Optimal. Leaf size=88 \[ -\frac{a^3 \sin ^3(c+d x)}{15 d}+\frac{a^3 \sin (c+d x)}{5 d}-\frac{i a^3 \cos ^3(c+d x)}{15 d}-\frac{2 i a \cos ^5(c+d x) (a+i a \tan (c+d x))^2}{5 d} \]
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Rubi [A] time = 0.0710485, antiderivative size = 88, 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 = {3496, 3486, 2633} \[ -\frac{a^3 \sin ^3(c+d x)}{15 d}+\frac{a^3 \sin (c+d x)}{5 d}-\frac{i a^3 \cos ^3(c+d x)}{15 d}-\frac{2 i a \cos ^5(c+d x) (a+i a \tan (c+d x))^2}{5 d} \]
Antiderivative was successfully verified.
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Rule 3496
Rule 3486
Rule 2633
Rubi steps
\begin{align*} \int \cos ^5(c+d x) (a+i a \tan (c+d x))^3 \, dx &=-\frac{2 i a \cos ^5(c+d x) (a+i a \tan (c+d x))^2}{5 d}+\frac{1}{5} a^2 \int \cos ^3(c+d x) (a+i a \tan (c+d x)) \, dx\\ &=-\frac{i a^3 \cos ^3(c+d x)}{15 d}-\frac{2 i a \cos ^5(c+d x) (a+i a \tan (c+d x))^2}{5 d}+\frac{1}{5} a^3 \int \cos ^3(c+d x) \, dx\\ &=-\frac{i a^3 \cos ^3(c+d x)}{15 d}-\frac{2 i a \cos ^5(c+d x) (a+i a \tan (c+d x))^2}{5 d}-\frac{a^3 \operatorname{Subst}\left (\int \left (1-x^2\right ) \, dx,x,-\sin (c+d x)\right )}{5 d}\\ &=-\frac{i a^3 \cos ^3(c+d x)}{15 d}+\frac{a^3 \sin (c+d x)}{5 d}-\frac{a^3 \sin ^3(c+d x)}{15 d}-\frac{2 i a \cos ^5(c+d x) (a+i a \tan (c+d x))^2}{5 d}\\ \end{align*}
Mathematica [A] time = 0.447172, size = 55, normalized size = 0.62 \[ \frac{a^3 (-6 i \sin (2 (c+d x))+9 \cos (2 (c+d x))+5) (\sin (3 (c+d x))-i \cos (3 (c+d x)))}{30 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.06, size = 126, normalized size = 1.4 \begin{align*}{\frac{1}{d} \left ( -i{a}^{3} \left ( -{\frac{ \left ( \cos \left ( dx+c \right ) \right ) ^{3} \left ( \sin \left ( dx+c \right ) \right ) ^{2}}{5}}-{\frac{2\, \left ( \cos \left ( dx+c \right ) \right ) ^{3}}{15}} \right ) -3\,{a}^{3} \left ( -1/5\,\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{4}+1/15\, \left ( 2+ \left ( \cos \left ( dx+c \right ) \right ) ^{2} \right ) \sin \left ( dx+c \right ) \right ) -{\frac{3\,i}{5}}{a}^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{5}+{\frac{{a}^{3}\sin \left ( dx+c \right ) }{5} \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 ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.20822, size = 142, normalized size = 1.61 \begin{align*} -\frac{9 i \, a^{3} \cos \left (d x + c\right )^{5} + i \,{\left (3 \, \cos \left (d x + c\right )^{5} - 5 \, \cos \left (d x + c\right )^{3}\right )} a^{3} - 3 \,{\left (3 \, \sin \left (d x + c\right )^{5} - 5 \, \sin \left (d x + c\right )^{3}\right )} a^{3} -{\left (3 \, \sin \left (d x + c\right )^{5} - 10 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )} a^{3}}{15 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.1175, size = 131, normalized size = 1.49 \begin{align*} \frac{-3 i \, a^{3} e^{\left (5 i \, d x + 5 i \, c\right )} - 10 i \, a^{3} e^{\left (3 i \, d x + 3 i \, c\right )} - 15 i \, a^{3} e^{\left (i \, d x + i \, c\right )}}{60 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.727356, size = 117, normalized size = 1.33 \begin{align*} \begin{cases} \frac{- 24 i a^{3} d^{2} e^{5 i c} e^{5 i d x} - 80 i a^{3} d^{2} e^{3 i c} e^{3 i d x} - 120 i a^{3} d^{2} e^{i c} e^{i d x}}{480 d^{3}} & \text{for}\: 480 d^{3} \neq 0 \\x \left (\frac{a^{3} e^{5 i c}}{4} + \frac{a^{3} e^{3 i c}}{2} + \frac{a^{3} e^{i c}}{4}\right ) & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.50648, size = 1254, normalized size = 14.25 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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