Optimal. Leaf size=90 \[ -\frac{i a^6}{6 d (a-i a \tan (c+d x))^3}-\frac{i a^5}{8 d (a-i a \tan (c+d x))^2}-\frac{i a^4}{8 d (a-i a \tan (c+d x))}+\frac{a^3 x}{8} \]
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Rubi [A] time = 0.0667252, antiderivative size = 90, 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} \[ -\frac{i a^6}{6 d (a-i a \tan (c+d x))^3}-\frac{i a^5}{8 d (a-i a \tan (c+d x))^2}-\frac{i a^4}{8 d (a-i a \tan (c+d x))}+\frac{a^3 x}{8} \]
Antiderivative was successfully verified.
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Rule 3487
Rule 44
Rule 206
Rubi steps
\begin{align*} \int \cos ^6(c+d x) (a+i a \tan (c+d x))^3 \, dx &=-\frac{\left (i a^7\right ) \operatorname{Subst}\left (\int \frac{1}{(a-x)^4 (a+x)} \, dx,x,i a \tan (c+d x)\right )}{d}\\ &=-\frac{\left (i a^7\right ) \operatorname{Subst}\left (\int \left (\frac{1}{2 a (a-x)^4}+\frac{1}{4 a^2 (a-x)^3}+\frac{1}{8 a^3 (a-x)^2}+\frac{1}{8 a^3 \left (a^2-x^2\right )}\right ) \, dx,x,i a \tan (c+d x)\right )}{d}\\ &=-\frac{i a^6}{6 d (a-i a \tan (c+d x))^3}-\frac{i a^5}{8 d (a-i a \tan (c+d x))^2}-\frac{i a^4}{8 d (a-i a \tan (c+d x))}-\frac{\left (i a^4\right ) \operatorname{Subst}\left (\int \frac{1}{a^2-x^2} \, dx,x,i a \tan (c+d x)\right )}{8 d}\\ &=\frac{a^3 x}{8}-\frac{i a^6}{6 d (a-i a \tan (c+d x))^3}-\frac{i a^5}{8 d (a-i a \tan (c+d x))^2}-\frac{i a^4}{8 d (a-i a \tan (c+d x))}\\ \end{align*}
Mathematica [A] time = 0.538751, size = 109, normalized size = 1.21 \[ \frac{a^3 (-9 \sin (c+d x)-12 i d x \sin (3 (c+d x))+2 \sin (3 (c+d x))-27 i \cos (c+d x)+2 (6 d x-i) \cos (3 (c+d x))) (\cos (3 (c+2 d x))+i \sin (3 (c+2 d x)))}{96 d (\cos (d x)+i \sin (d x))^3} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.062, size = 156, normalized size = 1.7 \begin{align*}{\frac{1}{d} \left ( -i{a}^{3} \left ( -{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{4}}{6}}-{\frac{ \left ( \cos \left ( dx+c \right ) \right ) ^{4}}{12}} \right ) -3\,{a}^{3} \left ( -1/6\, \left ( \cos \left ( dx+c \right ) \right ) ^{5}\sin \left ( dx+c \right ) +1/24\, \left ( \left ( \cos \left ( dx+c \right ) \right ) ^{3}+3/2\,\cos \left ( dx+c \right ) \right ) \sin \left ( dx+c \right ) +1/16\,dx+c/16 \right ) -{\frac{i}{2}}{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{6}+{a}^{3} \left ({\frac{\sin \left ( dx+c \right ) }{6} \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}{16}}+{\frac{5\,c}{16}} \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.69633, size = 142, normalized size = 1.58 \begin{align*} \frac{6 \,{\left (d x + c\right )} a^{3} + \frac{6 \, a^{3} \tan \left (d x + c\right )^{5} + 16 \, a^{3} \tan \left (d x + c\right )^{3} + 12 i \, a^{3} \tan \left (d x + c\right )^{2} + 42 \, a^{3} \tan \left (d x + c\right ) - 20 i \, a^{3}}{\tan \left (d x + c\right )^{6} + 3 \, \tan \left (d x + c\right )^{4} + 3 \, \tan \left (d x + c\right )^{2} + 1}}{48 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.21181, size = 151, normalized size = 1.68 \begin{align*} \frac{12 \, a^{3} d x - 2 i \, a^{3} e^{\left (6 i \, d x + 6 i \, c\right )} - 9 i \, a^{3} e^{\left (4 i \, d x + 4 i \, c\right )} - 18 i \, a^{3} e^{\left (2 i \, d x + 2 i \, c\right )}}{96 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.731826, size = 133, normalized size = 1.48 \begin{align*} \frac{a^{3} x}{8} + \begin{cases} \frac{- 512 i a^{3} d^{2} e^{6 i c} e^{6 i d x} - 2304 i a^{3} d^{2} e^{4 i c} e^{4 i d x} - 4608 i a^{3} d^{2} e^{2 i c} e^{2 i d x}}{24576 d^{3}} & \text{for}\: 24576 d^{3} \neq 0 \\x \left (\frac{a^{3} e^{6 i c}}{8} + \frac{3 a^{3} e^{4 i c}}{8} + \frac{3 a^{3} e^{2 i c}}{8}\right ) & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.35198, size = 617, normalized size = 6.86 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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