Optimal. Leaf size=120 \[ \frac{a^4 \sin ^5(c+d x)}{21 d}-\frac{10 a^4 \sin ^3(c+d x)}{63 d}+\frac{5 a^4 \sin (c+d x)}{21 d}-\frac{2 i \cos ^7(c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{21 d}-\frac{2 i a \cos ^9(c+d x) (a+i a \tan (c+d x))^3}{9 d} \]
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Rubi [A] time = 0.100777, antiderivative size = 120, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {3496, 2633} \[ \frac{a^4 \sin ^5(c+d x)}{21 d}-\frac{10 a^4 \sin ^3(c+d x)}{63 d}+\frac{5 a^4 \sin (c+d x)}{21 d}-\frac{2 i \cos ^7(c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{21 d}-\frac{2 i a \cos ^9(c+d x) (a+i a \tan (c+d x))^3}{9 d} \]
Antiderivative was successfully verified.
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Rule 3496
Rule 2633
Rubi steps
\begin{align*} \int \cos ^9(c+d x) (a+i a \tan (c+d x))^4 \, dx &=-\frac{2 i a \cos ^9(c+d x) (a+i a \tan (c+d x))^3}{9 d}+\frac{1}{3} a^2 \int \cos ^7(c+d x) (a+i a \tan (c+d x))^2 \, dx\\ &=-\frac{2 i a \cos ^9(c+d x) (a+i a \tan (c+d x))^3}{9 d}-\frac{2 i \cos ^7(c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{21 d}+\frac{1}{21} \left (5 a^4\right ) \int \cos ^5(c+d x) \, dx\\ &=-\frac{2 i a \cos ^9(c+d x) (a+i a \tan (c+d x))^3}{9 d}-\frac{2 i \cos ^7(c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{21 d}-\frac{\left (5 a^4\right ) \operatorname{Subst}\left (\int \left (1-2 x^2+x^4\right ) \, dx,x,-\sin (c+d x)\right )}{21 d}\\ &=\frac{5 a^4 \sin (c+d x)}{21 d}-\frac{10 a^4 \sin ^3(c+d x)}{63 d}+\frac{a^4 \sin ^5(c+d x)}{21 d}-\frac{2 i a \cos ^9(c+d x) (a+i a \tan (c+d x))^3}{9 d}-\frac{2 i \cos ^7(c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{21 d}\\ \end{align*}
Mathematica [A] time = 0.706612, size = 111, normalized size = 0.92 \[ \frac{a^4 (-42 \sin (c+d x)-135 \sin (3 (c+d x))+35 \sin (5 (c+d x))-168 i \cos (c+d x)-180 i \cos (3 (c+d x))+28 i \cos (5 (c+d x))) (\cos (4 (c+2 d x))+i \sin (4 (c+2 d x)))}{1008 d (\cos (d x)+i \sin (d x))^4} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.069, size = 233, normalized size = 1.9 \begin{align*}{\frac{1}{d} \left ({a}^{4} \left ( -{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{6}}{9}}-{\frac{ \left ( \cos \left ( dx+c \right ) \right ) ^{6}\sin \left ( dx+c \right ) }{21}}+{\frac{\sin \left ( dx+c \right ) }{105} \left ({\frac{8}{3}}+ \left ( \cos \left ( dx+c \right ) \right ) ^{4}+{\frac{4\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}}{3}} \right ) } \right ) -4\,i{a}^{4} \left ( -{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{9}}-{\frac{2\, \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{63}} \right ) -6\,{a}^{4} \left ( -1/9\,\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{8}+{\frac{\sin \left ( dx+c \right ) }{63} \left ({\frac{16}{5}}+ \left ( \cos \left ( dx+c \right ) \right ) ^{6}+6/5\, \left ( \cos \left ( dx+c \right ) \right ) ^{4}+8/5\, \left ( \cos \left ( dx+c \right ) \right ) ^{2} \right ) } \right ) -{\frac{4\,i}{9}}{a}^{4} \left ( \cos \left ( dx+c \right ) \right ) ^{9}+{\frac{{a}^{4}\sin \left ( dx+c \right ) }{9} \left ({\frac{128}{35}}+ \left ( \cos \left ( dx+c \right ) \right ) ^{8}+{\frac{8\, \left ( \cos \left ( dx+c \right ) \right ) ^{6}}{7}}+{\frac{48\, \left ( \cos \left ( dx+c \right ) \right ) ^{4}}{35}}+{\frac{64\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}}{35}} \right ) } \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.06109, size = 244, normalized size = 2.03 \begin{align*} -\frac{140 i \, a^{4} \cos \left (d x + c\right )^{9} + 20 i \,{\left (7 \, \cos \left (d x + c\right )^{9} - 9 \, \cos \left (d x + c\right )^{7}\right )} a^{4} -{\left (35 \, \sin \left (d x + c\right )^{9} - 90 \, \sin \left (d x + c\right )^{7} + 63 \, \sin \left (d x + c\right )^{5}\right )} a^{4} - 6 \,{\left (35 \, \sin \left (d x + c\right )^{9} - 135 \, \sin \left (d x + c\right )^{7} + 189 \, \sin \left (d x + c\right )^{5} - 105 \, \sin \left (d x + c\right )^{3}\right )} a^{4} -{\left (35 \, \sin \left (d x + c\right )^{9} - 180 \, \sin \left (d x + c\right )^{7} + 378 \, \sin \left (d x + c\right )^{5} - 420 \, \sin \left (d x + c\right )^{3} + 315 \, \sin \left (d x + c\right )\right )} a^{4}}{315 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.09222, size = 267, normalized size = 2.22 \begin{align*} \frac{{\left (-7 i \, a^{4} e^{\left (10 i \, d x + 10 i \, c\right )} - 45 i \, a^{4} e^{\left (8 i \, d x + 8 i \, c\right )} - 126 i \, a^{4} e^{\left (6 i \, d x + 6 i \, c\right )} - 210 i \, a^{4} e^{\left (4 i \, d x + 4 i \, c\right )} - 315 i \, a^{4} e^{\left (2 i \, d x + 2 i \, c\right )} + 63 i \, a^{4}\right )} e^{\left (-i \, d x - i \, c\right )}}{2016 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.26361, size = 230, normalized size = 1.92 \begin{align*} \begin{cases} \frac{\left (- 176160768 i a^{4} d^{5} e^{10 i c} e^{9 i d x} - 1132462080 i a^{4} d^{5} e^{8 i c} e^{7 i d x} - 3170893824 i a^{4} d^{5} e^{6 i c} e^{5 i d x} - 5284823040 i a^{4} d^{5} e^{4 i c} e^{3 i d x} - 7927234560 i a^{4} d^{5} e^{2 i c} e^{i d x} + 1585446912 i a^{4} d^{5} e^{- i d x}\right ) e^{- i c}}{50734301184 d^{6}} & \text{for}\: 50734301184 d^{6} e^{i c} \neq 0 \\\frac{x \left (a^{4} e^{10 i c} + 5 a^{4} e^{8 i c} + 10 a^{4} e^{6 i c} + 10 a^{4} e^{4 i c} + 5 a^{4} e^{2 i c} + a^{4}\right ) e^{- i c}}{32} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.88701, size = 1902, normalized size = 15.85 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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