Optimal. Leaf size=121 \[ \frac{2 \sin ^5(c+d x)}{21 a^3 d}-\frac{20 \sin ^3(c+d x)}{63 a^3 d}+\frac{10 \sin (c+d x)}{21 a^3 d}+\frac{4 i \cos ^5(c+d x)}{21 d \left (a^3+i a^3 \tan (c+d x)\right )}+\frac{i \cos ^3(c+d x)}{9 d (a+i a \tan (c+d x))^3} \]
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Rubi [A] time = 0.11338, antiderivative size = 121, 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 = {3502, 3500, 2633} \[ \frac{2 \sin ^5(c+d x)}{21 a^3 d}-\frac{20 \sin ^3(c+d x)}{63 a^3 d}+\frac{10 \sin (c+d x)}{21 a^3 d}+\frac{4 i \cos ^5(c+d x)}{21 d \left (a^3+i a^3 \tan (c+d x)\right )}+\frac{i \cos ^3(c+d x)}{9 d (a+i a \tan (c+d x))^3} \]
Antiderivative was successfully verified.
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Rule 3502
Rule 3500
Rule 2633
Rubi steps
\begin{align*} \int \frac{\cos ^3(c+d x)}{(a+i a \tan (c+d x))^3} \, dx &=\frac{i \cos ^3(c+d x)}{9 d (a+i a \tan (c+d x))^3}+\frac{2 \int \frac{\cos ^3(c+d x)}{(a+i a \tan (c+d x))^2} \, dx}{3 a}\\ &=\frac{i \cos ^3(c+d x)}{9 d (a+i a \tan (c+d x))^3}+\frac{4 i \cos ^5(c+d x)}{21 d \left (a^3+i a^3 \tan (c+d x)\right )}+\frac{10 \int \cos ^5(c+d x) \, dx}{21 a^3}\\ &=\frac{i \cos ^3(c+d x)}{9 d (a+i a \tan (c+d x))^3}+\frac{4 i \cos ^5(c+d x)}{21 d \left (a^3+i a^3 \tan (c+d x)\right )}-\frac{10 \operatorname{Subst}\left (\int \left (1-2 x^2+x^4\right ) \, dx,x,-\sin (c+d x)\right )}{21 a^3 d}\\ &=\frac{10 \sin (c+d x)}{21 a^3 d}-\frac{20 \sin ^3(c+d x)}{63 a^3 d}+\frac{2 \sin ^5(c+d x)}{21 a^3 d}+\frac{i \cos ^3(c+d x)}{9 d (a+i a \tan (c+d x))^3}+\frac{4 i \cos ^5(c+d x)}{21 d \left (a^3+i a^3 \tan (c+d x)\right )}\\ \end{align*}
Mathematica [A] time = 0.225129, size = 98, normalized size = 0.81 \[ \frac{\sec ^3(c+d x) (-378 i \sin (2 (c+d x))+216 i \sin (4 (c+d x))+14 i \sin (6 (c+d x))-567 \cos (2 (c+d x))+162 \cos (4 (c+d x))+7 \cos (6 (c+d x))-210)}{2016 a^3 d (\tan (c+d x)-i)^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.103, size = 207, normalized size = 1.7 \begin{align*} 2\,{\frac{1}{d{a}^{3}} \left ({\frac{{\frac{23\,i}{3}}}{ \left ( \tan \left ( 1/2\,dx+c/2 \right ) -i \right ) ^{6}}}-{\frac{2\,i}{ \left ( \tan \left ( 1/2\,dx+c/2 \right ) -i \right ) ^{8}}}+{\frac{9/4\,i}{ \left ( \tan \left ( 1/2\,dx+c/2 \right ) -i \right ) ^{2}}}-{\frac{{\frac{59\,i}{8}}}{ \left ( \tan \left ( 1/2\,dx+c/2 \right ) -i \right ) ^{4}}}+4/9\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) -i \right ) ^{-9}-{\frac{34}{7\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) -i \right ) ^{7}}}+{\frac{35}{4\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) -i \right ) ^{5}}}-{\frac{19}{4\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) -i \right ) ^{3}}}+{\frac{57}{64\,\tan \left ( 1/2\,dx+c/2 \right ) -64\,i}}-{\frac{i/32}{ \left ( \tan \left ( 1/2\,dx+c/2 \right ) +i \right ) ^{2}}}-1/48\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) +i \right ) ^{-3}+{\frac{7}{64\,\tan \left ( 1/2\,dx+c/2 \right ) +64\,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 = 2.3048, size = 289, normalized size = 2.39 \begin{align*} \frac{{\left (-21 i \, e^{\left (12 i \, d x + 12 i \, c\right )} - 378 i \, e^{\left (10 i \, d x + 10 i \, c\right )} + 945 i \, e^{\left (8 i \, d x + 8 i \, c\right )} + 420 i \, e^{\left (6 i \, d x + 6 i \, c\right )} + 189 i \, e^{\left (4 i \, d x + 4 i \, c\right )} + 54 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + 7 i\right )} e^{\left (-9 i \, d x - 9 i \, c\right )}}{4032 \, a^{3} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.59516, size = 267, normalized size = 2.21 \begin{align*} \begin{cases} \frac{\left (- 811748818944 i a^{18} d^{6} e^{28 i c} e^{3 i d x} - 14611478740992 i a^{18} d^{6} e^{26 i c} e^{i d x} + 36528696852480 i a^{18} d^{6} e^{24 i c} e^{- i d x} + 16234976378880 i a^{18} d^{6} e^{22 i c} e^{- 3 i d x} + 7305739370496 i a^{18} d^{6} e^{20 i c} e^{- 5 i d x} + 2087354105856 i a^{18} d^{6} e^{18 i c} e^{- 7 i d x} + 270582939648 i a^{18} d^{6} e^{16 i c} e^{- 9 i d x}\right ) e^{- 25 i c}}{155855773237248 a^{21} d^{7}} & \text{for}\: 155855773237248 a^{21} d^{7} e^{25 i c} \neq 0 \\\frac{x \left (e^{12 i c} + 6 e^{10 i c} + 15 e^{8 i c} + 20 e^{6 i c} + 15 e^{4 i c} + 6 e^{2 i c} + 1\right ) e^{- 9 i c}}{64 a^{3}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.17019, size = 231, normalized size = 1.91 \begin{align*} \frac{\frac{21 \,{\left (21 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 36 i \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 19\right )}}{a^{3}{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + i\right )}^{3}} + \frac{3591 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{8} - 19656 i \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} - 56196 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6} + 95760 i \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 107730 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 79464 i \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 38484 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 10944 i \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1615}{a^{3}{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - i\right )}^{9}}}{2016 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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