Optimal. Leaf size=174 \[ -\frac{8 \sin ^7(c+d x)}{143 a^4 d}+\frac{168 \sin ^5(c+d x)}{715 a^4 d}-\frac{56 \sin ^3(c+d x)}{143 a^4 d}+\frac{56 \sin (c+d x)}{143 a^4 d}+\frac{16 i \cos ^7(c+d x)}{143 d \left (a^4+i a^4 \tan (c+d x)\right )}+\frac{9 i \cos ^5(c+d x)}{143 a d (a+i a \tan (c+d x))^3}+\frac{i \cos ^5(c+d x)}{13 d (a+i a \tan (c+d x))^4} \]
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Rubi [A] time = 0.158102, antiderivative size = 174, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {3502, 3500, 2633} \[ -\frac{8 \sin ^7(c+d x)}{143 a^4 d}+\frac{168 \sin ^5(c+d x)}{715 a^4 d}-\frac{56 \sin ^3(c+d x)}{143 a^4 d}+\frac{56 \sin (c+d x)}{143 a^4 d}+\frac{16 i \cos ^7(c+d x)}{143 d \left (a^4+i a^4 \tan (c+d x)\right )}+\frac{9 i \cos ^5(c+d x)}{143 a d (a+i a \tan (c+d x))^3}+\frac{i \cos ^5(c+d x)}{13 d (a+i a \tan (c+d x))^4} \]
Antiderivative was successfully verified.
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Rule 3502
Rule 3500
Rule 2633
Rubi steps
\begin{align*} \int \frac{\cos ^5(c+d x)}{(a+i a \tan (c+d x))^4} \, dx &=\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 \operatorname{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*}
Mathematica [A] time = 0.79145, size = 139, normalized size = 0.8 \[ -\frac{i \sec ^4(c+d x) (-6006 i \sin (c+d x)-25740 i \sin (3 (c+d x))+14300 i \sin (5 (c+d x))+1365 i \sin (7 (c+d x))+99 i \sin (9 (c+d x))-24024 \cos (c+d x)-34320 \cos (3 (c+d x))+11440 \cos (5 (c+d x))+780 \cos (7 (c+d x))+44 \cos (9 (c+d x)))}{183040 a^4 d (\tan (c+d x)-i)^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.11, size = 306, normalized size = 1.8 \begin{align*} 2\,{\frac{1}{{a}^{4}d} \left ({\frac{{\frac{825\,i}{256}}}{ \left ( \tan \left ( 1/2\,dx+c/2 \right ) -i \right ) ^{2}}}-{\frac{4\,i}{ \left ( \tan \left ( 1/2\,dx+c/2 \right ) -i \right ) ^{12}}}-{\frac{{\frac{1375\,i}{64}}}{ \left ( \tan \left ( 1/2\,dx+c/2 \right ) -i \right ) ^{4}}}+{\frac{31\,i}{ \left ( \tan \left ( 1/2\,dx+c/2 \right ) -i \right ) ^{10}}}+{\frac{{\frac{i}{64}}}{ \left ( \tan \left ( 1/2\,dx+c/2 \right ) +i \right ) ^{4}}}+{\frac{{\frac{465\,i}{8}}}{ \left ( \tan \left ( 1/2\,dx+c/2 \right ) -i \right ) ^{6}}}+{\frac{8}{13\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) -i \right ) ^{13}}}-{\frac{150}{11\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) -i \right ) ^{11}}}+52\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) -i \right ) ^{-9}-{\frac{279}{4\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) -i \right ) ^{7}}}+{\frac{6291}{160\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) -i \right ) ^{5}}}-{\frac{1207}{128\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) -i \right ) ^{3}}}+{\frac{233}{256\,\tan \left ( 1/2\,dx+c/2 \right ) -256\,i}}-{\frac{{\frac{11\,i}{256}}}{ \left ( \tan \left ( 1/2\,dx+c/2 \right ) +i \right ) ^{2}}}-{\frac{{\frac{135\,i}{2}}}{ \left ( \tan \left ( 1/2\,dx+c/2 \right ) -i \right ) ^{8}}}+{\frac{1}{160\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) +i \right ) ^{5}}}-{\frac{5}{128\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) +i \right ) ^{3}}}+{\frac{23}{256\,\tan \left ( 1/2\,dx+c/2 \right ) +256\,i}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Fricas [A] time = 2.48283, size = 435, normalized size = 2.5 \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.60521, size = 369, normalized size = 2.12 \begin{align*} \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}\: 4321909805122299299574579200 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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.17725, size = 336, normalized size = 1.93 \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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