Optimal. Leaf size=271 \[ -\frac{64 \sin ^3(c+d x)}{12155 a^8 d}+\frac{192 \sin (c+d x)}{12155 a^8 d}+\frac{128 i \cos ^3(c+d x)}{12155 d \left (a^8+i a^8 \tan (c+d x)\right )}+\frac{16 i \cos (c+d x)}{2431 a^2 d \left (a^2+i a^2 \tan (c+d x)\right )^3}+\frac{112 i \cos (c+d x)}{12155 d \left (a^2+i a^2 \tan (c+d x)\right )^4}+\frac{168 i \cos (c+d x)}{12155 a^3 d (a+i a \tan (c+d x))^5}+\frac{24 i \cos (c+d x)}{1105 a^2 d (a+i a \tan (c+d x))^6}+\frac{3 i \cos (c+d x)}{85 a d (a+i a \tan (c+d x))^7}+\frac{i \cos (c+d x)}{17 d (a+i a \tan (c+d x))^8} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.312019, antiderivative size = 271, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {3502, 3500, 2633} \[ -\frac{64 \sin ^3(c+d x)}{12155 a^8 d}+\frac{192 \sin (c+d x)}{12155 a^8 d}+\frac{128 i \cos ^3(c+d x)}{12155 d \left (a^8+i a^8 \tan (c+d x)\right )}+\frac{16 i \cos (c+d x)}{2431 a^2 d \left (a^2+i a^2 \tan (c+d x)\right )^3}+\frac{112 i \cos (c+d x)}{12155 d \left (a^2+i a^2 \tan (c+d x)\right )^4}+\frac{168 i \cos (c+d x)}{12155 a^3 d (a+i a \tan (c+d x))^5}+\frac{24 i \cos (c+d x)}{1105 a^2 d (a+i a \tan (c+d x))^6}+\frac{3 i \cos (c+d x)}{85 a d (a+i a \tan (c+d x))^7}+\frac{i \cos (c+d x)}{17 d (a+i a \tan (c+d x))^8} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3502
Rule 3500
Rule 2633
Rubi steps
\begin{align*} \int \frac{\cos (c+d x)}{(a+i a \tan (c+d x))^8} \, dx &=\frac{i \cos (c+d x)}{17 d (a+i a \tan (c+d x))^8}+\frac{9 \int \frac{\cos (c+d x)}{(a+i a \tan (c+d x))^7} \, dx}{17 a}\\ &=\frac{i \cos (c+d x)}{17 d (a+i a \tan (c+d x))^8}+\frac{3 i \cos (c+d x)}{85 a d (a+i a \tan (c+d x))^7}+\frac{24 \int \frac{\cos (c+d x)}{(a+i a \tan (c+d x))^6} \, dx}{85 a^2}\\ &=\frac{i \cos (c+d x)}{17 d (a+i a \tan (c+d x))^8}+\frac{3 i \cos (c+d x)}{85 a d (a+i a \tan (c+d x))^7}+\frac{24 i \cos (c+d x)}{1105 a^2 d (a+i a \tan (c+d x))^6}+\frac{168 \int \frac{\cos (c+d x)}{(a+i a \tan (c+d x))^5} \, dx}{1105 a^3}\\ &=\frac{i \cos (c+d x)}{17 d (a+i a \tan (c+d x))^8}+\frac{3 i \cos (c+d x)}{85 a d (a+i a \tan (c+d x))^7}+\frac{24 i \cos (c+d x)}{1105 a^2 d (a+i a \tan (c+d x))^6}+\frac{168 i \cos (c+d x)}{12155 a^3 d (a+i a \tan (c+d x))^5}+\frac{1008 \int \frac{\cos (c+d x)}{(a+i a \tan (c+d x))^4} \, dx}{12155 a^4}\\ &=\frac{i \cos (c+d x)}{17 d (a+i a \tan (c+d x))^8}+\frac{3 i \cos (c+d x)}{85 a d (a+i a \tan (c+d x))^7}+\frac{24 i \cos (c+d x)}{1105 a^2 d (a+i a \tan (c+d x))^6}+\frac{168 i \cos (c+d x)}{12155 a^3 d (a+i a \tan (c+d x))^5}+\frac{112 i \cos (c+d x)}{12155 d \left (a^2+i a^2 \tan (c+d x)\right )^4}+\frac{112 \int \frac{\cos (c+d x)}{(a+i a \tan (c+d x))^3} \, dx}{2431 a^5}\\ &=\frac{i \cos (c+d x)}{17 d (a+i a \tan (c+d x))^8}+\frac{3 i \cos (c+d x)}{85 a d (a+i a \tan (c+d x))^7}+\frac{24 i \cos (c+d x)}{1105 a^2 d (a+i a \tan (c+d x))^6}+\frac{168 i \cos (c+d x)}{12155 a^3 d (a+i a \tan (c+d x))^5}+\frac{16 i \cos (c+d x)}{2431 a^5 d (a+i a \tan (c+d x))^3}+\frac{112 i \cos (c+d x)}{12155 d \left (a^2+i a^2 \tan (c+d x)\right )^4}+\frac{64 \int \frac{\cos (c+d x)}{(a+i a \tan (c+d x))^2} \, dx}{2431 a^6}\\ &=\frac{i \cos (c+d x)}{17 d (a+i a \tan (c+d x))^8}+\frac{3 i \cos (c+d x)}{85 a d (a+i a \tan (c+d x))^7}+\frac{24 i \cos (c+d x)}{1105 a^2 d (a+i a \tan (c+d x))^6}+\frac{168 i \cos (c+d x)}{12155 a^3 d (a+i a \tan (c+d x))^5}+\frac{16 i \cos (c+d x)}{2431 a^5 d (a+i a \tan (c+d x))^3}+\frac{112 i \cos (c+d x)}{12155 d \left (a^2+i a^2 \tan (c+d x)\right )^4}+\frac{128 i \cos ^3(c+d x)}{12155 d \left (a^8+i a^8 \tan (c+d x)\right )}+\frac{192 \int \cos ^3(c+d x) \, dx}{12155 a^8}\\ &=\frac{i \cos (c+d x)}{17 d (a+i a \tan (c+d x))^8}+\frac{3 i \cos (c+d x)}{85 a d (a+i a \tan (c+d x))^7}+\frac{24 i \cos (c+d x)}{1105 a^2 d (a+i a \tan (c+d x))^6}+\frac{168 i \cos (c+d x)}{12155 a^3 d (a+i a \tan (c+d x))^5}+\frac{16 i \cos (c+d x)}{2431 a^5 d (a+i a \tan (c+d x))^3}+\frac{112 i \cos (c+d x)}{12155 d \left (a^2+i a^2 \tan (c+d x)\right )^4}+\frac{128 i \cos ^3(c+d x)}{12155 d \left (a^8+i a^8 \tan (c+d x)\right )}-\frac{192 \operatorname{Subst}\left (\int \left (1-x^2\right ) \, dx,x,-\sin (c+d x)\right )}{12155 a^8 d}\\ &=\frac{192 \sin (c+d x)}{12155 a^8 d}-\frac{64 \sin ^3(c+d x)}{12155 a^8 d}+\frac{i \cos (c+d x)}{17 d (a+i a \tan (c+d x))^8}+\frac{3 i \cos (c+d x)}{85 a d (a+i a \tan (c+d x))^7}+\frac{24 i \cos (c+d x)}{1105 a^2 d (a+i a \tan (c+d x))^6}+\frac{168 i \cos (c+d x)}{12155 a^3 d (a+i a \tan (c+d x))^5}+\frac{16 i \cos (c+d x)}{2431 a^5 d (a+i a \tan (c+d x))^3}+\frac{112 i \cos (c+d x)}{12155 d \left (a^2+i a^2 \tan (c+d x)\right )^4}+\frac{128 i \cos ^3(c+d x)}{12155 d \left (a^8+i a^8 \tan (c+d x)\right )}\\ \end{align*}
Mathematica [A] time = 0.900893, size = 139, normalized size = 0.51 \[ -\frac{i \sec ^8(c+d x) (-24310 i \sin (c+d x)-55692 i \sin (3 (c+d x))-56100 i \sin (5 (c+d x))-51051 i \sin (7 (c+d x))+6435 i \sin (9 (c+d x))-194480 \cos (c+d x)-148512 \cos (3 (c+d x))-89760 \cos (5 (c+d x))-58344 \cos (7 (c+d x))+5720 \cos (9 (c+d x)))}{3111680 a^8 d (\tan (c+d x)-i)^8} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.118, size = 306, normalized size = 1.1 \begin{align*} 2\,{\frac{1}{d{a}^{8}} \left ({\frac{{\frac{19109\,i}{5}}}{ \left ( \tan \left ( 1/2\,dx+c/2 \right ) -i \right ) ^{10}}}+{\frac{784\,i}{ \left ( \tan \left ( 1/2\,dx+c/2 \right ) -i \right ) ^{14}}}+{\frac{{\frac{1793\,i}{256}}}{ \left ( \tan \left ( 1/2\,dx+c/2 \right ) -i \right ) ^{2}}}-{\frac{2692\,i}{ \left ( \tan \left ( 1/2\,dx+c/2 \right ) -i \right ) ^{12}}}-{\frac{{\frac{7937\,i}{64}}}{ \left ( \tan \left ( 1/2\,dx+c/2 \right ) -i \right ) ^{4}}}-{\frac{64\,i}{ \left ( \tan \left ( 1/2\,dx+c/2 \right ) -i \right ) ^{16}}}-{\frac{{\frac{10241\,i}{4}}}{ \left ( \tan \left ( 1/2\,dx+c/2 \right ) -i \right ) ^{8}}}+{\frac{{\frac{13313\,i}{16}}}{ \left ( \tan \left ( 1/2\,dx+c/2 \right ) -i \right ) ^{6}}}+{\frac{128}{17\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) -i \right ) ^{17}}}-{\frac{1376}{5\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) -i \right ) ^{15}}}+{\frac{21400}{13\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) -i \right ) ^{13}}}-{\frac{38954}{11\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) -i \right ) ^{11}}}+{\frac{6847}{2\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) -i \right ) ^{9}}}-{\frac{12799}{8\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) -i \right ) ^{7}}}+{\frac{57083}{160\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) -i \right ) ^{5}}}-{\frac{4351}{128\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) -i \right ) ^{3}}}+{\frac{511}{512\,\tan \left ( 1/2\,dx+c/2 \right ) -512\,i}}+{\frac{1}{512\,\tan \left ( 1/2\,dx+c/2 \right ) +512\,i}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 2.66626, size = 452, normalized size = 1.67 \begin{align*} \frac{{\left (-12155 i \, e^{\left (18 i \, d x + 18 i \, c\right )} + 109395 i \, e^{\left (16 i \, d x + 16 i \, c\right )} + 145860 i \, e^{\left (14 i \, d x + 14 i \, c\right )} + 204204 i \, e^{\left (12 i \, d x + 12 i \, c\right )} + 218790 i \, e^{\left (10 i \, d x + 10 i \, c\right )} + 170170 i \, e^{\left (8 i \, d x + 8 i \, c\right )} + 92820 i \, e^{\left (6 i \, d x + 6 i \, c\right )} + 33660 i \, e^{\left (4 i \, d x + 4 i \, c\right )} + 7293 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + 715 i\right )} e^{\left (-17 i \, d x - 17 i \, c\right )}}{6223360 \, a^{8} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 3.49903, size = 369, normalized size = 1.36 \begin{align*} \begin{cases} \frac{\left (- 143500911498201343931187200 i a^{72} d^{9} e^{82 i c} e^{i d x} + 1291508203483812095380684800 i a^{72} d^{9} e^{80 i c} e^{- i d x} + 1722010937978416127174246400 i a^{72} d^{9} e^{78 i c} e^{- 3 i d x} + 2410815313169782578043944960 i a^{72} d^{9} e^{76 i c} e^{- 5 i d x} + 2583016406967624190761369600 i a^{72} d^{9} e^{74 i c} e^{- 7 i d x} + 2009012760974818815036620800 i a^{72} d^{9} e^{72 i c} e^{- 9 i d x} + 1095825142349901171838156800 i a^{72} d^{9} e^{70 i c} e^{- 11 i d x} + 397387139533480644732518400 i a^{72} d^{9} e^{68 i c} e^{- 13 i d x} + 86100546898920806358712320 i a^{72} d^{9} e^{66 i c} e^{- 15 i d x} + 8441230088129490819481600 i a^{72} d^{9} e^{64 i c} e^{- 17 i d x}\right ) e^{- 81 i c}}{73472466687079088092767846400 a^{80} d^{10}} & \text{for}\: 73472466687079088092767846400 a^{80} d^{10} e^{81 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^{- 17 i c}}{512 a^{8}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.17698, size = 336, normalized size = 1.24 \begin{align*} \frac{\frac{12155}{a^{8}{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + i\right )}} + \frac{6211205 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{16} - 55791450 i \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{15} - 303072770 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{14} + 1091397450 i \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{13} + 2909561798 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{12} - 5901218466 i \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{11} - 9405145178 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{10} + 11877161010 i \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{9} + 12017308160 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{8} - 9710430158 i \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} - 6263238566 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6} + 3172666718 i \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 1247921210 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 365303990 i \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 77883902 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 10498214 i \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 982907}{a^{8}{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - i\right )}^{17}}}{3111680 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]