Optimal. Leaf size=171 \[ -\frac{i a^6}{32 d (a-i a \tan (c+d x))^4}-\frac{i a^5}{16 d (a-i a \tan (c+d x))^3}-\frac{3 i a^4}{32 d (a-i a \tan (c+d x))^2}+\frac{i a^4}{64 d (a+i a \tan (c+d x))^2}-\frac{5 i a^3}{32 d (a-i a \tan (c+d x))}+\frac{5 i a^3}{64 d (a+i a \tan (c+d x))}+\frac{15 a^2 x}{64} \]
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Rubi [A] time = 0.106948, antiderivative size = 171, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {3487, 44, 206} \[ -\frac{i a^6}{32 d (a-i a \tan (c+d x))^4}-\frac{i a^5}{16 d (a-i a \tan (c+d x))^3}-\frac{3 i a^4}{32 d (a-i a \tan (c+d x))^2}+\frac{i a^4}{64 d (a+i a \tan (c+d x))^2}-\frac{5 i a^3}{32 d (a-i a \tan (c+d x))}+\frac{5 i a^3}{64 d (a+i a \tan (c+d x))}+\frac{15 a^2 x}{64} \]
Antiderivative was successfully verified.
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Rule 3487
Rule 44
Rule 206
Rubi steps
\begin{align*} \int \cos ^8(c+d x) (a+i a \tan (c+d x))^2 \, dx &=-\frac{\left (i a^9\right ) \operatorname{Subst}\left (\int \frac{1}{(a-x)^5 (a+x)^3} \, dx,x,i a \tan (c+d x)\right )}{d}\\ &=-\frac{\left (i a^9\right ) \operatorname{Subst}\left (\int \left (\frac{1}{8 a^3 (a-x)^5}+\frac{3}{16 a^4 (a-x)^4}+\frac{3}{16 a^5 (a-x)^3}+\frac{5}{32 a^6 (a-x)^2}+\frac{1}{32 a^5 (a+x)^3}+\frac{5}{64 a^6 (a+x)^2}+\frac{15}{64 a^6 \left (a^2-x^2\right )}\right ) \, dx,x,i a \tan (c+d x)\right )}{d}\\ &=-\frac{i a^6}{32 d (a-i a \tan (c+d x))^4}-\frac{i a^5}{16 d (a-i a \tan (c+d x))^3}-\frac{3 i a^4}{32 d (a-i a \tan (c+d x))^2}-\frac{5 i a^3}{32 d (a-i a \tan (c+d x))}+\frac{i a^4}{64 d (a+i a \tan (c+d x))^2}+\frac{5 i a^3}{64 d (a+i a \tan (c+d x))}-\frac{\left (15 i a^3\right ) \operatorname{Subst}\left (\int \frac{1}{a^2-x^2} \, dx,x,i a \tan (c+d x)\right )}{64 d}\\ &=\frac{15 a^2 x}{64}-\frac{i a^6}{32 d (a-i a \tan (c+d x))^4}-\frac{i a^5}{16 d (a-i a \tan (c+d x))^3}-\frac{3 i a^4}{32 d (a-i a \tan (c+d x))^2}-\frac{5 i a^3}{32 d (a-i a \tan (c+d x))}+\frac{i a^4}{64 d (a+i a \tan (c+d x))^2}+\frac{5 i a^3}{64 d (a+i a \tan (c+d x))}\\ \end{align*}
Mathematica [A] time = 0.488739, size = 138, normalized size = 0.81 \[ \frac{a^2 (-120 i d x \sin (2 (c+d x))+30 \sin (2 (c+d x))+32 \sin (4 (c+d x))+3 \sin (6 (c+d x))+30 (4 d x-i) \cos (2 (c+d x))+16 i \cos (4 (c+d x))+i \cos (6 (c+d x))-80 i) (\cos (2 (c+2 d x))+i \sin (2 (c+2 d x)))}{512 d (\cos (d x)+i \sin (d x))^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.059, size = 141, normalized size = 0.8 \begin{align*}{\frac{1}{d} \left ( -{a}^{2} \left ( -{\frac{\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{8}}+{\frac{\sin \left ( dx+c \right ) }{48} \left ( \left ( \cos \left ( dx+c \right ) \right ) ^{5}+{\frac{5\, \left ( \cos \left ( dx+c \right ) \right ) ^{3}}{4}}+{\frac{15\,\cos \left ( dx+c \right ) }{8}} \right ) }+{\frac{5\,dx}{128}}+{\frac{5\,c}{128}} \right ) -{\frac{i}{4}}{a}^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{8}+{a}^{2} \left ({\frac{\sin \left ( dx+c \right ) }{8} \left ( \left ( \cos \left ( dx+c \right ) \right ) ^{7}+{\frac{7\, \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{6}}+{\frac{35\, \left ( \cos \left ( dx+c \right ) \right ) ^{3}}{24}}+{\frac{35\,\cos \left ( dx+c \right ) }{16}} \right ) }+{\frac{35\,dx}{128}}+{\frac{35\,c}{128}} \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.65595, size = 155, normalized size = 0.91 \begin{align*} \frac{15 \,{\left (d x + c\right )} a^{2} + \frac{15 \, a^{2} \tan \left (d x + c\right )^{7} + 55 \, a^{2} \tan \left (d x + c\right )^{5} + 73 \, a^{2} \tan \left (d x + c\right )^{3} + 49 \, a^{2} \tan \left (d x + c\right ) - 16 i \, a^{2}}{\tan \left (d x + c\right )^{8} + 4 \, \tan \left (d x + c\right )^{6} + 6 \, \tan \left (d x + c\right )^{4} + 4 \, \tan \left (d x + c\right )^{2} + 1}}{64 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.11037, size = 309, normalized size = 1.81 \begin{align*} \frac{{\left (120 \, a^{2} d x e^{\left (4 i \, d x + 4 i \, c\right )} - i \, a^{2} e^{\left (12 i \, d x + 12 i \, c\right )} - 8 i \, a^{2} e^{\left (10 i \, d x + 10 i \, c\right )} - 30 i \, a^{2} e^{\left (8 i \, d x + 8 i \, c\right )} - 80 i \, a^{2} e^{\left (6 i \, d x + 6 i \, c\right )} + 24 i \, a^{2} e^{\left (2 i \, d x + 2 i \, c\right )} + 2 i \, a^{2}\right )} e^{\left (-4 i \, d x - 4 i \, c\right )}}{512 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.00391, size = 272, normalized size = 1.59 \begin{align*} \frac{15 a^{2} x}{64} + \begin{cases} \frac{\left (- 8589934592 i a^{2} d^{5} e^{14 i c} e^{8 i d x} - 68719476736 i a^{2} d^{5} e^{12 i c} e^{6 i d x} - 257698037760 i a^{2} d^{5} e^{10 i c} e^{4 i d x} - 687194767360 i a^{2} d^{5} e^{8 i c} e^{2 i d x} + 206158430208 i a^{2} d^{5} e^{4 i c} e^{- 2 i d x} + 17179869184 i a^{2} d^{5} e^{2 i c} e^{- 4 i d x}\right ) e^{- 6 i c}}{4398046511104 d^{6}} & \text{for}\: 4398046511104 d^{6} e^{6 i c} \neq 0 \\x \left (- \frac{15 a^{2}}{64} + \frac{\left (a^{2} e^{12 i c} + 6 a^{2} e^{10 i c} + 15 a^{2} e^{8 i c} + 20 a^{2} e^{6 i c} + 15 a^{2} e^{4 i c} + 6 a^{2} e^{2 i c} + a^{2}\right ) e^{- 4 i c}}{64}\right ) & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.25628, size = 462, normalized size = 2.7 \begin{align*} \frac{120 \, a^{2} d x e^{\left (8 i \, d x + 4 i \, c\right )} + 240 \, a^{2} d x e^{\left (6 i \, d x + 2 i \, c\right )} + 120 \, a^{2} d x e^{\left (4 i \, d x\right )} + 8 i \, a^{2} e^{\left (8 i \, d x + 4 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) + 16 i \, a^{2} e^{\left (6 i \, d x + 2 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) + 8 i \, a^{2} e^{\left (4 i \, d x\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) - 8 i \, a^{2} e^{\left (8 i \, d x + 4 i \, c\right )} \log \left (e^{\left (2 i \, d x\right )} + e^{\left (-2 i \, c\right )}\right ) - 16 i \, a^{2} e^{\left (6 i \, d x + 2 i \, c\right )} \log \left (e^{\left (2 i \, d x\right )} + e^{\left (-2 i \, c\right )}\right ) - 8 i \, a^{2} e^{\left (4 i \, d x\right )} \log \left (e^{\left (2 i \, d x\right )} + e^{\left (-2 i \, c\right )}\right ) - i \, a^{2} e^{\left (16 i \, d x + 12 i \, c\right )} - 10 i \, a^{2} e^{\left (14 i \, d x + 10 i \, c\right )} - 47 i \, a^{2} e^{\left (12 i \, d x + 8 i \, c\right )} - 148 i \, a^{2} e^{\left (10 i \, d x + 6 i \, c\right )} - 190 i \, a^{2} e^{\left (8 i \, d x + 4 i \, c\right )} - 56 i \, a^{2} e^{\left (6 i \, d x + 2 i \, c\right )} + 28 i \, a^{2} e^{\left (2 i \, d x - 2 i \, c\right )} + 50 i \, a^{2} e^{\left (4 i \, d x\right )} + 2 i \, a^{2} e^{\left (-4 i \, c\right )}}{512 \,{\left (d e^{\left (8 i \, d x + 4 i \, c\right )} + 2 \, d e^{\left (6 i \, d x + 2 i \, c\right )} + d e^{\left (4 i \, d x\right )}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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