Optimal. Leaf size=89 \[ \frac{\sin ^5(c+d x)}{7 a^2 d}-\frac{10 \sin ^3(c+d x)}{21 a^2 d}+\frac{5 \sin (c+d x)}{7 a^2 d}+\frac{2 i \cos ^5(c+d x)}{7 d \left (a^2+i a^2 \tan (c+d x)\right )} \]
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Rubi [A] time = 0.0585017, antiderivative size = 89, 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 = {3500, 2633} \[ \frac{\sin ^5(c+d x)}{7 a^2 d}-\frac{10 \sin ^3(c+d x)}{21 a^2 d}+\frac{5 \sin (c+d x)}{7 a^2 d}+\frac{2 i \cos ^5(c+d x)}{7 d \left (a^2+i a^2 \tan (c+d x)\right )} \]
Antiderivative was successfully verified.
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Rule 3500
Rule 2633
Rubi steps
\begin{align*} \int \frac{\cos ^3(c+d x)}{(a+i a \tan (c+d x))^2} \, dx &=\frac{2 i \cos ^5(c+d x)}{7 d \left (a^2+i a^2 \tan (c+d x)\right )}+\frac{5 \int \cos ^5(c+d x) \, dx}{7 a^2}\\ &=\frac{2 i \cos ^5(c+d x)}{7 d \left (a^2+i a^2 \tan (c+d x)\right )}-\frac{5 \operatorname{Subst}\left (\int \left (1-2 x^2+x^4\right ) \, dx,x,-\sin (c+d x)\right )}{7 a^2 d}\\ &=\frac{5 \sin (c+d x)}{7 a^2 d}-\frac{10 \sin ^3(c+d x)}{21 a^2 d}+\frac{\sin ^5(c+d x)}{7 a^2 d}+\frac{2 i \cos ^5(c+d x)}{7 d \left (a^2+i a^2 \tan (c+d x)\right )}\\ \end{align*}
Mathematica [A] time = 0.19092, size = 95, normalized size = 1.07 \[ \frac{i \sec ^2(c+d x) (-70 i \sin (c+d x)+63 i \sin (3 (c+d x))+5 i \sin (5 (c+d x))-140 \cos (c+d x)+42 \cos (3 (c+d x))+2 \cos (5 (c+d x)))}{336 a^2 d (\tan (c+d x)-i)^2} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.09, size = 174, normalized size = 2. \begin{align*} 2\,{\frac{1}{{a}^{2}d} \left ({\frac{i}{ \left ( \tan \left ( 1/2\,dx+c/2 \right ) -i \right ) ^{6}}}-{\frac{5/2\,i}{ \left ( \tan \left ( 1/2\,dx+c/2 \right ) -i \right ) ^{4}}}+{\frac{{\frac{23\,i}{16}}}{ \left ( \tan \left ( 1/2\,dx+c/2 \right ) -i \right ) ^{2}}}-2/7\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) -i \right ) ^{-7}+2\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) -i \right ) ^{-5}-{\frac{55}{24\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) -i \right ) ^{3}}}+{\frac{13}{16\,\tan \left ( 1/2\,dx+c/2 \right ) -16\,i}}-{\frac{i/16}{ \left ( \tan \left ( 1/2\,dx+c/2 \right ) +i \right ) ^{2}}}-1/24\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) +i \right ) ^{-3}+3/16\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) +i \right ) ^{-1} \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 = 2.32369, size = 244, normalized size = 2.74 \begin{align*} \frac{{\left (-7 i \, e^{\left (10 i \, d x + 10 i \, c\right )} - 105 i \, e^{\left (8 i \, d x + 8 i \, c\right )} + 210 i \, e^{\left (6 i \, d x + 6 i \, c\right )} + 70 i \, e^{\left (4 i \, d x + 4 i \, c\right )} + 21 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + 3 i\right )} e^{\left (-7 i \, d x - 7 i \, c\right )}}{672 \, a^{2} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.22719, size = 233, normalized size = 2.62 \begin{align*} \begin{cases} \frac{\left (- 176160768 i a^{10} d^{5} e^{19 i c} e^{3 i d x} - 2642411520 i a^{10} d^{5} e^{17 i c} e^{i d x} + 5284823040 i a^{10} d^{5} e^{15 i c} e^{- i d x} + 1761607680 i a^{10} d^{5} e^{13 i c} e^{- 3 i d x} + 528482304 i a^{10} d^{5} e^{11 i c} e^{- 5 i d x} + 75497472 i a^{10} d^{5} e^{9 i c} e^{- 7 i d x}\right ) e^{- 16 i c}}{16911433728 a^{12} d^{6}} & \text{for}\: 16911433728 a^{12} d^{6} e^{16 i c} \neq 0 \\\frac{x \left (e^{10 i c} + 5 e^{8 i c} + 10 e^{6 i c} + 10 e^{4 i c} + 5 e^{2 i c} + 1\right ) e^{- 7 i c}}{32 a^{2}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.1495, size = 196, normalized size = 2.2 \begin{align*} \frac{\frac{7 \,{\left (9 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 15 i \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 8\right )}}{a^{2}{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + i\right )}^{3}} + \frac{273 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6} - 1155 i \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 2450 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} + 2870 i \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 2037 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 791 i \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 152}{a^{2}{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - i\right )}^{7}}}{168 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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