Optimal. Leaf size=67 \[ -\frac{4 \sin ^3(c+d x)}{15 a d}+\frac{4 \sin (c+d x)}{5 a d}+\frac{i \cos ^3(c+d x)}{5 d (a+i a \tan (c+d x))} \]
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Rubi [A] time = 0.0519919, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {3502, 2633} \[ -\frac{4 \sin ^3(c+d x)}{15 a d}+\frac{4 \sin (c+d x)}{5 a d}+\frac{i \cos ^3(c+d x)}{5 d (a+i a \tan (c+d x))} \]
Antiderivative was successfully verified.
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Rule 3502
Rule 2633
Rubi steps
\begin{align*} \int \frac{\cos ^3(c+d x)}{a+i a \tan (c+d x)} \, dx &=\frac{i \cos ^3(c+d x)}{5 d (a+i a \tan (c+d x))}+\frac{4 \int \cos ^3(c+d x) \, dx}{5 a}\\ &=\frac{i \cos ^3(c+d x)}{5 d (a+i a \tan (c+d x))}-\frac{4 \operatorname{Subst}\left (\int \left (1-x^2\right ) \, dx,x,-\sin (c+d x)\right )}{5 a d}\\ &=\frac{4 \sin (c+d x)}{5 a d}-\frac{4 \sin ^3(c+d x)}{15 a d}+\frac{i \cos ^3(c+d x)}{5 d (a+i a \tan (c+d x))}\\ \end{align*}
Mathematica [A] time = 0.151753, size = 72, normalized size = 1.07 \[ -\frac{\sec (c+d x) (40 i \sin (2 (c+d x))+4 i \sin (4 (c+d x))+20 \cos (2 (c+d x))+\cos (4 (c+d x))-45)}{120 a d (\tan (c+d x)-i)} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.084, size = 141, normalized size = 2.1 \begin{align*} 2\,{\frac{1}{ad} \left ({\frac{-i/2}{ \left ( \tan \left ( 1/2\,dx+c/2 \right ) -i \right ) ^{4}}}+{\frac{3/4\,i}{ \left ( \tan \left ( 1/2\,dx+c/2 \right ) -i \right ) ^{2}}}+1/5\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) -i \right ) ^{-5}-5/6\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) -i \right ) ^{-3}+{\frac{11}{16\,\tan \left ( 1/2\,dx+c/2 \right ) -16\,i}}-{\frac{i/8}{ \left ( \tan \left ( 1/2\,dx+c/2 \right ) +i \right ) ^{2}}}-1/12\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) +i \right ) ^{-3}+{\frac{5}{16\,\tan \left ( 1/2\,dx+c/2 \right ) +16\,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 = 2.03251, size = 200, normalized size = 2.99 \begin{align*} \frac{{\left (-5 i \, e^{\left (8 i \, d x + 8 i \, c\right )} - 60 i \, e^{\left (6 i \, d x + 6 i \, c\right )} + 90 i \, e^{\left (4 i \, d x + 4 i \, c\right )} + 20 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + 3 i\right )} e^{\left (-5 i \, d x - 5 i \, c\right )}}{240 \, a d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.855722, size = 197, normalized size = 2.94 \begin{align*} \begin{cases} \frac{\left (- 30720 i a^{4} d^{4} e^{12 i c} e^{3 i d x} - 368640 i a^{4} d^{4} e^{10 i c} e^{i d x} + 552960 i a^{4} d^{4} e^{8 i c} e^{- i d x} + 122880 i a^{4} d^{4} e^{6 i c} e^{- 3 i d x} + 18432 i a^{4} d^{4} e^{4 i c} e^{- 5 i d x}\right ) e^{- 9 i c}}{1474560 a^{5} d^{5}} & \text{for}\: 1474560 a^{5} d^{5} e^{9 i c} \neq 0 \\\frac{x \left (e^{8 i c} + 4 e^{6 i c} + 6 e^{4 i c} + 4 e^{2 i c} + 1\right ) e^{- 5 i c}}{16 a} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.15397, size = 161, normalized size = 2.4 \begin{align*} \frac{\frac{5 \,{\left (15 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 24 i \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 13\right )}}{a{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + i\right )}^{3}} + \frac{165 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 480 i \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 650 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 400 i \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 113}{a{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - i\right )}^{5}}}{120 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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