68 \[ \int \frac {\cos ^5(c+d x)}{(a+i a \tan (c+d x))^4} \, dx=\frac {{\left (-143 i \, e^{\left (18 i \, d x + 18 i \, c\right )} - 2145 i \, e^{\left (16 i \, d x + 16 i \, c\right )} - 25740 i \, e^{\left (14 i \, d x + 14 i \, c\right )} + 60060 i \, e^{\left (12 i \, d x + 12 i \, c\right )} + 30030 i \, e^{\left (10 i \, d x + 10 i \, c\right )} + 18018 i \, e^{\left (8 i \, d x + 8 i \, c\right )} + 8580 i \, e^{\left (6 i \, d x + 6 i \, c\right )} + 2860 i \, e^{\left (4 i \, d x + 4 i \, c\right )} + 585 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + 55 i\right )} e^{\left (-13 i \, d x - 13 i \, c\right )}}{366080 \, a^{4} d} \]
[In]
[Out]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 367 vs. \(2 (153) = 306\).
Time = 0.50 (sec) , antiderivative size = 367, normalized size of antiderivative = 2.11 \[ \int \frac {\cos ^5(c+d x)}{(a+i a \tan (c+d x))^4} \, dx=\begin {cases} \frac {\left (- 1688246017625898163896320 i a^{36} d^{9} e^{54 i c} e^{5 i d x} - 25323690264388472458444800 i a^{36} d^{9} e^{52 i c} e^{3 i d x} - 303884283172661669501337600 i a^{36} d^{9} e^{50 i c} e^{i d x} + 709063327402877228836454400 i a^{36} d^{9} e^{48 i c} e^{- i d x} + 354531663701438614418227200 i a^{36} d^{9} e^{46 i c} e^{- 3 i d x} + 212718998220863168650936320 i a^{36} d^{9} e^{44 i c} e^{- 5 i d x} + 101294761057553889833779200 i a^{36} d^{9} e^{42 i c} e^{- 7 i d x} + 33764920352517963277926400 i a^{36} d^{9} e^{40 i c} e^{- 9 i d x} + 6906460981196856125030400 i a^{36} d^{9} e^{38 i c} e^{- 11 i d x} + 649325391394576216883200 i a^{36} d^{9} e^{36 i c} e^{- 13 i d x}\right ) e^{- 49 i c}}{4321909805122299299574579200 a^{40} d^{10}} & \text {for}\: a^{40} d^{10} e^{49 i c} \neq 0 \\\frac {x \left (e^{18 i c} + 9 e^{16 i c} + 36 e^{14 i c} + 84 e^{12 i c} + 126 e^{10 i c} + 126 e^{8 i c} + 84 e^{6 i c} + 36 e^{4 i c} + 9 e^{2 i c} + 1\right ) e^{- 13 i c}}{512 a^{4}} & \text {otherwise} \end {cases} \]
[In]
[Out]
Exception generated. \[ \int \frac {\cos ^5(c+d x)}{(a+i a \tan (c+d x))^4} \, dx=\text {Exception raised: RuntimeError} \]
[In]
[Out]
none
Time = 0.78 (sec) , antiderivative size = 249, normalized size of antiderivative = 1.43 \[ \int \frac {\cos ^5(c+d x)}{(a+i a \tan (c+d x))^4} \, dx=\frac {\frac {143 \, {\left (115 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 405 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 575 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 375 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 98\right )}}{a^{4} {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + i\right )}^{5}} + \frac {166595 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{12} - 1409265 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} - 6232655 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{10} + 17535375 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 34610004 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} - 49771722 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 53349582 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 42730974 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 25431835 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 10954229 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 3278067 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 614627 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 60094}{a^{4} {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - i\right )}^{13}}}{91520 \, d} \]
[In]
[Out]
Time = 7.93 (sec) , antiderivative size = 262, normalized size of antiderivative = 1.51 \[ \int \frac {\cos ^5(c+d x)}{(a+i a \tan (c+d x))^4} \, dx=\frac {\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {15049\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{128}-\frac {4513\,\sin \left (\frac {3\,c}{2}+\frac {3\,d\,x}{2}\right )}{32}+\frac {4513\,\sin \left (\frac {5\,c}{2}+\frac {5\,d\,x}{2}\right )}{32}-\frac {15461\,\sin \left (\frac {7\,c}{2}+\frac {7\,d\,x}{2}\right )}{64}+\frac {3941\,\sin \left (\frac {9\,c}{2}+\frac {9\,d\,x}{2}\right )}{64}-\frac {183\,\sin \left (\frac {11\,c}{2}+\frac {11\,d\,x}{2}\right )}{32}+\frac {183\,\sin \left (\frac {13\,c}{2}+\frac {13\,d\,x}{2}\right )}{32}-\frac {99\,\sin \left (\frac {15\,c}{2}+\frac {15\,d\,x}{2}\right )}{256}+\frac {99\,\sin \left (\frac {17\,c}{2}+\frac {17\,d\,x}{2}\right )}{256}+\frac {\cos \left (\frac {3\,c}{2}+\frac {3\,d\,x}{2}\right )\,3003{}\mathrm {i}}{32}-\frac {\cos \left (\frac {5\,c}{2}+\frac {5\,d\,x}{2}\right )\,3003{}\mathrm {i}}{32}+\frac {\cos \left (\frac {7\,c}{2}+\frac {7\,d\,x}{2}\right )\,7293{}\mathrm {i}}{32}-\frac {\cos \left (\frac {9\,c}{2}+\frac {9\,d\,x}{2}\right )\,1533{}\mathrm {i}}{32}+\frac {\cos \left (\frac {11\,c}{2}+\frac {11\,d\,x}{2}\right )\,103{}\mathrm {i}}{32}-\frac {\cos \left (\frac {13\,c}{2}+\frac {13\,d\,x}{2}\right )\,103{}\mathrm {i}}{32}+\frac {\cos \left (\frac {15\,c}{2}+\frac {15\,d\,x}{2}\right )\,11{}\mathrm {i}}{64}-\frac {\cos \left (\frac {17\,c}{2}+\frac {17\,d\,x}{2}\right )\,11{}\mathrm {i}}{64}\right )\,2{}\mathrm {i}}{715\,a^4\,d\,{\left (\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )+\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,1{}\mathrm {i}\right )}^{13}\,{\left (\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )+\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )\,1{}\mathrm {i}\right )}^5} \]
[In]
[Out]